Digital image processing for system identification

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these w ill be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, M l 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DIGITAL IMAGE PROCESSING FOR SYSTEM IDENTIFICATION by Hung-Chi Chung A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirement for the Degree DOCTOR OF PHILOSOPHY (CIVIL ENGINEERING) December 2001 Copyright 2001 Hung-Chi Chung Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U M I Number 3065772 U M I* U M I Microform 3065772 Copyright 2002 by ProQuest Information and Learning Company. A ll rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, M l 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNI VERSI TY PARK LOS ANGELES. CALIFORNIA 90007 This dissertation, written try under die direction of h i* Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of r e quirements for die degree of DOCTOR OF PHILOSOPHY Dent of Graduate Studies DISSERTATION COMMITTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I would like to express my sincere appreciation to the benevolent research seniors who gave me their support and encouragement to help me to accomplish this dissertation. 1 am deeply grateful to my advisor, Professor Masanobu Shinozuka, for his constant supervision and guidance from the beginning to the end. I would also like to thank Professor Jianwen Liang of Tianjin University, and Professor Shigeru Kushiyama of Hokkai-Gakuen University for their enthusiastic support and encouragement while we were collaborating the projects and performing the digital image experiments. I am also grateful to Professor Ce Liu at Houston University and his associates for their assistance in the preliminary implementation of image processing for remote sensing. I am especially grateful to my wife, Jui-Yen, for her patience, understanding, for taking good care of my sweet baby daughter, Haley, and the support that only such a companion can offer. Last but not least, I would like to thank my parents and parents-in-law who gave me everything, endless love and motivation to pursue knowledge in life. This dissertation is dedicated to them. This research is sponsored by the National Science Foundation under Grant INT-9604614 and also by the Multidisciplinary Center for Earthquake Engineering Research under Grants R92632 and R92250-B. I am deeply grateful for their physical support and encouragement. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS............................................................................ ii LIST OF TABLES............................................................................................. vi LIST OF FIGURES............................................................................................ vii LIST OF SYMBOLS......................................................................................... riii ABSTRACT ...................................................................................................... xv PART I INTRODUCTIONS .................................................................... 1 1. Introduction .................................................................................................. 2 1.1 Motivation ....................................................................................... 2 1.2 Background .................................................................................... 4 1.3 Research Objectives ....................................................................... 5 1.4 Research Approach ......................................................................... 6 1.5 Structure o f Dissertation ................................................................. 9 2. Digital Image Processing: Techniques and Applications ......................... 10 2.1 Introduction ................................................................................... 10 2.2 Fundamentals of Digital Image Processing ................................... 14 2.2.1 Enhancement of digital images ....................................... 16 2.2.2 Acquisition of motion pictures and segmentation ........... 20 2.2.3 Image registration and geometrical modification ............ 23 2.2.4 Correlation analysis and recognition of image difference.. 26 2.3 Applications of Image Processing .................................................. 28 2.4 Application to System Identification .............................................. 29 2.5 Summary.......................................................................................... 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PART n PROOF-OF-CONCEPT EXPERIMENTS ............................. 33 3. A Pendulum System with Coulomb Damping ........................................... 34 3.1 Introduction .................................................................................... 36 3.2 Implementation of DIP Method ..................................................... 37 3.3 A Pendulum System with Coulomb Damping ............................... 42 4. Numerical Experiments; Cases 1 and 2 of a Pendulum System ........... 44 4.1 Numerical Experiments for Case 1 ............................................... 44 4.1.1 Observation of pendulum response .................................. 44 4.1.2 Inverse analysis for friction coefficients identification 47 4.1.3 Forward analysis verification of pendulum displacement.. S O 4.2 Numerical Experiments for Case 2 ............................................... 53 4.3 Findings and Discussion ................................................................ 64 4.3.1 Findings ............................................................................ 64 4.3.2 Discussion ........................................................................ 66 5. Relative Displacement between Base-Isolated Structure and Shaking Table .................................................................................... 68 5.1 Introduction .................................................................................. 68 5.2 Proposed Study and Preliminary Result ...................................... 71 5.2.1 Proposed study and current scheme .............................. 71 5.2.2 Obstacle of using NCREE’s video record ...................... 74 5.2.3 Preliminary result ............................................................ 75 5.3 Discussions .................................................................................. 79 6. Nonlinear Elastomer Membrane.............................................................. 81 6.1 Introduction .................................................................................. 81 6.2 Rubber Elasticity .......................................................................... 83 6.3 Mooney-Rivlin Model ................................................................ 88 6.4 Procedures of System Identification ............................................ 89 6.5 FEM for Nonlinear Elastic Elastomer Membrane ....................... 91 7. Numerical Experiment; Nonlinear Elastomer Membrane ................. 96 7.1 System Setup ............................................................................... 96 7.2 Extraction of Tracked Points ....................................................... 97 7.3 Geometrical Reconstruction ........................................................ 100 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.4 Measurement of Deformation ...................................................... 102 7.5 Regression Analysis to Determine Stress-Strain Relationship 110 7.6 FEM Verification ........................................................................... 113 7.6.1 FEM model and boundary conditions ............................... 113 7.6.2 Verification by FEM ........................................................ 114 7.6.3 Error analysis .................................................................... 132 7.7 Findings and Discussions .............................................................. 133 7.6.1 Findings ........................................................................... 133 7.6.2 Discussions ...................................................................... 135 8. Conclusions and Recommendations ......................................................... 138 8.1 Conclusions .................................................................................. 138 8.2 Recommendation for Future Studies .......................................... 140 PART III REFERENCES ......................................................................... 142 References ........................................................................................................ 143 PART IV APPENDICES ........................................................................... 147 Appendix A: Source Codes for Pendulum Experiments ............................. 148 A.I Interfacial Programs for CCD, Frame grabber and MATLAB ............................................................................ 148 A.H Program for Positioning Coordinates of Pendulum Weights Center (MATLAB Codes) ........................................... 168 A.III Program for Smoothing the Data Obtained by CCD Method (FORTRAN Codes) ....................................................... 169 A.IV Runge-Kutta 4th Method for Forward Verification Analysis (MATLAB Codes) ....................................................... 170 Appendix B: Source Codes for Elastomer Membrane by (Still) Digital Camera Method ........................................... 171 B.I Program for Cmputing Grid’s C enter......................................... 171 B.II Program for Nonlinear Regression by the Mooney-Rivlin Equation ..................................................................................... 172 B.III FEM for the Verification of Nonlinear Elastic Elastomer Nodal Displacements and Determination of Stresses & Strains ... 174 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Page Table 4-1 Results of numerical experiment (Case I) ................................. 47 Table 4-2 Results of numerical experiment (Case 2) ................................ 61 Table 7-1 Measurement of total elongation (between two fixed ends along the x-axis) ............................... 112 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Page Fig. 2-1 Continuous image field and its representation o f image Intensity ....................................................................................... 1 1 Fig. 2-2 Digital image field and its representation of image Intensity ........................................................................................ 12 Fig. 2-3 Rubber band method for image intensity adjustment.................... 18 Fig. 2-4 Histogram equalization method for image intensity adjustment ............................................................................... 19 Fig. 2-5 Method of hit-or-miss mask operation ......................................... 20 Fig. 2-6 Three stages for motion analysis ................................................. 21 Fig. 2-7 Segmentation by using image intensity ..................................... 24 Fig. 2-8 Polynomial warping for geometrical modification ..................... 25 Fig. 2-9 Correlation analysis algorithms ..................................................... 27 Fig. 3-1. Remote sensing of a vibration system using image processing method .................................................................... 37 Fig. 3-2. Configuration of digital image system .......................................... 38 Fig. 3-3. Flowchart of identification procedures ......................................... 39 Fig. 3-4. Experimental set-up of pendulum system for use in two cases 42 Fig. 4-1. Displacement observation data from DIP when 0=7.73 degree (Case 1) ..................................................... 46 Fig. 4-2. Displacement observation data from DIP when ^=13.47 degree (Case 2) ................................................... 46 Fig. 4-3. Frictional coefficient estimated from inverse analysis and averaging when 0=7.73 degree (Case 1 )............................. 48 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4-4. Frictional coefficient estimated from inverse analysis and averaging when >5=13.47 degree (Case 1) ............................ 49 Fig. 4-5 Modes of sliding frictions ............................................................. 50 Fig. 4-6 Concepts of stick and slip phenomenon ..................................... 50 Fig. 4-7 Verification by forward analysis for P = 7.73° .......................... 52 Fig. 4-8 Verification by forward analysis for P = 13.37° ........................ 52 Fig. 4-9(a)(b) Displacement of metal board observed by DIP ........................ 54 Fig. 4-9(c)(d) Displacement of metal board observed by DIP ........................ 55 Fig. 4-10(a)-(b) Relative displacement of pendulum and its foundation ........... 56 Fig.4-10(c)-(d) Relative displacement of pendulum and its foundation ............... 57 Fig. 4-11. Frictional coefficient estimated from inverse analysis and averaging when /M1.25 degree (Motion 1, Case 2) ............. 59 Fig. 4-12 Frictional coefficient estimated from inverse analysis and averaging when y3=8.25 degree (Motion 2, Case 2) ............... 59 Fig. 4-13. Frictional coefficient estimated from inverse analysis and averaging when/£=13.47 degree (Motion 3, Case 2) ............ 60 Fig. 4-14. Frictional coefficient estimated from inverse analysis and averaging when /*=13.47 degree (Motion 4, Case 2) ............ 60 Fig. 4-15 (a) Forward analysis of Case 2 studies ........................................... 62 Fig. 4-15 (b,c) Forward analysis of Case 2 studies ............................................ 63 Fig. 4- 15(d) Forward analysis of Case 2 studies ........................................... 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5-1 Base-isolation test for the busing transformer of electric System (Courtesy of National Center for Research of Earthquake Engineering, Taiwan, ROC) ..................................... 70 Fig. 5-2 Flowchart o f system identification using digital image Processing ................................................................................ 72 Fig. 5-3 Digitization o f video images from videocassettes .......................... 73 Fig. 5-4 Using circular marks for tracking motion of the second floor’s and shaking table’s displacement........................................................ 76 Fig. 5-5 Displacement sensed by digital image processing ...................... 77 Fig. 5-6 Relative displacement of supported structure and Foundation o f shaking table; NCREE’s and DIP method’s results ............................................................................ 77 Fig. 6-1 Configuration of rubber membrane experiment ............................ 83 Fig. 6-2 Cross-linked network in rubber material (micro vision)................. 85 Fig. 6-3 Analysis flowchart for membrane system identification .............. 90 Fig. 6-4 Estimation o f £2* by the Mooney-Rivlin equation ........................ 94 Fig. 6-5 Flowchart o f iterative schemes of FEM analysis .......................... 95 Fig. 7-1 System configuration for membrane study .................................. 96 Fig. 7-2 Segmentation of multiple tracked points on the membrane surface ...................................................................... 98 Fig. 7-3 Fast Fourier Transform for distinguishing textures ..................... 99 Fig. 7-4 Sensed points in a binary image .................................................... 100 Fig. 7-5 Method of quarterly averaging ..................................................... 102 Fig. 7-6 Grid number and markings on the membrane surface ................. 104 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7-7 Deformation of membrane with stretching of 1.745 lb (observation data by DIP) ........................................................... 104 Fig. 7-8 Deforamtion of membrane by the stretching force of 2.74 lb ... 105 Fig. 7-9 Deforamtion of membrane by the stretching force of 3.29 lb .... 105 Fig. 7-10 Deforamtion of membrane by the stretching force of 4.29 lb ... 106 Fig. 7-11 Deforamtion of membrane by the stretching force of 4.66 lb .... 106 Fig. 7-12 Deforamtion of membrane by the stretching force of 5.13 lb .... 107 Fig. 7-13 Deforamtion of membrane by the stretching force of 6.66 lb .... 107 Fig. 7-14 Deforamtion of membrane by the stretching force of 8.50 l b 108 Fig. 7-15 Deforamtion of membrane by the stretching force of 9.31 lb 108 Fig. 7-16 Nonlinear regression of Monney-Rivlin relationship ................... 112 Fig. 7-17 FEM model and boundary conditions .......................................... 114 Fig. 7-18 Verification of the DIP and FEM results of membrane deformation with stretching force 1.74 lb (Case 1) ................... 115 Fig. 7-19 Verification of the DIP and FEM results of membrane deformation with stretching force 2.74 lb (Case 1) ................... 116 Fig. 7-20 Verification of the DIP and FEM results of membrane deformation with stretching force 3.29 lb (Case I) ................... 116 Fig. 7-21 Verification of the DIP and FEM results of membrane deformation with stretching force 4.29 lb (Case 1) ..................... 117 Fig. 7-22 Verification of the DIP and FEM results of membrane deformation with stretching force 4.74 lb (Case 1) ...................... 117 Fig. 7-23 Verification of the DIP and FEM results of membrane deformation with stretching force 5.13 lb (Case 1) .................... 118 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7-24 Verification o f the DIP and FEM results o f membrane deformation with stretching force 6.66 lb (Case 1) .................... 118 Fig. 7-25 Verification of the DIP and FEM results of membrane deformation with stretching force 8.50 lb (Case 1) ..................... 119 Fig. 7-26 Verification of the DIP and FEM results of membrane deformation with stretching force 9.31 lb (Case I) .................... 119 Fig. 7-27 Verification of the DIP and FEM results of membrane deformation with stretching force 1.74 lb (Case 2) ..................... 120 Fig. 7-28 Verification of the DIP and FEM results of membrane deformation with stretching force 2.74 lb (Case 2) ................... 120 Fig. 7-29 Verification of the DIP and FEM results of membrane deformation with stretching force 3.29 lb (Case 2) .................... 121 Fig. 7-30 Verification of the DIP and FEM results of membrane deformation with stretching force 4.29 lb (Case 2) ................... 121 Fig. 7-31 Verification of the DIP and FEM results of membrane deformation with stretching force 4.74 lb (Case 2) .................. 122 Fig. 7-32 Verification of the DIP and FEM results of membrane deformation with stretching force 5.13 lb (Case 2) .................. 122 Fig. 7-33 Verification of the DIP and FEM results of membrane deformation with stretching force 6.66 lb (Case 2) .................. 123 Fig. 7-34 Verification of the DIP and FEM results of membrane deformation with stretching force 8.50 lb (Case 2) .................... 123 Fig. 7-35 Verification of the DIP and FEM results of membrane deformation with stretching force 9.31 lb (Case 2) ..................... 124 Fig. 7-36 Verification of the DIP and FEM results of membrane deformation with stretching force 1.74 lb (Case 3) .................... 124 Fig. 7-37 Verification of the DIP and FEM results of membrane deformation with stretching force 2.74 lb (Case 3) .................. 125 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7-38 Verification o f the DIP and FEM results o f membrane deformation with stretching force 3.295 lb (Case 3) .................... 125 Fig. 7-39 Verification of the DIP and FEM results of membrane deformation with stretching force 4.29 lb (Case 3) .................. 126 Fig. 7-40 Verification of the DIP and FEM results of membrane deformation with stretching force 4.74 lb (Case 3) ...................... 126 Fig. 7-41 Verification of the DIP and FEM results o f membrane deformation with stretching force 5.135 lb (Case 3 ) ..................... 127 Fig. 7-42 Verification of the DIP and FEM results of membrane deformation with stretching force 6.66 lb (Case 3) ...................... 127 Fig. 7-43 Verification of the DIP and FEM results of membrane deformation with stretching force 8.5 lb (Case 3) ......................... 128 Fig. 7-44 Verification of the DIP and FEM results of membrane deformation with stretching force 9.3 lb (Case 3) ...................... 128 Fig. 7-45 Displacements for Cases 1,2 and 3 at the boundary where the external forces were applied ................................................... 130 Fig. 7-46 Root-mean-squares difference in nodal displacements for Cases 1,2 and 3 ....................................................................... 131 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF SYMBOLS CCD Couple Charged Device DIP Digital Image Processing FEM Finite Element Method A = Stretching A times in the crosslinked network movement a = Step length of Rubber Crosslinked Network Node in 3-D B = Shrinking B times in the crosslinked network movement P = Slant Angle between the Metal Board and the Gravitational Direction F = Helmhotz Free Energy f n = Natural Frequency of System /„ = Mean Value of Measured Natural Frequency g = Gravitational Acceleration Constant k = Boltzmann’s constant L = Cable Length of Pendulum System A . = Extension Ratio of l-D Elastomer Fiber Lx = Extension Ratio in x Direction A # y = Extension Ratio in y Direction kz = Extension Ratio in z Direction m = Mass of Pendulum Weight Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N = Total Number of Crosslinked Network Movement n Number of Chains Per Unit Area M = Frictional Coefficient M s = Sliding (Dry) Friction Coefficient P = Step distance of rubber crosslinked network node movement r = Total deformation of crosslinked network link e = Angle of Motion of Pendulum Weight s = Thermodynamic ratio of free energy on absolute temperature a = Stress T = Absolute temperature t = Time Te = Measured Period u = Entropy of rubber elastomer material W = Total ways of rubber crosslinked network movement ( O n = Natural Frequency Xc = Horizontal Displacement of Pendulum Weight X = Displacement of rubber crosslinked network node movement xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Usefulness of digital image processing techniques demonstrated and further suggests a variety of applications in many disciplines of engineering, science, and medicine. The objectives of this study are to identify and develop the underlying principle that permits nonlinear system identification with the aid of digital images sensed and developed non-intrusively and remotely. In particular, for the purpose of the proof-of-concepts, this study applies the principle to system identification problems that often arise in the field of engineering mechanics. More specifically, the following example problems are solved for demonstration of the efficacy of digital image processing techniques. First, a nonlinear oscillating pendulum system with Coulomb damping was studied for identification of friction coefficient. This is one of the most fundamental and yet difficult problems to solve. Intricate micromechanics interpretation of friction phenomena is available in the area of tribology, and highly sophisticated analysis o f friction phenomena is performed involving contact problems with the aid of continuum mechanics. These approaches are not immediately useful from the view point of the practice in structural dynamics dealing with friction issues, where Coulomb friction model is widely used for its mathematical expedience and ease of application. This study successfully implemented the digital image processing technique for the identification of values of the friction coefficient on the basis of images of pendulum motion captured by CCD camera. Second, this study xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. demonstrated that the algorithms and procedures can be extended to identify, in near real-time, the friction coefficient and the relative motion between the model structure and shaking table in a shaking table test of a structure base-isolated by a pendulum friction sliding system. Third, a digital still camera was used to observe finite in plane deformation of a thin nonlinear elastomeric membrane for the purpose of identifying its constitutive relationship. The relationship was postulated to follow the Mooney-Rivlin’s stress function model. The digital image processing schemes developed were verified with iterative nonlinear analysis making use of finite element programs that are valid in the finite deformation range. These proof-of-concept experiments did indicate significant potential of digital image processing technique to be used for the purpose of system identification in the broad field of engineering mechanics. It is to contribute this experience on discovering smart structures and design of smart systems in future task of civil engineering. This study laid a basic foundation for other applications of digital image technologies in civil engineering in relation to advanced technologies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PARTI INTORDUCTIONS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION 1.1 Motivation Digital image processing method is utilized to demonstrate its application in engineering mechanics with specific regard to the area of system identification. Using optical devices with remote sensing techniques to study engineering mechanics problems has rarely been done. It was not practical to use image methods due to their hardware and technological limitations in the past few years. With rapid progress in the development of optical technologies and digital computing capabilities, this is no longer the case. By taking these advantages, image method is implemented and introduced as key research issue to perform its proof-of-concept in doing engineering analysis such as to solve a system identification problem without assistance of traditional sensors. To investigate typical studies on system identification problems, the sensors play key roles in measurement and sensing; traditional approaches used lots of physical sensors that need to be configured in the system in order to perform system identifications. There have many kinds of sensors that are commonly used in civil engineering the in-house or in-situ studies. As usual, the sensors need to be physically attached on structural surface or imbedded in structural members. This 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. presents a major obstacle to perform system identification, particularly of complex systems, because they require cumbersome wiring for data acquisition, rooms for installment of sensors in members, and designing for their size and material in order to avoid intrusion. Quite recently, digital image processing method has been introduced and demonstrated its effectiveness in system identification remotely and non-intrusively. With the aid of digital image devices and implementation of software and hardware issues, the obstacles and inconvenience of traditional approaches is alleviated by infusing digital images processing with remotely sensing techniques. 1.2 Background Identification problems are acknowledged as inverse problems applying to many fields of engineering mechanics. The purpose of identification is to obtain the parameters, representing the characteristics of system, which the system function is in terms of. Generally speaking, inverse problems are concerned with the determination of system characteristics by interpretation from both the input and output. Mathematically, it needs a system which is represented as an unknown function that need to be determined or ruled out from the input and output been charged into this system. However, such problems are usually ill-conditioned and have to be overcome through cumbersome development of computational schemes. Objective functions that may satisfy the relationship of system response in input and 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. output are candidates of solution. The effort of experimental procedures is also needed in order to obtain the data for conducting inverse analysis. The application of using digital image processing techniques for system identification especially for the civil infrastructure systems is because of some successful evidence that some efforts having been done by electrical engineers. The first attempt using a digital camera to remotely sense a vibration system was carried out by Shinozuka at the University of Southern California, collaborating with Liu at the University of Houston, in 1998. A CCD camera was used for the purpose to capture images. This task is intended to explore the possibility of using a CCD camera to remotely monitor the structural motion. The camera is assumed fixed relative to the object to be monitored. An experimental setup in conjunction with Visual C++ source codes, to activate the camera and MATLAB software, and this were discussed, however, in an unpublished internal report. Due to the hardware limitation o f the CCD they used, the experiment did not succeed to sense a 49-Hz antenna in vibration. On this hand, the sense of physics in dealing with this experiment is not carefully considered, but a prototype o f image system is illustrated due to their in-house practicing. A prototype of the remote digital image sensing system is configured in a PC with Microsoft Window operating system. It was proof o f a great breakthrough that the digital computer succeeded in performing massive computations in dealing with the digital images, and overcome the timing demands in read-and-write phases. At that time, the Hitachi built CCD (KP160) was capable to capture as fast as 30 frames 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. per second with an optimum configuration matched with an image frame grabber (Imagenation built, PX-S10, NTSC mode) adapters. The highest sensitivity of that CCD with PX510 could achieve 786 pixels and 488 pixels (enhanced VGA mode) picture elements. Just recently, the digital image technologies were improved in storage size and floating calculating power, which make the digital computer more suitable to satisfy the computational power and memory demands of image processing. Image sensitivity gets into super VGA, true or high color mode with a digitizing-and-capturing rate promoted to be more than 1000 frames per second. A high-performance camera that can achieve much higher image sensitivity and capturing rate such as 1,000, 2,000 frames per second is affordable. In this sense, digital image technologies have matured and are ready to perform their enhanced applications for more general engineering uses. 13 Research Objectives The objectives of this research are, (1) to develop the underlying digital image processing principle that permits non-intrusive and remote system identification, with the aid of digital image devices and, (2) to apply the principle to system identification by demonstration of proof-of-concept experiments. With these respects, on the one hand, digital image processing techniques present their applications in motion analysis and remote sensing in studying a nonlinearly oscillating pendulum system. The scheme and algorithm applies to a study on the system identification of a base-isolated structure system designed for seismic 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. resistance. On the other hand, the technique is capable of managing highly nonlinear in-plane finite deformation problem o f continuum systems. For these purposes, image methods are implemented with analytical system identification algorithms. It also demonstrates that the capability to develop an automated, near real-time remote sensing system for dynamic system identification. In dynamic system identification, accelerations caused more concerns than did displacements in the system identification and control. As an alternative, this study attempts to identify relative motion problems with an on-line digital imaging system and a digital computer for the purpose of the proof-of-concept experiment. Digital imaging instruments and cameras now have better sensitivity, resolution, higher storage capacity, and computational speed. Digital image systems are inexpensive and easy to set up. They have great potentials for applications of studying nonlinear system identification problems. 1.4 Research Approach This study is to develop the principle and methodology of using digital imaging technique and apply them to solve system identification problems arisen in engineering mechanics and civil engineering. New instruments to measure displacement and velocity of structural vibration systems have been developed. Comparing with the traditional approach, the instruments were replaced by a combination of digital imaging and digital computing systems. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Without the assistance of traditional motion sensors installed in systems but with the aid of digital image devices, the motion of nonlinearly oscillating systems were measured remotely and non-intrusively by making use of digital image processing techniques, including contrast manipulation, segmentation, and near real time system analysis. The fundamentals of digital image processing techniques were employed adaptively in order to implement an appropriate imaging system in a near real time processing. The hardware support, software assistance and computer codes were necessarily integrated to achieve this goal by considering the complexity of the studied system’s characteristics and the quality of sensed images. Sensing is the key research issue in the system identification studies. Disobeying traditional sensing means, displacement of the targets in a dynamic system can be sensed by making use of an optically imaging system. With the aid of the image devices, the time history of the motion was recorded by following the time sequence in pictures. To extract the data from the pictures that display the motion, it needs the systematic analysis on each frame of pictures and efficient operation procedures in order to avoid the program down in the process of system identification. To acquire the data from images is one thing, and to analyze the data for system identification is another. Both of acquisition and analysis of data are important and interfered. This must be avoided in the sensing process that programs would be critically down if that image processing has taken all the memory space of the computer without evacuating rooms for data collection, or vice versa. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. System identification problems were also studied in the way of off-line analysis. The motion of the studied systems could be recorded in video images. The video images could be stored in media and carried over everywhere. To use video image processing to study system identification problems is also an interesting topic. For many engineering purposes, video images are the very handy and helpful tools in the practice of motion estimations. There are two major issues associated with any system-identification process. First, it needs to assume a mathematical model that could be completely determined by a finite set of parameters and the model should be able to anticipate the behavior of the system within an acceptable tolerance (Ghanem and Shinozuka, 1995). Second, it needs to identify these parameters based on the observed behavior of the system by carrying out the loop of system identification analysis of both inverse identification and forward verification. In this study, system identification procedures are comprised of image processing unit and system analysis unit. Sometimes, regression analysis would merge from system analysis unit for the needs o f further dealing with curve fitting of acquired data. In the image-processing unit, it is without saying that the major task is to apply imaging and processing technique to accomplish the measurement. In the system analysis unit, the parametric study of the parameters is performed by iterative schemes in finding the values of parameters and some of numerical analysis techniques are needed to do the nonlinear system analysis in solving the nonlinear system identification problem. For the example, Runge-Kutta 4th Method is a very 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. effective approach in solving the nonlinear differential equation of the pendulum problem and the Finite Element Analysis Programs are useful in studying nonlinear finite in-plane deformation problem of thin membrane problems. 1.5 Structure of Dissertation This dissertation is divided to four parts; namely, Introductions, Proof-of- concept Experiments, References and Appendices. In Part I, Chapter I introduces the motivation, background of technologies and research approaches. Chapter 2 lists out the functions and implementations o f digital image technique used for this study. The useful tools of image processing method were summarized by their state-of-the- art capabilities with practical applications. In Part II, three proof-of-concept experiments were studied and demonstrated by analytical and numerical investigations. In Chapters 3 and 4, a nonlinear oscillating pendulum system (one degree-of-freedom image system) was respectively discussed analytically and performed in numerical experiments. Chapter 5 infused a case study of the relative displacement between the base-isolated structure and a shaking table by processing video images. In Chapters 6 and 7, a 2-D elastomer membrane was examined its nonlinear constitutive relationships by means of digital image processing and FEM analysis. Finally, Chapter 8 concludes this study by a summary of proof-concept experiments. In Part III, References are listed. In the last, Part IV, Appendices of source codes developed are presented. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 DIGITAL IMAGE PROCESSING: TECHNIQUES AND APPLICATIONS 2.1 Introduction "All o f us have done some sort o f image analysis. In fa ct our eyes do it fo r us all o f the time. Getting the computer to evaluate an image in some sort o f similar fashion can be difficult. ” (Prestridge, 1993) It reminds us that some human being’s intuitive ability exists and capable of managing visualization and judgement of images better than computers do. In the 1960s, investigators devoted their enthusiasms for finding out the size and shape of objects’ features by implementing image analysis. Concepts o f digital image processing had been merged as to its orthodox at that time just by the motivation of the satisfactory of the “vision” effect. Digital image processing is based on the conversion of a “continuous” image field to equivalent “digital” form. At the time of image method’s emergence, there were two arguments for telling the properties of images; the for use of processing images were disturbed by two major issues, namely, are images deterministic and that can we use statistical approach to make images studies deterministic? By now, these arguments are out of concerns any more because images are deterministic in digital form, and the display o f such 2-D digital image planes can fully represent the 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. image properties originated from the continuous ones. A concept of continuous and digital image intensity level is shown in Fig. 2-1,2-2 where the images are described as continuous and discrete field on its illumination intensity domain with continuous and discontinuous integer image intensity values. The usage of digital images (Fig. 2-2) is not only for the replacement of continuous images (Fig. 2-1) but also convince a legal mean to achieve better operation performance of image methods that have been richly developed by modem digital technologies. I I Total Arm "5 § I 3 E a u 3 Image Intensity Level Fig. 2-1 Continuous image field and its representation of image intensity 1 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. II m $ * E e o 3 E 3 U a Image Intensity Level Fig. 2-2 Digital image field and its representation of image intensity In the early 1960s, digital images were appeared in typical course and used as digital planar presentation more often. They were split from the practices in signal processing, telecommunications, home video entertaining, radar technologies and astronomic surveys. With years’ improvement, these techniques were sharpened and developed for constructing modem digital image processing methods and getting matured and aggressively powerful for use in the application such as the artificial intellectual instruments and robotics. Mostly, digital image processing means all sort of operations of digital images but, more precisely, digital image operations fall into two categories; namely, image processing or image analysis. Image processing is defined as an 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. operation of turning one image into another image. For instance, one image is undergoing image processing with operation of image histogram equalization or edge sharpening, etc, to gain the better resolution or contrast of image. There is no image analysis being running feature identification until the feature is extracted with concerns by assigning the targets and focus on parts of the image. The image method in a digital image processing approach is focusing primarily on digital image that represents continuous natural image. A general image method concept includes all kinds of image processing such as analogue and digital transformation, image producing and reconstruction, and advanced processing such as image enhancement, histogram equalization, adjustment of contrast and illumination intensity, and geometrical correction, etc. Breaking through the analogue to digital and digital to analogue processing, digital image processing method is presenting its convenience of processing by digital computers. Image methods are implemented aggressively by current researchers’ efforts and becoming a popular tools in some advanced technologies like robotics, astronomic discovery, medical assistance, artificial intellects, and remote sensing, etc. Application of image methods in system identification has not been aggressively introduced. There were not enough evidences to show their capability to engineering application and they are ease of application to solve the obstacles from system complexity and old-fashioned sensing techniques. Image methods have been showing their ease of application in motion analysis and system identification 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by using their fundamental and extensive principles abstracted from digital image processing techniques. The power of vision encourages implementations of image methods for investigating 2D or 3D objects, although they could be presented only in a way of 2- D image. With a certain mathematical computation and mostly non-sophisticated algorithms, 2-D images can be a profile presentation of 3D objects by compiling characteristics and phases of visible information. The image technologies are very useful and immediately applicable to study scientific and engineering problems. 2.2 Fundamentals of Digital Image Processing Digital image processing includes image digitization, enhancement, coding, restoration & reconstruction, edge detection, segmentation, registration and geometrical modification, motion estimation, correlation analysis, texture analysis, nonlinear filtering, compiling and archiving, etc. Digitization is a common function of digital image devices that they can automatically make image acquisition in digital forms. Enhancement of images is comprised of more areas such as contrast manipulation, noise removal, and histogram adjustment, etc. They are the basic techniques necessary for reconditioning images into a satisfactory state. The others like edge detection, segmentation, motion estimation, etc., are mostly used for specific purposes and practical applications. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Utilization of digital image processing here is emphasized in the following function with its purposes: (1) Before performing engineering application, for example of system identification, it needs to obtain the best image condition in contrast and luminance of image pixels. Image enhancement techniques are so that infused. (2) To acquire motion picture and extract the object form sensed images, segmentation techniques are used to carry out this goal. (3) The information sensed by image methods is in relation to the physical quantities of problem. Coordinates of objects are necessary to be registered in time and spatial domains. Therefore, image registration technique involves the correction of false geometrical positions or distortion and the assignment of coordinates for studied object in images. (4) Some texture recognition technique is needed to assist in segmentation processes. Due to the complexity of image luminance intensity distribution, the characteristics of image map are possible to be ruled out by division of textures. To recognize the difference of textures is advantageous for undergoing segmentation processing. (5) Images are laid out frame by frame in digital computer. They are necessary to correlate to other images frame by frame in order to distinguish the displacement of motion objects. Correlation analysis on different image frames is therefore needed. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is quite important and critical to execute the image processing before performing system analysis on this subject. The procedures are to minimize the noises encountered in the sensing and insure the result of further identification processing. They are abstracted in the following sections. 2.2.1 Enhancement of digital images This procedure is especially tedious but important before performing system identification. An image been taken from a camera and through the work of A/D digitization of hardware application needs calibration on its color and luminance between image pixels by invoking three major image operations, contrast manipulation, histogram modification and noise cleaning. Poor contrast, very common defect of images, happened when the pictures are taken in poor illuminating environment or missing the focus. Usually, manipulation of contrast is done by adjustment of image luminance. The feature of contrast of an image can be sensed by its image histogram representation. For example, the numbers of some intensity values denser around a range of luminance and looser around some others would not be with good contrast. The schemes of contrast manipulation are to redistribute image histogram as equally as possible around overall luminance domain. Mathematically speaking, manipulation of intensity can be carried out by using one-on-one transformation method just depending on the determined transfer function. There are many candidates of transfer function capable being used but, however, none of them is superior to any others at all times. Fig. 2-3 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and Fig. 2-4 show images that before and after the image intensity adjustment by using the rubber band transfer function and histogram equalization method. The others like square root, cube root or Gaussian error transfer functions are all with their characteristics and they are without especially meaningful contribution on the contrast manipulation in practice. Rubber Band method is quite often used and convenient for manipulation in programming source codes. As the definition of histogram equalization, this method is to attempt to distribute the histogram equally by means of using the accumulative function of original histogram for transfer function. In the proof-of-concept experiments, the digital images are reconditioned by simply using rubber band method primarily, and histogram equalization as an alternative one in case of complicate image combination like the experiment of shaking table tests. Image noises means that the unexpected discontinuity of intensity of neighboring pixels or unordinary black outs of pixels on the image plane. The phenomenon image noise might happen due to the defects of image system or damage of sensors of the camera. In image processing, the hit-or-miss mask operation (Fig. 2-5) is quite straightforward and efficient for the reduction of unexpected blackouts or clusters caused by the defects from hardware problems. The others like low pass filters are also valid to reduce the white noise and for purposes such as smoothing the edges, etc. However, hit-or-miss mask operation method is simple to program and adaptive for many purposes of image processing. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Image (b)bnage wo iso ( c) Histogram n m 1 * : i : 1 0 0 0 6 0 0 l i f 6 0 0 I1 ■ i C O 1 ' 2 0 0 li 0 pgr 100 150 200 (d) Histogram i 09 0 8 07 06 05 04 03 0 2 0 1 0 01 02 03 0 4 05 06 0 7 08 09 (e) Transfer function ■ 09 08 • 07 06 ■ 05 04 / 03 02 01 0 Qt 0 2 03 04 OS 06 07 Q8 09 (0 Transfer function Before adjustment After adjustment Fig. 2-3 Rubber band method for image intensity adjustment (The image is digitized from the videotape by the Courtesy of National Center for Research of Earthquake Engineering, Taipei, Taiwan, ROC) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 04 (a) Inage (b) Image C O O , 3500 3000 2500 Tim ! tsoo taoo a n i too IS O ( c ) Histogram 3 ) Histogram ai 02 aa 04 as as or as 09 (e) Transfer function 09 0 6 06 04 03 02 09 (0 Transfer function Before adjustment After adjustment Fig. 2-4 Histogram equalization method for image intensity adjustment 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. No Yes Match window patterns? Miss Hit No action Execute operation Image window scanning Fig. 2-5 Hit-or-miss mask operation 2.2.2 Acquisition of motion pictures and Segmentation It needs the function for motion analysis to digitize the motion video or continuous signal sensed from image system into pictures. The pictures recorded the motion in discrete 2-D image maps representing the states of different time. They need to be stored and processed in digital computers. Generally speaking, acquisition of motion pictures from image system is to transform the motion video, a continuous signal field in the time and spatial domain, into pictures that each picture indicates the state that was sensed at the different time. There are three stages for pursuing motion analysis using image processing shown in Fig. 2-6. The first stage is to sample continuous signal into discrete and digital image format. The second 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. one is to segment the information from digital images. The third is to register coordinates of image attributes and determine the physical quantities. In Fig. 2-6, Fc (x,y,t) represents continuous image signals from motion sensing to be digitized. The C/D is a converter to digitize continuous images to digital ones indicated by Fd (ijjc). Where x and y are the spatial domains in a continuous image field and, i and j represent discrete integer spatial domain in a digital image held. The symbol t is time and, k is the sampled time index numbered in integer. r ----------------------------------- Stage I. Acquisition of digital images Fc (x, y, t)----- C/D Fd (i, j, k) Stage II. Segmentation of motion objects Fd(i, j, k)_ Segmentation G b(i,j,k) i --------------------------- Stage m . Identification of physical attributes c b(M ,ia+ Registration Coordinates Fig. 2-6 Three stages for motion analysis 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Stage n , digital images are processed by segmentation. The motion attributes on the images are extracted and built into binary images, Gb(ij,k). In the last stage, the motion attributes of binary images are interpreted under registration processing. For example, the targets of motion objects are positioned with registered physical coordinates. Segmentation is to separate image into several regions with similar attributes to extract interested objects from an entire image map. The major issue of segmentation operation hereby is to extract the singular or multiple objects from where the image they were lying and their positions and displacements can be identified their corresponding pixel coordinates. Most of the time, segmentation processes have to be adaptive to the situation encountered in the image analysis. It can be a very simple or difficult way to carry out segmentation. The simplest method for the extraction and segmentation is to use the intensity segmentation by the analysis o f image histogram and intensity distribution. By this method, the attributes could be extracted from the difference of luminance, image intensity, of image pixels. For example, the luminance of a monotonic image such as black-and-white image could be classified into 256 gray levels. By the assignment of range of luminance values, the image could be separated into two intensities, namely, 0 and 1, which if the luminance value is within this range the intensity will change into 1 and the other way that the intensity is out of this range will be changed into 0 (Fig. 2-7). For most of the optical image 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. processing, the image feature is simply determined by its intensity properties, therefore, the segmentation using intensity segmentation is quite useful and convenient to the practical purposes. Some other methods like detection of image edges, and contouring of image boundaries, transformation method, etc, are seen in the digital image processing texts and they are not superior in solving the segmentation issues but more sophisticated most of the time than intensity segmentation, especially, for our purpose of doing motion analysis and system identification. However, for some of the pattern identification and texture recognition issues they did show their superiority in supporting the solutions on those subjects. 2.2.3 Image registration and geometrical modification Image registration has to be discussed with geometrical modification in motion analysis. It was originally defined as a mapping between the image and map coordinates of topological features. The task of registration is to map the attributes such as the moving objects with shape and size, into a physical sense. This function is frequently used in geographical information science and remote sensing. In this study, registration technique is adopted for the purpose of identifying coordinates, size and boundary of objects. It assists to position coordinates and to evaluate displacements, velocities and accelerations of moving objects. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1500 1000 500 0 50 T O O 150 200 250 (a) Original image (b) Histogram (c ) Binary image (d) Coordinates at object center Fig. 2-7 Segmentation by using image intensity Image registration has two major issues; namely, to recover the distorted dimension of picture frames and to correctly position moving objects in each picture frame. To relief the issues with solution, the image warping technique is possible to carry out. Once the image is in correctly coordinated framework, the positions of targeted objects could be identified at ease. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Polynomial warping technique (Fig. 2-8), sometimes, it is called rubber sheet stretching, is a handy tool for carrying out this geometrical modification (Pratt, 1994). The algorithm of spatial warping processing is basically a one-to-one coordinate transformation. ^ One-to-one coordinate transformation X Acknowledged points with coordinates Fig. 2-8 Polynomial warping for geometric modification Images are distorted due to the reflection and projection processes while they are constructed through optical lens and imaging instruments. The most common distortion of images is such as an object with its geometric boundary changed from its original shape. With acknowledgement of some pixels in relation to their correct coordinates, these techniques are effective to resolve the distortion and registration 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. issue. Pictures from imaging devices are also possibly deformed and skewed or distorted due to the angle between the face of image plane and camera shooting line. Warping technique is capable of managing geometric modification to restore image errors in geometry. 2.2.4 Correlation analysis and recognition of difference Optical measurement in experimental mechanics employed correlation techniques more often than segmentations. The goal of using correlation and segmentation techniques is to study those physical quantities embedded in a picture that correlates to the others taken at the identical phenomenon but at different times. Image segmentation is to extract the object from whole picture and present a local interest on a 2-D plane. Image correlation is to review pictures in time sequential and compare each of them to the others in order to identify the difference and changes between two or among a set of pictures. The algorithm of correlation analysis is illustrated in Fig. 2-9. The quantification o f correlation analysis of digital images is to compute the correlation coefficient, R, by formula of p . - t q . - c (A(i + p,j+q)-nA X B(k + p,l + q )-iiB ) n2 crA c rB 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Image A Image B R >Threshold Finish Full Scanning Stop No Marks Marks R = C O R R 2 D (A if Fig. 2-9 Correlation analysis algorithms where A(i, j) and B(i, j) (i, j =l,2,...n) mean two selected windows, from two different pictures, brought into correlation analysis. Window size is n by n pixels (n is an odd number). The t value is (n-l)/2. /Ia and fig are the average value of 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. brightness over the window cells of window A and B. a A and < tb are the image brightness standard deviation of window A and B. A threshold, namely, T, is selected for the criteria condition to mark on those positions with significant changes, that is the value of R is greater than T. 23 Applications of Image Processing A number of evidences showed that digital image processing techniques have been developed and have widespread applications in many disciplines. For instances: some of surgery had been operated by medical doctors remotely through thousand miles faraway by high-speed telecommunication services with the aid of digital image techniques (Computerworld news, on September 26, 2001); transportation engineers used the technologies in the assessment of damage of highway and surface distress in-situ surveys (Gian et al.. 1993, Lee, 1993, and Kelvin and Wang, 2000); the sewer-system engineers used video imaging to locate and monitor the damage of underground pipes (Makar, 1999); weather prediction officials and meteorological scientists implement these techniques in detecting and tracking the area of storms and in estimating runoffs (Kreseski, et al., 1994; Draper and Rao, 1986); and more examples can be listed. In civil engineering, there were found many image processing applications introduced in traditional civil engineering areas like materials science, transportation engineering, structural mechanics, construction investigation and management, and 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. others (Lee and Chou, 1994). These applications indicated the capabilities of digital image techniques are useful in solving traditional engineering problems in material characterization (Prestridge, 1993), vehicle detection (Chou and Sethi, 1993), highway crack distress detection (Kelvin and Wang, 1999) and geotechnical engineering (Lee and Chou, 1993). There were a common place of these applications that implemented image processing methods present usefulness to estimate the size, shape and deformation from in-situ investigations. However, there was not significant amount of research efforts in relation to the topic of system identification solved by making use of digital image processing techniques. To circulate those tasks accomplished and developed by electrical engineers and computer scientists in digital image processing conferences (International Conferences on Image Processing: ICIP-1994; ICIP-1998 and ICIP-1999), it would be found that their interests were specifically focused on the subjects of compression, textures identification, recognition and segmentation, etc. Just recently, digital image processing had been proposed to system identification, which is an area that had not been brought up enough concerns (Shinozuka, et al., 2000). 2.4 Application to System Identification This study challenges to solve the nonlinear and dynamic problems in system identification by making use of digital image processing technique. The studies of system identification are referred to inverse problems in the field of engineering 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mechanics. General speaking, systems, in the classification of motions, are divided into two types, the static and dynamic. They can also be categorized into linear and nonlinear systems depending on their material properties and constructed structures. Image processing in the dynamic system identification problems is to deal with motion analysis concerned the targeted objects on an image map by studying their temporal correlation. Applications of motion analysis by using digital image processing could be classified into the following subjects: (1) calibration of intensity and contrast, (2) Extraction of tracked target(s) from the background where the targefs) were located, (3) measurement of the displacement of target(s) based on the extraction result, (4) correlation analysis on both spatial and temporal domains, and (5) evaluation of the physical quantities such as velocity, acceleration and their orientations, etc. These subjects are still in a lack of systematic organization in implementation of concepts and methodologies. A proof-of-concept experiment of using digital image processing method was demonstrated in its successful performance on identification of a nonlinearly oscillating pendulum system identification problems (Shinozuka, et al., 2000; Shinozuka, et al., 2001). The studies indicated the hardware and software implementations and system identification procedures. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.S Summary Digital image processing can be summarized into the following outlines with the stated applications: (1) Image enhancement: A general, fundamental and all-purpose function in digital image processing. This technique is used when the images were poorly contrasted or images were bothered by noises. It is a general tool and can be used in many occasions. (2) Correlation analysis: The correlation among two or more images to distinguish the difference between pixels or textures. This is used in remote sensing, medical diagnosis, in land management planning, and rehabilitation. (3) Image segmentation: The most complicated application of digital image studies. It is the use of segmentation techniques to separate useful from non-useful subjects on an image map. The important information is extracted for further analysis or other purposes. This processing is widely used in remote sensing, object searching, targeting and tracking and is especially important in system identification applications. (4) Motion analysis: A combination of segmentation and texture recognition techniques. It is used in general motion problems to identify positions, and estimate velocities and accelerations based on images. It also studies the frequency of appearance in the sensed environment. This process is 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. used in surveillance, monitoring, and system identification, especially of dynamic systems. (5) Image compression: A coding and decoding technique. Commonly used in information and communication technologies. This is a key research issue in enhancing the capabilities and performance of digital imaging technologies. However, it is only listed for information here, as it is beyond the scope of this research and application. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PART II PROOF-OF-CONCEPT EXPERIMENTS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 PENDULUM SYSTEM WITH COULOMB DAMPING 3.1 Introduction The image system used for this proof-of-concept experiment is based on the hardware introduced by Shinozuka, in which a CCD camera and digital image grabber system were conceived and utilized as a major source of data sensor that is opposite to the traditional motion sensing means. There are many effective traditional motion sensors that can achieve high precision performance such as strain gauges, accelerometers and displacement sensors. Most of these traditional motion sensors, however, are intrusive, at least to the extent that they have to be physically attached to the object's surface and often require extensive wiring for data acquisition. Depending on the circumstances, these requirements make it extremely difficult to pursue the purpose of system identification. In this study, the problem of identification of the frictional coefficient involved in a single-degree-of-ffeedom nonlinear vibration system with frictional damping is revisited to show that digital image processing techniques are, in principle, effective in the identification of nonlinear systems by means o f an inverse analysis. Identification of the friction coefficient is carried out with the aid of the inverse analysis to determine the friction force. Thus, the friction coefficient 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. involved in the governing nonlinear differential equation of the pendulum under vibration. Using the friction force determined from the inverse analysis, the forward analysis computes, by numerical integration of the nonlinear differential equation, the angular displacement to be compared with the observed data. The process is repeated with readjustment of the value of the friction coefficient in each iteration stage, until a satisfactory agreement is achieved between the observed displacement and the forward analysis result. It appears that this is the first attempt in which the dynamic friction coefficient is estimated with the aid of a non-intrusive method. Moreover, experiments are also carried out under the condition that the slant plane of a solid board, on which the weight of the pendulum slides, is moved back and forth (horizontal in its original plane) with no other forces applied externally to the weight. This condition simulates a building isolated by a sliding base-isolator. Algorithms can be developed to identify the friction coefficient and the motion of the plate under this condition. Such algorithms are extremely useful for identification of sliding behavior of building isolated by sliding base-isolators. 3.2 Implementation of DIP Method This research is begun with an idea to employ some developed technologies, for example, the digital image technologies, which have practical techniques for advanced applications such as system identification of civil infrastructure systems 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without the help o f the traditional sensors. Digital image processing (DIP) techniques are widely used in many areas such as astronomic monitoring, medical diagnosis, product quality control and weather forecast. The techniques have been applied not only to the industries but also in a way of enriching the lifestyle of the general public through their products such as those developed in communications and audio/video appliances. Because of the rapid development and improvement in current computational and digital technologies, applications of imaging methods for general mechanics purposes is no longer deterred by the difficulties associated with data storage capacity and computational speed. The sensing with digital image processing method extracts image data through digital image systems. An image system acquires an image in a digital format and then transfers it to a digital computer for processing and storage. Image processing begins with capturing of an image and ends with obtaining data from the digital image processed by several steps. These steps may include digitization, noise filtering and feature identification. An image system, developed by Shinozuka and Liu, consisting of a CCD camera, automatic focusing lens and frame grabber operating at the rate o f 30 fps (frames per second) integrated with a personal computer, successfully demonstrated potential applications of remote sensing technology for structural vibration problems as shown in Fig. 3-1. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vibration system Digital computer P I U Imaging System Fig. 3-1. Remote sensing of a vibration system using the image processing method The configuration of the digital image processing device can be divided into three units shown in Fig. 3-2; namely, the imaging unit, storing unit, and processing unit. The digital image processing method consists of three major procedures. In the first procedure, the imaging unit digitizes the optical signal and outputs digital image of 256-color bitmap. Simultaneously, the second procedure starts activating the digital computer to store the digital image signals into a storage unit such as a memory disk by operating codes. The third procedure is then implemented to process the identification of image features using image processing algorithms coded in C++ computer language and MATLAB computer codes. The details of source codes are shown in Appendix I. The size of the picture frame is carefully chosen for the necessary resolution and consideration of computer storage capacity. The processing unit is combined with hardware and software. For the hardware used for this study, personal computer with a Pentium 166-MMX processor is the minimum 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. requirement and the random access memory must be larger than 80 Mbytes. For operating platform, a Window 95 or NT 4.0 is recommended. The interface codes are programmed using C language and complemented with MATLAB software to carry out the image processing. The processing unit used includes a band-pass filter with adjustable bandwidth to transform the digital image into binary form and subsequently compute the x-y coordinates of points of interest. CCD camera Frame grabber Memory disk RAM No Source codes in C++ Ianguage, Matlab with ♦ image processing toolbox Finished? Yes Data acquisition Process unit Imaging unit Storage unit Fig 3-2. Configuration of digital image system The methodology of identification used in this experiment could be divided into three major procedures; image processing, data analysis and regression analysis. The image processing procedure is mentioned earlier in this paper and illustrated in Fig. 3-1. The picture of the motion is digitized through CCD and an image grabber 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. device and then recorded in the random access memory (RAM) drive virtually configured in the Micro Soft Window system. The deployment of the RAM drive in digital image processing is necessitated by the fact that current hardware and software performance is critically limited in real time processing capability unless the RAM drive is used to substantially reduce the processing time from image capturing to image storage. The stored sequential images will then be processed through bandpass filtering to obtain the displacement time history data. A flowchart depicting the overall procedure of this experiment is shown in Fig. 3-3. Image processing Image grabbing Regressive analysis Image Estimation of friction coefficient (ft) System analysis Noise filtering Inverse analysis Observed data Verification Stop Forward analysis Fig. 3-3. Flow chart of identification procedures 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 A Pendulum System with Coulomb Friction Proof-of-concept experiments were originally implemented by Shinozuka et al. and demonstrated that some types of nonlinear system identification problems can be solved by means of digital imaging techniques with relative ease. A pendulum system, simple but not necessarily easy to deal with, is used as a nonlinear system for identification. The physical quantity to identify is the Coulomb friction coefficient (//) between the weight of the pendulum and the slant board on which the weight slides. The experiments are performed under two different conditions (Fig. 3-4); one is the case where the board is stationary and the other is where the board is allowed to move. The board maintains an angle ( 3 with its vertical support whether it is stationary or in motion. A weight of metal, with mass m, is hung from a fixed point on the board and can be set into either (1) a free vibration mode under gravitational acceleration (g) when the pendulum is energized with a potential energy provided by the initial displacement, or (2) a forced vibration modes, due to the motion of the board (horizontal and in the initial plane of the board) and the gravitational acceleration continuously acting on the weight. First, the free vibration on the stationary board is considered. In this case, the weight will start to move if the static friction between the weight and board is overcome by the component of the gravitational force acting on the mass in the tangential direction to the circular trace of the motion o f the mass. A CCD camera seating in front of the pendulum system will capture the instantaneous image of 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. motion sequentially at an acquiring rate of 30 fps. The capturing process continues until the weight stops, because the force component resulted from the action of gravity on the mass is exceeded by the friction force. The final position of the weight is not necessarily at the lowest position it could geometrically occupy. Through digital image processing, the sequential image of motion can be transformed into digital data of displacement records. The center of the weight is targeted by a colored mark to be distinguished and highlighted from the background of the picture. Image processing is performed to digitally lay out the reference coordinates of the target with its high illumination (or brightness) intensity by applying a bandpass filter. By continuously processing each digital sequential image, the displacement time history of the weight is determined. In image processing, the displacement of a targeted object can be sensed in real time. Displacement data may then be used to calculate its derivatives to obtain the velocities and accelerations in a time history format. The physical quantity may be fully observed or obtained in each time instantly, through substitution of the equation of motion. Therefore, the friction coefficients can be easily obtained. The equation of motion when the board is stationary (Case 1) is + mgL cos P sin 0 = -m gfi sin P (3-1) and when the board is moving along the x direction (Case 2) is — —■ L cos Q+ mgL cos P sin 6 = -m gfi sin p d r (3-2) 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Case 1- Backboard is stationary mgsin/? mg cos/? mg. (b) Case 2- Backboard is moving 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 Fig. 3-4 Experimental set-up of pendulum system (front/side view) For use in two different cases For the definition of the variables and parameters in Eqs. 3-1 and 3-2, the reader is referred to Fig. 3-4. The second term of the left hand side of Eq. 3-2 represents the effect of the motion of the board. Also, the term at the right hand side of both Eqs. 3-1 and 3-2 indicates the frictional resistance, which always acts in an inverse direction to motion of the weight. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The natural frequency of the pendulum moving on the board is c o H = lg ? os & when the angle 0 is very small. However, if p is very small, or if the plane of the board is very close to the vertical plane, then cos/?*l and the natural frequency is image system's fps capability. If the fps is much larger than/,, say fps-/, > 10, then the sequential images are taken frequently enough to accurately trace the motion of the weight in the time domain. The identification of frictional coefficient is practically useful for the purposes o f investigating the performance of base-isolator in civil infrastructure systems used for seismic resistance. The pendulum experiment carried out in-house is tentative to proceed the analysis of combination of image processing and mechanical problem in its development of algorithms and sensing technologies. There is a significant meaning that this in-lab proof-of-concept experiment gives a very insightful idea for system identification of system such as base-isolated foundation installed in the civil infrastructure systems. The natural frequency/, should be compared with the 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 NUMERICAL EXPERIMENTS; CASE 1 AND 2 OF A PENDULUM SYSTEM 4.1 Numerical Experiments for Case 1 A metal disk, weighing 0.13 kg, was used as a weight of pendulum system. The procedures of images taking and processing for identifying the metal central displacement while it is moving are discussed in Chapter 2, the section of image segmentation and Chapter 3 for camera, digital image devices configuration and computer language implementations. 4.1.1 Observation of pendulum response The CCD camera stands in front of the metal board where the pendulum weight is sitting and captures the images for segmentation of the position of the weight’s image from the whole picture. The method of segmentation using image intensity is used. In this experiment, the pendulum weight is painted as dark luminance (black) that is easily been separated from the lighter colored background (gray). By the manipulation of the gain of low brightness or intensity of luminance less than about 30 over 256 (from 0 to 255, 256 gray scales), the shape of the pendulum weight is clearly separated without too much disturbance on of the other 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. areas within the entire picture. The processing is programmed by MATLAB and called by main computer source codes written by C++ language. The central displacement of the pendulum weight is recorded as data in a file with ASCII (*.TXT) format which is most compatible with many other software and easy to be programmed for purposes like on-line off-line analyses. This time, the digital data is used for off-line analysis. The weight is pulled up to the position which is higher than its gravity equilibrium position, and with a gain of potential energy due to gravity, then release the weight at that point and the weight moves down and the momentum increases. However, the friction is applied all the times while the pendulum is moving, and the momentum decreases due to the energy loss because of the work done by friction. Based on the observation, the weight starts to move, when the initial displacement is such that the inertia force (mgcosfi sin#) overcomes the static frictional resistance (jistngsmfJ). With a consideration of comparison of two different cases in identifying frictional coefficients when using different slant angles, = 0.125 and 0.235 radians (Case 1-A; 7.73 and Case l-B; 13.47 degrees) are used. The observations of Case 1- A&B studies using two different slant angles are shown in Figs. 4-1 and 4-2. The result is also summarized in Table 3-1. The length of wire of pendulum is fixed to 0.365m, thus the natural frequencies can be estimated as 0.82 Hz (7= 1.22 sec) and 0.81 Hz (T=l.23 sec) in the slant angles 0.125 and 0.235 radians, respectively. By the plots of two straight lines approaching to to the peak values of response curves, it 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shows that those two straight lines act as a displacement response enveloped in both cases, which is a feature unique to the observation of Coulomb damping. | 40 S > 20 Fig. 4-1. Displacement observation data from DIP when >3=7.73 degree (Case 1) 40 5 20 - 2 0 -4 0 - e o Fig. 4-2. Displacement observation data from DIP when ^=13.47 degree (Case 1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4-1 Results o f Case 1 Numerical Experiments Case 1 Numerical Experiments Case 1-A 1-B P 7.73° 13.47° 0 0 54.80° 52.16° L(m) 0.365 0.365 f(H z) 0.82 0.81 T (sec) 1.22 1.23 0.24 0.24 4.1.2 Inverse analysis for friction coefficients identification The frictional coefficients are obtained by carrying out inverse analysis based on Eq. 3-1. First, the data acquired from digital image processing was recorded as discrete temporal displacement in angular coordinate (1-D, function of time). The displacement is retrieved and put into filtering processing by applying low pass filter to screen out the high frequency noise due to image processing. The filtering operation can maintain the continuities of the first derivative and second derivatives of displacement curves found in digital image processing acquisition. By applying the finite difference scheme, the first and second derivatives of displacement are estimated approximately but closely to the theoretical value and the frictional coefficients at each instant moment are computed accordingly. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To sketch the friction coefficients through the entire time history, the temporal fluctuation of the frictional coefficients is plotted in Figs. 4-3 (/?=7.73 deg) and 4-4 ifi = 13.47 deg) together with the time history of velocity to show that the velocity does not have a significant relationship with the frictional coefficient. However, in Fig. 4-3 and 4-4, it still show that the friction coefficient has an approximate relationship with the velocity in such a way that, when the velocity reaches to a certain value, the friction can be taken as a constant, and when the velocity is lower than that certain value, the friction coefficient is proportional to the velocity. * $ I 5 3 4 (i from Inverse Analysis 3 — n from Averaging . 0.5 2 . 0.25 1 0 05 1.5 1 -2 -3 Time (sec) o e o S o Fig. 4-3. Frictional coefficient estimated from inverse analysis and averaging when >9=7.73 degree (Case 1) 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. li from Inverse Analysis x Vei — li from Averaging I S r 0.5 0.25 • w s S , $ 1.0 3.0 Time (sec) Fig. 4-4. Frictional coefficient estimated from inverse analysis and averaging when >9=13.47 degree (Case 1) The fluctuation is believed to be that because of the stick-and-slip effect (Fig 4-5). The stick-and-slip phenomenon is conceptually understood arising from the relative motion of two surfaces in contact with the friction forces under pressure (Fig. 4-6). In the process of stick, two surfaces are interlocked by the micromechanical roughness of both surfaces and it creates more resistance to move. And, in the process of slip, the extent of surface interlocking is released and become weakened presumably by the sustained application of the opposite forces to induce relative motion. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Steady Sliding (b) Periodic stick-sllp motion Fig. 4-5 Modes of sliding frictions ( c ) Chaotic motion Stick Slip Moving direction Stick on surfaces A & B Slip on surfaces A & B Fig. 4-6 Concepts of stick and slip phenomenon 4.1.3 Forward analysis verification of pendulum displacement In this section, the friction coefficient identified by digital image processing is substituted into the equation of motion to verify the vibration displacement of the metal weight in the pendulum system. In forward analysis, the displacement is the time function which is the solution of the differential equation of motion (Eq. 3-1). 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. While the frictional coefficient is identified through the inverse analysis proposed previously, the equation of motion becomes solving problem of a second ordinary differential equation. However, equation of pendulum system is a nonlinear equation without handily method to get the displacement results. Therefore, numerical method is preferred in carrying out the solving problem of this nonlinear differential equation. In many engineering applications, a constant friction coefficient is used and substituted into the calculation of friction forces. For the purpose of verifying if this is a reasonable approximation; (a) regression analyses are performed to estimate the mean frictional coefficient which turn out to be J i =0.24; (b) the mean value obtained from regressive analysis is used in forward analysis, the results of inverse and forward analysis from (a) and (b) are put into the verification with time-varying // value, to complete system identification’s analysis loop (see Fig. 3-3, flowchart of system identification), including the inverse and forward investigations. The comparison shows that the time-varying frictional coefficient would achieve better accuracy, but the approximation based on J i is quite reasonable. The verification is performed by applying the Runge-Kutta 4M method to obtain 0 solved by applying Eq. 3-1 in Chapter 3. The figures verifying the displacement acquired by the image processing and result from forward analysis are shown in Figs. 4-7 and 4-8. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.4 0.5 2.5 3.5 beta=7.73(deg) -0.4 Verified by Forward Analysis -0.8 Time (sac) Fig. 4-7 Verification by forward analysis for P=7.73° ? * E a a ■ o c £ * 5 E o « a m 0.8 0.6 0.4 0 1.5 3.5 -0.4 beta=13.47(deg) Verified by Forward Analysis__________ •0.6 •0.8 -1 Time (sec) Fig. 4-8 Verification by forward analysis for P= 13.47° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 4.2 Numerical experiment for Case 2 Case 2 studies are experiments simulating the ground motion and identifying the Coulomb friction coefficients while the pendulum system is under the combined vibration conditions of supported system and foundation. The type of friction studied in this case is sliding friction that is not exactly the same as in ordinary base- isolator devices. However, with practical point of views, this study can illustrate the simplified version of base-isolation studies. In Case 2, the support of the pendulum, which is the backboard in this case, is moved back and forth along the horizontal axis (x) to simulate the ground motion. Fig. 4-9 (a)-(d) represents four examples of moving metal board for Case 2 studies. Base-isolation is widely discussed for purpose of seismic damage reduction of buildings and other structural systems. The pendulum experiment with moving support does not provide an exact simulation of this base-isolator model. However, Case 2 experiments provide a very insightful idea to demonstrate ways in which digital image processing method may be introduced to solve problems arising from, as an example, sliding base-isolators. In this case, the friction force related to relative motion between the structure and the base that moves with the ground in a complex manner. However, the relative-motion-model for sliding base-isolation is conceptually the same as the pendulum structures considered here. Thus, the proposed digital image system addresses the same sensing issues that the base- isolation devices present in principle. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The result of relative displacement between metal board and the pendulum weight in the proof-of-concept experiments is shown in Fig. 4-10 (a)-(d) with different motion of the board. (a) Moti on -1 0.1 E, 0.05 O -0.05 - 0.1 T im a(M C ) ( b) Moti on • 2 0.1 0.05 -0.05 -0.1 Fig. 4-9 (a)-(b) Displacement o f metal board observed by DIP 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c ) Motion - 3 0 .1 E 0.05 c 1 • o m a « a -0.05 - 0.1 i j (d) Motion-4 0.1 E | 0.05 • o a a » 5 -0.05 - 0.1 Tim * (m c) Fig. 4-9 (c)-(d) Displacement of metal board observed by DIP Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Motion-1 0 .6 0.4 0.0 -0.4 - 0.6 (b) Motion-2 0.6 0.4 2- -0.2 -0.4 -0.6 Fig. 4-10 (a)-(b) Relative displacement of pendulum and its foundation 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 .6 0.4 -0.4 - 0.6 (d) M o tio n .4 0.6 0.4 S -0.2 -0.4 - 0.6 Tim * (m c ) Fig. 4-10 (c)-(d) Relative displacement o f pendulum and its foundation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4-9 shows four types of board motion lasting 5.5 seconds. Motion 1, as in Fig. 4-9(a), is moving slower than motions 2, 3 and 4. Figs. 4-11— 14 show the results of regression of frictional coefficients. In motion 1, the friction force shows much similarity to that obtained in the Case 1 study. However, its values tend to be greater than those in Case 1. Motions 2 and 3 acts at a much higher speed and the resulting friction forces were greater but still averaged less than the value estimated in Motion 1. Their peak friction values tended to be larger than in the case of the board being stationary. This occurrence is an indication that shaking the base increases the resistance to motion, not only by the effect of sliding friction but also because the friction exceeds the simple product of weight times the reasonable frictional coefficient. The corresponding damping effect of the system may occur to affect the accuracy and precision of the friction forces. The results from the inverse analysis by the MS-Excel spreadsheet operations are shown in Table 4-2. Observation on Figs. 4-11-4-14 and Table 4-2, it is quite apparent that the determination of the friction coefficients n for Case 2, using a regression analysis, requires more care than exercised for Case 1 . This is primarily because of the higher frequency content involved in the board motion and the resulting complexity in which the stick-slip process proceeds (Fig. 4-5). 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Motion -1 1.0 0.9 0.8 0.7 0.6 3.0.5 0.4 0.3 0.2 0.1 0.0 Observed mu — Regressive Mu Averaged - 0.255 0.5 1.5 2.5 3.5 4.5 5.5 Time (sec) Fig. 4-11. Frictional coefficient estimated from inverse analysis and averaging when >9=10.25 degree (Motion 1, Case 2) Motion - 2 0.9 0.8 0.7 0.6 = L 0.5 0.4 0.3 0.2 0.1 Observed mu regressive mu Average |i = 0.22 0.5 1.5 2.5 3.5 4.5 5.5 Tim e(sec) Fig. 4-12. Frictional coefficient estimated from inverse analysis and averaging when 0=1 1.05 degree (Motion 2, Case 2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Motion - 3 0.5 -•-observed mu — regressive mu average *i=0.20 0.6 a. 0.5 1.5 4.5 2.5 3.5 Time (sec) Frictional coefficient estimated from inverse analysis 5.5 Fig. 4-13. and averaging when >9=9.94 degree (Motion 3, Case 2) Motion - 4 1 0.9 0.8 0.7 0.6 a. 0.5 0.4 0.3 0.2 0.1 0 observed mu regressive mu average ji = 0.250 0.5 1.5 2.5 3.5 4.5 5.5 Time(sec) Fig. 4-14. Frictional coefficient estimated from inverse analysis and averaging when >9=10.25 degree (Motion 4, Case 2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4-2 Results o f Case 2 Numerical Experiments Case 2 Numerical Experiments Case M-l M-2 M-3 M-4 P (sec) 10.25 11.05 9.94 10.25 Oo(sec) 0 0 0 0 L(m) 0.405 0.405 0.405 0.405 Ave. T(sec) (Shaking) 1.46 0.72 0.66 0.86 Ave. T(sec) (Relative Displacement) 0.33 0.72 0.64 0.87 Ave. fi 0.25 0.22 0.20 0.25 The frictional coefficients obtained from the work of spreadsheet of calculation by following the same schemes used in Case I, however, the values of frictional coefficients in each of the four motion types are quite different. To justly to verify the frictional coefficients in the dynamic foundation Case 2 studies, the Runge-Kutta 4th method had been performed by MATLAB codes. The results of verification of displacements in image observation and Runge-Kutta approximation are compared in Figs. 4-15 (a)— (d). The method of Runge-Kutta approximation for solving this nonlinear problem has to suffer the accumulation of errors in the expression of time history results. In the beginning 2 seconds of approximation of pendulum weight displacement in Motions 1, 2 3 and 4, the displacements are met 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. quite closely, however, in the following seconds, the errors were accumulative due to this Runge-Kutta numerical approximation. Proo f-o f-concept experiments in simulation of ground motion are performed the efficacy of proposed DIP method with implemented effective algorithms. The schemes are useful for system identifications, especially for the nonlinear vibration systems there are difficulties for large displacement measurement. The motions of the board were applied different types of motions (stationary or non-stationary, slow or fast), but the experiments are all stable and successful. This means that method of using digital image processing is capable of identifying the relative displacement of a multi-degree-of-freedom system. (a) Motion * 1 os 0.4 ~T 0 2 3 -0 2 hnago OtMMvation Fonw d Vorfficatton - 0 6 Timo(soc) Fig. 4-15 (a) Forward analysis of Case 2 studies 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) Motion-2 0.6 0 .4 0.2 I 3 O -0.2 9 £ - 0.4 - - Fofwarf Verification -0 .6 Tim* (sec) Fig. 4-15 (b) Forward analysis of Case 2 studies (c) Motion-3 0 8 0 4 02 •0.4 * - forward Verification - 0 6 Fig. 4-15 (c) Forward analysis of Case 2 studies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (d) Motion-4 0.6 0.4 -04 1 mag Processing * * ■ Forward Verification -0.6 Fig. 4-15 (d) Forward analysis o f Case 2 studies 4.3 Findings and Discussions 4.3.1 Finding The same material was used in Case 1 and Case 2 studies, but Case 1 and Case 2 were based on two different motion states. In Case 1, the weight of the pendulum was placed on a stationary metal board. The weight was simply acted by gravitational force. Because the metal board and weight were attached each other, Coulomb damping effect happened during the motion. So it is a Coulomb-damped free vibration problem. Coulomb damping appears in the contact problem with friction between the media's interfaces. When the weight moved along the path of 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the weight's movement, the friction would occur between their touching surfaces. Finally, the weight would stop because the friction breaks the conservation of energy system. When the board is stationary, identified friction coefficient, ft, is not constant during the motion. In regression analysis, the computed mean value of ft over each half velocity-cycle is 0.24, independent of the velocity and slant angles. To use the mean value obtained for forward analysis, can always reproduce the angular displacement as the observed by digital image processing. In Case 2, the metal board was moving by the act of shaking. The board was under an unsteady moving state. The momentum created by the shaking changed the characteristic of reaction force between the interface of weight and the metal board. Due to that the weight was still attached on the board and the friction was acting on their contacting surfaces. As the motion of the board continued, the motion of pendulum would keep going. The change of the metal board’s momentum created the excitation forces that would act on the pendulum weight because the weight was still attached on the metal board while the board was moving. The weight was continuously oscillating as long as the metal board was moving. When the board moves, the frictional coefficients were observed and varied with larger fluctuations than the case of stationary board. The computed mean values of ft in four types of motions also differed from each other. To use the ft values for forward analysis can still reproduce the angular displacement and give good fit to the original observations. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The motion style of Case 2 study can be used to simulate the motion of friction pendulum devices in the application of seismic resisting base-isolators under ground motions. With the modification of the experimental setups, it is conceptually possible to apply this experience in the seismic studies. However, if use the mean value of // for forward analysis, it does not give good fit to that observed by digital image processing. The pendulum experiment was used for the proof-of-concept. The pendulum is a system that behaves as a nonlinear vibrating system. This case study proved that digital image processing technique is useful and effective to perform the system identification for the similar nonlinear vibration systems. It has the merit to resolve the problems that traditional methods have failed to deal with, such as that the nonlinearly vibrating pendulum problems. 4.3.2 Discussions Digital image processing with CCD camera is effective and quite suitable for system identification of the vibration systems, especially of the nonlinear systems. Traditional sensing methods were usually inconvenient and ineffective to perform system identification, particular of nonlinear systems. For examples, most traditional motion sensors need to be physically attached to structures and require cumbersome wiring for data acquisition. This presents a major obstacle in performing system identification. Besides, the traditional methods were usually unable to perform real time analysis at the mean time of sensing. In this study, the traditional motion 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sensors were omitted and the technique of remote sensing by digital image method was used instead. The proof-of-concept experiment of the pendulum study demonstrated that digital image processing technique is capable of acquiring accurate sensing and performing system analysis in near real-time. To perform accurate measurement and analysis, "speed" is a key issue. Hardware and software plays an important role. The hardware comprised of the CCD camera, image grabber adapter and an at-least equivalent Pentium digital computer. Software included the image processing algorithms and system analysis codes. Software and hardware have to match perfectly each other. In the study, the oscillating frequency of the pendulum was 1~2 Hz and the capability of the CCD camera and image grabber could achieve 30Hz. We almost obtained 30 frames at each cycle of the pendulum oscillation. The accurate system analysis results were achieved. To promote a higher resolution for the study, it needs to exchange the camera and image grabber and, however, the capability of the digital computing system is also necessary to take into account to prevent analysis from breaking down, particularly for the real-time analysis. In real-time processing, pictures shooting and processing have to be carried out at the same time. When the image is shot and stored in the digital computer, the analysis runs immediately. After the analysis procedure is finished, the image would be deleted to save the memory space. Therefore, the goal of real-time processing can be carried out. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 RELATIVE DISPLACEMENT BETWEEN BASE- ISOLATED STRUCTURE AND SHAKING TABLE S.l Introduction This study employs digital image processing technique in identification of relative displacement between the base-isolated structure and shaking table. The concept of base-isolation has been applied in many seismic resistance designs for civil infrastructure systems. The devices of base-isolation, which were built by using this concept are such as friction pendulum bearings, does show their validness and effectiveness in the mechanisms and economics for the purpose of prevention seismic hazard during earthquakes. In the United States, some of the world’s largest and most critical buildings, bridges and industrial facilities, comprising huge amount of construction expenses were installed the seismic bearings. The lead-rubber bearing is world famous for being the seismic isolation system which successfully protected the University Hospital from damage during the 1994 Northridge earthquake while the ten nearby hospitals were so badly damaged they had to be evacuated. This seven-story hospital (the University of Southern California Teaching Hospital) underwent ground 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. accelerations of 0.49g, while the rooftop acceleration was only 0.2 lg, that is a reduction by a factor of 1.8. The Olive View Hospital, nearer to the epicenter of the earthquake, underwent a top floor acceleration of 2.3lg compared with its base acceleration o f 0.82g, a magnification by a factor of 2.8. A comparison between the hospital seismically isolated with lead-rubber bearings, the University Teaching Hospital, and the un-isolated building, the Olive View Hospital, shows an advantage by a factor o f 5 (approximation from 1.8 times of 2.8) in favor of the isolated hospital. Further support for the lead rubber bearing isolation system occurred in the January 199S Great Hanshin Earthquake, Kobe, where a building isolated with the lead-rubber bearing system in the affected zone survived without damage or disruption to services. For this building, the Computer Center of the Ministry of Post and Telecommunications, preliminary results indicate a maximum ground acceleration of 0.40g while the sixth floor acceleration had a maximum of 0.13g, that is an attenuation by a factor of 3. These examples of the behavior of non-isolated and isolated buildings in real earthquakes, clearly illustrate the advantages of using bearing isolation systems for seismic resistance. When the seismic hazard happens, the bearings of the structure will reduce the energy that absorbed from the ground motion seismic force and increase the rigidity of the supported structures by isolating the supported structures and foundation. This study adopted a videocassette recording a shaking table test for video image processing to identify the relative displacement between a base-isolated 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. structure and a shaking table. The test and analysis had been conducted by the National Center for Research of Earthquake Engineering (NCREE) in Taipei, Taiwan, for study of seismic performance of a base-isolated electric power distribution system (Fig. 5-1). According to their result, base-isolation applied in this shaking table test did take effect to reduce the seismic hazard to the base-isolated electric power distribution system. Due to the difficult of acquiring good resolution of image data, relative displacement was the current concern as the preliminary goal for showing the proof-of-concept demonstrating image method capable of its application in earthquake engineering. Fig. 5-1 Base-isolated bushing transform of electric system for shaking table test (Courtesy of National Center for Research of Earthquake Engineering, Taiwan, ROC) 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S.2 Proposed Study and Preliminary Result 5.2.1 Proposed study and current scheme The research using video image processing for identification of a shaking table test is proposed to show efficacy o f image method in applications of earthquake engineering. Although the current research is focused on a simplified problem, a simulated configuration of real earthquakes, this experience can point out a principle that can be applied to identification o f systems involving a base-isolated structure and a shaking table. It disobeyed traditional methods in performing this kind of system identification. For examples, (1) sensors used are different from those usually need to be wired and imbedded in structural members; (2) displacement data are acquired directly from the measurement not from the computation of accelerations and velocities; (3) the system analysis uses more differential methods than the integrals, etc. To use image method for shaking table system identification, we need to setup one and two cameras for studying one and two directional shakings, respectively. By the principles used in the remote sensing, processing digital images with the aid of cameras can identify displacement and further compute the velocity and acceleration. In the procedures o f system analysis, acceleration and velocity are equally important with displacement data to do mechanical calculation. The demands of acceleration, velocity and displacement accuracy are also critical important. If all of the above requirements were satisfied to the needs in system analysis, the identification would be carried out at ease with highly reliable results. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The system identification procedures are illustrated in a proposed flowchart shown in Fig. 5-2. System analysis proceeds after finishing data acquisition from digital image processing. It needs to create a model involving the stiffness and damping characteristics of such a system for inverse analysis. In this study, the proof-of-concept is focused on the implementation of image method, and it is recommended to refer other papers (e.g. Skinner, et al., 1993) for details of shaking table system equations. Image processing Image grabbing Regression analysis Image storage Evaluation of system characteristics Noise filtering System analysis Observed data Ok? Stop Verification Forward analysis | Fig. 5-2 Flowchart o f system identification using digital image processing 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This study adopted the video record made from NCREE in a videocassette. The videocassette had been playing in a digital computer and digitized by the frame grabber that captured images frame by frame in a constant time interval. The sketch of the system setup using video image method is shown in Fig. 5-3. Based on the images digitized in the previous method, the segmentation algorithm used in the study of pendulum system was not working in this case. It requires a more advanced consideration to remove the background where large areas of non-uniform intensities were located. TV tuner adapter Personal com puter Video cam era Fig. 5-3 Digitization of video images from videocassette In the NCREE’s shaking table test, there were two parts in configured base- isolated structure; namely, the distribution device and stand of electric power bushing transformer. The distribution device and stand of this electric distribution system was assumed a rigid body and they were firmly bolted and welded. Based on 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the experimental setup, the system complexity within the base-isolated structure was simplified and image method could just take advantage of this simplification in separation of the object with a piece of structure pattern from the original picture. The goal of current scheme is feasibly achieved by means of image method. 5.2.2 Obstacle of using NCREE’s video record In this study, images were obtained through the digitization of the NCREE’s record in a videocassette. There were several reasons that make processing these images more difficult. First, the illumination was not fully served because this experiment was not for image processing purpose in the beginning. Second, the camera was probably fixed when the experiment was undergoing shaking tests. Images were only obtained from one source of the shaking table test record. The previous two points were crucial to deter the processing of system analysis for performing system identification. This pointed out the possible limitation by using the video for extensive study that needs better resolution. Segmentation of digital image processing is a main concern and critical process in identification of relative displacement between the base-isolated structure and shaking table. This study showed that using straightforward image segmentation technique was not appropriate to manage this problem with the case. The resolution of intensity distribution within image pixels was poor because this record did not have well-illuminated background during the test. The enhancement of image intensity and contrast would take limited effect in the improvement of this situation. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Segmentation usually needs to apply highpass filter in order to identify the position of edges between pixels and to construct objective boundary lines. The process of applying highpass filter, in this case, was not successful by all means. The fast Fourier Transform method made it more difficult to interpret outcome in a complicated 2-D mapped layout. Temporal resolution was also poor because the speed of using TV-tuner card (Hauppauge WinTV Model 401) to digitize images was only capable of acquiring images by 15 frames per second. To improve this obstacle, it needs to invest in hardware promotion for using a higher image-grabbing device. 5.2.3 Preliminary result Displacements of base-isolated structure and shaking table can be measured by segmentation of circular marks determined by regional scanning for deploy of positions. First, it needs to decide the depth and width o f the scope of scanning area that the whole area of a level was covered. Second, using highpass filter (Pratt, 1994), we can determine boundary of the area occupied by image pixels belonging to super structure. At the same time, the comer of shaking table could be identified by finding the intercept of two edge lines which they could be determined by regional scanning and highpass filter. Once the tracked points are chosen, it became a straightforward routine to determine the coordinates o f tracked points. Fig. 5-4 shows the tracked points of base-isolated structure and shaking table by a circular symbol. Fig. 5-4 (b) and (c) are presenting the segmentation of circular symbol 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. marks from the image background. Fig. 5-4(d) means that the coordinates of circular marks’ centers are determined by image scanning. In this study, the position being tracked on the base-isolated structure is at the second floor of configured mass blocks. The relative displacement is calculated by subtraction the displacement of shaking table from the sensed displacement of the second floor. Result of the second floor and shaking table and relative displacement are plotted in Fig. 5-5. This result is also compared with that (smoothed and using acceleration sensors) from the NCREE’s data in Fig. 5-6. (a ) Original image ( b ) Circular marks painted (c ) Segmented circular marks (d) Coordinates determination Fig. 5-4 Using circular marks for tracking motion of the second floor and shaking table 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Displacement (Pix) 2 0 0 150 100 — 2nd floor — Shaking table j Relative displacement j 50 0 5 10 15 20 Time (Sec) Fig. 5-5 Displacements measured by digital image processing; the 2n d floor’s, shaking table’s and relative displacement 0.3 — NCREE Result - 0.1 -0.3 0 6 3 9 12 15 Time (sec) Fig. 5-6 Relative displacement of supported structure and foundation (shaking table); NCREE’s and DIP method’s results 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The relative displacement between the base-isolated structure and shaking table has been successfully identified by using digital image processing with the video digitized from the tape that was recorded by NCREE. In this experiment, there are the following major findings. First, the video was originally for the purpose of backing up an experimental record and not for an image experiment. It does not have satisfactory in resolution of both spatial and time domains in these image resources. Second, hardware is crucial to affect the experimental result. The videotape has been put into a VCR for playing with the connection to a digital computer. The speed of digitization by employing a TV-tuner card with VCR digitizing the video into individual frame of pictures could achieve only 15 frames per second. This makes it difficult to evaluate the velocity and acceleration. With poor evaluation of velocity and acceleration, it would not be capable of performing good system analysis to determine the system characteristics. Third, image processing involves complicated segmentation algorithm that has been overcome by off-line image modification. This experiment illustrated an idea of indefinite target for system identification using image method. The segmentation pattern would vary when difference structures are used. However, according to the preliminary result from this video, we can still achieve a very close plotting in comparison with the NCREE’s result, smoothed and computed from the method that acceleration sensors were used. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Discussions Digital image processing method was used to identify the base-isolated structure incompletely because we were encountered with the following difficulties: (1) the source of images was obtained from the National Center for Research of Earthquake Engineering (NCREE), Taipei, ROC, and it was not for the purpose of image-system identification originally. The image quality and resolution were average poor and it was disturbing to proceed system identification. In this study, the author successfully captured the images of motion of base-isolated structures and the shaking table. The best resolution for digitization is 15 frames per second. If we want to proceed the system analysis, such resolution discouraged the computation of velocities and accelerations. So, we cannot achieve a finer time domain displacement curve. (2) The base-isolated structure and the shaking table were composed into a 3-D system. The images were digitized from a videocassette that had been recorded by only one camera. The vision was limited to one visible angle of the site. It is very difficult to describe the 3-D movement of the structure and shaking table. In the study, the author simply took out a time section that has 1-D motion pictures within the videos, and used them into the digitized images to proceed the motion analysis. In the comparison with the results from NCREE and image processing, the one from image processing was quite close to that from NCREE's method, according to Fig. 5-6. However, for further proceeding the 3-D analysis of the leftover videos, the current algorithm is inappropriate. (3) The image 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. segmentation scheme is complicated. In image processing, it is necessary to extract the positions of base-isolated structure and the shaking table for estimation of their relative displacement. It was not successfully accomplished. In the experience of segmentation processing, extraction of the structure and the shaking table was very complicated. The main problem was due to the poor contrast and dark illumination when the videos were taken. The photographer did not consider the in-situ illumination and reduction of the background's complexity. If the background was arranged with more monotonic illumination and colors, the difficulties of extraction steps would be largely reduced and a better experiment could be carried out. With the stated three major problems, this experiment was only used for the identification, as a proof-of-concept, of the relative displacement between the base-isolated structure and shaking table, and the 3-D system analysis was not carried out. To extend the current study to 3-D analysis, it needs to add one or two more cameras by taking more visible angles of the site and to consider appropriate sensing instrumentation to promote the accuracy and reduce the complexity of digital image processing. 8 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 NONLINEAR ELASTOMER MEMBRANE 6.1 Introduction Digital image processing (DIP) method is utilized to study on constitutive relation's identification of a 2D elastomeric membrane system. The motives of this study is to implement a non-traditional sensing method for system identification especially in the measurements of a 2D membrane's in-plane deformation and further use these measurements to proceed the identification of nonlinear characteristics of the studied material. To pursue the system identification of such nonlinear elastomeric membranes, it needs to identify finite in-plane deformation of the studied specimen, a rubber membrane, that has nonlinearly elastic characteristics when it is under acting of loads. A distill digital camcorder, Sony make Model TRV11, was used to take images of the specimen when it is stretched under the loads. The finite in-plane deformation was sensed by making use of equally spaced 143 bullet markings (arranged by 11 rows and 13 columns) painted on the membrane surface before the images were shot. With this approach, traditional displacement sensors were discarded and replaced with a digital image device. The data sources of system analysis were supported on the basis of digital images. The fundamental and extensive image 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. processing techniques were importantly infused in the system analysis of this experiment. There have been some problems about the traditional sensing techniques in dealing with finite deformation problems of such elastomeric membrane systems: (1) displacement sensors are difficult to be selected and installed for measuring finite deformation of these specimens; (2) existed displacement sensors are limited in the scale of measurement range. It needs to look for a suitable sensor in order to measure a displacement that it will not exceed the limit of measurement capacity. If the deformation is excessively large and more than the tolerable capacity of sensors's measurement, it will fail to acquire the accurate data; (3) it needs a massive amount of sensors installed for studying a 2D membrane problem. To achieve good accuracy, numerous sensors are necessary to be arranged around the surface. The more number of sensors are installed, the more intrusions to measurement are made. While the traditional sensors are installed on the thin-and-soft membrane, the detected membrane's surface displacement may be d im in ish ed because of the intrusions made from the sensors. And, more restrictions of applying traditional sensors could be listed. In this dissertation, existed techniques of remote sensing were employed by implementing computer programs of digital image processing to facilitate the acquisition of membrane's finite in-plane deformation based on the images captured by still digital cameras. Traditional displacement sensors such as strain gauges that usually are necessarily wired to signal receiving devices were completely discarded. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Digital image devices were used instead. An illustration of using digital image device for studying membrane's problem is shown in Fig. 6-1. Within the configuration, a membrane gripped by the two edging areas can be used as a specimen for a proof-of-concept experiment, to identifying its nonlinear material characteristics. The finite in-plane deformation of the tested specimen is capable of obtaining by making use of a digital camera. In consequence, by undergoing analysis of the sensed pictures recording the deformation when the specimen were under loads, we can interpret the stress-strain relationships based on the developed system analysis schemes. Digital camera Fixed end Rubber membrane Fig. 6-1. Configuration of rubber membrane experiment 6.2 Rubber Elasticity A rubber membrane is used for the specimen of this experiment. To study material properties of rubbers, we must review their historical investigations from both empirical and theoretical approaches. One of the most famous studies in rubber 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elasticity researches was done by Mooney (Mooney, 1948). The material characteristics of rubbers were concluded by an empirical stress-strain equation— Mooney-Rivlin's stress function. Rubbers are made from polymers. According to those discoveries in 1980’s, investigators found out that the elasticity and plasticity condition o f polymers is changed with respect to its age, temperature, humidity and light, as well as the environment where it is exposed. Rubbers have their amorphous flexibility because, in a micro-scale, the molecules are structured and bonded as chains and those chain- structure networks control the properties of stretching and retracting. The connecting style of chains are so-called “cross-linked networks” (Fig. 6-2). They are named so because the chains are usually not along a singular plane and are crossed out each other in a 3-dimensional space. When rubber is stretched by force, the equilibrium of the chains will be disturbed and complicated molecular rearrangement will start at the same time. When the force is released, the chains will turn back to the original equilibrium. “Entropy forces” occur in the state of molecular rearrangement and adjustment. Such forces are in favor of retraction, and return the network to its original equilibrium. The gain of entropy means that rubber is suffered in the action of stretching. The loss of entropy means the release of stretching action to create retraction force for recovery when external stress is removed. This phenomenon was observed in the stretching-and-extracting test of rubber. Thus, rubber is thought to be an elastic material, and it is, however, a nonlinear elastic material. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6-2 Cross-linked network in rubber (3D, micro vision) To explain the elasticity of rubber, it requires the development of the concept of entropy forces. Based on the statistical mechanics, consider ID random walk one end of a rubber chain within a cross-linked network for doing a stretching-and- retracting movement. The end has moved to distance x. By traveling JV times comprising of A times forward and B times backward and with each step length is assumed to be p at each time movement, the distance o f a ID random walk is expressed as x = ( A - B ) p (6-1) By all possible combinations, the travelling distance, x, can be achieved in W ways, where W = N l/(A !xB !). By applying Stirling’s approximation and solve for A and B in terms of N, p and x. x l \nW = N \ n l — — - (6-2) 2 N p ’ 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The relationship of entropy was established by Boltzmann who introduced the Boltzmann constant (k) and proved that entropy is proportional to the logarithm of the occurrence probability of a random walk. The entropy is shown that S - k ( x l } iVln2— (6-3) 2 NP') where k is the Boltzmann’s constant This can be generalized to the 3D case S = U ( r 2 > N \ n l - (6-4) 2 Npl ) where r is the 3D random walk and it can be expressed by r 2 = k \x l + X 2 y2 + X 2 z 2 (6-5) that Xx, Xy, and Xz are defined as the elongation ratio along the x, y and z direction, respectively. Equation 6-4 is concluding expression of entropy force of a cross-link’ s chain and S is representative to the entropy force of a single chain of the cross- linking network. This has to be reminded, to study the entropy force of an entire cross-linking network is necessary to transfer the terminology of S into the thermodynamic ratio of entropy of the entire network. Continuing to discuss the change o f thermodynamic ratio in the stretching test of a rubber tissue, the change of thermodynamic entropy can be expressed as -<>*’ +(K - l ) / +(K- m ' l (6-6) By assuming that the rubber is isotropic, homogeneous, and incompressible, elongation ratios are eligibly simplified by using the following relationships. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. And to substitute Eq. 6-7 into Eq. 6-6 and multiply with number, n, the total number of fiber chains per volume, it gets A5(X) = y [ X I - ^ - 3 J (6-8) Recall that the Helmholtz Free Energy theory, Free Energy (F) is expressed as F = U -T S (6-9) where U, bond distortion, is usually negligible if the absolute temperature, T, is constant. This assumption is always true in small change of entropy. For small change of entropy, the change of free energy AF can also be shown as AF = AU-TAS. (6-10) Entropy force (f) is then capable of deriving from Eqs. 6-8 & 6-9, and it is < r = f(X) = ^ = okT(X ~ ) (6-11) dk X z Eq. 6-11 indicates a one dimensional Hookeian Elastic relationship in terms of stress and elongation in x direction. For an isotropic material, a simplification of 3D elastomeric solid, a rubberlike continuum media, under stretching force is concluded as so. Equation 6-11 can be rewritten as p r r " kT ( 6 - l 2 ) Where it is granted that X = 1+e. The stress (a) and strain (e) relationship of rubberlike material specimen under uniaxial stretching movement is determined by 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. so. Based on the derivation o f Eq. 6-12, the stress is related to the strain nonlinearly constant. No matter that in large or small deformation problems, the relationships of Eq. 6-12 would keep if material property of elastomer is isotropic, homogeneous and incompressible. There is no guarantee that the relationship will be correct in a case for the ai so tropic, nonhomogeneous or compressible specimens. However, by numerical approaches, Eq. 6-12 could give very good concept and experience in solving the elastomer material mysteries. The successful practice of using such theory in presentation of rubber material properties had also been proved by a famous result— Mooney-Rivlin stress function. 6.3 Mooney-Rivlin Stress Function According to the equation proposed by Mooney and Rivlin (1948), an expression for rubber elasticity appears to be the replacement of conclusion in Eq. 6- 12. The parameter n, k and T are vanished in this expression. Instead of showing the thermodynamic parameters, n, k and T, Eq. 6-12 becomes a function purely of the deformation ratio X. Such relationship is like because o f the infusion of the term A .'2. The parameter n, k, and T are usually o (6-13) andean be rewritten as, (6-14) 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where a is the stress and Ci and Cz are constants. Constant Ci was proved to be associated with the shear modulus (Marie and Lai, 1982, pp. 397). The Mooney- Rivlin Equation is applied to the case when rubber is under tension test. However, this relationship may also be applied to compression case (Dusek and Prins, 1969). This study attempts to compare the derived relationship based on assumptions of thermodynamics with that from Mooney’ s result. The tension tests of rubber were run in order to do so. To approach by different stress functions, the regression method to manage the nonlinear curve fitting is necessary. 6.4 Procedures of System Identification The methodology of identification used in this study can be divided to two major procedures; namely, image processing and system analysis. The image processing procedure includes the earlier mentioned operations in previous proof-of- concept experiments and a more complicated attributes-extraction technique in the procedure of measuring and sensing in-plane multiple targets. In the procedure, the images were taken by a still digital camera and stored in a digital computer for service of data. Multiple image data were sensed by the image device, the Sony make digital camcorder, and stored for waiting for attribute-extraction operations. Images were reconditioned by applying low-pass filters for noise removal and brightness and contrast enhancement. Then, to use FFT transform mask to dig out the interested area from image surface where the specimen's deformation information 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was mixed with other backgrounds. The segmented image pixels from grid- segmentation operation were registered with coordinates system. Then the grids were positioned by image-registration operation. A flowchart depicting the two procedures with listed operations of this experiment is shown in Fig. 6-3. System analysis Image processing Image grabbing lj(*\ y,fP ) Initialization for Mooney-Rivlin —^ Coefficients C, and C; Regression Analysis Image enhancement Noise filtering Grids segmentation Image registration displacement 2D FEM nonlinear elastic analysis Observed displacement Upfx. y j u n de r (f,) Stop Fig. 6-3 Analysis flowchart for membrane study 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The measured displacements of specimen's surface grids would be used for performing identification of material characteristics. Regression analysis was infused prior to the curve fitting of observed data with Mooney-Rivlin's equation. The constants C| and C2 were eligible be initialized by regression analysis by using the nonlinear Mooney-Rivlin's equation. In consequence, the iterative finite element program was utilized to verify the displacement calculation with image observation for optimizing the values of Ci and C2. The procedures of system analysis were completed as long as the constants Ci and C2 were optimized. 6.5 FEM for Nonlinear Elastic Elastomer Membrane The finite element program was implemented to employ the Mooney-Rivlin Equation to be the rubber membrane’s stress-strain model in analysis of its constitutive relationship. Analytically, the conclusion of Mooney-Rivlin's equation is only valid for applying to 1-D deformation problem. To perform this nonlinear elastic analysis upon the nonlinear elastic material, iteration analysis which it adjusts the material stiffness by using “piecewise linearity” of elasticity is utilized to manage such kind of nonlinear elastic stress-strain relationship. Although elastomer material like rubber is essentially incompressible, in this study, it is dealt as plane stress problem by taking account of the boundary conditions and the shape of specimen that has been used. Also, to perform 2-D analysis, the use of coordinate transformation in the stress and strain plane, Eq. 6-14 would be eligible to determine 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the tensile stress and strain relationship along the principal direction that indicates the uni-axial loads for the stretching test. In reference to the elastic, isotropic and homogeneous material and on the assumption of plane-stress problem, the stress-strain relationship is expressed as 1 — i) * 1 * 1. 1 u o 1 0 0 0 0 l - o 2 J (6-15) Translating Eq. 6-17 to the principal direction, the relationship of the principal stress and strain may be written as _ E f l o lje .l l - o 2|o lJKj (6-16) where x\2 and y\2 are both equal to zero and vanished in this equation. Assuming that the principal tensile stress is in the 2-direction, so that tensile stress 0 2 is can be recalled as in Eq. 6-16 and expressed as 1 o 2 — 2 C, 0 + e * )- ( l + e 2 > J + 2 C, 1 a * * ; ) 1} (6-17) And in consequence, . ve, + e, ez = ------ 1 - v 1 (6-18) 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where e2* is equivalent uni-axial tensile strain (see Eq. 6-18). Assuming that the Poisson’s ratio after nonlinear deformation is equal to the value that v= 0.25 in elasticity, it is derived that £z*=£2- In the process of finite element analysis, the iterative scheme is used to compute estimated equivalent strain, e2 *. Fig. 6-4 illustrates the scheme of equivalent linear elastic analysis using Secant Modulus. The estimated equivalent Secant Modulus (Es) in each element and iteration is computed by Et= - ^ - , where t 82 is the temporary approximate value of e2 * in each iteration. If a 2 , which is evaluated at the center point in each element, FEM analysis using the above method, t 82 are capable to be computed from the inverse calculation of Eq. 6-15. In this study, it is assumed that the iterative procedure stops if all of the elements satisfy the following criteria | E, As* | <0.1 (psi) (6-19) and that, Ae*z = A-A (6-20) where e€2, the intermediate result of equivalent uni-axial strain at each iteration along direction-2, is calculated within the FEM procedure by the Eq. 6-18. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O j i-l Computation o f equivalent Uni-axial strain by secant Young's Mudulus Es M ooney-Rivlin curve F i ; i ■ ^Estimation o f by l i calculation o f o=ffi inverse o=f(E) Iteration stops when it satisfies t£” -> z2* if | EAe2*| < 0.1 (psi) W here SE2*=fi2- ^ 2 Fig. 6-4. Estimation of £2* by the Mooney-Rivlin Equation Recalling that, according to the Mooney-Rivlin equation, we could introduce the initial value of the secant elastic modulus Es by the derivative on e , such that En = — = 2 C ,jl+ — — —— de l [ (1 +e) 6C, (1+e)4 (6-21) By taking the limit of that e is approaching to zero, the tangent elastic modulus can be expressed by Eo, such that E0 = lim ^0 Em = 6(C, + C 2) (6-22) 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In FEM procedures, the secant elastic modulus Eo is used initially by the tangent value at s = 0 for the equivalent elastic modulus. Details of the iterative scheme are illustrated by the flowchart of Fig. 6-5. Using this scheme, accumulation of error is eligible to be avoided and solution at arbitrary loading stages can be obtained directly without executing the calculation without making incremental error. i = i+ / No Iterative schemes of FEM analysis Yes Stop Start Ei = e2* if v = constant Ei=-v e2 FEM evaluation of stress Evaluation of equivalent uni-axial strain by inverse calculation o f Mooney-Rivlin Equation by Eq. 6-19 (£ 2 « e2*). Fig. 6-5. Flowchart of iterative finite element analysis 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 NUMERICAL EXPERIMENT; NONLINEAR ELASTOMER MEMBRANE 7.1 System Setup A Sony make digital-8 camcorder, model TRV-11, was used to image the in plane deformation of rubber membrane with the surface dimension shown in Fig. 7- 1. There were totally 143 points (13 by 11) marked on the grids’ intercepts on membrane surface for symbols of tracking. Two edge areas were gripped perfectly 3" Force m 0 2 5 Gripped area Rubber membrane (thickness = 1/8”) Marked grid Gripped area Fig. 7-1 System configuration for membrane study 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with paste glue and claimed by two pieces o f metal in order to apply uniform tension forces. The gripped areas along the loading edges were assumed completely closed and the tension force was uniformly distributed. The positions of the tracked points in the middle area o f the membrane surface would be recorded by the digital camera in nine different loading-stage stretching tests. This study is, at first, to observe the targeted points by the image system, then to use their positioned coordinates to compute the displacement, and the last, to perform system analysis involving assumptions of constitutive model to identify material characteristics. The specimen was laid stationary when the camera was shooting to the membrane surface in each loading stage. At each stage of loading, it became a static problem because the equilibrium is maintained. Thus, the displacement extraction would simply concern on dealing with single frame of picture at each loading stage. 7.2 Extraction o f Tracked Points The tracked points were marked on the membrane surface before taking image snapshot. They were necessarily extracted from the image map by a systematic way. Usually, it is very difficult to do so. In this study, Fast Fourier Transform (FFT) method is used to extract those points by area amputation from the original entire image map (Fig. 7-2). By the use of FFT, it is quite convenient to cut off the partial area on the image map and keep the area matched with the FFT 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transformed texture features non-related to the interested area. The characteristics of texture can be distinguished by calculating first or second moment of FFT’s amplitude from their complex values. In the experiment, three major textures were found in the picture with their amplitude profiles of the transformed result (Fig. 7-3). These textures can be directly distinguished by human beings’ eyes without difficulty but, however, for digital computers, it needs more labor dedication to develop a systematic program to make it work. etc. Classification of textures T, * Texture extraction by F F T { T |(x ,y )} method Binary image Keep images if matched with Texture 7 % Fig. 7-2 Segmentation of multiple tracked points on the membrane surface 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Z=VRe‘+ lm ‘ A ^ X,Y Fig. 7-3 Fast Fourier Transform (FFT) for distinguishing textures As it turned out from the method shown in Fig. 7-2, the tracked points were extracted in a format of clusters group by group on the binary images. The black clusters are areas of grid marks of intensity 0 and background of intensity 1 in this binary layout. In the binary image, there still has some fractured image pixels with 0 intensity. These fractured image pixels were not the representative clusters o f the tracked points. They could be removed by applying hit-or-miss transformation method (Chapter 2). Because the fractal noises are usually with 1 or 2 pixels only and without other pixels nearby, the mask of hit-of-miss filter is very easy to define. After the noise removal by filters, the estimation of gird center of each cluster area is reconstructed by computing the mass center of clusters nearby and within an 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. appropriate-sized window by using first moment method. Fig. 7-4 shows sensed points of the targeted position on the membrane surface. The points are plotted into a binary image with 0 (clusters) and Is (background). The coordinates of the center of the clusters are computed by finding the geometrical center o f grouped clusters by making use of mask scanning. O Mass center of clusters T Clusters Image pixel Fig. 7-4. Sensed points in a binary image 73 Geometrical Reconstruction The reading of membrane surface coordinates extracted from the images was not all geometrically correct, because images might be taken by the camera with an 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. inclined angle or because of some other factors to affect the quality of image construction such as fringe defect. The distortion from the fringe can be treated at ease by mathematical operation; the fringe defects can be recovered by simply doing polynomial warping transformation (Chapter 2). In this study, the defect from fringe was pretty mild. Compared with fringe phenomenon, the misplacement of specimen's position was more disturbing to give a bad measurement in this experiment. No matter that the error of geometry was made from fringe or misplacement of system setup, the previously mentioned polynomial warping method was useful to resolve the combined defects in the reading of grids' geometry. At the same time of doing polynomial warping, physical coordinates were introduced by giving values in boundary points. Thus, the coordinates of tracked points on the membrane’ s surface can be "registered". There were 143 points involved in the modification of polynomial warping in order to reconstruct a registered geometry of the image. The result from warping operation was still presenting illness to demonstrate a well-registered geometry. Thus, quarterly averaging procedure was necessary to average the noises from that illness. The method of quarterly averaging used the geometric center of the 143 grids as an origin and then took the average mapping of grid's coordinates, in each quarter that are associated with the same address if they were mapped to the first quarter. This operation can reduce the sampling white noise by averaging and save the labor o f computation by making one forth of the original finite elements. It also reduced the complication of FEM analysis. The scheme is illustrated in Fig. 7-5. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With quarterly averaging of each quarter, a neat and symmetric result could be layout in the presentation of measurement. However, this procedure also erased the features of the specimen’ s non-homogeneity. The membrane specimen might not be homogeneously deformed. In this study, however, the non-homogeneity problem of rubber elastomer was neglected and the measurement of rubber specimen is assumed and simplified to be ideally homogeneous. >=> Fig. 7-5. Method of quarterly averaging 7.4 Measurement of Deformation The deformation of the membrane was measured based on the observation of grids' positions on its surface. The deformation was caused by applying seven stages of loading forces; that is, 1.74, 2.74, 3.29, 4.29, 4.74, 5.13, 6.66, 8.50 and 9.31 pounds of forces were applied to the membrane stretching test. The camera took one image at each loading stage. The image was then processed to extract tracked points’ positions. The data of coordinates have been observed by the methods mentioned in 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the previous sections. Yet, the data for original grid's coordinates was missing because there was no image taken at the occasion when rubber was without applying any force. Thus, the coordinates of when the rubber was without applying any force had to be initiated by extrapolation of the identified positions on those images in the early stages of loading. In Fig. 7-6, it shows the extrapolated result of the coordinates of grid marks by numbering the grid points. There were 143 dot markings being the targets on membrane surface. The 143 dots that marked on the rubber membrane surface had been reconstructed through the processing of geometrical modification showing the symmetry along the x and y directions, because the quarterly averaging were applied already. The result from the geometrical correction reduced the error made from misplacement of the camera and system setup. On the other hand, it modified the errors that were made in the beginning from the marking positions without straightly lined up. Figs 7-7~ 7-14 show the result of deformed positions on the membrane surface at those 9 loading stages. 1 0 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1.5 1 0.5 1 0 >- -0.5 -1 -1.5 -2 - 3 - 2 - 1 0 1 2 3 X(in) Fig. 7-6 Grid number and markings of membrane measured 2 15 1 0.5 1 0 > - -0.5 - 1 -1.5 -2 - 3 - 2 - 1 0 1 2 3 X On) Fig. 7-7 Deformation of membrane with the stretching of 1.74 lb (observation data by DIP) Fx =1 745(lb) and Fy=0 . I 1 " I ~ I Original shape of rubber membrane 1 o T “ M « « •1 Oft — T 1 17 O B i M o * * SI Ift * no to IB U2 « * V B B * » K B m oi U l 0 B m * B IS ■ m li* or M O I a X U li P 00 to oi n 1 » * ■ 7 B 0 ■ » to OS n r B B B 3 B ■ m □a or I II tt «l ■ n la p to IB O I s m it * * B « IB O I i 17 » O ■ » B no 1 3 1 Oft ' i n » 0 S 0 n la 107 IB C O i ts B ftl S f t * » D to o 1 11 K B " t u 0 B P * M B to OI ------------ 1 -------- . 4 _ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 15 1 05 1 0 > - ■05 - 1 -15 -2 -3 -2 -1 0 1 2 3 Xfin) Fig. 7-8 Deformation o f membrane by stretching of 2.74 lb Fx =2 740b) and Fy=0 I I 1 1 1 Fx =3.295(lb) and Fy =0. 2 15 1 05 0 •05 1 1 5 -2 3 • 2 0 1 2 3 1 X (in) Fig. 7-9 Deformation of membrane by stretching of 3.29 lb 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F =4 29(lb) and F =0 * T 05 c > - -05 X (in) Fig. 7-10 Deformation of membrane with stretching of 4.29 lb F =4 66(lb) and F =0 * y > * -05 X(in) Fig. 7-11 Deformation of membrane with stretching of 4.66 lb 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fx =5.135(lb)andFy=0. 05 c > - •05 X (in) Fig. 7-12 Deformation of membrane with stretching of 5.14 lb Fx =6 66(lb) and Fy=0 0 5 C > - •1.5 X p n ) Fig. 7-13 Deformation of membrane with stretching o f 6.66 lb 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fx =8 51 O b) and Fy=0 05 c > - -05 X(in) Fig. 7-14 Deformation of membrane with stretching of 8.51 lb F x =9 310b) and F y=0 05 > - -05 -15 X f m ) Fig. 7-15 Deformation of membrane with stretching of 9.31 lb 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The grids shown in Figs. 7-7-7-14 were registered by making use of physical coordinates, (x, y), instead of image pixels, (i, j). Recalling that in a digital image, the coordinates were presented by a set of integer numbers, (i, j). Such image coordinates represented the row and column number of the seat in an image map. For example, the image coordinates (i, j) mean that the position in the i-th row and j- th column of an image map. The ratio of image coordinates to physical coordinates was an index of measurement resolution. In this study, as an example in Fig. 7-7, there were 180 pixels along the direction of stretching. And the length of membrane was measured 2.7 in. Therefore, there were 66 pixels in one inch in length and this is not bad at all for being the measurement resolution. Observing the performance of specimen under tension (Figs. 7-7-7-15), the maximum elongation was approaching to 1+3/4 times of the length when rubber is at rest without applying any force. The elongation of robber membrane along the stretching direction and forces were listed in Table 7-1. In the table, forces were measured by counting the hung weights. There was no stress sensors installed on the membrane surface, and there was also no way to measure the forces in-plane by using any sensors. In order to proceed using the data as what had been observed in Figs. 7-7-7-15, the average stress and strain were necessary introduced in order to initialize the values of Mooney-Rivlin’ s constants C| and C2 as we discussed in Chapter 6. The average stress is easy to be calculated from the measure weight of the loading dividing by the membrane cross-sectional area. The average strain was 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. calculated by taking the total elongation along the stretching direction dividing by the original length that the rubber is without applying any force. The average stress and strain had been plotted and is shown as the circles in Fig. 7-16. The Mooney- Rivlin's constants were initialized by making use of regression analysis doing curve fitting on the points of the average stress and strain plots. 7.5 Regression Analysis to Determine Stress-Strain Relationship Regression analysis was introduced in order to initialize the values o f the Mooney-Rivlin constants, Ci and C2. To do so, it needs to know the objective function for curve fitting. There were three functions chosen for doing regression analysis for comparison. They were selected based on the derived stress function from thermodynamic theories and the empirical equation such as Mooney-Rivlin’ s equation. In Fig. 7-16, it showed a plot of experimental measurements and the average strain (observed elongation divided by the original length) and the average stresses (stretching forces divided by the edge boundary cross-sectional area, say, Area = 3” x 1/32” ) in the boundary. There are three case studies with different assumptions. In Assumption 1, we used the Mooney-Rivlin equation to undergo nonlinear regression to directly determine two constants Ci and C2. In Assumption 2, nonlinear regression was performed by combination of second-order polynomials, and then to undergo curve fitting by the Mooney-Rivlin equation to determine Ci and 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cz- In the last, Assumption 3 used only Ci constant (Ci=nkT) of the Mooney-Rivlin equation (i.e. to make C2 = 0) based on the equation derived (Eq. 6-12). The regression results were presented complete different sets of Ci and C2 parameters, but curve fitted significantly well in all of these cases. The difference of these sets of constants could be illustrated when the stress is less than 80 lb/in2 , the curve of Assumption 2 is slightly flying over those curves of Assumptions 1 and 3. It means that the stiffness by Assumption 2 is stronger. In contrast, when the stress is larger than 80 lb/in2 , the stiffness of Assumption 3 is larger than that of Assumptions 1 and 2. However, Assumption 2 was showing the best curve fitting result (with a correlation coefficient o f r = 0.99). The Mooney-Rivlin's constants were initialized based on the regression result. The stress-strain relationship was determined due to the initialization of the Mooney-Rivlin’ s constants. For the preparation of further robust iterative analysis to optimizing the values of those constants, the finite element analysis would be infused to carry out this task. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7-1. Measurement of total elongation (between two fixed ends along the x-axis) _______________ Lp=2.6875 (in) No. Force (lb) Elongation (in) Average strain 1 1.74 0.187 0.0695 2 2.74 0.338 0.1257 3 3.29 0.437 0.1626 4 4.29 0.625 0.2325 5 4.74 0.749 0.2786 6 5.13 0.971 0.3612 7 6.66 1.187 0.4416 8 8.50 1.749 0.3555 9 9.31 2.062 0.7671 140 120 100 £ f (0 ! I Mooiwy R Nfcl R h w m Icb (C 1 « 1 M 1 .C 2 ^M 0 ), Boom y RMi n R agresai on | C 1 < 1 U i, C2«30. 14) . rt.tM ■M oonay-fllvfin Ragresaion (Cl “36.82. C2*0), i-OJM 0 0. 0 0.4 1 S t rain Fig. 7-16 Nonlinear regression of Mooney-Rivlin equation (Assumptions 1,2 and 3) 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.6 Verification by Finite Element Methods (FEM) 7.6.1 FEM model and boundary conditions Using MATLAB programming to simplify the original problem, the dimension of membrane specimen has been simplified into quarter size of its original area (grids are reduced from 143 to 42) to produce iso-parametric meshing finite elements procedure shown in Fig. 7-17. The simplification is according to the symmetrical conditions of the membrane in its geometry and boundary conditions. The nodes are renumbered due to the smaller size finite element meshing. The coordinates are redefined into X | and X 2, which previously were referred to y and x, respectively, according to observation results, previously. It is assumed that the rubber is an isotropic and homogeneous material in plane stress field and symmetrically deformed when the loading is applied in the direction of X 2. Boundary nodes, such as 1, 2, 3, 4, 5,6, and 7, are restrained along the X 2 direction, and nodes 1, 8,15,22,29,36 and 43 are restrained in the xi direction. The elements of the metal pieces embracing the rubber edges were introduced as in nodes 36 through 49 of the membrane system. It is assumed that the metal stiffness was 100 times of the others’ rubber elemental stiffness. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to M etal it>- Rubber ip: iP - P *l Figure 7-17 FEM model and boundary conditions 7.6.2 Verification bv Finite Element Method The FEM procedure was executed by three different cases based on Assumptions 1, 2 and 3 as indicated in Section 7.5. By utilizing Mooney-Rivlin's nonlinear stress-strain relationship, the displacements of the rubber membrane through iterative finite element method (equivalent linear elastic FEM analysis) were resolved as in the Case 1 by using the Money-Rivlin constants Ci =19.21 and C2=25.90. Because the iterative schemes in the indicated FEM procedure does not accumulate the errors, the result plots will present the goodness of this digital image processing matters and equivalent linear elasticity to approximate the indicated 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nonlinear stress-strain relationship of such rubber-like materials. The Case l ’s verification plots of DIP and FEM’s results are shown in Fig. 7-18-7-26 due to that there were nine loading stages. As in Case 2, C|=12.85 and C2=36.14 were used for the constants in the Mooney-Rivlin’s nonlinear stress-strain relationship. Through the equivalent linear elastic analysis, displacement of the grids on the membrane surface was computed. The comparison of FEM result with the DEP’ s are shown in Figs. 7-27-7-35. For Case 3, values of C|=36.62, C2=0 were used. The grids’ displacement by the comparison of FEM's with DIP's are shown in Figs. 7-36- 7-44. 3 2.5 2 'c g 5 1.5 g t* >r i 0 5 0 0 05 1 1 5 2 2.5 3 X, direction (in) Fig. 7-18 The DIP and FEM result of membrane deformation with stretching force of 1.74 lb 115 Comparison of nodal displacements. Fx2=t .745(lb) ----- Image Result FEM Result ----- Original Shape • DEI =0.12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 C 15 0.5 Comparison of nodal displacements. F -=2.74(lb) 1 Image Result FEM Result Original Shape D EI =0.17 0.5 1 15 2 X , direction (in) 2.5 Fig. 7-19 Verification of the DIP and FEM results of membrane deformation with stretching force 2.74 lb (Case 1) Comparison of nodal displacements. F ^ S 2950b) 2.5 0.5 Image Result FEM Result Onginal Shape D E I = 0.26 0.5 1 1.5 2 X, direction (in) 2.5 Fig. 7-20 Verification of the DIP and FEM results of membrane deformation with stretching force 3.29 lb (Case 1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements. F](2 =4 29(lb) 3 25 2 C * e o Z 15 s “ 6 x T * 1 0 5 0 0 0.5 I 1 5 2 2 5 3 X , direction (in) Fig. 7-21 Verification of the DIP and FEM results of membrane deformation with stretching force 4.29 lb (Case 1 ) 3 2.5 2 C O c 1.5 • 5 1 0.5 0 0 0.5 1 1.5 2 2 5 3 X , direction (in) Fig. 7-22 Verification of the DIP and FEM results of membrane deformation with stretching force 4.74 lb (Case 1 ) Companson of nodal displacements ^ = 4 74(lb) Image Result FEM Result Ongmal Shape D EI =0.07 Image Result FEM Result - Ongmal Shape D EI =0.18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements. ^ = 5 135(tb) 2.5 G 1.5 0.5 Image Result FEM Result Ongmal Shape D EI =0.04 0.5 25 1 15 2 X, direction (in) Fig. 7-23 Verification of the DIP and FEM results of membrane deformation with stretching force 5.135 lb (Case 1) Comparison of nodal displacements. G6(lb) 3 2.5 2 T c o Z 1.5 S s i 0.5 0 0 0.5 1 15 2 2.5 3 X, direction (in) Fig. 7-24 Verification of the DIP and FEM results of membrane deformation with stretching force 6.66 lb (Case I) Image Result FEM Result Ongmal Shape O EI = 0.05 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements ^=8.5(11}) 25 Image Result FEM Result Ongmal Shape 0.5 2 5 X , direction (in) Fig. 7-25 Verification of the DIP and FEM results of membrane deformation with stretching force 8.5 lb (Case 1 ) Comparison of nodal displacements. ^ = 9 30b) 2.5 Image Result FEM Result Ongmal Shape 0.5 2 5 X, direction (in) Fig. 7-26 Verification of the DIP and FEM results of membrane deformation with stretching force 9.3 lb (Case I) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 U 15 > T 0.5 Comparison of nodal displacements F ^ l 745(tb) Image Result FEM Result Ongmal Shape DEI =0.11 0.5 1 15 2 X , direction (in) 25 Fig. 7-27 Verification of the DIP and FEM results of membrane deformation with stretching force 1.74 lb (Case 2) 2.5 c 1.5 * 6 x T * 0.5 Companson of nodal displacements Fj2=2 74(lb) Image Result FEM Result Original Shape DEI =0.14 0.5 1 15 2 X , direction (in) 25 Fig. 7-28 Verification of the DIP and FEM results of membrane deformation with stretching force 2.74 lb (Case 2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements ^ = 3 29S(tt>) 3 ----------- ------------ ------------ ------------ ------------ ,------ 2.5 5 1.5 0.5 Image Result FEM Result Original Shape D EI = 0.23 H 0.5 t 15 2 X , direction (in) 25 Fig. 7-29 Verification of the DIP and FEM results of membrane deformation with stretching force 3.29 lb (Case 2) Comparison of nodal displacements Fj2=4 29(Ib) 3 2 5 2 C * e o n 15 1 0.5 0 0 0.5 1 15 2 2.5 3 X , direction (in) Fig. 7-30 Verification of the DIP and FEM results of membrane deformation with stretching force 4.29 lb (Case 2) Image Result FEM Result Ongmal Shape DEI =0.16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 O 1.5 05 Comparison of nodal displacements. F =4 74(lb) r m Image Result FEM Result Ongtnal Shape DEI =0.09 0.5 1 15 2 X , direction (in) 2.5 Fig. 7-31 Verification of the DIP and FEM results o f membrane deformation with stretching force 4.74 lb (Case 2) Comparison of nodal displacements F^sfj 135(|b) 3 25 2 C 15 s • 6 i 0.5 0 0 0 5 1 1.5 2 25 3 X , direction (in) Fig. 7-32 Verification of the DIP and FEM results of membrane deformation with stretching force 5.13 lb (Case 2) Image Result - FEM Result Original Shape DEI =0.05 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements. ^ = 6 660b) 3 25 2 O -a »r i 0.5 0 0 0.5 1 15 2 2 5 3 X , direction (in) Fig. 7-33 Verification of the DIP and FEM results of membrane deformation with stretching force 6.66 lb (Case 2) Image Result FEM Result Ongmal Shape J — I - i I 'i Companson of nodal displacements. ^ = 8 50b) 3 2.5 2 C 15 S ■ 5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 X , direction 0n) Fig. 7-34 Verification of the DIP and FEM results of membrane deformation with stretching force 8.50 lb (Case 2) Image Result FEM Result Ongmal Shape DEI = 0.10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements. F ^ ^ S p b ) 2 5 Image Result FEM Result Original Shape 0.5 0.5 25 X , direction O n) Fig. 7-35 Verification of the DIP and FEM results of membrane deformation with stretching force 9.31 lb (Case 2) Comparison of nodal displacements. F ^ l 7450b) 3 2 5 2 C * c o c 15 S " Q i 0.5 0 0 0.5 1 15 2 25 3 X, direction O n) Fig. 7-36 Verification of the DIP and FEM results of membrane deformation with stretching force 1.74 lb (Case 3) Image Result - FEM Result Ongmal Shape DEI = 0.20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 C 1 5 s * 6 i 0.5 Campanson of nodal displacements F =2 74(lb) Image Result FEM Result Ongmal Shape DEI =0.35 0.5 25 1 15 2 X, direction (in) Fig. 7-37 Verification of the DIP and FEM results of membrane deformation with stretching force 2.74 lb (Case 3) Companson of nodal displacements. Fjf2=3 295(lb) J 2.5 ----- Image Result FEM Result 2 ■ = • ----- Onginal Shape • c o Z 15 DEI =0.41 ■o 1 I 1 0.5 0 0 0.5 1 15 2 2 5 3 X , direction (in) Fig. 7-38 Verification of the DIP and FEM results of membrane deformation with stretching force 3.29 lb (Case 3) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of nodal displacements. F)£ 2 =4 29(lb) 3 25 2 “ e * c o Z 1 5 s * 5 >r i 0.5 0 0 0.5 1 1 5 2 2 5 3 X , direction (in) Fig. 7-39 Verification of the DIP and FEM results of membrane deformation with stretching force 4.29 lb (Case 3) 3 2.5 2 C 1 5 s - 6 i 0.5 o 0 0.5 1 15 2 2.5 3 X, direction (in) Fig. 7-40 Verification of the DIP and FEM results of membrane deformation with stretching force 4.74 lb (Case 3) Companson of nodal displacements F]t2=4 74(!b) Image Result FEM Result Original Shape - DEI = 0.14 I / I I I I i ^ — i I I Image Result FEM Result Ongmal Shape DEI =0.28 / i 1 • -k. I _ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Companson of nodal displacements. Fx2=5.135(lb) 3 2.5 2 Z 15 s - 5 i 05 0 0 0.5 1 15 2 2 5 3 X , direction (in) Fig. 7-41 Verification of the DIP and FEM results of membrane deformation with stretching force 5.13 lb (Case 3) 3 2.5 2 ¥ C 15 S ■ 5 >r i 0.5 0 0 0.5 1 1 5 2 2 5 3 X , direction (in) Fig. 7-42 Verification of the DIP and FEM results of membrane deformation with stretching force 6.66 lb (Case 3) Companson of nodal displacements ®Hto) Image Result FEM Result Original Shape I DEI = 0.06 Image Result FEM Result Ongmal Shape DEI =0.11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Companson of nodal displacements. ^=8.5(11)) 2.5 Image Result FEM Result Original Shape DEI =0.14 0.5 0.5 2.5 X , direction (in) Fig. 7-43 Verification of the DIP and FEM results of membrane deformation with stretching force 8.50 lb (Case 3) Companson of nodal displacements Fx2=9.3(lb) 2.5 Image Result FEM Result Ongmal Shape DEI =0.15 0.5 0.5 25 X, direction (in) Fig. 7-44 Verification of the DIP and FEM results of membrane deformation with stretching force 9.31 lb (Case 3) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cases 1, 2 and 3 were different in their regression analysis objective functions. Observing from the results in Cases 1 and 2 and comparing with Case 3, it showed that Case 3 was more difficult to agree with the observation by digital image processing. To discuss this point by the stiffness point of view, the material property assumption in Case 3 seemed more rigid than that in Cases 1 and 2. Cases 1 and 2 were very closely tied in the approaching the observation by image processing. Observing the results of Case 1 (Figs. 7-18 through 7-26) and Case 2 (Figs. 7-27 through 7-35), the nonlinear FEM analysis using equivalent linear elasticity method did show the validness in the solution of this membrane study. They demonstrated the agreement in points between those in FEM’ s result and image processing observation. The method used in equivalent linear FEM analysis is based on the assumption of material nonlinearity. This meant that to calculate the displacement of the grids on the membrane surface by assuming that this rubber is material non-linear is appropriate. But this is not to conclude that the approach of geometrical nonlinearity is not appropriate. This is still beyond the scope of this study yet. In this study, FEM analysis was just playing a role to show the convergence of this method in its performance of dealing with nonlinear continuum system identification. The FEM is coded handily by using MATLAB program in order to be integrated in a prototype of toolkit doing system identification by digital image processing. In typical steps of analysis on the similar problems, the large- 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. deformation and nonlinearity of geometries could be handled by further infusing fancy commercial FEM programs. In FEM analysis, the output of the displacement in nodes was importantly concerned because we need to proceed verification in forward analysis. In this study, the displacement at the boundary points would be used to achieve this goal more often because of recalling that the Mooney-Rivlin's constants were initialized by using boundary points. The plot of the applied forces and displacement at the boundary points was shown to compare with that observed from image processing in Fig. 7-45. 10 9 6 7 6 5 4 A Image Observation 3 Mooney-Rivlin M odal, C1*19.6, C2*2&9 2 Mooney-Rivlin (Parabola), C1*12.8, C2-36.1 — Mooney-Rivlin, C1-36.6, C2-0 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Displacement (in) Fig. 7-45 Displacement at the boundary where external forces were applied 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The displacement at the boundary points was verified in an agreement with that result in the regression analysis. However, the error estimation in the mean time of analysis could not be ignored. The plot to show errors in image pixels is shown in Fig. 7-46. The error of nodal displacement on the surface was also evaluated by computing the root-mean-squares of nodal displacements at the mean time of analysis. Error analysis was a key issue controlling the major analysis mechanism. 1 0 Caaa 1: 01*19.6, 02*25.9 C I 8 * 7 a - ■ C * f 2: 01*12.6, 02*36.1 Cat* 3: 01*36.6, 02*0.00 6 5 4 3 2 1 0 0 2 1 3 5 8 4 7 6 9 10 Fore* (lb) Fig. 7-46 Root-Mean-Squares difference in nodal displacements for Cases 1,2 and 3 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.6.3 Error analysis In the nonlinear finite element analysis, the evaluation of strain satisfies the adaptive tolerance according to the range of the stress scale, and the change of the strains between two iterations. The error of stress is usually larger than that of strain. To maintain the estimation of error of stress for each loading stage, the adaptive strain error criteria was implemented in the finite element analysis of the rubber membrane. It is certain that the adaptive criteria for the stress error is less than 0.1 lb/in2 . However, the efficiency of finite element procedures would be improved by avoiding unnecessary iterations. By observation of the initial and final loading cases of the 2-D membrane experiment, strain (£2) of elastomer was found to be about 0.06 and 0.8 respectively. To adjust the tolerance criteria condition in strain by 0.01 would result in the increased error in the stress estimation when the stress is smaller. The error of stress was likely to approach 2.26 psi with the strain 0.06 and 0.664 psi when the stress corresponds to strain 0.8. Instead of controlling error of strain only, error on the stress was mainly controlled by using 0 .1 lb/in2 for tolerance criteria. Therefore, the adaptive error tolerance of strain would become 0.0044 and 0.015 for the cases of £2=0 .0 6 and £2=0 .8 , respectively. To further investigate the stability of error analysis on the finite element procedures, a strain error index (SEI) is introduced by F E M j e . D IP * (7-1 ) 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where FEM.e2 is the strain obtained by finite element analysis and DIP.e2 is the result from the digital image processing. Number n is the number of elements of the FEM meshing model. For the convenience, the displacement error index (DEI) is more often used instead of using SEI. The DEI is as According to the definition of error index, SEI is always greater than DEI. The Mooney-Rivlin’s model is used as the key equation to determine the constitutive relationship in the procedures of finite element analysis. For the convenience, the assumption of isoparametric elements has been used and the goodness of the Monney-Rivlin’s constants (Q and C2 ) estimated by DIP is verified by displacement error index (DEI). In consequence, the displacements obtained by images are compared with those from FEM intermediately by computing the root- mean-squares of nodal displacements between two methods. This procedure was to reconfirm whether or not it was made by good approximation to the FEM calculation. Therefore, the stress and strain were eligible to be computed. 7.7 Findings and Discussions 7.7.1 Findings This study gives an insightful experience of digital image processing dealing with identification for a nonlinear elastomeric membrane. This experience can be abstracted from the visualization of images to system analysis. Visualization of the (7-2) 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. marking nodes on the membrane surface is a key issue in this study. In the test, the nodes were shifted and increased by sizes due to the loads of forces. In order to accomplish the image taking, the position of camera has to be placed in a position that can take complete pictures covering the whole specimen's area. The robust image processing techniques such as segmentation, registration, and modification of geometry were employed. In the process of system analysis, the inverse analysis procedure is restricted with the limited input data. The inverse analysis includes the regression procedure and finite element verification. In the analysis, the parameters C| and Cj of the key equation (Mooney-Rivlin equation) has to be initialized by means of nonlinear curve fitting. The stress-strain relationship model is determined if Ci and C2 are determined. In the regression analysis, three cases were announced by using three different curve features. In them, Case 1 and Case 2 could achieve the result with errors of 1~2 pixels in forward verification. But, in Case 3, the error could approach more than 3 pixels especially in the last several stages of loads. This can be concluded by its regression result that the curve presented higher stiffness when the strain was larger enough, say, larger than 0.6. When the strain increases, the stress in Case 3 is much larger than in Case I and 2. By the observation o f Cases 1,2 and 3, the stress-strain relationship based on Mooney-Rivlin type is superior than that derived from Helmholtz's free energy theory. The error in this study achieved 5 pixels (~2 inches) in the last stage of loading. Such error represented the loss of the accuracy in estimating the strain on 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the membrane surface. Therefore, Case 3 has to be omitted and the assumption of objective curve is rejected. But, it is still hard to game between Case 1 and Case 2, since they can produce almost the same accuracy in the system analysis. However, C| and C2 values are completely different in Case 1 and Case 2 (Case 1: C|=19.6, C2=25.9 and Case 2: C,=12.9, C2=36.1). In addition to the care of finding an objective curve for regression, in this study, we must notice that the nonlinear FEM analysis is very sensitive in iterative routines. In the process of FEM analysis, the convergence criteria of computation used relative strain error by using the threshold of stress error within 0.1 psi. In the FEM analysis of seven respective loading stages, the error would not accumulate by the increment of loading. Because the analysis for each loading stage is independent of the other stages, the error of strain for the previous procedures would be accumulated in the current computation. The threshold of using Ao = 0.1 psi in the FEM analysis is quite appropriate and this study was presented with satisfied results. 7.7.2 Discussions Digital image processing was firstly applied to the system identification on the nonlinear continuum system, although it had been applied to many engineering mechanics applications. Traditional methods like using motion sensors and gauges are not suitable and inconvenient to perform system identifications, especially of membrane systems. With the limitation in measurement and dimension of sensor’ s size, the results of system identification would be affected by intrusion of sensors. Most of the time, to achieve certain accuracy, it needs a great amount of sensors 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. being installed on the system. It makes troubles in the sensing and loses the accuracy of measurement. It also needs complicated installing procedures. Sometimes, the sensors would fail to acquire data because of the damage o f wires. Compared with the traditional methods, image methods are less troublesome and without those phenomena. The image method can be assisted by remote controlling schemes and proceed the sensing in a non-intrusive and remotely way. It creates fewer obstacles for performing system identifications. This study is hard to apply the traditional sensing technique for this rubberlike membrane displacement measurement, because we may not be able to find appropriate sensors. The sensors need to be light and capable of measuring large deformation. Mot o f the time, the physical sensors would be intrusive in the measurement and affects the accuracy of displacement measurement. The more sensors installed, the worse measurement will be read. However, digital image method does not have the same problem. It is more appropriate to deal with this problem for sensing because image method is non-wired, non-touched, but easily controlled. In the proof-of-concept experiment, digital images stored the most realistic deformation geometry of the membrane surface. The observation could be used for many practical applications such as the studies of material homogeneity, incompressibility, etc. In the study, the material of membrane system was assumed to be homogeneous, isotropic with constant v = 0.24. It reduces the complexity to carry out the nonlinear FEM analysis. However, It causes two problems; the 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. analytical model for FEM analysis is too idealized and geometrical non-linearity is not in the consideration. With such simplification, FEM process was employed easier to perform nonlinear elastic system analysis by using the Mooney-Rivlin equation to invoke the material non-linearity in the middle o f analysis. The preliminary result showed that the simplification is quite reasonable and the assumption makes agreement of the results. It should be noted that Mooney-Rivlin equation is an empirical formula and it is derived for uni-axial stress-strain relationship. In order to evaluate stress-strain relationship in a 2-D system, it needs more rigorous attempt and to consider the material is non-homogeneous. As a proof-of-concept experiment, especially for the case of nonlinear continuum membrane system identifications, digital image processing demonstrated its convincing efficacy in studying the engineering mechanics problems. However, this method requires more robust hardware practicing and software implementation. The technique used in this study still has the room for improvement such as stated image segmentation method. It needs more care of classifying image features and distinguishing textures. There is necessity in the development of the segmentation technique, especially applying to more general cases. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions In this study, objectives have been achieved by fulfilling the proof-of- concepts of digital image processing and system identification. Based on the experiments, we can conclude to the following outlines: (1) Digital image processing method is useful for system identification, especially of nonlinear systems. The provided proof-of-concept experiments successfully accomplished the analysis loops of identifications; that is, inverse and forward analysis were carried out in order to verify the results o f identification. (2) Hardware is critical in the studies on the image-and-system identification problems. For example, the CCD camera was capable, when digitizing motion pictures, of achieving at most 30 frames per second. With this hardware (CCD and image grabber), the application will be limited according to its capability and performance. (3) Analytical model is crucial in the mechanical or structural system analysis. With respect to analytical analysis o f system-identification process, there are two major issues crucial to achieve the goals. First, it needs a correct model that can be fulfillment of the system by a mathematical model completely determined by 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a finite set of parameters. This model should be able to anticipate the behavior of the system within an acceptable tolerance. Second, it needs verification of these parameters based on the observed behavior of the system. Loop of system identification analysis has to be completed if both inverse identification and forward verification can be committed. (4) Developed algorithms are eligible to integrate with computer programs and instrumentation to perform identification in real-time. With modification of the current prototype of experimental configuration, it is possible to carry out nonlinear system identification with real-time, remote, and non-intrusive processing. (5) This study can contribute concepts in the following areas: (a) Digital image technologies. The DIP techniques were summarized and demonstrated to practically advanced applications such as in system identifications. (b) Information technologies. The study laid a basic foundation for other applications of digital image technologies in civil engineering in relation to robotics, artificial intelligence, communication and information technologies. (c) Cooperation in different disciplines. The carried out proof-of-concept experiments identified specific electrical engineering and computer science knowledge-base required for system identification of civil engineering systems. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6) These experiments provided insightful experience for further applications of digital image technologies in civil engineering. 8.2 Recommendations for Future Studies This research is an innovative process compared with traditional civil engineering approaches. It has great potential for applications such as in development of unsupervised change-detection systems, robotics, information technologies, etc. Yet, there is the room to improve the current studies with the applicable extensions: (1) The CCD camera is not the only image device capable of sensing in this subject. CCD cameras are designed as arrayed sensors for continuous signal gaining and transmission. This method is popular in applications of astronomic surveillance and security monitoring. For the application of system identification, digital and video cameras are also capable o f carrying out with equivalent effects. For example, some of the motion scopes used for industrial purpose can achieve approximately 1000-2000 frames per second within packaged hardware and software combinations. These products are currently the state-of-the-art in image technologies. They are recommended for system identifications. (2) Using digital image processing cooperated with mechanical analytical methods can supply the concise in-lab instrumentation for motion sensing and analysis. It still needs effort to carry out this idea into more in-situ and realistic applications. 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For in-situ instrumentation, it arises more concerns in hardware profiles. Resolution of optical devices and speed of digitization are all crucial. It needs further investigations on the feasibility study on the implementation of in-situ imaging systems for system identification. (3) In the base-isolation experiment, image processing involves complicated segmentation algorithm and extraction procedures that have been skipped by off line image modification. This requires further effort in the development of better algorithms and improvement in technique. (4) In the study of 2-D elastomer membrane problem, it is recommended to develop a more robust analytical method for the FEM analysis in investigating geometrical non-linearity. (5) The study of 3-D problems has not been carried out yet. It requires introducing more image devices to reconstruct a 3-D system with images. This will be a challenging topic in future studies. (6) Digital image technologies are bond to be integrated with information system to enlarge the extents of applications. The capability of current information technology leads into a wider thinking of its impacts to traditional and sophisticated theories. The development o f an integrated, automatic and unsupervised system is the trend. Applying advanced technologies to solve traditional obstacles will become a hot topic in future studies. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PART in REFERENCES Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES Barron, D., (1992), “Performance of Optical Flow Techniques,” the Proceedings o f IEEE Conference on Computer Vision and Pattern Recognition, Champain, IL, 236- 242. Bathe, K. J., (1996), Finite Element Procedures, Prentice-Hall, Inc. 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M., Watson, R., (1991), “Displacement Control Device for Base-Isolated Bridges,” Earthquake Spectra, VoL 7,179-200. Cook, R.D., Malkus, D.S., Plesha, M.E., (1989), Concepts and Applications o f Finite Element Analysis, 3rd Ed. John Wiley & Sons. Dezler, J. and Paulus, D., (1994), “Active Motion Detection and Object Tracking,” the Proceedings o f IEEE International Conference on Digital Image Processing, Nov. 13-16, Austin, TX, III, 635-639. Draper, S.E. and Rao, S.G., (1986), “Runoff Prediction Using Remote Sensing Imagery,” Water Resource Bulletin, Vol. 22,942-949 Environmental Systems Research Institute (ESRI), (1998), Using ArcView Image Analysis, Manual of ArcView Image Analysis Extension, ESRI. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Erman, B. and Mark, J.E., (1997), Structures and Properties o f Rubberlike Networks, Oxford University Press, NY, US Filatrault, A. and Cherry, S., (1987), “Performance Evaluation of Friction Damped Braced Steel Frame under Simulated Earthquake Loads”, Earthquake Spectra, VoL 3,57-8?. Howe, R. and Clemena, G.G., (1998), “An Assessment of the Feasibility of Developing and Implementing an Automated Pavement Distress System Incorporating Digital Image Processing”, Virginia Transportation Research Council Rep. No. VTRC 98-R1. Jahne, B., (1995), Digital Image Processing - Concepts, Algorithms, and Scientific Applications, 3rd Ed., Springer-Verlag. Kelvin, L.R., Transue, D. J., Schuller, M. P., (2000), “Acoustic Tomographic Imaging of Concrete Infrastructure”, Journal o f Infrastructure Systems, ASCE, Vol. 6., 15-23 Kreseski, D., Mercer, R.E., Barron, J. L., Joe, P., Zhang, H., (1994), “Storm Tracking in Doppler Radar Images”, the Proceedings o f IEEE International Conference on Image Processing, November 13-16, Austin, TX, VoL III, 226-230. Lai, J. and Mark, J. E., (1986), Advanced in Elastomers and Rubber Elasticity, Plenum Publishing Corporation, 393-406. Lee, H., (1993), “Fundamental Pavement Crack Imaging Algorithms”, Proceedings, EF/NSF Conference on Digital Image Processing: Techniques and Applications in Civil Engineering, Konda, Hawaii, 195-202 Lee, H. and Chou, E., (1993), “Survey of Image Processing Applications in Civil Engineering”, Proceedings, EF/NSF Conference on Digital Image Processing: Techniques and Applications, Konda, Hawaii, Feb. 28-March 5. Makar, J.M., (1999), “Diagnostic techniques for sewer systems”, Journal o f Infrastructure Systems, ASCE, VoL 5,69-78. MATLAB, (1998), Image Processing Toolbox, Mathwork Inc. Meirovitch, L., (1986), Elements o f Vibration Analysis, 2n d Ed., McGraw-Hill, 21-34. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Murota, N., (2000), “Earthquake Simulator Testing of Base Isolated Transformer/Bushing Systems”, M. S. Thesis, Department of Civil and Environmental Engineering, University of California, Irvine, CA. Pall, A.S., and Marsh, C., (1982), “Response of Friction Damped Braced Frames”, Journal o f Structural Division, ASCE, VoL 108, 1313-1323. Pall,A.S., Vezina, S., Proulx, P., Pall, R., (1993), “ Friction dampers for seismic control o f Canadian space agency headquarters”, Earthquake Spectra, VoL 9, 547- 557. Pekau, O. A. and Guimond, R., (1991), “Controlling seismic response of eccentric structures by friction dampers”, Earthquake Engineering and Structural Dynamics. VoL 20,505-521. Persson, B.J., (1998), Sliding Friction -Physical Principles and Applications, Springer, NY, 45-85. Pratt, W. K., (1991), Digital Image Processing, 2n d Ed., John Willey & Sons, Inc., NY. Press, W.H., Teulolsky, S.A., Vetterling, W. T., Flannery, B. P., (1992), Numerical Recipes in Fortran - The Art o f Scientific Computing, 2n d edition, Cambridge. Prestridge, E., (1993), “Digital Image Processing and Analysis in Material Science,” Proceedings, EF/NSF Conference on Digital Image Processing: Techniques and Applications in Civil Engineering, Konda, Hawaii. Rejaiesh, A., (2001), “Urban damage assessment using remotely sensed images”, the Proceedings o f the & h International Conference o f Structural Safety and Reliability, June 17-21, Newport Beach, CA. Rao, R.S., Gergely, P., White, R.N., (1995), Retrofit o f Non-Ductile Reinforced Concrete Frames Using Friction Dampers, Technical Report, NCEER-95-0020. National Center o f Earthquake Engineering Research, SUNY, Buffalo, New York Reismann, H. and Pawlik, P. S., (1980), Elasticity- Theorey and Applications, A Wiley-Interscience Publication, John Wiley & Sons, New York. Rossie, M., and Bozzoli, A., (1994), “Tracking and counting moving people”, the Proceedings o f IEEE International Conference on Digital Image Processing, Nov. 13-16, Austin, TX, II, 212-216. 145 of the copyright owner. Further reproduction prohibited without permission. Shinozuka, M., Chung, H.C., Liang, J., (2000), “Digital Image Processing for System Identification”, the Proceedings o f the 7 th International Symposium o f Society o f Photonics and Instrument Engineering on Smart Structure and Materials, Newport Beach, California, March 4-8. Shinozuka, M., Chung, H.C., Ichitsubo, M., Liang, J., (2001), “System Identification by Video Image Processing”, the Proceeding o f the International Symposium o f the Society o f Photonics and Instrument Engineers on Smart Structure and Materials, Newport Beach, California, March 3-8. Tanaka, M. and Dulikravich, G.S., (1998), Inverse Problems in Engineering Mechanics, Elsevier. Treloar, L., (1949), The Physics o f Rubber Elasticity, Oxford at the Claredon Press,UK. Wang, K.C., (2000), “Design and Implementations of Automated Systems for Pavement Surface Distress Survey”, Journal o f Infrastructure Systems, ASCE, Vol. 6,24-32. Wang, Y. P., Chung, L. L., Liao, W.H., (1998), “Seismic Response Analysis of Bridge Isolated with Friction Pendulum Bearings”, Earthquake Engineering and Structural Dynamics, 27,1009-1039. Williams, J. A., (1994), Engineering Tribology, Oxford Science Publication, NY. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PART IV APPENDICES Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDICES APPENDIX A. C ++Source Codes for Pendulum Experiments A.I Interfacial Programs for CCD, Frame grabber and MATLAB START OP PROGRAM - - - /I**.****.***.*******.**.****.****.***** PXGDI3-C T h is p ro g ram p r o v i d e s an e x a m p le o f d i s p l a y i n g v id e o i n a w indow . You s h o u ld s e t y o u r d i s p l a y d r i v e r t o 256 c o l o r s . T h is sam p le c o p i e s a c o m p le te fr a m e o f v id e o t o a b u f f e r a n d c a l l s S e tO I B its t o c o p y t h e b u f f e r t o a w indow . T he b u f f e r i s w r i t t e n u p s i d e down, l i n e f o r l i n e . S ee t h e f u n c t i o n G etlm ag e. I t u s e s G etRow O t o c o p y t h e im age u p s i d e down f o r S e tD I B its . t i n c l u d e < w indow s.h> # i n c l u d e < s t r i n g .h > # i n c l u d e < s t d l i b . h > # i n c l u d e < s t d i o .h > # i n c l u d e <com m dlg.h> # i n c l u d e < tim e .h > # i n c l u d e < m ath.h> # i n c l u d e "w p x 5 _ 9 5 .h " # i n c l u d e " c : \ m a t l a b \ e x t e r n \ i n c l u d e \ e n g i n e . h" # d e f i n e TEST_TIMKR 1 # d e f i n e ON 1 # d e f i n e OFF 2 / * - - - ....................................... T h is b u tto n s t r u c t u r e i s u s e d t o c r e a t e t h e f u n c ti o n b u t t o n s a s c h i l d w indow s. T h e r e a r e o t h e r w ay s t o do t h i s . W e f i g u r e d t h i s w o u ld b e t h e b e s t way t o do i t f o r o u r e x a m p le . */ # d e f i n e NBUTTONS 5 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ty p e d e f s t r u c t tagBUTTON { H W N D hw nd; lo n g s t y l e ; c h a r " t e x t ; i n t i d ; } BUTTON; / • T h e s e v a l u e s a r e u s e d t o c o n t r o l t h e p r o c e s s i n g o f t h e W M _C O M M A N D m e s s a g e . '/ # d e f in e # d e f in e # d e f in e # d e f in e # d e f in e # d e f in e ID_FBASE ID_START ID_STOP ID_WRITE ID_TIME ID EXIT 101 ID_FBASE+0 ID_FBASE+1 ID_FBASE+2 ID_FBASE+3 ID FBASE+4 BUTTON F u n c tio n s [NBUTTONS] = { {NULL, BS_PUSHBUTTON, * S t a r t " {NULL, BS_PUSHBUTTON, " S to p " , {NULL, BS_PUSHBUTTON, " P r o c e s s {NULL, BS_PUSHBUTTON, " T im e r " {NULL, BS_PUSHBUTTON, " E x i t " , }; ID_START>, ID_STOP} , " , ID_WRITE}, ID_TIME}, ID _EX IT}, s t a t i c u n s ig n e d B u tto n T e x tL e n ; s t a t i c i n t B u tto n H e ig h t, B u tto n W id th ; s t a t i c i n t V ie w O ffs e t; BOOL A pplnit(H IN STA N CE h l n s t , HINSTANCE h P r e v , LPSTR szC m dL ine, i n t s w ) ; LONG WINAPI C tlP r o c (HWND, UINT, HP ARAM, LPARAM) ; H W N D C reateC ontrolW indow (H IN ST A N C E h P re v , LPSTR szC m dL ine, i n t sw) ; v o id R egisterW indow C lasses(H IN S T A N C E h P r e v , LPSTR szC m dL ine, i n t s w ) ; v o id A p p E x it (v o id ) , * BOOL A p p l d l e ( v o i d ) ; v o id AppPaint(HW ND hwnd, HDC h d c ) ; v o id G etG lobals(H W N D hw nd, LPARAM lP a r a m ) ; v o id G e t Im ag e (FRAMEHANDLE f r h ) ; v o id DrawButtons(HW ND hw nd, LPARAM lP a ra m , i n t d r a w ) ; 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v o i d S etB itM ap H ead (v o id ) ; i n t A llo c B u f f e r (v o id ) ; v o i d D o T est (HW ND hwnd) ,- i n t M a tP ro c e ssIm a g e (HWND) ; / * t e x t d a t a f o r d ra w in g t e x t * / s t a t i c i n t cx C h ar, / * a v e r a g e c h a r a c t e r w id th */ c y C h a r; /* c h a r a c t e r h e i g h t * / s t a t i c HINSTANCE h A p p In s t; s t a t i c H W N D h w n d C tl; i c c h a r szAppName[] = "Im ag e P r o c e s s in g " ; s t a t i c c h a r s z T i t l e B a r [100] ; s t a t i c c h a r s z T i t l e [ ] = "Im ag e G r a b b e r (32 b i t s ) " ; s t a t i c i n t iB o a rd R e v ; s t a t i c BOOL f S t a r t ; s t a t i c BOOL p x 5 1 0 _ 6 1 0 ; / / i f f a l s e , t h i s i s a PX500 b o a r d . s t a t i c i n t ImageMaxX, ImageMaxY, WindowX, WindowY; s t a t i c i n t S t a r t T e s t , T e s t i n g , F ram eC o u n t, E r r o r C o u n t; s t a t i c FGHANDLE f g h ; s t a t i c FRAMEHANDLE f r h [ 2 ] ; s t a t i c tagQ [2],* s t a t i c i n t f r h l d x ; /* f r h l d x , in d e x e s b o th f r h [ ] a n d ta g Q [] * / / * ------------------------------------- < GDI F u n c tio n s >------------------------------------------------- */ v o id C r e a te G r a y P a le t te (v o id ) ; s t a t i c HP ALETTE h p a l e t t e ; s t a t i c HANDLE h B u f; s t a t i c BYTE * g p B its ; s t r u c t { BITMAPINFOHEADER h e a d ; 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RGBQUAD c o l o r s [2 5 6 ]; } m aphead; / * -------------------------------------------------- < f i l e d a t a >--------------------------------------------- * / i n t pxW riteFile(H W N D , FRAMBHANDLE) ; v o id I n itia liz e F ile S a v e ( H W N D ) ; s t a t i c OPENFILENAME s f n ; s t a t i c c h a r s z F i l t e r S e l [] = " T e x t F i l e s ( * . t x t ) \ 0 * . t x t \ 0 " ; # d e f in e MAXPATH 128 ttd e fin e MAXDIR 66 i d e f i n e MAXEXT 5 s t a t i c c h a r szSaveFileNam e[M AXPATH] ; s t a t i c c h a r szS av eF ileT itle[M A X P A T H ] ; s t a t i c c h a r szS a v e F ile D ir[M A X D IR ]; s t a t i c c h a r szSaveExt[M A X EX T]; / * -------------------------------------------------- my V a r i a b l e s * / s t a t i c c h a r tm p [2 0 1 ]; s t a t i c lo n g m y G lo b a lC o u n te r; s t a t i c c l o c k _ t m y C lo c k [3 0 * 6 0 * 6 0 * 2 4 ]; * Name: W inMain * * D e s c r i p t i o n : M ain w indow p ro g ra m a n d m e ssa g e p r o c e s s i n g lo o p . * i n t PASCAL WinMain(HINSTANCE h l n s t , HINSTANCE h P re v , LPSTR szC m dL ine, i n t sw) { M SG m sg; h A p p In st = h l n s t ; /* s a v e f o r l a t e r * / /* C a l l i n i t i a l i z a t i o n p r o c e d u r e */ i f ( ! A p p I n i t ( h l n s t , h P re v , szC m d L in e, s w )) { A p p E x it( ) ; r e t u r n FALSE; } /* ---------------- P o l l i n g m e s s a g e s from e v e n t q u e u e 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. */ fo r(;;) { i f (P eek M essag e (&msg, NULL, 0 , 0,PM_REMOVE) ) { i f (m sg .m e ssag e == WM_QUIT) b r e a k ; T ra n s la te M e s s a g e (&msg) , - D isp a tc h M e ssa g e (& m sg ) ; } e l s e { i f (A p p ld le {)) W a itM e ssa g e ( ) ; } } A p p E x it( ) ; r e t u r n m sg.w P aram ; } /****************.*****.*********************************.** . Name: A p p ln it * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . * * * * * * * * * * * / BOOL A p p l n i t (HINSTANCE h l n s t , HINSTANCE h P r e v , LPSTR szC m d L in e, i n t sw) { i n t v id e o L in e s , v B la n k L e n g th ; f g h = 0 ; f S t a r t = FALSE; hB u f = NULL; T e s t i n g = FALSE; m y G lo b a lC o u n te r = 0 ; / * -------------- i n i t i a l i z e t h e l i b r a r y - * / i f ( ! I n i t L i b r a r y () ) { M e ssag e B o x (0 , " I n i t L i b r a r y F a i l e d " , szA ppN am e, MB_OK) ; r e t u r n FALSE; } /* ---------- a l l o c a t e an y fra m e g r a b b e r . / fg h = A l l o c a t e F G ( - l ) ; i f ( I f g h ) { M e ssag e B o x (0 , " A llo c a te F G F a i l e d " , szAppN am e, MB_OK); 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r e t u r n FALSE; } R e s e tF G (f g h ) ; / / d e t e r m i n e i f b o a r d i s a PX500 o r P X 510/610 px510_610 * R ead C o n f i g u r a t i o n (fg h ) & PXC_CUSTOM_HW; s w itc h (V id eo T y p e ( f g h ) ) { c a s e 1 : /* NTSC, same f o r a l l m o d e ls */ ImageMaxX - 3 2 0 ; ImageMaxY = 2 4 0 ; S e t Im ag es i z e ( f g h , ImageMaxX, 1 2 8 , 0 , 0 , ImageMaxX, ImageMaxY, 8) ; b r e a k c a s e 0 : / ‘ no v i d e o * / d e f a u l t : / ‘ unknown o r n o n - i n t e r l a c e d v i d e o * / c a s e 2 : /* CCIR * / i f ( !px510 610) { ImageMaxX = 6 4 0 ; ImageMaxY = 5 1 2 ; S e t Im ag es i z e ( f g h , ImageMaxX, 2 5 6 , 0 , 0 , ImageMaxX, ImageMaxY, 8) ; } e l s e / / t h i s i s a PX510 o r PX610 b o a r d { ImageMaxX = 6 4 0 ; ImageMaxY = 5 1 2 ; / / e x t r a s t e p n e c c e s a r y f o r PX 510/610 / / i n o r d e r t o d i s p l a y t h e e n t i r e 572 l i n e s , / / n e e d t o u s e USER SYNC v id e o L in e s = 2 8 8 ; v B la n k L e n g th = 2 4 ; S e tV id e o F o rm a t ( f g h , v id e o L in e s , v B la n k L e n g th , USER_SYNC) ; S e t Im ag es i z e ( f g h , ImageMaxX, 2 5 6 , 1 5 , 0, ImageMaxX, ImageMaxY, 8) ; } b r e a k ; } iB o ard R ev = R e a d R e v is io n (f g h ) ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /* - — ................. a l l o c a t e tw o fra m e b u f f e r s ■*/ f r h [ 0 ] - A l l o c a t e B u f f e r (ImageM axX, ImageMaxY, 8) ; i f ( ! f r h [ O l ) { M e ssa g e B o x (0 , " A llo c a te B u f f e r F a i l e d o n f r h [ 0 ] " , szAppName, MB_OK); r e t u r n FALSE; } f r h [ l ] = A l l o c a t e B u f f e r (ImageMaxX, ImageM axY, 8) , - i f ( ! f r h [1 ]) { M e ssag e B o x (0 , " A llo c a te B u f f e r F a i l e d o n f r h f l ] " , szAppName, MB_0K) ; r e t u r n FALSE, - } S e tB itM a p H e a d O ; i f ( S A llo c B u f f e r ()) { M e ssa g e B o x (0 , " F a il e d t o a l l o c a t e s c r a t c h b u f f e r " , szAppName, MB_0K) ; r e t u r n FALSE; } /* c r e a t e windows */ R e g is te rW in d o w C la s s e s (h P re v , szC m d L in e, sw) ; h w n d C tl = C re a te C o n tro lW in d o w (h P re v , szC m dL ine, sw) ; i f ( lh w n d C tl) { M essag eB o x (0, " C a n n o t c r e a t e c o n t r o l w indow ", szAppN am e, MBJDK) , - r e t u r n FALSE; } Show W indow (hw ndC tl, s w ); E n a b le W in d o w (F u n c tio n s [ID_STOP-ID_FBASE] .hw nd, 0) ; E nableW indow (F u n c t io n s [ ID_WRITE - ID_FBASE] . hw nd, 0) ; S e tT im e r (h w n d C tl, TEST_TIMER, 2 0 0 0 ? NULL) ; / * 2 - s e c o n d ti m e r */ C r e a t e G r a y P a l e t t e () ; I n i t i a l i z e F i l e S a v e ( h w n d C t l ) ; r e t u r n TRUE; } /******************************************************************* * Name: A p p E x it * 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v o i d A p p E x it(v o id ) { i n t i ; / * w a it f o r t h e g r a b s t o s t o p b e f o r e d e l e t i n g t h e b u f f e r s */ f o r { i = 0 ; i < 2 ; + + i) { i f ( t a g Q t i ] ) w h i l e ( ! I s P i n i s h e d ( f g h , t a g Q [ i ] ) ) W a itV B (fg h ); } f o r ( i = 0 ; i< 2 ; + + i) { i f ( f r h [ i ] ) F reeF ram e ( f r h [ i j ) ; } if ( h B u f ) { G lo b a lU n lo c k (h B u f) ; G lo b a 1F r e e ( h B u f ) ; } i f ( h p a l e t t e ) D e l e t e O b j e c t ( h p a l e t t e ) ; i f ( f g h ) F re e F G ( fg h ); E x i t L i b r a r y ( ) ; } . * Name: A p p ld le * BOOL ^ j p l d l e ( v o i d ) { HDC h d c ; i f ( I s l c o n ic ( h w n d C t l ) ) r e t u r n TRUE; i f (G e tK e y S ta te (VK_LBUTTON) >=0) { / * swap t h e in d e x fro m 0 t o 1 o r v i c e v e r s a * f f r h l d x = ( f r h l d x == 1) ? 0 : 1 ; /* co p y t h e v i d e o ram to a memory b u f f e r * / i f ( t a g Q t f r h l d x ] ) { w h i l e ( ! I s F i n i s h e d ( f g h , t a g Q [ f r h l d x ] )) W a itV B (fg h ); i f ( C h e c k E r r o r ( f g h ) ) { + + E rro rC o u n t; } 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e l s e { G e t l m a g e ( f r h [ f r h l d x ] ) ; m yC lock [m y G lo b a lC o u n te r] = c lo c k () ,- + + F ram eC o u n t; } t a g Q [ f r h I d x ] = 0 ; } if(T R U E ) { t a g Q [ f r h I d x ] = G r a b ( f g h , f r h [ f r h l d b t ] , QUEUED) , - h d c * G e tD C (h w n d C tl); A p p P a in t(h w n d C tl, h d c ) ; R e le a se D C (h w n d C tl, h d c ) ; } i f ( f S t a r t ) i f ( ip x W r ite F ile ( h w n d C tl, f r h f f r h l d x ] )) f S t a r t = FALSE; r e t u r n FALSE; } e l s e { r e t u r n TRUE; / / b a c k g ro u n d a p p ; n o t h i n g t o d o . } } * . * Name: A p p P a in t * **************** **********************************************/ v o id A p p P a in t (HW ND hwnd, HDC h d c ) { i f ( h p a l e t t e ) { S e l e c t P a l e t t e ( h d c , h p a l e t t e , TRUE); R e a l i z e P a l e t t e ( h d c ) ; } S e tD I B its T o D e v ic e ( h d c , 0 , V ie w O f f s e t, ImageMaxX, ImageM axY, 0 , 0 , 0 , ImageMaxY, g p B i t s , (LPBITMAP INFO) im a p h e a d , DIB RGB COLORS); } " ' / * * * * * * * * * * * * * * * * * * * * * * * * Name: C tlP r o c * 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * D e s c r i p t i o n : M ain w indow p r o c e d u r e . * LONG WINAPI CtlProc(HW ND hw nd, OINT m e s s a g e , W PA RA M w Param , LPARAM IP a ra m ) { / /MINMAXINFO *lpm m i; HDC h d c ; PAINTSTRUCT p s ; RECT R e c t; d o u b le d e ltT im e , myFPS; s w i t c h (m essage) { c a s e WM_CREATE: G e tG lo b a ls (h w n d , IP a ra m ); G etW indow R ect(hw nd, & R e c t); S etH indow P os (hw nd, NULL, R e c t . l e f t , R e c t . t o p , WindowX, WindowY, SWP_SHOWWINDOW) , - D ra w B u tto n s(h w n d , IP a ra m , TRUE); r e t u r n OL; / * i f y o u w ish t o l o c k t h e window s i z e , un-com m ent t h i s c o d e */ c a s e WM_GETMINMAXINFO: //lp m m i = (MINMAXINFO * ) IP aram ; //lp m m i-> p tM in T ra c k S iz e .x = WindowX; / / lp m m i-> p tM in T ra c k S iz e .y = WindowY; / / lp m m i-> p tM a x T ra c k S iz e .x = WindowX; / / lpmmi - >ptM axT rackS i z e . y = WindowY ; / / lpmmi -> p tM a x S iz e .x = WindowX; / / lp m m i-> p tM a x S iz e .y * WindowY; r e t u r n OL; c a s e WM_SIZE: In v a lid a te R e c t( h w n d , NULL, TRUE); D ra w B u tto n s(h w n d , IP a ra m , FALSE); r e t u r n OL; c a s e WM_PAINT: h d c = B e g in P a in t (hw nd, &ps) , - A p p P a in t(h w n d ,h d c ); E n d P a in t(h w n d ,& p s ); r e t u r n OL; c a s e WM_PALETTECHANGED: In v a lid a te R e c t( h w n d , NULL, FALSE); r e t u r n OL; c a s e WM_TIMER: i f ( T e s t i n g | | S t a r t T e s t ) D o T e st(h w n d ); r e t u r n OL; 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. c a s e W M _COM M AND: sw itc h (w P a ra m ) { d e f a u l t : b r e a k ; c a s e ID_START: m y G lo b a lC o u n te r = 0 ; f S t a r t = TRUE; EnableW indow ( F u n c tio n s [ID_START-ID_FBASE] .h w n d , 0) ; E n a b le W in d o w (F u n c tio n s [ID_STOP-ID_FBASE] .h w n d , 1) ; r e t u r n OL; c a s e ID_ST0P: f S t a r t = FALSE; /******** r e p o r t t h e r e s u l t s * * * * * * * * * * * * * * * * / d e ltT im e = (d o u b le ) (m yC lock [m y G lo b a lC o u n te r-1 ]- m y C lo c k [0 l) tim e : %d", d e ltT im e = d e ltT im e / (d o u b le ) CLOCKS_PER_SEC; myFPS = m y G lo b a lC o u n te r / d e ltT im e ; w s p r i n t f (tm p, (LPSTR) "F ra m es p e r s e c o n d : % d \n R eco rd in g (in t)m y F P S , ( i n t ) d e l t T i m e ) ; M essageB ox(hw nd, tm p , " R e c o rd in g R e s u l t s " , MB_OK); EnableW indow ( F u n c tio n s [ID_START-ID_FBASE] .hw nd, 1) ; E n a b le Window ( F u n c tio n s [ID_STOP-ID_FBASE] .h w n d , 0) ; E nableW indow ( F u n c tio n s [ID_WRITE- ID_FBASE] . hw nd, 1) ; r e t u r n OL; c a s e ID_WRITE: i f (M a tP ro c e ssIm a g e ( h w n d C tl)) { //E n a b le W in d o w (F u n c tio n s [ID_WRITE-ID_FBASE] .hw nd, 0 ) ; r e t u r n OL; } b r e a k ; c a s e ID_TIME: S t a r t T e s t = TRUE; r e t u r n OL; c a s e ID_BXIT: D estroyW indow (hw nd) ; r e t u r n OL; } b r e a k ; c a s e WM_DESTROY: K illT im e r(h w n d , TEST_TIMER); P o s tQ u itM e s s a g e ( 0 ) ; 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r e t u r n OL; } r e t u r n (D efW indow Proc (hw nd, m e s s a g e , w Param , I P a r a m ) ) ; } ♦ * Name: D ra w B u tto n s * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / v o i d D ra w B u tto n s (H W N D hw nd, LPARAM IP aram , i n t d ra w ) { i n t i ; i n t rem ; RECT R e c t ; G e tC l i e n t R e c t (hwnd, (LPRECT) & R ect) ; B u tto n W id th = R ect.rig h t/N B U T T O N S ; rem = R e c t . r i g h t % NBUTTONS; i f ( d r a w ) { / * — - ............ c r e a t e t h e f u n c ti o n b u t t o n s a s c h i l d w indow s - * / f o r ( i = 0 ; i<NBUTTONS; i+ + ) { i f ( i == NBUTTONS-1 ) F u n c tio n s [ i ] .hw nd = C re a te W in d o w (" b u tto n " , F u n c t i o n s [ i] . t e x t , WS_CHILD I WS_VISIBLE | F u n c tio n s [ i] . s t y l e , B u tto n W id th * i, 0 , B u tto n W id th + re m , B u tto n H e ig h t, hwnd, (HMENU) F u n c t i o n s t i ] . i d , h A p p In s t, NULL) , - e l s e F u n c t io n s [ i] .h w n d = C re a te W in d o w (" b u tto n " , F u n c t i o n s [ i ] . t e x t , WS_CHILD | WS_VISIBLE | F u n c tio n s [ i] . s t y l e , B u tto n W id th * i, 0, B u tto n W id th , B u tto n H e ig h t, hw nd, (HMENU) F u n c t i o n s [ i ] . i d , h A p p In s t, NULL); } } e l s e { / * ----------------------------------------------------- r e s i z e t h e f u n c ti o n b u t t o n s */ f o r ( i = 0 ; i<NBUTTONS; i+ + ) { i f ( i == NBUTTONS-1 ) S etW indow P os ( F u n c tio n s [ i ] .h w n d ,N U L L ,B u tto n W id th * i, 0 , B u tto n W id th + rem , B u tto n H e ig h t, SWP_SHOWWINDOW) ; e l s e S etW indow P os ( F u n c tio n s [ i ] .h w n d ,N U L L ,B u tto n W id th * i,0 , 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ButtonWidth, ButtonHeight, SWP_SHOWWINDOW) ; } } } * * Name: C r e a te G r a y P a le t te * v o id C r e a te G r a y P a le t te (v o id ) { i n t i ; s t r u c t { W O RD V e rs io n ; W ORD N u m b e rO fE n trie s ; PALKTTtlENTRY a E n t r i e s [256] ; } P a l e t t e ; i f ( h p a l e t t e ) D e l e t e O b j e c t ( h p a l e t t e ) ; P a l e t t e . V e r s i o n = 0 x 3 0 0 ; P a l e t t e .N u m b e rO fE n trie s = 2 5 6 ; f o r (i= 0 ; i< 2 5 6 ; + + i) { P a l e t t e . a E n t r i e s [ i] .p e R e d = i ; P a l e t t e . a E n t r i e s [ i ] . p e G r e e n = i ; P a l e t t e . a E n t r i e s [ i ] . p e B l u e = i ; P a l e t t e . a E n t r i e s [ i ] . p e F l a g s = PC_NOCOLIAPSE; raap h ead . c o l o r s [ i ] . rgbR ed = i ; m ap h ead . c o l o r s [i ] . rg b G reen = i ; m a p h e a d .c o l o r s [ i ] . rg b B lu e = i ; m a p h e a d .c o lo r s [ i ] .r g b R e s e r v e d = 0 ; } h p a l e t t e = C r e a t e P a l e t t e ( (LOGPALETTE * ) & P a l e t t e ) ; } /*************** * * Name; G etIm age * T h is f u n c ti o n c o p i e s t h e fra m e b u f f e r t o a memory b u f f e r u p s id e * dow n. S t r e t c h D I B i t s e x p e c ts t h e im ag e t o b e r e v e r s e d l i n e f o r * l i n e . I f y o u w a n t t h e im age t o b e r i g h t s i d e up i n t h e b u f f e r you * n e e d t o m o fify t h e d a t a i n t h e c a l l t o S tr e t c h D I B i t s a n d i n th e * m aphead s t r u c t u r e . * 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v o id G et Im age (FRAMEHANDLE f r h ) { i n t i ; BYTE *p; p = g p B i ts ; f o r (i=Im ageM axY-1; i> = 0 ; - - i ) { G e tR o w (frh , p , i ) ; p += ImageMaxX; } } * * Name: SetB itM apH ead * v o id S etB itM ap H ead (v o id ) { /* s e t up b itm a p h e a d e r */ maphead.head.biSize^sizeof(BITM APINFOHEADBR) ; m aphead. h e a d . biw idth=Im ageM axX ; m aphead. h e a d . biH eight=Im ageM axY ; m aphead. h e a d . b i P l a n e s = l ; m aphead. h e a d . b iB itC o u n t= 8 ; m aphead. h e a d . biC om pression=B I_R G B ; m aphead. h e a d . b is iz e lm a g e = 0 ; m aphead. h e a d . b iX P e ls P e rM e te r= 0 ; m aphead. h e a d . b iY P e ls P e rM e te r= 0 ; m aphead. h e a d . b iC lrU s e d = 0 ; m aphead. h e a d . b i d r Im p o rt a n t =0 ; } * Name: R e g is te rW in d o w C la s s e s * v o id R eg iste rW in d o w C la sse s (HINSTANCE h P rev , LPSTR szC m dL ine, i n t sw) { WNDCLASS wc; memset(&wc,0 , s i z e o f ( w c ) ) ; /* c l e a r s t a c k v a r i a b l e * / /* r e g i s t e r t h e c o n t r o l window c l a s s */ i f (!h P rev) { 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w c . s t y l e = CS_BYTEALIGNWINDOW | CS_VREDRAW | CS_HREDRAW CS_DBLCLKS ; w c. lpfnW ndProc w c. c b C ls E x tr a w c. cbW ndExtra w c .h l n s t a n c e w c. h l c o n w c. h C u rs o r w c. h b rB ack g ro u n d w c. lpszMenuName w c. lp szC lassN am e (WNDPROC)CtlProc; 0 ; 0 ; h A p p In s t; NULL; L o ad C u rso r (MULL, IDC_ARROW) (HBRUSH) (COLOR_WINDOW + 1) ; NULL; s zAppName; R e g i s t e r C l a s s (&wc); * Name: C reateC o n tro lW in d o w * **** **********************************************************/ H W N D CreateControlW indow(HINSTANCE h P rev , LPSTR szC m dL ine, i n t sw) { HW ND hwnd; w s p r i n t f ( s z T i t l e B a r , "%s - - R e v is io n : %X", s z T i t l e , (U IN T )iB oardR ev); = CreateW indow (szAppName, / / window c l a s s name s z T i t l e B a r , WS_OVERLAPPEDWINDOW, / / window s t y l e CW_USEDEFAULT, / / i n i t i a l x p o s i t i o n CW_USEDEFAULT, / / i n i t i a l y p o s i t i o n CW_USEDBFAULT, / / i n i t i a l x s i z e CW_USEDEFAULT, / / i n i t i a l y s i z e NULL, / / p a r e n t w indow h a n d le NULL, / / window menu h a n d le h A p p In s t, / / p rogram i n s t a n c e handle NULL); / / c r e a t i o n p a r a m e t e r s r e t u r n hwnd; } * * Name: G e tG lo b a ls * v o id GetGlobals(HWND hwnd, LPARAM IParam ) 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. { i n t i ; HDC h d c ; TEXTMETRIC tm,* / * t e x t m e t r i c s t r u c t u r e * / /* g e t t h e maximum b u t t o n l e n g t h * / B u tto n T e x tL en = 0; f o r ( i = 0 ; i<NBUTTONS; i++) if ( B u tto n T e x tL e n < s t r l e n ( F u n c t i o n s [ i ] . t e x t ) ) B u tto n T e x tL en = s t r l e n ( F u n c t i o n s [ i ] . t e x t ) ; hdc = GetDC (h w n d C tl); S e l e c t O b j e c t (hdc, G e tS to c k O b je c t (SYSTEM_FONT)) ; G e tT e x tM e tric s (h d c, &tm) ,* R eleaseD C (hw ndCtl, h d c ) ; cx C h ar = tm .tm A veC harW idth; cy C h ar = tm .tm H e ig h t + tm .tm E x te rn a lL e a d in g ; B u tto n H e ig h t = 4 * c y C h a r /3 ; V ie w O ffs e t = B u tto n H e ig h t ,- /* c a l c u l a t e th e i n i t i a l window s i z e * / if(GetSystemM etrics(SM _CXSCREEN) > 6 4 0 .0 ) { WindowX = ImageMaxX + (G etS y stem M etrics (SM_CXFRAME) *2) ; WindowY * ImageMaxY + B u tto n H e ig h t + G e tS y ste m M e tric s (SM_CYCAPTION) + G e tS y s te m M e tric s (SM CYFRAME)*2 ; }" e l s e { WindowX = G e tS y s te m M e tric s (SM_CXSCREEN) * 3 /4 ; WindowY = G e tS y s te m M e tric s (SM_CYSCREEN) * 3 /4 ; } } * Name: A llo c B u f f e r * i n t A llo c B u f f e r ( v o id ) { hBuf = G lo b a lA llo c (GMEM_FIXED, ImageMaxX * ImageMaxY) ; i f ( h B u f == NULL) r e t u r n FALSE; g p B its = (BYTE *) G lo b a lL o c k (h B u f) ; r e t u r n TRUE; } 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * Name: D oTest * v o id DoTest(HNND hwnd) { s t a t i c i n t i , T im er; s t a t i c c l o c k _ t S ta r t T i m e ; s t a t i c d o u b le d T e s tT im e , dFPS; i f ( S t a r t T e s t ) { f o r ( i = 0 ; icNBUTTONS; ++i) EnableWindow ( F u n c tio n s [i] .hwnd, 0) ; EnableWindow ( F u n c tio n s [ID_TIME-ID_FBASE] .hwnd, 1) ; Timer = 9; FrameCount = 0; E rro rC o u n t = 0; S t a r t T e s t = FALSE; T e s tin g = TRUE; S ta rtT im e = c l o c k 0 ; } e l s e { --T im e r; E nable Window ( F u n c tio n s [ID_TIME-ID_FBASE] .hwnd, Timer%2) ; if ( T im e r < 0) { dT estT im e = (d o u b le ) ( c lo c k ( ) - S ta rtT im e ) / (double) CLOCKS_PER_SEC; dFPS = Fram eC ount / dT estT im e; T e s tin g s FALSE; w s p r in tf ( tm p , (LPSTR)"Frames p e r s e c o n d : % d\nT otal E r r o r s : %d", (i n t ) dFPS, E r r o r C o u n t) ; M essageBox(hwnd, tm p, " T e st R e s u l t s " , MB_OK) ; f o r ( i = 0 ; i <NBUTTONS; ++i) EnableW indow ( F u n c tio n s [i] .hwnd, 1) ; } } * * Name: p x W r it e F i l e * i n t p x W r ite F ile (HWND hwnd, FRAMEHANDLE f r h ) { i n t r e t ; c h a r te m p F ile [5 0 ] = " c : \\T e m p \\tm p " , b u f [ 1 0 ] ; s t a t i c c h a r " s z W r i t e E r r o r = \ 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "An e r r o r was e n c o u n te r e d w r i t i n g d a t a t o t h e file.\n\ The f i l e i s i n c o m p le te and t h e d a t a i s i n v a l i d . " ; /* w r i t e t h e f i l e t o a v i r t u a l h a r d d i s k * / _ i t o a (m y G lo b alC o u n ter, b u f , 1 0 ) ; s t r c a t ( te m p F ile , b u f ) ; r e t = W riteB M P (frh , te m p F ile , 1) ; i f ( r e t != 0) { w s p r i n t f ( tm p , "W rite e r r o r : %d", r e t ) ; M essageBox(hw nd, s z W r ite E r r o r , tm p, MB_OK); r e t u r n FALSE; } + + m yG lobalC ounter; EnableW indow (F u n c tio n s [ ID_START - ID_FBASE] .hwnd, m yG lobalCounter% 30 >= 15) ; r e t u r n TRUE; } i n t M atP ro cessIm ag e (HW ND hwnd) { i n t r e t ; lo n g i ; d o u b le * x P tr, * y P t r , re lT im e ; c h a r te m p F ile [ 2 0 ] , b u f [50], ♦ e n g S trin g « ” [X,Y]* m y M atG etL o catio n ( ' " ; mxArray *x= NULL, *y= NULL; E ngine "m yM atEngine; FILE " m y F ile P tr ; /* c a l l m atled) e n g in e t o p r o c e s s th e image */ i f ( IG etSaveF ileN am e (&sfn) ) r e t u r n FALSE; i f ( ! ( m y F ile P tr= f o p e n ( s f n . I p s t r F i l e , "w "))) { M essageB ox ( (HWND)NULL, (LPSTR) " C a n 't Open F i l e f o r W r ite !", (LPSTR) " O p e n F ile " , MB_OK); r e t u r n FALSE; } i f (I (myM atEngine = engOpen (NULL))) { M essageB ox ((HWND)NULL, (L PST R )"C an't s t a r t MATLAB e n g i n e !" , (LPSTR) "M atE n g in e", MB_0K); r e t u r n FALSE; } f o r ( i = 0; i < m y G lo b alC o u n ter; i++) { _ i t o a ( i , b u f , 1 0 ); l s t r c p y ( t e m p F i l e , " c :\\T e m p \\tm p " ) ,- l s t r c a t (te m p F ile , b u f ) ; l s t r c p y ( b u f , e n g S t r i n g ) ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. l s t r c a t ( b u f , te m p F il e ) ,- l s t r c a t ( b u f , " ' ) ; " ) ; r e t = e n g E v a lS tr in g (myM atEngine, b u f) ; i f ( r e t != 0) { w s p r i n t f ( tm p , "M atE ngine e r r o r : %d", r e t ) ; M essageBox (hwnd, "M atEngine i s no l o n g e r r u n n i n g ! " , tm p, MB_OK) ; r e t u r n FALSE; } x = e n g G e tA rra y (myMatEngine, " X "); y = e n g G e tA rra y (myM atEngine, "Y") ; i f (x==NULL | | y==NULL) { w s p r i n t f (tmp, "M atE ngine e r r o r : %d” , 9 9 ); M essageBox (hwnd, "e n g G e tA rra y f a i l e d ! " , tmp, MB_0K); r e t u r n FALSE; } e l s e { x P tr = m x G etP r(x ); y P tr = mxGetPr (y ) ,- r e lT im e * (m yC lockti] - m y C lo ck [0 ]) / (d o u b le) CLOCKS_PER_SEC; f p r i n t f (m y F ile P tr, "% -8. 4 f \ t % - 5 . I f \ t % - 5 . l f \ n " , r e lT im e , * y P tr, * x P t r ) ; } / / D e l e t e F i l e ( (LPCTSTR ) t e m p F i l e ) ; /* Debug S e c t i o n */ / / w s p r i n t f (tm p, "Loop T e s t : i = %d", i) ; //M essageB ox(hw nd, tmp, "D ebug", M B OK) ; } m x D estro y A rray (x) ,* m x D e s tro y A rra y (y ) ; e n g C lo se (m y M a tE n g in e ); f c l o s e ( m y F i l e P t r ) ; r e t u r n TRUE; } ♦ * Name: I n i t i a l i z e F i l e S a v e * * D e s c r i p t i o n : I n i t i a l i z e th e f i l e o p e n /s a v e s t r u c t u r e s ******** ******************************************************/ v o id In itia liz e F ile S a v e (H W N D hwnd) { s f n . I p s t r T i t l e = "S av e Image i n F i l e " ; s f n . F l a g s = OFN_OVERWRITEPROMPT | OFN_HIDEREADONLY; s f n . I S t r u c t S i z e = s i z e o f (OPENFILENAME) ; 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s f n . hwndOwner = hwnd; s f n . l p s t r F i l t e r = s z F i l t e r S e l ; s f n .n M a x C u s tF i l te r = 0; s f n . n F i l t e r l n d e x = 21; s fn .n M a x F ile = MAXPATH; s f n . n M a x F i l e T i t l e = MAXPATH; s f n . l p s t r F i l e = szSaveFileN am e; s f n . I p s t r l n i t i a l D i r = s z S a v e F ile D ir ; s f n . I p s t r F i l e T i t l e = s z S a v e F i l e T i t l e ; s f n . l p s t r D e f E x t = szS aveB xt; } END OF PROGRAM (Appendix A. I) ---- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A.II Program for Positioning Coordinates of Pendulum's Weight Center (MATLAB Codes) f u n c t i o n [ x , y ] = m y M a t G e t L o c a t io n ( f i l e N a m e ) g l o b a l I BW L X i Yi x y I = i m r e a d ( f i l e N a m e ) ; BW = r o i c o l o r d , 0 , 1 0 0 ) ; V = B W (100, 1 : 3 2 0 ) ; [X i, Y i] = f i n d ( V ) ; x = (m a x (X i) + r a i n ( X i ) ) / 2 ; y = (m a x (Y i) + m i n ( Y i ) ) / 2 ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AJII. Program for Smoothing the Data Obtained by CCD Camera (FORTRAN Codes) subroutine smooth(nfold,g,d£band) dimension g(nfold),w(500),gl(5000),g2(5000) t=1.0/df udf=1,854305/band*df lmax=ifix( 2.0/udf)+1 w(l)=0.75*udf do 110 l=2,lmax dif= 1.570796*real(l-1 )*udf w(l)=w( I )*(sin(dif)/dif)**4 110 continue U=lmax*2-1 In=U-l+nfold lt=(ll-l)*2+nfold le=It-Imax+l do 150k=l,It gl(k)=0.0 ISO continue do 160 k=l,nfold gl(U-l+k)=g(k) 160 continue do 180k=lmax,le s=w(l)*gl(k) do 170 l=2,lmax s=s+w( l)*(g 1 (k-l+1 )+g 1 (k+I-1)) 170 continue g2(k)=s 180 continue do 190 I=2,lmax g2(U+l-1 )=g2(U+l-1 )+g2(ll-l+1) g2(ln-I+1 )=g2(ln-l+1 >+-g2(ln+l-l) 190 continue do 200 lc= Unfold g(k)=g2(ll-l+k) 200 continue return end Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AJV. Runge-Kutta 4t h Method for Forward Analysis (MATLAB codes) d t = .0 3 3 ; c k = 0 .5 0 2 3 /0 .0 2 1 3 2 ; f d x c c = w k lr e a d ( ' a c r h s 2 . w k l') ; % xcc= w klread( ' x c c . w k l') ; [n n o ] = s i z e ( f d x c c ) ; f o r i = l : n t ( i ) = ( i - 1 ) * d t ; p ( i ) = f d x c c ( i , 2 ) / 0 .0213; c k l ( i ) = f d x c c ( i , 1 )* 0 .1 3 * 0 .4 0 5 /0 .0 2 1 3 ; end tm a x = (n - 1 )* d t y l (1 )= 0 .3 4 4 ; y2 (1 )= 0 .1 1 7 5 2 4 ; f o r n l = l : (n -1 ) k l l = d t * y 2 ( n l ) ; k 2 1 = d t* ( ( p ( n l ) + p ( n l + l ) ) / 2 - c k * s i n ( y l ( n l ) ) - c k l ( n l ) * c o s ( y l ( n l ) ) ) ; k l2 = d t* ( y 2 ( n l) + k 2 1 /2 ) ; k 2 2 = d t* ( ( p ( n l ) + p ( n l + l ) ) / 2 - c k * s i n ( y l ( n l ) + k l l / 2 ) - c k l (n l) * co s ( y l (n l) + k l l / 2 ) ) ; k l3 = d t* ( y 2 ( n l) + k 2 2 /2 ) ; k 2 3 = d t* ( ( p ( n l ) + p ( n l + l ) ) / 2 - c k * s i n ( y l ( n l ) + k l 2 / 2 ) - c k l ( n l ) * c o s ( y l ( n l ) + k l 2 / 2 ) ) ; k l4 = d t* ( y 2 (nl)+k23) ; k 2 4 = d t* ( (p ( n l) + p ( n l+ 1 ) ) / 2 - c k * s i n ( y l ( n l ) + k l 3 ) - c k l (nl) *cos ( y l (n l) + k l3 ) ) ; y l ( n l + 1 ) = y l (nl) + ( k ll+ 2 * k l2 + 2 * k l3 + k l4 ) /6 ; y 2 ( n l+ 1 ) = y 2 ( n l) + (k21+2*k22+2*k23+k24) / 6 ; end y l = y l •; y 2 = y 2 •; w k lw rite ( ’D lS x 2 s .w k l' ,y l) ; w k l w r i t e ( ’D lS v 2 s . w k l' , y 2 ) f i g u r e p l o t ( t , y l , t , y 2 ) ; x l a b e l ( ' T i m e ( s e c ) ') y l a b e l ( 'D is p la c e m e n t and V e l o c i t y * ) t i t l e ( ' U s e R u n g e -K u tta Method f o r I n v e r s e C o m p u ta tio n ’ ) g r i d on Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B. Source Codes for Elastomeric Membrane Experiment B.I. Program for Computing the Grids* Center (MATLAB Codes) f o r i * 1 :13 f i l e n a m e = s t r c a t ( 'B I N 2 0 0 ', i n t 2 s t r ( i ) , ' . r a w ’ ) ,- f i l e i d = f o p e n ( f i l e n a m e , ’r b ' ) ,- B W = f r e a d ( f i l e i d , [480 6 4 0 ] ) ; s t a t u s = f c l o s e ( f i l e i d ) ,- [xs y s l= s iz e ( B W ),- BW( : , :) = (255-BW(: , : ) ) / 2 5 5 ; w inx= 5; w iny=5; pnum=0; f o r j = l : w i n x : (x s-w in x + 1 ) f o r k = l:w in y : (y s-w in y + 1 ) v=BW(j : (j+ w in x -1 ) , k : (k + w in y -1 )) ; i f -is e m p ty ( f i n d ( v ) ) [x y] = f in d ( v ) ; i d x = r o u n d ( s i z e ( x ) / 2 ) ; x c = j + x ( i d x ) ; y c = k + y ( id x ) ; c l e a r x ; c l e a r y ; i f B W (x c,y c)* = l v=BW( ( x c - 1 2 ) : (x c + 1 1 ), (y c-1 2 ) : (y c+ 1 1 )) ; [x y] = f i n d ( v ) ; xc2= xc-12+ round(m eeua(x)) ; y c 2 = y c -1 2 + ro u n d (m e a n (y )) ; pnum=pnum+l; c o o r (pnum, 1) =pnum; coor(pnum , 2 ) = x c 2; c o o r (pnum, 3) =yc2 ; end end end end file n c u n e 2 = 8 trc a t ( ' p t d a t a ' , i n t 2 s t r (i) , ' . w k l ') ; w k lw r i t e ( f i l e n a m e 2 ,c o o r ) ; end Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.II. Program for Nonlinear Regression by the Mooney-Rivlin’s Equation % % c u r f i t _ s e . m C u rv e f i t t i n g f o r S t r e s s and S t r a i n %% % % f u n c t i o n c u r f i t _ s e c l e a r ; o d a ta = w k lr e a d {•d a t a 4 c f .w k l') d a t a = o d a t a ; [xs y s ] = s i z e ( d a t a ) ; i f ys>2 d i s p ( 1 E r r o r i n in p u d a t a ' ) ; c u r f i t _ s e = -1 ; r e t u r n end % % s t a r i n g p o i n t c l= 1 5 , c2=15; c l= 1 5 ; c2= 15; p c l= 0 ; pc2=0; f o r i = l : 5 0 0 f o r j = 1 :XS z (j ,1 ) = ( 2 * (1 + d a ta (j , 1 ) ) A3 -2 ) / (1 + d a ta ( j,l))^ 2 ,* z ( j ,2 ) = ( 2 * ( l + d a t a ( j , l ) P 3 - 2 ) / ( 1 + d a t a ( j , 1 ) ) A3 ; end Z D =z'*z; f o r j= l:X S d ( j ) = d a t a ( j , 2 ) - c l * 2 * ( ( 1 + d a t a ( j , 1 ) ) - 1 / ( 1 + d a t a ( j , 1 ) ) * 2 ) - . . . 2 * c 2 * ( 1 - 1 / ( 1 + d a t a ( j , l ) ) A3 ) ; end CD =pinv(ZD )* z ' * d ' ; p c l = c l ; p c2 = c 2 ; Cl=Cl+CD(1) ; c2=c2+CD(2) ; %if ( s q r t ( ( p c l - c l ) x2+ (p c2 -c2 ) *2) / s q r t ( c l A2+c2A2 ) ) <0 .00000001 % [ c l c2 ] % i % x p = l i n s p a c e (0 ,1 ,1 0 0 ) ; % f o r p i = l :100 % a l f = l + x p ( p i ) ; % y p ( p i) = 2 * c l* ( a l f - l / a l f ) + 2 * c 2 * ( 1 - 1 / a l f A3) ; % en d % p l o t ( x p , y p ) , a x i s ([0 1 0 1 2 0 ] ) ; % h o ld o n ; % p l o t ( o d a t a ( : , 1 ) , o d a t a ( : ,2 ) , ' . ’ ) % r e t u r n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % e n d a n s = s q r t ( ( p c l - c l ) A2 + ( p c 2 - c 2 ) A2) end x p = l i n s p a c e ( 0 . 1 ,1 0 0 ) ; f o r p i = l :100 a l f = l + x p ( p i ) ; y p ( p i ) = 2 * c l * ( a l f - l / a l f a2)+2* c2 * ( 1 - 1 / a l f a3 ) ; en d p l o t ( x p , y p , ' r ' ) . a x i s ([0 1 0 120] ) ; h o ld o n ; p l o t ( o d a t a ( : , 1 ) , o d a t a ( : , 2 ) , ' * ' ) [ c l c2] 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B .m Finite Elem ent Methods for the Verification of Nonlinear Elastic Elastomer Nodal Displacements MAIN PROGRAM---- % N o n lin e a r e l a s t i c p ro b le m on a p a l n e r u b b e r % "m o d n l_ ru b b er.m " : m ain program % 2 _ d im e n s io n a l p l a t e a n a l y s i s on t h e b a s i s e q u i v a l e n t e l a s t i c c a l i c u l a t i o n . % % -------- n o t a t i o n ---------------- % eO i n i t i a l v a l u e o f y o u n g 's m odulus % po : p o is o n r a t i o % x l : t o t a l l e n g t h i n x d i r e c t i o n , u n i t : i n c h % y l : t o t a l l e n g t h i n y d i r e c t i o n % n b x ,n b y : num ber o f d e s c r e t i z a t i o n % nkou : num ber o f e x t e r n a l boundary p o i n t s % nnd : num ber o f e le m e n t % n le n g : num ber o f unknown d is p la c e m e n ts % nband : band w id th % n i : one d i m e n s io n a l a r r a y on t r i a n g l e o f band*band % a f : f o r c e v e c t o r | d is p la c e m e n t v e c t o r % u : d is p la c e m e n t v e c t o r f o r r e p e a t e d c a l c u l a t i o n % g l o b a l ek % global expux e x p u y tO = clo ck ; % ********************* in p u t d a ta **************************** nbx=6; nby=5; %C1=20.8 7 1 1 ; C 2= 23.9 8 3 2 ; %%Testl % C 1 = U .12;C 2= 38.5; %%Test2 %C1=36.9 6 8 2 ;C 2= 0; %%Test3 % New V e rs io n %C1=19. 6131;C2=25 .9 0 2 8 ; c a s e t a g = 'n c l ' ; %C1=12 .8 5 2 1 ;C 2=36.1 4 0 6 ; c a s e t a g = ' nc2 ' ; C l= 3 6 .6 2 1 4 ;C 2 = 0 ;c a s e ta g = 'n c 3 1; e0=6*(C1+C2); % e l a s t i c v a lu e f o r Mooney R iv lin e E q u a tio n %e0=209.09 p o = 0 .2 5 ; % e l a s t i c v a l u e x l = l .5 ; y l = l .25; %yl = 43 /3 2 ; % 4 3 /3 2 = 2 .6 8 7 5 /2 t = l / 3 2 ; % t : t h i c k n e s s o f p l a t e 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % f in a lf o r c e = 9 . 30; % preload=0,- %1.745 % force= ( ( f i n a l f o r c e - p r e l o a d ) / 2 ) / n b x ; % f i n a l f o r c e u n i t ( l b ) ; f o r c e / 2 i n th e c a s e o f i x = l & ix=nbx+1 % ndelf=10; % t h e d e v id e d num ber o f f o r c e % d e lf = f o r c e / n d e l f ; % in c re m e n t f o r c e f o r c e = [1 .7 4 5 2 .7 4 3 .2 9 5 4 .2 9 4 .7 4 5.135 6 .6 6 0 8 .5 0 0 9 .3 0 0 ]; f o r c e = ( f o r c e / 2 ) / n b x ; n d e l f = le n g th ( f o r c e ) ; % nkou=2* (nbx+1) + (nby-1) +1+ (n b x + 1 ),- % (nbx+1) :ap p en d m e ta l a l= x l/n b x ; b l= y l/n b y ; % append m e t a l e le m e n t nby=nby+l; y l= y l+ b l; % ******************** f o r ix = l:n b x f o r i y = l : nby e ( i x , i y ) = e 0 ; e i ( i x , i y ) = e 0 ; p o r ( i x . i y ) = p o ; end end » % append m e ta l e lem e n t f o r ix = l:n b x iy = n b y ; e ( i x , iy ) = e 0 * 1 0 A 4 ; end % * * * * * * * * * * * * * * * * * * * nnd=nbx*nby; nleng=2* (nbx+1) * (nby+1) ; nband=2* (nbx+3) ; n i= (nb£mdA2+nband) /2 ; in=numb (nbx, n b y ) ; % num bering n f ix=bound ( n b x ,n b y ) ; % b o u n d a ry c o n d i t i o n % nsyo= 0; a f o r c e ( l ) =0; a d i s p ( l ) =0; % - --------- % 1 I t e r a t i o n c o r r e s p o n d in g t o lo a d s t e p num ber 1 % ........................................................................ f o r f l o o p = l : n d e l f % f o r c e in c re m e n t % s te p f = flo o p * d e lf ; s t e p f = f o r c e ( f l o o p ) ; e l o o p = l ; w h ile e l o o p f o r i y = l .-nby f o r ix= 1 : nbx ie = i x + ( i y - 1 ) * n b x ; 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s q e l m ( a l , b l , t , p o r , e , i x , i y , ie ) ; % elem e n t s t i f f n e s s m a t r i x end end a f = f v e c ( n l e n g , n b x , n b y , s t e p f ) ; % f o r c e v e c t o r a f = g a u s s ( n l e n g , n i ,n k o u , n s y o ,n b x , n b y ,n n d , n b a n d ,e k ,n f i x , a f ) ; % s o l v e r by g a u s s %nsyo=l; u = af ; i c = l ; f o r iy = l:n b y + l f o r ix = l:n b x + 1 u x ( i x , i y ) = a f ( i c ) ; s t e p u x ( i x , i y , f l o o p ) = u x ( i x , i y ) ; ic = ic + 2 ; end en d ic= 2 ; f o r iy = l-.n b y + l f o r ix = l:n b x + 1 u y ( i x , i y ) = a f ( i c ) ,- s te p u y ( i x , i y , flo o p ) = u y ( i x , i y ) ,- ic = ic + 2 ; end en d % s t r a i n a t e a c h e le m e n t c e n t e r p o i n t f o r iy = l:n b y f o r ix = l:n b x i e = i x + ( i y - l ) * n b x ; f o r j e = l : 4 e u x ( j e ) = a f (1+2*( i n ( i e , j e ) - 1 ) ) ; % x d is p la c e m e n t a t ea c h node i n a e le m e n t eu y (j e ) = a f ( 2 * i n ( i e , j e ) ) ,-% y d is p la c e m e n t a t e a c h node i n a e le m e n t end e p x ( i x , i y ) = (- e u x ( l) + e u x ( 2 ) -e u x (3) +eux (4)) / (2 * al) ; epy ( i x , iy ) = ( - e u y ( l) -e u y (2 ) +euy (3) +euy (4)) / (2*bl) ; g a m x y ( ix ,iy ) = ( - e u x ( l ) - e u x ( 2 ) + e u x (3 )+ e u x ( 4 ) ) / ( 2 * b l ) . . . + ( - e u y ( 1 ) + eu y ( 2 ) - e u y ( 3 ) + e u y ( 4 ) ) / (2 * al) ; s i g x ( i x , i y ) = e ( i x , i y ) * ( e p x ( i x , i y ) + p o r ( i x , i y ) * e p y ( i x , i y ) ) / ( l - p o r ( i x , i y ) A 2 ) ; s i g y ( i x , iy ) = e ( ix , iy ) * ( p o r ( i x , iy) * e p x ( ix , iy ) +epy ( i x , i y ) ) / ( l - p o r ( i x , iy ) *2) ; ta u x y ( i x , iy ) = e ( i x , iy) *gamxy ( i x , iy ) / (2* (1+por (ix , i y ) )) ; % i f ep x ( i x , i y ) ==epy ( ix , iy ) i f gamxy ( i x , iy ) <0 t e i t a ( i x , i y ) = - p i / 4 ; e l s e i f gamxy ( i x , i y ) ==0 t e i t a ( i x , i y ) =0; e l s e 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t e i t a ( i x , i y ) = p i / 4 ; end e l s e t e i t a ( i x , iy) = atan (g am x y ( i x , iy ) / ( e p x ( ix , iy ) - e p y ( i x , i y ) ) ) / 2 ; e n d « c c = c o s ( t e i t a ( ix , i y ) ) A2 ; s s = s i n ( t e i t a ( i x , i y ) ) A2 c 2 - c o s ( 2 * t e i t a ( i x , i y ) ) s 2 = s i n ( 2 * t e i t a ( i x , i y ) ) e p l ( ix , iy ) = c c * e p x ( ix , iy ) +ss*epy ( i x , iy ) + s2*gam xy(ix, iy ) /2 ,- ep2 ( i x , iy ) = s s * e p x ( ix , iy ) + cc* ep y ( ix , iy ) - s 2 * g a m x y ( i x ,iy ) /2 ; g l 2 ( i x , iy ) = -s 2 * e p x (ix , i y ) + s2 * ep y ( ix , iy ) +c2*gamxy ( i x , iy) ; s i g e l ( ix , iy ) =e ( i x , iy ) * (e p l (ix , iy ) + p o r ( i x , iy ) *ep2 (ix , i y ) ) / (1- p o r ( i x , i y ) A2) ; s ig e 2 ( ix , iy ) = e ( i x , iy ) * ( p o r ( ix , iy ) * e p l ( i x , iy ) +ep2 ( ix , i y ) ) / (1 - p o r ( i x , i y ) A2) ; t a u e ( i x , iy ) = e ( ix , iy ) * g l2 ( i x , iy ) / (2* (1+por ( i x , iy ) )) ; en d %ix end %iy % td ra w in g o u t (n b x , n b y , u x , u y , t e i t a , e p x , e p y , gam xy, e p l , e p 2 , g l 2 , s i g x , s i g y , ta u x y , s i g e l , s i g e 2 , ta u e ) % r e t u r n f o r i y = l : n b y - l % -l:m e ta l f o r i x = l : nbx s i g 2 ( i x , iy ) = sige2 ( ix , iy ) ; i f s i g 2 ( i x , i y ) >0 a l= 0 ;a 2 = 5 ; g l o o p = l ; w h i l e g lo o p a s e p - l in s p a c e ( a l , a 2 , 10) ,- f o r i i s l :10 G ( i i ) = s i g 2 ( i x , i y ) - 2 * C 1 * ( ( 1 + a s e p ( i i ) ) - 1 / ( l + a s e p ( i i ) ) A2) . . . -2*C 2*( 1 - 1 / (1 + a s e p (i i ) ) A3 ) ; % % % % f o r Mooney R i v l i n E q u a t i o n % G ( i i ) = (- 1 1 1 . 4 9 * a s e p ( i i ) A2 + 2 0 9 .0 9 * a s e p ( ii ) ) - s i g 2 ( i x , i y ) ; % % % f o r P a r a b o la e q u a tio n i f i i > l & s i g n ( G ( i i ) * G ( i i - l ) )<0 n o w = G (ii); a l = a s e p ( i i - 1 ) ; a 2 = a s e p ( i i ) ; ep22 ( i x , iy ) = asep ( i i ) ; end en d 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i f g lo o p > l 6 ab s ( (pre-now ) /now ) <0 .0001 b r e a k end pre=now ; g lo o p = g lo o p + l; end tg lo o p en d e n d % i x en d %iy t r e t u r n % if e lo o p = = l % p o r = - e p l . / e p 2 ; %end f o r i y = l : n b y - l % - l:m e ta l f o r i x = l :nbx a l s l ; a 2 s - l ; a lo o p = l; w h ile a lo o p a s a l f = l i n s p a c e ( a l , a 2 , 10) ; f o r i i = l :10 i f ( 1 - p o r ( i x , i y ) / a s a l f ( i i ) ) -=0 % s i g l ( i x , i y ) = e ( i x , i y ) * ( - p o r ( i x , i y ) *ep2 ( i x , i y ) ) ,- s i g l ( i x , i y ) =- e p 2 2 ( i x , i y ) * p o r ( i x , i y ) * e ( i x , i y ) / ( 1 - p o r ( i x , i y ) / a s a l f ( i i ) ) ; F a l f ( i i ) = s i g l ( i x , iy ) - a s a l f ( i i ) * s i g 2 ( i x , i y ) ; i f i i > l & s i g n ( F a l f ( i i ) * F a l f ( i i - 1 ) )<0 n o w = F a l f ( i i ) ; a l s a s a l f ( i i - 1 ) ; a 2 = a s a l f ( i i ) ; a l f ( i x , iy ) = a s a l f ( i i ) ,- end end end i f a lo o p > l & a b s ( (p re-n o w )/n o w )< 0 .0 0 0 1 b re a k end pre=now; a lo o p = a lo o p + l; en d %aloop e n d end % s t e p d f o r i y = l : n b y - l % - l : m e t a l f o r i x = l :nbx e p l l ( i x , iy ) =-ep22 ( i x , iy ) *por ( i x , i y ) ; s i g l ( ix , iy ) = (e ( i x , iy ) / (1- p o r ( i x , iy ) * 2 )) * ( e p l l ( ix , iy ) + p o r ( i x , iy ) *ep22 ( i x , i y ) ) ; % sig l ( ix , iy ) = - e ( i x , iy ) *por (ix , iy ) *ep22 ( i x , iy ) / (1- p o r ( ix , iy ) / a l f ( i x , i y ) ) ; % e p l l ( i x , i y ) = ( s i g l ( i x , i y ) / e ( i x , i y ) ) * ( 1 - p o r ( ix , iy ) / a l f ( i x , i y ) ) ; e s ( i x , iy ) = sig 2 ( i x , iy ) /ep 2 2 (ix , iy ) ; 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. %e ( i x , iy ) = es ( i x , iy ) ,- e ( i x , i y ) = ( 0 . 7 + e s ( i x , i y ) + 0 . 3 * e ( i x , i y ) ) ; end e n d % p o r= -e p ll . /e p 2 2 ; % r e t u r n % s t e p e i f l a g = 0 ; i f e lo o p ~ = l f o r i y = l : n b y - l % -l:m e ta l f o r ix = l:n fa x e _ ta n = 2 * C l* ( 1 + 2 / ( l + e p 2 2 ( i x , i y ) )* 3 )+ 6 * C 2 / ( l + e p 2 2 ( i x , i y ) ) A 4; C R = l/e _ ta n ; i f a b s ( e p 2 ( i x , i y ) - e p 2 2 ( i x , i y ) ) <CR | (eloop>3 & . . . ab s (ep2 (ix , iy ) - e p 2 2 ( i x , i y ) ) / e p 2 2 ( i x , i y ) <0.055) | elo o p > 9 % i f l a g = i f l a g + l ; end end end en d d i s p ( i f l a g ) i f ifla g = = n n d - 6 % nnd=number o f e le m e n t - 6 -.metal e le m e n ts b re a k e n d e lo o p = e lo o p + l en d % w h ile e lo o p a d i s p ( f l o o p + l ) = m e a n ( u y ( l : n b x + l , n b y + l - l ) ) ; % - 1 :m etalap p en d a f o r c e (flo o p + 1 ) = f o r c e (f lo o p ) *nbx*2,- % afo rce (flo o p + 1 ) = ( ( f i n a l f o r c e - p r e l o a d ) / n d e l f ) * flo o p en d % flo o p % o u t _ t a b l e ( n b x , n b y , f l o o p , s t e p u x , s t e p u y , t e i t a , e p l l , e p 2 2 , s i g l , s ig 2 ) o u t p u t f i g ( a d is p , a f o r c e , nbx, nby, n d e l f , s t e p u x , s te p u y , x l , y l , c a s e ta g ) r e t u r n % **************** en d **************** END OF MAIN ------- N um bering n o d es ---------- f u n c t i o n in= num b(nbx,nby) % N um bering o f e le m e n t number & node num ber % "numb.m" f o r i i = l : n b y % i i : r o w f o r j j = l : n b x % j j : c o l u m n f o r j e = l : 2 n n ( i i , j j , j e ) = j j + (nbx+ 1)* ( i i - l ) + j e - l ; 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. end f o r j e = 3 : 4 n n ( i i , j j , j e ) = j j + ( n b x + 1 ) * i i + j e - 3 ; end end end f o r i i = l : n b y f o r j j = l : n b x i e = j j + ( i i - 1 ) * n b x ; % i e : e l e m e n t number f o r j e = l : 4 % j e : l o c a l node number i n ( i e , j e ) = n n ( i i , j j , j e ) ; % i n ( ) m o d e num ber o f a w hole s t r u c t u r e en d end end End o f n um bering n o d es -------- BOUNDARY CONDITIONS------ f u n c ti o n n fix = b o u n d (n b x , n b y ) % B oundary c o n d i t i o n % "bound.m" n f i x ( l ) =1; i c = l ; f o r j j * l : n b x + 1 i c = i c + l ; n f i x ( i c ) = 2 * j j ; % bottom l i n e i n y d i r e c t i o n end f o r j j = l m b y - l % l e f t s i d e i n x d i r e c t i o n i c = i c + l ; n f i x ( i c ) = 2 * j j * ( n b x + 1 ) +1; end f o r j j = l : n b x + l % u p p e r l i n e i n y d i r e c t i o n i c = i c + l ; n f i x ( i c ) =2* (nbx+1) *nby+2* (j j -1) +1; end f o r j j = 2 : n b x + l % u p p e r-1 l i n e i n y d i r e c t i o n : ap p en d m e ta l i c = i c + l ; n f i x ( i c ) = 2 * (nbx+1) * (nby-1) +2* (j j -1) +1; end END OF BOUNDARY CONDITIONS------- E lem en t s t i f f n e s s m a tr ix o f f i n i t e e le m e n ts m odels f u n c ti o n s q e l m ( a l , b l , t , p o r , e , i x , i y , i e ) % s t i f f n e s s m a t r i x o f a s q u a r e / r e c t a n g u l a r FEM e le m e n t g l o b a l ek 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a = a l / 2 ; b = b l / 2 ; c o e = a * b * t * e ( ix , iy) / (4* ( 1 - p o r ( i x , iy ) *2)),- a d = 2 / ( 3 * a * 2 ) ; b d = ( l - p o r ( i x , i y ) ) / (3*bA2) ; c d = ( 1 + p o r ( i x , i y ) ) / (2*a*b) ; d d = ( l - 3 * p o r ( i x , i y ) ) / (2*a*b) ; ed= (1 - p o r (i x , iy) ) / ( 3 * a A2) ; f d = 2 / ( 3 * b * 2 ) ; % e k ( l , 1 , i e ) =2*ad+2*bd; e k ( l , 2 , i e ) = c d ; e k ( l , 3 ,ie ) = - 2 * a d + b d ; e k ( l , 4 , i e ) = - d d ; e k ( l , 5 , ie )= a d -2 * b d ; e k ( l , 6 , ie ) = d d ; e k ( l , 7 , ie ) = -a d -b d ; e k ( l , 8 , i e ) = - c d ; e k ( 2 , 2 , i e ) =2*ed+2*fd; e k ( 2 , 3 , i e ) = d d ; e k ( 2 , 4 , ie ) = - 2 * e d + f d ; e k ( 2 , 5 , i e ) = -d d ; e k ( 2 , 6 , i e ) = e d -2 * fd ; e k ( 2 , 7 , i e ) = - c d ; ek (2, 8 , i e ) = - e d - f d ; e k ( 3 , 3 , i e ) =2*ad+2*bd; e k ( 3 , 4 , i e ) = - c d ; e k ( 3 , 5 , i e ) = - a d - b d ; e k ( 3 , 6 , i e ) = c d ; e k ( 3 ,7 , i e ) = a d - 2 * b d ; e k ( 3 , 8 , i e ) = - d d ; e k ( 4 , 4 , ie ) =2*ed+2*fd; e k ( 4 , 5 , i e ) = c d ; e k ( 4 , 6 , i e ) = - e d - f d ; e k ( 4 , 7 , i e ) = d d ; e k ( 4 , 8 , ie ) = e d - 2 * f d ; e k ( 5 ,5 ,ie ) = 2 * a d + 2 * b d ; e k ( 5 , 6 , i e ) = - c d ; e k ( 5 , 7 , i e ) =-2*ad+bd; e k ( 5 , 8 , ie ) = d d ; e k ( 6 , 6 , ie )= 2 * e d + 2 * fd ; e k ( 6 , 7 , i e ) = - d d ; ek (6, 8 , i e ) = -2 * ed + fd ; ek ( 7 , 7 , i e ) =2*ad+2*bd; e k ( 7 , 8 , i e ) = c d ; e k ( 8 ,8 ,ie ) = 2 * e d + 2 * f d ; % f o r i i = l : 8 f o r j j = i i :8 e k ( j j , i i , i e ) = e k ( i i , j j , i e ) ; en d en d f o r i i = l : 8 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f o r j j » l : 8 e k ( i i , j j , i e ) = c o e * e k ( i i , j j , i e ) ; en d end End o f e le m e n t s t i f f n e s s m a tr ix F o rc e V e c to rs ------- f u n c t i o n a f = f v e c ( n l e n g , n b x ,n b y , s t e p f ) % M aking o f F o rce v e c t o r % " fv e c .m " % u n i t :1b a f ( l : n l e n g ) =0; d e c l= 0 ; f o r j j = l :n b x + 1 k = 2 * jj+ 2 * n b y * (n b x + 1 ); a f ( k ) = ( l- 3 * d e c l+ d e c l* (j j - 1 ) ) ‘ s t e p f ; i f j j = = l | j j ==nbx+1 a f ( k ) = a f ( k ) / 2 ; en d end End o f F o rce V e c to r s -------- G auss Methods f o r S o lv e r o f L in e a r E q u a tio n s -------- f u n c t i o n a f= g a u ss (n le n g , n i , nkou, n sy o , nbx, n b y , nnd, nband, e k , n f i x , a f ) % S o l u t i o n by g a u s s on t h e b a s i s o f e x t e r n a l f i l e a c c e s s % nleng=50; * t o t a l unknown number %ni: o n e d im e n s io n a l a r r a y on t r i a n g l e o f band*band %nkou: num ber o f b o u n d a ry p o i n t s %nsyo: n e e d od d o n 't n e e d o f e l i m i n a t i o n % nbx,nby: d iv id e d num ber %nnd: num ber o f e le m e n ts % n b a n d : b an d w id th % n f i x ( i i ) : b o u n d ary p o i n t % k k (1 )= 1 ; f o r i i = 2 :nband k k ( i i ) = k k ( i i - l ) + n b a n d - i i + 2 ; end % s w itc h n s y o c a s e {0} % i n t h e c a s e o f nsyo=0 f i d = f o p en ( 1 cktem p - b i n ' , ' w1) ,- f r e w i n d ( f i d ) % s t a t u s = f s e e k ( f i d , 0, -1) f o r i i = l : n i c k ( i i ) =0; 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. end % A ssem ble o f s t i f f n e s s -------- kng=0; f o r iy = l.-n b y f o r i x = l : n b x i e = i x + ( i y - l ) *nbx; i t u = 0 ; f o r i i = l : 8 i f ii= = 5 itu = 2 * (n b x + 1 )-4 ; end n c = k k ( i i + i t u ) ; f o r j j = i i :8 %if iy = = l & ix = = l %xxx=nc %end c k ( n c )= c k (n c ) + e k ( i i , j j , i e ) ; i f jj= = 4 is u = 2 * (n b x + 1 )-4; e l s e is u = 0 ; end n c = n c + l+ isu ; en d end % n n l row s e l i m i n a t i o n ------------ n n l= 2 ; i f iy= = nby & ix==nbx n n l= n b a n d - l; end i f iy -= n b y & ix==nbx n n l =4; end %xxx»nnl f o r k = l : n n l k n g= kng+ l; f o r ii= l-.n k o u i f k n g = = n f ix ( ii) c k ( k k ( k ) ) =eibs ( c k ( k k ( k ) ) + 1.) * 10^20 ; % WRITE(6 ,* ) kng en d en d i f c k ( k k ( k ) )==0 c k ( k k ( k ) ) = a b s ( c k ( k k ( k ) )+ 1 .)* 1 0 * 2 0 ; en d n l= n b em d -k + l; f o r j j = 2 : n l l x = k k ( k ) + j j -1 ; i f c k ( lx ) - = 0 .0 k j =kng+j j -1 ; te m p = c k ( lx ) / c k ( k k ( k ) ) ; 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a f ( k j ) = a f ( k j ) - a f ( k n g ) * te m p ; f o r i i = 2 : j j i k = k + i i - l ; j i = j j - i i + l ; k x = k k ( ik ) +j i - 1 ; j x = k k ( k ) + i i - l ; c k ( k x ) * c k ( k x ) - c k ( jx ) * te m p ; en d e n d % i f e n d % j j end % k % % S ave t h e r e s u l t o f e l i m i n a t i o n i n t o f i l e - - i f iy = = n b y & ix= = nbx c o u n t = f w r i t e ( f i d , c k ( l : n i ) , ’ f l o a t 6 4 ’) ; % for i i — 1 :1 4 % y = c k ( i i ) %end e l s e % m 2 = n n l* n b an d -n n l* (n n l-1 ) /2 ; c o u n t = f w r i t e ( f i d , c k ( l : m 2 ) , 'f l o a t 6 4 ') ; %if ix= = 4 & iy = = l % f o r i i = l : 5 0 % y = c k ( i i ) % e n d %end % % **** s h i f t o f t r i a n g l e & n e x t p r e p a r a t i o n f o r i i = l :n b a n d -n n l f o r j j = l : n b a n d - i i - ( n n l - 1 ) % x x x = k k ( ii) + jj-1 % y y y = k k ( ii+ n n l) + jj-1 c k ( k k ( i i ) + j j - 1 ) = c k ( k k ( i i + n n l ) + j j - 1 ) ; e n d end f o r i i = l :n band k = n b a n d - i i - ( n n l -2) ; i f k<=0 k - 1 ; en d f o r j j = k : n b a m d - i i+ l % WRITE(6,112) k k ( i i ) + j j -1 * 112 FORMAT(1H ,1015) c k ( k k ( i i )+ j j -1 )= 0 ; end en d % i i end % i f en d % ix e n d %iy s t = f c l o s e ( f i d ) ; c a s e {1} % i n t h e c a s e o f n s y o = l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % ------------ i n th e c a s e o f c h a n g in g o n ly f o r c e v e c t o r — % f i d = f o p e n { ’ck tem p . b i n ' , ' r ') ; f r e w i n d ( f i d ) kng=0 f o r iy = 1 : n b y f o r i x = l :nbx n n l = 2 ; i f iy= = nby & ix==nbx n n l= n b a n d - l; en d i f iy -= n b y & ix==nbx n n l = 4 ; en d m 2 = n n l* n b a n d - n n l* ( n n l- 1 ) /2 ; [ c k , c o u n t ] = f r e a d ( f i d , m 2 ,’f l o a t 6 4 ') ; k f =0 , * f o r i i = l : n n l kng= kng+ l; m c c = n b a n d -ii+ l; f o r jj = l :m c c k f= k f + l; i f (j j -1 )> 0 kngj =kng+j j - 1 ; a f ( k n g j) * a f ( k n g j ) - a f (kng) *ck ( k f ) / c k (k f 1) e l s e k f l = k f ; end e n d % j j en d % i i end % i x end % i y end % s w itc h c a s e % retu rn % % Backword S u b s t i t u t i o n ------------------------ i f nsyo==0 [ f id ,m e s s a g e ] = f o p e n ( 'c k t e m p . b i n ', ' r ') ; end k n g = n le f o r i y * n b y .- - l :l f o r i x = n b x : - l : l n n l =2; i f iy~=nby & ix==nbx n n l= 4 ; e n d i f iy==nby & ix==nbx % **** s t a g e 1 **** n f = - 8 * n i; s t a t u s = f s e e k (f i d , 0, ' e o f ' ) ; s t a t u s = f s e e k ( f i d , n f , 0 ) ; % b a c k s p a c e ( f i d ) [ c k , c o u n t ] s f r e a d ( f i d , n i , ' f l o a t 6 4 ' ) ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m2 = n i ; a f (nleng) = af (n len g ) / c k (kk (nband) ) ; f o r k = l n b a n d -1 k n g = k n g -l; f o r j j = 2 : k+1 kmc=kng+j j -1 ; a f (kng) = af (kng) - c k ( k k ( n b a n d - k ) + j j - l ) *af(km c) ; en d a f (kng) = a f ( k n g ) /c k ( k k ( n b a n d - k ) ) ,- end e l s e % **** S ta g e 2 **** m 2 = n n l* n b an d -n n l* ( n n l- l) /2 ; n f =-8* (prem2+m2) ; %xxx=prem2+m2 s t a t u s = f s e e k ( f i d , n f , 0) ; % b a c k s p a c e ( f i d ) kf=m 2+l; [ck, c o u n t] = f r e a d ( f i d , m2, ' f l o a t 6 4 ’ ) ; f o r i i = l : n n l ; kng= kng-1; m cc = n b an d + ii- n n l ; f o r j j =mcc: - 1 : 1 k f = k f - l ; i f j j - = l km c= k n g + jj- 1 ; a f (kng) = a f (kng) - c k ( k f ) * a f (kmc) ; e l s e a f (kng) = a f (kng) / c k ( k f ) ; end end % j j en d % i i en d % i f prem2=m2; end * i x end % i y % %for i i = l : n l e n g % u ( i i ) = a f ( i i ) ; %end s t = f c l o s e ( f i d ) ; End o f G auss M ethod ---------- O u tp u t t o T a b le ------- f u n c ti o n o u t _ t a b l e (nbx, nby, f lo o p , s te p u x , s te p u y , t e i t a , e p l l , e p 2 2 , s i g l , s ig 2 ) % d ra w in g o u t.m d i s p ( ' ') 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d i s p C Nodal D is p la c e m e n t i n X D i r e c t i o n ’ ) d i s p ( ' ') d i s p ( ' 1 2 3 4 5 6 7 ') f o r iy = n b y + l: - 1 : 1 s = s p r i n t f ( • %2u %7.3f % 7.3f % 7.3f * 7 .3 f % 7 .3 f % 7.3f %7.3f ’ , i y , s t e p u x ( l : n b x + l , i y , f l o o p ) ) ; d i s p ( s ) end d i s p ( ' ') d i s p C Nodal D is p la c e m e n t i n Y D i r e c t i o n ’ ) d i s p ( ' ') d i s p ( ’ 1 2 3 4 5 6 7 ') f o r iy = n b y + l: - 1 : 1 s = s p r i n t f ( ’ %2u * 7 .3 f % 7.3f % 7.3f %7.3f % 7.3f % 7.3f % 7 .3 f' , iy ,s te p u y ( l- .n b x + 1 , i y , f l o o p ) ) ; d i s p ( s ) end % d i s p C ') d i s p ( ' P r i n c i p a l D i r e c t i o n i n e a c h e l e m e n t ') d i s p ( ' ') d i s p ( ' 1 2 3 4 5 6 ') f o r iy = n b y : - 1 : 1 s = s p r i n t f ( ' %2u \ t %7.3f % 7.3f % 7.3f % 7.3f % 7 .3 f %7. 3 f ’ , i y , t e i t a ( : , i y ) ) ; d i s p ( s ) end % d i s p ( ' ') d i s p ( ' P r i n c i p a l S t r a i n i n e a c h e l e m e n t ') d i s p ( ' ') d i s p ( ' 1 2 3 4 5 6 ') f o r i y = n b y - l : - 1 : 1 % - l i m e t a l s = s p r i n t f ( ’%2u \ t %7.3f % 7.3f % 7.3f % 7.3f % 7 .3 f % 7 .3 f' , i y , e p l l ( : , i y ) ) ; d i s p ( s ) s = s p r i n t f ( ' \ t %7.3f % 7.3f % 7.3f % 7.3f % 7.3f % 7 .3 f' , e p 2 2 ( : , i y ) ) ; d i s p ( s ) end % d i s p ( ’ ') d i s p ( ' P r i n c i p a l S t r e s s i n e a c h e le m e n t•) d i s p ( ' ') d i s p ( ' 1 2 3 4 5 6 ') f o r i y = n b y - l : - l : l % -l:m etal s = s p r i n t f ( ' %2u \ t %7.3f % 7.3f % 7.3f % 7.3f % 7 .3 f % 7 .3 f' , i y , s i g l ( : , i y ) ) ; d i s p ( s ) s = s p r i n t f ( ’ \ t %7.3f % 7.3f % 7.3f % 7.3f % 7.3f % 7 .3 f' , s i g 2 ( : , i y ) ) ; d i s p (s) end 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. End of Output Table O u tp u t F ig u r e and R e s u lt --------- f u n c ti o n o u t p u t f i g ( a d is p , a f o r c e ,n b x , nby, n d e l f , s te p u x , s t e p u y , x l , y l , c a s e ta g ) % global ex p u x ex p u y % E x te r n a l F o rc e & D isp la c e m e n t R e r a t i o n s h i p on e x p e r im e n t f o r c e s [0 1 .7 4 5 2 .7 4 0 3.295 4 .2 9 0 4 .7 4 0 5.135 6 .6 6 0 8 .5 0 0 9 .3 0 0 ] ; % f o rc e = fo rc e -1 .7 4 5 ? % disp=[2 .6 8 7 5 2 .8 7 5 0 3 .0625 3 .1 2 5 3 .1 3 7 5 3 .3125 3 .3 7 5 0 3.4375 3.4375 3 .8 1 2 5 4 .7 5 0 ] ; % d is p = (d is p -2 .6 8 7 5 ) / 2 ; d i s p = [0 0 .0 7 1 0 .1 4 8 0.162 0 .2 4 6 0 .3 1 0 2 0.360 0 .5 2 1 0 .7 9 3 0 .9 5 5 ]; f i g u r e (1) box; e l f p l o t ( d i s p , f o r c e ) t i t l e ( 'C o m p ariso n o f FEM A n a ly s is a n d E x p e rim e n ta l D a t a ') x l a b e l ( ' D is p la c e m e n t ( i n c h ) ' ) , a x i s ( [0 1 .2 0 1 0 ] ) ; y l a b e l ( ' F o rc e ( l b ) 1) g r i d on h o ld on p l o t ( a d i s p , a f o r c e , ' r ' ) h o ld o f f w k lw rite ( 'a d i s p - t e m p . w k l ' , a d i s p ' ) % f o r iy = l:n b y + 1 f o r ix = l;n b x + 1 f o r f l o o p = l : n d e l f s t e p u x ( i x , iy , f lo o p ) = s te p u x ( ix , iy , f lo o p ) ;%- s t e p u x ( i x , i y , 1 ) ; % a d j u s t t o com pare w ith e x p e rim e n t d a t a s t e p u y ( i x , i y , f l o o p ) = s t e p u y ( i x , i y ,f l o o p ) ;%- s t e p u y ( i x , i y , 1) ; end end end % n o d e x = lin s p a c e ( 0 , x l , nbx+1) ,- n o d e y = lin s p a c e ( 0 , y l , n b y + 1 ); % e x p e rim e n ta l d a t a e x p d a ts w k lr e a d ( ' d i s p i n 3 a .w k l ') ; % % % c a l c u l a t i o n o f r . m . s . v a lu e f o r i r m s = l : (nbx+1) f o r j r m s = l ; (nby+1) p t i = i r m s + ( j r m s - 1 ) * (n b x + 1 ); s t e p u v (2 * p t i -1 , :)= s t e p u x (i r m s , j rm s , :) ; s t e p u v ( 2 * p t i , : ) = s te p u y (ir m s , j r m s , ; ) ; end end Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f e ra u v d : 8 4 , 1 : n d e lf ) = s t e p u v ( l : 8 4 , 1 :n d e l f ) ,- w k lw r ite ( 'RM S-temp.wkl ■, ferauv) ; f i n a r a e = s t r c a t ( ' RMS-temp' , c a s e t a g , ' . w k l') ; w k lw r ite (fin am e, femuv) ; % % en d o f rms a n a l y s i s expda t =e x p d a t ' ; [ m ,n ] = s iz e ( e x p d a t) ; f o r i s t a g e = l : m % i g n o r e l a s t 4 s t a g e d a t a ; o n ly u s e 9 lo a d c a s e s i c = l ; f o r iy = l:n b y + 1 -1 % - l: m e t a l f o r ix = l:n b x + 1 e x p u x ( ix t i y , i s t a g e ) = ex p d at ( i s t a g e , ic ) ,- ic = ic + 2 ; en d en d i c = 2 ; f o r iy = l- .n b y + l- l % - l : m e a t l f o r ix = l:n b x + 1 expuy ( ix , i y , i s t a g e ) » e x p d a t ( i s t a g e , ic ) ; ic = ic + 2 ; en d end end % % % b e g in m aking c o m p a ris o n g r a p h o f FEM and E x p e r im e n ta l d a ta f o r i i = l : n d e l f f i g u r e ( i i + 1 ) e l f h o ld on a x i s e q u a l ; a x i s ([0 3 0 3 ] ) ; box; f o r iy = l:n b y + 1 -1 % - l i m e t a l f o r i x = l : nbx x ( l) = n o d e x ( ix ) ; x ( 2 ) = nodex(ix + 1 ) ; y (1) * n o d e y ( iy ) ; y (2) = y (1 ); p l o t ( x , y , ' b ' ) % o r i g i n a l sh ap e end end p l o t ( [ 1 . 7 5 ; 2 . 0 ] , [ 2 ; 2 ] , ' b ' ) t e x t ( 2 . 1 5 , 2 , 'O r i g i n a l S h a p e ') ; p l o t ( [ 1 . 7 5 ; 2 . 0 ] , [ 2 . 5 ; 2 . 5 ] , ' k ' ) t e x t ( 2 . 1 5 , 2 . 5 , ' Image R e s u l t ') ; p l o t ( [ 1 . 7 5 ; 2 . 0 ] , [ 2 . 2 5 ; 2 . 2 5 ] , ' r : ') t e x t ( 2 . 1 5 , 2 . 2 5 , 'FEM R e s u l t ' ) ; f o r i y = l : n b y - l % -l:m e ta l f o r ix = l:n b x + 1 x ( l ) = n o d e x ( i x ) ; x (2) = x (1 ); y ( 1 ) = n o d e y ( iy ) ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y (2) =nodey (iy + 1 ) ; p l o t ( x , y , ' b ' ) % o r i g i n a l sh ap e end e n d s t r c a p t = s t r c a t ( ’ C o m p ariso n o f n o d a l d i s p l a c e m e n t s . F _x_2= ' , n u m 2 s t r ( f o r c e ( i i + 1 ) ) , ' (lb ) ') ; t i t l e ( s t r c a p t ) x l a b e l ( 'X _ l d i r e c t i o n (in )') y l a b e l C X 2 d i r e c t i o n (in )') % n o d e = l; f o r i y = l : n b y + l - l % - l:m e ta l f o r ix s l:n b x + 1 cgx (n o d e) = nodex (ix ) + step u x (i x , i y , i i ); cg y (n o d e) =nodey (iy ) + step u y ( ix , i y , i i ) ; node=n o d e+1 ,- end en d f o r iy = l:n b y + 1 -1 % - l:m e ta l f o r ix = l:n b x n o d e * i x + ( i y - 1 ) * (nbx+1) ; x ( l ) = c g x ( n o d e ) ; x ( 2 ) = c g x ( n o d e + l) ; y (1 )= c g y ( n o d e ) ; y (2 ) = c g y ( n o d e + 1 ); p l o t ( x ,y , ' r : ') % deform ed sh ap e en d en d f o r i y = l : n b y - l % - l:m e ta l f o r ix = l:n b x + 1 n o d e = ix + ( iy - 1 ) * (nbx+1) ,- x (l) = cg x (n o d e) ; x (2) =cgx (node+nbx+1) ; y (1 )= c g y ( n o d e ) ; y (2 )= c g y (node+ nbx+ 1) ; p l o t ( x , y , ' r : ') % deform ed sh ap e en d en d n o d e = l; f o r i y = l : n b y + l - l % -l.-m etal f o r ix = l:n b x + 1 cgx (node) = n o d e x (ix )+ e x p u x ( ix , i y , i i ) ; cg y (node) = n o d ey (iy ) +expuy (i x , i y , i i ) ; node= node+ l ,- en d en d f o r iy = l:n b y + 1 -1 % - l:m e ta l f o r ix = l:n b x node=ix+ ( i y - 1 ) * (nbx+1) ,- x (1) =cgx (n o d e ) ,- x (2 ) = c g x (n o d e + 1 ); y (1) =cgy (node) ,- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y (2) = cg y (node+1) • plot(x,y,’k') % deformed shape of experiment end end f o r i y = l : n b y - 1 % -l:m e ta l f o r i x s l : n b x + l n o d e = ix + (iy -1 ) * (nbx+1) ; x ( l ) = cg x (n o d e) ; x (2) = c g x (node+nbx+1) ; y ( 1 ) = cg y (n o d e) ; y (2) = cg y (node+nbx+1) ; p l o t ( x , y , ' k ' ) % d efo rm ed s h a p e o f e x p e r im e n t en d end h o ld o f f end % % o f i i End o f O u tp u t F ig u re s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Asset Metadata

Creator Chung, Hung-Chi (author)

Core Title Digital image processing for system identification

Contributor Digitized by ProQuest (provenance)

School Graduate School

Degree Doctor of Philosophy

Degree Program Civil Engineering

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

Tag applied mechanics,Computer Science,engineering, civil,OAI-PMH Harvest

Language English

Advisor Shinozuka, Masanobu (committee chair), Ching, T.E. (committee member), Masri, Sami F. (committee member), Wellford, L. Carter (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-192589

Unique identifier UC11334439

Identifier 3065772.pdf (filename),usctheses-c16-192589 (legacy record id)

Legacy Identifier 3065772.pdf

Dmrecord 192589

Document Type Dissertation

Rights Chung, Hung-Chi

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 au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA