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Optimal design, nonlinear analysis and shape control of deployable mesh reflectors

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Optimal design, nonlinear analysis and shape control of deployable mesh reflectors

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Content Optimal Design, Nonlinear Analysis and Shape Control of Deployable Mesh Reflectors by Hang Shi A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) August 2013 Copyright 2013 Hang Shi ii To my parents and my wife iii Acknowledgements First, my thesis advisor, Prof. Bingen Yang must be acknowledged foremost. His guidance, encouragement and vast knowledge have been invaluable to my entire graduate study. Many gratitude thanks go to Dr. Houfei Fang previously at Jet Propulsion Laboratory for his long-term cooperation and for his feedbacks and comments on the project. A debt of gratitude is owed to Prof. Paul Newton in Aerospace and Mechanical Engineering Department at University of Southern California for his enthusiastic help regarding the nonlinear programming methodology and to Prof. Henryk Flashner from the same affiliation for his suggestion of introducing switching control into my work. Thanks are due to Prof. P. Frank Pai at the University of Missouri for constructive discussions on the nonlinear modeling of highly flexible structures, to Prof. Michael Safonov in Electrical Engineering Department at the University of Southern California for his instructions on structured uncertainties and LMI solver implementations, to Dongyuan Zhan for the enlightening discussion on the system design formulation and to Yangmin Xie from University of Illinois at Urbana-Champaign for the cooperation on the feedback shape control research. Many thanks to Mark Thomson from Jet Propulsion Laboratory for his comments on a paper draft for AIAA SDM Conference, to Prof. Weihua Su at University of Alabama and Dr. Huimin Song from National Renewable Energy Laboratory for discussions on FEM simulations. Special thanks go to my labmates Shibing Liu, Youngchul Song, Dean Bergman, Homin Choi. Financial supports from the Jet Propulsion Laboratory and the Department of Aerospace and Mechanical Engineering at the University of Southern California are gratefully acknowledged. Los Angeles, May 2013 Hang Shi iv Abstract This thesis presents the research on optimal design, nonlinear analysis and shape control of the perimeter truss deployable mesh reflector (DMR) as a type of state-of-the-art space structures, all three portions of which construct the Active Shape Control architecture. To design the shape of DMR, a new mesh generation approach is developed to automatically determine the mesh geometry of the working surface of DMRs, which is among very few of pseudo-geodesic mesh geometry design methods. Once the desired geometry of the mesh facets is generated, the optimal design method is presented to determine the structural parameters such as undeformed length of members and the external loads so that not only the mesh facets are deformed onto the desired working shape exactly but also the best tension distribution is assured in the reflector structure. In the academic literature such method is the first to include the external loads as the design variables to design deployable trusses. The nonlinear static analysis is carried out based on a new static truss model using a nonlinear programming solving technique formulated in the thesis. Since the static model does not need the initial configuration of the reflector, both the initial shape and the deformed shape are obtained by the solving approach. The natural frequencies and the mode shapes are addressed on the linearized dynamics derived at the nonlinear static equilibrium solution and this linear dynamic system is used in the following shape control study. As the final part of the architecture, both the static and dynamic shape control strategies are proposed in the thesis. The static shape control approach provides one of few thermal compensation solutions regarding the fully nonlinear statics of reflector structures. The dynamic shape control use the feedback of the nodal coordinates and is the first feedback shape control on DMRs. Both shape control methods have demonstrated their advancements on improving the surface accuracy of DMRs for the space thermal environment in orbiting missions. v Table of Contents ACKNOWLEDGEMENTS ........................................................................................................... iii ABSTRACT ................................................................................................................................... iv LIST OF TABLES ......................................................................................................................... ix LIST OF FIGURES ........................................................................................................................ x CHAPTER 1 INTRODUCTION ................................................................................................ 1 1.1 Background on Deployable Mesh Reflectors ................................................................. 1 1.2 DMR in Consideration ................................................................................................... 3 1.3 Objective of the Research............................................................................................... 4 1.4 Outline of the Thesis ...................................................................................................... 6 CHAPTER 2 DESIGN OF SURFACE MESH GEOMETRY ................................................... 8 2.1 Introduction .................................................................................................................... 8 2.2 Problem Statement ....................................................................................................... 11 2.3 Basic Design Steps ....................................................................................................... 12 2.3.1 The proposed design steps ............................................................................................... 12 2.3.2 Surface accuracy evaluation ............................................................................................. 32 2.4 Additional Design Considerations................................................................................ 34 2.4.1 Design formulation when designing the “effective” region of the working surface ........ 34 2.4.2 Compensation for the deviation of the best-fit surface from the desired surface ............. 39 2.5 Design Criteria for Optimal Parameter Selection ......................................................... 42 2.5.1 Requirement on the force balancing equilibrium at designated deformed working status .............................................................................................................................. ..................... 42 vi 2.5.2 The surface accuracy requirement based on the operating frequency .............................. 43 2.6 The Influence of Design Parameters and Performance Evaluation .............................. 45 2.6.1 The influence of the design parameters on pseudo-geodesic property and the best-fit RMS error ..................................................................................................................... ............ 46 2.6.2 Performance evaluation of the generated geometry ......................................................... 51 2.7 Complete Design Procedure via Optimal Parameter Selection .................................... 57 2.8 Comparative Studies of Different Design Methods ..................................................... 61 2.8.1 Comparisons on configurations of design mesh facets .................................................... 62 2.8.2 Comparisons on generated mesh geometries at the same configuration .......................... 64 2.9 Design Results .............................................................................................................. 67 2.9.1 Quick view of design results without tuning the design parameters ................................ 67 2.9.2 Improvement on the JPL mesh geometry ......................................................................... 68 2.9.3 Design of an extremely large mesh reflector ................................................................... 72 2.10 Conclusions ................................................................................................................ 75 CHAPTER 3 OPTIMAL DESIGN OF STRUCTURAL PARAMETERS .............................. 77 3.1 Introduction .................................................................................................................. 77 3.2 Force Balance Model at Equilibrium ........................................................................... 79 3.3 System Model and its Solution ..................................................................................... 87 3.4 Optimization Formulation ............................................................................................ 96 3.4.1 Objective 1: To minimize the member tension band of the truss ..................................... 96 3.4.2 Objective 2: To unify the member tensions as a desired one ........................................... 97 3.4.3 Constraint 1: Specific element strain requirement ........................................................... 98 3.4.4 Constraint 2: Specific cross-section area requirement ..................................................... 98 3.4.5 Constraint 3: Specific member tension range .................................................................. 99 3.4.6 Constraint 4: Specific external load range ....................................................................... 99 3.5 Design Procedure & Remarks .................................................................................... 100 3.6 Extended Applications ............................................................................................... 105 3.6.1 Weight minimization ...................................................................................................... 105 3.6.2 Layered design strategy .................................................................................................. 106 3.7 Design Results and Discussions ................................................................................. 110 vii 3.7.1 A Simple 3D truss .......................................................................................................... 110 3.7.2 A 25-element truss structure .......................................................................................... 113 3.7.3 Deployable mesh reflectors ............................................................................................ 116 3.8 Concluding Remarks .................................................................................................. 127 CHAPTER 4 NONLINEAR STATICS AND STATIC SHAPE CONTROL........................ 128 4.1 Introduction ................................................................................................................ 128 4.2 Nonlinear Model of Truss Structure ........................................................................... 130 4.3 Algorithm of Solving the Nonlinear Model ............................................................... 134 4.4 Numerical Examples on Nonlinear Static Analysis ................................................... 138 4.4.1 Condition analysis of different initial shapes ................................................................. 138 4.4.2 Verification on Initial shape profile and deformed shape profile ................................... 141 4.4.3 Verification of the optimal parametric solution on a DMR ........................................... 142 4.5 Static Shape Control Under Thermal Distortions....................................................... 145 4.5.1 Static shape control strategy ........................................................................................... 145 4.5.2 Nonlinear static shape control implementations ............................................................ 146 4.6 Conclusion .................................................................................................................. 150 CHAPTER 5 COUPLED ELASTIC-THERMAL DYNAMICS ........................................... 151 5.1 Introduction ................................................................................................................ 151 5.2 Nonlinear Dynamic Model ......................................................................................... 151 5.3 Linearized Model and Vibration Analysis ................................................................. 155 5.4 Numerical Analysis and Discussion ........................................................................... 157 5.4.1 A six-element truss ......................................................................................................... 157 5.4.2 A sample of deployable mesh reflector .......................................................................... 159 5.5 Quasi-static Strategy in the Space Mission ................................................................ 160 5.6 Numerical Results and Discussion on the Quasi-static Strategy ................................ 162 5.6.1 Simulations on a simple example of truss structure ....................................................... 163 5.6.2 Dynamic analysis on a sampled deployable mesh reflector ........................................... 167 5.7 Conclusion .................................................................................................................. 170 CHAPTER 6 ONGOING WORK AND CONCLUDING REMARKS ................................. 171 viii 6.1 Ongoing Research on Feedback Shape Control ......................................................... 171 6.1.1 Introduction .................................................................................................................... 171 6.1.2 System modeling with uncertainties .............................................................................. 172 6.1.3 Controller design ............................................................................................................ 176 6.1.4 Case study and performance discussion ......................................................................... 179 6.1.5 Conclusion .............................................................................................................. ........ 185 6.2 Concluding Remarks .................................................................................................. 185 BIBLIOGRAPHY ....................................................................................................................... 188 APPENDIX A DERIVATIONS OF , θ i j AND , α i j TO OBTAIN THE NODAL COORDINATES ........................................................................................................................ 199 APPENDIX B CALCULATION OF THE CURVATURE OF THE FIRST CONVERGING RING ........................................................................................................................................... 202 APPENDIX C CALCULATION OF NODAL COORDINATES AFTER APPLYING THE CONVERGING TREATMENTS ............................................................................................... 204 APPENDIX D UNIVERSAL MAPPING FUNCTION .......................................................... 207 APPENDIX E EVALUATION OF THE BEST-FIT SURFACE OF THE MESH GEOMETRY FOR AN OFFSET-FEED PARABOLIC REFLECTOR ............................................................ 211 APPENDIX F FORMULATION OF B w LOWER BOUND ................................................ 213 ix List of Tables Table 2.1 Specifications of various design configurations .................................................................... 62 Table 2.2 Actual surface accuracy comparison under best-fit compensation ........................................ 66 Table 2.3 Specifications of the generated mesh geometries .................................................................. 71 Table 2.4 Tension performance of the designs ...................................................................................... 71 Table 2.5 Specifications of designated mesh geometries ...................................................................... 74 Table 3.1 Nodal constraint matrices at node i ....................................................................................... 85 Table 3.2 Nodal coordinates of desired configuration ......................................................................... 111 Table 3.3 Nodal loads of desired configuration ................................................................................... 112 Table 3.4 Element profiles .............................................................................................................. ..... 113 Table 3.5 Loading conditions on nodes ............................................................................................... 114 Table 3.6 The optimal design of the truss parameters ......................................................................... 115 Table 3.7 Tension distribution in the structure (lbs.) ........................................................................... 120 Table 3.8 Tension specifications in the structure (lbs.) ....................................................................... 124 Table 4.1 Comparison of element lengths (m). ................................................................................... 139 Table 4.2 Comparison of element lengths (m). ................................................................................... 140 Table 4.3 Comparison of element lengths (m). ................................................................................... 141 Table 4.4 Verification of force balance (N). ........................................................................................ 141 Table 5.1 Nodal coordinates ............................................................................................................. ... 158 Table 5.2 Natural frequencies of linearized model .............................................................................. 158 Table 5.3 Nodal coordinates in initial and deployed configurations ................................................... 164 Table 5.4 From thermal deformation to rebalanced deformation ........................................................ 165 Table 5.5 Natural frequencies of linearized model .............................................................................. 165 Table 5.6 Influence of inputs on modes ............................................................................................... 167 Table 5.7 External forces in different equilibriums ............................................................................. 168 Table D.1 Two critical indices........................................................................................................... ... 209 x List of Figures Figure 1.1 The deployment process of a deployable reflector. (Courtesy of M. Thomson’s presentation at Large Space Apertures Workshop, Nov. 2008) ................................................................................1 Figure 1.2 Diverse types of DMRs installed on various applications. (Courtesy of ISAS, JAXA, Harris Corporation and TRW Astro Aerospace) .............................................................................................2 Figure 1.3 A typical state-of-the-art DMR considered in the research (Courtesy of Ref. [101]) .............3 Figure 1.4 R&D of Active Shape Control (ASC) Architecture ................................................................5 Figure 1.5 The engineering model for the working surface of DMRs .....................................................5 Figure 2.1 Desired working shape of a deployable mesh reflector: (left) a center-feed case or an offset- feed spherical case; (right) an offset-feed parabolic case. .................................................................. 11 Figure 2.2 A reference sphere for the designated shape. ........................................................................ 13 Figure 2.3 The working shape of offset parabolic reflectors in XZ plane. ............................................. 14 Figure 2.4 Subdivision separation of the reference surface. ................................................................... 16 Figure 2.5 Geodesic rings and subdivision lines on the reference surface. ............................................. 16 Figure 2.6 The facet edge in the great circle. .......................................................................................... 17 Figure 2.7 Rings’ intersections on one subdivision. ............................................................................... 18 Figure 2.8 Nodes on the rings within one subdivision. ........................................................................... 19 Figure 2.9 Nodes inside one subdivision. ............................................................................................... 20 Figure 2.10 Topology of member connections within one subdivision. ............................................... 22 Figure 2.11 The member connections in the boundary layer. ............................................................... 24 Figure 2.12 The mapping between the spherical reference and the ellipsoidal surface in XY plane. .. 27 Figure 2.13 The creation of the mapping function. .............................................................................. 29 Figure 2.14 Nodal projection in the plane passing z-axis. .................................................................... 30 Figure 2.15 The effective region on the center-feed parabola (XZ plane). .......................................... 36 Figure 2.16 The effective region on the offset-feed parabola (XZ plane). ........................................... 37 Figure 2.17 The surface deviations in XZ plane. .................................................................................. 39 xi Figure 2.18 The influence of B c on the reference sphere at 9 r n = , 0.3 c τ = , 2.0 ρ= , 1 B w = . ...... 47 Figure 2.19 The influence contour of c τ andρ on , rms bf δ at 10 r n = . .................................................. 49 Figure 2.20 The influence contour of c τ andρ on , rms bf δ at 20 r n = . ................................................. 49 Figure 2.21 The influence contour of c τ andρ on total l at 10 r n = . ...................................................... 50 Figure 2.22 The influence of c τ andρ on total l at 20 r n = . ................................................................... 50 Figure 2.23 The selection of r τ versus various rms D δ and / FD . .................................................. 51 Figure 2.24 Tension distribution vs B c on the effective region (left) and the entire surface (right). .... 52 Figure 2.25 Tension distribution vs B w on the effective region (left) and the entire surface (right). ... 54 Figure 2.26 Tension distribution vs s n on the effective region (left) and the entire surface (right). .... 54 Figure 2.27 Tension distribution vs r n on the effective region (left) and the entire surface (right). .... 55 Figure 2.28 Top-view of mesh geometry at /2 FD= and 10 r n = for the ultra-minimum geometry (left) and proposed geometry (right). ......................................................................................... ......... 56 Figure 2.29 Performance of total length in the worst case scenarios. ................................................... 56 Figure 2.30 Design Core of the proposed method. ............................................................................... 57 Figure 2.31 The flowchart of the optimal design method via systematic tuning. ................................. 58 Figure 2.32 Configuration comparison on the total nodal numbers (left) and the total numbers of elements (right). .................................................................................................................................. 63 Figure 2.33 The best-fit surface RMS errors compared in three case studies. ..................................... 65 Figure 2.34 Generated mesh geometries on the same configuration with reduced nodes on the aperture rims. ................................................................................................................ ...................... 66 Figure 2.35 Quick view of mesh geometries at diverse configurations. ............................................... 68 Figure 2.36 3D view of surface mesh geometry of JPL’s design. ........................................................ 69 Figure 2.37 The designed mesh geometries: (a) JPL’s design (left); (b) Modified design (middle); (c) Improved design (right). ...................................................................................................... ............... 70 Figure 2.38 The proposed design satisfying the operating frequency. ................................................. 72 Figure 2.39 The top-view of design for the extremely large DMR (Strategy 1). ................................. 74 Figure 2.40 The top-view of design for the extremely large DMR (Strategy 2). ................................. 75 Figure 3.1 Optimal Parametric Design Method. ................................................................................... 101 Figure 3.2 Multi-optimal-solution cases: (a) finite solutions (left); (b) infinite solutions (right). ........ 104 xii Figure 3.3 Topology of layer separation strategy. ................................................................................ 106 Figure 3.4 The deformed shape of a simple 3D truss. ........................................................................... 110 Figure 3.5 The deformed configuration of 25-element truss. ............................................................... 113 Figure 3.6 The working configuration of a center-feed DMR. ............................................................. 117 Figure 3.7 Tension distribution (left) and load distribution (right) at the optimal design. .................... 118 Figure 3.8 Layer separation of the DMR surface. ................................................................................. 119 Figure 3.9 The working configuration of a center-feed DMR. ............................................................. 120 Figure 3.10 The working configuration of an offset-feed parabolic DMR. ........................................ 121 Figure 3.11 The top view of an offset-feed parabolic DMR. .............................................................. 122 Figure 3.12 The side view of an offset-feed parabolic DMR. ............................................................ 123 Figure 3.13 The undeformed profile and the element strains for cable net. ....................................... 125 Figure 3.14 The undeformed profile and the element strains for tension ties. ................................... 125 Figure 3.15 The undeformed profile and the element strains for supporting truss. ............................ 126 Figure 3.16 The consistent normalized tension distributions regarding various minimum tension constraints. .................................................................................................................. ...................... 127 Figure 4.1 A truss element in undeformed and deformed configurations. ............................................ 130 Figure 4.2 Nodal displacements and force balance in local coordinate. ............................................... 131 Figure 4.3 Initial shape when ,, org j ini j L l = . .......................................................................................... 138 Figure 4.4 Initial shape when ,, org j ini j L l < . ........................................................................................... 140 Figure 4.5 Initial shape when ,, org j ini j L l > . .......................................................................................... 140 Figure 4.6 Initial shape and deformed shape. ....................................................................................... 142 Figure 4.7 Initial shape and deformed shape based on optimal external loads. .................................... 143 Figure 4.8 Algorithm performance when solving initial shape. ............................................................ 143 Figure 4.9 Algorithm performance when solving deformed shape. ...................................................... 144 Figure 4.10 Tension forces of members. ............................................................................................ 144 Figure 4.11 Four status of static shape control strategy. ..................................................................... 145 Figure 4.12 Initial and desired (deformed) shapes.............................................................................. 147 Figure 4.13 Deformed shapes with and without thermal loading. ...................................................... 148 Figure 4.14 Thermally distorted shape and rebalanced shape. ........................................................... 148 Figure 4.15 Desired (deployed) and rebalanced shapes. ..................................................................... 148 Figure 4.16 Algorithm performance when solving initial shape. ....................................................... 149 Figure 4.17 Algorithm performance when solving deformed shape. ................................................. 149 xiii Figure 4.18 Algorithm performance when solving deformed shape under thermal effects................ 149 Figure 4.19 Algorithm performance when solving rebalanced shape. ............................................... 150 Figure 5.1 Initial (undeformed) shape of the example structure .......................................................... 158 Figure 5.2 The first four mode shapes of the truss ............................................................................... 159 Figure 5.3 A sampled deployed mesh reflector................................................................................... 159 Figure 5.4 The first four mode shapes of the mesh reflector. ............................................................... 160 Figure 5.5 Temperature separation during orbiting missions. .............................................................. 161 Figure 5.6 Initial shape of the example structure ................................................................................. 163 Figure 5.7 Initial and desired (deployed) shapes................................................................................... 163 Figure 6.1 The Controller structure ....................................................................................................... 176 Figure 6.2 System formulation for H ∞ synthesis .................................................................................. 177 Figure 6.3 The Working Shape of DMR ............................................................................................... 179 Figure 6.4 Temperature change in one cycle ........................................................................................ 181 Figure 6.5 Comparison of step forces input between nonlinear system and linearized system ............ 182 Figure 6.6 Structured singular value for ,11 aug P     in closed loop system of the 8th section ................. 182 Figure 6.7 Comparison of the responses to impulse forces disturbance ............................................... 183 Figure 6.8 Comparison of shape error norm between closed system and open system ........................ 184 Figure A.1 Nodes inside one subdivision. ............................................................................................. 200 Figure B.1 First converging ring within one subdivision in projected XY plane. ................................. 202 Figure C.1 Nodes on converging rings within one subdivision in projected XY plane......................... 205 Figure D.1 The relative chord errors when 6 s n = and at (a) 0.7 ee ba = (left); (b) 7 i = (right). .. 208 Figure D.2 The relative chord errors for cr ii = and 0.7 ee ba = . ......................................................... 209 Figure E.1 The best-fit surface for an offset-feed parabolic reflector in XZ plane. .............................. 211 Figure F.2 The connections at single node on boundary rim in XY plane. ........................................... 213 1 Chapter 1 Introduction 1.1 Background on Deployable Mesh Reflectors A reflector is a structural device that receives and reflects electromagnetic signals, which normally has a dish shape as working surface and is supported by another structure (commonly a truss) behind. Unlike reflectors on the ground, a reflector installed onto a satellite or a space shuttle must consider many crucial requirements, one of which requires that the reflector has to be deployable. Because the size of a space reflector is usually much larger than the spacecraft that carries it, the reflector must be first folded into a small volume on the ground that can be stowed inside the spacecraft, and then be deployed into the space after the spacecraft has been launched onto the designated orbit (Figure 1.1). After the deployment is completed, the reflector will produce and automatically maintain a working surface (aperture) with tolerant surface errors. Due to this particular feature, such structural devices are called deployable reflectors [81]. Figure 1.1 The deployment process of a deployable reflector. (Courtesy of M. Thomson’s presentation at Large Space Apertures Workshop, Nov. 2008) 2 Figure 1.2 Diverse types of DMRs installed on various applications. (Courtesy of ISAS, JAXA, Harris Corporation and TRW Astro Aerospace) As one of many types of deployable reflectors, large deployable mesh reflectors (DMR) have brought continuously interest in research and industry in past decades and have achieved 3 most significant development among diverse types of deployable reflectors against solid surface reflectors and inflatable reflectors for radio-frequency (RF) space applications (Figure 1.2). Deployable mesh reflectors have very wide applications and have been used in renown space projects with different functions, such as MBSAT [96] for global broadcasting, ETS VIII [54] for satellite communication, in SMAP mission [11] for remote sensing, “NEXRAD in Space” [15] mission for weather forecasting, Thuraya and Inmarsat 4 [101] for mobile communication and HALCA satellite used in Space VLBI mission [100, 29] for radio astronomy observation. 1.2 DMR in Consideration Figure 1.3 A typical state-of-the-art DMR considered in the research (Courtesy of Ref. [101]) As one type of state-of-the-art DMRs, the DMR with an aperture rim (Figure 1.1) that has been considered by NASA engineers is studied in our research and Figure 1.3 illustrates the structure design for such reflector. The reflector is supported by a stiff and stable deployable flat truss and its working surface (the front net) is constructed by a mesh (network) of flat triangular facets, as well as the rear net. The edges of facets are elastic cables interconnected at facet nodes. The nodes of the front net and the rear net are connected by tension ties where actuators are installed. In the deployment, the supporting flat truss deploys the reflector from the folded status into its working configuration and those actuators properly adjust the length of the tension ties, so as to generate the desired shape of the working surface. After the deployment the actuators are also capable of maintaining the front net at the desired working shape during in-space missions. 4 Therefore in its working status (fully deployed or deformed status) as shown in Figure 1.3, the highly elastic mesh is stretched and the flat facets in the front are deformed into an approximation of required RF surface. Mathematically, DMRs of this type including the supporting truss, the front and rear nets and the tension ties, are engineered as deployable truss structures of which elements are assembled at the nodes with axial strains and no bending. Such structure has geometric and material nonlinearities. Two crucial factors are used to assess the performance of deployable mesh reflectors: the aperture size of the reflector (mostly in term of the diameter) and the root-mean-square (RMS) value of the surface error. According to the antenna theory, larger-sized reflectors are capable of transmitting greater amount of data with higher power, and the smaller surface RMS error implies broader frequency bandwidth of the transmitted signals. The characteristic ratio, which is defined as the ratio of the reflector diameter to surface RMS error, is one of the key parameters to evaluate the performance of the mesh surface. Obviously, to increase the characteristic ratio, one can either increase the diameter of the reflector or decrease the surface RMS error; both of which, however, will enlarge the number of mesh facets and increase the complexity and difficulty in design and manufacturing of this kind of reflectors. Therefore, development of large-sized deployable reflectors with small surface RMS errors, although in urgent demand, due to the stringent requirements on the surface performance, has been a challenge for years [26]. Previous investigations [31, 32, 36] have shown the performance limitation due to thermo-elastic strains and manufacturing errors of materials in passive structure and have suggested that active surface shape control becomes necessary and most promising to improve the surface performance of DMRs for future space missions. 1.3 Objective of the Research To further advance the development of DMRs, this research is to develop an “Active Shape Control (ASC)” architecture for extremely large light-weight in-space deployable mesh reflectors for high RF applications. The R&D structure of ASC architecture is shown in Figure 1.4. The key techniques in the architecture start with an optimal method to design the proper shape geometry and structural parameters of the DMR, which uniquely considers the form finding problem as a pure design task. A nonlinear static shape control approach is then proposed 5 by utilizing nonlinear optimization techniques upon a new static deployable truss model. Once the coupled elastic-thermal dynamics of DMRs is investigated for the space orbiting missions, the linearized model is analyzed and ready for the controller design purpose. Finally the first feedback shape control strategy is applied to dynamically compensate thermal distortions and to adapt environment disturbances. Figure 1.4 R&D of Active Shape Control (ASC) Architecture Figure 1.5 The engineering model for the working surface of DMRs The shape design portion of the ASC architecture considers the full DMR structure in Figure 1.3. However, to simplify the research problem and improve the computational efficiency 6 in the nonlinear analysis and shape control part of the architecture, those studies will focus on the working surface of the DMR of which the shape determines the reflector’s performance, and their numerical results are obtained. In this case, the working surface can be considered to be fixed at the boundary nodes. Because of the symmetry of the front net and the rear net, the external loads applied on the surface by tension ties are always normal to the aperture plane and are used to control the shape of the facet mesh in the working surface. Therefore, by focusing on the cable net of the working surface, the boundary truss, the rear net and the tension ties are removed, and its engineering model is considered as a 3D truss with geometry and elasticity nonlinearity and is shown in Figure 1.5. 1.4 Outline of the Thesis In Chapter 2, a new design method is presented to optimally generate the mesh geometry of the mesh facets in the working surface. The generated geometry can always guarantee the satisfactory of the operating frequency in the reflector applications. The designed mesh facets are of pseudo-geodesic property which results in the minimum total length of the cable net and the most competitive surface accuracy of the working surface comparing to other existing approaches. In Chapter 3, an optimal design method is proposed to determine the structure parameters of the DMR once the mesh geometry is obtained. The design approach includes the external forces into the design variable for the first time in the academic literature and provides the best possible tension distribution at the design results. It is also capable of minimizing the weight of the structure and besides DMR it can be applied for other deployable truss structures such as deployable boom and deployable mast. In Chapter 4, the nonlinear static analysis is carried out using a new nonlinear static truss model for the reflector. A solving technique is proposed for the solution of the static deformation using nonlinear programming techniques. Then the nonlinear static shape control strategy is formulated upon the static reflector model to compensate the thermal distortion during orbiting mission and uses the same solver to determine both the static control inputs and the optimal deformed shape under the static shape compensation. 7 In Chapter 5, the nonlinear dynamics of the reflector is investigated considering both the elastic deformation and the thermal expansion of the material on the orbit. After the mode shapes are identified and the natural frequencies are evaluated, the linearized dynamic model is readily available for the further development of the dynamic shape controller via feedback. In Chapter 6, the ongoing research presents the first feedback shape controller for DMRs, which combine the switching control and robust control to reject the environment disturbance and compensate the thermal distortion of reflector structures. Then the conclusion is followed to summarize major contribution of the thesis and gives suggestions on the remaining work and the further research. In Appendix A ~ F, it provides the detailed formulation which are skipped in Chapter 2 as to design the mesh geometry of the mesh facets in the DMR’s working surface. 8 Chapter 2 Design of Surface Mesh Geometry 2.1 Introduction Starting from 1980s, many successful researches have been conducted to investigate fundamental design requirements and preliminary design approaches of DMR. Hedgepeth [31] regulated the potential reflector size and surface accuracy due to the fabricating errors and thermal strains, based on a proposed equivalence principle [33] of evaluating the fabrication impact and surface errors. Fager [12] suggested the stiffness requirement especially on tetrahedral space truss. Hedgepeth [32] also concluded frequency ratio requirement, achievable characteristic ratio and allowable member slenderness of large space structure. Agrawal, et al. [1] introduced the approximate approach of surface error evaluation on facet surface in the preliminary design and Mikulas, et al. [56] addressed solutions to preliminary truss reflector design considering structural weight, natural frequency, packaging volume and assembly time. Lake, et al [47] summarized a set of analytical design expressions connecting structural mass, stiffness, damping and deformation, the lattice geometry and the fabrication errors to critical performance metrics. Upon all above studies which built up the foundation of DMR development, many diverse design concepts [74, 26] have been proposed, engineered and some eventually realized, among which the following concepts should be recognized: wrap-rib design [52, 3] by JPL and Lockheed, rigid-rib antenna [14] and radial-rib antenna [26] by Harris Corporation, hoop-column concept [94, 4] by NASA Langley and Harris Corporation and currently the folding rib reflectors from Harris Corporation [16]. In recent decades, the truss-type reflectors have attracted 9 significant attentions: deployable tetrahedral truss concept was presented by [35, 5, 77], Pactruss reflector concept was proposed and evaluated in [34, 75], tension-truss concepts were developed in [59] and [115], truss-supported mesh modules was structured in [53] and the tensegrity concept is also introduced by [105]. As one of the most renowned truss-type design, the AstroMesh reflector [103] from Northrop Grumman use the perimeter truss design, which along with the previous folding rib design is the current state-of-the-art on the development of DMRs. For any design concept of DMRs, the mesh plays a dominant role in determination of the total weight and surface accuracy of the reflector and therefore optimal design of the mesh geometry for the working surface is essentially important to the performance of the mesh reflector [64, 105]. As stated previously, a typical working surface of DMRs is formed by a mesh of triangle facets, with their edges being viewed as structural members. In most design cases, the mesh facets of a DMR is required to have geodesic or pseudo-geodesic property, by which the structural members and nodes of the reflector are either on or very close to the geodesic lines of the designated surface curvature (see Ref. [102] for instance). This geodesic or pseudo-geodesic mesh geometry leads to a minimum total length of members, which in turn renders a minimized total weight of the structure. Early work on mesh geometry design for mesh reflectors was performed by Nayfeh, et al [64] in 1979, in which the triangle facets were first generated on the pyramid and then projected onto the designated surface of a DMR. Miura and Miyazaki [59] used this method in development of the concept of tension truss reflectors. Bush, et al. [5] used an arc division approach to achieve unified facet geometry for the surface of deployable truss reflectors. Kenner, et al. [43] designed the facet geometry of tetrahedral truss reflectors, which later on was adapted by Mikulas, et al. [56]. While these investigations focused on achieving the uniform of facets size and facet edges, the mesh geometries generated are not geodesic, and they all have difficulties to compromise the aperture rim. Tibert, et al. [105] proposed an approach similar to that by Nayfeh [64], which draws triangle facets on the XY plane before the vertical projection, and in a follow-up work [106], Tibert used a force density approximation to revise the facet geometry on the XY plane with minimum member length configuration. Morterolle, et al [62] adapted a similar force density approach and vertically projected the minimum total length geometry of the XY plane onto the 10 offset-feed parabolic reflector surface while maintaining the uniform tensions of the members. Although the XY plane geometry is geodesic, the design methods proposed in these studies do not guarantee the geodesic property on the three-dimensional working surface. A mesh design method recently proposed by Liu, et al [51] is focused on the uniform member tensions and uniform facets of a mesh configuration, but pays no attention to geodesic property of mesh geometries. Thomson, et al. [104] presented a pseudo-geodesic design for a perimeter-truss DMR that was used for AstroMesh deployable reflectors [101, 102]. However, its detailed description of the geometry design method is not seen in these articles. In the open literature, few research works are available on systematic design of surface geometries with geodesic or pseudo-geodesic property for deployable mesh reflectors. In the current study, a new design method is developed for optimal design of the geometry of mesh facets for DMRs with aperture rims, among which the perimeter-truss DMR is a type of state-of- the-art space structures. The proposed design method systematically generates the surface geometry of a DMR with assured pseudo-geodesic property, and simultaneously fulfills the operating frequency requirements for the reflector. This method is applicable to both spherical and parabolic working surfaces, with either center-feed configuration or offset-feed configuration. This chapter proposes the optimal mesh geometry generation for the working surface of DMRs which is the first block in the Active Shape Control architecture (as in Figure 1.4). It will present the design methodology and the detailed formulation which leads to seven design parameters first. Then the complete optimal design procedure will be presented via systematic tuning of seven design parameters and the generated mesh geometry of DMR’s working surface after deployment will guarantee the pseudo-geodesic property and the satisfactory of surface accuracy requirement according to the operating frequency in the reflector application, and a comparative study will address the strength of the suggested method among the existing design approaches. The remainder of the chapter is arranged as follows. The problem considered in this study is stated in Section 2.2. The basic steps of the proposed design method and the surface accuracy evaluation are presented in detail in Section 2.3. Section 2.4 addresses the additional concerns on the formulation focusing on the design of the reflector’s effective region and the compensation of best-fit surface deviation and the design criteria the proposed method considers 11 are presented in Section 2.5. Discussions of the influence of seven design parameters on the performance of the resulting reflector are given in Section 2.6. Then the complete optimal design procedure is proposed in Section 2.7 via systematic tuning the seven parameters, followed by the comparison of the current effort with previous research in Section 2.8. The proposed method is illustrated on three design examples in Section 2.9. Finally Section 2.10 summarizes the contributions and applications of the work. 2.2 Problem Statement In design of the shapes of DMRs, two design tasks are to be carried out. After the deployment the working surface (the front net) of reflector is deformed onto a desired working shape. Define the mesh geometry to be the nodal coordinates and member connectivity of the mesh facets. Then in the first task, the proper mesh geometry at the desired working shape need to be designed. Thereafter, the undeformed profile of the DMR is determined in the second task, which when fully deployed has a deformed surface that is exactly at the desired working shape designed previously. Figure 2.1 Desired working shape of a deployable mesh reflector: (left) a center-feed case or an offset-feed spherical case; (right) an offset-feed parabolic case. This chapter will resolve the optimal solution regarding the first design task. The desired shape of the deployed working surface of reflector is defined in Figure 2.1. This working shape can have either a center-feed configuration or an offset-feed configuration, where F ( p F ), D ( p D ) and off e are the required (parent) focal length, the (parent) aperture diameter and the offset distance, respectively. The objective of this research is to develop an optimal design method that systematically generates the mesh geometry of the working surface of DMRs after deployment. 12 The designed geometry guarantees the surface accuracy requirement according to the operating frequency of the reflector applications and is always of pseudo-geodesic property. 2.3 Basic Design Steps In this section, the methodology of generating the mesh geometry of a desired working surface is first presented in 9 design steps and the surface accuracy evaluation of DMRs is then introduced. In surface mesh geometry designs for offset-feed reflectors, the central hub configuration [46] is adopted. 2.3.1 The proposed design steps 1. Determine the surface root-mean-squire (RMS) error at the desired working shape. The surface accuracy of DMRs is described by the root-mean-squire (RMS) error between the mesh facets of the reflector surface and the desired working shape. It has been known that the allowable RMS error of the working surface of a mesh reflector is restricted by the operating frequency or equally the wavelength λ of the reflector. Depending on applications, different budgets for surface accuracy related to the facet geometry have been suggested [101, 60, 107], such as λ δ = rms N (2.1) with N = 50, 75, 100, 150 or 200. 2. Determine a reference sphere One key in the proposed method is to define a spherical reference surface that has the same focal length and aperture diameter as the desired working surface of the mesh reflector to be designed. The mesh geometry (the nodal coordinates and member connectivity) of the reference sphere is first determined, and it is then projected onto the desired reflector surface. If the desired working shape is also spherical, no geometry projection is needed. In this case, by placing the aperture of the designed surface on XY plane with the vertex A on the positive z axis 13 (Figure 2.2), the reflector should face down along negative z direction. It follows that radius and height of the spherical working shape and its reference sphere are calculated by () 2 22 2 2 2 2, 4 ++ + − = ==−− ss sss xy z R H R D RF H R R (2.2) If a center-feed parabolic reflector is to be developed, the geometry of the corresponding reference sphere is still given by Eq. (2.2). The working surface of the reflector is described by () 2 22 1 , 416 −=− + = pp D zH x y H FF (2.3) Because the focal ratio / FDof a typical center-feed parabolic DMR is usually above 0.25, the paraboloid is always within its reference sphere and < p H H . Figure 2.2 A reference sphere for the designated shape. If an offset-feed parabolic reflector is considered, the parent paraboloid (for the working surface) and the corresponding reference sphere are defined in different coordinate systems. In Figure 2.3, the parent surface is defined in the global coordinates g gg x yz and xyz is the local coordinates in which the design approach is implemented. The parent parabolic surface is placed vertically down with its vertex on the origin g O of the global coordinate system with the parent diameter p D and the parent focal length p F given. The offset reflector is cut off from the parent surface by a cylinder which has a radius c R and an offset distance off e from z axis and the offset reflector has the same focal length with the parent reflector as = p F F . 2 = ca c D R denotes the 14 diameter of the circular aperture of an offset reflector and the focal ratio of offset parabolic reflector is usually defined as / ca F D . The points s P and r P are the intersections of parent parabola and the cutting cylinder in XZ plane, and the point O is the middle of the chord     s r PP . The contour of the reflector has the minimum distance z e from x axis at s P . The aperture of the reflector is elliptical and coplanar within local XY plane. Since the parent diameter p D , the focal length p F and the offset distance off e are known, the offset ratioγ off is defined as 2 γ = off off p e D (2.4) Figure 2.3 The working shape of offset parabolic reflectors in XZ plane. With the geometry shown in Figure 2.3, the reflector parameters are obtained as follows 222 11 ,, 4 2 16 16 γ =− = = poffp cp off g z pp DD RD e H e F F (2.5) () 0 1 2     +     =          −+  ooffc o o zg xeR y z eH (2.6) 15 tan( ) 2 ϕ − = g z off c He R (2.7) where g H is the height of the parent surface, (, , ) oo o x yz are the global coordinates of point O, and ϕ off is the rotation angle between the local coordinate xyz and the coordinate system ˆˆˆ xyz . Now consider the coordinate transformation () ϕ         =+       g o g y off o g o x xx yR y y zzz (2.8) where () ϕ y off R is the rotation matrix around y axis. With the parent parabolic surface described in the global coordinates () 22 1 4 =− + g gg p zxy F (2.9) once applying the coordinate transformation, the desired working surface in the local coordinates xyz is given by () () () () () 2 2 1 sin cos cos sin 4 ϕϕ ϕ ϕ   −+ +=− + ++     off off o off off o p x zz x z x y F (2.10) and the elliptical reflector aperture described in the local coordinates is by 22 22 1 += ′′ ee xy ab (2.11) where () cos ϕ ′ = c e off R a and ′ = ec bR . By setting 0 = x and 0 = y in Eq. (2.10), the height p H at the center of the reflector can be obtained by solving the equation 2 22 11 sin ( ) cos( ) sin( ) 0 42 4 ϕϕ ϕ  ++ ++=    o off p off off o p o pp p x HxHz FF F (2.12) 16 with the condition 0 ≥ p H . The reference surface is defined to pass the XY plane circle that is externally tangent to the reflector aperture at l P and r P . The reference surface is also defined to share the same focal length with the reflector as shown in the equation below. Although the focal ratio / pp FD of the parent surface may be small [107], the focal ratio of the reference sphere / FD is usually larger than 0.25, as has been seen in many design cases. This assures the existence of the reference surface by the following equation with < p H H . () 2 22 2 2 2 2, , 2 4 ++ + − = ′ == = =− − s s eps ss xy zR H R Da F FR F D HR R (2.13) 3. Generate geodesic curves on the reference surface. Figure 2.4 Subdivision separation of the reference surface. Figure 2.5 Geodesic rings and subdivision lines on the reference surface. In the proposed method, a mesh geometry is first generated on the reference surface. This is done by dividing the reference into subdivisions and radial layers using geodesic curves. The reference surface is first divided into subdivisions by geodesic curves (subdivision lines) passing the vertex A; the subdivisions are axis-symmetric with respect to the z axis, as shown in Figure 17 2.4. The number of subdivisions s n is an even number, such as 4, 6 and 8. A subdivision is further divided into geodesic rings; see Figure 2.5. The portion of a ring inside every subdivision has two intersection nodes with two subdivision lines and is geodesic of the reference surface passing those intersections. The size of a facet is bounded by the distance between the two neighboring rings, and because of this, the edge of a facet that ends at two neighboring intersection nodes on a subdivision line must be shorter than the allowable facet length. In general, facets are non-uniform throughout the reflector surface. With the assumption of equilateral triangle facets on the designated spherical surface, the maximum allowable length f l of the flat facets under the surface accuracy requirement is determined by 4 415 δ = f rms l F DDD (2.14) This formula is proposed in Reference [1] and confirmed in References [101, 31]. Although all facets in the current investigation are flat and without mesh saddling, one can design mesh geometry with mesh saddling effect by using a facet length formulation in [30] and adapting the method proposed in this chapter. Figure 2.6 The facet edge in the great circle. Considering the facet edge as a chord in the great circle of the surface (see Figure 2.6), the radian φ f of the relative arc for the allowable chord length can be computed by 18 sin 22 φ = f f s l R (2.15) Similarly denote φ t as the total radian for half of the reflector surface in the great circle, which is calculated by sin 2 φ = t s D R (2.16) Then it follows that the minimum number of the rings r n is given by φ φ  =    t r f n round (2.17) In other words, the actual number of the rings on the spherical surface r n must be larger than r n . φφ tf in Eq. (2.17) is derived below and it is the function of D F andδ rms D . 1 1 4 1 sin 4 2sin 15 φ φ δ − −    =    t f rms D F D DF (2.18) Figure 2.7 Rings’ intersections on one subdivision. Thus, the surface is divided into r n layers by r n concentric rings from the center to the outer of the reflector. Figure 2.7 shows such layer separation on one subdivision line. Since the 19 subdivision lines are on the great circles of the reference sphere, the radian of the half of the reflector surface on the subdivision is also φ t . Define () φ i as the radian of the arc on the subdivision between the vertex A and ith ring and then it is the total radian of the first i layers along the subdivision. The radian of each layer on the subdivision presents its thickness and we use () wi to describe the relative radian, or in other words the relative thickness of every layer. By default, all layers share the same thickness (radian) and their relative thickness are all 1 as shown below ()11,2, ==  r wi i n (2.19) As the total radian of all r n layers in the half of the reflector surface is φ t , the radian of i layers () φ i can be calculated by () () () 1 1 1, 2, , φ φ = = ==    r i t r n i i iwii n wi (2.20) 4. Generate nodal indices and coordinates. Figure 2.8 Nodes on the rings within one subdivision. Nodes of the mesh geometry are defined on the rings (Figure 2.8). On the portion of ith ring inside each subdivision, including the nodes on both subdivisions, nodes equally separate the ring portion and hence the relative chords or the facet edges are equal. Because of the symmetry of the subdivisions, all the facet edges with their vertices on the same ring are of the same length and this assures that all the nodes of the surface are on the geodesics. According to the topology of Figure 2.8, within each subdivision the outer ring has one more node than its inner neighbor ring. Therefore, on each ring the total number of nodes is 20 1, 2, , ==  is r nin i n (2.21) where s n is the number of subdivisions of the reflector, and the total number of nodes of the reflector is 1 1 11 2 = + =+ = +  r n r total s r s i n nin nn (2.22) Figure 2.9 Nodes inside one subdivision. For convenience of analysis and discussion, the nodes are labeled by a pair of index numbers { } , , = ij Ni j (2.23) where i presents the ith ring and j the jth node on the ring. The vertex A can be denoted by 0,1 N . With above index system, the coordinates () ,, , ,, ij ij ij x yz of the nodes on the geodesics are determined as ,, , ,, , ,, sin cos 1, 2, , 1 sin sin 1, 2, , cos θα θα θ = =− = = =−+   ij s i j i j r ij s i j i j i ij s i j s xR in yR j n zR R H (2.24) ()() 0,1 0,1 0,1 , , 0,0, = x yz H (2.25) 21 ,, ,, , cos 2 sin 1, 2, , 2 0 α α = == =  rr rr r nj nj nj n j r s nj D x D yj nn z (2.26) where , θ ij and , α ij are two variables shown in Figure 2.9. It is easy to see that () , 2 1 π α =− r nj rs j nn (2.27) To obtain the values of , θ ij and , α ij for 1, 2, , 1 =−  r in , two different types of nodes should be considered separately: the nodes on the subdivision lines and the nodes which are not. The detailed formulas are given in Appendix A based on the previous three steps. 5. Construct the converging treatments on the facet size and ring curvature Although all the nodes are on the geodesics, from the top-view of the reflector structure the curvatures of the geodesic rings are very different from the curvature of the outside aperture. This commonly results in the larger facet areas and surface RMS errors in the outer region of the surface. To resolve this problem, two converging treatments are necessary. The first treatment is to adjust the curvatures of the rings and modify the nodal coordinates of the nodes which are not on the subdivision lines. For the rings closed to the aperture which are defined as converging rings, their curvatures after projected onto XY plane are increased from inside to outside gradually and gently converge to the aperture curvature. Set the largest geodesic ring as the 1 − c n th ring and all the outer rings except the outside rim as the converging rings. Then the c n th ring is the first converging ring from inside and its curvature in XY plane is1/ c n r where c n r is calculated in Appendix B. By considering that the curvatures of the converging rings in XY plane increase from inside to outside as a power function, the radius of each converging ring i r in XY plane is defined below with the available range of c n as11 ≤≤ − cr nn . () ,1, , 1 11 1 1 as 1 1 sin ρ φ  =+ −   − =− +     ≤≤ − −     cc cc r c cr is n r c n in n n in nn rR i r n n r (2.28) 22 with ρ as a proper positive number chosen. The second treatment is due to the non-uniform of the facets and it adjusts the nodal coordinates of the nodes on the subdivision lines. Without the converging rings, all nodes are exactly on the geodesics of the reference sphere and all facets are quite uniform. By introducing the converging rings, the nodal positions are slightly adjusted and facets in the outer layer intend to possess larger size than facets in the inner layer. The compensation to reject this increasing trend on facet length is necessary and therefore the following treatment is presented. Define the ratio of maximum member length in outmost layer of the working surface versus the maximum member length in the first layer isς . Since the outmost layer is r n th layer, a power function is defined to describe the converging thickness weight of each layer and the relative thickness functions in Eq. (2.19) is modified as below. Note the power functions defined in this step share the same exponential coefficientρ . () () 1 1, 2, 1 2 1 11 , 1, , 2 ρ ς ς ς +  =−   =   − −  +− + = +   −     c c cc r rc in wi in in n n nn (2.29) Once the above converging functions are available, those treatments must be applied onto the calculation of the nodal coordinates () ,, , ,, ij ij ij x yz for the mesh geometry on the spherical reference and their detail derivations can be found Appendix C. 6. Generate the connections of members at nodes Figure 2.10 Topology of member connections within one subdivision. Once all the nodal coordinates are obtained, the member of the reflector can be defined by the connections of the nodes and the topology of such connections is shown in the figure 23 above. Due to the different locations, the members are of two categories. In the first category, members are inside the layers. Since the rings divide the reference into multiple layers, when the ends of members are not on the same ring, they are called to be inside the layers. Inside the first layer, the members are all connected to the vertex A and the connections are defined below by nodal indices of two member ends. { } 0,1 1, ,1,2,, ==  kj s M NN j n (2.30) Starting from second layer and without repetitively counting the members, each node on ith ring has three members connected inside the 1 + i th layer if it is on the subdivision line or two members when it is not. Since the first node on ith ring is always on the subdivision lines, we have its connection k M within 1 + i th layer defined by { } {} {} ,1 1,( 1) ,1 1,1 ,1 1,2 , ,1,2,,1 , ++ + +    ==−      s ii in ki i r ii NN MNN i n NN (2.31) For the node on the subdivision lines, its index satisfies 1( 1) 1,2, , =+ − =  s j qi q n (2.32) Therefore, within 1 + i th layer every node satisfying the above equation connects three members defined in Eq. (2.33) while other nodes are with two members with the connections described in Eq. (2.34). { } {} {} ,1, ,1,1 ,1,2 , 1, 2, , 1 ,2,3,, .(44) , + ++ ++  =−    ==   ∈      ij i s r kijis s ij i s NN in M NN j in jEq NN (2.33) {} {} ,1, ,1,1 1, 2, , 1 , 2,3, , , .(44) + ++  =−   ==   ∉     r ij i s ks ij i s in NN M jin NN jEq (2.34) 24 where 1, + is N is the second end index of the last member connection at the previous node on the same ring. For example when considering the member connections of the second node in ith ring, the last member connection of the first node is { } ,1 1,2 , + ii NN and hence 2 = s . Members in the second category are those on the rings. On each ring, the members are connected between every two neighbor nodes. Then their connections can be described as { } {} ,,1 ,,1 , 1, 2, , 1, 2, , 1 , +  =   =  =−      i ij ij r k i in i NN in M jn NN (2.35) To calculate the number of members, it should be counted by layers. For ith layer, all members include the members inside this layer and the members on ith ring. Therefore the number of members of ith layer is () 31 − s in and the total number of members is () 1 31 31 2 = + =− =  r n r total s r s i n min nn (2.36) It can also be counted that each layer has () 21 − s in facets and then the total number of facets is () 2 1 21 = =− =  r n total s r s i f in nn (2.37) 7. Reduce the number of nodes in the boundary layer Figure 2.11 The member connections in the boundary layer. 25 When only inner portion (effective region) of the reflector’s mesh surface is in function of transmitting signals, the outside boundary layer can be specially designed. For the practical purpose, the number of nodes on the aperture rim needs to be smaller than the number of facets in the boundary layer to enable the high frequency applications; fewer nodes on the boundary rim would reduce the complexity of the reflector’s supporting structures and the difficulty of deployment. Therefore, in our method the member connections in the boundary layer are specially designed in this step. Assuming i n nodes are on the aperture rim and each node has B c connections to B c nodes on the inner ring. B c is an odd number and larger than 1 because the number of nodes can’t be reduced if B c is 1. These connections follow two rules: first, along all connected nodes on 1 − r n th ring, the node positioned in the middle is the only one that connects one node on the aperture rim or r n th ring; second, other 1 − B c nodes must connect two neighbor nodes from r n th ring. The configuration of such connection inside boundary layer is illustrated in Figure 2.11.It can be counted that the number of nodes is reduced to be () 21 + B c of the number of nodes on 1 − r n th ring, or mathematically () () 21 1 − == + rs ir B nn nin c (2.38) Hence the total number of nodes in the entire surface is (2.39) Similar to the derivation in Eq. (2.26) and (2.27), the nodal coordinates are obtained by the following equation () () ,, ,, , cos 2 21 sin 1,2, , 21 0 α α = − == + =  rr rr r nj nj rs nj n j B nj D x nn D yj c z (2.40) where 26 () () () , 1 1 1 π α + =− − r B nj rs c j nn (2.41) While the members on the boundary rim is still described by Eq. (2.35), instead of Eq. (2.33) and (2.34) the members within the boundary layer are redefined as { } {} {} ,1 1,( 1) 1 ,1 1, 1 ,1 1,1 , 1 ,1,2,, 2 , −− −+ −+ −   −  ==      rr r s rr rr nn n nk B kn nk nn NN c MN N k NN (2.42) () () () ,1 1 1, 1 22 21 2,3, , 1 , 1, 2, +− −−−+ −  =   + =    =    rB B r rs B knj c c nj k B nn j c MN N kc (2.43) And the number of members in the boundary layer is obtained below and is independent with B c . () 21 =− Br s mn n (2.44) The number of facets in the boundary layer is () 2 1 1 =− + B Br s B c f nn c and it is B c times of the number of the nodes on the aperture rim. From Eq. (2.36) and (2.44), the total number of members for the entire reflector is then () , 32 1 2 + =− r total B r s n mnn (2.45) 8. Map the geometry from of the spherical reference onto an ellipsoidal intermediate surface The purpose of this step is to build up a bridge between the circular aperture of the reference sphere and the elliptical aperture of the desired working surface for an offset-feed parabolic reflector. (This step is not needed for either spherical reflectors or center-feed parabolic reflectors.) First, an ellipsoid is defined as an intermediate surface which has the same origin 27 with the spherical reference and projects circles onto XZ plane and ellipses onto XY plane. To ensure the ellipsoidal surface passes the reflector aperture rim, it is defined as () 2 22 22 2 1 2 , +− ++ = ′ == s ee e s ese e zR H xy ab a R aRb b D (2.46) Then a set of mapping functions is designed to perform two tasks: in the same horizontal plane, it should uniquely project the points from the circle of the spherical reference to the ellipse of the ellipsoidal surface; if all pairs of neighbor points on the circle result in equal chords (equally separate the circle), the mapped points on the ellipse should do the same (as shown in Figure 2.12). Figure 2.12 The mapping between the spherical reference and the ellipsoidal surface in XY plane. Using this function, the nodes of spherical reference are horizontally mapped onto the ellipsoid while the z coordinates of nodes remain unchanged. Since all nodes of the reference sphere are on rings and different rings have different numbers of the nodes, ideally one mapping function is needed to map the nodes of each ring from the sphere to the ellipsoid. To build up the mapping function for ith ring, the following problem is first to solve. Assume in XY plane s in nodes (the 28 same number of nodes on ith ring) are equally separate the circle of which the center is at the origin, the ellipse which nodes are mapped onto is internally tangent to the circle at two ends of its major axis and has the same ellipticity ee ba with the reflector aperture, and the chords between every two neighbor nodes on the ellipse are all equal. The normalized coordinates in XY plane of the mapped nodes () ,, , , , υυ xij yi j are derived as () () ,,1 , ,1 ,, , , , , 1 ,, 1 1, 0 cos 2 , sin 2 , 2, υυ υπηυ πη ηε − =   ==   == =     =    xi y i e xij i j yi j i j s e j ij ik k b j in a (2.47) with , ε ik as normalized polar angle increments at the kth chord and , η ij to be the dimensionless polar angle of jth mapped node. Applying the equal chords assumption, , ε ik can be obtained by solving the following 1 − s in equations, and so does , η ij . ()( )( )( ) ()( )( )( ) 22 2 2 ,, 1 , , , , 1 ,, ,, 2 , , 1 ,, 2 , , 1 22 2 2 ,, , , 1 , , , , 1 ,, ,,1 , , , ,1 ,1, , 2 υυ υ υ υ υ υ υ υ υ υυ υ υ υυ ++ ++ ++ −−  −+ − = − + − = −    −+ − = − + −   ss s s s s xi j x i j y i j y i j xi j x i j y i j y i j s x i in x i in y i in y i in x i in x i y i in y i jin (2.48) Define , η ij to be the mapping function value and the number set 1 − s j in as its independent variable which presents dimensionless polar angle of nodes on the circle. The mapping function for the ith ring , mi f is expressed as a cubic spline by the curve fitting techniques (shown in Figure 2.13) and is constructed by a finite number of piecewise third-order polynomials. 29 Obviously () ,, (1 ) η=− ij mi s fj in . Denote , α ij to be the polar angle of nodes on the ellipsoid surface in XY plane, we have , ,, 2( ) 2 α απ π = ij ij mi f (2.49) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (j-1)/in s η i,j mapping function (fitted) Figure 2.13 The creation of the mapping function. Since the mapped node () ,, , ,,   ij ij ij x yz on the ellipsoid of Eq. (2.46) is at the same height of the node on the spherical reference before mapping, the nodes of the secondary reference are calculated by () () 2 2 ,, , 2 2 ,, , ,, cos sin α α =− +− =−+− =      ij e i j s ij e ij e i j s ij e ij ij xa z RH b yazRH a zz (2.50) ()() 0,1 0,1 0,1 , , 0,0, =   x yz H (2.51) For the mesh geometry design with r n rings, r n mapping functions are needed. Practically, however, the mapping function at the certain ring has the capacity to accurately present most of 30 the other mapping functions and such uniform property reduces the number of the mapping function and therefore the complexity of the design calculations. The detailed discussion of this can be seen in Appendix D. 9. Project nodes of the reference sphere onto the desired working surface This step is needed when the desired working surface is parabolic and when pseudo-geodesic property of the parabolic geometry is to be ensured. Once the nodes and members are generated on the reference surface, all the nodes should be projected onto the desired working surface (shown in Figure 2.14) while all the member connections remain unchanged. Assume the projection center is () 0,0, p z with < pp zH and the projected node on the desired surface is () ,, , ,, ij ij ij x yz . For both the center and offset cases, the desired working surface is closed to the reference surface which is determined by the focal ratio and the offset ratio of typical DMRs. Therefore, the projection will adjust the geometry of the working surface in a gentle manner. The proper projecting center p z is obtained using the projection method [64] briefly described below. From the equation of straight line, we have Figure 2.14 Nodal projection in the plane passing z-axis. 31 ,, , , ,, , ,, , , ,, , center-feed offset-feed ξ ξ − == = − − == = −   ij ij ij p ij ij ij ij p ij ij ij p ij ij ij ij p xy z z xy z z xy z z xy z z (2.52) Substituting Eq. (2.52) into Eq. (2.3) and (2.10), two second order equations are obtained for the center-feed and offset-feed cases respectively, () () ,, 22 2 ,, , 1 0 center-feed 4 ξξ ++− +−= ij ij ij ij p i j p p xy z z z H F (2.53) 2 2, 1 , 0 0 offset-feed ξξ ++=   ij ij bb b (2.54) in which () ()( ) () () () ( )( ) () ()( ) () () () () () () 2 2 2, , 1, , , , 2 0 1 cos sin 4 1 sin cos cos sin sin 2 1 cos sin 4 ϕϕ ϕϕ ϕ ϕ ϕ ϕϕ =+ −+ =− + − + + − + =++ +        off i j off i j p o p off i j off i j p off i j off i j p off p o p off p o off p o p bx zzy F bx zz x zz zx F bzz zx F (2.55) Solving above equations is trivial but two solutions will be obtained. However, only the solution which results in the same signs between the nodal coordinates of the reference node and the projected node can be considered as the solution. From Eq. (2.52) , ξ ij must be larger than zero to ensure X or Y coordinates of nodes sharing the same signs before and after the projection. Combining the condition that , 0 ≥ ij z , we have the following criteria to choose the solution: () , ,, 0 0 ξ ξ >    −+ ≥   ij ij i j p p zz z (2.56) Particularly, when the projection center is in the negative z domain, the solution always results in being larger than zero and only the second condition of Eq. (2.56) is needed. Based on above 32 derivations, one may scan the projection center along the z-axis and the final selected projection center is denoted as , prms z which minimizes the upper bound of the surface RMS errorδ rms defined as () 2 max 1 16 15 δ = i rms l F (2.57) with i l being a length of a facet edge. For each chosen p z , the projected mesh geometry on the desired surface can be calculated in Eq. (2.52). Once the above 9 steps are carried out, the nodes on the working surface are indexed by , ij N and their coordinates are calculated as () ,, , ,, ij ij ij x yz . The connectivity of all the members is described by the matrix k M . The entire reflector surface is constructed by the projection of the geodesic rings and the converging rings closed to the aperture. Although only the projection formulation for parabolic shape is presented in this chapter, the proposed method is generally applicable to any surface of a revolving solid. 2.3.2 Surface accuracy evaluation Once the mesh geometry of reflector is generated, the surface accuracy which dominates the performance of DMR must be measured. Since many DMRs only have the inner portion of the working surfaces used for the signal transmission, the mesh facets within such portion are denoted as effective facets. Then the actual surface accuracy , δ rms act is defined to be the RMS error between the effective facets and the desired working surface. Due to the complexity of the measurement on the actual surface accuracy, the best-fit surface RMS error is used to evaluate the surface accuracy of the generated mesh facets. To describe this error, the best-fit surface is 33 first defined to be a surface with the shape that leads to the minimum RMS error between the effective facets and such surface. Thus this minimum RMS error is named as the best-fit surface RMS error, or mathematically , δ rms bf . In this work, the best-fit surface RMS error is evaluated by the general approach presented in the appendix of Ref. [1] which rigorously calculates , δ rms bf through exact equations of geometries without approximations. Note that for the offset parabolic reflector, the global coordinates of the effective facets should be used when calculating the best-fit RMS error. One must know that the best-fit surface RMS error does not include the surface deviation between the best-fit surface and the desired working surface where the facet vertices are located and according to the definitions, , δ rms act is not equal to , δ rms bf . However, after applying the best-fit compensation technique which will be seen immediately in Section IV, the best-fit surface by the generated effective facets will accurately present the desired working surface and , δ rms act is evaluated by its best-fit RMS error. By overviewing the proposed method, it is found that seven parameters need to be determined to fully design the surface mesh geometry: the number of rings r n , the number of subdivisions s n , the index of first converging ring c n , the power coefficient ρ of curvature convergence functions, the number of members B c which the node on the reduced-node boundary rim connects, the relative thickness of boundary layer B w and ς presenting the member length difference between the outmost layer of the working surface and the first layer. As tuning these seven parameters, the generated mesh geometry will be modified by following through the design steps from Step 2 to Step 9. 34 2.4 Additional Design Considerations Two important issues are considered besides the basic design steps stated before: 1. the mesh surface of reflector may not be fully used to transmit signals and only part of the surface is effective, which requires different design formulation; 2. the best-fit surface of the generated mesh facets always has deviations from the desired working surface, when the vertices of the mesh facets are located on the desired working surface as proposed by the basic design steps in the previous section. 2.4.1 Design formulation when designing the “effective” region of the working surface In many applications, only a major portion of the reflector’s working surface (namely the effective region) serves for signal transmission and the mesh facets within the effective region are the effective facets. As such, the design objective is to meet the operating frequency requirements on the effective region of a working surface with specified effective diameter effective D ( , p effective D ), the effective focal length effective F ( , p effective F ) and the effective offset distance effective e . The boundary layer attached outside of the aperture of the effective portion can be designed by adding one additional ring as the boundary rim, where a node-reducing technique described in Step 7 can be utilized. To integrate this consideration in the design process, we formulate the boundary layer by extending the actual size of the desired reflector while remaining the geometry within the effective aperture. This is achieved by first finding proper D , p D , F , p F and off e for a regular design procedure to begin with. 35 First, it should be noted that attaching the additional layer outside the effective aperture will not change the focal length of the surface and = effective FF , , = p p effective FF . Also the offset reflector has the same focal length with its parent reflector as , = effective p effective FF . Because one boundary layer is added outside the effective region, instead of using Eq. (2.17) and (2.18), the number of layers of the effective region is obtained by the following equations () 1 1 φ φ  − =+    r r f n n round (2.58) () 1 1 4 1 sin 4 1 2sin 15 φ φ δ − −    −  =     effective effective r f effective rms effective effective D F n D DF (2.59) Although the diameter and number of layers of the reflector have been changed, the consistence of the mesh geometry in the effective region is guaranteed as long as the following two conditions are met: (i) the diameter of 1 − r n th ring of the surface remains as effective D ( , p effective D ); and (ii) () wi of first 1 − r n layers remain the same after r n th layer is added as the boundary, which ensures the radians of those layers to be consistent. The second condition can be fulfilled by separately assigning the thickness of the boundary. Then the relative radian of each layer () wi is redefined below by introducing a design variable B w to describe the thickness of the boundary layer. () 11,2, 1 =−  =  =   r Br in wi win (2.60) And () φ i is evaluated by substituting Eq. (2.60) into Eq. (2.20). 36 The formulation of the first condition is addressed into three cases which leads to the derivation of D , p D and off e . The simplest case is to design a spherical reflector. According to Figure 2.7, we have () () () () 1 1 1 1 φ φ − = = − =   r r n r i n r i wi n n wi (2.61) The above equation adds the boundary layer with its thickness proportional to the total thickness of all the layers in the effective region and therefore does not affect the relative thickness of each layer within the effective aperture. Since the diameter of the effective portion is effective D as stated in the first condition, similar to Eq. (2.16) we have, () () () () sin 1 , sin 44 φφ −= = effective rr effective D D nn FF (2.62) Therefore, () () 1 1 1 1 4sin sin 4 − = − =     =         r r n effective i n effective i wi D DF F wi (2.63) Figure 2.15 The effective region on the center-feed parabola (XZ plane). 37 In the second case, the center-feed parabolic reflector is concerned. Since the parabolic surface is closed to the reference sphere (in Figure 2.15), 4 effective D F is a very good approximation of () () sin 1 φ − r n and Eq. (2.61) and (2.62) are also valid for the center-feed parabolic design. Therefore D is calculated by Eq. (2.63). In fact, the effective aperture on the spherical reference based on such formulation is slightly larger than the desired effective aperture of the paraboloid. Thus after the projection later in Step 9, it guarantees that the designed mesh geometry always has a slightly larger effective diameter than the one required. Figure 2.16 The effective region on the offset-feed parabola (XZ plane). The final case is to design the offset-feed parabolic surface (shown in Figure 2.16). The mathematical relationship between the effective parent diameter , p effective D , the effective offset distance effective e and the effective reference diameter effective D have been expressed from Eq. (2.4) to (2.13),where the effective offset angleϕ effective is also calculated. Now the offset surface is transformed into the local coordinates and the formulation of the center-feed parabolic case can be used. Then the diameter of entire reference surface D is obtained by applying the equations from Eq. (2.58) to (2.63). Since the reflector is enlarged slightly, the offset angle of the actual reflectorϕϕ ≈ off effective and the effective reflector aperture is parallel to the actual reflector aperture. According to the geometry of Figure 2.16, we have 38 ()( ) , cos ϕ =+− p p effective effective effective DD D D (2.64) () cos 2 ϕ − =− effective off effective effective DD ee (2.65) Thus, the diameters of circular aperture ca D and the effective circular aperture , ca effective D are 2− poff De and , 2− p effective effective De respectively. Once D , p D , F , p F and off e are obtained in any of three design cases, the designated mesh geometry which focuses on the design of the effective region will be generated by following the regular design steps of Step 2, 4, 6 ~ 8 and modified Step 3, 5, 9. The above formulation on the surface parameters has stated the modification of Step 3, which is to use Eq. (2.58), (2.59) and (2.60) to replace Eq. (2.17), (2.18) and (2.19) respectively. In Step 5, if the design task is to focus on the effective region of the reflector surface, the converging function of Eq. (2.28) is modified by the following equation, the maximum available c n is 2 − r n and the curvature of 1 − r n th ring converges to a circular curvature. () ,1, , 1 11 1 1 as 1 2 sin 1 ρ φ  =+ −   − =− +     ≤≤ − −−     cc cc r c cr is n r c n in n n in nn rR i r n n r (2.66) Note that as the reflector mesh always has more than two rings including the boundary ring, r n is guaranteed to be larger than 2 and the above equation exists. For the case of the effective region, the converging function is especially important because it will ensure the XY- plane projection of the effective aperture ( 1 − r n th ring) on the reference to converge to an exact circle. Similarly with Eq. (2.29), the relative thickness function in Eq. (2.60) is modified as below. () () 1 1, 2, 1 2 1 11 , 1, , 1 12 ρ ς ς ς +  =−     − − =+ − + = + −   −−    =     c c cc r rc Br in in wi i n n n nn win (2.67) 39 in which ςis redefined as the ratio of maximum member length in 1 − r n th layer versus the maximum member length in the first layer. When the effective region is the design objective, the projection of Step 9 should avoid any shrinks on the effective aperture of reflector and the vertical projection is preferred as =−∞ p z . From Eq. (2.3) and (2.10), the projected nodal coordinates are calculated by Eq. (2.68). Note that the projected z-axis coordinates of the offset-feed reflector is the positive (larger) solution of the second order equation of Eq. (2.10). () ,, 22 ,,, , , 2 11 20 ,,, , , 2 1 center-feed 4 4 offset-feed 2 == =− ++ −+ − == =  ij i j ij ij ij i j ij p ij ij ij i j i j xx y y z x y H F bb bb xx y y z b (2.68) where () () () () ( ) () () () () 2 2 1, 2 2 0, , , 1 sin 4 1 cos sin cos 2 1 cos sin 4 ϕ ϕϕ ϕ ϕϕ = =+ + =++− +    off p off i j o off off p off i j o i j off i j o p b F bxx F bxxy xz F (2.69) 2.4.2 Compensation for the deviation of the best-fit surface from the desired surface Figure 2.17 The surface deviations in XZ plane. 40 As stated previously, the best-fit surface of mesh facets is a surface with the shape that minimizes the RMS error between the effective facets and such surface. It is known that the best- fit surface has deviation from the desired shape even when the facet vertices are on the desired surface (Figure 2.17). Such deviation can be expressed by a best-fit focal length and a vertical shift of the desired working surface [1]. To compensate this deviation, a straight-forward strategy is proposed as follows. Using the geometry of the effective facets, the exact geometry derivation in Ref. [1] will calculate its best-fit RMS error , δ rms bf and the best-fit surface is expressed by the vertical shift Δ bf H from the desired working surface and the focal length bf F of the best-fit surface. In the offset parabolic case, the best-fit surface is obtained in the global coordinates and its vertical deviation and focal length are for parent surface as , Δ pbf H and , pbf F . Then the diameter of the best-fit surface is () () () () 2 2 ,, ,, 2 4 2 spherical working shape 4 parabolic center-feed shape parabolic offset-feed shape 1 2 =− −+Δ =−Δ =− − =+ − bf bf bf bf bf bf p bf pbf p ca ca bf off bf off ca ca bf DF FHH DFH H DD D D ee D D (2.70) where , ca bf D is the circular aperture of the best-fit surface derived in Appendix E. For deviation compensation, a modified working surface (as shown by Figure 2.17) is defined in a similar way as in Figure 2.1 with its parameters given by () 22 2 ,, , , , , 1 2 == ==+− =− − mod bf mod bf p mod p p bf p mod p ca mod ca off mod off ca mod ca FFF D DD FFF D DD D e e D D (2.71) where 2 ,, = ca mod ca ca bf DDD . Thus the same set of the design parameters is applied onto this modified surface, which generates the modified mesh geometry as the final design result. The numerical investigations are carried out to evaluate the compensation results. First the best-fit surface by the modified mesh facets in the effective region is obtained with its diameter, focal length and offset distance as , mod bf D ( ,, pmod bf D ), , mod bf F ( ,, p mod bf F ) and ,, off mod bf e 41 respectively and the best-fit RMS error , δ rms mod of the effective mesh facets is computed. Then, the following six parameters can be used to quantify the deviation of the best-fit surface away from the original desired working shape, ,, ,, , ,, ,, ,, ,, ,, ,, δδ δ −− − == = −− − == = mod bf mod bf rms mod rms bf DF rms rms bf p mod bf p p mod bf p off off mod bf Dp F p eoff pp off DD F F rr r DF DD F F ee rr r DF e (2.72) Since the best-fit RMS error only measures the surface accuracy of the effective region and the boundary layer will not affect the geometry of the effective region, in this discussion the reflector has its whole surface as the effective region for convenience. The numerical studies are carried out under the same set of the design parameters ( 0.2 τ = c , 3.5 ρ= , 6 = s n , 0 = B c , 1 = B w andς is calculated by its definition) when the desired focal ratio /[0.3,3] ∈ FD , the desired parent focal ratio / [0.17,1] ∈ pp FD , the desired offset ratio [0.05,0.99] γ ∈ off and the total number of rings 10 ≥ r n . The results show to design a spherical surface, 0 0.0002% << D r ; F r is zero analytically; the decrease of rms r is dominated by increasing r n with 0.38% < rms r . As the parabolic case, F r and , Fp r are very consistent as zero with largest uncertainty at 0.0035% for center-feed designs and 6 2.4 10 % − × for offset-feed ones; D r , , D p r , eoff r and rms r are proportional to / FD , / pp F D , 1γ off and 1 r n with 00.075% << D r , , 0 0.0026% << Dp r , 0 0.052% << eoff r and maximum rms r at 5.56% and 3.00% for center-feed and offset-feed cases respectively. Regarding most of center-feed parabolic reflectors with their focal ratios less than 1, rms r is reduced below 1.8%. Obviously after applying the compensation strategy, the best-fit surface by the modified mesh geometry has almost the same focal length as required. The diameter of the best-fit surface is either almost exact to or slightly larger than the target diameter; the offset distance is sufficiently closed to or slightly less than the required, thus always resulting in either an exactly desired surface or a slightly larger best-fit working surface better than the desired. Therefore, such compensation strategy is demonstrated to be very successful and the best-fit surface by the 42 modified mesh facets accurately presents the desired shape and fulfills the application requirements. Furthermore, the RMS error of such best-fit surface , δ rms mod describes the RMS difference between the generated effective facets and the desired working surface, and is the same with the actual surface accuracy , δ rms act according to its definition. Denote the maximum value of rms r as rms r and we have 0.4% spherical surface 5.6% center parabolic surface & / 1 1.8% center parabolic surface & / 1 3.0% offset parabolic surface   >  =  ≤    rms FD r FD (2.73) Additionally, the numerical studies shows that when 15 ≥ r n and /1 ≤ FD for center parabolas or in offset-parabolic cases /1 ≤ ca FD , 1% < rms r . This implies that as long as () , 1 δδ<− rms bf rms rms r , the actual surface RMS error of final designed mesh facets , δ rms act is always smaller than the required, or mathematically ,, δδ δ =< rms act rms mod rms (2.74) 2.5 Design Criteria for Optimal Parameter Selection In the proposed design method, two design criteria need to be fulfilled regarding the deployed equilibrium and the surface accuracy of the deformed mesh facets. 2.5.1 Requirement on the force balancing equilibrium at designated deformed working status After all the nodal coordinates and member connections of a desired working surface are determined, the designed mesh facets must be statically indeterminate without mechanisms at its deployed working configuration. To this end, the global force balance equation of the deformed mesh facets of the working surface (the front net of the reflector) is constructed as follows [114] by assuming that the aperture nodes are fixed in space and the external loads are applied onto all the facet nodes. 43 [ ]{ } { } = MTQ (2.75) where [ ] M is the equilibrium matrix, { } T is the vector of member tensions and { } Q is the external load vector. Set e p to be the number of equilibrium equations and e q to be the unknown member tensions. Therefore the matrix [ ] M has dimension of × ee pq . Denote the rank [ ] M of as e r . Then it is preferred [71, 7] to have 0 −→ ee pr (2.76) 0 −> ee qr (2.77) When − ee pr is positive, the system is kinematically indeterminate and mechanism occurs. This situation has never been observed in use of the proposed method and − ee pr is always zero so that the system does not suffer any mechanism. The condition 0 −> ee qr guarantees infinite solutions of Eq. (2.75) when it is solvable. This property is very helpful when designing the undeformed profile of the reflector, and it is capable of delivering uniform tension distributions. 2.5.2 The surface accuracy requirement based on the operating frequency The effective region, as described previously, is the portion of the reflector surface serving for the signal transmission. After the mesh geometry of the working surface is generated, the surface accuracy of its effective region must be measured and regulated according to certain operating frequency. As defined, the actual surface accuracy , δ rms act is the RMS error between the effective facets (the mesh facets within the effective region) and the desired working surface. Denote the allowable surface accuracy restricted by the operating frequency as δ rms and it is calculated by Eq. (2.1). Then the actual surface accuracy of the generated effective facets should satisfy the following, , δδ < rms act rms (2.78) Due to the complexity of the measurement on the actual surface accuracy, previous researchers have proposed other surface RMS errors of the flat mesh facets. The simplest 44 evaluation of the surface RMS errors considers the RMS of nodal displacements of the facet vertices away from the desired working surface, or namely “nodal surface RMS error”. However, the nodal surface RMS error does not consider surface deviation of the facets and therefore could not provide the accurate measurement of the surface accuracy. Another surface RMS error which has been widely observed in the literature is defined by 2 , 1 16 15 δ = i rms apx l F (2.79) under the assumption of equilateral triangle facets throughout the entire mesh, and is decided by the lengths of the facet edges. However, although in the generated mesh geometry the facets are similar, they are not exactly uniform and their edges are not equal either. Therefore such surface RMS error presents no more than a rough approximation of the surface accuracy and has only been suggested to fast estimate the surface RMS error in the preliminary design stage before the mesh geometry of the working surface is determined [102, 31]. To accurately measure the surface error caused by the mesh facets, the best-fit surface RMS error is introduced to evaluation the surface accuracy. As described in Section 2.3.2, the best-fit surface RMS error is the RMS error between the designed effective facets and their best- fit surface. To calculate this error, Ref. [107] has suggested the formulation of “axial surface RMS error” which provides a closed-to-exact estimation for the best-fit RMS errors. However, the current study aims the accurate measurement of the surface accuracy and therefore the exact best-fit surface RMS error in the effective region denoted as , δ rms bf is evaluated by the rigorous calculating approach presented in the appendix of Ref. [1] using the geometry of the effective facets. Note that the global coordinates of the effective facets should be used to calculate the best-fit RMS errors of offset parabolic reflectors. According to the discussion in the Section 2.4, once the deviation between the best-fit surface and the desired working surface is compensated by the proposed technique, the modified mesh facets are generated on the modified working surface. The best-fit surface obtained by these mesh facets in the effective region accurately presents the desired working surface and therefore we have, 45 ,, δδ = rmsmod rmsact (2.80) Based on the discussion in the numerical investigations of the compensation technique in Section 2.4, the design result will always satisfy the operating frequency requirement in Eq. (2.78) as long as the best-fit RMS error by the effective facets before the best-fit compensation validates the following condition. Otherwise the design parameters may need to be adjusted. () , 1 δδ<− rms bf rms rms r (2.81) in which 0.4% spherical surface 5.6% center parabolic surface & / 1 1.8% center parabolic surface & / 1 1.0% center parabolic surface & 15 & / 1 3.0% offset parabolic surface 1.0% offset parabolic surface > ≤ = ≥≤ rms r FD FD r nFD & 15 & / 1          ≥≤   rca nFD (2.82) 2.6 The Influence of Design Parameters and Performance Evaluation As mentioned in the previous section, seven design parameters are involved. In this section, the influence of these parameters on the outcome and the performance of the designed mesh geometry are discussed. In the proposed method, without tuning any design parameter, one surface mesh geometry design is obtained by going through 9 design steps twice. The first trial is achieved by the assigned design parameters with the member length ratio ςas 1, and the generated mesh geometry of the first trial is used to calculate the final value of ς according to its definition. Then the second trial uses the same design parameters except the final ς just calculated, and the result of the second trial is the meth geometry of this design. Therefore ς is automatically calculated and does not need to be specified. Due to the fact that the geometry of parabolic shape is projected from the spherical reference or the ellipsoid surface which is uniquely mapped from the sphere, once the mesh 46 geometry on the reference sphere is designed, the parabolic surface geometry is determined. Similarly, the modified mesh geometry is also uniquely decided by the mesh geometry before best-fit deviation compensation. Hence this section only need to address the necessary discussion upon the reference sphere and the best-fit compensation is not used in the investigation. Note that a boundary layer outside the effective region is always considered in this section to be able to discuss B w . 2.6.1 The influence of the design parameters on pseudo-geodesic property and the best-fit RMS error To obtain a guide on proper selection of the seven parameters, the influence of these parameters on the pseudo-geodesic property and the best-fit surface RMS error of designated mesh facets are examined. Instead of concerning the number of rings r n , τ r is defined by Eq. (2.83) to describe the ratio of r n over the minimum number of rings r n . Similarly, c n is calculated by Eq. (2.84) where τ c presents the relative location at which the first converging ring occurs. Then seven design parameters are all dimensionless and independent with specific designs. () τ = rrr n round n (2.83) () τ = ccr n round n (2.84) The number of subdivisions s n is usually determined by the reflector installation or the symmetry requirement of application in the conceptual design and is always pre-selected to be 4, 6, or 8. In this section unless specified, s n is chosen to be 6. Two parameters, the number of nodal connections within boundary layer B c and the relative thickness of the boundary layer B w , have impacts mainly on the outer layer with no effects on the surface RMS error of the working surface, and therefore they are quite independent with other design parameters and can be discussed separately. When discussing B c , it must be aware of that different B c may result in 47 fractional number of nodes on the boundary rim. Then the number of rings 9 = r n to ensure () 1 − rs nn being divided by () 12 + B c exactly. 0 1 2 3 4 5 6 7 500 520 540 560 580 600 620 c B l total (m ) F/D = 0.3 F/D = 0.5 F/D = 1 F/D = 2 F/D = 3 Figure 2.18 The influence of B c on the reference sphere at 9 r n = , 0.3 c τ = , 2.0 ρ= , 1 B w = . Case studies in Figure 2.18 show the total length of members total l is proportional to B c and / D F when nodal reduction is used. Without nodal reduction ( 0 B c = ), total l is always between the ones at 3 B c = and 5 = B c regardless the focal ratio. Therefore to sufficiently reduce the nodes on the boundary rim while maintain minimized total l , 3, 5 B c = is preferred unless being specifically requested by the application. Another advantage of using 5 B c = is that the number of nodes on the boundary rim is always an integer for any r n as 6 s n = . When the number of rings is large and the mesh is very complex, then 7 B c = is also acceptable to further reduce the number of node and the number of facets of boundary layer. Note that the discussions in the numerical studies below use 5 B c = . In general, the relative thickness B w of boundary layer should be as small as possible to reduce the total length of members. Without nodal reduction in the boundary layer, B w does not need to be specified and could be assigned as 1. When nodes on the boundary rim is reduced, B w then has the lower bound to avoid the member connections from intruding the previous layer 48 and the inequalities below derived in Appendix F must be satisfied. Further discussion of B w will be seen in the immediate following. Without being specific, 1 B w = in this discussion section. () () () 1 1 1 1 4 1 sin 1 1 1 cos sin 21 4 π − − = −       >−≥      −        −     r effective n effective B B i effective B rs effective D F wwiandw Dc nn F (2.85) As defined in Eq. (2.84), the ratioτ c describes the percentage of area where pure geodesic rings (non-converging rings) take place. The power function’s coefficientρcontrols the converging speed on the curvature of the converging rings from inside to outside. Since the converging treatment is the main source which courses the non-uniform of the facets and the only source moving nodes away from geodesics of the reference sphere, these two parameters could have large impacts on the best-fit RMS errors and observable influences on the total element length. Due to the coupling nature of two parameters, they should be discussed together. Figure 2.19 presents two contour plots of the impact on the actual working surface RMS error by systematic facets , δ rms act at/0.5 = FD (left) and/3 = FD (right) with 10 = r n , in which ρ varies from 0.4 to 7 and τ c from 0.1 to the largest possible value at 2 =− cr nn . Figure 2.20 presents the contour results at 20 = r n . The numerical results show that τ c has stronger impact on the surface accuracy than ρ does. According to the figure, the valley area, which is preferred by resulting in the smallest , δ rms act , grows from a small region with center at 0.55 τ = r and 1.5 ρ= to nearly the left-bottom half of the plot when increasing either / FD or r n . Both inside and outside the preferred region, the contour lines are closed-to-parallel to the diagonal line of the figures which is an important guidance for choosing proper parameter values. Moreover B c and B w do not affect the effective facets of the reflector. Because of such independency, the ultimate selections of parameter values to minimize the best-fit surface RMS error are 0.10 ~ 0.75 τ = c and 1.2 ~ 6.0 ρ= . 49 ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F/D = 0.5 ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F/D = 3 Figure 2.19 The influence contour of c τ andρ on , rms bf δ at 10 r n = . ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F/D = 0.5 ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F/D = 3 Figure 2.20 The influence contour of c τ andρ on , rms bf δ at 20 r n = . The numerical cases in Figure 2.21 and Figure 2.22 also show that the total length of the reflector decreases for larger τ c andρ universally, implying that fewer converging rings lead to less total length of members. Although such selection does not overlap with the preferred region for minimum surface error, it provides the clue on adjustingτ c andρ to reduce the total element length once the surface RMS error is sufficiently small. 50 ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F/D = 0.5 ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F/D = 3 Figure 2.21 The influence contour of c τ andρ on total l at 10 r n = . ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F/D = 0.5 ρ τ c 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F/D = 3 Figure 2.22 The influence of c τ andρ on total l at 20 r n = . The number of the rings r n or equivalently τ r may be the most important parameter to design the mesh geometry, which plays the dominant role of impacting the best-fit surface RMS error , δ rms bf and the total length of members total l . Apparently, increasing r n increases the number of the facets and reduces the best-fit RMS error of effective mesh facets, but on the contrary it also increases the number of members and the total length of members. Since minimum number of rings r n is dependent on the reflector characteristic ratioδ rms D and the ratio of focal length versus diameter / FD , the relationship betweenτ r , δ rms D and / FD are presented in Figure 2.23 51 at 0.3 τ = c , 2.0 ρ= which are within their preferred region for the minimum best-fit RMS error. Note that the Y axis of Figure 2.23 denotes the smallest τ r which satisfies the δ rms requirement. Because δ rms D and / FD studied in the figure covers most of current mesh reflectors and the ones in demand, we can claim that the ultimate τ r in the proposed design method to fulfill the surface error requirement is below 1.5. After the improvement by properly adjusting other design parameters in the specific design task, τ r used is actually smaller. When the aperture of the reflector surface is a circle, τ r is usually larger than 1. For the offset parabolic surface with the ellipticity ee ba closed to 0.7, τ r could be below 1. Ultimately 0.8 ~ 1.5 τ = r which is sufficiently large to result in () , 1 δδ<− rms bf rms rms r . 0.5 1 1.5 2 2.5 3 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 F/D τ r δ rms /D = 10 -6 δ rms /D = 10 -5 δ rms /D = 10 -4 Figure 2.23 The selection of r τ versus various rms D δ and / FD . By including the boundary layer in the discussion, the parameterςis the ratio of the maximum member length in 1 − r n th layer versus the length in the first layer and is used to compensate the non-uniform facet edges. As stated previously, ς is automatically calculated. The numerical studies have been carried out to understand howς being affected by other parameters, in which τ c andρ have the major influence on the calculatedς and the effects of r n and / FD are much smaller. In general, ς usually falls in the range from 1.3 to 2.5. 2.6.2 Performance evaluation of the generated geometry 52 The tension distribution and the total length of the members are evaluated once the mesh geometry is generated under the proper values of the design parameters as discussed above. The tension distribution of a designed mesh facets at 0.3 τ = c , 2.0 ρ= , 7 = r n , 6 = s n , 5 = B c and 1 = B w is estimated by a method given in our previous work [114]. This method yields an optimal tension distribution of the mesh surface, which minimizes the tension band band T under given constraints on external forces and member tensions. In this discussion the external forces are applied on all facet nodes and restricted to the positive z-coordinate direction, which usually is the loading constraint on the reflector surface of such type of DMRs. The ratio of minimum tension min T versus maximum tension max T is defined as the uniform ratio unif r of tension distribution, or namely min max = unif rT T . One reason of not using uniform ratio as optimization objective is that because of () max 1 =− band unif TT r the uniform ratio can’t constrain the maximum tension and the tension band separately, and they may be too large to be desired or exceed the affordable tension of members even when the uniform ratio is closed to 1. The second reason may be even more important: the uniform ratio obtained by optimizing the tension band is numerically verified to be independent with the minimum tension constraint and is universally a constant as long as the minimum tension is positive. This fact implies that the uniform ratio minimizing the tension band is a consistent factor to evaluate the tension distribution at a given mesh geometry. 0.5 1 1.5 2 2.5 3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F/D r unif No reduction c B = 3 c B = 5 c B = 7 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 F/D r unif No reduction c B = 3 c B = 5 c B = 7 Figure 2.24 Tension distribution vs B c on the effective region (left) and the entire surface (right). 53 Figure 2.24 shows the tension distribution for different boundary connection types. Regarding the tension distribution within the effective region, without boundary reduction the tension uniform ratio is below 0.6. The uniform ratio decreases when the focal ratio increases and the trend is much more obvious when/0.8 < FD . By introducing the reduction, the tension distribution is more uniform with larger / FD and reaches the saturation at about/0.8 = FD . When 3 = B c the tension uniform ratio is always the largest and is above 0.6 except the case where the focal ratio is very closed to 0.3. 5 = B c results in 0.05~0.1 smaller uniform ratio than the ratio at 3 = B c . When/0.5 < FD , the uniform ratio is below the ratio without reduction. Therefore we can conclude that the tension distribution improvement of the working surface is very promising at B c being 3 and 5, which are preferred in the geometry design. When the focal ratio is closed to 0.3, 3 = B c is more desirable. 7 = B c gives the worst uniform ratio under nodal reduction both on the effective region and the entire surface which provides another reason not to choose it unless the geometry is extremely complicated and the reduction on the number of the nodes at boundary rim is so important. Although the tension distributes more unevenly by introducing the boundary reduction for the entire surface, however, this non-uniform distribution only occurs inside the boundary layer which is not that important as long as all tensions of members are within the requirement range. Figure 2.25 shows the influence of B w on the uniform ratio. In general increased B w could enlarge the uniform ratio both on the effective region and the entire reflector surface, because of larger relative angles between boundary elements and the effective aperture. Particularly, the improvement on the working surface converges as B w grows above 1.2. Regarding the fact that larger B w results in larger total length of members, B w should be balanced between the demand of minimum total length and the uniform tension distribution. If certain increase of the total length can be tolerant, 1.2 ~1.4 = B w . Otherwise, B w has to be the smallest number satisfying Eq. (2.85). 54 0.5 1 1.5 2 2.5 3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 F/D r unif w B = 1 w B = 1.2 w B = 1.4 w B = 1.6 w B = 1.8 w B = 2 0.5 1 1.5 2 2.5 3 0.2 0.25 0.3 0.35 0.4 0.45 0.5 F/D r unif w B = 1 w B = 1.2 w B = 1.4 w B = 1.6 w B = 1.8 w B = 2 Figure 2.25 Tension distribution vs B w on the effective region (left) and the entire surface (right). 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 F/D r unif n s = 4 n s = 6 n s = 8 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 F/D r unif n s = 4 n s = 6 n s = 8 Figure 2.26 Tension distribution vs s n on the effective region (left) and the entire surface (right). Figure 2.26 presents the tension performance at different surface symmetry. Although more symmetry axes higher the uniform ratio is for entire surface, 6 = s n leads to the best tension distribution at the effective region as/0.5 ≥ FD . 8 = s n brings most uniform distribution for/0.5 < FD when the uniform ratio by 6 = s n is only slightly smaller. This provides one reason why six symmetric axes are used most often in the mesh geometry design of DMRs. According to Figure 2.27, increasing r n reduces the uniform ratio within the effective region and increases the overall tension uniform. By only taken the performance of the effective region into account, one can conclude that it is harder to achieve uniform tension performance within effective region 55 for a reflector with more complex mesh geometry, which places an additional emphasis on designing surface geometry with as less mesh facets as possible. 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 F/D r unif n r = 7 n r = 9 n r = 12 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 F/D r unif n r = 7 n r = 9 n r = 12 Figure 2.27 Tension distribution vs r n on the effective region (left) and the entire surface (right). The following discussion is going to quantitatively address the pseudo-geodesic performance. The numerical investigation is based on the parameter set of 5 = B c , 5 = B w , 10 = r n , 6 = s n , 0.3 τ = c and 2 ρ= when they are constant. Among 9 design steps, the only one preventing all nodes from sitting on geodesics is when introducing the converging treatments into the mesh geometry. Considering the design with no converging treatment on rings, hence all the nodes are on the geodesics of the reference sphere (Figure 2.28). Although this geometry can not be accepted practically due to the significant reduction of diameter of effective aperture, it leads to the minimum total length of members for all the design at the same configuration. Therefore this mesh geometry is called the ultra-minimum geometry and its total length is named the ultra-minimum length ultra l . Then the performance of the total length is evaluated by the total ultra ll , where total l is obtained from the design using the proposed method in the chapter. Numerical investigations show that the ratio total ultra ll decreases as τ c orρ increases. Therefore the worst case scenarios are at 0.1 τ = c and 1.2 ρ= within their ultimately preferred region, as shown in Figure 2.29. Hence after the converging treatment which compromises the facet mesh to the aperture rim, the pseudo-geodesic mesh geometry sacrifices less than 0.8%~3.0% increase 56 from the ultra-minimum ultra l under proper design parameters. Also, it can be seen that larger r n reduces total l further towards the ultra-minimum length at the ultra-minimum geometry. Figure 2.28 Top-view of mesh geometry at /2 FD= and 10 r n = for the ultra-minimum geometry (left) and proposed geometry (right). 0.5 1 1.5 2 2.5 3 1.01 1.015 1.02 1.025 1.03 F/D l total /l ultra n r = 7 n r = 10 n r = 14 n r = 18 n r = 22 n r = 25 Figure 2.29 Performance of total length in the worst case scenarios. According to the first design criterion, all the designs have to satisfy Eq. (2.76) and Eq. (2.77). Although it would be very hard to mathematically prove the validation of the criterion, however, for all the design cases involved in this chapter, this design criterion has never been violated. 57 2.7 Complete Design Procedure via Optimal Parameter Selection Upon above discussion, we will first summarize the basic design steps and the additional considerations in the chapter into a design core and then propose the complete optimal design procedure via systematic tuning of seven parameters of the design core. Figure 2.30 Design Core of the proposed method. The flowchart of the design core is shown in Figure 2.30. The input of the design core is the selected seven design parameters and the desired working surface. The desired working surface could either be defined by the set of the (parent) focal length F ( p F ), the (parent) diameter of reflector D ( p D ) and the offset distance off e of the entire reflector surface or the set of the effective surface parameters , p effective F , , p effective D , effective F , effective D and effective e for the effective region of the reflector surface. If the set of the effective surface parameters are provided, D , p D , F , p F and off e should be first obtained according to the Section 2.4 and then the mesh geometry is obtained by the step 2, 4, 6 ~ 8 and the modified step 3, 5, 9. Otherwise as the output of the design core the mesh geometry is generated by the regular design steps 2 ~ 9. Obviously, this design core included all the formulations in Section 2.3 – 2.4 of the chapter except the design step 1 and the best-fit compensation technique. 58 Figure 2.31 The flowchart of the optimal design method via systematic tuning. 59 Using the design core, the complete optimal design procedure for the mesh geometry generation is proposed as follows and the flowchart is presented in Figure 2.31. To design a mesh geometry of a spherical or parabolic deployable mesh reflector by given either the set of D , p D , F , p F and off e , or the set of , p effective F , , p effective D , effective F , effective D and effective e , the optimal design procedure contains the following stages: Stage 0: initiate the design task. From Step 1, the allowable surface RMS errorδ rms is calculated. s n is determined by the design requirement which is commonly 4, 6 or 8. B c is preferred to be 5 unless specified by the requirement. When / FD is below 0.5, 3 = B c is also an option. r n is evaluated either by Eq. (2.17) or (2.58). Once the previous three parameters are decided, they remain constant in the whole design procedure. ς is determined within the design procedure automatically and not need to be selected. When the demand of minimum length is not very stringent, 1.2 ~ 1.4 = B w . Otherwise, for every ς , B w is also automatically calculated to be the smallest value fulfilling Eq. (2.85). Then only three parameters are left unknown and going to be selected in the following stages. The ultimate values for the preferred region ofτ r ,τ c and ρ are: 0.8 ~ 1.5 τ = r , 0.10 ~ 0.75 τ = c and 1.2 ~ 6.0 ρ= . Usually, 1.2 τ = r , 0.3 τ = c and 2.0 ρ= are usually a nice set of initial attempt. In the following stage 1 ~ 3, for each set of design parameter tuned, the designed mesh geometry is obtained by implementing the design core twice (as previously described in Section 2.6). In the first implementation of design core, set 1 ς = . If B w is not preselected as 1.2 ~ 1.4, the corresponding B w is also calculated in Eq. (2.85). Then ς is updated by its definition using the results of the first implementation, as well as B w if necessary. By running the design core for the second time, the geometry of the mesh facets generated is the design result for this tuning iteration. Stage 1: find proper τ c and ρ at minimum , δ rms bf . By varying τ c and ρ at the tuning increments of 0.03 and 0.2 respectively, the minimum , δ rms bf can be always obtained within their preferred region according to the discussion in Section III. Sinceτ c plays more important role on 60 surface accuracy in general, it is suggested to adjust τ c first and ρ secondly to further narrow down the selections. Stage 2: finalize optimal τ r for the simplest configuration. Using τ c and ρ from stage 1, τ r is tailored by the increments of 0.02 ~ 0.05 (depending on r n ) to find the smallest τ r satisfying the design criterion of Eq. (2.81), which is its final design value. After this stage, the configuration of the mesh facets is determined and is the simplest for the design task considered. Stage 3: determine the optimal selection of the rest parameters to minimize total l . Based on the final τ r , adjustτ c andρ to minimize total length while still fulfill Eq. (2.81). As the same with stage 1, observing the performance by first varyingτ c and then ρ in the next would make the selection much easier. Due to the limited influence on the total length fromτ c andρ , one may not expect large variation on total l by changing those parameters. Once the minimum total l is found, the related τ c ,ρ ,ς and B w are at their optimal design values. Stage 4: achieve the optimal mesh geometry. First apply the best-fit compensation technique and modify the desired working surface if it is required by the design task. Then by using the optimal design parameters selected, the optimal mesh geometry is generated by implementing the design core regarding the modified working surface. The following remarks address the important properties of the proposed optimal design method: Remark 1: This design approach provides the optimal generation of mesh geometry (the designated configuration with nodal coordinates and member connectivity of mesh facets) satisfying both design criteria without the need of manual adjustments on specific nodes or the boundary meshes. The method is developed for deployable mesh reflectors of which the reflector surface has an aperture rim. Although seven design parameters are to determine during the design procedure, two of them are pre-determined, two others are automatically calculated and only three are undetermined of which the selections are restricted within a very limited region based on the performance investigation. Remark 2: The design method guarantees the pseudo-geodesic property of the mesh geometry for the designated surface. On the spherical reference, the pseudo-geodesic 61 performance has been quantitated to only endure 0.8~3.0% increase from the ultimate minimum total length sharing the same configuration. The parabolic mesh geometry designed is projected by the selected projection center to ensure the minimum surface RMS error. In Stage 2, such mesh design always leads to the smallestτ r and hence the simplest configuration to fulfill the operating frequency requirement, plus the enforcement of minimum total member length when tuning the parameters in Stage 3. Therefore after the optimal design parameters selected, the generated mesh geometry is assured at the minimum total member length for certain design task, or in another word, being pseudo-geodesic for designated operating frequency. Remark 3: The deviation of best-fit surface are successfully compensated, which assures the best-fit surface of the designed mesh geometry always being the desired working surface. The facet edges on the effective aperture rim and the outer rim are of equal length, allowing the rim ultimately to be deployable from practical purpose. The boundary nodal reduction results in less rim nodes than the facets in the boundary layer, which is required for high frequency applications. Moreover, the generated mesh facets always enjoy promising uniform tension distribution and no mechanisms, which are highly desired by deployable reflector applications. Remark 4: The proposed method evaluates the actual surface RMS error between the mesh facets and desired working surface and optimally tunes the design parameters to achieve the required surface accuracy. In the contrary the previous approaches commonly estimate the ring numbers of the mesh directly by the maximum allowable member length in Eq. (14) of the part 1 without systematic tuning, and therefore easily bring much larger (usually 1.2~2.5 times larger depending on the aperture ellipticity) best-fit surface RMS errors than the requirement. 2.8 Comparative Studies of Different Design Methods In this section, the proposed method is compared with the existing methods mentioned in the introduction of the chapter. A mesh geometry design method for DMRs should achieve two goals: (i) it defines a configuration of the mesh facets that determines the complexity of the reflector surface at the desired working status; and (ii) it determines the proper coordinates of all nodes, which eventually decides the total element length and the best-fit RMS error of the reflector. Hence, the comparison of different design methods herein is focused on the diverse 62 configurations and the various generations of mesh geometries of reflector surfaces. In this work, only the mesh surfaces consisting of triangular facets are considered. 2.8.1 Comparisons on configurations of design mesh facets Not all the previous methods have proposed specific configuration designs. All the existing mesh geometry designs (including this work) share the same configuration in the effective region of reflector surfaces which is first presented by Ref. [64]. It is the specifically designed boundary configurations that differs the design results from each other. Besides our nodal reduction technique on the boundary rim, the research in [106] also designed three other different configurations for the outer rim to reduce the boundary nodes, which were used in Ref. [62, 51] for case studies. Therefore these configurations and ours with nodal reduction will be discussed here. Table 2.1 shows the specifications in the different configurations given in Ref. [106] and the method proposed in the current chapter. Here, “Regular” configuration denotes the one without any special boundary design [64]; “AM1”, “AM2”, “AM3” are the configuration names in Ref. [106]; 3 = B c , 5 = B c and 7 = B c are for the mesh geometries generated by the proposed method at different B c values. Since “AM1”, “AM2”, “AM3” configurations use six subdivisions, for comparison purposes, 6 = s n is used in the proposed method. From the table, the design configuration at 5 = B c (which is the preferred parameter value based on the previous discussion) has fewer boundary nodes than the other previous designs, and this number can be reduced further by choosing 7 = B c . Table 2.1 Specifications of various design configurations Topologies No. of boundary nodes Total nodal number t n Total element number t m Regular 6 r n 2 33 1 ++ rr nn 2 93 6 −− rr nn AM1 3( 1) + r n 2 39 11 +− rr nn 2 918 45 +− rr nn AM2 3 r n 2 36 5 +− rr nn 2 99 24 +− rr nn 63 AM3 2( 1) + r n 2 35 3 +− rr nn 2 97 20 +− rr nn 3 = B c 3( 1) − r n 2 32 − r n 2 93 6 −− rr nn 5 = B c 2( 1) − r n 2 31 −− rr nn 2 93 6 −− rr nn 7 = B c 1.5( 1) − r n 2 31.5 0.5 −− rr nn 2 93 6 −− rr nn 5 10 15 20 25 30 35 40 0.7 0.8 0.9 1 1.1 1.2 n r r node AM1 AM2 AM3 c B = 3 c B = 5 c B = 7 5 10 15 20 25 30 35 40 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 n r r elem AM1 AM2 AM3 Proposed Figure 2.32 Configuration comparison on the total nodal numbers (left) and the total numbers of elements (right). Define node r as the total nodal number at the configuration with particular boundary design versus the one at the regular topology, similarly node r as the ratio of those total element numbers. Figure 2.32 presents the comparison of the geometric complexity among the configurations in Table 1. It is obvious that total nodal number of our designed topology is always smaller (4% ~ 8% less at 40 = r n ) than the cases of all other design configurations. The total numbers of elements in the proposed design remain equivalent to the one at the regular design when “AM1”, “AM2”, “AM3” configurations suffer noticeable increase (2.7% ~ 5.6% at 40 = r n ) on t m by reducing the boundary nodes. Therefore the design method in this chapter always provides the simplest configuration among the above research. Since fewer elements on the certain surface intend to have shorter total elemental length, our method is more capable of reducing the total weight of reflectors. Moreover, according to the previous discussion on tension 64 distribution, such simplicity will make the reflector surface easier to pursue the uniform tensions within members. 2.8.2 Comparisons on generated mesh geometries at the same configuration Each of the previous research proposed an algorithm to generate the nodal coordinates. However, the design algorithms in Ref. [64, 5 and 43] are specially developed for the configurations which is unable to compromise the aperture rim in practice and therefore are excluded from the comparison. The research works in [106] and [62] which presented different algorithms for offset parabolic surfaces are denoted as “equal-force-density” algorithm (E-FD for short) and the “equal-tension” algorithm (E-T) respectively. Another “equal-length” algorithm is shown in Ref. [51], which however is even not preferred in their case studies due to minimum improvement on the total length and the surface RMS error comparing to either E-FD or E-T approach. Therefore for comparison, E-FD algorithm, E-T algorithm and our approach will be used to design the nodal coordinates on the same mesh configuration proposed by this chapter. Here the mesh geometry of our method is generated at the same set of design parameters as 0.2 τ = c , 3.5 ρ= , 6 = s n , 1 = B w without any further tuning. First the regular configuration which uses entire surface as the working region with no boundary reduction on the rim ( 0 = B c ) are considered. Same with previous research, the case studies are carried out on offset-feed reflectors. The desired working surfaces share the same parent focal ratios and offset ratios with Aerospatiale, BAe/Surrey and AstroMesh reflectors and their focal ratios of circular apertures are 0.38, 0.60, 0.67 respectively [107]. The mesh geometries are generated on the desired surfaces and the best-fit RMS errors are evaluated without considering the best-fit surface deviations for now. The results show the designs from all the algorithms have very consistent total lengths with less than 0.1% difference regardless r n and their focal ratios. Define error r as the surface RMS error ratio between either E-FD or E-T algorithm and our method. Figure 2.33 demonstrates that the best-fit surface RMS errors in the designs by E-FD and E-T approaches are always larger than ours with a growing trend for the mesh geometries with more rings and smaller focal ratios of circular apertures. This implies that to achieve the same surface accuracy more rings are need for the design by E-FD or E-T method, 65 resulting in higher structural complexity. Particularly, E-FD method leads to very closed best-fit RMS error with ours if the reflector is shallow, and then the benefit of the proposed method is not obvious. However, in the case of Aerospatiale shape when the reflector is deep, E-FD method results in around 6% increase on best-fit RMS error and to achieve the same surface accuracy with our design it needs at least one more ring of mesh facets. In the design already with 20 rings, one more ring can enlarge the total number of nodes by 10% and the increase of the total member length will be very obvious. Considering the E-T method, the proposed method has large advantages even for the shallow reflectors. In the case of AstroMesh shape design with 20 rings, the same configuration of E-T method has 16% more best-fit RMS error and the mesh facets with the same surface accuracy requires at least two more rings, equivalently about 20% more facet nodes in the surface. 10 15 20 25 1 1.5 2 2.5 n r r error Aerospatiale, E-FD Aerospatiale, E-T BAe/Surrey, E-FD BAe/Surrey, E-T AstroMesh, E-FD AstroMesh, E-T Figure 2.33 The best-fit surface RMS errors compared in three case studies. To evaluate the actual surface accuracy of generated mesh facets, the best-fit surface compensation are applied in our designs and the actual surface RMS errors are compared in Table 2.2 for an offset parabolic reflector with 14.3 = p Dm and 2.3 = off em. The best-fit surfaces of the generated mesh facets by the proposed approach always present perfect desired working surfaces with exactly desired p F and less than 0.00063% relative errors on p D . Clearly our 66 designs always provide higher surface accuracy between the mesh facets and the desired working surface and the RMS error difference also has the same growing trend with error r . Table 2.2 Actual surface accuracy comparison under best-fit compensation r n 6 8 10 10 10 / ca F D 1.3268 1.3268 1.3268 0.8846 1.7691 RMS errors (mm) in Ref. [62] 0.75 0.48 0.33 0.82 0.21 RMS errors (mm) in our design 0.721 0.402 0.256 0.385 0.197 error r 1.040 1.194 1.289 2.130 1.066 Figure 2.34 Generated mesh geometries on the same configuration with reduced nodes on the aperture rims. Another type of configurations is to have boundary nodal reductions (at 5 = B c in the discussion) outside the effective region. A set of typical designed geometries are shown in Figure 67 2.34. E-FD and E-T approaches lead to obvious shrink on the diameter of the effective apertures, or equivalently result in much larger reflector rims to support the same size of the effective aperture. Therefore those methods should not be applied directly onto these configurations. It has been proposed to design the inner geometry separately to enable the usage of E-FD or E-T algorithm plus manual calculation of the boundary rim geometry. However, this strategy is essentially of designing the regular configuration and based on the previous discussion such proposal is still not able to compete with the method in the chapter regarding the minimum total length and the surface accuracy of designed mesh geometry. In summary, the comparative study demonstrates the strength of the proposed method: it always provides the simplest mesh facets and the most competitive surface RMS errors simultaneously for both the regular configuration and the configuration with reduced boundary nodes, which together return the smallest total length of designated mesh geometry under certain operating frequency requirement and therefore can be viewed as the behalf of its pseudo- geodesic property. 2.9 Design Results After a quick view of the generated mesh geometry with various configurations, the proposed design method is applied in two perimeter-truss deployable mesh reflectors: a model for the DMR currently being developed at JPL, and a design example for an extremely large DMR. 2.9.1 Quick view of design results without tuning the design parameters To show the capability of the proposed design method, diverse mesh geometries on the sphere with D = 4 m and F = 2 m are generated; see Figure 2.35. The geometries of different configurations are generated by the proposed method with varying numbers of the subdivisions and nodal connections on the outside rim. Other parameters are selected as 11 = r n , 1.2 = B w , 4 = c n , 1.5 ρ= . Particularly, when 5 = B c , set 10 = r n to ensure the number of the node on the outside rim is integer. Here the best-fit compensation technique is not applied. Some mesh geometries generated may not be axis-symmetric as the parameters are selected rather arbitrarily. Note that the red circles present the effective apertures of the mesh geometries. 68 4, no reduction = s n 4, 3 == sB nc 4, 5 == sB nc 4, 7 == sB nc 6, no reduction = s n 6, 3 == sB nc 6, 5 == sB nc 6, 7 == sB nc 8, no reduction = s n 8, 3 == sB nc 8, 5 == sB nc 8, 7 == sB nc Figure 2.35 Quick view of mesh geometries at diverse configurations. 2.9.2 Improvement on the JPL mesh geometry Here we show that when the proposed method is used, the current JPL design of mesh geometry can be improved. The design objective is to generate the mesh geometry for the center- feed parabolic surface with 4 GHz operating frequency, diameter 30 m = effective D and the focal ratio / effective effective FD at 0.9187. Choose the surface accuracy to be 1/150. The allowable surface RMS error then is 0.50 mm and 5 1.67 10 δ − =× rms effective D . The current JPL geometry design is shown in Figure 2.36(3D view) and Figure 2.37(a) (top-view) with 835 nodes (24 nodes on the outer rim), 2526 members and 17 rings. The best-fit RMS error of the working surface , δ rms bf is 1.28 mm and this error does not address the deviation between the best-fit surface and the 69 desired working surface. The total member length is 2981.8 m. Apparently the best-fit surface error from the systematic facets is larger thanδ rms because the number of the rings in the design is straightly determined by the maximum allowable member length without any further adjustment. -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 0 1 2 x y z Figure 2.36 3D view of surface mesh geometry of JPL’s design. First we use the design method in the chapter to modify the JPL design, under the condition that the number of rings r n and the nodal reduction type B c remain unchanged. To make a fair comparison, the diameter of the outer rim D is also the same with JPL’s design and hence B w is obtained by Eq. (2.86). As discussed in Step 10, =−∞ p z to ensure that the effective aperture always has the designated diameter. () 1 1 1 1 sin ( ) 4 1 sin 4 − − = −    =−          r n effective B i effective effective D F wwi D F (2.86) Therefore only two parameters are to be adjusted:τ c andρ . Choosing 0.1 τ = c and 6.8 ρ= to intentionally have smaller , δ rms bf than the original design, the modified design is plotted in Figure 2.37(b) To be comparable to the JPL’s result, this design does not compensate the best-fit deviation either. The geometry has 841 nodes and 2544 members, surface RMS error at 1.197 mm, and 2994.0 m total length. After manually removing six extra nodes at the crosses between the 16th ring and six symmetry axes, the modified design has the same configuration with the 70 original design, leading to the same numbers of nodes and members. Then the total length of the reflector reduced to 2982.1 m while the surface accuracy in the effective region remains consistent. The final design specs are listed in the Table 4. It can be seen that with the slightly higher surface accuracy, the modified geometry has 0.11% increase of the total member length from the original design. Considering the fact that this geometry is achieved by only allowing two parameters to vary, our design method has demonstrated very competitive performance on the generated mesh geometry. Figure 2.37 The designed mesh geometries: (a) JPL’s design (left); (b) Modified design (middle); (c) Improved design (right). In order to show how much the JPL’s design can be improved, another geometry design is obtained by fully utilizing the optimal design procedure except the best-fit surface compensation. Here, instead of using 0.50 mm as surface error requirement, we attempt to design the geometry with the best-fit surface RMS error as closed as possible to the JPL’s geometry. The design parameters are tuned to be 0.75 τ = c , 1.6 ρ= and 16 = r n with B c , s n and B w being preselected as 5, 6 and 1 respectively. ςis automatically calculated as 1.9488. The generated mesh geometry is shown in Figure 2.37(c) and the specs are presented in Table 4. While the best- fit surface RMS error is almost the same with the JPL’s design, the major advantage of the proposed design is that it only uses 16 rings to construct the mesh geometry. By using one less ring it significantly reduces the mesh complexity of the reflector surface by 84 nodes and 276 members. This new design also reduces the thickness of the boundary layer and therefore the diameter of the entire reflector by 1.8414 m while maintaining the same desired size of effective aperture. Those improvements together dramatically cut down the total member length by 341.3 71 m (11.45% of the JPL design) leading to very obvious influence on the total weight reduction of the reflector. Table 2.3 Specifications of the generated mesh geometries Design Case t n t m r n , δ rms bf (m m) total l (m) JPL 835 2526 17 1.28 2981.8 Modified 835 * 2526 * 17 1.20 2982.1 * Improved 751 2250 16 1.31 2640.5 Table 2.4 Tension performance of the designs The tension distributions of the three different designs are also investigated. From the reflector application requirements, the minimum tension constraint is 2 lbs., the maximum tension is 9 lbs. and the external loads are in the positive z-axis direction only. The optimal tension distribution is calculated by minimizing the tension band of the entire reflector and the performance specs are shown in Table 2.4. The member tensions of all three cases are satisfied with the tension constraints. The modified design has more uniform tension distributions than the JPL’s geometry for both the effective surface and the entire surface. The improved design has much smaller effective tension band comparing with the other two cases and the effective uniform ratio , unif eff r is above 0.6. The entire tension band is obviously larger, but this only implies that all the non-uniform tensions are within the boundary layer outside the effective region which we do not care much. Note that the band of this non-uniform tension distribution in the boundary layer can be narrowed down by choosing slightly larger B w such as 1.2 if a small increase on the total length of members is tolerable. It also can be observed that the uniform Design Case Effective Tension Band (lbs.) Entire Tension Band (lbs.) , unif eff r , unif entire r JPL 1.8458 2.1791 0.520 0.4786 Modified 1.7295 2.0943 0.536 0.4885 Improved 1.2130 3.6409 0.623 0.3546 72 ratios from the modified design and the improved design are consistent with the varying trend discussed in Section 2.6. Finally, a new design of the mesh geometry is achieved by fully implementing the complete optimal design procedure. The optimal design parameter set is 6 = s n , 5 = B c , 1 = B w , 1.45 τ = r , 0.75 τ = c , 2.4 ρ= and 2.0434 ς= . The diameter D of the desired working surface after adding one boundary rim is 30.7735 m. To compensate the best-fit deviation, the modified surface has a diameter of 30.8146 m and a 27.5610 m focal length. Using the final design parameter set on the modified surface, the optimal mesh geometry is shown in Figure 2.38. The proposed design has 25 rings, 1849 nodes (48 on the outer rim) and 5544 members. The diameter and the focal length of the best-fit surface only have 0.00016% and 0.00065% relative errors with the ones at the modified surface respectively, demonstrating the accurate presentation of the desired working surface by the best-fit surface. The actual surface error from the desired working surface is 0.498 mm and the total length of members is 3990.9 m. Figure 2.38 The proposed design satisfying the operating frequency. 2.9.3 Design of an extremely large mesh reflector After the proposed method has been validated by the previous examples, the current task is to design an offset parabolic mesh geometry for the deployable mesh reflector with , 100 = p effective Dm of effective parent surface. The focal ratio of effective circular aperture 73 , /0.8 = effective ca effective FD , the effective circular aperture is 40 m (equivalently 10 m as the effective offset distance) and the operating frequency is 40 GHz. Then by choosing the surface accuracy requirement as 1/50, 0.150 mm δ = rms and 6 3.40 10 δ − =× rms effective D with the effective reference aperture effective D as 44.1765 m. Two design strategies are proposed here. First we use the full reflector surface as the effective region and therefore the aperture nodes can not be reduced ( 0 = B c ). 6 = s n which has the best tension distribution. Due to the reflector is extremely large already, the total member length can not suffer any further increase by selecting larger B w of better tension performance. Instead B w is automatically calculated to be the smallest value satisfying Eq. (2.85). The optimal selection of other design parameters are 1.26 τ = r , 0.25 τ = c , 5.0 ρ= , 2.0452 ς= and 1 = B w . Applying the compensation strategy, the parent diameter, the parent focal length and the offset distance of the modified surface are 100.0043 m, 32 m and 9.9979 m respectively. The designed mesh geometry is presented in Figure 2.39 which is designed on the modified surface at the final design parameters. The best-fit surface of the mesh facets is exactly at the desired working surface with less than 6 10 − m uncertainty on the parent diameter, the parent focal length and the offset distance. The second strategy is to add an extra boundary rim outside the mesh geometry where the rim nodes can be reduced. Select 5 = B c to reduce the rim nodes by a factor of 3 while r n is. The vertical projection is used to ensure the size of effective aperture as discussed. In Stage 1, the combination of 0.10 τ = c and 2.0 ρ= offers the smallest best-fit RMS error for the effective region and τ r is found to be 1.26 in Stage 2. Finally the third stage leads to the design parameter values as 0.25 τ = c , 5.0 ρ= , 1.9665 ς = , 1 = B w . The final desired working surface has 100.5443 = p Dm , 32 = p Fm and 9.7278 = off em due to the additional boundary ring. After the modified working surface (100.5485 m, 32 m and 9.7252 m for parent diameter, parent focal length and offset distance respectively) is calculated, Figure 2.40 shows the final designated mesh geometry from the top-view and its best-fit surface exactly presents the final desired working shape with 7 10 − m of error magnitude on p D , p F and off e . 74 Table 2.5 Specifications of designated mesh geometries Strategy t n t m r n , δ rms mod (mm) total l (m) 1. 8269 24492 52 0.149 11048.5 2. 8373 25116 53 0.149 11509.5 Besides the design specifications shown in Table 2.5, several other results should be mentioned. Although the first design has fewer nodes, its outer rim has 312 nodes while the second design only has 104 boundary nodes. If the boundary layer of the second design is removed, its effective surface has the same configuration. It can be summarized that the first strategy provides the less complexity of the reflector surface and the smaller weight while the second design offers the advantage of significant reduction on the boundary nodes and the benefit of easier deployment. -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 x y Figure 2.39 The top-view of design for the extremely large DMR (Strategy 1). 75 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 x y Figure 2.40 The top-view of design for the extremely large DMR (Strategy 2). 2.10 Conclusions To the knowledge of the authors, this chapter proposes one of the few implementations of geodesic mesh geometry design on deployable mesh reflectors for both center-feed and offset- feed configurations. The main results are described below: First, the design core which the proposed optimal design method is built upon, is developed for any reflector surface with an aperture rim and has 9 basic design steps. The design core generates the pseudo-geodesic mesh geometry (nodal coordinates and member connections) for the desired working surface. It introduces the nodal reduction design in the boundary layer and reduces the number of the nodes on the aperture rim, and suggests a unique mapping technique to resolve the geometry complexity caused by elliptical aperture of offset parabolic 76 reflectors. It also considers the design focus on the effective portion of the reflector surface by slightly modifying the design steps. The mesh geometry generated by the design core is fully determined by seven design parameters which are optimally selected in the complete design procedure. The actual surface accuracy between the mesh facets and the desired working surface is used to evaluate the performance of the design results in this work. It is demonstrated to be the same with the best-fit surface RMS error after compensating the best-fit surface deviation from the desired working surface and is evaluated by the exact RMS error calculation without approximation. The complete optimal design procedure systematically determines the seven design parameters and guarantees the pseudo-geodesic property of designated mesh geometry in certain design task. The design parameters are tuned according to the discussion of their influence on the performance of the design outcome which shows promising uniform tension distribution in the designed geometry. Two proposed design criteria are always satisfied by the designed mesh geometry and therefore the mesh facets are statically indeterminate without mechanisms and fulfill the operating frequency requirement of reflector application. The advantages of the proposed method are demonstrated in the comparative study to enjoy the simplest surface mesh configuration and the smallest surface RMS errors simultaneously and therefore the minimum total length of designated mesh geometry under certain surface accuracy requirement. The presented method is applied on the JPL’s reflector model and improves its performance of surface accuracy and the total member length. The design result for an extremely large reflector example proves this method as an efficient tool for development of deployable mesh reflectors. 77 Chapter 3 Optimal Design of Structural Parameters 3.1 Introduction This chapter presents the optimal parametric design method for the shape design of DMRs, which is the second block of the design portion in the Active Shape Control architecture (Figure 1.4). As described in Chapter 1, the truss-type deployable mesh reflector adapts a dish surface as the working shape, which is constituted by the mesh (network) of facets (mostly triangular) with cable edges. DMRs of this type are engineered as deployable truss structures. Once the reflector design concept with the mesh geometry of the working surface (equivalently the deformed shape of reflector’s truss model) is determined in Chapter 2, the next design level is to choose the proper structural parameters including undeformed element lengths, cross- section areas of elements and applied nodal loads under certain constraints, which are crucial for the structural performance of the reflector. For past 40 years, the parametric design problem of truss structures in general has been heavily investigated and a large amount of research articles are available in various types of publications. The majority of the design problems are to minimize the total weight of trusses by properly determining cross-section areas of elements based on undeformed configurations and given external loads. Diverse design formulations and solving methods have been reported on such design mission, involving the constraints on areas of cross-sections [109, 76], on element strains [109, 78], on axial stresses [90, 72, 9] and on nodal displacements [109, 90, 78-79]. Other researches presented the design approaches under the same problem setup with additional considerations: [25] suggested to simultaneously achieve the design goal and the static analysis 78 of the structure; loads with bounded uncertainty were considered in [10]; the buckling and the plastic behaviors are introduced into the truss design by [92]; [57] added the elastic nonlinearity into design formulation; [39] focused on the snap-through behavior in the truss design problem. After weight reductions have been demonstrated by considering prestresses in the unloaded configuration [37, 13], several design approaches [93, 28, 48] have been proposed to solve the prestressed truss design in the linear programming formats, although all these methods only handled linear geometric trusses. In addition to the areas of cross-sections, the nodal coordinates of the unloaded truss were added into the design variables to improve the weight reduction in [95, 73]. [68] accessed this prestressed design solution by utilizing the snap-through behaviors. However, the parametric design task of deployable truss structure (for instance, deployable mesh reflectors) is entirely different from the traditional problem setup. While the deployable mesh reflector has very stringent requirements on the structural performance of the deployed working shape, its application has very little concerns on the initial unloaded shape of the truss structure. This leads to the necessity of solving the parametric design based on the deformed configuration. Surprisingly, by reviewing the previous research works, little has been done regarding such design mission. Moreover, the applied nodal loads which are always predefined in the past investigations have not been counted into the design variables yet. In this chapter, a new optimal design method is presented to rigorously solve the parametric design problem of deployable truss structures. For a designated deployed shape of the deployable truss, the method will obtain the solutions of structural parameters (cross-section areas, element tension distribution, applied loads and undeformed lengths of elements) to optimize certain structural performance (uniform axial tension, minimum total weight or minimum tension band) under the design constraints on elemental strains, inner tensions, external loads and areas of cross sections according to specified application requirements. Unlike the previous research, the problem is formulated on the deformed shape of the truss and both the nonlinear geometry and elasticity are considered. The design solution is obtained by numerical optimization techniques and is guaranteed to provide the optimal set of parameters for the given deformed shape. As the applied loads are treated as the design variables for the first time in the academic literature, the design result is optimized among all possible loading cases on such deployed shape and the solution is solely dependent on the deformed shape geometry (nodal 79 coordinates and member connectivity at the deformed status) itself, which implies the structural performance of the design results can be considered as a natural property of the deformed shape. Due to the same engineering model in general, this method can be applied on any truss-type structures such as deployable truss, deployable cable nets and deployable tensegrity structures, and in diverse space applications including deployable boom, deployable mast and deployable mesh reflectors. The remainder of chapter is arranged as following: The truss model, the solution of the model and the optimization formulation on the parametric design problem are presented in Section 3.2, 3.3 and 3.4 respectively. Section 3.5 summarizes the entire design procedure with the important remarks discussing the significance and performance of the method. Section 3.6 extends the problem formulation for broader design considerations. Finally three design examples are illustrated in Section 3.7 and the conclusion of the research work are addressed in Section 3.8. 3.2 Force Balance Model at Equilibrium To mathematically formulate the design problem, the force balance model of a truss at equilibrium is first presented in this section, followed by the system model in the next section. In the chapter it is assumed that cross-section area of an element does not vary at different axial position of the element. Due to in-space applications of deployable truss structures, the gravity of the elements is ignored. The term “undeformed profile” in this work is defined to be the vector of undeformed element lengths. Note that because of the prestress, the undeformed profile of truss may not be the same with the element profile calculated from the initial unloaded status of the structure. In a truss structure, the elements are one-dimensional bars which only afford stress in axial direction, and are connected at two ends, namely nodes. Mostly, the elements are connected freely without any constraints. However, if the nodes have certain physical constrains in displacement, these nodes are said to be on the boundary of the structure and provide the boundary conditions. The external forces of the truss structure are only applied on the nodes, no matter whether they are on the boundary or not. Let n and m be the total number of the nodes and elements respectively. Then two nodes of j th element in the deformed shape are expressed as, 80 12 3 1 2 3 , =+ + = + + ≠        ii i i ii i i rxe ye ze r xe ye ze i i (3.1) where 12 3 ,,   ee e are the base vectors of a global coordinate system xyz. The longitude direction of the member (deformed) is expressed by the unit vector () ,1 , 2 , 3 1 γααα =−= + + −       j ixj yj zj i i i rr e e e rr (3.2) where the direction cosines are given by ,, , ,, αα α −− − == = ii i ii i xj y j z j jj j xxyy zz ll l (3.3) with j l being the member length ()()() 22 2 =− + − + − ji i i ii i lxx yy zz (3.4) Then we have ( ){} {} 12 3 , , , , with γγ γααα   ==      T jxjyjzj jj ee e (3.5) Define the axial tension of jth element being j T , the tension vector of all members as { } g T and the global nodal coordinates of the structure is { } g Y , in which { }{} 11 1 =  T giiinnn Yx y z x y z x y z (3.6) and { }{} = T gi i i i Yxyz as the nodal coordinates of ith node. Inside each element, the force contributed onto the i th node is { } γ j j T and its force contribution on i th node is { } γ − j j T . In total, each element will contribute its tension onto force balance at two nodes, which is expressed by { } {} {} {} {} { } {} {} {} {} γγ ∂∂ ∂∂ − ∂∂ ∂ ∂ gg jj g g jj gg g g ii YY TT TT YT Y T (3.7) 81 Considering the entire structure and assembling all the elements, then the force contribution of all elements on all nodes is { } {} {} {} { } {} {} {} {} 1 γγ =   ∂∂∂∂   −   ∂∂ ∂ ∂    m ggjj g jj j gg g g ii YY TT T YT Y T (3.8) Because of the force balance of the structure at each node, the summary of the inner forces and external loads (including the applied loads { } g Q and reaction forces { } g R ) should be zero and then we have, { } { } { } 0  ++ =  gg g g MT Q R (3.9) with {} {} {} {} {} {} {} {} 1 γγ =  ∂∂ ∂∂    =−   ∂∂∂∂   m jj g jj j gg ii TT YY M YY TT (3.10) { } { } ,1 ,1 ,1 , , , , , , =  T gx y z xi yi zi xn yn zn Qq q q q q q q q q (3.11) { } { } ,1 ,1 ,1 , , , , , , =  T gx y z xi yi zi xn yn zn Rr r r r r r r r r (3.12) Then Eq. (3.9) is the force balance equation without any boundary conditions. The following is to discuss the formulation based on the boundary conditions which has four types in our consideration: Type 1. ball-and-sock joint: At ith node with such joint, the boundary constraint will provide the support forces in three directions, the external load will not affect the structure and the force balance at this node is not worth to discuss. Therefore in the matrix  g M , the rows related to the ith node, the numbers of which are32,3 1,3 −− ii i , and { } ,, , ,, xiyi zi qq q in the vector { } g Q and { } ,, , ,, xiyi zi rr r in the vector { } g R need to be indexed out and the number of system dimension is reduced by 3. Specifically, if two ends of the elements are both ball-and-sock joints, the element which only influents the reaction forces, is not deformable due to the external loads 82 and always satisfies the equilibrium no matter what deformation it has. Therefore such element is out of the interest of static equilibrium and should be indexed out from the vector { } g T . Type 2. slotted roller joint: Since the constrain at this joint is described as {} { } ,, , , , , = T llxi lyi lzi i cc c c (3.13) with { } 1 = l i c , the only direction of the force balance at the node is along the direction of constraint and the node has one dimensional displacement. Therefore, all the forces at this node, both inner and external, should be projected to the vector { } l i c and only one coordinate of the projection need to be indexed for the force balancing. Mathematically the projection of any force vector { } i F at this joint is calculated by { } {}{}{ } = T cl l l ii i i F cc F (3.14) and the system dimension of the force balance is reduced by 2 for each constraint of this type as shown below, ,, ,, , , ,, , , 0 remove dimensions at 3 -1, 3 0 & 0 remove dimensions at 3 - 2, 3 0 & 0 remove dimensions at 3 - 2, 3 1  ≠  =≠   == −  lxi lxi l y i lxi l yi cii cc i i cc i i (3.15) Type 3. plane roller joint: this constrain restricts the displacement of the node in the plane which has the normal vector of {} { } ,, , , , , = T nnxi nyi nzi i cc c c (3.16) with { } 1 = n i c . Then the projection of the forces onto the plane should be balanced. For any force vector { } i F , its projection onto the constraint plane with the normal vector { } n i c is {} [] {}{} () {} =− T cn n n iiii F Ic c F (3.17) 83 and only two dimension of { } cn i F need to be considered for force balance. Therefore, each joint of this type will reduce the system dimension by 1 and the dimension reduction is expressed as. ,, ,, , , ,, , , 0 remove dimension at 3 0 & 0 remove dimension at 3 -1 0 & 0 remove dimension at 3 - 2  ≠  =≠   ==  nz i nz i n y i nz i n y i ci cc i cc i (3.18) Type 4. short link: this boundary constraint will maintain the end of the element at the constant distance , ci l away from the node and the direction of short link is described as {} { } ,, , , , , = T eexi eyi ezi i cc c c (3.19) with { } 1 = e i c . In this case, the nodal coordinate of the end of the element is no longer the nodal coordinate { } i Y of the constraint node, but the nodal coordinate {} { } * ** * = T ii i i Yx y z at the connection between the element and the short link, which is obtained as {} {} { } * , =+ ci e ii i YY l c (3.20) Also this type of constraint provides the support force along the direction of the short link, and the force balance in the plane normal to the short link should be discussed. Similar to the force balance at the constraint of type 3, the plane projection of the force vector { } i F is {} [] {}{} () {} =− T ce e e iiii F Ic c F (3.21) and two dimension of this projection is indexed into calculation. Therefore, the system dimension will decrease by 1 for each type 4 constraint (Eq. (3.22)). ,, ,, , , ,, , , 0 remove dimension at 3 0 & 0 remove dimension at 3 -1 0 & 0 remove dimension at 3 - 2  ≠  =≠   ==  ez i ez i e y i ez i e y i ci cc i cc i (3.22) Besides the boundary conditions, not all the nodes can always be actuated by applied external forces and the loads may have directional constraints, which may lead to dependent 84 entries of { } g Q . Let the vector { } Q with the dimension of 1 × d presents the independent external loads applied at the selected nodes.     g B is a3 × nd coefficient matrix and is used to describe the transform from { } Q to { } g Q . If the external force at ith node is restricted to the specific direction and j th element j Q of the load vector { } Q is used to present this independent force, as the entries of   g B , 32, −ij B , 31, − ij B and 3, ij B are the directional cosines of the applied force at this node in the global coordinates. In the second case, the external force of i th node is restricted to the specific plane which is parallel to the plane ,, =+ xiyi zc x c y , and j Q and 1 + j Q are the independent force components at this node. Then in the matrix  g B , 32, 1 − = ij B , 31, 1 1 −+ = ij B , 3, = x ij B c and 3, 1 + = y ij B c . As the third case, if the force is in the yz plane presented by j Q and 1 + j Q , then 31, 1 − = ij B and 3, 1 1 ij B + = . Similarly, as the force is in the xz plane, 32, 1 − = ij B and 3, 1 1 + = ij B . In the last case, when the external force is a predefined none-zero force or a unknown three-dimensional free force at ith node expressed by independent load components j Q , 1 + j Q and 2 + j Q , 32, −ij B , 31, 1 −+ ij B , 3, 2 + ij B are all set to 1. After   g B is formulated by setting the rest entrees as zeros, the vector { } g Q presenting the external loads in all directions of all nodes can be obtained from { } Q by the following, { } { }  =  gg QB Q (3.23) Particularly, if no external forces are applied on the entire structure { } Q is set to a zero vector with 3 = dn and  g B is a 33 × nn identity matrix. To build up the complete force balancing model, four boundary conditions and the loading restrictions addressed above must be included into the Eq. (3.9). Here we propose the modeling approach as follows. The system modeling procedure: 1. Preliminary treatment on nodes. If type 4 boundary nodes exist, instead of calculating the matrix  g M , replace the boundary node of type 4 { } i Y by the modified nodal coordinates 85 {} * i Y in Eq. (3.20). Then calculate the matrix *     g M based on the new set of the structure nodes with modified nodal coordinates for all type 4 boundary nodes. If no type 4 boundary conditions occur in the structure, then *   =   g g MM . After the first step the system of equation is { } { } { } * 0  ++ =  gg g g MT Q R (3.24) 2. Apply the boundary conditions of the trusses. Construct n 3-by-3 nodal constraint matrices [ ] d i C for i th node according to Table 1 in which [] I denotes the identity matrix. Use n matrices [ ] d i C as the diagonal entries to construct a 3n-by-3n matrix [ ] D C as below, [] [ ] [] [] 1 0 0         =           d d D i d n C C C C (3.25) Define the dimension reduction operation in Table 3.1 as the function of () * ℜ . Implement this operation on [ ] D C , or [ ] () ℜ D C in mathematical expression. Obviously [ ] () ℜ D C has full rank in rows. After applying the boundary conditions onto the structure as multiplying [ ] () ℜ D C onto Eq. (3.24), the system model is rewritten below since [ ] (){ } { } 0 ℜ= Dg CR . [] () { } [] (){ } *  ℜ=−ℜ  D gg D g CMT C Q (3.26) Table 3.1 Nodal constraint matrices at node i Type Description Constraint Matrix Dimension Reduction on [ ] D C Not B.C. Free move in three dimensions [ ] [ ] 33 × = d i CI No operations Type 1 Fixed node, unmovable [ ] [ ] 33 0 × = d i C Remove rows at 86 3i-2, 3i-1, 3i Type 2 Movable in one dimension [ ] {}{} = T dl l ii i Cc c Eq. (3.15) Type 3 Movable in a two- dimensional plane [ ] [ ] {}{} 33 × =− T dnn ii i CI c c Eq. (3.18) Type 4 Movable on the contour of a sphere [ ] [ ] {}{} 33 × =− T dee ii i CI c c Eq. (3.22) 3. Treatment on elements. If two ends of j th element are all on the type 1 joints, then the element can not be affected by external loads. It must be index out of the tension vector { } g T as { } T and the corresponding jth column of the matrix *     g M should also be index out of the matrix to obtain the modified matrix [ ] * M . If no elements are fixed with both ends of type 1 nodes, then[] * *  =  g MM and{ } { } = g TT . Hence the system equation becomes [ ] ()[ ]{} [ ] (){ } * ℜ=−ℜ D Dg CM T C Q (3.27) 4. Add the external loading restrictions. After considering all boundary conditions into the model, the loading conditions at the actuated nodes need to be included by substituting Eq. (3.23) into Eq. (3.26). Then we have [ ] ()[ ]{} [ ] () {} *  ℜ=−ℜ  DDg CM T C B Q (3.28) Therefore, the system equation of force balance can be written as []{ } []{ } =− MTCQ (3.29) in which [ ] [ ] ()[ ] [] [ ] () * =ℜ   =ℜ   D Dg M CM CC B (3.30) 87 In the above equation, the matrix [ ] M is a × pq matrix, { } T is a 1 × q vector, [ ] C is a × pd matrix and { } Q is the external force vector with dimension of d with 12 3 4 1 33 2 =− − − − =− cc c c c p nn n n n qm m (3.31) in which 1 c n , 2 c n , 3 c n and 4 c n are the number of the boundary nodes of type 1, 2, 3, and 4 respectively and 1 c m is the number of elements with both type 1 nodes as ends. It should be noticed that p is also the degree of freedom of the structure. Note that j th column of [ ] C presents the impact of independent load component j Q on specific degree of the system. If the columns of [ ] C are linearly independent, [ ] [ ]{} { } + =− CM T Q and the size of [ ] [ ] + CM is × dq. According to Ref. [7, 70], here the Maxell’s rule for the general truss structure with boundary conditions, loading restrictions and undeformable members is = dq (3.32) Define s p as number of the pre-stress states, m q as number of infinitesimal mechanisms including the rigid and flexible mechanisms, and r is the rank of the matrix [ ] [ ] + CM , then =− =− s m p dr qq r (3.33) 3.3 System Model and its Solution Define d S to be the set of element indices for deformable elements and u S being the set of element numbers for undeformable elements of which two ends are type 1 boundary nodes. According to the modeling procedure of Eq. (3.29), the entry of { } T is j T when ∈ d j S , namely { } { } =∈ j d TTjS (3.34) 88 Similarly define { } L to be the undeformed profile of the deformable elements, or { } { } =∈ j d L Lj S (3.35) In many applications, the designate configuration of truss structure works under different temperatures from the material’s nominal temperature. The induced static thermal distortion is described by the thermal strain , ε th j and , εα =Δ th j j j T (3.36) withα j as the coefficient of thermal expansion of jth element and Δ j T being the temperature difference from nominal temperature. Hence the total strain of the truss member is the summary of elastic strain and thermal strain. Setting , () εε + j jthj A as the area of cross-section of element which is also the function of axial strain, the element tension is , () εε σ =+ j jj thj j TA (3.37) withε j as the elastic strain and σ j as the stress of the element. For jth element when ∈ d j S , the elongation of element and the element stress under geometric and elastic nonlinearity of material are , εε − += j j jthj j lL L (3.38) () σεε = j jjj E (3.39) with j l as the deformed length and () ε j j E being the nonlinear Young’s modulus. After including the force balance of the system, we have the claim below, []{ } []{ } =− MTCQ (3.40) , ( ) ( ) for εεεε += ∈ j jj j thj j j d EATjS (3.41) 89 , for εε − += ∈ jj j th j d j lL j S L (3.42) Claim: The solution of the system of Eq. (3.40) - (3.42) always exists, if and only if the deformed shape of the truss structure exists in reality. As the design problem of deployable truss structures, the undeformed profile is the solution under the given external forces. Since the above equation system includes the force balance, the material elasticity and the elongation geometry of the truss structure, it fully describes the statics of the structure and offers the sufficient and necessary condition on the existence of the configuration. Because the existence condition of Eq. (3.42) which is 1 ε >− j can be included into the conditions for Eq. (3.40) and Eq. (3.41) shown in the immediate following, we shall only need to discuss the existence conditions of first two equations and then their solution formulation. As known, when the solution of Eq. (3.40) exists, it can be expressed as below where r is the rank of [ ] M , { } h T is the homogenous solution and { } p T is the particular solution. {} {} { } {}  +<  =  =   hp p TT rq T Trq (3.43) with { } [ ] [ ]{} + =− p TMCQ (3.44) Denote { } k n as the null vectors of the matrix [ ] M , and the null space [ ] N of the matrix [ ] M is [] {} {} 1 −   =    qr Nn n (3.45) Set { } β being an arbitrary matrix constructed in the following {} 1 ββ β −   =    T qr (3.46) Then the general form of a homogenous solution of the Eq. (3.43) is 90 { } [ ]{ } β = h TN (3.47) Therefore the solution of the system Eq. (3.40) can be summarized by the following lemma. Lemma 1: When Eq. (3.40) has a solution, such solution is obtained by {} []{}[][]{} [][]{} β + +  −<  =  −=   NMCQrq T M CQ r q (3.48) From matrix theory, the vector { } p T can always be obtained by Eq. (3.44) regardless the existence of the Eq. (3.40). If Eq. (3.40) does not have any solution, { } p T calculated by Eq. (3.44) is not the exact solution but the least square approximate solution of the system. Substitute Eq. (3.44) into Eq. (3.40), we have [][][] () []{} 0 + −= IMM CQ (3.49) Then Eq. (3.49) is valid if and only if Eq. (3.40) has a solution. As [] [ ][ ] () [] + − I MM C is a p- by-d matrix, the rank of[] [ ][ ] () [] + − I MM C , namely Q r , has the relationship of ≤ Q rd . Define  Q N to be the null space of the matrix [] [ ][ ] () [] + − I MM C and { } β Q as an arbitrary ()1 −× Q dr real vector. Therefore the sufficient and necessary condition to validate Eq. (3.49) is {} {} { } {}  +<  =  =   hp Q pQ QQ r d Q Qrd (3.50) in which { } 0 = p Q is the particular solution, { } h Q is the homogenous solution of Eq. (3.49) and { } { } β  =  hQ Q QN (3.51) In summary, we shall have the following lemma: Lemma 2: The equation of Eq. (3.40) has a solution if and only if 91 {} { } {} 0 β  <   =  =   QQ Q Q Nrd Q rd (3.52) Particularly, if [ ][ ] [ ] + = MMI , Eq. (3.40) always has a solution for any { }∈ d QR . The lemma reveals the fact that except the special case, the existence of { } T in Eq. (3.40) purely depends on the choice of the external load instead of the rank of the matrix [ ] M . Therefore, as long as the external load vector is to be determined instead of predefined, Eq. (3.40) always has a solution. In the design problem, the external loads could be either pre- defined or under determined. If the external loads are predefined as { } pre Q including the case when no loads are presented, the loads must satisfy the following consistency theorem for linear systems to validate the Eq. (3.40). [ ] [ ] [ ]{ } () ( )   =−   pre rank M rank M C Q (3.53) Therefore it can be treated as a preliminary exam of any truss design. If Eq. (3.53) is not valid when the loads are predefined, the design is impossible to be realized. By adding the predefined case into the Lemma 2, the external forces can be written as {} { } {} {} 0 β    ==    <   pre Q QQ Q Q predefined Qrd Nrd (3.54) Define {} {} () {} {} {} {} & or & & β β ββ  <=    ==<    <<   T Q T QQ T T T QQ r q predefined r d Xrqrd rq r d (3.55) 92 [] [] () [][] [] [ ][] & or & & + +  <=    =− = <      −<<     Q QQ QQ N r q predefined r d KMCN rqrd NM CN rqr d (3.56) {} [][]{ } [] & 0 +  −<  =    pre c M C Q r q predefined K others (3.57) Then according to Lemma 1 and 2, the solution of Eq. (3.40) can be concluded by the following Lemma: Lemma 3: Under the satisfaction of Eq. (3.53) for the case of predefined external loads, the sufficient and necessary solution { } T of Eq. (3.40) is {} [][]{ } {} []{} { } & 0& +  −=   ===   +   pre Q c M C Q r q predefined Trqrd KX K others (3.58) The existence condition of Eq. (3.41) is based on the fact that the elongation of an element of the truss is always bounded. The elastic element can only shrink or extend within certain strain limitation before it is buckled, broken or yields, mathematically * min, max, 1εεε −< ≤ ≤ j jj (3.59) as * min, ε j and max, ε j are the lower bound and upper bound of the elastic strain due to the material properties respectively. In order to ensure the positive original and deformed lengths of the element, from Eq. (3.42) it is sufficient and necessary to have , 1 εε >− − j th j (3.60) Therefore considering the thermal distortion, the modified lower bound of the elastic strain min, ε j is * min, , min, max( 1 , ) εεε =−− j th j j (3.61) 93 Hence we can define the following, () () min, , min, max, max, , inf ( ) ( ) , sup ( ) ( ) ε ε εεε ε εε ε εεε ε =+   ∈   =+ j j jjjjjthjj jj j jjjjjthjj TEA TEA (3.62) Obviously we have, Lemma 4: Eq (3.41) and (3.42) have a solution if and only if the followings hold: { } { } { } min max ≤≤ TT T (3.63) { } { } {}{} min min, max max, =∈ =∈ jd jd TT jS TT jS (3.64) As long as () ε j j E , , () εε + j jthj A are known functions of ε j , min, j T and min, j T can be calculated numerically. Since usually the stress of the material is or closed to a monotonically increasing function of the strain and the area of the cross section of element does not vary much, it is a good estimation of value range of { } () ε T by: { } { } {}{} min min, min, , min, max max, max, , max, ()( ) ()( ) εε εε εε εε =+ ∈ =+ ∈ jj j j thjj d jj jj thjj d TE A jS TE A jS (3.65) Therefore we may have the following corollary based on Lemma 4. Corollary 4.1: When the Young’s modulus is a monotonically increasing function of the strain with constant cross-area for each element of truss structure, Eq. (3.63) and (3.65) provide the sufficient and necessary formulation to solve Eq. (3.41) and (3.42). Similarly, we have, Corollary 4.2: If the longitudinal rigidity of the material for each deformable element of truss structure is linear, Eq. (3.63) and (3.66) are the sufficient and necessary condition of the existence of solution for Eq. (3.41) and (3.42). { } { } {}{} min min, max max, ε ε =∈ =∈ jj j d jj j d TEA jS TEA jS (3.66) 94 Corollary 4.3: If the element material has constant cross-section area and linear cable elasticity of which Young’s modulus is defined by Eq. (3.67), Eq. (3.63) and (3.68) are the sufficient and necessary condition for the existence of the solution in Eq. (3.41) and (3.42). 0 () 00 ε ε ε ≥  =  ≤  jj jj j E E (3.67) { } { } {}{} min min, max max, max(0, ) ε ε =∈ =∈ jj j d jj j d TEA jS TEA jS (3.68) In addition, the areas of cross-section of elements can also be unknown variables in many engineering problems and are to be determined in the design. The areas of cross section as design variables are defined below, { } { } | =∈ j d A Aj S (3.69) Then similar with Lemma 4, the following lemma is formulated for the design task when { } A is under determination. Lemma 5: When the area of cross section { } A is one of the design variables, Eq. (3.41) and (3.42) have a solution if and only if the following equations and Eq. (3.63) hold, in which [] min σ and [ ] max σ are diagonal matrices with the diagonal entries to be min, σ j and max, σ j for ∈ d j S respectively. { } [ ]{ } { } [ ]{ } min min max max ,σσ == TAT A (3.70) () () min, min, max, max, inf ( ) , and sup ( ) ε ε σεε εε ε σεε =  ∈∈  = j j j jjj jj j d j jjj E j S E (3.71) Corollary 5.1: If the Young’s modulus is a monotonically increasing function and { } A is the design variable, Eq. (3.63), (3.70) and (3.72) provide the sufficient and necessary formulation to solve Eq. (3.41) and (3.42). 95 min, min, min, max, max, max, () () σεε σεε = ∈ = jj j j d jj j j E j S E (3.72) Corollary 5.2: If the Young’s modulus of the material for each deformable element of truss structure is linear and { } A is the design variable, Eq. (3.63), (3.70) and (3.73) are the sufficient and necessary condition of the existence of solution for Eq. (3.41) and (3.42). min, min, max, max, σε σε = ∈ = jj j d jj j E j S E (3.73) Corollary 5.3: If the Young’s modulus of each element is defined by Eq. (3.67) and { } A is the design variable, Eq. (3.63), (3.70) and (3.74) are the sufficient and necessary condition on the existence of the solution in Eq. (3.41) and (3.42). min, min, max, max, max(0, ) σε σε = ∈ = jjj d jj j E j S E (3.74) Combining all above lemmas and corollaries, the solution of the system model with Eq. (3.40) - (3.42) can be concluded into the following theorem: Theorem: For the first two cases of Eq. (3.58), { } T is the solution of Eq. (3.40) - (3.42) if and only if it satisfies the Eq. (3.53) and (3.63). In the third case, when cross-section areas are not the design variables, the solution is sufficient and necessary to satisfy the inequality below, [ ]{ } { } < aX b (3.75) with [] [ ] [] {} { } { } {}{ } max min ,  −   ==    −+ −    c c TK K ab TK K (3.76) If { } A is considered as design variable, the following inequality is the sufficient and necessary condition of solution. 96 [ ]{ } [ ]{ } { } +< A A aX a A b (3.77) in which [] [ ] [] {} { } {} max min , σ σ  − −  ==       c AA c K ab K (3.78) In the different structure setup, { } min T , { } max T , min, σ j and max, σ j in the above theorem can be formulized by Lemma 4, 5 or their corollaries accordingly. In whichever cases of the theorem, the undeformed profile { } L and the load { } Q can always be obtained from feasible { } X , { } T and { } A . When Eq. (3.53), (3.63), (3.75) or (3.77) is infeasible, based on Claim, such structure configuration does not exist and can not be realized. In addition to the theorem, if Eq. (3.65) and (3.72) are the approximations of their limits, the formulation resulting in Eq. (3.53), (3.63), (3.75) and (3.77) only offers the sufficient conditions. 3.4 Optimization Formulation Based on the solution of the truss model, the optimization formulations are discussed in this section regarding different design objectives and various design constraints, all of which share the generality of the quadratic programming format. It should be pointed out that the following six optimizing formulations are not isolated cases. Any combination of one objective and four types of constraints by the different design requirements can be freely merged into the same optimization format in general. 3.4.1 Objective 1: To minimize the member tension band of the truss In many space applications, the tension are expected to distributed within a range as narrow as possible. As the first two cases of Eq. (3.58), the element tensions are constant, resulting the only one possible tension band without need of optimization. To minimize the band of tension distribution for the third case, we define {} [ ]{} * = T TBT (3.79) 97 [] 11 11 −   −     =     −   −      T q qq B q qq (3.80) Hence () {} {} ** 12 2 1 2 μμ ∞ −≤ ≤ upper lower TT T T (3.81) where 1 μ and 2 μ are positive constants, and max( ) = upper j TT and min( ) = lower j TT for ∈ d j S . Therefore minimizing the tension band − upper lower TT can be achieved by {} 2 * 2 1 min( ) 2 T . This leads to the following quadratic programming problem as the design objective function, {} {}[]{} { }{ } 1 min( ) 2 + TT bb X XH X f X (3.82) where by Eq. (3.58) and (3.79), [] [ ][] () [][] () {} { }[][ ][] = = T bT T T TT bc T T H BK B K f KB B K (3.83) 3.4.2 Objective 2: To unify the member tensions as a desired one Another design objective frequently considered is to achieve a designated uniform or closed-to-uniform tension distribution in the structure. Similarly the optimization is only required for the third case of Eq. (3.58). Denote the desired uniform tension is des T . Then the different between the actual tensions with the desired one can be defined as: { } { } 1 − des q TT (3.84) in which { } 1 q is a unity vector with dimension of q. Hence obtaining uniform tension distribution is to minimize the above different, or to minimize the norm of the difference as the following, 98 {} {} 2 2 1 min 1 2  −   des q TT (3.85) Since {} {} {}{} {}{} {}{ } 2 2 2 11 1 11 11 22 2 −= − + TT T desq desq desq q TT T T T T T (3.86) Substituting Eq. (3.86) and dropping the constant term {}{} 2 1 11 2 T des q q T and { }{} 1 − T des q c TK , the design objective function of the above optimization is then, {} {}[]{} { }{} 1 min 2  +   TT uu X X HX f X (3.87) in which [ ] [ ] [ ] = T u H KK (3.88) {} [ ] [ ] [ ] { } 1 =− TT T uc desq fK K KT (3.89) 3.4.3 Constraint 1: Specific element strain requirement As discussed previously, the solution formulation of truss model considers the strain constraint for each element. Therefore, as the first two case of the solution { } T , Eq. (3.63) should be satisfied. Otherwise Eq. (3.75) or (3.77) should be added into the optimization formulation as constraints. 3.4.4 Constraint 2: Specific cross-section area requirement When the cross-section area is the design variable, it is commonly restricted with a selection range. Denote the min, j A and max, j A are the minimum and maximum area the element could have respectively, and { } { } { } { } min min, max max, || =∈ = ∈ jdjd A AjS A A jS (3.90) 99 Then for any case of the solution { } T , in addition to Eq. (3.63) and (3.77), the inequality below has to be satisfied. { } { } { } min max << AA A (3.91) 3.4.5 Constraint 3: Specific member tension range Many designs restrict the inner tension range of the elements directly by certain predefined lower and upper bound vector for each element as { } l T and { } h T respectively. Then besides Eq. (3.63) to be satisfied, the following additional inequality should also be included, { } { } { } << lh TT T (3.92) In the first two cases of Eq. (3.58), tension solution is unique and Eq. (3.63) and (3.92) must be valid to ensure the existence of the design. If the optimization formulation is need as the third case, besides Eq. (3.75) or (3.77), one can simply add the following constraint equation into optimization constraints, [ ]{ } { } < tt aX b (3.93) with [] [ ] [] {} { } { } {} { } ,  −   ==    −+ −    hc tt lc TK K ab TK K (3.94) 3.4.6 Constraint 4: Specific external load range The external loads applied on the structure are usually generated by actuators or other structure elements and the magnitude of the loads has certain limitation, either lower bound, upper bound or both. Let { } min Q and { } max Q to describe the lower limit and the upper limit of the external force vector respectively, then Eq. (3.95) need to be fulfilled in the design. { } { } { } min max << QQ Q (3.95) 100 When the external forces are directly obtained in the first two cases of Eq. (3.58), the above constraint can be verified directly upon the satisfactory of Eq. (3.63). In the third case, the external force restriction will be considered in the optimization by adding the following inequality into the constraints including Eq. (3.75) or (3.77), [ ]{ } { } []{} { } fmax fmax fmin fmin < < aX b aX b (3.96) in which [] [ ] []  () fmax fmin 0& & ×−    <<   =− =     =<    dq r QQ QQ Nrqr d aa Nrqrd (3.97) { } { } {} { } fmax max fmin min = =− bQ bQ (3.98) 3.5 Design Procedure & Remarks Within this core section of the chapter, we will summarize the entire design procedure of optimal parametric design method for deployable truss structures based on the discussions above including all the objectives and constraints mentioned. Moreover, important remarks will be followed addressing the unique features and key characteristics the proposed method possesses. The flow chart in Figure 3.1, presents the optimal parametric design method proposed in the chapter. The goal of the design is to determine the optimal set of structural parameters, including cross-section areas, external loads and/or undeformed profile for certain deformed configuration of a space truss structure which fulfills the design objective under the design constraints. 101 Figure 3.1 Optimal Parametric Design Method. The very first step of the procedure is to build up the equilibrium equation in Eq. (3.29) of the structure with boundary conditions and formulate the matrices [ ] M and [ ] C . This involves all the derivations in Section III and need to follow the modeling procedure proposed in the section. Once the system model is ready, its solvability must be verified. When the external loads are predefined, Eq. (3.53) must be valid if and only if Eq. (3.29) is solvable; otherwise it means that the desired truss configuration is impossible to be realized and the design algorithm is ended. If the loads are unknown and need to be determined, Lemma 1 guarantees the system equation to always have a solution. After the feasibility of the system is verified, all the constraints as Eq. (3.63), (3.91), (3.92) and (3.95) are formulated according to the design requirements. If the elemental tensions 102 fall into first two cases of Eq. (3.58), then the tensions and the induced loads must satisfy the constraints obtained. In the case that not all the constraints are fulfilled, the design is not achievable. When the design is feasible, the rest of the design variables such asε k , k A and k L , can be calculated from the unique solution of { } T and { } Q . In the third case of Eq. (3.58), the optimization is needed, and the objective function as Eq. (3.82) or (3.87) and optimization constraints such as Eq. (3.75), (3.77), (3.91), (3.93) and/or (3.96) could be formulated according to the design task. Only the feasible solution of the optimization validates the design and this solution is the optimal design for the element tensions { } T and the external loads { } Q . Note that the optimization has the standard format of the quadratic programming problem and many successful solving algorithms are available already. Under the design constraints on strains and areas of cross sections, the final design of the other parameters asε j , j A and j L can be obtained by solving the system of Eq. (3.41) and (3.42). Since the solution existence of this system is guaranteed by the Theorem, practically one can separately solve Eq. (3.41) under the relevant constraints first and then the rest of two equations later. By overviewing the design procedure, it is obvious that the optimal design method proposed in the flowchart of Figure 3.1 always provides the sufficient and necessary formulation to obtain the design solution. According to the theorem, lemmas and corollaries mentioned and the above discussion of the design procedure, we can address the following important remarks: Remark 1: The chapter presents the unique optimal design approach regarding the deformed configuration of deployable trusses. It handles the truss design under thermal distortions and the constraints on element tensions, external loads, element strains and cross- section areas. The formulation of the design method considers the static deformation of flexible trusses with nonlinear elasticity and nonlinear geometry and such generality allows it to be applied onto other truss-type structures such as cable networks and tensegrity structures in diverse applications. The method solves force balancing, deformation geometry and stress-strain relations simultaneously by deriving the solution of the first one and transforming the latter two into the constraints of the design, which avoid the complexity of solving three system equations together. The undeformed profile and the cross-section areas obtained by the method may lead to 103 either zero-tension or pretension initial status without any external loading, which is not concerned in the deployable structure applications. Remark 2: The proposed method innovatively includes the external loads into the design variables for the first time in the academic literature and therefore transforms the design result into the natural property of the deformed shape geometry (nodal coordinates and member connections) of truss structures. The method is able to design all the parameters together (external loads, inner tensions, areas of cross sections and undeformed profile) on the deformed truss. Therefore under the same structural setup including boundary conditions, loading restrictions and constraints from application requirements, the design solution is solely dependent on the deformed geometry and its structural performance can be considered not only as one set of design results but essentially as the property of the certain deformed shape. Remark 3: The design algorithm guarantees the “Best or None” solution of the design task by providing either the global optimal truss parameters from all possible choices or the infeasible solution which violates the design constraints least. This is first because of the sufficient and necessary design formulation; secondly, the Hessian matrix of the objective function in the design optimization is always semi-definite and therefore the quadratic programming algorithm always has the global convergence and a global optimal. In some unusual cases, there may be more than one global optimal sharing the same value of objective function and the existing solving algorithm always attempts to return a feasible one. Hence the feasible solution is always the best result from all possibilities on the design task. On the other side, the infeasible solution means the task is impossible to be achieved, leading to either the fault design requirements or the ill structure shape. Due to this nature, it can be used as a criterion of truss design for the working (deformed) configuration. Since the derived member tensions are always the solution of the force balance equation regardless the satisfaction of design constraints, this implies that the infeasible design can be treated as the guidance to determine the most aggressive limits and constraints which the structure configuration is tolerable with. Therefore when it is allowed, the design requirement may be adjusted to ensure the feasible solution. This is another key feature of the proposed method and provides a significant advantage in practical implementations. 104 Remark 4: In the final step of determining ε j and j A by solving Eq. (3.41) under their constraints, it is possible to result in multiple solutions. This could happen due to the nonlinearity of the Young’s modulus and the non-consistence of cross-section areas as shown in the left of Figure 3.2. In the special case, when the cross-section areas are the design variables and the inner tensions fall into the first two cases of Eq. (3.58), there may be infinite sets of solutions within the constraints (as in Figure 3.2(b) where j E is constant, 0 Δ= j T and ε = jjj j A TE ). However, all those solutions are optimal by sharing the same design perspective and validate every design requirement. Engineers could choose any set of solutions based on other secondary design considerations such as less deformations or uniform cross-section areas. Figure 3.2 Multi-optimal-solution cases: (a) finite solutions (left); (b) infinite solutions (right). Remark 5: Considering the common types of nonlinear materials, the inequalities of Corollary 4.1 and 5.1 are fairly good estimations of the available range of { } T . Using this assession, the feasible design from the proposed design formulation is the second best solution which could be much closed to the globally optimal one and usually adequate for the design mission. The infeasible solution can not deny the design case although the existence could be very unlikely. However, infeasible case often implies that it would be very hard to find a set of parameters validating the design task. 105 3.6 Extended Applications In this section, the above design method is extended to achieve other design tasks. However, due to either the non-global convergence of the optimization or only sufficient formulation of the problem, the extended design approach may not provide the “Best or None” solution and Remark 3 of Section VI no longer holds. 3.6.1 Weight minimization Weight is always a crucial factor for space structures. In the following, it shows that the optimal parameters of space truss structure with minimum weight design can be achieved by slightly modifying the proposed method. Define ρ j as the density of the material. To minimize the weight of the structure is to minimize the total weight of the deformable members, mathematically, min (0) ρ ∈  j jj d A LjS (3.99) When { } T is one of the first two cases of Eq. (3.58), it is unique and should satisfy Eq. (3.63) to ensure the existence of the system. Since j T is calculated by j A and j L from Eq. (3.41) and (3.42), the optimization should also include those equations. Then we have the optimization formulation of this case to be {}{} {}{}{ } , , , min max min (0) .. ( ) ( ) ρ εεεε εε ∈ += − += ≤≤ jjj d LA jj jj thjj j jj jthj j AL j S st E A T lL L TT T (3.100) According to the Theorem in the third case of Eq. (3.58), Eq. (3.75) and (3.77) are the existence condition of the system. Similarly, the formulation of the weight minimization for such case is 106 {}{ }{} []{} {} []{}[]{} { } ,, , , min (0) .. ( ) ( ) or ρ εεε ε εε ∈ += − += <+< jjj d LX A jj jj thjj j jj jthj j AA AL j S st E A T lL L aX b a X a A b (3.101) Simply replacing the final “Determine Design Variables” step in Figure 3.1 by the relevant optimization of Eq. (3.100) or (3.101) and combining other constraints into this optimization according to the design requirements, the proposed method can be used to design the optimal parameters of the deformed truss structure with minimum the total structure weight. Since both of optimizations are the nonlinear programming problem, the design solution obtained by common solving technique could only be local optimal. However, if the global optimal can be achieved by specially designed solving algorithm, Remark 3 on the design method will still be valid. 3.6.2 Layered design strategy Figure 3.3 Topology of layer separation strategy. Besides considering the truss structure as a whole, it is usually able to be separated into multiple layers and pay more attention on certain important layers in some design task. The topology of the layer generation is presented in Figure 3.3. Assume a truss is divided into s layers. As the criterion of separation, only the k th and k+1 th layers can have intersection nodes to agree the topology of Figure 3.3. The borders of layers which are the intersections between layers are set on nodes, namely the border nodes, and the members with two ends at the border nodes belong to the next layer. Assume in kth layer there are k d load components and k q deformable members. { } k Q presents the independent external force components applied on kth 107 layer and { } k T is the vector of tensions for deformable elements in the layer. Since the system model of the entire structure is []{ } []{ } =− MTCQ After regrouping the tensions and external forces by layers, the system equation can be rewritten into Eq. (3.102), or alternatively Eq. (3.103). Note that the separation in rows for [ ] M and [ ] C presents the force balance inside each layer. Because only the tensions and loads applied onto the layer’s nodes have contributions to the force balance of the layer, [ ] M will become a Jordan matrix and [ ] C is reformed into a diagonal matrix. [ ] [] [ ] [] [] [] [] {} {} {} {} [] [] [] [] {} {} {} {} 1,1 11 1 2,1 2,2 22 2 11 1 1, 2 1, 1 ,1 , 0 0 0 0 −− − −− −− −             =−                    ss s ss s s ss s ss s s M CTQ MM CTQ C TQ MM C TQ MM (3.102) [ ] { } [ ] { } { } , =− + kkk kk k M TCQ D (3.103) with {} { } [] {} 1 ,1 01 2 − −  =  =  −≥   k k kk k D MT k (3.104) By substituting the particular solution of { } k T as[ ] [ ]{} { } () , + −+ kk kk k MCQ D , similarly with Lemma 1, the sufficient and necessary condition of validity of Eq. (3.103) is shown below, from which { } k Q can be solved by Eq. (3.106), [][] [] () []{} [] [ ] [ ] () {} ,, ,, ++ −=− kk kk k k k k k k k I MM C Q I M M D (3.105) {} { } {} {} { } , , β    ==   +<    pred k pQkk k k QQ p Qk k kk k Q predefined QQ rd NQ rd (3.106) 108 in which { } [][] [] () {} ,, + +  =−  pQ k kk kk kk QM I M M D (3.107) [][] [] () [] ,, + =−  Q kk kk k k M IM M C (3.108) with , Qk r as the rank of  Q k M and     Q k N being the null space of     Q k M . By defining the rank of [ ] , kk M as k r and its null space as [ ] k N , the solution of Eq. (3.103) is derived as the following, {} [] {} [] { } () [] {} []{} () []{} { } , , , & & + +  −=   =− = =   +   pre k k k kk k k pkkQkk kk kk k k c kk k M D C Q r q predefined TM D CQ rqr d KX K others (3.109) with [] [] () [] [] [] [ ] [] , , , , , & or & & + +  <=    =− = <       −<<    kk Qk k k QkkQkk kkkk k QkkQkk kkkk k N r q predefined r d KMCN rqrd NM CN rqr d (3.110) {} [] {} [] { } () [] {} []{} () , , & + +  −<  =  −   pre k k k kk k k c k p k kk k k M D C Q r q predefined K M D C Q others (3.111) {} {} () {} {} {} {} , , , & or & & β β ββ  <=    ==<    <<   T kk Qk k k T QkkQkk k k T T T QkkQkk k k r q predefined r d Xrqrd rq r d (3.112) 109 It is obvious that { } k D contains the external forces and tensions from previous layer. Therefore after recursively obtaining { } k T and { } k Q from the first layer to the last one, all the tensions and loads can be obtained on the entire structure. The design procedure of the layered strategy can be fitted into the procedure in Figure 3.1 simply and nicely. As the very first step, the system model of Eq. (3.29) must be built and the existence of the solution shall be confirmed before further operations. Upon this, the structure is divided into layers and the entire design procedure proposed in Section VI should be placed into iteration and implemented once per layer. Besides the force balancing and its solution replaced by Eq. (3.103), (3.106) and (3.109), each design iteration only needs the modification by adding a subscript k to present the variables for kth layer, without losing the generality of the optimization formulation. As such, multiple design implementations are carried out recursively and propagated throughout layers, generating the optimal design for the entire truss structure. This layered strategy only provides the sufficient condition of the formulation and Remark 3 of the Section VI is no longer valid. Two reasons are behind this conclusion. First, the solvability of Eq. (3.103) is not guaranteed even though the whole structure model is solvable. Therefore it is crucial to confirm the existence of Eq. (3.29) at the beginning. As long as Eq. (3.29) is solvable, by varying the layer separation, the proper separation strategy can be found to ensure the solvable model at each layer. Second, the solution of Eq. (3.103) may not validate the constraints set by design requirements while the whole structure is feasible. It is often caused by improperly separated layers where < kk qp and very likely a unique solution occurs. To avoid this case and enable the usage of the optimization in the design, besides regenerating layers, one way is to combine the kth layer with its neighbor layers as a new layer, of which the system model always has more unknowns than equations, or >≥ kk k qp r for [ ] , kk M . Comparing with the design using whole structure model, the major advantage of layered strategy is the capability to set different design objectives and/or constraints on different layers depending on the design requirements. For instant, a deployable mesh reflector may need narrower tension band distribution in the center than in the boundary area and then more strict constraints may be placed when designing the central region of the structure. 110 3.7 Design Results and Discussions To demonstrate the proposed design method, three examples are presented which cover all the design tasks discussed previously: the first example is a simple 3D truss, which focuses on the modeling procedure with different boundary conditions and thermal distortions; in the second example a 25-bar truss structure shares the same geometry with the ones in the previous researches and it will address the weight minimization problem under various constraints; Deployable mesh reflectors are considered in the final part of this section and the optimal design results under different design focuses will be presented and discussed. 3.7.1 A Simple 3D truss -2 -1 0 1 2 -2 0 2 -3 -2.5 -2 -1.5 -1 -0.5 0 y 3 (3) 6 1 (4) x (1) (6) 4 (5) 5 (2) 2 z Figure 3.4 The deformed shape of a simple 3D truss. A simple 3D truss is constructed by 6 elements and is plotted in the Figure 3.4. Among six nodes of the truss, the first three nodes are boundary nodes of type 3, 2, and 4 respectively which is defined below and all elements are connected by node 4, 5, and 6. {} { } {} { } {} { } 1 2 ,3 3 Node 1: 0 0 1 Node 2: 0.6667 0.6667 0.3333 Node 3: 0.0976 0.1952 0.9759 with 0.2 = = =− = T n T l T ee c c cl (3.113) 111 In the figure, type 2 boundary node is marked as a cross, a triangular for type 3 and a circle as type 4 boundary. The nodal coordinates of the truss is listed in the Table 3.2. Set the longitudinal rigidity of element to be 1000 N as a constant and assume all nodes are applied by the same external forces {} { } 21 10 =−− T i Q . The working temperature of the truss is 80 o C lower than the nominal temperature of element material and the thermal coefficient of expansion is 41 10 −− K . The design goal is to obtain a proper undeformed profile such that the elements in the working configuration have uniform tension as 200N. Table 3.2 Nodal coordinates of desired configuration Node Coordinates (m) i x i y i z 1 0 3 0 2 -2 -2 0 3 2 -1 0 4 -1 1 -1 5 -1 -1.5 -2 6 1.5 -0.5 -3 According to the proposed design method, the first step is always to build up the system model of force balance achieved by following the procedure in Section III. Since the structure has the type 4 boundary node at node 3, its nodal coordinates at () 1.9805, 0.9610, 0.1952 − should be updated by Eq. (3.20) to calculate *     g M . From the boundary condition of Eq. (3.113), [ ] d C is derived by Eq. (3.25) and Table 1. After the dimension reduction on [ ] d C , four rows are removed (two at node 3 and one each at node 1 and 2). Because no element has type 1 boundary nodes as its two ends, [] * *   =   g MM , { } { } = g TT and Eq. (3.27) is formulated. Once     g B is generated as all nodes are applied by the external loads, the system model of the structure is formulated by Eq. (3.29), with 14 = p , 6 = q and 18 = d . 112 Since the external load is predefined by the design requirement, it is necessary to verify Eq. (3.53) to ensure the existence of the solution for Eq. (3.29). However, the rank of [ ] M is 6 and the right hand side of Eq. (3.53) is 7, which implies that the desired configuration can not exist and it ends the design procedure. However, if the external forces are not assigned but to be determined, the solution of Eq. (3.29) always exists and 8 = Q r . Considering 0.3 0.3 ε −< < j as the elastic strain limit of the design and Eq. (3.75) can be formulated as the optimization constraint. Because the design goal does not restrict the member tension and external loads and < Q rd , no other design constraints need to be considered and the undeformed profile is solved by the optimization in Eq. (3.87). It must be pointed out that this example reaches a very special case in which the solution of { } X can be derived directly rather than solving the optimization. In the third case of Eq. (3.58), {} [ ] {} { } () [ ][ ] [ ] when ++ =− = c X KT K KK I (3.114) which can be verified by substituting it into Eq. (3.58). Therefore instead of setting the minimum tension band as the design goal, the element tensions can be assigned arbitrarily. As long as they are feasible to all the tension limits, the original profile calculated later will be valid and fulfill the design requirements. This particular case is not specified in the design procedure of Section VI because it can be solved by the optimization although unnecessarily. In the optimization, the Hessian matrix is semi-definite and hence the solution of optimization is still global optimal but no long unique, which reconciles the arbitrary assignment of element tensions in the direct derivation. Table 3.3 Nodal loads of desired configuration Applied Node External load (N) , x i Q , y i Q , zi Q 1 81.6497 163.2993 0 2 -19.4036 -19.4036 -9.7004 3 50.0415 -67.6387 18.5129 4 -223.0710 107.2488 105.7656 113 5 -86.7905 -211.6828 -179.2214 6 284.3201 14.6919 -390.1001 Since all member tension is chosen as 200N, the external loads are calculated and listed in the Table 3.3. According to the design procedure, the optimal undeformed profile is solved from Eq. (3.41) and (3.42) and shown in Table 3.4 with the deformed profile in the working configuration for comparison. Table 3.4 Element profiles Element Member length (m) Undeformed j L Deformed j l 1 2.0549 2.4495 2 1.9222 2.2913 3 2.5857 3.0822 4 2.9661 3.5355 5 2.2589 2.6926 6 2.4096 2.8723 3.7.2 A 25-element truss structure -2 -1 0 1 2 -2 0 2 0 1 2 3 4 5 7 3 x 1 10 6 4 8 2 5 y 9 z Figure 3.5 The deformed configuration of 25-element truss. 114 A 25-bar truss configuration (Figure 3.5) which is well recognized by previous works [78, 9, 13], is considered as the deformed configuration of the structure in the second example. The nodal sets of () 3, 4,5, 6 and () 7,8,9,10 construct two co-centered parallel squares with 2- meter and 5-meter edges respectively. The top element with ends at node 1 and 2 is placed on the symmetry of the structure along x-axis direction with 2-meter length. The distances between two squares and the top element are both 2.5 meters. The bottom four nodes are fixed on the ground. Each element of the structure has the same linear Young’s modulus at 10 2 110 / × Nm and the density at the undeformed status is 3 3000 / kg m . The nodes 1, 2, 3 and 6 are applied by undetermined external forces with directional constraints. The external forces at nodes 1 and 2 are defined by the directional constraints in Table 3.5 and the loads on nodes 3 and 6 are restricted within the planes 0.7071 0.7071 =− + zx y and 0.7071 0.7071 =− − zx y respectively. Therefore the loading set does not have the same symmetry with the truss structure. The design task is to obtain the optimal parameter set of external loads, cross-section areas of elements and the undeformed profile which minimizes the total weight of the structure. As the design requirement, the external forces, the areas of cross sections and the element strains are bounded by the following equations. 0.02 0.02 ε −≤≤ i (3.115) 42 4 2 410 9 10 −− ×≤≤× i mA m (3.116) 55 12 210 , 410 ×≤ ≤× NQQ N (3.117) 55 ,3 ,3 ,6 ,6 110 , , , 2 10 ×≤ ≤× xy x y NQ Q Q Q N (3.118) Table 3.5 Loading conditions on nodes Applied Node Directional Cosines of Applied Forces xy z 1 -0.6 0.7 0.3873 2 -0.4 0.3428 0.85 115 Once the truss model is formulated, the structure configuration and the solution of the optimization are testified to be feasible according to the proposed method. Therefore as discussed in the Section VII, the design variables are obtained by Eq. (3.101) in the final step of Figure 3.1. Particularly by considering the linear material property in this example, Eq. (3.101) can be simplified into the optimization formulation below before adding the constraints of Eq. (3.91) and (3.96) into the optimization. {}{} []{}[]{} { } 2 , min .. ρ ∈ + +<  jj j j d XA jj j AA EAl j S EA T st a X a A b (3.119) The final optimal design solution satisfies all design requirements and is presented in Table 3.6. The external forces applied on node 1 and 2 are both 5 210 × N . The planar loads at node 3 and 6 are calculated and presented as () () 55 3 31 2 3 55 5 41 2 3 1.0239 10 1 10 1.6881 10 1 10 1 10 1.4142 10 =× +× − × =× +× − × Qe e eN Qe e eN (3.120) The minimum total weight of the structure is135.04kg . From previous discussion, the design results are local optimal and dependent with the initial guess currently assigned as a uniform vector for { } X and { } { } () min max 2 + AA for { } A . Table 3.6 The optimal design of the truss parameters Member Ends i L (m) i A ( 42 10 − m ) Member Ends i L (m) i A ( 42 10 − m ) (1, 2) 2.0271 4 (7, 3) 3.2144 9 (1, 3) 2.6398 8.9175 (7, 4) 4.4980 4 (1, 4) 3.3008 4 (7, 6) 4.4659 4.8345 (1, 5) 3.4226 8.8089 (8, 3) 4.4659 9 (1, 6) 2.6962 4 (8, 4) 3.2144 6.6491 (2, 3) 3.2883 5.6592 (8, 5) 4.6178 4 116 (2, 4) 2.6398 4.6494 (9, 4) 4.5446 4 (2, 5) 2.7475 4.4783 (9, 5) 3.3456 9 (2, 6) 3.2883 5.5315 (9, 6) 4.6350 4 (3, 4) 2.0069 4 (10, 3) 4.5300 4 (4, 5) 1.9917 4 (10, 5) 4.6482 4.3239 (5, 6) 1.9650 4 (10, 6) 3.3456 5.1921 (6, 3) 2.0385 4 3.7.3 Deployable mesh reflectors While the proposed method is capable of designing all truss-type DMRs which can be modeled as trusses, here it is implemented on the perimeter truss DMRs (Figure 1.3) introduced in Chapter 1. The complete shape design of this DMR begins with the geometry design of DMR’s working shape after the deployment which is well resolved by Chapter 2 (equivalently Ref. [82, 83]). As stated in the introduction the next design stage is to design the structural parameters of DMR utilizing the proposed method. Therefore, the particular design mission in this example is to find the optimal tie-down forces generated by the deformation of the tension ties and the undeformed profile of the entire reflector structure, so that in the deformed working configuration the tension band of the cable nets is minimum and the cable tensions are as uniform as possible. Define the member indices of the cable nets, the supporting truss and the tension ties as C S , S S and T S respectively. The longitudinal rigidity of each reflector truss member is defined as 25000 0 00 ε ε ≥  =∈  <  j jjC j lbs EA j S (3.121) 125000 =∈ jjS EAlbsjS (3.122) 40000 0 00 ε ε ≥  =∈  <  j jjT j lbs EA j S (3.123) As the design constraint, all the cable tensions are bounded by 117 29 ≤≤ ∈ j C lbs T lbs j S (3.124) In the following, two different reflector setups, a center-feed DMR and an offset-feed DMR, are investigated based on different design focuses: 1. Optimal parameters for reflector working surfaces The reflector performance is determined by the structural performance of the working surface. As the tension distribution of the tension ties and the supporting flat truss is unrestricted, the design task may only focus on investigating the optimal parameter set for the cable network of the working surface as described in Section 1.3 and illustrated in Figure 1.5. Therefore the design subject is modified to be the reflector cable-net surface with fixed boundary nodes and applied vertical external loads. This engineering model is applied on a center-feed parabolic reflector currently under development at JPL, of which the deformed surface is shown in Figure 3.6, with 33.267 m in diameter, 27.56 m in focal length and 2.5097 m in height. Among 835 nodes, 24 nodes on the aperture are type 1 boundary nodes and the rest nodes are applied by vertical external loads which need to be determined. Then the design task is to solve the optimal external loads and the undeformed profile of the cable nets constructed by 2526 cable-elements which minimize the inner tension band defined by Eq. (3.82). The loads are constrained in vertical direction and the element tension is restricted by Eq. (3.124). -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 0 1 2 x y z Figure 3.6 The working configuration of a center-feed DMR. 118 First, the regular design approach in Figure 3.1 is implemented. The tension distribution and the external load distribution are plotted in Figure 3.7. Darker lines represent the larger inner tensions and darker dots are for the larger magnitudes of applied loads on the related nodes. The elements and nodes on aperture rim are not shown due to zero tension or none external loads applied. From the figures, the tensions of members are quite uniform within the working region of the surface as preferred and non-uniform tensions only occur within the boundary layer; the loading distribution also has the uniform behavior in the center area of the surface and the irregular loads are near six symmetric axes in the outer region of the working surface. The tension specifications of the design results in Table 3.7 also confirms the above distribution by much smaller tension band of inside surface comparing to the tension band of the whole structure. Figure 3.7 Tension distribution (left) and load distribution (right) at the optimal design. The minimum tension band for the full surface presents the optimal solution for all possible loading cases and is only dependent on the shape geometry of the deployed reflector mesh for the specific setup of boundary types, loading constraints and tension constraints. Therefore 2.1791 lbs. minimum tension band is essentially one of the structural properties at the deployed mesh surface. The necessary and sufficient formulation and the global convergence of the optimization provide the “Best or None” property for the design solution: among any set of structural parameters, the reflector’s cable network enjoys 2.1791 lbs. tension band as the 119 minimum; as one design criterion, any design requirement leading to the smaller tension band than 2.1791 lbs. will only cause the fault design and can never be realized. Hence for instance, one may conclude that once replacing the upper tension constraint by 4 lbs., no feasible design results can be achieved. Figure 3.8 Layer separation of the DMR surface. Another approach is to utilize the layered design strategy. The reflector is divided into 9 layers with the first layer in the center and the ninth one at the boundary, and the top view of the layer separation is shown in Figure 3.8 which should be interpreted by the defined topology in Figure 3.3. In this separation, each layer is solvable and the design solution is feasible. The structure design is then achieved iteratively as discussed in the Section VII by propagating the layered design from the center to the entire structure. The design results are listed in the Table 3.7 and in Figure 3.9 the layered tension bands are compared with the one in the design of the overall structure model. In this case, all member tensions lie in the designated range of Eq. (3.124) and the tension band is slightly wider than the optimal design using the whole structure model. From Figure 3.9, it is obvious that by both approaches the maximum tension bands occur at the boundary layer. The tension bands of inside 120 layers by layered strategy are narrower because this strategy pays more attention on local performance and puts more efforts to narrow down the tension bands of inside layers first. Once inside ones have been determined, the tensions in the boundary layer don’t have much space of variety resulting in the larger band. The overall model always ensures the tension band of entire structure is at minimum, but sometime meaning the sacrifice on the inside tension bands. Table 3.7 Tension distribution in the structure (lbs.) Layered Strategy Overall Model Average 2.2012 2.2410 Maximum 4.3492 4.1791 Minimum 2.0000 2.0000 Band of structure 2.3492 2.1791 Band of inside layers 0.1384 0.6438 1 2 3 4 5 6 7 8 9 0 0.5 1 1.5 2 2.5 Layer number Tension Band (lbs) Layered Strategy Overall Model Figure 3.9 The working configuration of a center-feed DMR. According to the previous discussion, the layered strategy only provide the sufficient design condition and therefore it loses “Best or None” property comparing to the overall design strategy: on one side, the design result is optimal only for such layer separation and other separations may result in either better or worse tension distribution; on the other hand, if the 121 design result at this layer separation was infeasible to the design constraints, it could not deny the existence of the feasible design and a different layer strategy was then needed. However, in practice the layered strategy still has some advantages. First, due to its local focus, it always leads to more uniform tension distribution of inside area of the reflector which is the major portion of the working surface as shown in this example. This can be more desirable than the result at the overall model when the element tensions in the outer region do not attract much concern. Second, the layered strategy only needs the local structural models, resulting in a tremendous save on the computational costs. Considering the numerical errors when calculating the null space, the rank and the general inverse of a large matrix, smaller layered models could lead to much more accurate results as the numerical benefit by practical means. 2. Design of full deployable mesh reflector structures -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 0 0.5 1 x y z Figure 3.10 The working configuration of an offset-feed parabolic DMR. Finally, a full deployable mesh reflector (Figure 1.3) is considered in this example for the same design mission (minimizing tension band on the cable net surface) under the same design setup by Eq. (3.121) - (3.124) An offset-feed parabolic DMR has been designed by the author’s previous research [82, 83] and shown in Figure 3.10. As stated, the upper net is the parabolic working surface of the reflector and the bottom dish is the rear net. Two cable nets are symmetric to the central plane of the reflector and all cable elements are illustrated in blue lines with normal width. The thick blue lines are the supporting truss serving as the boundary rim. The tension ties connect the nodes between the front net and the rear net and are drawn in light yellow lines. Due to the symmetry of the reflector, all the tension ties are always perpendicular to the reflector 122 central plane. The reflector surface (Figure 3.11) is constructed by triangle facets with vertices on an offset parabolic shape having 6 m effective circular aperture, 4.8 m focal length and 3 m effective offset distance defined in Chapter 2 with a coplanar elliptical aperture rim on the boundary. Note that the reflector surface is not symmetric to the center of the aperture (as shown in Figure 3.12). Between two apertures the height of the reflector is 1.2 m. Each cable net is constituted by 573 nodes on 14 rings (547 nodes inside the net and 26 nodes on the aperture rim) and 1690 cable elements. For the reflector’s working surface, the actual best-fit surface RMS error from systematic facets is 0.329 mm and is designated to serve the transmission task of 18 GHz signal. 547 tension ties are used and the supporting truss has 104 elements, of which 26 elements are on each surface rim, 26 vertical elements connect two elliptical apertures and 26 diagonal elements actuate the deployment of the reflector structure. Since the DMR is usually installed onto the satellite boom in the space mission, at the engineering model of this example no boundary condition is assigned on any nodes of the structure. The loads of the reflector are generated by the deformations of the tension ties and therefore the external nodes are considered to be predefined as zero. Figure 3.11 The top view of an offset-feed parabolic DMR. 123 Figure 3.12 The side view of an offset-feed parabolic DMR. Since the cable elements, which are part of the elements, are considered in the design objective and restricted in Eq. (3.124), the following equation is used to modify {} * T in the design formulation, in which [] I is the identity matrix at the scale of C m as the number of the cable elements. {} [ ]{} * * 0  =  T TB I T (3.125) in which * 1 1 1 1 −   −     =    −   −      C CC T C CC m mm B m mm (3.126) Hence [ ] b H and { } b f in the objective function Eq. (3.82) are derived as [] [ ][] () [][] () {} { }[] [][] ** ** 00 00   =     =   T bT T T T TT bc T T HBI K BI K f KI B B I K (3.127) Then the optimal design results are obtained through the regular implementation of our proposed method with tension specs listed in Table 3.8. The optimal tension band is 5.8680 lbs. and all cable tensions satisfy the requirement constraint. The tensions of ties are one order of magnitude smaller than the cable tensions on average while the supporting truss tensions are approximately ten times larger. 124 Table 3.8 Tension specifications in the structure (lbs.) Cable nets Tension ties Supporting Truss Average 2.9825 0.1799 -34.5679 Maximum 7.8680 0.4444 0 Minimum 2.0000 0.0872 -71.1549 Band of mesh surface 5.8680 N/A N/A Band of working region 5.6266 0.3571 N/A Due to the linear and piecewise linear elasticity of the reflector materials, the undeformed profile can be uniquely calculated from the tension distribution, which is presented together with the related strains in the following three figures for cable elements, tension ties and supporting truss respectively. The member indices of both the cable elements and tension ties are counted from inside to outside. As shown in Figure 3.13, the cable lengths are fairly consistent within the center area of the surface, and become significantly larger in the outer region of the net. The band of cable strains tends to gradually increase from inside to outside in general and the largest tension strain also occurs in the boundary layer of the mesh reflector. The lengths of tension ties periodically grow by rings due to the increasing distance between two nets from the center to the aperture (Figure 3.14) and the strains of tension ties oscillates versus the element indices under the same ring cycle. In Figure 3.15, the left portion of element indices represents the supporting truss on two aperture rim, lengths of which are all equal. Their strains have four varying cycles implying the reflector aperture is symmetric to x and y axes. The middle set of the truss vertically connects the rim nodes on two mesh surface. Unlike the tensile cables and ties, the above truss elements are under compression. The diagonal elements in the supporting truss are presented in the right portion of the plot with zero strains under the working status, implying that after the reflector is fully deployed by the diagonal truss, those truss elements do not contribute any inner tensions at the static deformed configuration. 125 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 0.8 Original length (m) Member No. 200 400 600 800 1000 1200 1400 1600 0 1 2 3 4 x 10 -4 Elemental strain Figure 3.13 The undeformed profile and the element strains for cable net. 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Original length (m) Member No. 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 2 3 4 5 6 7 8 9 10 11 12 x 10 -6 Elemental strain Figure 3.14 The undeformed profile and the element strains for tension ties. 126 3930 3940 3950 3960 3970 3980 3990 4000 4010 4020 4030 0.5 1 1.5 Original length (m) Member No. 3930 3940 3950 3960 3970 3980 3990 4000 4010 4020 4030 -5 0 x 10 -4 Elemental strain Figure 3.15 The undeformed profile and the element strains for supporting truss. Similar to the whole design strategy of the previous JPL’s example, the optimal results above are solely dependent on the deployed shape geometry of the mesh reflector and enjoys the “Best or None” property as stated in Remark 3. Therefore the minimum tension band is also one of the structural properties of the reflector. The further investigation has been carried out to numerically address the relationship between the optimal tension distribution and the minimum tension limit , lj T which is the only active constraint based on the design results. By proportionally varying the minimum tension limits, different optimal design can be achieved for every constraint case, resulting in different optimal tension performance, loading status and undeformed profile. Then the normalized tension distribution is calculated and shown in Figure 3.16 where six sets of optimal solutions are purposely moved away from each other by 0.02 in y axis. In despite of the fact that only the cable nets are considered in the design objective and the constraints, the normalized tension distributions of the full reflector structure remain numerically consistent for all six design solutions at different lower tension limits with the uncertainty being at the magnitude of 14 10 − . Therefore the structure performance obtained by the design formulation of our method is essentially universal for any positive minimum tension constraint. The uniform ratioυ unif as the ratio of the minimum cable tension versus the maximum, is universally a constant (0.2542 in this design example) regardless the tension constraints as long 127 as they are positive. Such independence of the tension distribution not only suggests this uniform ratio as one of the structural characteristics of the reflector, but also implies that instead of being obtained by the optimal design directly towards the largest uniform ratio of the structure, the uniform ratio υ unif minimizing the tension band is the consistent factor and should be used to evaluate the tension distribution performance for a given shape of the deployable truss. 0 500 1000 1500 2000 2500 3000 3500 4000 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Member No. Normalized Tension Distribution T l,j = 2 lbs T l,j = 5 lbs T l,j = 10 lbs T l,j = 20 lbs T l,j = 50 lbs T l,j = 100 lbs Figure 3.16 The consistent normalized tension distributions regarding various minimum tension constraints. 3.8 Concluding Remarks In this chapter, the optimal design method is proposed to determine the structural parameters of deployable truss structures at their working (deformed) configuration, which is formulated based on the static equilibrium considering both nonlinear elasticity and geometry. As the external loads, cross-section areas of elements, element tensions and undeformed profile being the design variables, the optimal design result can be obtained under diverse constraints and is suggested as one of the structural property of the deformed shape geometry. The design results of three examples (two small trusses and one case of deployable mesh reflectors) are presented and demonstrate that the method is readily for diverse space applications using truss- type structures, such as deployable mast, deployable booms and deployable mesh reflectors. 128 Chapter 4 Nonlinear Statics and Static Shape Control 4.1 Introduction The research regarding two blocks, nonlinear static modeling and static shape control, in the Active Shape Control architecture (Figure 1.4) is presented in this chapter. Due to the flexibility of the large scale space structure, it is not easy to precisely maintain the desired surface shape, not mention to adjust it for variant tasks. The shape control is therefore originated from the research on flexible spacecraft structures, especially on the space reflectors which has the most stringent requirement on surface accuracy. The technique term “shape control” is first introduced into academic journals by Haftka et al. [23] in 1985, which analytically studied the problem using the force or thermal actuators for the linear flexible structures. Combining the research in Ref. [24], it elaborated the actuator placement problem for the static shape control. Balas theoretically studied the optimal control of reflectors on the quasi-static case with discussion on model reduction and controller convergence [2]. Upon all above researches which lay down the foundation, the shape control problem has been investigated on very diverse flexible structure applications. The control solutions of the static beams were presented by Irschik, et al. [17, 19] and the finite element analysis on static composite plates can be found in [110, 40]. For the dynamic shape control, Chandrashekhara and Varadarajan [8] proposed an adaptive algorithm for laminated composite beams and Varadarajan, et al. [108] developed an optimal shape control solution on composite plate using piezoelectric actuators and position sensors. A robust controller validated on the flat plate is 129 introduced in [42] by Kashiwase while Hu and Vukovich suggested a rigorous treatment on the robust stability of circular plate [41]. A general solution of static and dynamic control problems was presented by Ziegler [116] for discrete structures based on the flexibility matrix. Koconis and Kollar [45] formulated a shape control method for composite beams, plates and shells with piezoelectric actuators embedded. One may also find the literature survey in [18] regarding the development of shape control theory, especially on applications using piezoelectric actuators. Although the space reflector applications motivated the foundation of shape control theory and many optimal controllers are proposed for different flexible structures, the public research on the surface shape control of Deployable Mesh Reflector (DMR) is still limited. A series of studies have been done on the truss reflectors particularly for MUSES-B mission [97], which considers the linear static shape control. Regarding the most recent DMR design concept as cable-truss or tension-truss design, Mitsugi et al. presented a shape control approach on linear static model [58]; Tanaka and Natori additionally proposed the force density approximation of nonlinear static deformation [98, 99]. Wang et al. presents an active shape adjustment regarding static disturbances [111]. But unlike as they claimed, the transformation matrix between local and global coordinates is constant and hence the reflector structure model used in their study was still under linear geometry assumption. Based on the above survey, the static shape control regarding the fully nonlinear structure model has been rarely addressed. To implement the nonlinear static shape control, a new nonlinear static truss model is proposed in Section 4.2. The model fully considers nonlinear geometry and elasticity and unlike the model used in large amount of previous research [38, 22, 49] it does not use the information on the nodal coordinates of the initial configuration. Then the solving technique is discussed to obtain the solution of the nonlinear model in Section 4.3. The algorithm uses quadratic cost function and computes exact solution of trust region formulation within each iteration. The numerical examples are shown in Section 4.4 to validate the truss model and the solving approach. Section 4.5 presents the nonlinear static shape control strategy based on the proposed reflector’s truss model and the solving algorithm, followed by the numerical implementation on a sampled DMR. Finally Section 4.6 concludes this chapter. 130 4.2 Nonlinear Model of Truss Structure Figure 4.1 A truss element in undeformed and deformed configurations. Assume that the truss structure has n nodes and m members and one typical truss member that is connected in the structure has nodes i and j as its two ends. The deformation process of this element is shown in Figure 4.1.  i R and i r  are defined as the vector of node i under undeformed and deformed situation respectively. Then the nodes in deformed configuration can be expressed by position vectors as Eq. (3.1), where 12 3 ,,   ee e are the unit vectors of the global coordinate system and ,, ii i x yz are the position coordinates of node i. Therefore the node positions of deformed element can be described as { } { } , T def j i i i ii i yxyzxyz = (4.1) with j presenting the index number of the element. Then it is obvious that the deformed element is defined by the vector below ,1 , 2 , 3 j ixj yj zj i lr r le l e le =− = + +      (4.2) in which ,, , ,, xjiyj izj i ii i lx x l y y l z z =− = − = − (4.3) 131 and the unit vector in the longitudinal direction of the deformed member is ,1 , 2 , 3 1 j jxj yj zj j le e e l ββ β β == + +     (4.4) with ,, , , , , /, /, / x j x j j y j y j j zj zj j ll l l l l ββ β == = (4.5) while ,, , ,, xjyj zj ββ β are the direction cosines, and j l is deformed length of the element which is calculated by 22 2 ,, , j xj y j z j ll l l =+ + (4.6) Figure 4.2 Nodal displacements and force balance in local coordinate. In local coordinate of the element, the relationship between nodal displacement and forces is shown in Figure 4.2. From the geometry, the elongation of the element is j jj L lL Δ= − (4.7) in which j L is the original length of element. According to elasticity theory, the relation between stress and strain is j jj E σε = (4.8) with j j j N A σ = , , j j jth j l l εε Δ =− (4.9) 132 while j N is the axial force of the element, , j th ε is the thermal strain and j A is the area of cross- section. In general j E could be nonlinear and a function of strain as () j jj EE ε = (4.10) and j jj j NEAε = (4.11) From the force balance of the element, the nodal force vector { } j q in local coordinates is {} , , 1 1 ij j j ij q qN q  −   ==       (4.12) Hence, {} ()()() 22 2 , 1 1 1 ii i ii i j jj jth j xx y y z z qEA L ε  −+ − + − −   =−−      (4.13) By applying standard coordinate transformation, the nodal forces of the element in the global coordinate xyz are obtained as { } { } , T j jj QT q β  =  (4.14) where ,, , , ,, , 000 00 0 xj y j z j j xjyj zj T β ββ β ββ β   =      (4.15) {} { } ,, , ,, , T kxi yi zi xi y i z i Qq q q q q q = (4.16) 133 It can be noticed that the right side of Eq. (4.14) has two independent variables: original length j L and node positions in deformed shape { } , def j y , and can be rewritten as a nonlinear function of them: { } { } () { } { } ,, , T j jj defj j j Q f L y Tq β  =≡  (4.17) Finally, force balance at all the nodes of the structure leads to the equilibrium equation in global coordinates, {} { } { } { } (,) = org def QfL y (4.18) in which {} { } ,1 ,1 ,1 , , , , , , =  T xy z xi yi zi xn yn zn Qq q q q q q q q q (4.19) { }{} 1 =  T org i m LL L L (4.20) { }{} 11 1 T def i i i n n n yxyz xyz xyz =  (4.21) The nonlinear model of Eq. (4.18) has two advantages. It has considered the geometry and elasticity nonlinearity at the same time which results in the generality of the model. Furthermore, the equation of system describes the relationship between original lengths of elements, deformed shape profile and external forces, without requiring any information of initial shape profile where the structure is static under zero external loads. This may provide significant convenience when the shape of the reflector is geometrically complex and unknown. Let { } ini l is defined as the element length vector in initial profile of the structure, when { } { } = org ini L l , meaning each element is fully extended without any inner tension, the formulation of the equilibrium equation in this special case could be simplified from above and results in the following model of structure, which is discussed in the work of Ref. 113. [ ]{ } { } { } { } ++ = NL NL NL K DU T Q (4.22) 134 4.3 Algorithm of Solving the Nonlinear Model After the model of the reflector is derived, the solution of Eq. (4.18) needs to be obtained. In the case that the original length vector and deformed shape profile of the structure are known, the external force vector is trivial to calculate, by plugging in the value of { } org L and { } def y into the right hand side of Eq. (4.18). However, when some known external loads are applied on the structure, how to obtain original lengths of elements or the deformed shape profile can be troublesome and therefore comes this algorithm of nonlinear solver. Depending on the objectives of the calculation, either original length vector { } org L or the deformed shape profile { } def y , it has two slightly different formulations on the problem. Case 1: { } def y is unknown while { } org L and { } Q are given. Define the function F to be the 2-norm of the Eq. (4.18), {}{ } {} {} 2 1 (,) 2 =− org def FfL y Q (4.23) Then to solve Eq. (4.18) is to solve the following unconstraint optimization problem { } min ( ) def Fy (4.24) In this chapter, the framework of trust region method is adapted. The updating policy of trust region radius is suggested by Ref. [67] and the only unspecified aspects left in the framework are the cost function formulation and the solution computed for each step of iteration according to Ref. [88]. By using quadratic model, the sub-problem of the Eq. (4.24) in each step of iteration is to minimize the following cost function, {} {}{} {} { } {} 2 1 () () 2  ++ ∇  T kdef y y y k def y kk k Fy g p p F y p (4.25) where { } y k g is a 13 × nrow vector as the gradient function and { } 2 ()  ∇  kdef k Fy is the 33 × nn Hessian matrix in th k iteration. The gradient function at each step can be obtained by 135 {} { } {} {} {} {} () , () ()   == −   T kdef T k ydef defy k k k def k dF y gfyQJ dy (4.26) in which ,   def y k J is the Jacobin matrix of the structure at th k iteration defined by Eq. (4.27) { } { } {} , ()  =  def k def y k def k df y J dy (4.27) and can be linearly assembled from the local Jacobin matrices that are derived from Eq. (4.28) at each element. { } { } {} ( ) {} () {} {} ,, , ,, , ,,,, ,, ,, () 11 1 11 1 β ββ ε εε  =   −− −       =+ +         idef k def y i k def k T i TT k ik ik ik ik ik ik i i k i ik i k kk def def def kk k df y J dy dT dE A d EA T T EA dy dy dy (4.28) To avoid calculating the second derivative of the objective function, a symmetric matrix is introduced in Levenberg-Marquardt method (Ref. [61]) as the approximation of Hessian, { } 2 ,, ( T y def y def y k def kk k k HJ J Fy       =≈∇       (4.29) Since { } () kdef k Fy is constant when { } y p varies, finally the cost function is {} { }{} {} {} 1 () 2 T yy y y y y y kk mp g p p H p  =+  (4.30) Case 2: { } org L is unknown while { } def y and { } Q are given Comparing to Eq. (4.24), the optimization problem is changed to treat { } org L as optimizing variable, { } min ( ) org FL (4.31) 136 Hence, following the similar derivation of Eqs. (4.25) - (4.30), the cost function can be obtained as {} { }{} {}[]{} 1 () 2 T L LL L L L L k k mp g p p H p =+ (4.32) where {} { } {} {} {} {} () , () ()   == −   T korg T k Lorg defL k k k org k dF L gfLQJ dL (4.33) [] ,, T L def L def L k kk HJ J    =    (4.34) { } { } {} , ()  =  org k def L k org k df L J dL (4.35) and the Jacobin matrix is assembled from the local Jacobin matrices calculated below { } { } {} ( ) {} {} {} ,, , ,, ,, , , , , , , , () 11 1 11 1 β ββ ε εε  =   −− −       =+ +         iorg k def L i k org k T i TT k ik ik ik ik ik i i k i k i ik ik kk org org org kk k df L J dL dT dE d EA T A T E A dL d L dL (4.36) Although the cost functions are different for different cases, the same approach is used to obtain the exact step solution. The sub optimization problem always has a global minimum. However, since [] k H may not always be positive definite, the solution which minimizes Eq. (4.30) and Eq. (4.32) may not be unique. Based on previous work by Sorensen and More (Ref. [91]), an exact global solution of the trust region problem for each step can be concluded into the following four cases and for simplicity of the expression, the subscripts of previous vectors and matrices will be dropped in the Eqs. (4.37) - (4.40). Set Δ to be the trust region radius in each step, λ andγ to be undetermined scalars. The exact solution { } * p for the optimization Eq. (4.30) and Eq. (4.32) is 137 i. when [ ] H is positive definite and [] {} 1 T Hg − −≤Δ , then { } [ ] {} 1 * T p Hg − =− (4.37) ii. when [ ] H is positive definite and [] {} 1 T Hg − −>Δ , then { } [ ] [ ] {} *1 () T p HI g λ − =− + with [0, ) λ∈+∞ (4.38) iii. when [ ] H is indefinite and { }{ } 1 0 ≠ gq where { } 1 q is the eigenvector corresponding to the smallest eigenvalue 1 λ , or when [ ] H is indefinite, { }{ } 1 0 = gq and [] [] {} 1 1 () T HI g λ − −+ ≥Δ , { } [ ] [ ] {} *1 () T p HI g λ − =− + with 1 [, ) λλ ∈− +∞ (4.39) iv. when [ ] H is indefinite, { }{ } 1 0 = gq and [] [] {} 1 1 () T HI g λ − −+ <Δ {} [] [] {} { } {} 1 *1 1 1 () T q pH I g q λγ − =− + + (4.40) whileλ andγ satisfy the equation { } * =Δ p Under the assumption that F is twice continuously differentiable and bounded, and [ ] H is also bounded, the Ref. [88] has implied that by calculating exact solutions of iterations, the algorithm of nonlinear solver will always converge to the first order stationary point where { } 0 = g . It should be noticed that the mapping from { } Q to { } def y or from { } Q to { } org L defined by Eq. (4.18) is not always a one-to-one mapping. In Eqs. (4.30) and (4.32) when [ ] H is positive definite, the sub-problem always has only one solution and the Eqs. (4.24) and (4.31) will have the unique global minimizer. However, when [ ] H is semi-positive definite which sharps the problem to be non-convex, the sub-problem may have more than one 138 solution and the algorithm only leads to local minima. This latter case is especially common while the norm of { } Q is zero or small enough. 4.4 Numerical Examples on Nonlinear Static Analysis All the examples discussed here are only applied by the vertical external forces which is the same loading setup as the reflector structure shown in Figure 1.5. Due to this fact, when n is large, { } Q can be huge and sparse. Similar to the reflector cables, in the examples the relation between stress and strain is a piecewise linear function and the nonlinear elasticity is described by () ε E below, 0 () 00 ε ε ε >  =  ≤  E E (4.41) Though the nonlinear elasticity of the structure results in the discontinuity of the Eq. (4.23), it can be demonstrated that by smoothing the curve of () ε E at origin well enough, practically function F can be treated as a twice continuously differentiable function and the convergence of the algorithm still holds. 4.4.1 Condition analysis of different initial shapes -3 -2 -1 0 1 2 3 -2 0 2 4 -3 -2 -1 0 x Initial Shape Profile y z Figure 4.3 Initial shape when ,, org j ini j L l = . 139 Table 4.1 Comparison of element lengths (m). Member No. { } org L { } ini l 1 5 5.0000 2 5 5.0000 3 5 5.0000 4 6.9282 6.9282 The elements of the structure in initial shapes may have three different types of tensions. The first case is special, which assumes one element at initial shape has the same length with its original length. In this case, no inner force occurs at this element. If all elements of the structure are in first case, it will lead to the equilibrium equation as Eq. (4.22). When an element is originally shorter than its length in the initial shape, then the element will be extended and positive pretension occurs. As the third case, if the original length of the element is larger than the deformed length, the element will encounter pressure with negative pretension, or in our case will dangle freely with zero tension due to Eq. (4.41). A simplest example is used to show the different initial shapes due to three cases discussed above, of which the elements construct a triangular pyramid with three vertexes fixed in X-Y plane and one unconstrained. First, the original lengths of the elements are designed to ensure that { } { } = org ini L l . Figure 4.3 and Table 4.1 confirm that the initial shape obtained from our algorithm agrees with the theoretical prediction and the inner forces of elements are zero. The initial shape of the second case is shown in Figure 4.4. From Table 4.2, every element has been extended with positive tension because of larger deformed length than original length. Since a very small external load (less than 9 10 − N ) is applied on the unconstrained node to generate the initial shape in our algorithm, the displacement in vertical direction of the node is in the magnitude of 13 10 − m . 140 -4 -2 0 2 4 -2 -1 0 1 2 3 4 -3 -2 -1 0 x 10 -13 x Initial Shape Profile y z Figure 4.4 Initial shape when ,, org j ini j L l < . Table 4.2 Comparison of element lengths (m). Member No. { } org L { } ini l 1 2.5 3.4717 2 3 4.0649 3 3.5 4.5335 4 6.9282 6.9282 -4 -3 -2 -1 0 1 2 3 4 -2 0 2 4 -6 -5 -4 -3 -2 -1 0 x 10 -14 x Initial Shape Profile o y z Figure 4.5 Initial shape when ,, org j ini j L l > . 141 Table 4.3 Comparison of element lengths (m). Member No. { } org L { } ini l 1 2 2.5193 2 3.5 4.4088 3 9 6.0739 4 6.9282 6.9282 In the third case, one element has larger original length than the length in initial shape (member No. 3 in Table 4.3 which is marked red in the Figure 4.5), due to the nonlinear elasticity of Eq. (4.41) it has zero tension and dangles between two nodes. Similar with the previous case, the vertical deformation of the node is very small and the initial shape in Figure 6 is treated as a two-dimension plane. All the three cases demonstrate that though in our model the initial shape information is not gathered, our algorithm is able to obtain the initial profiles for all different length conditions. 4.4.2 Verification on Initial shape profile and deformed shape profile A second example is used to verify the correctness of proposed model and algorithm. The example structure has seven unconstraint nodes and 24 elements, which construct a single bay in the center. The initial shape and deformed shape are plotted together in Figure 4.6 and force balance at each direction of coordinate at each node is checked and listed in Table 4.4. It shows clearly that the force balance error at each nodal direction is small enough and its magnitude is at least down to the order of -11, therefore providing a proof on the correctness and accuracy of our algorithm. Table 4.4 Verification of force balance (N). Initial deformed Initial deformed 1 x 4.0657e-20 -1.1368e-13 4 z -4.5797e-12 -9.2086e-12 1 y 3.8160e-11 0 5 x -9.3001e-12 -1.6370e-11 1 z -1.3672e-12 -7.9580e-12 5 y .9506e-11 -1.1368e-13 2 x 9.3001e-12 1.6370e-11 5 z -5.6363e-13 -6.5938e-12 142 2 y 3.8047e-11 0 6 x -1.5576e-11 1.3415e-11 2 z -5.6363e-13 -6.5938e-12 6 y -2.1187e-11 -7.9580e-12 -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - deformed; dotted - initial y z Figure 4.6 Initial shape and deformed shape. 4.4.3 Verification of the optimal parametric solution on a DMR According to Chapter 2 and 3, the desired geometry of the mesh facets, the undeformed lengths of the reflector structure and the external loads applied are optimally obtained. Using the nonlinear static truss model and its solving technique, it is the time to obtain the initial configuration of the reflector and verify the actual deformed shape of the working surface being the desired geometry exactly. A sampled DMR model is considered in this verification, of which the mesh surface has 37 nodes, 90 members and a spherical working shape with diameter D = 30 m and height H = 11.18 m. The longitudinal rigidity of cable members is 1.1121e+005 N when the elastic strain is large than 0. The element tension bounds are specified as 2, 9 σσ == lh lbs lbs (4.42) By given the undeformed lengths of members using the approach in Chapter 3, the initial configuration is obtained in the left of Figure 4.7 at a set of small external loads 143 ( { } 5 10 QN − < and { }() 10 10 i i QEA − < ). The red lines in the graph indicate that there are six elements dangling in the initial shape which are six-fold symmetric to the origin of X-Y plane. The actual-deformed shape is also obtained under optimal external forces and shown in the right of Figure 4.7. It should be pointed out that the initial shape and deformed shape are plotted separately since two graphs will cover each other due to small deformation of the reflector. It has been verified that the error of nodal coordinates between the desired mesh geometry and the actual deformed geometry is -15 8.2530 10 × and its relative error is -16 2.3410 10 × . -10 -5 0 5 10 -10 -5 0 5 10 -4 -2 0 y o o o Initial shape x o o o z -10 -5 0 5 10 -10 -5 0 5 10 -4 -2 0 x Deformed configuration y z Figure 4.7 Initial shape and deformed shape based on optimal external loads. 0 5 10 15 20 25 0 10 20 30 No. of iteration F norm 0 5 10 15 20 25 0 1 2 3 4 x 10 4 No. of iteration Gradient 0 10 20 30 40 50 60 0 2 4 6 8 10 x 10 13 No. of sub-iteration Multiplier 0 10 20 30 40 50 60 0 1 2 3 4 5 x 10 -10 No. of sub-iteration Norm of error Figure 4.8 Algorithm performance when solving initial shape. 144 0 2 4 6 8 10 12 14 0 10 20 30 40 No. of iteration F norm 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 x 10 5 No. of iteration Gradient 0 5 10 15 20 25 0 1 2 3 4 x 10 13 No. of sub-iteration Multiplier 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 x 10 -9 No. of sub-iteration Norm of error Figure 4.9 Algorithm performance when solving deformed shape. Figure 4.8 and 4.9 show the performance of the algorithm. Apparently, while the norm of the calculation error within sub-iterations are always small enough, the cost function and its gradient are reduced to be within the tolerance as the solving iterations proceed. Along with the theoretical convergence proof discussed previously, this figure demonstrates the actual convergence of the solving algorithm. Furthermore, by checking every member tension shown in Figure 4.10, it is obvious that all the tension forces are in the designate range stated in Eq. (4.42). 10 20 30 40 50 60 70 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 Element number Internal tension (lbs) Verification of Constraint Figure 4.10 Tension forces of members. 145 4.5 Static Shape Control Under Thermal Distortions Upon the nonlinear static model of the reflector truss structure and the solving technique, this section presents the nonlinear static shape control approach to compensate the thermal distortions caused by the large temperature difference in the space environment during orbiting missions. 4.5.1 Static shape control strategy Combining the deployment process, we categorize the static deformations of DMR into 4 statuses (as shown in Figure 4.11). Before the reflector is deployed, it is folded in the spacecraft and has a virtual initial surface which the reflector will be deformed to under zero external forces. In this status, namely 0 S , the reflector is in certain nominal temperature 0 T when no thermal distortion occurs at each member of structure. Figure 4.11 Four status of static shape control strategy. Up-left: 0 S ; Up-right: 1 S ;Bottom-left: 2 S ; Bottom-right: 3 S In the status of 1 S , which is the deployment process, the proper external loading status 1 Pwhich is predetermined by the approach in Chapter 3, are applied on the nodes of the 146 reflector and the deformation of the structure will generate the expected working surface. In this step, the reflector is still considered in the nominal temperature 0 T . However, as it is shown in the next status of 2 S , the surface temperature 1 T determined by the space environment where the reflector is deployed and by Sun’s ray on the surface, is always not the nominal temperature and the structure will statically distort under thermal loads into a new equilibrium with the external loading condition remain as 1 P . Finally, the external forces are adjusted in status of 3 S to ensure the surface of the reflector deforms back to the desired working shape under the current surface temperature 1 T . It should be noticed that the geometry of the mesh facets in this new working surface may not be the same with the mesh geometry in the desired surface, but the facet vertices in both surface should locate on the same desired working surface and hence share the exact same shape, which will maintain the minimum change on the surface accuracy from its original design. The nonlinear truss model of the reflector and the solving technique are directly applicable for the status 0 S , 1 S and 2 S . In 3 S the deformed shape after applying the static shape control is rebalanced at the desired working surface under the external loading status 1 c PP + . To obtain such rebalanced mesh surface, two modification is needed: first the z dimension of the nonlinear static truss model is removed due to the vertical loading; then the modified nonlinear static system equation and the geometry function of the desired working shape construct the equation system which need to be evaluated in the solving approach and 2-norm of this equation system is the cost function to minimize. 4.5.2 Nonlinear static shape control implementations The above nonlinear static control strategy is applied onto the same DMR example as shown in Section 4.4.3 except using 1000 N as the its longitudinal rigidity, by which the elastic deformation and the thermal distortion of the DMR are large enough to be noticeable on the figure. The environment temperature in 2 S and 3 S is assumed to be 300K lower than the nominal temperature of the reflector structure. 147 The numerical results for the four statuses are presented in the following figures. Figure 4.12 shows the initial shape of the reflector and the desired deployed shape as described in status 0 S and 1 S . In the initial shape, six members have longer undeformed length than the distance between their two ends and are dangling. After deployed, all cable members are properly stressed and the reflector surface is at its desired shape. In 2 S the reflector is thermally distorted away from the desired shape and this deformation can be observed in Figure 4.13. After applying the static shape control of 3 S , the reflector is pulled back to its desired shape and the thermal distortion is compensated. The difference between the thermal distorted shape and the rebalanced shape is shown in Figure 4.14. The mesh facets of the working surface in the rebalanced shape locate their vertices exactly on the desired surface with the nodal coordinate errors at magnitude of 13 10 − . The rebalanced shape and the originally desired shape are plotted together in Figure 4.15 for comparison. Please note that the nodal deviations between two shapes are in millimeters and therefore are not noticeable in the figure due to the large scale of the reflector itself. The performance of the algorithm are also presented when solving initial shape, desired shape, thermally distorted shape and the rebalanced shape in Figure 4.16 to Figure 4.19, all of which demonstrate the algorithm’s convergence which has been proved theoretically, the calculation accuracy and the promising converging speed. -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - Deployed; dotted - Initial y z Figure 4.12 Initial and desired (deformed) shapes. 148 -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - Thermal; dotted - Deployed y z Figure 4.13 Deformed shapes with and without thermal loading. -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - Rebalanced; dotted - Thermal y z Figure 4.14 Thermally distorted shape and rebalanced shape. -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - Rebalanced; dotted - Deployed y z Figure 4.15 Desired (deployed) and rebalanced shapes. 149 0 10 20 30 40 0 10 20 30 No. of iteration F norm 0 10 20 30 40 0 500 1000 1500 No. of iteration Gradient 0 10 20 30 40 50 60 70 0 5 10 15 x 10 10 No. of sub-iteration Multiplier 0 10 20 30 40 50 60 70 0 2 4 6 8 x 10 -9 No. of sub-iteration Norm of error Figure 4.16 Algorithm performance when solving initial shape. 0 2 4 6 8 10 12 0 10 20 30 40 No. of iteration F norm 0 2 4 6 8 10 12 0 5000 10000 15000 No. of iteration Gradient 0 5 10 15 20 25 0 2 4 6 8 x 10 10 No. of sub-iteration Multiplier 0 5 10 15 20 25 0 1 2 3 4 x 10 -10 No. of sub-iteration Norm of error Figure 4.17 Algorithm performance when solving deformed shape. 0 10 20 30 40 50 0 10 20 30 40 50 No. of iteration F norm 0 10 20 30 40 50 0 500 1000 1500 2000 No. of iteration Gradient 0 20 40 60 80 0 2 4 6 8 10 x 10 10 No. of sub-iteration Multiplier 0 20 40 60 80 0 1 2 3 4 5 x 10 -10 No. of sub-iteration Norm of error Figure 4.18 Algorithm performance when solving deformed shape under thermal effects. 150 0 5 10 15 20 0 2 4 6 8 No. of iteration F norm 0 5 10 15 20 0 50 100 150 No. of iteration Gradient 0 5 10 15 20 25 30 0 1 2 3 4 x 10 10 No. of sub-iteration Multiplier 0 5 10 15 20 25 30 0 1 2 3 4 5 x 10 -10 No. of sub-iteration Norm of error Figure 4.19 Algorithm performance when solving rebalanced shape. 4.6 Conclusion The study in this chapter proposes one of few methods that enable the active shape control strategy upon the nonlinear static model of the reflector structure. The nonlinear static truss model developed does not need the initial configuration information and considers both geometric and elastic nonlinearity. The optimization algorithm which solves the static model and the nonlinear static shape control problem has the strong convergence, fast converging speed, and is ready to be implemented on the large and complex mesh reflector problems. To further speed up the optimization and reduce the computational cost, two-dimensional subspace spanning method [6] may be used to avoid calculating the factorization of Hessian approximation matrix when computing the step solution; since the Hessian approximation matrix may be badly scaled, a scaling technique need to be adapted for even higher accuracy of results. 151 Chapter 5 Coupled Elastic-Thermal Dynamics 5.1 Introduction This chapter presents the dynamic analysis portion of the Active Shape Control architecture (Figure 1.4). As stated in Chapter 4, considering the performance limitation in passive structure and the manufacturing tolerance [31, 33], it has been suggested that active shape control must be introduced on the deployable space reflectors [20] to the compensation of reflect surface errors caused by thermal loads. To develop an active surface control technique, a control-orientated dynamic model of deployable mesh reflectors is necessary. Following the previous effort on the shape design and the static analysis of mesh reflectors in Chapter 3 and 4 (also see [84, 114]), this chapter first presents the formulation to build up a control-orientated coupled elastic-thermal dynamic truss model for the reflector structure. The model is based on the structural dynamic theories [55, 112, 69, 65], and is useful for free vibration analysis and the design of feedback shape control laws in the Chapter 6. Then the quasi-static strategy is proposed to analyze the on-orbit dynamics of the reflector structure under cyclic thermal effects in Section 5.5, with its implementation on a sampled DMR in Section 5.6. 5.2 Nonlinear Dynamic Model As it mentioned previously, the deployable mesh reflector in consideration is viewed as a 3-D truss structure. For the jth element of the truss of initial length j L (given undeformed length), let the nodes (ends) of the element after deformation be described by (, , ) ii i x yz and (, , ) ii i x yz , 152 where , ii are the indexes of the ends of the jth element. The vector of nodal coordinates of the element can be written as {} i i i d j i i i x y z y x y z           =            (5.1) Since the temperature changes is one of the major factors which affect the surface accuracy of mesh reflectors, besides investigating the nonlinear dynamic vibrations of structure, the dynamic thermal distortion also need to be considered. For a single truss element, the thermal strains , j th ε under temperature changes is defined as , j th j j T εα =Δ (5.2) whereα is the thermal coefficient of the material. Then the total strain of kth element of the structure is the summary of thermal strain , j th ε and elastic strain j ε by assuming no expansions due to other factors such as piezoelectric effects, etc.: ,, jj tj jth j j lL L εεε − == + (5.3) in which j l is the deformed length of the element and j L is the original (undeformed) length. Because of the in-space working environments of the reflector, the gravity is the ignored and the potential energy all comes from the elastic energy of the material. The following formulation in this section continues the previous derivation, and the elastic energy of single element under thermal effects is 153 () ( ) () ( ) ,, , 00 , 00 , , tj j yh L jjjj tj j L jjj j jth VE A sdds E Asdds ε εε εε ε ε εε ε ε ε − = =+   (5.4) with () j j E ε to be Young’s modulus and () , , tj A s ε as the cross-section area at axial location s and a nonlinear stress-strain relation () j jjj E σεε = has been adopted. Hence we have the variation of the elastic energy as () ()()() {} {} , ,, , , , 0 , L tj j j tj jth t j jth tj d j d j VE A sds y y ε δεε εε ε δ ∂ =− − ∂  (5.5) The kinematic energy is =+ tr TT T (5.6) where t T is the kinetic energy due to translation, and r T due to rotation about the longitudinal axis of the element. Because the moment of inertia of the element about the longitudinal axis is small, 0 ≈ r T . The center of mass of the element is () 0 ,, L j cj c j j mssds sL M α ==  (5.7) where j M is the total mass of the element. Then the velocity of the center of mass is obtained as () () () ,, , ,, , ,, , cj cj cj i i cj cj cj i i cj cj c j i i x xl x y yl y zzl z αα αα αα =+− =+− =+−       (5.8) which in a matrix format is { } [ ] { } cd j j j yy α =  (5.9) 154 with [] ,, ,, ,, 10 0 00 01 0 0 0 00 1 0 0 cj cj cj cj j cj cj αα αα α αα  −  =−   −  (5.10) {} , , , cj ccj j cj x yy z     =       (5.11) Therefore, the kinematic energy of the element is () 22 2 ,,,, 1 2 j tj j c j c j c j TT M x y z == + +   (5.12) Under the virtual work of external nodal forces at the kth element is {} { } , T nc j d j j Wy F δδ = (5.13) For the entire structure, we have {} {} () ()()() , ,, ,, , 11 0 , T L mm T tj jdef jtj jth tj jth tj def VV y E A sds y ε δδ δ εε εε ε  ∂  == − −  ∂     (5.14) {} {} {} {} 11 T mm T c j jdef j c j def y TT y M y y δδ δ  ∂  ==−  ∂     (5.15) {}{} , 1 m T nc nc j def WW y Q δδ δ ==  (5.16) where m is the number of elements of the truss, { } def y is the global coordinate vector of all nodes (after deformation), and { } Q is the vector of the external forces applied at the nodes. By the extended Hamilton principle 155 () 2 1 0 δδ δ −+ =  T nc T WV Tdt (5.17) It follows that the nonlinear equations of motion of the deployable mesh reflector under thermal load are in the matrix form {} {} {} {} () ()()( ) {} , ,, , , , 11 0 , TT L mm c jtj jc jtjjthtjithtj j def def y M yE AsdsQ yy ε εε ε ε ε    ∂ ∂    +−− =    ∂∂          (5.18) From Eq. (5.8) and (5.9), it can be concluded that{ } c j y  is a linear function of { } def y , { } [ ] { } cdef j j yy γ =   (5.19) where [] [ ] { } {} d j j def y y γα ∂ = ∂ and it is a constant matrix. Thus, the equation of motion can be rewritten as []{} {} () ()()( ) {} , ,, ,, , 1 0 , T L m tk mdef k tj jth tj jh tj def MyE AsdsQ y ε εε εε ε  ∂  +−− =  ∂      (5.20) with[ ] [] [] 1 m T mj jj MMγγ  =    . 5.3 Linearized Model and Vibration Analysis To develop the feedback shape controller on deployable mesh reflectors, it is very common to linearize the nonlinear model [112] at the nonlinear static equilibriums. The static equilibrium of the model { } def y is the solution of the following equation by setting { } 0 =  def y in the equation of motion. {} () ()()() {} , ,, ,, , 1 0 , T L m tj jtj jth tj jth tj def E Asds Q y ε εε εε ε  ∂  −− =  ∂     (5.21) 156 In particular, considering the nonlinear equilibrium of a single element under the assumption of constant Young’s modulus, uniform cross-section in geometry and negligible Poisson’s effect, the above equilibrium equation can be verified to be the same with the result in Ref. [84], which future supports this nonlinear dynamic model. Since { } { } { } =+Δ def def def yy y where { } Δ def y is a small perturbation, according to the perturbation techniques, nonlinear dynamic equation can be linearized as [ ]{ } [ ]{ } { } m def therm def therm My K y P Δ+ Δ =  (5.22) where [] {} {} () ()()() {} L , ,, ,, , 0 , def T tj therm j t j j th t j j th t j def def y KE Asds yy ε εε ε ε ε    ∂ ∂    =−−    ∂∂        (5.23) {}{} {} () ()()() {} L , ,, ,, , 1 0 , def T m tj therm j tj jth t j jth tj def y PQ E A sds y ε εε εε ε   ∂   =−− − −   ∂      (5.24) It can be seen that the coefficients of the second order derivative are the same, which reveals the fact that the thermal distortion does not affect the mass matrix. Therefore the eigenvalue problem of the linearized model is to solve [ ] [ ] (){ } 0 im therm i MK λμ −+ = (5.25) where { } μ i is the eigenvector and the corresponding eigenvalueλ i is obtained from [ ] [ ] () det 0 therm i m KM λ−= (5.26) Moreover, by adapting the analysis method in Ref [27], it is also able to indicate the ability to control the structure modes by different inputs. First rewrite the linearized system into the state space form, { } [ ]{ } [ ]{ } =+  x Ax B u (5.27) 157 in which {} { } {} {} { } [] [] [][ ] [] [ ] 1 1 0 ,, , 0 def therm m mtherm def y I xuPA BM MK y − −  Δ   == = =   Δ      (5.28) where { } u is the vector of linearized external forces and is considered as the input of system. Defineλ Ai to be the ith eigenvalue of [ ] A and the vector { } j B is the input direction vector corresponding to the j th input of { } u . Then for each λ Ai and { } j B , the matrix in Eq. (5.29) can be formulated and the minimum singular value ij σ of each matrix will be calculated. Since [ ] A can have many eigenvalues and the system may have multiple inputs, we can generate a 2-D table of the minimum singular values ij σ , in which one dimension is λ Ai and the other one is the input element j u . According to the reference [27], the minimum singular values in the table indicate how much influence the different inputs j u have on controlling the different vibration modes of the structure. [ ] [ ] { } Ai j IA B λ   −   (5.29) 5.4 Numerical Analysis and Discussion The proposed nonlinear dynamic model and linearized model are applied to two examples in numerical simulation: a six-element truss and a 90-element spherical reflector structure. For now the thermal effects are not included in the numerical examples in this section. 5.4.1 A six-element truss The first example is a simple truss structure of six nodes and six elements, which is under external forces in the vertical (z) direction. For simplicity, it is assumed that truss elements have linear stress-strain relation and uniform geometry. As shown in the figure, nodes 1, 2 and 3 are fixed, and nodes 4, 5 and 6 are movable. The truss is subject to unity external loads at movable nodes 4, 5, 6 in the vertical down direction (negative z direction). The coordinates of the nodes are given in Table 5.1. All the elements have the same longitudinal rigidity EA = 10000 N. 158 -2 0 2 4 -2 0 2 4 -1.5 -1 -0.5 0 x 1 (1) 4 (4) (5) 2 6 y (2) (6) 5 (3) 3 z Figure 5.1 Initial (undeformed) shape of the example structure Table 5.1 Nodal coordinates Node Number Initial configuration (m) Deformed configuration (m) , iini x , iini y , iini z i x i y i z 1 0 4 0 0 4 0 2 -2.8284 -2.8284 0 -2.8284 -2.8284 0 3 3.8730 -1 0 3.8730 -1 0 4 -2 2 -2.1231 -0.055862.5944 -2.9210 5 -1 -3 -1.8729 -1.9808 -2.1293 -2.7113 6 2 0.5 -2.5552 2.4379 -0.8577 -3.4695 With the linearized model described based on the unity loads, the eigenvalue problem of the truss is solved. The first nine natural frequencies are listed in Table 5.2, and first four mode shapes of the truss are plotted in Figure 5.2. Table 5.2 Natural frequencies of linearized model Mode Number ω i (rad/s) Mode Number ω i (rad/s) Mode Number ω i (rad/s) 1 0.2987 4 21.0312 7 45.3164 2 0.3536 5 32.5099 8 52.5608 3 0.6261 6 34.9730 9 67.3926 159 -3 -2 -1 0 1 2 3 4 -2 0 2 4 -2.5 -2 -1.5 -1 -0.5 0 x solid - mode shape; dotted - equilibrium y z -2 0 2 4 -2 0 2 4 -3 -2 -1 0 x solid - mode shape; dotted - equilibrium y z Mode 1 Side wing Mode 2 Nodding -2 0 2 4 -3 -2 -1 0 1 2 3 4 -2 -1 0 x solid - mode shape; dotted - equilibrium y z -2 0 2 4 -2 0 2 4 -3 -2 -1 0 x solid - mode shape; dotted - equilibrium y z Mode 3 Twisting Mode 4 Up & down Figure 5.2 The first four mode shapes of the truss 5.4.2 A sample of deployable mesh reflector -10 -5 0 5 10 -8 -6 -4 -2 0 2 4 6 8 -3 -2 -1 0 y x z Figure 5.3 A sampled deployed mesh reflector 160 The second example uses the same setup with the one in Section 4.5.2, which has 37 nodes and 90 members. Its cable members have longitudinal rigidity as 1.1121e+005 N. The deployed configuration is shown in Figure 5.3. The natural frequencies of linearized model at the deployed equilibrium are calculated. The range ofω i is from 13.6346 rad/s to 195.6108 rad/s, with the first four being 12 13.6346 rad/s ωω == , 34 14.7368 rad/s, 16.0460 rad/s ωω == . Due to the axis symmetry of the reflector, repeated natural frequencies appear in pairs (for instance, the first two). The first four mode shapes of the reflector are plotted in Figure 5.4, where the solid lines portray mode shapes and dotted lines represent the equilibrium configuration of the truss. Figure 5.4 The first four mode shapes of the mesh reflector. 5.5 Quasi-static Strategy in the Space Mission While the reflector is orbiting Earth in space, the amount of sunlight which shoots on the structure determines the surface temperature of the reflector. For one orbiting cycle, the range of the temperature variations could be from around 250K to 350K depending on the altitude of the different orbit, while the period time of each cycle varies from one hour to 24 hours. Therefore 161 the speed of temperature variation is from around 0.004K/s to 0.07K/s, which can be considered to be very slow. By assuming the uniform temperature distribution on reflector elements, the temperature of the reflector gradually changes in a cycle. Therefore it is practicable and reasonable to treat the structural dynamic thermal distortion as a quasi-static process. Hence, we propose the following strategy on the dynamic analysis of the deployable mesh reflector in space missions: 0 0.5 1 1.5 2 2.5 3 -200 -150 -100 -50 0 50 100 150 time (h) T em perature v ariation (K ) t 0 t 1 t 2 t n-2 t n-1 t n Δ T Δ T Δ T Δ T Figure 5.5 Temperature separation during orbiting missions. In the orbiting mission of the reflector, the dynamic thermal distortion is approximated as a quasi-static process. If the time period of an orbiting cycle starts at the time 0 t and ends at the time n t , inside the cycle it is able to find n-1 moments, 1 t , 2 t ,…, 2 − n t , 1 − n t , so that between every two moments, 1 − i t and i t , the reflector surface has the same temperature variationΔT . Then all these n-1 moments divide each orbiting cycle into n sections (Figure 5.5). Due to the very slow speed of temperature changes on the reflector surface, it is assumed that inside each section, the surface temperature of reflector remain unchanged, and between two neighbor sections the temperature varies byΔT . Based on this separation of each cycle, the mesh reflector in space missions is analyzed as following: 1. Static deformation: 162 Between different sections, the deformation of the reflector is treated in a purely static manner. The same with the status 2 S and 3 S described in Figure 4.11 of Chapter 4.5, first the reflector surface will deform away from the desired surface under temperature 1 T which is not the nominal temperature 0 T of the structure. Then by adjusting the external loading status from 1 P to 1 c PP + , the reflector mesh facets are rebalanced back at the desired shape. During the orbiting mission, the reflector will experience 2 S and 3 S each time when it moves from one section to another. When entering the next section, the surface temperature of the reflector is considered to change into a new constant, such as 1 T and the reflector will deform under the thermal effects. Then in the next status of 3 S , with properly adjusted external loading condition 1 c PP + , the reflector is rebalanced and deformed again, and the shape is maintained in the designated surface. 2. Dynamic process: Inside every section, the temperature is considered to remain constant. Therefore we only consider the dynamics of the reflector due to mechanical vibrations and disturbance, while the dynamic behavior by the actual temperature changes is ignored. Upon the nonlinear static equilibrium after rebalance in 3 S which is calculated above, the linearized model proposed in this chapter is applied on the structure. The dynamic analysis such as the natural frequencies and mode shapes is investigated based on this linear model and will provide the important guidance for the further design of feedback surface controller. 5.6 Numerical Results and Discussion on the Quasi-static Strategy In this section, the simulation regarding the thermal effects in the orbiting mission is first carried out on a simple example of truss structure, which also provides the guidance on the detailed procedure of the quasi-static strategy. Then the proposed strategy is applied on a sample of deployable mesh reflector with simulation results presented. In the first part, a simple truss is used as the study case and it has 6 nodes (3 are on the boundary) and 9 elements and the external forces are restricted in the vertical direction. Besides all the assumptions made in the Section 5.2 and 5.3, it is further assumed that each element has linear elasticity and uniform geometry along 163 its axial direction without Poisson’s effects for simplicity. Moreover, in the initial shape of the structure no pre-tension occurs and each element is at its original length, as shown in Figure 5.6. -4 -2 0 2 -3 -2 -1 0 1 2 3 4 -1.5 -1 -0.5 0 3 (6) (5) x (9) 6 5 (4) (1) (8) (7) 2 (3) 1 (2) 4 y z Figure 5.6 Initial shape of the example structure 5.6.1 Simulations on a simple example of truss structure 1. Static deformation in the quasi-static strategy: -4 -3 -2 -1 0 1 2 3 -2 0 2 4 -1.5 -1 -0.5 0 y x solid - Deployed; dotted - Initial z Figure 5.7 Initial and desired (deployed) shapes For the simple truss, the longitudinal rigidity EA of each element is 1000 N. With the nodes 1, 2, and 3 fixed on the boundary and nodes 4, 5, and 6 movable, the structure is designed to deploy to the a spherical working surface of diameter D = 5 m and height H = 3 m. The initial shape and the desired shape after deployment are shown in Figure 5.7. In the deformed shape, the predetermined external forces, which are in the vertical down direction with magnitude of 10 N, are applied on nodes 4, 5, and 6 and generate the desired working surface. The nodal coordinates in initial and deformed shapes are displayed in the Table 5.3. 164 Table 5.3 Nodal coordinates in initial and deployed configurations Node Number Initial configuration (m) Desired configuration (m) , iini x , iini y , iini z i x i y i z 1 0 4 0 0 4 0 2 -3.8730 -1 0 -3.8730 -1 0 3 2.8284 -2.8284 0 2.8284 -2.8284 0 4 -2.0241 1.9926 -0.9449 -2 2 -1.1231 5 -1.0017 -3.0220 -0.4362 -1 -3 -0.8729 6 1.9538 0.4781 -1.4283 2 0.5 -1.5552 In the orbiting mission, when the structure enters the following section of the orbiting cycle with 10 K of temperature increase at the surface, the structure is considered to deform to another shape under the effects of static thermal distortion. To maintain the surface of the structure in the desired working shape against the thermal deformation, the external forces are adjusted statically, the structure is rebalanced into a new equilibrium and the surface deforms back to the designated shape. The above process is corresponding to the status of 2 S and 3 S in Figure 5.5. -4 -2 0 2 -2 0 2 4 -1.5 -1 -0.5 0 y solid - Thermal; dotted - Deployed x z -4 -3 -2 -1 0 1 2 3 -2 0 2 4 -1.5 -1 -0.5 0 y solid - Rebalanced; dotted - Thermal x z Figure 5.8 Static deformation process. Left: Thermal deformation; Right: Rebalance deformation In the Table 5.4, the adjusted external loads, the nodal coordinates of the shape of thermal deformation and the rebalanced configuration are presented. Figure 5.8 shows the static shape 165 deformation for these two statuses. As already shown in Chapter 4, by comparing the nodal coordinates of rebalanced configuration in Table 5.4 with the deployed configuration in Table 5.3, it is confirmed that even though two shapes are on the consistent working surface, the nodal coordinates are not necessarily the same. Table 5.4 From thermal deformation to rebalanced deformation Node Number Shape of thermal deformation (m) Adjusted loads (N) Rebalanced configuration (m) , iini x , iini y , iini z i x i y i z 1 0 4 0 N/A 0 4 0 2 -3.8730 -1 0 N/A -3.8730 -1 0 3 2.8284 -2.8284 0 N/A 2.8284 -2.8284 0 4 -2.0163 2.0198 -1.2012 -19.7582-2.0265 2.0161 -1.1022 5 -1.0124 -3.0341 -0.9576 -18.5886-1.0138 -3.0470 -0.8324 6 2.0203 0.5014 -1.6432 -13.1722 2.0135 0.5022 -1.5490 2. Dynamic process of the strategy: As the structure is rebalanced to the desired shape, the nonlinear dynamic model is applied on the structure and is then linearized at the rebalanced equilibrium. The natural frequencies of the structure are shown in Table 5.5; the first three mode shapes, are plotted in Figs. 5.9 to 5.11. Table 5.5 Natural frequencies of linearized model Mode ω i (rad/s) Mode ω i (rad/s) Mode i ω (rad/s) 1 λ 2.4323 4 λ 5.1705 7 λ 13.4770 2 λ 3.4226 5 λ 5.6653 8 λ 14.6022 3 λ 4.1855 6 λ 12.3475 9 λ 15.5461 166 -4 -3 -2 -1 0 1 2 3 -2 0 2 4 -1.5 -1 -0.5 0 y solid - mode shape; dotted - equilibrium x z -4 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 4 x solid - mode shape; dotted - equilibrium y Figure 5.9 Mode shape 1 with structure equilibrium. Left: 3-D view; Right: Top view -4 -2 0 2 -2 0 2 4 -2 -1.5 -1 -0.5 0 y solid - mode shape; dotted - equilibrium x z -4 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 4 x solid - mode shape; dotted - equilibrium y Figure 5.10 Mode shape 2 with structure equilibrium. Left: 3-D view; Right: Top view -4 -2 0 2 -2 0 2 4 -1.5 -1 -0.5 0 y solid - mode shape; dotted - equilibrium x z -4 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 4 x solid - mode shape; dotted - equilibrium y Figure 5.11 Mode shape 3 with structure equilibrium. Left: 3-D view; Right: Top view 167 Also by calculating the minimum singular values of the matrix in Eq. (5.29) for each input and every mode, a table of the singular values is obtained and only the results for the first three modes are shown below. From Table 5.6, it can be concluded that the second input of the system which is the vertical load at node 5, has the largest influence when controlling the first mode of the structure at the current equilibrium. Similarly, for the second mode and third mode, the third input (the load at node 6) and the first input (the load at node 4) have most control impact respectively. Table 5.6 Influence of inputs on modes System input element Mode 1 u 2 u 3 u 1 λ 0.0193 0.0650 0.0003 2 λ 0.0029 0.0005 0.0439 3 λ 0.0436 0.0318 0.0160 5.6.2 Dynamic analysis on a sampled deployable mesh reflector Now the quasi-static strategy in the chapter is applied on the model of the deployable mesh reflector considered in Section 5.4.2. Following the similar procedure above for the first example, the reflector is on the orbit separated into quasi-static sections by temperature of 10 K. When the structure travels between different predetermined sections of the orbiting cycle (as 2 S ), the surface of the reflector generates a new equilibrium under thermal effects. In the next status of 3 S , the external loads are adjusted, the structure is rebalanced and the surface of the reflector is pulled back to the desired working shape. The external forces in 2 S and 3 S as entering one orbiting section are presented in the Table 5.7. Note that some external forces are repeated on different nodes because of the symmetry of the structure and only the distinct loads with relevant nodes are shown in the table. It can be seen that after rebalancing the reflector to ensure the surface in the desired working shape, the vertical external forces are increased and the elements of the structure are further tightened. 168 Table 5.7 External forces in different equilibriums Node Number External forces in 2 S (N) External forces in 3 S (N) 1 -6.7799 -14.5709 2, 3, 4, 5, 6, 7 -7.3163 -15.7293 8,10,12,14,16,18 -8.6915 -18.7904 9,11,13,15,17,19 -8.6797 -18.7177 Since the reflector has been statically pulled back to the working surface under the temperature variation, the dynamics of the structure inside the section should be investigated and the natural frequencies and mode shapes are calculated. The obtained ω i is up to 195.6173 rad/s with 12 13.6584 rad/s ωω == , 3 14.7843 rad/s ω = and 4 16.0913 rad/s ω = (shown in Figure 5.12). -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - mode shape; dotted - equilibrium y z -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - mode shape; dotted - equilibrium y z -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - mode shape; dotted - equilibrium y z -10 -5 0 5 10 -10 -5 0 5 10 -4 -3 -2 -1 0 x solid - mode shape; dotted - equilibrium y z Figure 5.12 First four modes of the mesh reflector. First: up-left; Second: up-right; Third: bottom-left; Forth: bottom-right 169 Besides the analysis for the control impact of inputs on different dynamic modes by Eq. (5.29), it is very important to investigate the relationship between the natural frequencies of the linearized model and the variations of the parameters, such as the temperature of the reflector surface. Each time as the temperature varies, we need to consider two different static status of the reflector: 2 S , when the structure is distorted away from the working surface under temperature changes; 3 S , when the structure is rebalanced by the external loads under thermal effects and the working shape of the surface is maintained. Then based on these two nonlinear static equilibriums, the linearized models can be derived with calculated natural frequencies. Considering the overall range of temperature variation to be 100 K, the dynamics of the structure is analyzed for every 5 K changes of temperature and the results are plotted in Figure 5.13 and 5.14. In Figure 5.13, it indicates that naturally the natural frequencies of the system will increase when the temperature of the structure increases and this increasing curve may not be smooth. However, Figure 5.14 shows that after the structure is rebalanced to the working shape at each equilibrium, the natural frequencies decrease smoothly when the structure temperature increases. -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 13 14 15 16 17 18 19 20 Temperature Variation (K) Natural Frequncy (rad/s) ω1 ω3 ω4 ω6 ω7 ω9 ω10 Figure 5.13 Variation of natural frequencies at 2 S status 170 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 13 14 15 16 17 18 19 20 21 Temperature Variation (K) Natural Frequncy (rad/s) ω1 ω3 ω4 ω6 ω7 ω9 ω10 Figure 5.14 Variation of natural frequencies at 3 S status 5.7 Conclusion A nonlinear elastic-thermal model of space deployable mesh reflectors has been developed. The linearized model enables free vibration analysis of deployable mesh reflectors, which provides important information on the dynamic properties of the reflector structure. With this model, a quasi-static strategy to analyze the dynamics of deployable mesh reflector in the space mission is proposed. The examples illustrate the guidance and the procedure of modeling and dynamic analysis of this type of space structures, and show the temperature dependence of system natural frequencies. The nonlinear model, its linearization, and the quasi-static strategy shall be utilized for the further development of dynamic shape control of DMRs via feedback. 171 Chapter 6 Ongoing Work and Concluding Remarks 6.1 Ongoing Research on Feedback Shape Control This section presents the ongoing study on the dynamic shape control problem of deployable mesh reflectors via feedback [80], which is the last portion of the proposed Active Shape Control architecture. The controller developed will focus on dynamically compensate the thermal distortion due to the orbital temperature, reject force disturbance and reduce the mechanical vibration induced. 6.1.1 Introduction As shown in the literature review given in Chapter 4, from the achievable resources the dynamic shape control of DMR has not been much observed. The DMR system that the dynamic shape controller is developed on is shown in Figure 1.5 which is modeled as a 3D truss structure. While orbiting Earth, the surface shape of DMR is influenced by external disturbances, such as temperature variations and environmental forces. Temperature variation changes properties of the truss material and thus the corresponding static and dynamic behavior of the system. As a result, the nodal positions change along with the temperature even though the same amount of tension loads are applied on each node. On the other hand, environmental forces, such as the solar pressure, the aerodynamic drags and gravity gradient in LEO mission, also have significant impact on the surface deformation of the structure. The two kinds of disturbances are addressed in different ways. Firstly, the temperature is taken into consideration when modeling the system. Therefore, the model is nonlinear and 172 temperature dependent. The change of the temperature is respectively slow (usually with period longer than 1 hour and changing rate smaller than 0.1°C/s) compared to the mechanical dynamics (usually higher than 1Hz). Then we can divide the temperature range to multiple sections and obtain quasi-static model with respect to different nominal temperatures for each temperature section. Robust control is proposed to ensure the stability within the sections and deal with the model uncertainty caused by deviation of temperature with respect to the nominal value. On the other hand, to decrease the shape deviation and structural vibration by disturbance forces, we pursue good disturbance rejection performance with a state feedback controller, which is enabled by installing sensors and actuators on DMR. Optical sensors are attached onto the reflector feed, which instantly measure the current position of each node. Tension ties (cables) are connected to each node, and by varying the deformation of the cables, the actuators installed onto the tension ties provide the external loads in vertical direction (Figure 1.5) as control inputs to actively adjust the surface shape. Therefore, the control framework proposed in this chapter is a supervisory control with robust controllers designed for different temperature section. Since the temperature changes gradually, the mechanical vibration settles quickly enough for each switch, which guarantees the switching stability [50]. State feedback controller based on H ∞ optimal control theory is designed for each section to satisfy control performance and ensures the robust stability when model uncertainty presents. 6.1.2 System modeling with uncertainties Upon the fixed boundary on the aperture, the nodal positions of DMR vary due to the dynamic force loads and temperature changes, inducing tensile and compression of flexible elements. The thermal-elastic coupled truss dynamic model is proposed to handle the thermal loads and mechanical vibrations in Chapter 5. The equation of motion is written as Eq. (6.1). 173 [ ]{ } { } {} () () ()() {} , ,, 1 ,, , 0 , k m def damp def T tj jtj jth m def L j tj jth t j My C y E Q y Asds ε εε εε ε =   ++      ∂    −  =  ∂         −         (6.1) { }{} 11 1 T def n n n yxyz xyz =  (6.2) where m is the number of the elements, n is the number of the nodes and j denotes the variables of the th j truss element; j L is the original (undeformed) length for th j truss element; ,, ii i x yz are the x, y, z coordinates of th i node after deformation ; j E is Young’s modulus; , j th ε is thermal strain; , tj ε is the strain considering both thermal and elastic effects; () , , tj A s ε as the cross- section area at axial location s along the truss elements; { } Q is the external forces applied on the nodes; [ ] m M is the mass matrix of the structure; damp C     is the engineering damping matrix and is diagonal. As it is mentioned above, this temperature dependent model needs to be transformed to quasi-static model with respect to the temperature, so that we can treat it as LTI model during a certain temperature range. Denoting the temperature of the DMR as () Tt , and the temperature range for the whole orbiting cycle as [ ] min max ∈ DMR TT T , then DMR T can be divided into p sections as shown in the following expression. ,min ,max qq q TT T   ∈   (6.3) where 1,2... =qp , () max min ,min min 1 − =+ − q TT TT q p , and max min ,max min − =+ q TT TT q p . For each section, choose a nominal temperature as ,min ,max , 2 + = qq qnor TT T , and the model in th q section is then taken as the one at () , = q nor Tt T . The corresponding static equilibrium of the 174 system is denoted as { } , qdef y , which can be obtained by solving Eq. (6.1) under the conditions { } 0 def y =  , { } 0 def y =  and , ,, , qnor qjth jth TT εε = = . The actual { } , qdef y can be written as { } { } { } ,, , qdef qdef qdef yy y =+Δ , where { } , qdef y Δ is a small perturbation with respect to { } , qdef y . According to the perturbation techniques, Eq. (6.1) is linearized as Eq. (6.4). It can be noticed that the thermal distortion does not affect the mass and damping matrices and therefore [ ] , qm m M M =  , , q damp damp CC   =   . { } { } { } { } ,, , , , , , qm qdef qdamp q def q therm q def q therm My C y K y P      Δ+ Δ+ Δ =        (6.4) where 1,2... qp = {} {} () () ()() {} , , ,,, , , L , ,,, , 0 , k qdef T tj jtj qjth qdef qtherm qdef tj q jth tj y E y K y Asds ε εε εε ε    ∂    −    ∂ ∂   =   ∂     −       {} {} {} () () ()() {} , , ,,, , , L 1 ,,, , 0 , k qdef T tj jtk qjth m qdef qtherm j tk q j th t j y E y PQ Asds ε εε εε ε =   ∂   −   ∂   =−   −     Rewrite the linear system Eq. (6.4) into the state space form in Eq. (6.5). Since the number of the nodes is n, the position vector { } , qdef y Δ has dimension 31 × n as well as { } , qdef y Δ  . { } { } { } qq q q q x Ax B u   =+    (6.5) 175 where {} { } {} , , qdef q qdef y x y  Δ  =  Δ    , { } { } , qqtherm uP = , [ ] 33 3*3 11 ,, , , 0 nn n n q q m q therm q m q damp I A MK M C × −−     =                    , 3 1 , 0 nn q qm B M × −   =      . { } q x is the deviation of the system states from its equilibrium point and can be measured by sensors; { } q u is the vector of linearized external loading forces and is considered as the input of the system. The external forces are treated as disturbances { } q w and the output { } q y is the positions of the nodes { } , qdef y Δ . As a result, the complete system can be written as Eq. (6.6). Following the same way, p LTI state space nominal models can be obtained for the p temperature sections. For further details of the modeling process, the reader is referred to Chapter 5. { } { } { } { } {} { } , q q qq qqw qq q x Ax B u B w yC x     =+ +      =   (6.6) where 31 , 31 0 × ×  =    n qw n B I and [] 31 31 0 ×× =  qn n CI . When the temperature () Tt deviates from the nominal temperature , q nor T , there is model uncertainty between the nonlinear model and the nominal model, which is induced by the thermal effect on the structure. To estimate the model uncertainty range within one section, we use the difference between the nominal model and models linearized at the temperature boundaries to obtain the largest possible model uncertainty in th q section (Eq. (6.7)). It can be noticed that the temperature only influences the value of elements in () qq A T   , but not in other coefficient matrices. It is also worth to mention that the authors focus on model uncertainty by the temperature deviation since it dominates the resources of model uncertainty among all the factors. Other model uncertainties such as the uncertainty induced by linearization are not discussed in detail here. However, to compensate their effect on system dynamic, we actually need larger stability margin than it is required by Eq. (6.7) when designing the feedback controller. 176 () , ,max, , , , max , qij qij qij qij qij AAAAA Δ= − − (6.7) where , qij A and , qij A are the elements of () qq A T     when ,max qq TT = and ,min qq TT = respectively. 6.1.3 Controller design The control strategy used in this chapter is to design feedback controllers for each temperature section, and switches the operating controller when the temperature () Tt changes from one section to another. The temperature of the DMS is assumed to be pre-known or measured, and therefore a supervisor can switch among controllers accordingly (Figure 6.1). In th q section the temperature dependent nonlinear system is approximated as a LTI model as shown above. Figure 6.1 The Controller structure To design the robust state feedback controller, the closed loop system is restructured as Figure 6.2. The upper block stands for the structured model uncertainty by pulling out the element uncertainties in Eq. (6.7) by Eq. (6.8). ,max ,2 ,2 qq qq AB C       Δ= Δ       (6.8) where ,max, () qqij diag A  Δ= Δ  for all ,max, 0 qij AΔ≠ . Denote the dimension of q  Δ  as vv × , then ,2 q B   and ,2 q C   has dimension as 6nv × and 6 vn × respectively with ,2, 1 qir B = and ,2, 1 qrj C = 177 where r is the th r element at the diagonal of q   Δ   corresponding to ,max, qij A Δ . All the other elements in ,2 q B   and ,2 q C   are zeros.  q Δ Figure 6.2 System formulation for H ∞ synthesis The lower block is the nominal closed loop system. For the purpose of disturbance attenuation, we want small magnitude value of the transfer function from the disturbance { } q w to the output { } q y ; considering control sensitivity, we need small magnitude value of the transfer function from the disturbance { } q w to the input { } q u . If the frequency property of the disturbance is known, the weight transfer functions () 1 Ws and () 2 Ws can be used to shape the closed loop Bode plots in specific frequency range accordingly. In this chapter, however, we choose 1 WI = and 2 WI γ = to examine the performance over all frequency range. γ works as a weight factor to adjust the tradeoff between disturbance attenuation and input sensitivity. By the definition above the corresponding H ∞ optimal control is to find a feedback gain p K     to stabilize the system and minimize , , qyd qud T T ∞         . Define {} { } { } { } T TT qq q zy u = , and it can be expressed as Eq. (6.9). 178 { } { } { } qq q q q zC x D u   =+     (6.9) where 0 q q C C    =      ,and [] 0 q D I γ   =       . According to [21], q K  ∃  stabilizes the system in Eq. (6.6) and yields the close-loop transfer function satisfying the inequality (6.10), if and only if 6n q X S  ∃∈  and 6 nn q LR × ∈  such that the LMI (6.11) and (6.12) are satisfied. And a stabilizing control gain is given by 1 qq q KL X −     =     . , , qyd qud T T μ ∞   <   (6.10) where μ is a positive scalar. [ ] 0 X > (6.11) [] [] [] [] , , 00 0 T T TTT T qq q q q q q q qw q q q T qw qq q AX X A BL L B B X C L D BI CX D L I μ μ                    ++ + +                       −<        +−            (6.12) Then the H ∞ optimal control is to calculate the minimal μ for which the LMI (6.11) and (6.12) are satisfied. Once the state feedback controller is obtained, the augmented plant for the closed loop system can be written as Eq. (6.13) according to Figure 6.2. [] [] [] [] ,2 , ,2 () : 00 00 qq q qqw aug q qq q AB K BB Ps C CD K       +                =          +          (6.13) 179 As it is mentioned, the changing rate of the temperature is very slow compared to the mechanical dynamics in the system. The inequality condition which can ensure the robust stability of the system in th q section using the obtained q K     is shown below: ,11 1 qPaug q μ σ Δ < (6.14) where ,11 q Paug μ is the structured singular value for system in Eq. (6.15), and q σ Δ is the largest singular value of q  Δ  (in this case it is the largest absolute value of the elements in the diagonal matrix q  Δ  ). [] ,2 ,11 ,2 () : 0 qq q q aug q AB K B Ps C      +           =        (6.15) 6.1.4 Case study and performance discussion -5 0 5 -10 -5 0 5 10 -3 -2 -1 0 x (m) y (m) z (m) Figure 6.3 The Working Shape of DMR A small DMR example is used as the case study, which has 7 movable nodes where vertical loads are applied, 6 fixed nodes on the boundary rim, 24 deformable elements and 21 degrees of freedom. Each element is assumed to have the uniform geometry with linear elasticity. In the following simulation, the longitudinal rigidity EA of each element is 5 10 N and 180 the coefficient of thermal expansion is 51 10 o C −− . The designated working shape of the DMR surface is presented in Figure 6.3 and obviously it is a-six symmetric. The coordinates of nodes is shown in Eq. (6.16) with three entries of the same row presenting the x, y, and z coordinates respectively for one node. The accuracy of the surface shape is highly relevant to the position of the 7 movable nodes. In the following discussion, we focus on analyzing the deviation of the nodes from the desired surface rather than calculating the surface error itself. {} 0 0 3.8197 5.7711 0 2.6650 2.8855 4.9979 2.6650 2.8855 4.9979 2.6650 5.7711 0 2.6650 2.8855 4.9979 2.6650 2.8855 4.9979 2.6650 8.6603 5 0 010 0 8.6603 5 0 8.6603 5 0 010 0 8.6603 5 0 def y −     −     −   −−     −−   −− −     = −−       −   −−  −   −            (6.16) For this case study, we assume that the DMR is used in orbiting condition similar to [23], where the temperature range in one orbiting cycle is [ ] min max DMR TT T ∈ with min 80 TC =− ° and max 50 TC =° , and the period for one cycle is about 5200 seconds. To simplify the formulation of the surface temperature, we assume the temperature changes as a sinusoid wave in one cycle, and then the temperature curve in one cycle is shown as Figure 6.4. If we want the temperature variance within one section to be 10 C ° , the number of section would be 13 p= . We need to design state feedback controller for all of the 13 sections separately. In the following analysis, the feedback control for 8th section is used as an example to show the controller design process, where 8,min 10 TC =− ° and 8,max 0 TC =° . The nominal temperature in this section is 8, 5 nor TC =−° . One of the time zone corresponding to this section in Figure 6.4 is [ ] 51 193 ts ∈ . 181 . 0 1000 2000 3000 4000 5000 -80 -60 -40 -20 0 20 40 60 Time (second) Temperature ( o C) Figure 6.4 Temperature change in one cycle To examine whether the dynamics of the linearized system fits the nonlinear system at the nominal temperature, their response to step forces are compared in Figure 6.5. The step forces applied on the seven nodes are shown in Eq. (6.17), and Figure 6.5 shows the response of the central node, which tends to be most sensitive to disturbances among all the nodes, to such forces in three directions. By comparison, the dynamical behaviors of the two systems are mostly consistent and therefore the linearized system is qualified for the purpose of controller design. {} 71 0 81 81 070 2040 70100 0 T step ts F ts ×  <  =  ≥ −− −− −   (6.17) Using the approach introduced in Section III, a state feedback controller can be obtained for the linear system of the 8th section. Choose the weight factor γ as 0.001, the resulting optimal μ is 3.08, and a state feedback controller q K     is obtained. This is a relatively conservative design to avoid aggressive force actuation on the structure. The model uncertainty by the temperature variation can be estimated by Eq. (6.7) to generate the structured uncertainty as Eq. (6.8). The largest singular value of [ ] 8 Δ is 8 0.6231 σ Δ = . The structured singular value of ,11 () aug Ps     is shown in Figure 6.6. It can be seen that 8, 11 1 Paug μ < , and the robust stability condition in Eq. (6.14) is not only satisfied but with extra stability margin to cover the effects of other sources of model uncertainties. 182 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 85 -5 0 5 x 10 -3 x (m ) 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 85 -5 0 5 x 10 -3 y (m) 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 85 -3.9 -3.8 -3.7 Time (s) z (m) Nonlinear system response Linearized system response Figure 6.5 Comparison of step forces input between nonlinear system and linearized system 10 -2 10 -1 10 0 10 1 10 2 10 3 -100 -80 -60 -40 -20 0 Frequency (rad/s) magnitude (dB) Figure 6.6 Structured singular value for ,11 aug P     in closed loop system of the 8th section To examine whether the feedback controller helps to attenuate the disturbance, we use the state feedback controller q K   to work on the nonlinear system in Eq. (6.1). Disturbances are chosen as impulses as Eq. (6.18). The responses of the closed loop system and open loop system are both shown in Figure 6.7. The desired location of the central nodes is shown in Eq. (6.19). It can be seen that the open loop system keeps vibrating for a long time after the action of the impulse disturbance. This is due to the flexibility and lightly damped nature of the DMR. This phenomenon is avoided by utilizing the state feedback control. By inspection, not only the closed loop system is robust stable to the model uncertainties, but also it quickly suppresses the 183 vibration induced by the disturbances within a couple of seconds. It may also worth to mention that there are vibrations in z direction even before the appearance of the impulse disturbances. This is due to the influence of temperature variance, which shows high sensitivity of the open loop system to external environmental factors. According to the smooth response in the first second of the third plot in Figure 6.7, such vibration is also addressed by the closed loop controller. The mechanical vibration is no longer easy to be stimulated by the varying temperature with the closed loop controller. () { } ( ) 0 700 200 400 700 1000 0 81 T Ft t δ δ = −−−−− − (6.18) ()( ) 00 3.82 des des des xy z=− (6.19) 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 85 -1 0 1 x 10 -3 y (m) 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 85 -3.85 -3.8 Time (s) z (m) 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 85 -2 0 2 x 10 -3 x (m) Open loop system Closed loop system Figure 6.7 Comparison of the responses to impulse forces disturbance The discussion above show the benefit we can obtain using a robust state feedback control to actively control the node positions for a DMR within the 8th temperature section. For the other 12 sections, the way to obtain the feedback controllers are identically the same, and the control performances for each section are also similar. In the following contents, we examine the possible error induced by controller switching between sections. As we show in Figure 6.1, when the temperature varies from one section to another, we switch to the corresponding controller to ensure the stability; the transient dynamics caused by the switches settles in 1 second, which is far before the next switch happens. Therefore the relatively long section duration between switches by the supervisory controller ensures the switching stability. 184 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 6 x 10 -3 Time (s) Error norm (m) Open loop system Closed loop system Figure 6.8 Comparison of shape error norm between closed system and open system To quantify the error, a norm is defined as Eq. (6.20). It gives the space deviation of the nodes from their designated surface. During the time period [ ] 51 471 ts ∈ in Figure 6.4, the system goes across three sections. Therefore, within this time period two controller switches occur. The error norm of the DMR in this process is calculated and shown in Figure 6.8. Here no force disturbance is added into the system and the surface error is purely induced by the temperature variation in the orbiting cycle. The error of the closed loop system is compared with the error of open loop system. The results show significant improvement on decreasing the error norm by the state feedback controllers. The errors of the closed loop system are very small at the nominal points, relatively larger at the boundaries. As for the open loop system, the constant tension load is designed to work under , 5 nor open TC =− ° , which is corresponding to the system equilibrium at 125 ts = in Figure 6.4; therefore, the further the temperature is away from its value at this transient, the larger the error would be. It can be expected that when the temperature keeps going up along the curve in Figure 6.4 after 471 ts = , the error of the open loop system would increase along. Moreover, the open loop system vibrates all the time, which can also be observed from Figure 6.7. Clearly, closed loop control shows large benefits in shape maintaining and vibration suppression. It can be noticed, though, when we switch from one controller to another there is a spike of the error norm. This is due to the sudden change of surface error when switching between state feedback controllers, which induces vibration of the structure. To avoid this, bumpless transfer can be used in the future work[21]. 185 () () { } norm Et dt ∞ =Δ (6.20) where {}{ } 12 T n dd d d Δ=Δ Δ Δ  and k d Δ is the deviation of the th k node from the designated shape geometry in current section. 6.1.5 Conclusion To the knowledge of the authors, this section proposes one of the first implementations of feedback control strategy on dynamic shape control of the DMRs. The proposed method is to compensate the thermal distortion and attenuate external disturbances on the surface shape. The closed-loop strategy is implemented by measuring nodal coordinates of the structure and applying external loads on movable nodes. Then the overall temperature variation in one orbiting cycle of DMR is segmented, one robust state feedback controller is design by H ∞ synthesis for each temperature section and the supervisory switching logic is used to adapt proper controllers according to the current temperature. The controller of a simple DMR structure is designed in the case study and the time- domain performance is validated on the nonlinear coupled thermal-elastic model. The simulation results demonstrate the considerably improving ability against external force disturbance. The surface error by temperature variance is limited within 2 mm and is significantly smaller than the open loop response, which implies the potentiality to have reflector of larger size and higher surface accuracy once the active control strategy is utilized. 6.2 Concluding Remarks This thesis research has developed the Active Shape Control (ASC) architecture for the type of state-of-the-art deployable mesh reflectors. As presented in Figure 1.4, the ASC architecture has performed the following three missions: 1. The optimal shape design of DMRs. The current work views the traditional form finding problem on the DMR shapes as a pure design problem, which contains two stages: the optimal generation of surface mesh geometry for the working surface and the optimal parametric design of the reflector structure based on the designated mesh facets. The first design stage provides the pseudo-geodesic mesh geometry with 186 minimum total length and smallest surface error among other design approaches. The second stage integrates the external loads of trusses into design variables for the first time and ensures the best possible design solution on the reflector structure parameters. In the unique merging of two design stages together, the actual deployed shape from the design results not only exactly matches the desired one but also completely satisfies the equilibrium equations of the reflector structure with geometric and material nonlinearities. 2. The nonlinear static shape control of DMRs. The research suggests a new nonlinear static truss model for the reflector and such model does not involve the initial configuration of the structure where no external load presents. Both the initial shape and the deformed shape of the reflector truss are obtained by the presented solving technique which is the gradient-based trust region nonlinear programming approach with global convergence. Upon the static model and the solving algorithm, a nonlinear static shape control method is constructed as one of few static shape control methods handling the fully nonlinear DMR structure model. This shape control method has successfully compensated the thermal distortion of the reflector which is commonly observed in the orbiting mission and the reflector mesh facets are always maintained on the desired working surface without any deviations. 3. The feedback dynamic shape control of DMRs. The thesis work proposes a theoretical but the first feedback dynamic shape controller for DMR applications. The controller is developed on the coupled elastic-thermal dynamic model of the reflector truss structure from which the natural frequencies are evaluated and the mode shapes are identified. The feedback shape control method combines the switching control strategy with robust controller design formulation to compensate the quasi-static thermal deformations of DMRs during orbiting. The scaled simulation also shows the promising rejection of external disturbances and the fast attenuation of mechanical vibrations. To further improve the ASC architecture, the sharp shift at the switching boundary of the feedback shape controller has to be reduced or removed and the time response of the system needs be bumpless. 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After the first three steps of the design, D and F of the reference sphere, () wi and () i φ are available and will be used to calculate the nodal coordinates in this section. The nodes of the geometry are categorized into the following two types and the derivations are formulated separately: 1. The nodes on the subdivision lines: The indices of the nodes on the subdivision lines satisfy the following 1( 1) =+ − j qi (A. 1) in which 1, 2, , s qn =  and 1, 2, , 1 r in =−  . Therefore all the nodes on the subdivision lines are denoted as , ij N . From the geometry in Figure 2.7 and Figure 2.9, we have . . (A. 2) () , 2 1 π α =− ij s q n (A. 3) According to Figure 2.9, the coordinates of such nodes are 200 ,, , ,, , ,, sin cos 1, 2, , 1 sin sin 1, 2, , cos θα θα θ = =− = = =−+   s ij i j ij r s ij ij ij s ss ij ij xR in yR qn zR R H (A. 4) 2. Nodes not on the subdivision lines: Because of the symmetry of the subdivision, the derivations in one subdivision are necessary. Considering the portion of the ith ring inside one subdivision, it is projected onto a XY plane which is passing the origin of the reference sphere (in Figure A.1). According to previous discussion, 1 i+ nodes are on this ring portion and two nodes at the ends of the portion are on the subdivision lines. Denote one end node as , ij N and then the other node is , ij i N + . By the symmetry of the subdivision lines, two end nodes are at the same height and the chord ,, ij ij i NN +   is parallel to the XY plane. The nodes , ij N and , ij i N + with their projected nodes , ij N′ and , ij i N + ′ have constructed a rectangular. Since the xy coordinates of , ij N′ and , ij i N + ′ are obtained in Eq. (A. 4), by the geometry in the Figure A.1, we have Figure A.1 Nodes inside one subdivision. ()() 22 ,, ,, , , , , ++ + + ′′ == − +− ij i j i i j i j i ij i j i i j i j i NN N N x x y y (A. 5) 22 ,, , ′=+ ij ij ij oN x y (A. 6) 201 ,, cos 2 γ + = ij i j i s NN R (A. 7) sinγ = s oa R (A. 8) 2 β π γ =− (A. 9) 1, 2, , 1 β β== −  l ll i i (A. 10) () sinγβ = + l l oa ob (A. 11) () ,, ,, cos 2 γβ + ′′== − + ij ij i ll l l ij ij NN bN bN ob (A. 12) ,, , ′′ ==+− ll s i j ij ij bb N N z R H (A. 13) , cosθ + ′ = ll ij l l bb ob (A. 14) , sinθ + ′= ll ij l ob ob (A. 15) ()()() 22 2 ,, , cos 2 ϕ ′′ ′′ +− = ′′ ll ij ij l l ij ob oN b N ob oN (A. 16) () , 2 1 π αϕ + =+ − l ij l s q n (A. 17) Once , ij l θ + and , ij l α + are calculated for 1, 2, , 1 li =−  , together with , ij θ and , ij α from Eq. (A. 2) and (A. 3), the coordinates of nodes can be obtained by Eq. (2.24). 202 Appendix B Calculation of the curvature of the first converging ring Figure B.1 First converging ring within one subdivision in projected XY plane. This appendix section describes the curvature calculation of the first converging ring in the converging treatments introduced in Step 5 in Chapter 2. Figure B.1 presents the XY plane projection of the c n th ring which is the first converging ring from inside. A′is the projected vertex A and two end nodes on the subdivision are projected as ' ,1 c n N and ' ,1 cc nn N + with the coordinates () ,1 ,1 , cc nn xy and () ,1 , 1 , cc cc nn nn xy ++ respectively. The coordinates of middle point on the geodesic arc are, 203 () () ,/21 ,/21 sin cos sin sin φα φα + + = = ccc ccc ns c nn ns c nn xR n yR n (B. 1) in which ,/21 cc nn α + and /2 c n ϕ are calculated by setting c in = , 1 q= and /2 c ln = in the formulation of Appendix A. Then the distance c n H between () , cc nn x y and '' ,1 , 1 ccc nnn NN +    is 12 2 1 1 −+ = + cc c nn n cx y c H c (B. 2) ,1 ,1 12,11,1 ,1 ,1 , + + − ==− − cc c cc cc c nn n nn nn n yy ccycx xx (B. 3) Denote the curvature of c n th ring in XY plane as 1/ c n r , where the radius c n r satisfies () 2 2 ,1 , 1 2 2 +  ′′ +− =    ccc cc c nnn nn n NN rH r (B. 4) 204 Appendix C Calculation of nodal coordinates after applying the converging treatments This appendix presents the calculation of nodal coordinates on the spherical reference when two converging treatments are applied in the Step 5 in Chapter 2. Practically, the second treatment shall be implemented first to adjust the nodes on the subdivisions, and then the nodes not on the subdivisions but on the converging rings are modified by the first treatment based on the subdivision nodes just obtained. To apply the second treatment, the relative thickness function () wi is replaced by Eq. (2.29) or (2.67) depending on whether the entire reflector surface is effective. Then () i φ is calculated in Eq. (2.20) and the subdivision nodes are obtained by Eq. (A. 2), (A. 3) and (A. 4) no matter whether those nodes are on the converging rings or not. Among rest of the nodes, the coordinates of those not on the converging rings are calculated through the derivations in Appendix A. Then the nodes which are on the converging rings but not on the subdivisions will be modified according to the first treatment. Since the second index of the subdivision nodes on the converging rings is expressed as 1( 1) =+ − j qi (C. 1) where 1, 2, , s qn =  and ,1, , 1 cc r in n n =+ −  .In Figure C.1, A′ is the XY plane projection of the vertex A. , ij N′ and , ij i N + ′ are the projected ends of ith ring within one subdivision and their coordinates are already obtained when implementing the second converging treatment. Following the Appendix B, the curvature of the first converging ring 1/ c n r is then calculated and therefore the radius of each converging ring i r is available by Eq. (2.28) and (2.66). , ij l N + ′ is the 205 projected node on this converging ring which equally separate the ring’s projection on XY plane. According to the geometry of the figure, we have Figure C.1 Nodes on converging rings within one subdivision in projected XY plane. ,, , 2 + ′′ ′′ = ij i j i ij NN aN (C. 2) , sin 2 μ ′′ = ij i aN r (C. 3) 22 ,, , ′′=+ ij ij ij ANx y (C. 4) , , sinσ ′′ = ′′ ij ij aN A N (C. 5) μ μ = l l i (C. 6) 22 ,, 22cosμ + ′′=− ii l ij ij l NN r r (C. 7) ()() 22 ,,, , ,, , 2cos 22 πμ μ σ ++ + − ′′ ′ ′ ′′ ′ ′ ′′ =+ − −+   l i j l i j i j l ij ij ij l i j AN N N AN N N AN (C. 8) 206 ()()() 22 2 ,, ,, ,, cos 2 σ ++ + ′′ ′′ ′ ′ +− = ′′ ′′ ij l i j i j i j l l ij l i j AN AN N N AN AN (C. 9) () , 2 1 π ασ + =+ − l ij l s q n (C. 10) Calculating , ij l α + for 1, , 1 li =−  , 1, 2, , s qn =  and combining above formulation with Eq. (2.24), the nodal coordinates on the converging rings but not on the subdivisions are calculated below, ,, , ,, , 22 2 ,,, cos ,1, , 1 sin 1, , 1 1, 2, , α α ++ ++ +++ ′′ = =+ − ′′==− = =− − −+    ij l i j i j i cc r ij l i j i j i s ss ij l i j l ij l xAN in n n yAN l i qn zRx y RH (C. 11) 207 Appendix D Universal mapping function The uniform property of the mapping function which is defined in Step 8 in Chapter 2 is investigated here and used to improve the efficiency of the proposed step. In the design of a specific DMR, the number of the subdivision and the aperture ellipticity are kept constant. . As such the mapping function , mi f for the ith ring is determined by the number of the nodes used in the formulation (i.e., the ring index i), To see if , mi f is capable of presenting the mapping functions of other rings, assume that s kn nodes (ki ≠ ) are on the circle of the spherical reference (the same number of nodes on the kth ring) and that the nodes equally separate the circle. It can be shown that the normalized polar angles , kj η for the mapped nodes on the ellipse of the ellipsoid reference is given by ,, 1 () 1,, η − ==  kj mi s s j f jkn kn (D. 1) Then both the coordinates of the mapped nodes and the lengths of the chords between all neighbor nodes can be obtained. If , mi f is an accurate universal mapping function, all the chords should share the same length. To measure the accuracy of , mi f , the relative chord error () c k ε for the ellipse with s kn nodes is defined as the maximum length difference between chords versus the radius of the spherical reference. This radius varies from the magnitude of 1 10 − m to 1 10 m as the diameter of the reference aperture is always within 1~200 m. In practice, the absolute chord error is preferred to have the magnitude of 0.1 mm or less. Thus, the mapping function , mi f is 208 accurate enough and is said to be uniform for the kth ring if () c k ε is in the order of 5 10 − . After implementing the above procedure, the results are shown in Figure D.1. Note that the mapping function is fully defined by the aperture ellipticity and the number of the nodes on the ring and therefore these results are generic for different design tasks. This investigation reveals three things: () c k ε decreases when using the mapping function of the outer ring; () c k ε also goes down when the aperture is more closed to a circle; by properly choosing i, () c k ε remains in the order of 5 10 − once k is large enough. Then it can be concluded that the uniform mapping function always exists in all the outer rings starting from certain index number. 5 10 15 20 25 30 35 40 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 k ε c (k) i = 1 i = 3 i = 7 i = 10 i = 15 5 10 15 20 25 30 35 40 10 -5 10 -4 10 -3 10 -2 10 -1 k ε c (k) b e /a e = 0.7 b e /a e = 0.8 b e /a e = 0.9 Figure D.1 The relative chord errors when 6 s n = and at (a) 0.7 ee ba = (left); (b) 7 i = (right). Let cr i be the smallest ring index such that , cr mi f is uniform in all the outer rings. Also define cr k as the largest ring index where , cr mi f is not accurate enough to use. Since ee ba is observed to be larger than 0.7 [107], two critical indices cr i and cr k are obtained numerically in Table D.1 for different s n by setting 0.7 ee ba = . In Figure D.2 which is also generic for various design tasks, the relative chord errors are evaluated at the critical mapping function , cr mi f with the worst case ee ba to be 0.7, which presents the indices of the rings where the mapping function is 209 uniform and demonstrate cr k in Table 1. It can be seen that with more subdivisions of the geometry, the rings on which , cr mi f is uniform start early. Table D.1 Two critical indices cr k cr i 4 = s n 10 13 6 = s n 7 9 8 = s n 5 7 5 10 15 20 25 30 35 40 10 -4 10 -3 10 -2 k ε c (k) n s = 4 n s = 6 n s = 8 Figure D.2 The relative chord errors for cr ii = and 0.7 ee ba = . In the proposed design method, on the ith ring when cr ik ≤ the mapping function , mi f is obtained individually. For the rings fall into the uniform region (its index is larger than cr k ), one universal mapping function , cr mi f is needed. If the boundary layer is defined and the nodal reduction is applied, then its mapping function has two cases. When rcr nk ≤ or () 0.5 1 1 rcrB nkc ≤++ resulting in less nodes of the boundary rim than cr k th layer, the mapping function is calculated individually as , r mn f . Otherwise, the boundary rim has sufficient number of nodes and using the universal mapping function , cr mi f is appropriate. In total, the design method will require 12, 9 and 7 mapping functions at most when s n is 4, 6, 8 respectively. Moreover, since neither the scale nor the focal ratio of the reflector is directly used to determine 210 the mapping functions, it implies the set of mapping functions can be shared between the different design missions with the consistent ellipticity of the aperture and the same number of subdivisions. 211 Appendix E Evaluation of the best-fit surface of the mesh geometry for an offset-feed parabolic reflector Figure E.1 The best-fit surface for an offset-feed parabolic reflector in XZ plane. The evaluation approach for the best-fit surface of the offset-feed parabolic mesh geometry is needed in Section 2.4 of Chapter 2 and presented in this part of the appendix. As shown in Figure E.1, the parent paraboloid and its best-fit surface are in the global coordinates. , ca bf D is the circular aperture of the reflector’s best-fit working surface which is the portion of the parent best-fit surface within the offset aperture. , pbf F is the parent best-fit focal length obtained in Ref. [1] as well as the vertical deviation , pbf H Δ . , off bf e is the offset distance of the best-fit working surface. tan( ) ϕ′ is the slope of the best-fit parabola at the point intersecting with the parent aperture in XZ plane. , pbf H Δ is always sufficiently small to assume that the circular aperture of the best-fit working surface is co-centered with the circular aperture of the offset parabola. From the geometry, we have, 212 () ,, 4 ′=−Δ p p bf g p bf DF H H (E. 1) , tan( ) 4 ϕ ′ ′ = p pbf D F (E. 2) () 1 tan( ) 8 γ ϕ + = p off off p D F (E. 3) , 11 tan( ) tan( ) 1 tan( ) ϕϕ ϕ − ′′ − = − pp off ca ca bf off DD DD (E. 4) Or () , tan( ) tan( ) tan( ) ϕ ϕϕ ′ ′ =− − ′ − ca bf ca p p off DD D D (E. 5) 213 Appendix F Formulation of B w lower bound Figure F.2 The connections at single node on boundary rim in XY plane. This appendix is to derive the lower bound of the design parameter B w , which is mentioned in the Section 2.6 of Chapter 2. The member connections at one boundary node are shown in Figure F.2. Here , r nj AN ′′ is not necessarily perpendicular to the chord and then , r nj N′ presents any boundary node. To avoid the intersection between members in the boundary layer and the ones on the 1 r n − th ring, we have , ′′ ′ > r nj ANAD (F. 1) 214 As the 1 r n − th ring is always a circle due to the converging treatment in Step 5 in Chapter 2 described in the first part of the chapter, 1, r nj ANAC − ′′ ′ = . Since one boundary node connects () 12 B c + nodes on the 1 r n − th ring, then () () 1, 1 cos 21 π − ′′  − ′ =<   ′′ −  r nj B rs AN c AC nn AD AD (F. 2) Substituting Eq. (F. 1) to have () () 1, , 1 cos 21 π − ′′  − <   ′′ −  r r nj B rs nj AN c nn AN (F. 3) According to the geometry in Figure 2.7, () () () () 1, , sin 1 sin φ φ − ′′ −= ′′ = r r s rnj sr nj Rn AN Rn AN (F. 4) Then () () () () () 1 sin 1 sin 1 cos 21 φ φ π −   −  >   −    −   r r B rs n n c nn (F. 5) Combining Eq. (2.20) and (2.62), we can derive the lower bound of B w as () () () 1 1 1 1 4 1 sin 1 1 cos sin 21 4 π − − = −       >−      −        −     r effective n effective B i effective B rs effective D F wwi Dc nn F (F. 6)

Abstract (if available)

Abstract This dissertation presents the research on optimal design, nonlinear analysis and shape control of the perimeter truss deployable mesh reflector (DMR) as a type of state-of-the-art space structures, all three portions of which construct the Active Shape Control architecture. ❧ To design the shape of DMR, a new mesh generation approach is developed to automatically determine the mesh geometry of the working surface of DMRs, which is among very few of pseudo-geodesic mesh geometry design methods. Once the desired geometry of the mesh facets is generated, the optimal design method is presented to determine the structural parameters such as undeformed length of members and the external loads so that not only the mesh facets are deformed onto the desired working shape exactly but also the best tension distribution is assured in the reflector structure. In the academic literature such method is the first to include the external loads as the design variables to design deployable trusses. ❧ The nonlinear static analysis is carried out based on a new static truss model using a nonlinear programming solving technique formulated in the thesis. Since the static model does not need the initial configuration of the reflector, both the initial shape and the deformed shape are obtained by the solving approach. The natural frequencies and the mode shapes are addressed on the linearized dynamics derived at the nonlinear static equilibrium solution and this linear dynamic system is used in the following shape control study. ❧ As the final part of the architecture, both the static and dynamic shape control strategies are proposed in the thesis. The static shape control approach provides one of few thermal compensation solutions regarding the fully nonlinear statics of reflector structures. The dynamic shape control use the feedback of the nodal coordinates and is the first feedback shape control on DMRs. Both shape control methods have demonstrated their advancements on improving the surface accuracy of DMRs for the space thermal environment in orbiting missions.

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Asset Metadata

Creator Shi, Hang (author)

Core Title Optimal design, nonlinear analysis and shape control of deployable mesh reflectors

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 01/29/2014

Defense Date 05/30/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag deployable mesh reflector,dynamic analysis,OAI-PMH Harvest,optimal design,shape control,static analysis

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Yang, Bingen (Ben) (committee chair), Jin, Yan (committee member), Wellford, L. Carter (committee member)

Creator Email hangshi@usc.edu,shwillcn@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-305403

Unique identifier UC11294474

Identifier etd-ShiHang-1878.pdf (filename),usctheses-c3-305403 (legacy record id)

Legacy Identifier etd-ShiHang-1878.pdf

Dmrecord 305403

Document Type Dissertation

Format application/pdf (imt)

Rights Shi, Hang

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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