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An approach to dynamic modeling of percussive mechanisms

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An approach to dynamic modeling of percussive mechanisms

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Content AN APPROACH TO DYNAMIC MODELING OF PERCUSSIVE MECHANISMS by Samuel D. Goldman A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) August 2021 Copyright 2021 Samuel D. Goldman Acknowledgments This work was only possible with the persistent guidance of my advisor, Dr. Henryk Flashner, and my co-advisor Dr. Bingen Yang who developed the underlying theory used throughoutthisdissertation. I’dalsoliketoacknowledgemyundergraduateresearchmentor, Dr. Yun Kang, for providing me with the necessary tools to conduct thoughtful research, which allowed me to enter this program in the first place. I cannot begin to express my thanks to Dr. Kris Zacny for originally offering the opportu- nity to conduct my research with Honeybee Robotics, especially at a time when I considered leaving school. Additionally, I’d like to extend my gratitude to Gale Paulsen and Zach Mank as well for patiently providing crucial mentorship in mechanical design, and allowing to me to work on some incredible, challenging programs despite my inexperience. My younger self would never believe I could end up playing a role in the development of technologies which directly aid human understanding of our place in the solar system. More generally, I’d like to express my appreciation for all of the folks at Honeybee Robotics, whose ingenuity, enthusiasm, and scintillating presence made this experience far more enjoyable than it should have been. Lastly, this could not have been accomplished without the continued support and encour- agement of my parents, and my beloved fianceé, Sara. ii Contents Acknowledgments ii List of Tables vi List of Figures vii Abstract xv 1 Background and Motivation 1 1.1 Background on Percussive Drilling . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Review of Existing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Survey of Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Longitudinal Waves in Three Dimensions in Elastic Rods . . . . . . . 8 1.3.3 Longitudinal Waves in One Dimension in Slender Elastic Rods . . . . 9 1.3.4 Impact of a Cylindrical Hammer on a Rod . . . . . . . . . . . . . . . 13 1.3.5 Impact of Rigid Mass on Rod . . . . . . . . . . . . . . . . . . . . . . 14 2 Application of Distributed Transfer Function Method 19 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Generic Bar with Boundary Forcing . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Governing Equation and Boundary Conditions . . . . . . . . . . . . . 20 2.2.2 Determination of Natural Frequencies and Transient Response . . . . 23 2.3 Nonuniform Bar with Longitudinal Boundary Forcing . . . . . . . . . . . . . 24 2.4 Integration of Theoretical Pulse Waveforms . . . . . . . . . . . . . . . . . . . 26 2.4.1 Rectangular Pulse Boundary Forcing . . . . . . . . . . . . . . . . . . 26 2.4.2 Sine Pulse Boundary Forcing . . . . . . . . . . . . . . . . . . . . . . 27 2.4.3 Cosine Pulse Boundary Forcing . . . . . . . . . . . . . . . . . . . . . 28 2.4.4 Decaying Exponential Boundary Forcing . . . . . . . . . . . . . . . . 30 2.4.5 Adjusted Decaying Exponential Boundary Forcing . . . . . . . . . . . 30 3 Experimental Measurement of the Percussive Wave Produced by TRI- DENT 33 3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Preliminary Drop Test . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.2 Response of Prototype Percussion Subassembly . . . . . . . . . . . . 35 iii 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 DTFM Transient Response for Uniform Bar 41 4.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Rectangular Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.1 Spring Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Fixed Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Single-cycle Cosine Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Exponential Decay Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Adjusted Exponential Decay Forcing . . . . . . . . . . . . . . . . . . . . . . 51 5 DTFM Transient Response for Non-Uniform Bar 55 5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Rectangular Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.1 Spring Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Single-cycle Cosine Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.4 Exponential Decay Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.5 Adjusted Exponential Decay Forcing . . . . . . . . . . . . . . . . . . . . . . 63 5.6 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 A Technique to Incorporate Experimental Forcing 68 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2 Demonstration of Numerical Convolution Technique . . . . . . . . . . . . . . 70 6.3 Performance with Experimental Forcing . . . . . . . . . . . . . . . . . . . . 72 7 Dynamic Analysis of the Helical Compression Spring in the TRIDENT Percussion Mechanism 75 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2 TRIDENT Percussion System . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.3 Static Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.4 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.4.1 Discrete-element Dynamic Model . . . . . . . . . . . . . . . . . . . . 84 7.4.2 One-Dimensional Dynamic Models . . . . . . . . . . . . . . . . . . . 85 7.4.3 Distributed Transfer Function Method . . . . . . . . . . . . . . . . . 88 7.4.4 Multi-Dimensional Models of Spring Behavior . . . . . . . . . . . . . 91 7.4.5 Finite Element Model of Spring and Hammer . . . . . . . . . . . . . 92 7.5 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.5.1 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8 Application of DTFM to a Manual Soil Penetrometer Device 108 8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.2 Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 iv 8.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.4.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.4.2 Application of DTFM model . . . . . . . . . . . . . . . . . . . . . . . 119 8.4.3 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 121 References 123 v List of Tables 7.1 Relevant material properties of 17-7 PH and 302 series stainless steels, as well as the midline and amplitude shear stresses estimated by quasi-static theory. 82 7.2 Spring configurations used in the percussion subassembly testbed. 17-7 PH and 302 SS wire was .148" thick, while OTCS wire was .156" thick. . . . . . . 98 7.3 Operational life of various spring and spacer combinations. Both the neoprene spacer and the stronger OTCS material resulted in significant increases in cycle life. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 vi List of Figures 1.1 AprototypeunitofTheRegolithandIceDrillforExplorationofNewTerrains (TRIDENT), a state-of-the-art rotary-percussive drill designed by Honeybee Robotics Spacecraft Mechanisms Corporation. . . . . . . . . . . . . . . . . . 2 1.2 A compression spring that broke as a result of undesirable wave transmission in the percussion mechanism analyzed in this work. An unbroken spring from the same batch is shown for reference. . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Force balance of a 1D element in a slender elastic rod. . . . . . . . . . . . . . 9 1.4 An example of a sudden change in impedance due to a difference in area, density, and/or elastic modulus. . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Collinear impact of a cylindrical hammer on a long slender rod. . . . . . . . 13 1.6 Generation of a rectangular pulse resulting from the impact of a cylindrical hammer on a rod of equal area, density, and modulus. . . . . . . . . . . . . . 14 1.7 Rigid mass impacting a long slender rod. . . . . . . . . . . . . . . . . . . . . 15 1.8 An example of a 1D model of a percussive hammer and drill rod that divides eachcomponentintodiscretesegments,eachwithuniformmechanicalimpedance. 16 2.1 Depiction of rectangular pulse forcing. . . . . . . . . . . . . . . . . . . . . . 27 2.2 Depiction of half-wave sine pulse forcing. . . . . . . . . . . . . . . . . . . . . 28 2.3 Depiction of single-cycle cosine pulse forcing. . . . . . . . . . . . . . . . . . . 29 2.4 Depiction of exponential decay pulse forcing. . . . . . . . . . . . . . . . . . . 30 2.5 Depiction of corrected exponential decay pulse forcing. . . . . . . . . . . . . 32 vii 3.1 Illustration of the setup for drop testing. SG1 is the upper strain gauge and PLC is the Piezo Load Cell (or Piezo Force Transducer). . . . . . . . . . . . 35 3.2 The instrumented percussion testbed (left), and the prototype drill (right). . 36 3.3 Step down feature present on the top of the anvil to prevent interference due to deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Top view of the step down feature on the anvil, which sits centered in a bronze strike plate in the assembled testbed. . . . . . . . . . . . . . . . . . . . . . 37 3.5 Time traces of a force pulse produced from dropping a steel bar on a long slender rod, measured at near the top of the rod and the base of the rod. . . 38 3.6 Force waveform resulting from the percussion subassembly used in the TRI- DENTdrill. Thiswaveformwastakenwhenthemechanismlifewasatapprox- imately 3.6× 10 6 cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1 A basic model of the percussive drilling of rock where F (t) represents the prescribed force pulse due to a hammer impact. The boundary on the right is shown as a spring, and will also be replaced with a free and fixed boundary. 41 4.2 Transient response of a uniform bar subject to a rectangular pulse on the left with a spring boundary on the right. . . . . . . . . . . . . . . . . . . . . . . 42 4.3 The wavefront resulting from an approximate solution for the impact of two semi-infinite elastic rods with radial inertia and dispersion taken into account, developed by Skalak [49], taken from Graff [17]. . . . . . . . . . . . . . . . . 43 4.4 A detailed view of the wavefront produced from rectangular forcing using DTFM with 50 modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 Transient response of a slender uniform rod subject to rectangular forcing on the left with a fixed boundary on the right. . . . . . . . . . . . . . . . . . . . 45 4.6 Transient response of a slender uniform rod subject to cosine pulse forcing on the left with a fixed boundary on the right. . . . . . . . . . . . . . . . . . . . 46 4.7 Transient response of a slender uniform rod subject to cosine pulse forcing on the left with a free boundary on the right. . . . . . . . . . . . . . . . . . . . 47 viii 4.8 Asemi-theoreticalsingle-cyclecosineapproximationofthedroptestdescribed in Section 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.9 Transient response of a uniform rod with exponential pulse forcing on the left and a free boundary on the right. . . . . . . . . . . . . . . . . . . . . . . . . 49 4.10 Transient response of a uniform rod with exponential pulse forcing on the left and a fixed boundary on the right. . . . . . . . . . . . . . . . . . . . . . . . 50 4.11 A detailed view of the exponential pulse wavefront constructed with a varying number of modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.12 Simulated adjusted exponential pulse forcing on a uniform rod with a fixed boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.13 Simulated adjusted exponential pulse forcing on a uniform rod with a free boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.14 A detailed view of the adjusted exponential pulse wavefront constructed with a varying number of modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.1 Stepped bar system subject to longitudinal forcing on the left, with a spring boundary on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Transient response of a stepped bar system to rectangular pulse forcing on the left, supported by a spring on the right. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . 56 5.3 A detailed view of a complex segment of the transient response. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . 57 5.4 Transient response of a single-cycle cosine pulse acting on a nonuniform bar with a fixed boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.5 Transient response of a single-cycle cosine pulse acting on a nonuniform bar with a free boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ix 5.6 Transient response of an exponential decay pulse acting on a nonuniform bar with a fixed boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.7 Transient response of an exponential decay pulse acting on a nonuniform bar with a free boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.8 Transient response of an exponential decay pulse acting on a nonuniform bar with a spring boundary (k = 2500kN/mm). A sudden decrease in cross sec- tional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . 62 5.9 Transient response of an adjusted exponential decay pulse acting on a nonuni- form bar with a fixed boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . . . . . 63 5.10 A detailed view of an adjusted exponential decay pulse interacting with a sudden change in cross sectional area in a rod. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . 64 5.11 Transient response of an adjusted exponential decay pulse acting on a nonuni- form bar with a free boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. . . . . . . . . . . . . . . . . . . . . . . . 65 6.1 Percussive hammer used in TRIDENT. . . . . . . . . . . . . . . . . . . . . . 68 6.2 Complex pulse shape produced by TRIDENT cannot be easily modeled using simple forcing functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.3 Bar response to sinusoidal forcing which was applied using analytical convolu- tion and discrete convolution with sample periods of 1, 5, and 10μs. Response was constructed using 20, 40, and 100 modes which corresponds to vibrational periods of 20, 10, and 4μs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.4 Distributed system model of the dummy drill rod used in the experiment in Section 3.1.2 with an unknown boundary condition. . . . . . . . . . . . . . . 72 x 6.5 Experimentally captured incident pulse, padded with zeros to allow for com- putation of the transient response at times longer than the pulse duration. . 73 6.6 Bar response to experimentally measured forcing. Effects of fixed, free, and spring boundaries are compared with the experimental measurement of the reflected wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.7 Bar response to experimental forcing with a spring boundary compared to the experimentally measured reflection. Despite a delay, the spring boundary captures the dynamic behavior of the base of the testbed. . . . . . . . . . . . 74 7.1 A late-stage prototype TRIDENT unit. . . . . . . . . . . . . . . . . . . . . . 78 7.2 Rendering of the percussion mechanism in TRIDENT. The gears and cam (blue) are driven by a motor to rotate, causing the hammer (purple) to raise up and compress the spring (red). After a full rotation the hammer is released towards a drill bit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.3 A simplified depiction of the general operation of the percussion mechanism in TRIDENT. Typical hammering frequency is approximately 16Hz. . . . . . 79 7.4 Time-history force produced by the TRIDENT percussion mechanism. . . . 80 7.5 Frequency content of the impact produced by TRIDENT. . . . . . . . . . . . 80 7.6 Goodman diagram for two springs of two materials in the TRIDENT percus- sion configuration. The line for 302 series spring was generated using experi- mental data and known material properties. The line for the 17-7 PH spring was generated using a zero-max basis, assuming 10 7 cycle endurance strength in shear (S se ) equal to 30% of the material tensile strength. . . . . . . . . . . 83 7.7 Illustration of the general operation of the percussion mechanism in TRI- DENT with consideration into dynamic spring behavior post impact (Step 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.8 Discrete-element model of an impact occurring at the face of the hammer, composed of a massless spring and a lumped element mass equal to the mass of the hammer assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 xi 7.9 Illustration of the general operation of the percussion mechanism in TRI- DENT with consideration into dynamic spring behavior post impact (Step 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.10 Graphical depiction of a half-sine pulse from 0 to T/2. The shaded area under the curve is equal to the delivered impulse to the hammer. . . . . . . . . . . 87 7.11 Governing equations and diagramatic representation of Wittrick’s coupled wave model. Taken from [53]. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.12 Governing equations and diagramatic representation of the Timoshenko beam model used to validate Wittrick’s coupled wave model. Taken from [53]. . . . 92 7.13 Basic depiction of the model with constraints. . . . . . . . . . . . . . . . . . 93 7.14 Physical hammer assembly and its finite element model counterpart. Note that the shaft going through the middle of the spring is hollow. . . . . . . . 94 7.15 Spacer location with respect to the spring and hammer. . . . . . . . . . . . . 95 7.16 Standard deviation and average of spring actuator current over the lifetime of the spring. Spring failure can be retroactively determined using motor teleme- try, which appears as three abrupt drops in current standard deviation. 1st, 2nd, and 3rd spring fractures denoted at roughly 300, 600, and 650 thousand cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.17 Fragments of the initial spring used in this study which failed in three sequen- tial fractures. An in-tact spring from the same batch is shown for comparison. 97 7.18 Maximum shear stress in the middle coil during a .5J rebound after a hammer impact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.19 Twolocationsusedtomeasureshearstressovertimeinthefollowingsimulations.101 7.20 Maximum shear stress in the middle coil after a .5J pulse applied to the face of the hammer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.21 Maximum shear stress in the end coil after a .5J pulse applied to the face of the hammer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 xii 7.22 Maximum shear stress in the middle coil after a 1.5J pulse applied to the face of the hammer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.23 Maximum shear stress in the middle coil after a 1.5J pulse applied to the face of the hammer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.24 Repeated failure in the fixed-end coil was observed. . . . . . . . . . . . . . . 104 7.25 Goodman diagram showing permissible combinations of midline and ampli- tude shear stress for 17-7PH material to survive 10M cycles. Quasi-static techniques permit 10M cycles, but a 1.5J dynamic recoil can dramatically decrease the expected life. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.1 U.S.AirForceAirFieldAssesmentteamusingaDCPtomeasuresoilstrength at a military airfield in Afghanistan. Photo by U.S. Air Force Master Sgt. Tracy L. DeMarco (Dec. 2, 2009). Public Domain. . . . . . . . . . . . . . . . 109 8.2 Disassembled DCP that is being used in this work. Photo by Kessler Soils Engineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.3 Proposed 1D models of the Kessler DCP. The upper model is the simplified model containing only a rigid boundary on the left, the lower model is the full model which takes into account the upper segment as well as the masses of the anvil and the handle piece. . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.4 DCP used in this experiment. Strain gauges are placed a small distance below the anvil in order to delay reflection time for a clear measurement of the incident pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.5 Raw time-history force of a DCP impactor (8.12kg) being dropped from three heights(5cm, 10cm, and15cm). Strainwassampledat200kS/sandconverted to force using the known area and elastic modulus of the drill rod. . . . . . . 116 8.6 Frequency content of the raw 15cm drop response. An increase in amplitude is present around 50kHz and may indicate a band of noise. . . . . . . . . . . 117 xiii 8.7 Filtered and unfiltered time-history force resulting from a 15cm drop with the DCP. A 6th order Butterworth filter was used with a cutoff frequency of 30kHz. This filter was chosen as it shows significant smoothing of the quiescent region, with minimal loss of peak magnitudes. . . . . . . . . . . . . 117 8.8 Annotated incident pulse of a 15cm (11.97J) drop with the DCP used in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.9 Illustrationofthesectionoftime-historyforcethatwillbeusedasthe incident pulse to compare with modeling. . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.10 Incident pulse used in modeling efforts. This pulse resembles exponential decay, which is the theoretical result of a rigid mass impacting a slender rod. 120 8.11 Incident pulse used in modeling efforts. This pulse resembles exponential decay, which is the theoretical result of a rigid mass impacting a slender rod. 120 8.12 DTFM transient response for the 5cm and 10cm case compared with exper- imental data. An exponential pulse was applied to the full model in both cases, with a magnitude proportional to the relative velocity at impact. . . . 121 xiv Abstract The search for resources and extraterrestrial life in our solar system requires subsurface sampling on other worlds. Because exact surface and subsurface conditions are often not known until arrival, sampling mechanisms must be designed to handle the most difficult plausible conditions in the environment of interest, such as hard rocks. One proven method of breaking through rock is percussive drilling, whereby a hammer impact is used to generate a high-magnitude short-duration strain wave, which travels down the length of a drill rod to crush rock. Because hammer impacts can result in significant transient stresses in nearby sensitive components, these mechanisms must be carefully designed. While there exists a large body of work concerning the generation of stress waves for simple-geometry hammers, hammers with complex geometries remain difficult to model accurately. In this work, a combined experimental and analytical approach to understanding stress waves induced by percussion mechanisms with complex geometries is developed and applied to various drilling systems. An analytical framework to model the transient response of percussion mechanisms is developedusingtheDistributedTransferFunctionMethod(DTFM).Thistechniqueisshown to be capable of handling high-frequency transient responses of nonuniform distributed sys- tems with complex boundary conditions subject to various styles of boundary forcing. An experiment is conducted to characterize the percussive waves produced by a drilling device and a dynamic cone penetrometer. These waves are incorporated into modeling efforts as boundary forcing by means of numerical and analytical convolution techniques. The primary xv focus of this experiment is The Regolith and Ice Drill for the Exploration of New Terrains (TRIDENT), a lunar drill designed for use on multiple upcoming NASA missions. It is found that DTFM-based percussion analysis is useful in characterizing unknown boundary conditions in physical systems. It is also found that a critical helical compression spring in the TRIDENT percussion mechanism fails at <20% of its required life, despite infi- nite cycle life as predicted by quasi-static fatigue theory. Further experimentation involving this spring suggests the presence of high-frequency dynamic behavior resulting from hammer impact. A DTFM-based percussion analysis of the spring shows that constructive interfer- ence resulting from high-frequency loading due to impact explains this failure. xvi Chapter 1 Background and Motivation 1.1 Background on Percussive Drilling Finding resources and evidence of life requires the sampling of subsurface rock or regolith on other worlds. Unfortunately, the exact surface and subsurface conditions of other worlds are often not known until after arrival, which can lead to mission failure if not properly addressed. For this reason, sampling mechanisms must be designed to handle the most difficult drilling conditions conceivable in the environment of interest, such as hard rocks. One proven method of breaking through rock is rotary-percussive drilling. This type of drilling is comprised of percussion and rotation action which serve distinct purposes. Specifically, percussion is used to generate a high-magnitude short-duration strain wave, often from a hammer impact, which travels down the length of a drill rod and forces bit teeth into a rock, ideally exceeding the strength of the medium and crushing it into chips [58], although the actual mechanism of crushing is complex and highly dependent on the bit, as well as the drilled medium [36, 37, 31, 48, 47, 21, 22, 24, 23]. The ability to penetrate and crush the medium is also dependent on the magnitude and duration of the strain wave, which is dependent on the geometry and material properties of the hammer and drill rod [36, 37, 38, 7, 12, 21, 22, 24, 23, 46]. For simple geometry hammers (e.g. a solid cylinder or rigid mass), the resulting strain wave is predictable and can be estimated during the design process, but complicated hammer geometries, such as those necessary for space applications, produce strain waves that can be challenging to predict [7, 46, 12]. Percussion is often combinedwithrotaryactionwhichservestoremovechipsfromthebottomofthehole, aswell as provide a fresh surface for the bit teeth to interact with each blow. For brittle materials, rotary action synergistically works with percussive action to greatly improve efficiency over pure percussive or pure rotary action alone [18]. Besides its superior ability to penetrate 1 hard rock, another major advantage of rotary-percussive drilling is the low requirement for weight on bit [18]. This is especially important for space applications such as sampling on moons and asteroids, where low gravity does not allow downward forces to be simply reacted by lander or rover weight alone. Figure 1.1: A prototype unit of The Regolith and Ice Drill for Exploration of New Terrains (TRIDENT), a state-of-the-art rotary-percussive drill designed by Honeybee Robotics Spacecraft Mechanisms Corporation. Despite these advantages, percussive drilling still faces all of the challenges associated with space. This includes strict limitations for power, and limitations on mass, which typ- ically results in components with thinner, weaker structures. Such structures tend to have lower natural frequencies, making them susceptible to unintended vibration. A lack of under- standing of the complex behavior of percussion mechanisms can make it difficult to design with precision towards reducing these drawbacks. For example, uninformed hammer and drill rod designs can result in a large portion of impact energy being wasted by propagat- ing within the mechanism, rather than towards breaking rock. In this case, the resulting mechanism would require higher magnitude impacts to compensate for this inefficiency. This 2 in turn will necessitate more massive motors requiring more power, and upsizing of hous- ings and fasteners to deal with the structural needs associated with these more powerful impacts. Furthermore, unexpected or poorly understood wave reflection and transmission during hammering can damage essential delicate components, such as springs, sensors, and other electronics. An example of this shown in Figure 1.2. Figure 1.2: A compression spring that broke as a result of undesirable wave transmission in the percussion mechanism analyzed in this work. An unbroken spring from the same batch is shown for reference. A better understanding of wave propagation in these mechanisms can lead to numer- ous benefits. A significant amount of work has been done in developing effective models of percussive wave transmission in long rods for relatively simple circumstances, but the industry preferred method is 3D Finite Element analysis, which requires highly skilled users to accurately model high frequency dynamic behavior, with computation times upwards of several days. This research is intended to lay the groundwork for a series of analytical or semi-analytical percussive wave models using the Distributed Transfer Function Method (DTFM), which combines improved flexibility in modeling while maintaining computational simplicity with a sound theoretical basis. 3 1.2 Scope This work is intended to accomplish the following goals: • Develop a general modeling framework using DTFM that can describe a diverse class of percussion mechanisms • Demonstrate the utility of this technique by comparing to gold-standard numerical methods and experimental results collected from various percussion mechanisms • Identify and understand failure modes and performance limitations of the percussion mechanism in TRIDENT The remainder of Chapter 1 will include a review of existing analysis methods relevant to percussive drilling. In Chapter 2, DTFM is introduced and applied to the context of percussive drilling, and a variety of theoretical force pulses are presented for use in DTFM. In Chapter 3 an experimental procedure to measure the pulse produced by percussion mech- anisms is demonstrated. In Chapters 4 and 5, the transient response of models of varying complexity is shown and compared to a gold-standard computational method to demonstrate the capabilities and limitations of DTFM. In Chapter 6 a technique to incorporate exper- imentally measured forcing into DTFM models is provided and implemented into a model of the experiment in Chapter 3. Chapter 7 discusses a component failure in the TRIDENT percussion mechanism, which is investigated though a series of experiments combined with several transient dynamic analysis techniques. In Chapter 8, an experiment is conducted to measure the percussive wave produced by a dynamic cone penetrometer, which is incorpo- rated into a DTFM model using a representative analytical forcing function. 4 1.3 Review of Existing Techniques 1.3.1 Survey of Literature In its simplest form, percussive drilling is a result of collision, or impact, between solid bodies. Unfortunately, rigorous dynamic analysis of impact beyond the simplest of cases tends to result in discouragingly complex mathematics, thus approximations are often used. Stereomechanical impact [16] is one of the most basic impact response approximations which is based on the impulse-momentum law, and assumes that there is an instantaneous change from initial to final velocity of the colliding bodies. This method can be enticing for the designer of a percussive mechanism in that it is mathematically simple, and works well to describe certain types of collisions such as the impact between two elastic spheres [16]. However, this method also requires certain nonphysical assumptions such as infinitesimal contact time and neglects any form of local deformation or transient response of materials. This results in significant error for impacts involving solid bodies where significant energy is converted to vibration rather than rigid-body momentum transfer. Physically, these bodies are typically those with geometries resulting in relatively low natural frequencies, such as long slender rods, as well as beams and plates [16, 14], which tend to respond to impact via the propagation of elastic waves. Propagation of longitudinal waves in slender rods has been investigated for over a century using the classical theory of elasticity [32]. Rigorous analytical solutions to the 3D wave equation for longitudinal waves in cylindrical bars are complex and not tractable for practical cases. A prominent example of this class of models are the solutions independently produced by Pochammer and Chree in the late 19th century. This approach assumes a semiinfinite cylindrical elastic bar, with sinusoidal forcing of a single frequency for an infinite duration [41, 8, 44]. A frequency equation for this problem was derived and discussed in detail by Love [32] which implies that wave speed is dependent on the perturbation frequency, thereby resulting in dispersion for any forcing that contains a finite band of frequency content. In particular, it is expected that high frequency content of a longitudinal wave travels slower 5 than the lower frequency content that resulted from the same impact [44, 32]. Researchers such as Bancroft [3] have proposed methods that reduce the complexity of these solutions by assuming specific relationships between rod parameters, however additional complexities of 3D wave propagation analysis may be avoided if it is assumed there is negligible radial inertia, no dispersion, axial symmetry within the rod, and a planar wave front, which results in 1D wave theory. This approximation is acceptable for cases involving long slender rods transmitting longitudinal waves with wavelengths much longer than the rods diameter [16, 28]. In the early 20th century, Hopkinson [20] was considered the first to make basic measure- ments of longitudinal waves in steel bars, produced by the impact of a bullet and measured with a ballistic pendulum, allowing for estimation of durations and magnitudes of stress waves in the bar. Exact waveforms could not be resolved until later when instruments such as strain gauges, oscilloscopes of sufficient sensitivity and sampling rate, and amplifiers with sufficient bandwidth were available for experimental measurements of waveforms [14]. At this time Davies [9] used electronic methods to experimentally observe the behavior of longi- tudinal waves in a slender bar for wavelengths much longer than the radius of the bar. This experiment demonstrated that the response of the bar was in agreement with Pochhamer- Chree theory, and experimentally validated the use of 1D wave theory for slender rods. Donnell [11] was one of the first to use 1D wave theory to model rock drilling. These method were later improved by Fischer [14] who also expanded upon a graphodynamical method developed by De Juhasz [10]. Fischer also provided a series of applicable solutions to 1D longitudinal impact applied to rods for hammer drilling for a wide range of simple casesthatmaybeencountered, includingdissimilarinterfacesandsteppedbars, andprovided methods to theoretically predict the resulting shape of a strain wave generated from impact based on the geometry of the hammer and the rod, providing tools for cases with clear application to practice. Others including Dutta [12], Fischer [14], Simon [46], and Lundberg [38] studied the generation of pulse shapes in drill rods based on hammer geometry, and 6 provided computational methods to determine these pulse shapes using algorithms based on 1D wave theory. Hustrulid and Fairhurst [21, 22, 24, 23], Lundberg [36, 37], and Simon [48, 47] studied the efficiency of percussive drilling by applying 1D longitudinal wave propagation theory to the destruction of rock, with consideration of joints and complex rock boundary conditions modeled using a force-penetration relationship, often approximated as a bilinear spring. This was validated experimentally by Karlsson [27], Carlvik [7], and Lundberg [38]. At the same time, Carlvik [7] as well as Hawkes and Burks [19], studied bending resulting from longitudinalimpactpredictingthatupwardsof20%ofenergycanbeconvertedtoundesirable bending for poorly designed percussion mechanisms. Lundberg later refined an iterative computer simulation based on transmission and reflec- tion ratios of rod segments of desired lengths, which was validated experimentally as well [27]. Elias and Chiang [13] produced an alternative iterative computational method based on the impulse-momentum principle between sequential elements in an attempt to make forc- ing and boundary conditions more straightforward compared to Lundberg’s method, since hammers used in certain specialized applications tend to stray from the ideal cases, making it very difficult to model the impact forces in this way. Regardless, Lundberg’s computer simulation method still remains useful as it is convenient, intuitive, widely applicable for a variety of drilling machines, and shown to be accurate in practice [27, 13]. Asimplemethodofanalyticallydeterminingthetransferfunctionsofelasticcontinuaand their natural frequencies called the Distributed Transfer Function Method (DTFM) has been presented and refined by Yang and Tan [56]. Yang laid the framework to use this method to analytically compute the exact transient response of piecewise-uniform or “stepped” common mechanical systems (bars, shafts, and beams) subject to distributed or boundary forcing with added lumped elements such as springs and masses. While this can be attempted using traditional modal analysis techniques, it often results in severe mathematical complexity for all but the simplest cases. DTFM is preferable in these scenarios as it results in relatively simple solutions [54]. Additionally, Yang and Noh [55] have presented fundamental matrices 7 for a variety of nonuniform cross sections, allowing users to construct analytical models with continuously varying properties along the length of a continuum. DTFM is flexible in design and computationally simple, but has not yet been used in percussive drilling applications even though it appears to be a good candidate. Part of this work is intended to provide a framework for analytically simulating the propagation of stress waves as a result of impact on elastic continua using DTFM. Lundberg’s method will be used for comparison to illustrate the potential for DTFM to be used percussive drilling analysis. As motivation, an experiment is performed to measure the shape of the strain waveform from an existing percussive drill that is intended to be the primary sampling mechanism on NASA’s 2023 VIPER lunar mission, which was used as boundary forcing integrated in a DTFM model in this research. 1.3.2 Longitudinal Waves in Three Dimensions in Elastic Rods The equations of motion for displacement in three dimensions in an elastic solid were independently found by Pochhammer [41] and Chree [8], and can be derived using the clas- sical theory of elasticity. A derivation is given by Goldsmith [16], and the result is presented here for completeness. In cylindrical coordinates, (r,θ,z), it can be shown that the displace- ment u and rotation ω within a homogeneous isotropic elastic body of density ρ, elastic modulus E, Poisson’s ratio μ, is governed by ρ ∂ 2 ur ∂t 2 = (λ + 2G) ∂Δ ∂r − 2G r ∂¯ ωz ∂θ + 2G ∂¯ ω θ ∂z ρ ∂ 2 u θ ∂t 2 = (λ + 2G) 1 r ∂Δ ∂θ − 2G ∂¯ ωr ∂z + 2G ∂¯ ωz ∂r ρ ∂ 2 uz ∂t 2 = (λ + 2G) ∂Δ ∂z − 2G r ∂ ∂r (r¯ ω θ ) + 2G r ∂¯ ωr ∂θ (1.1) where Δ = ∂u r ∂r + u r r + 1 r ∂u θ ∂θ + ∂u z ∂z (1.2) 8 and G and λ are called the Lamé constants with G = 2E 1 +μ (1.3) and λ = μE (1 +μ)(1− 2μ) . (1.4) This set of equations has no closed-form solution for most practical cases, but provides the analytical basis for many techniques used to model wave propagation. 1.3.3 Longitudinal Waves in One Dimension in Slender Elastic Rods Consider a uniform elastic rod that is much longer than its diameter ( D L << 1) with negligible radial inertia. If we also assume cross sections that remain in a unified plane, wave propagation for wavelengths much longer than the rods diameter can be well described using the 1D wave equation[16, 28, 17, 38, 9]. A derivation of the 1D wave equation for longitudinal waves in a long slender bar can be found in texts by Goldsmith [16], Kolsky [28], and Graff [17]. Consider a force balance applied to a 1D differential element dx that has been displaced a distance u, as shown in Figure 1.3. Figure 1.3: Force balance of a 1D element in a slender elastic rod. 9 Using Newton’s second law, the force balance on the 1D element yields ΣF x =m ∂ 2 u ∂t 2 =N + ∂N ∂x dx−N = ∂N ∂x dx. (1.5) Using Hooke’s law, N =EA = ∂u ∂x EA (1.6) the sum of the forces can be rewritten as ρA ∂ 2 u ∂t 2 =EA ∂ 2 u ∂x 2 (1.7) or potentially more conveniently ∂ 2 u ∂t 2 = E ρ ∂ 2 u ∂x 2 , (1.8) where ρ is the material density, A is the cross sectional area of the rod, E is the elastic modulus, and = ∂u ∂x is the strain. The general solution of Equation (1.8) can be written in the d’Alembert form as u(x,t) =f(x−ct) +g(x +ct). (1.9) This solution implies two non-dispersive elastic waves that travel in the positive and negative directions, with a constant speed ofc = q E ρ . This solution can be differentiated with respect to position to reveal the distributed strain and velocity as (x,t) = ∂u(x,t) ∂x =f 0 (x−ct) +g 0 (x +ct). (1.10) Lundberg[38] provides a more application specific form of this solution using Hooke’s law, showing that the force along the length of the rod can be written as N(x,t) =EA ∂u(x,t) ∂x = [N p (x−ct) +N n (x +ct)] (1.11) whereN p andN n areforcewavestravelinginthepositiveandnegativedirectionsrespectively. 10 Equation (1.9) can also be differentiated with respect to time to reveal particle velocity along the length of the bar as v(x,t) = ∂u(x,t) ∂t =c[−f 0 (x−ct) +g 0 (x +ct)] (1.12) which can be written in terms of N p and N n as ∂u(x,t) ∂t = 1 Z [−N p (x−ct) +N n (x +ct)] (1.13) where Z = AE c =A q Eρ. (1.14) Z is called the characteristic mechanical impedance, which can be interpreted as the ratio between stiffness and velocity. Additionally, it can be shown [38] that the momentum P (t) and energy W (t) in a rod can be described as P (t) = 1 c Z (−N p +N n )dx (1.15) and W (t) =EA Z (N 2 p +N 2 n )dx. (1.16) It is useful in practice to consider a wave traveling only in the positive direction, N p (x− ct), that might interact with a sudden change in area, elastic modulus, or density in the longitudinal rod, shown in Figure 1.4. The resulting wave behavior can be predicted using the concept of characteristic mechanical impedance. 11 Figure 1.4: An example of a sudden change in impedance due to a difference in area, density, and/or elastic modulus. Here, the characteristic impedance of the left segment is Z 1 = A 1 √ ρ 1 E 1 , and similarly the right segment would have a characteristic impedance of Z 2 = A 2 √ ρ 2 E 2 . Using a force balance and assuming continuous velocity between these two segments, it can be shown [17, 38, 16] that the reflected (R) and transmitted (T) fraction of the wave travelling from left to right across this segment will be R = Z 2 −Z 1 Z 1 +Z 2 (1.17) T = 2Z 2 Z 1 +Z 2 . (1.18) This relationship between mechanical impedance and reflection and transmission coef- ficients is the basis for the theory surrounding 1D wave propagation in nonuniform rods, and is the fundamental concept used in Lundberg’s computational method, which will be described in greater detail in Section 1.3.5. Additionally, Graff [17] presents the fundamental result of waves interacting with free or fixed boundaries using the method of images. In the case of 1D longitudinal wave prop- agation, a wave interacting with a free boundary will be entirely reflected in the opposite direction with an opposite magnitude, causing it to destructively interfere with itself. A wave interacting with a fixed boundary will be entirely reflected in the opposite direction with the same magnitude, causing it to constructively interfere with itself. This basic behavior 12 is useful in understanding unknown boundary conditions and will be referenced frequently throughout this work. 1.3.4 Impact of a Cylindrical Hammer on a Rod The result of a longitudinal impact of two elastic rods is described in detail in numerous texts [16, 28, 38, 17], and is the basis for understanding percussive wave generation resulting from hammer impacts. The most basic example of this scenario involves an elastic cylindrical hammer (4> L D > 2) [14] with a rigid body velocity V impacting a quiescent long, slender rod ( L D >> 1) of equal material and cross sectional area. This is depicted in Figure 1.5. Figure 1.5: Collinear impact of a cylindrical hammer on a long slender rod. It can be shown [38] that the impact of the hammer on the drill rod will produce a rectangular compressive wave N p traveling in the positive direction, and a tensile wave N n traveling in the negative direction. These waves will have force magnitudes− 1 2 ZV in the positive direction and 1 2 ZV in the negative direction, where Z is the characteristic impedance of the rod. The lengths of these two waves will each be equal to the length of the hammer, L H , but because the negative wave is reflected off of the free boundary of the hammer, this results in one coherent rectangular wave of length 2L H with magnitude F 0 =− 1 2 ZV (1.19) and duration τ d = 2L H c (1.20) 13 traveling in the positive direction. This phenomenon is illustrated in Figure 1.6. Figure 1.6: Generation of a rectangular pulse resulting from the impact of a cylindrical hammer on a rod of equal area, density, and modulus. While this exact scenario rarely occurs in practice, it provides intuition on how force pulses are generated from hammer impacts. Cases where the characteristic impedance is not equal between the hammer and the rod will produce a force pulse with a longer duration and nonuniform magnitude. This is a result of a partial wave reflection occuring at the interface between the hammer and rod. Such cases are more complex, but are still predictable using the methods presented in this section. 1.3.5 Impact of Rigid Mass on Rod Modelingahammer impact asarigidmassimpactingarodmay beagood approximation for cases where the hammer is wider than the diameter of the rod, or could have a higher 14 elastic modulus than the rod, and may not be able to be well described as a slender rod itself [7, 17]. Figure 1.7: Rigid mass impacting a long slender rod. By applying the impulse-momentum principle to a rigid body of mass m travelling at speed V impacting an elastic rod [17], the strain distribution will be (x,t) = ∂u(x,t) ∂x = Θ(ct−x) −V c e −(ct−x) EA mc 2 (1.21) where E, A, and c are the elastic modulus, area, and wave speed of the elastic rod. Lundberg’s Computational Method Lundberg’s computational method unifies the content in Sections 1.3.3 through 1.3.5 in a computationally simple discrete numerical algorithm that is capable of dealing with nonuniform hammer and rod properties. It is described in detail by Lundberg [38], and is presented here in a basic form for context in comparisons made throughout this work. In its most basic form, Lundberg’s method requires that a hammer and drill rod are approximated by adjacent discrete 1D segments, each with a constant characteristic impedance, Z, based on its area, elastic modulus, and density. An illustration of this is given in Figure 1.8. 15 Figure 1.8: An example of a 1D model of a percussive hammer and drill rod that divides each component into discrete segments, each with uniform mechanical impedance. The total number of elements in this case would ben, and thus each segment would have a length of Δx = L total n (1.22) where L total is the combined length of the hammer and drill rod. Additionally, this means that the time increment of the resulting simulation is fixed at Δt = L total nc (1.23) where c is the wave speed. Thus, increasing the number of elements improves not only the spatial resolution of the simulation, but the temporal resolution as well. Now consider two adjacent segments at positionsi andi + 1, corresponding to character- istic impedances Z i and Z i+1 . At a given time kΔt, the force distribution can be written in terms of the previous time step [k− 1]Δt by considering the transmission and reflection of the positive and negative travelling force waves in adjacent segments. The resulting relation can be written as N p [i,k] =T p [i]N p [i,k− 1] +R n [i]N n [i + 1,k− 1] (1.24) 16 and N n [i,k] =R p [i]N p [i,k− 1] +T n [i]N n [i + 1,k− 1] (1.25) where R p [i] = Z i+1 −Z i Z i+1 +Z i (1.26) T p [i] = 2Z i+1 Z i+1 +Z i (1.27) R n [i] = Z i −Z i+1 Z i+1 +Z i (1.28) T n [i] = 2Z i Z i+1 +Z i (1.29) Thus, the net force distribution in the rod, N[k] =N n [k] +N p [k], (1.30) at time kΔt can be computed iteratively based on initial conditions. In this example, it is assumed that all segments in the hammer have a uniform velocity V with no initial force distribution, while the rod is initially quiescent. This initial condition can be applied to this model by considering that the particle velocity can be written v[i,k] = −N p [i,k] +N n [i,k] Z[i] (1.31) which implies that N p [i, 0] =− 1 2 Z[i]v[i, 0] (1.32) and N n [i, 0] = 1 2 Z[i]v[i, 0] (1.33) 17 wherek = 0 corresponds to the time immediately following the impact. In this specific case, v[i, 0] =          V, for 1≤i≤LH. 0, for LH + 1≤i≤n. (1.34) where LH is the index of the rightmost segment of the hammer, and n is the index of the rightmost segment of the rod, as shown in Figure 1.8. Additional details regarding specific cases can be found in [38], including methods to deal with boundary conditions and gapping between drill components. 18 Chapter 2 Application of Distributed Transfer Function Method 2.1 Motivation The Distributed Transfer Function Method (DTFM) is a systematic analytical technique which can determine exact natural frequencies, as well as the exact transient response, of distributed systems. It was chosen as the primary method of analysis in this research due to itscomputationalsimplicity, aswellasitsabilitytoeasilydealwithavarietyofboundaryand forcing conditions. By contrast, other analytical methods such as modal analysis result in complicated expressions that must be specifically derived for each set of boundary conditions or discrete elements. It has also been shown by Yang [54] that DTFM has great utility in dealing with stepped systems. In the context of this research, "stepped systems" specifically refers to a serial arrangement of uniform rods that may have differing area, elastic modulus, or density. By assigning additional constraint matrices between dissimilar sections, exact natural frequen- cies and transient behavior can be computed with a minimal increase in computational com- plexity. By comparison, in modal analysis stepped systems tend to be much more difficult to deal with except in the simplest of cases. Because percussive drilling systems are historically modeled as stepped distributed sys- tems with spring boundaries to represent rock, DTFM is a superior analytical method of analysis for this application. 19 2.2 Generic Bar with Boundary Forcing 2.2.1 Governing Equation and Boundary Conditions Consider a long slender bar undergoing longitudinal perturbations, f(x,t) such that the system can be described by the 1D wave equation, ρ L (x) ∂ 2 u(x,t) ∂t 2 − ∂ ∂x " EA(x) ∂u(x,t) ∂x # =f(x,t) (2.1) where ρ L (x) is the linear density, EA(x) is a distributed stiffness parameter, u(x,t) is dis- tributed longitudinal displacement, and f(x,t) is distributed forcing. We can also describe generic boundary conditions as m L ∂ 2 u(0,t) ∂t 2 +a 1 ∂u(0,t) ∂x +a 0 u(0,t) =γ L (t) (2.2) m R ∂ 2 u(L,t) ∂t 2 +b 1 ∂u(L,t) ∂x +b 0 u(L,t) =γ R (t) (2.3) where m L and m R are lumped masses at the left and right ends of the rod respectively, and γ L and γ R are boundary forcing functions. a 1 , a 2 , b 1 , and b 2 are coefficients of the displacement and first spatial derivative at respective ends of the bar. By taking the Laplace transform, our governing equation can be written as ρ L s 2 ˆ u(x,s)− ∂ ∂x " EA(x) ∂ˆ u(x,s) ∂x # = ˆ f(x,s) +ρ(x)(su(x, 0) + ˙ u(x, 0)) (2.4) Assuming an initially quiescent rod, such that u(x, 0) = 0 = ˙ u(x, 0) (2.5) the governing equation can be simplified to ρ L s 2 ˆ u(x,s)− ∂ ∂x " EA(x) ∂ˆ u(x,s) ∂x # = ˆ f(x,s) (2.6) 20 Additionally, by the assumption in Equation (2.5), the Laplace transform of the boundary conditions yields m L s 2 +a 0 ˆ u(0,s) +a 1 ∂ˆ u(0,s) ∂x =γ L (2.7) and m R s 2 +a 0 ˆ u(L,s) +a 1 ∂ˆ u(L,s) ∂x =γ R . (2.8) Next, define a spatial state vector ˆ η(x,s) =    ˆ u(x,s) ∂ˆ u(x,s) ∂x   , (2.9) which allows us to write our boundary equations in the form M b ˆ η(x,s) +N b ˆ η(x,s) =γ b (2.10) where M b =    m L s 2 +a 0 a 1 0 0    (2.11) and N b =    0 0 m R s 2 +b 0 b 1   . (2.12) and γ b =    ˆ γ L +m L (su 0 (0) +v 0 (0)) ˆ γ R +m R (su 0 (L) +v 0 (L))   . (2.13) Where u 0 and v 0 are the initial state of the position and velocity in the bar. With the assumption of quiescence in Equation (2.5) and assuming no lumped masses (m L =m R = 0), γ b =    ˆ γ L (s) ˆ γ R (s)   . (2.14) 21 The governing equation can then be written as ∂ ∂x ˆ η(x,s) =    0 1 ρ L (x) EA(x) s 2 − d dx EA(x) EA(x)    ˆ η(x,s) + ˆ f(x,s) EA(x)    0 1   . (2.15) It can be shown that Equation (2.15) can be solved using a state transition matrix[55], Φ, such as Φ(x,ξ,s) =U(x,s)U −1 (ζ,s) (2.16) and the fundamental matrix U can be found as U(x,s) =    u 1 u 2 ∂u 1 ∂x ∂u 2 ∂x   . (2.17) where u 1 and u 2 are the solutions to the homogeneous governing equation ρ L (x)s 2 ˆ u(x,s)− ∂ ∂x " EA(x) ∂ˆ u(x,s) ∂x # = 0. (2.18) Yang and Noh have found fundamental matrices for a variety of bars with nonuniform cross sections[55] which will be discussed in Section ??. For the current case of interest, it can be shown[55] that the state transition matrix of a uniform bar can be written as Φ(x, 0,s) =U(x,s) (2.19) where U(x,s) =    cosh(βx) 1 β sinh(βx) βsinh(βx)) cosh(βx)    (2.20) and β = s q EA/ρ . (2.21) 22 2.2.2 Determination of Natural Frequencies and Transient Response The state equation (2.15) can be manipulated to reveal the distributed transfer function of the bar, and it can be shown[56, 54, 55] that the natural frequencies are computed as the solution to det(M b +N b Φ(L, 0,jω k )) = 0 (2.22) where Z =M b +N b Φ(L, 0,s) (2.23) is called the boundary impedance matrix. Taking the inverse Laplace transform of the state equation [be specific], and simplifying this expression using transfer function residues, a compact form of the transient response can be derived [56, 54, 55] as    u(x,t) ∂u(x,t) ∂x    = 2Σ ∞ k=1 Φ(x, 0,jω k )Q k [I f,k (t) +I b,k (t) +I o,k (t)], (2.24) where jQ k =R k (2.25) and R k = adj(Z) d ds |Z(s)| s=jω k . (2.26) In general I f,k (t) accounts for the effect of external distributed loads, I o,k (t) accounts for the effects of nonzero initial conditions, and I b,k (t) accounts for boundary forcing. In the problems presented in this research, external distributed loads are neglected and initial conditions are assumed to be quiescent, meaning that I f,k (t) and I o,k (t) vanish. Thus, we are left with I b,k (t), which, assuming no discrete springs experiencing base excitation or intermediate lumped masses experiencing a direct load, can be written as 23 I b,k (t) = Z t 0 sin(ω k (t−τ))    γ L (τ) γ R (τ)   dτ. (2.27) 2.3 Nonuniform Bar with Longitudinal Boundary Forcing The method in Section 2.2 can be applied to a series of adjacent "stepped" rods, which is described by Yang in [54]. Assume we now have a series of n connected uniform rods with cross sectional area, elastic modulus, and density A i ,E i and ρ i respectively, where i = 1, 2,...n. For the preliminary cases that this work will deal with, it is assumed that there are no lumped masses or intermediate spring elements, which greatly simplifies the derivations in this section. A fully generalized description of this method can be found in [54], which may be of interest in future work as described in Section ??. In either case, Yang [54] shows that the fundamental matrix U described in Equation (2.20) to describe a uniform systems can be written for stepped systems as U(x,s) =          e F 1 (s)(x−x0) , for x 0 <x<x 1 . e F i (s)(x−x i−1 ) T i−1 e F i−1 (s)l i−1 ...T 1 e F 1 (s)l 1 , for x 1 <x<x i , for i = 1, 2,...n . (2.28) where F i (s) =    0 1 s 2 r 2 i 0    (2.29) with r 2 i = E i A i ρ i (2.30) 24 and under the assumption of no intermediate springs or lumped masses, T i =    1 0 0 EA i EA i+1   . (2.31) In this case, the boundary impedance matrix is computed as Z(s) =det(M b +N b Φ(x n ,x 0 ,s)) (2.32) with M b and N b given in Equations (2.11) and (2.12) respectively. Analogous to Equation (2.19) in the single uniform rod case, it can be shown that Φ(x,x 0 ,s) =U(x,s). (2.33) Thus natural frequencies can be computed as the roots of the characteristic equation Δ(ω) =det(M b +N b W (ω)) (2.34) where W (ω) =U(x n ,jω). (2.35) U(x n ,jω) is given in Equation (8.1). Again, similar to the uniform case, the transient response of the stepped system can be computed using the transfer function residues R k , where R k = adjZ(jω k ) jZ D (ω k ) . (2.36) Assuming no lumped masses, Z D (ω k ) is given by Z D (ω k ) =det(M b +N b d W (ω k )) +det(N b W (ω k )) +det(M b ). (2.37) 25 In this case, d W =E n (ω k )T n−1 e F n−1 (jω k )l n−1 ...T 1 e F 1 (jω k )l 1 +e Fn(jω k )ln T n−1 e F n−1 (jω k )l n−1 ...T 1 E 1 (ω k ) (2.38) where E i =     l i r i S ik −l i ω k C ik + r i ω 2 k S ik −l i ω k r 2 i C ik + 1 r i S ik l i r i S ik     (2.39) withS ik = sin( ω k l i r i ) andC ik = cos( ω k l i r i ). The transient response can be found from Equation (2.24) where Q k is given in Equation (2.25). 2.4 Integration of Theoretical Pulse Waveforms As shown in Section 2.2.2, any forcing function can be applied to the boundary of an elastic bar using Equation (2.27). It is common practice to model the forces resulting from an impact as a Dirac delta function when considering discrete elements. However, for an elastic continuum this results in a waveform that cannot be well represented with any less than an infinite number of modes, which is not useful in practice. For this reason, the most basic forcing function that will be analyzed is a rectangular pulse of finite duration and magnitude. 2.4.1 Rectangular Pulse Boundary Forcing Rectangular pulse forcing is the expected theoretical result of collinear impact between two rods of equal impedance, shown in Section 1.3.4. This model is nonphysical in that it has an instantaneous rise time and thus infinitely high frequency content. It can be derived in a form usable for DTFM by analyzing as the difference of two step functions with a delay, or F (t) =F 0 [θ(t)−θ(t−τ d )] (2.40) 26 where θ(t) is the Heaviside step function, τ d is the duration of the pulse, and F 0 is the magnitude of the pulse, which can be predicted analytically for ideal cases. A graphical depiction of this type of forcing is shown in Figure 2.1. Figure 2.1: Depiction of rectangular pulse forcing. From Equation (2.27), it follows that for a rectangular force pulse applied to the left side of the rod I b,k (t) = Z t 0 sin(ω k (t−τ))    F 0 [θ(t)−θ(t−τ d )] 0   dτ, (2.41) resulting in I b,k (t) =F 0 1− cos(ω k t) ω k − 1− cos(ω k (t−τ d )) ω k !    1 0    (2.42) 2.4.2 Sine Pulse Boundary Forcing A possible more physical pulse is a half-wave sine function of the form F (t) =F 0 sin(Ωt) [θ(t)−θ(t−τ d )] (2.43) whereF 0 is the peak force,τ d is the duration of the pulse, and Ω is the frequency of the sine wave such that τ d =π/Ω. This forcing is represented graphically in Figure 2. 27 Figure 2.2: Depiction of half-wave sine pulse forcing. In certain cases, this pulse approximation may be a good representation of impact due to its finite rise time, while still exhibiting an abrupt change at the beginning and end of the pulse. This forcing model could be a good approximation of a nonlinear contact condition during impact. Implementing this behavior as boundary forcing in a DTFM model can be done using Equation (2.27). Assuming forcing at the left boundary of the rod, we get I b,k (t) = Z t 0 sin(ω k (t−τ))    F 0 sin(Ωt) [θ(t)−θ(t−τ d )] 0   dτ, (2.44) resulting in I b,k (t) =F 0 ω k sin(Ωt)− Ω sin(ω k t) ω 2 k − Ω 2 − sin(ω k t−τ d (ω k + Ω)) + sin(Ωt) ω k + Ω (2.45) + sin(Ωt)− sin(ω k t +τ d (Ω−ω k )) ω k − Ω !    1 0    2.4.3 Cosine Pulse Boundary Forcing Single-cycle cosine pulse boundary forcing may be useful to represent certain impact events as it encapsulates the spring-like behavior of a sinusoid, while also being entirely smooth. This forcing can be described as 28 F (t) =F 0 (1− cos(Ωt)) [θ(t)−θ(t−τ d )] (2.46) where τ d is the duration, F 0 is the peak force, and Ω is the pulse frequency which in this case is Ω = 2π τ d . (2.47) This forcing is depicted in Figure 2.3. Figure 2.3: Depiction of single-cycle cosine pulse forcing. Using Equation (2.27), it follows that for a single-cycle cosine pulse applied to the left side of the rod results in I b,k (t) =F 0 " 1 2 ω k (cos(Ωt)− cos(ω k t) ω 2 k − Ω 2 + 1 4 cos(Ωt)− cos(ω k t−τ d (ω k + Ω)) ω k + Ω (2.48) − cos(ω k t +τ d (Ω−ω k ))− cos(Ωt) ω k − Ω + 1 2 1− cos(ω k t) ω k − 1 2 1− cos(ω k (t−τ d )) ω k #    1 0    29 2.4.4 Decaying Exponential Boundary Forcing The resulting longitudinal strain wave from the impact of a rigid massm with velocityV ontheendofarodwasdescribedinEquation(1.21), whichallowsustowritethecompressive forcing on the end of the rod (x = 0) for time t> 0 as EA(x,t) =EA V c e −(ct) EA mc 2 (2.49) which is shown in Figure 2.4 with F 0 =EAV/c. Figure 2.4: Depiction of exponential decay pulse forcing. Assuming forcing at the left side of the rod, Equation (2.27) results in I b,k (t) =F 0 ω k e −αt +α 0 sin(ω k t)−ω k cos(ω k t) ω 2 k +α 2 0 !    1 0    (2.50) where α = EA mc (2.51) 2.4.5 Adjusted Decaying Exponential Boundary Forcing Carlvik [7] points out that the instantaneous rise time resulting from exponential decay pulse forcing is nonphysical, and has presented a simple correction by adding additional time 30 dependence to the exponential decay function. This results in a forcing function that takes the general form F (t) =F 0 0 te −α 1 t (2.52) where F 0 0 can be interpreted as the initial linear loading rate. Differentiation of F (t) with respect to time yields d dt F (t) =F 0 0 e −α 1 t −α 1 F 0 0 te −α 1 t . (2.53) To compute the time of peak force (rise time), we have d dt F (t) =F 0 0 e −α 1 t −α 1 F 0 0 te −α 1 t = 0. (2.54) This shows that the rise time t =t r is t r = 1 α 1 . (2.55) Then, using Equation (2.52), this results in a peak force F 0 given by F 0 = 1 e F 0 0 α 1 . (2.56) The forcing in (2.56) is depicted graphically in Figure 2.5 below. 31 Figure 2.5: Depiction of corrected exponential decay pulse forcing. For forcing at the left side of the rod, Equation (2.27) results in I b,k (t) =F 0 0 ω k te −α 1 t α 2 1 +ω 2 k + 2α 1 ω k e −α 1 t + sin(ω k t)(α 2 1 −ω 2 k ) + 2α 1 ω k cos(ω k t) (α 2 1 +ω 2 k ) 2 !    1 0    (2.57) which is a form suitable for use in DTFM. 32 Chapter 3 Experimental Measurement of the Percussive Wave Produced by TRIDENT Due to the geometric irregularity of the percussion mechanism in this research, a two- part experiment was necessary to measure stress waveforms and also to show that existing methods do not easily describe the complex contact behavior of the hammer and anvil. The first part was based around a low complexity impact event (a dropped cylinder) to validate the use of DTFM as an experimentally based analytical method, and the second part was based around an impact resulting from the prototype TRIDENT drill percussion subassembly. 3.1 Methods A testbed was designed and assembled based on a prototype TRIDENT drill built by Honeybee Robotics, which is the primary sampling mechanism on the 2023 VIPER lunar mission. The first part of this testing was to measure strain waveforms from basic impact scenarios, such as cylinder impacts, to directly compare DTFM with physical data from a similar system. In addition, the second part of this testing was the measurement of the true strain waveform generated from the percussion mechanism as-designed for use in future analysis. Both parts involved the longitudinal impact of a cylindrical 17-7 stainless steel rod supported at the base with a steel plate and shock absorbing dampers to prevent excessive damage to the testbed and floor below. 33 3.1.1 Preliminary Drop Test Part one of the experiment involved the response of a .114m long, 19mm diameter steel cylinder, dropped from a height of .114m above the anvil. The resulting strain wave was measured and used to construct a preliminary DTFM model assuming a rigid boundary condition provided by the steel plate at the base. In this part of the experiment, the drill rod was originally chosen to be 690mm long, which would theoretically provide enough length to observe a distinct wave for a cylindrical hammer of approximate length 200mm, ideally allowing it to accommodate measurements for the prototype percussion subassembly used in part two of this experiment. The rod was cappedatthetopwithaprotectiveanvilmadeofcustom455precipitationhardenedstainless steel, cylindrically constrained with a lubricated bronze bushing. The anvil was fixed to the rod using an alloy steel threaded fastener, which was assumed to have a negligible effect on wave transmission in the rod, as shown by Lundberg [34]. The rod was instrumented with a full-bridge strain gauge configuration 30mm from the top of the anvil (referred to as SG1), and a piezo force transducer at the base, illustrated in in Figure 3.1. The full- bridge configuration was chosen as it is capable of measuring axial strain while negating the effects of bending strain and temperature sensitivity. Additionally, the gauge length was chosen to be 3mm, intended to be short enough to reduce averaging of measurements, considering that the expected length of the pulse was over an order of magnitude larger. Due to the high propagation speed of elastic waves in steel, typically around 5000m/s, high-speed amplification and data acquisition was necessary. A calibrated strain gauge amplifier with a bandwidth of 600kHz for 100x gain was chosen, which had a gain and excitation error less than .1%. and a digital oscilloscope capable of sampling faster than 1MS/s was used. Strain was computed from voltage as = −2V/V e K GF ((ν + 1)−V/V e (ν− 1)) (3.1) 34 where V e is the excitation voltage of 10V, V is the measured signal, K G F is the gauge factor of approximately 2, and ν is Poisson’s ratio. Figure 3.1: Illustration of the setup for drop testing. SG1 is the upper strain gauge and PLC is the Piezo Load Cell (or Piezo Force Transducer). 3.1.2 Response of Prototype Percussion Subassembly Parttwoofthe experiment was originallyconductedusing thesamerod asinthe previous section, the only difference being that forcing was produced from the prototype TRIDENT percussion subassembly. This resulted in uncertainty in the true duration and shape of the waveforms as the percussion subassembly produced impact durations longer than expected (estimated as twice the length of the hammer). Thus, the 690mm rod was replaced with a 1590mm rod of identical cross section and material, and stress waveforms were measured using two full-bridge strain gauge configurations at two distinct positions along the rod, 105mm and 386mm, from the top of the anvil. This allowed for later use of 2-point strain gauge algorithms, like those discussed by Lundberg [35, 34] and Carlsson [6], to separate incident and reflected waves if necessary. The resulting setup is shown in Figure 3.2 side by side with the full prototype TRIDENT drill. 35 Figure 3.2: The instrumented percussion testbed (left), and the prototype drill (right). A section view of the percussion mechanism is provided in Figures ?? and ??. In short, a motor drives a helical cam, which lifts the cam-follower against a compression spring until it is released, at which point the spring causes the hammer to accelerate towards the anvil. The spring was configured to have a full compression energy of 2J, resulting in an expected velocity of 2.53 m s at impact given a hammer mass of .626kg. It should be noted that the anvil was required to have nonuniform geometry with a “step down” feature to prevent mushrooming that could cause interference with the lubricated 36 bronze bushing, and likely caused the resulting waves to deviate from theoretical expecta- tions. This feature is present in the prototype drill as-built, and is shown in Figures 3.3 and 3.4. Figure 3.3: Step down feature present on the top of the anvil to prevent interference due to deformation. Figure 3.4: Top view of the step down feature on the anvil, which sits centered in a bronze strike plate in the assembled testbed. 37 3.2 Results Results of the initial drop tests in part one are shown in Figure 3.5, where measurements were made using a strain gauge configuration (SG1) placed 30mm from the top of the anvil, as well as a piezo force transducer located at the base of the rod. A convention of positive force representing compression was used. Figure 3.5: Time traces of a force pulse produced from dropping a steel bar on a long slender rod, measured at near the top of the rod and the base of the rod. Here it can be seen that the vibration damped plate behaves like a rigid boundary under this forcing condition. This is illustrated in part by the fact that the piezo force transducer measures forces approximately double that of the incident wave while maintaining a similar shape. This waveform will be approximated with a cosine pulse and implemented in a DTFM model in Section 4.3 for comparison. The strain wave generated by the TRIDENT percussion subassembly is shown in Figure 3.6, with a convention of positive force representing compression. SG1 was placed at 105mm from the top of the anvil and SG2 was placed 386mm from the top of the anvil. 38 Figure 3.6: Force waveform resulting from the percussion subassembly used in the TRIDENT drill. This waveform was taken when the mechanism life was at approximately 3.6×10 6 cycles. Here it can be seen that the force pulse produced by the prototype drill subassembly is complex, which is expected due to its irregular geometry. It contains multiple peaks of varying duration, with a maximum magnitude of approximately 7kN and an approximate total duration of 250 μs. The energy W contained in the incident pulse can be computed using Eq. (1.16). This results in a total measured energy of .55J. Given a spring energy of 2J, this suggest that only 28% of impact energy is converted into a productive percussive wave, while the rest may be converted to vibration within the mechanism. A potential explanation is considered in Chapter 7. This is considerably different from any of the force pulses derived in Section 2.4. Thus, incorporation into a DTFM model will require a technique that can handle arbitrary forcing, which is described in Chapter 6. A possible explanation for the distinct peaks may result from different components of the hammer assembly containing gaps between them while the hammer is accelerating towards the anvil. During impact, these multiple components may contact each other sequentially. This suggests that if these specific components are identified and redesigned, a more coherent pulse with a much higher magnitude could be generated, which would theoretically be more effective at breaking hard rocks. Additionally, it can be seen that the lower boundary appears to behave differently under this forcing compared to 39 the drop testing, shown in part by the fact that the reflected wave initially contains tensile force, which is only possible if the boundary exhibits behavior similar to a free or spring- like condition. This could be due to differences in strain rate between the drop test and the hammer assembly test, which could affect the response of the rubber dampers. This is investigated in Chapter 6. The noise present in these signals (particularly in SG2) was due primarily to electromagnetic interference from the motor, which could not be reduced any further without turning off the motor. 40 Chapter 4 DTFM Transient Response for Uniform Bar 4.1 System Description A one dimensional model of a percussive drill is provided in Figure 4.1 which is composed of a slender rod of length L, elastic modulus E, diameter D, and density ρ. In this section, this rod is forced on the left by a hammer that produces a prescribed contact force F (t) that takes the form of the pulses established in Section 2.4. A transient response was be computed using a varying number of modes for cases where the rod is supported on the right by a spring of stiffness K, as well as a free or fixed boundary. Figure 4.1: A basic model of the percussive drilling of rock where F(t) represents the prescribed force pulse due to a hammer impact. The boundary on the right is shown as a spring, and will also be replaced with a free and fixed boundary. 4.2 Rectangular Forcing While rectangular wave forcing is a convenient way to gain intuition on the response of a wave to various boundary conditions in Lundberg’s method, it is nonphysical, and generation of a true rectangular wave using DTFM would require an infinite number of modes. Thus, while the modeling in this section represents the simplest case for Lundberg’s method, it is one of the more difficult cases to construct using DTFM. 41 4.2.1 Spring Boundary The DTFM model was compared to Lundberg’s method while varying the number of modes. For this comparison, a spring boundary with stiffness 2500kN/mm was chosen as it produces a more complex reflection compared to a free or fixed end. F (t) was chosen to be a rectangular pulse of magnitude 12kN with an 80μs duration. Rod characteristics were set to values similar to those of the experiment with a .01905m diameter, 210GPa elastic modulus, and a density of 8050 kg m 3 . Figure 4.2 shows the transient response for a varying number of modes, which was compared to Lundberg’s method. Figure 4.2: Transient response of a uniform bar subject to a rectangular pulse on the left with a spring boundary on the right. 42 The interaction of the pulse with the spring boundary shows good agreement with Lund- berg’s method despite its finite frequency spectrum. While it is expected that the sharp corners present in a theoretical rectangular pulse cannot be exactly constructed using DTFM since there will always be higher frequencies that must be neglected, 3D wave theory pre- dicts that these high frequencies lag behind the wavefront, inadvertently causing the DTFM rectangular pulse to share certain characteristics with approximate 3D solutions, including a finite rise time and oscillatory behavior. A comparison is shown in Figures 4.3 and 4.4. Figure 4.3: The wavefront resulting from an approximate solution for the impact of two semi- infinite elastic rods with radial inertia and dispersion taken into account, developed by Skalak [49], taken from Graff [17]. 43 Figure 4.4: A detailed view of the wavefront produced from rectangular forcing using DTFM with 50 modes. It can also be seen that DTFM results in apparent small disturbances ahead of the wavefront, some of which are negative, implying tension. This is nonphysical and is also a result of an non-infinite number of modes, rather than dispersion. Additionally for DTFM the frequency of these oscillations depends on the highest frequency used to construct a transient response, which could be optimized to better fit the wavefront of the 3D solution if desired. 4.2.2 Fixed Boundary The transient response due to rectangular forcing was also computed for the case where the spring was replaced with a fixed boundary with the same configuration as described in the previous section. 44 Figure 4.5: Transient response of a slender uniform rod subject to rectangular forcing on the left with a fixed boundary on the right. The transient response using DTFM shows good agreement with Lundberg’s method when a fixed boundary is applied to the right end. As expected, a higher number of modes used to construct the response gives a result closer to Lundberg’s method. 45 4.3 Single-cycle Cosine Forcing The transient response of the bar described in Figure 4.1 was subjected to single-cycle cosine forcing, described in Section 2.4.3, with a duration of 80μs and a peak magnitude of 12kN. Figure 4.6: Transient response of a slender uniform rod subject to cosine pulse forcing on the left with a fixed boundary on the right. 46 Figure 4.7: Transient response of a slender uniform rod subject to cosine pulse forcing on the left with a free boundary on the right. Figures 4.6 and 4.7 show that the response constructed with 20 modes is nearly identical to the response constructed with 100 modes. Thus, this forcing does not require a very high number of modes in this case. However, this is dependent on the duration of the pulse, as well as the geometry of the bar. It is possible that more complicated bars with complex boundary conditions could result in complicated reflected waveforms that may require a higher number of modes. 47 This forcing shape was also used to model the drop testing discussed in Section 3 using the known duration and peak force from the strain gauge data, and setting the rod length to 690mm. A rigid lower boundary was used, and the transient response was computed specifically at the locations where the upper strain gauge and piezo force transducer were located. Results are shown in Figure 4.8. Figure 4.8: A semi-theoretical single-cycle cosine approximation of the drop test described in Section 3. The single-cycle cosine pulse shows good agreement with the drop testing, although it does not account for the fact that the experimental forcing is slightly asymmetrical. 4.4 Exponential Decay Forcing The exponential decay pulse discussed in Section 2.4.4 was applied to the uniform rod with a fixed and free boundary, constructed with a varying number of modes. The transient response with a free and fixed boundary is shown in Figures 4.9 and 4.10 respectively. 48 Figure 4.9: Transient response of a uniform rod with exponential pulse forcing on the left and a free boundary on the right. 49 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x [m] -6 -4 -2 0 F [N] 10 4 70 s 100 Mode 50 Mode 20 Mode 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x [m] -6 -4 -2 0 F [N] 10 4 140 s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x [m] -6 -4 -2 0 F [N] 10 4 210 s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x [m] -6 -4 -2 0 F [N] 10 4 280 s Figure 4.10: Transient response of a uniform rod with exponential pulse forcing on the left and a fixed boundary on the right. As expected, the free boundary resulted in an opposite-sign reflection at the free bound- ary, which destructively interferes with itself. Conversely, the fixed boundary resulted in an opposite-sign reflection at the free boundary, which destructively interferes with itself. In both cases, a relatively high number of modes is required to accurately represent the expo- nential decay pulse. Similar to the rectangular pulse, this is due to the instantaneous rise 50 time which requires an infinite number of modes to be perfectly captured. A detailed view of the wavefront is given in Figure 4.11. Figure 4.11: A detailed view of the exponential pulse wavefront constructed with a varying number of modes. 4.5 Adjusted Exponential Decay Forcing TheadjustedexponentialdecayforcingdiscussedinSection2.4.5providesamorephysical alternative to the forcing described in the previous section with additional time dependence, discussed in Section 2.4.5. The transient response of this forcing applied to a uniform bar with fixed and free ends, constructed with a varying number of modes, is show in Figures 4.12, 4.13, and 4.14. 51 Figure4.12: Simulatedadjustedexponentialpulseforcingonauniformrodwithafixedboundary. 52 Figure4.13: Simulated adjusted exponential pulse forcing on a uniform rod with a free boundary. 53 Figure 4.14: A detailed view of the adjusted exponential pulse wavefront constructed with a varying number of modes. The additional time dependence of this forcing results in a piecewise smooth function with a finite rise time, thus requiring far fewer modes to accurately construct compared to the unadjusted exponential decay pulse in the previous section. However, the non-smooth contact condition was still not well captured in the response constructed using 20 modes. 54 Chapter 5 DTFM Transient Response for Non-Uniform Bar 5.1 System Description In practical cases, drill rods tend to be more complex than a uniform elastic bar. In this section a model of a more complicated stepped bar system is developed, shown in Figure 5.1. Figure5.1: Stepped bar system subject to longitudinal forcing on the left, with a spring boundary on the right. For this system, E 1 = E 2 = 210GPa and ρ 1 = ρ 2 = 8050kg/m 3 , which are common values for the elastic modulus and density of steel. K was set to 2500kN/mm in order to elicit a relatively complicated response. A diameter change from 1.125in to .75in in the rod results in A 1 =.000641m 2 and A 2 =.000285m 2 . 5.2 Rectangular Forcing As previously stated, the propagation of rectangular forcing is intuitive and predictable. Despite this, DTFM is not well suited to handle rectangular forcing because it requires an infinite number of modes to perfectly represent. That being said, it is the ideal case to illustrate the limitations of this method, and in this section further complexity is introduced with a rod that contains a sudden change in diameter as well as a spring boundary, depicted in 5.1. The rectangular pulse in this section has duration 80μs with a magnitude of 12kN. 55 5.2.1 Spring Boundary Transient responses of the rectangular pulse acting on the stepped bar with a spring boundary were produced using DTFM, constructed with 50, 100, and 200 modes. The response was compared to Lundberg’s method as well, shown in Figures 5.2 and 5.3. Figure 5.2: Transient response of a stepped bar system to rectangular pulse forcing on the left, supported by a spring on the right. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. 56 Figure 5.3: A detailed view of a complex segment of the transient response. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. In general the DTFM response shows good agreement with Lundberg’s method in this more complicated case. However, additional reflections as well as the spring boundary tend to cause high frequency disturbances to arise, which was not well modeled with only 20 modes. 5.3 Single-cycle Cosine Forcing The nonuniform model shown in Figure 5.1 was subjected to single-cycle cosine forcing at the left end with fixed and free boundaries. The transient response was constructed with a varying number of modes and is shown in Figures 5.4, 5.5. 57 Figure 5.4: Transient response of a single-cycle cosine pulse acting on a nonuniform bar with a fixed boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. 58 Figure 5.5: Transient response of a single-cycle cosine pulse acting on a nonuniform bar with a free boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. In both cases, a partial reflection can be observed at the interface between the dissimilar sections. In this more complicated case of the nonuniform bar, a relatively low number of modes can be used to construct the transient response. 59 5.4 Exponential Decay Forcing The nonuniform model shown in Figure 5.1 was subjected to exponential decay forcing at the left end with fixed and free boundaries. The transient response was constructed with a varying number of modes and is shown in Figures 5.6, 5.7. Figure 5.6: Transient response of an exponential decay pulse acting on a nonuniform bar with a fixed boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. 60 Figure 5.7: Transient response of an exponential decay pulse acting on a nonuniform bar with a free boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. As expected, this behavior is similar to that of the uniform case with the complication of reflections occuring at the dissimilar interface. In addition to the fixed and free boundaries, this forcing was applied to the same stepped rod with a spring boundary. 61 Figure 5.8: Transient response of an exponential decay pulse acting on a nonuniform bar with a spring boundary (k = 2500kN/mm). A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. Here it can be seen that the spring boundary results in reflections that potentially exhibit higher frequency behavior than the inital pulse itself, requiring a higher number of modes to be well represented compared to the free or fixed case. 62 5.5 Adjusted Exponential Decay Forcing The nonuniform model shown in Figure 5.1 was subjected to adjusted exponential decay forcingattheleftendwithfixedandfreeboundaries. Thetransientresponsewasconstructed with a varying number of modes and is shown in Figures 5.9, 5.10, and 5.11. Figure 5.9: Transient response of an adjusted exponential decay pulse acting on a nonuniform bar with a fixed boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. 63 As expected, the incident wave was partially transmitted and partially reflected at the interface where the diameter suddenly decreases, which is particularly visible at 260μs. This region is shown in a detailed view in Figure 5.10. Figure 5.10: A detailed view of an adjusted exponential decay pulse interacting with a sudden change in cross sectional area in a rod. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. The transient response of this same distributed system with a free boundary on the right is shown in Figure 5.11. 64 Figure 5.11: Transient response of an adjusted exponential decay pulse acting on a nonuniform bar with a free boundary. A sudden decrease in cross sectional area is denoted by the dotted line at 1.2m. As expected, at the interface between the dissimilar rods the wave behaves identically to that of the fixed case. The interaction with the free boundary on the right results in an inverted reflected wave which destructively interferes with itself, which was expected as well. 65 5.6 Comparison of Methods The capability for DTFM to describe a variety of impact responses in slender rods has been shown, including systems with complications such as stepped bars and spring bound- aries. Lundberg’s method is superior at modeling a theoretical rectangular pulse response, and is easy to use for a variety of cases with practical applications. It also provides the capability to model the boundary as a bilinear spring to represent rock. In contrast, DTFM cannot model a rectangular pulse well except with a very high number of modes, and boundary elements are limited to free, fixed, and standard springs. However, the resulting transient response wavefront using DTFM shares some characteristics with 3D solutions in that it yields oscillatory behavior resulting from neglecting very high frequencies, although 3D solu- tions have a similar effect due to dispersion. Additionally, pulses that are piecewise smooth, which are often more realistic, can be constructed to a high degree of accuracy with a rela- tively small number of modes. BoundaryforcinginLundberg’smethodisdependentonhammerdesignwhichworkswell forsimplerhammers. However, asshownbytheexperimentinthiswork, theresultingforcing from more complicated hammers cannot be easily explained with this method. With DTFM one can experimentally measure a pulse waveform and construct a pulse approximation using the forcing derived in Section 2.4. Additionally, because the form of Eq. (2.27) resembles a convolution integral, experimental forcing may directly be incorporated into analytical DTFM models. Chapter 6 discusses this advantagesand limitations of this technique and it is applied to a physical percussive system. An additional advantage of DTFM is that it can systematically incorporate masses, springs, or additional dynamic systems to simulate more complicated systems, without need- ing to entirely redesign the modeling framework, which will be implemented in future work. Additionally, DTFM can be used to compute the force distribution at any given time with no increase in computational complexity since it is an analytical solution. By contrast, 66 Lundberg’s iterative method requires changing the spatial resolution in order to increase the temporal resolution. Lastly, as part of computing the transient response, DTFM also provides a method of computing exact natural frequencies of these complex, nonuniform systems, which can otherwise only be done using finite element methods. This is often a necessary analysis for validating space flight hardware. In short DTFM is shown to be a good candidate for modeling percussive mechanisms in that it is easy to make changes to model components, it inherently computes exact natural frequencies of complex systems, it is capable of handling pulse forcing, and it shows good agreement with Lundberg’s gold-standard method. 67 Chapter 6 A Technique to Incorporate Experimental Forcing 6.1 Motivation Transient pulse shapes are difficult to predict for hammers with complex geometry, such as that of TRIDENT shown in Figure 6.1. This can be seen experimentally as measurements of the resulting pulse shape from the percussion testbed, shown in Figure 6.2, do not resem- ble any of the primary pulse shapes outlined in previous chapters. Thus, incorporation of arbitrary or experimentally-captured forcing can be a valuable tool in modeling. Figure 6.1: Percussive hammer used in TRIDENT. 68 Figure 6.2: Complex pulse shape produced by TRIDENT cannot be easily modeled using simple forcing functions. Incorporation of experimental forcing is necessary in determining changes in percussion behavior resulting from design iterations, and allows for characterization of unknown bound- ary conditions. The expression for boundary forcing using DTFM resembles a convolution integral which suggests that numerical convolution may be one efficient way to incorporate prescribed arbi- trary forcing. In this chapter, a sampled and analytical sine wave are used as forcing in a DTFM model and their responses are compared to evaluate the performance of a numerical convolution algorithm. This technique is also applied to experimental data obtained from a hammer impact of the TRIDENT drill, which is used to characterize physical boundary conditions. 69 6.2 Demonstration of Numerical Convolution Tech- nique Numerical convolution was implemented in MATLAB as an alternative to using Eq. (2.27). Consider a vector F c containing a sequence of sampled force values, with n samples collected at a period T c . Discrete convolution can be written Ψ k = sinω k v c T c ∗ F c , v c = [1, 2,···n] (6.1) It follows that at a given time t =mT c I b,k [mT c ] = Ψ k [m]T c . (6.2) This technique was demonstrated using a known sinusoidal boundary forcing signal,F c = F 0 cos Ωt, where Ω = 25000πs −1 , F 0 = 6kN. This signal was sampled at 1, 5, and 10μs, and was used as boundary forcing on a uniform bar of equal properties to that in Figure 4.1, with a length of 1m. The response was computed using DTFM with 20, 40, and 100 modes. In this system, these modes corresponds to maximum natural frequencies of 50, 100, and 250kHz, or periods of 20, 10, 4 μs. Force distribution in the bar at 200μs for each case was compared to the analytical response using Eq. (2.27). Results are shown in Figure 6.3. 70 Figure 6.3: Bar response to sinusoidal forcing which was applied using analytical convolution and discrete convolution with sample periods of 1, 5, and 10μs. Response was constructed using 20, 40, and 100 modes which corresponds to vibrational periods of 20, 10, and 4μs. This exercise shows that for sufficiently small force sampling periods, the discrete convo- lution technique shows good agreement with the analytically computed response. However 71 it is clear that the allowable number of modes to construct a response depends on the sample rate of the force vector. Using modes that have associated vibrational periods smaller than the force sample rate results in aliasing, and thus an erroneous response. This can be clearly seen in the case of the 100 mode construction, where the smallest natural period of vibration is 4μs, which produces an erroneous response for force sampled with 5 and 10μs periods. For this reason, it is necessary that the vibrational period associated with the highest mode used to construct a response is greater than the force sample period. Thus, sample rates for collecting experimental force data should be carefully chosen such that they are higher than the highest natural frequency desired for constructing a response. 6.3 Performance with Experimental Forcing The numerical convolution technique was used to incorporate the incident pulse produced bytheTRIDENTpercussionmechanismintoananalyticalDTFMmodel. Thissystemmodel is shown in Figure 6.4 and is based off of the test fixture discussed in Section 3.1.2. Figure 6.4: Distributed system model of the dummy drill rod used in the experiment in Section 3.1.2 with an unknown boundary condition. In this system, E =193.14GPa, A =.000285m 2 , ρ L = 2.26kg/m. Because the physical test fixture is supported by a complex boundary condition composed of a metal plate and dampers, the transient responses with fixed, free, and spring boundaries (100kN/mm) were all modeled and compared. F (t) was chosen as the incident portion of the measured pulse from TRIDENT. This pulse is shown in Figure 6.5, where it is padded with zeros to allow for the transient response to be computed up to 1200μs despite the pulse only being roughly 280 μs long. 72 Figure 6.5: Experimentally captured incident pulse, padded with zeros to allow for computation of the transient response at times longer than the pulse duration. The transient response of this sytem was computed using 40 modes over 1200μs at a sim- ulated location where the original force measurement was made (105mm from the impacted end seen in Figure 6.4). Results for all boundary cases are compared with the experimental response, and are shown in Figure 6.6. Figure 6.6: Bar response to experimentally measured forcing. Effects of fixed, free, and spring boundaries are compared with the experimental measurement of the reflected wave. 73 The transient response at 500 time frames using 40 modes had a computation time of 83 seconds. It is clear that a fixed boundary model does not describe the physical system well, as the initial experimental reflection begins with tension, which should not exist if the incident pulse interacts with a rigid boundary. The 100kN/mm spring boundary shows better agreement with experimental results than the free boundary. This is expected due to the presence of supportive elements at this boundary in the experiment. The transient response of the spring boundary case is isolated in Figure 6.7. Figure 6.7: Bar response to experimental forcing with a spring boundary compared to the experi- mentally measured reflection. Despite a delay, the spring boundary captures the dynamic behavior of the base of the testbed. The spring boundary captures the behavior of the percussion testbed. This result demon- strates that incorporating a measured incident pulse via discrete convolution allows for char- acterization of unknown boundary conditions using DTFM, even in cases with complex inci- dent pulse shapes. Deviation from the experimental response is noted after around 600μs which appears to be a delay. This may be partially due to error in computing the theoretical wave speed in the rod, or due to the nonuniform anvil cap. This could additionally be due to internal damping, which was not accounted for in this model. 74 Chapter 7 Dynamic Analysis of the Helical Compression Spring in the TRIDENT Percussion Mechanism 7.1 Introduction Recent life testing of the TRIDENT percussion mechanism showed that the compression spring repeatedly failed before its desired life. Quasi-static fatigue analysis does not explain the unexpectedly short service life, less than 200,000 cycles, which is only 20% of the required mission life (1M cycles), and less than 2% the theoretically expected life (>10M cycles). Employing oil-tempered chrome-silicon spring steel (OTCS) increased the service life beyond desired limits in an ambient Earth environment. However, because this solution uses a non- stainless variety of steel, it is not well-suited for use in a lunar environment without the addition of exotic treatments and coatings. In addition, the exact cause of failure is still poorly understood, which poses a problem for potential future drill designs. Until this failure mode can be well explained, destructive life testing remains the only way to address this risk which is often time consuming and expensive. Based on experimentation described in this chapter,andproximitytoahigh-frequencymetal-on-metalimpactevent,wehypothesizethat the cause of failure in the spring is related to dynamic effects rather than simply problems associated with quasi-static fatigue or manufacturing techniques. A dynamic response of the spring could be elicited by the percussive wave generated during hammering in TRIDENT. A significant amount of work has been done in the prob- lem of wave propagation in percussive drilling. This is described in detail in Chapter 1. However due to the geometric complexity of drilling machines such as TRIDENT, standard longitudinal wave propagation techniques provide only a starting point for understanding 75 the shock induced vibration of complex nearby components, such as the compression spring in question. Helical compression springs are ubiquitous in modern machinery, and their dynamic behavior has been investigated in a wide variety of works. The problem of wave propagation in helical springs was first analyzed over a century ago by Love[33]. However, the result- ing system of twelve coupled governing equations is not tractible. In contrast, the simplest distributed system model one could conceive of might consist of only extensional waves, by modeling a spring as an equivalent soft elastic column. Among other things, this type of model neglects rotational expansion of coils, which can have a substantial effect on spring functionality depending on the application. For example, Stokes[50] analytically and experi- mentally investigated such radial expansion in axially impacted springs in recoil mechanisms with narrow housings. Accurate three-dimensional models of helical wires can be constructed by assuming the spring coil behaves as a curved Timoshenko[51] beam [53]. Such models are tractable, but require a system of twelve governing equations to relate forces and moments with displacements and rotations, and are thus mathematically complex. Wittrick [53] pro- vides a convenient reduced model that uses two wave equations for coupled rotational and extensional waves. This method has been the basis for practical applications, such as formu- lations for the frequency-dependent dynamic stiffening of springs, which has been used as an improvement over quasi-static spring models in an automotive suspension systems [30]. Wit- trick’s methods have been used as a basis to analytically and experimentally study coupling between torsional and flexural strain in axially excited helical springs, and have been shown to have utility in computing natural frequencies for springs of various geometries [29, 40]. Depending on the investigation, nonlinear finite element analysis (FEA) with sufficient capability to handle high frequency forcing is superior to the above analytical techniques. FEA can better handle nonuniform structures, and does not assume infinitely long structures via Saint-Venant’s principle. It also provides simplicity in prescribing transient forcing, as will be the forcing case in this work. 76 In this chapter, it was found that traditional quasi-static fatigue analysis cannot explain the results of limited life testing of the percussion mechanism. This was used as motivation for dynamic analyses of varying complexity. These techniques are presented and discussed in an attempt to better understand the underlying spring failure mechanism. Known impact characteristics were applied to a benchmark nonlinear finite element model of the exact compressionspringinTRIDENT,aswellasaone-dimensionalDistributedTransferFunction Method (DTFM) model that included only extensional waves. Both of these techniques were comparedtoarudimentarydiscrete-elementmodelcontainingonlyalumpedmassandspring element. In all cases the maximum shear stress in select coil regions was compared over time in an attempt to better understand the behavior of the spring with respect to failure. 7.2 TRIDENT Percussion System As mentioned in Chapter 1, TRIDENT is a rotary-percussive drill that has been under development by Honeybee Robotics over the past two decades, a recent prototype is shown in Figure 7.1. Its intended purpose is to drill as deep as 1 meter on the Moon, and transport cuttings to the surface from various depths for observation by scientific instruments, such as NASAs Near Infrared Volatile Spectrometer Subsystem (NIRVSS) [45]. TRIDENT relies on a cam-follower spring mechanism to produce percussion. A rotating helical cam rapidly compresses a hammer against a helical spring to a specified energy, then releases it towards an anvil atop a drill string to generate an impact. This impact produces a stress wave which travels down the length of the drill rod and interacts with the rock beneath. A rendering of this mechanism is given in Figure 7.2, and the principle of operation is diagramatically represented in Figure 7.3. 77 Figure 7.1: A late-stage prototype TRIDENT unit. 78 Figure 7.2: Rendering of the percussion mechanism in TRIDENT. The gears and cam (blue) are driven by a motor to rotate, causing the hammer (purple) to raise up and compress the spring (red). After a full rotation the hammer is released towards a drill bit. Figure 7.3: A simplified depiction of the general operation of the percussion mechanism in TRI- DENT. Typical hammering frequency is approximately 16Hz. The time-history force resulting from this impact, as well as its frequency content, are shown in Figures 7.4 and 7.5. 79 Figure 7.4: Time-history force produced by the TRIDENT percussion mechanism. Figure 7.5: Frequency content of the impact produced by TRIDENT. The pulse produced by TRIDENT produces forces as high as 7kN, and suggests a contact time between the anvil and hammer of approximately 250μs. Additionally, this pulse only accounts for .55J despite a percussion spring energy of 2J. This implies that 72% of the impact energy is not accounted for. Additionally, the frequency spectrum shows significant content above 10kHz, which is common in metal-on-metal impacts. 80 7.3 Static Models Typical quasi-static spring analysis for round-wire helical springs is described in basic mechanical design textbooks [43, 5], specialized texts [52], spring manufacturer handbooks [1], and publications of professional societies[25]. This type of analysis is commonly used in an industry setting due to its approachable mathematics and compatibility with commercial spreadsheet editing software. In quasi-static mdoels it is assumed that the entire spring bears the compressive load, equally distributed amongst its active coils. This quasi-static approx- imation allows for simple calculation of parameters of interest [52, 25], such as maximum torsional shear stress within coils and spring rate. Exact solutions exist to compute shear stress in an elastic coil [15], but tend to be mathematically complex. A commonly used approximation to relate local axial force and maximum shear stress is given by Wahl [52] and is given in Eq. (7.1). τ max = 16Pr πd 3 4c− 1 4c− 4 + .615 c , c = D d (7.1) P is the axial force, r is the mean coil radius, d is the wire diameter, D is the mean coil diameter, andc is the spring index. This also assumes a small helix angle and straining within elastic limits of the given material. This approximation is within 2% of the exact solution by Goehner[cite] for spring indices greater than 3, which is satisfied in this analysis (c = 5.55). Additionally, Castigliano’s theorem can be used to obtain an approximate spring rate k, assuming strain energy primarily in shear and torsion, and assuming load is equally distributed among all coils. This formulation is given in Eq. (7.2). k = Gd 4 8D 3 n (7.2) G is the material’s shear modulus, D is the coil diameter, d is the wire diameter, and n is the number of active coils. 81 Combined with known material properties and manufacturing techniques, these parame- ters also allow a designer to estimate the fatigue life and natural frequency of a given spring in certain environments. While fatigue failure is highly variable, the Goodman fatigue crite- rion [52, 25] is a simple and conservative method that is commonly used to predict fatigue failure for a cyclically loaded helical compression spring for a given number of cycles. Given a known torsional yield strengthτ y and cyclic endurance limitτ e , as well as expected midline torsional stressτ m and torsional stress amplitude about that midlineτ a for our configuration, the Goodman line can be written as τ a S se + τ m S sy = 1. (7.3) In the experiment presented in this work, a 302 series stainless steel spring and 17-7 series stainless steel spring were both used in the percussion mechanism in TRIDENT. Relevant properties for these particular configurations are described in Table 7.1. Table 7.1: Relevant material properties of 17-7 PH and 302 series stainless steels, as well as the midline and amplitude shear stresses estimated by quasi-static theory. Spring Type S se [ksi] S sy [ksi] τ a [ksi] τ m [ksi] 17-7 PH Stainless 54.6 115.3 28.9 56.2 302 Stainless 27.2 79.8 25.2 52.7 Experimental results [52, 59] exist showing that 304 series springs with equal wire diam- eter to the spring in this study survived 10 7 cycles when stressed between 20-65ksi in shear. These values were used to derive S se using Eq. (7.3). Alternatively, for the 17-7 PH spring the zero-max basis was used to define the Goodman line, assuming S se for 10 7 cycles is 30% of the tensile strength of 17-7 PH (181.9ksi). The resulting Goodman diagram is shown in Figure 7.6. 82 Figure 7.6: Goodman diagram for two springs of two materials in the TRIDENT percussion configuration. The line for 302 series spring was generated using experimental data and known materialproperties. Thelineforthe17-7PHspringwasgeneratedusingazero-maxbasis, assuming 10 7 cycle endurance strength in shear (S se ) equal to 30% of the material tensile strength. Figure 7.6 suggests that the 17-7 PH spring in the TRIDENT percussion mechanism should be capable of surviving 10 7 cycles, while the 302 series spring should fail at some point before this. Thus it does not explain the early failure of the 17-7 PH spring at less than 2x10 5 cycles. This fatigue model is based on the assumption that loading of the spring is purely sinu- soidal at a frequency far below the first natural frequency of the spring. In reality, a nearby high magnitude metal-on-metal impact occurs after every load cycle, which may result in higher stresses that cannot be described by Eq. (7.1-7.3). 83 7.4 Dynamic Models 7.4.1 Discrete-element Dynamic Model While convenient, the quasi-static fatigue theory only applies under the assumption that the spring is always in dynamic equilibrium, i.e. the frequency content of loading is far below the first natural frequency of the spring. However, in TRIDENT, a metal-on-metal impact occurs after very cycle, and roughly 72% of the impact energy is not accounted for in the stress wave propagating down the drill rod. It is likely that a significant portion of this unaccounted for energy is returned to the spring in the form of an impulse containing high frequencies, which cannot be ignored when considering potential failure modes. In essence, a single cycle of the spring may begin with the four events in Figure 7.3 which are described well with quasi-static theory, but in this work it is assumed that an impact will produce an additional fifth state that may result in non-negligible stress in the spring, shown in Figure 7.7. Figure 7.7: Illustration of the general operation of the percussion mechanism in TRIDENT with consideration into dynamic spring behavior post impact (Step 5). 84 The simplest dynamic model of post-impact dynamics can be constructed using discrete elements. This can be described using a spring and lumped mass element with a fixed base, shown in Figure 7.8. Figure 7.8: Discrete-element model of an impact occurring at the face of the hammer, composed of a massless spring and a lumped element mass equal to the mass of the hammer assembly. k is the spring constant computed using Eq. (7.2) as 46900 N/m, and m is the mass of the physical hammer assembly (.626kg). Assuming a near-instantaneous impact with energies .5J and 1.5J, initial velocity of the mass can be computed as 1.26m/s and 2.19 m/s respectively. Initial position can be computed using the known .003175m (.125") preload compression. Thus the response of this system can be described as the free response of an undamped oscillator, or x(t) =x 0 cosωt + v 0 ω sinωt, ω = s k m (7.4) This model assumes the system has a single natural frequency, and a massless spring, which is supported by the fact that the spring contains approximately 3% of the mass of the hammer assembly. 7.4.2 One-Dimensional Dynamic Models While the discrete-element dynamic model takes into account the inertia of the mass, an advantage over quasi-static techniques, it does not include consideration for the inertia of the coils. This can be a significant actor in the generation of stresses in springs experiencing high-frequency, high-magnitude loads[26, 50, 52, 25], such as in the percussion mechanism 85 in TRIDENT. It has evem been shown [26] that in some cases with relatively low-frequency excitation, springs can deviate from simple stiffness elements, and must be treated as dis- tributed systems. Industry-targeted technical guides[25] give basic formulations based on 1D wave theory for the additional stress as a result of high loading velocity, as occurs during an impact. This effect is roughly summarized by a rule of thumb stating that life of springs as determined by common cyclic loading tests may be reduced to as low as 2% under impact loading of the same range of deflection and frequency. Specifically, it is suggested that for an incident wave resulting from impact, one can expect an approximate stress increase of ΔS Δv = 1 10 3 q 2γG, (7.5) beforeanyreflection, whereS istheshearstress,v istheloadingvelocity,γ isthematerial density, and G is the shear modulus. In typical steels this results in an approximate peak shear stress increase of 5.1 ksi for each m/s increase in loading velocity[25]. However, it is also stated that the maximum stress in the spring will be the result of boundary reflections constructively interfering, which additionally depends on the dimensions of the spring and requires the construction of an appropriate model. Wavepropagationandreflectioninthespringinthisworkwasmodeledasalongitudinally elastic bar with approximately equivalent properties as the spring, fixed on one end with a lumped mass at one end to be impacted, shown in Figure 7.9. Figure 7.9: Illustration of the general operation of the percussion mechanism in TRIDENT with consideration into dynamic spring behavior post impact (Step 5). 86 In this case, E is the equivalent elastic modulus, A is the bar area, l is the length of the bar,ρ L is the linear density of the bar, andm is the mass of the end element. l was estimated as .032m, which is the preloaded length of the 4.6 active coils. m was left as .626kg, which is equivalent to the mass of the hammer. For simplicity, the impact forceF (t) is chosen to be in the form of a half-sine pulse, such as F (t) =F 0 sin( 2π T t) H(t)−H(t− T 2 ) . (7.6) Figure 7.10: Graphical depiction of a half-sine pulse from 0 to T/2. The shaded area under the curve is equal to the delivered impulse to the hammer. A duration ofT/2 = 250μs was chosen as it was approximately equal to the the measured pulse duration from previous impact testing. The magnitude F 0 of the half-sine pulse was chosen by determining the required impulse to deliver a specified kinetic energy to the hammer component. Rebound energies of .5J and 1.5J were chosen for this analysis. The .5J rebound represents the more benign case, where an energy equal to that measured in TRIDENT percussion is returned to the spring. The 1.5J rebound represents the most severe 87 possible case, which implies that all of the unaccounted for impact energy is returned to the spring.The impulse delivered by a half-sine force pulse is shown graphically in Figure 7.10 and can be computed as P = Z T/2 0 F 0 sin( 2π T t)dx = F 0 T π (7.7) Using the classical impulse momentum theorem, .5J and 1.5J kinetic energy in the ham- mer immediately after impact requires an impulse of .79Ns and 1.38Ns respectively. Given that T/2 =250μs, Eq. (7.7) gives required forcing magnitudes of 4.9kN and 8.8kN respec- tively. Additionally, a 146.8N (33lbs) preload was applied to the spring model to precompress before forcing occured, which is equal to the spring preload as assembled. 7.4.3 Distributed Transfer Function Method The transient response of the previously described model was computed analytically using the Distributed Transfer Function Method (DTFM). DTFM is a streamlined technique refined by Yang and Tan [56, 54, 55] to efficiently generate analytical models of distributed elastic systems, with optionality for combinations of nonuniform structures and discrete elements such as masses and springs. In the case of the system shown in Figure 7.9, the governing equation of the bar can be written as ρ L (x) ∂ 2 u(x,t) ∂t 2 − ∂ ∂x " EA ∂u(x,t) ∂x # =f(x,t) (7.8) where u(x,t) is the distributed longitudinal displacement, and f(x,t) is distributed forcing. We can also describe specific boundary conditions for this case as u(0,t) = 0, m R ∂ 2 u(L,t) ∂t 2 +EA ∂u(L,t) ∂x =γ R (t) (7.9) 88 wherem R is the mass on the right side, andγ R (t) is the chosen forcing at this position. The left and right boundaries can then be described by matrices such as M b =    0 1 0 0    N b =    0 0 m R s 2 EA    (7.10) respectively. Using the methods described by Yang [56, 54, 55], the transient response can be described by    u(x,t) ∂u(x,t) ∂x    = 2Σ ∞ k=1 U(x,jω k )Q k (jω k )I k (t). (7.11) Natural frequencies ω k can be computed as the solution to det(M b +N b U(L,jω k )) = 0 (7.12) U(x,s) =    cosh(βx) 1 β sinh(βx) βsinh(βx)) cosh(βx)   , β = s q EA/ρ . (7.13) Q k is related to the transfer function residues and can be computed as Q k = adj(Z) j d ds |Z(s)| s=jω k , (7.14) where Z is the boundary impedance matrix, described by Z =M b +N b U(L,s). (7.15) The forcing, I k (t), is then computed as I k (t) = Z t 0 sin(ω k (t−τ))    0 F (τ)   dτ. (7.16) 89 where F(t) was described in Eq. (7.6). Using Eq. (7.1), this technique was used to analytically compute the distribution of shear stress in the spring over 1200μs as a result of a .5J and 1.5J impact at the face of the hammer. Values of parameters EA and ρ L were assigned in two ways. The first method was to equate the wave speed of the soft column using known material density and stiffness in the physical spring. Specifically, the wave speed in the DTFM governing equation had stiffness parameter EA chosen to match a common analytical approximation [53], shown in Eq. (7.17). l 2 k m s = EA ρ L =c 2 (7.17) In this case, l is the length of the spring, k is the bulk spring constant, and m s is the mass of the spring. The other method was to iterate the stiffness parameter EA to match the first natural frequency with that of an analogous discrete spring-mass system. In this case, the target natural frequency was computed using Eq. (7.4), where m is the same as previously stated, and k was computed from Eq. (7.2). This resulted in a first naturalf requency of 274rad/s, requiring that EA = 1530Pa-m 2 . Using this technique,l was chosen as the the length of the active portion of the spring, which was computed as the full spring length minus two coil diameters, or .032m. The transient response of these models was computed in MATLAB using 50 modes, at 32 locations (.005m increment), at 240 times (.005ms increment). Typical run times were on the order of 93 seconds on a 2014 MacBook Pro with a 2.6 GHz dual-core i5 processor. 90 7.4.4 Multi-Dimensional Models of Spring Behavior 1D and discrete-element models can describe the dynamic behavior of helical springs well in certain cases, especially in systems with lower-frequency excitation and interfac- ing machine elements [29]. However, due to complex geometry of helical springs, multi- dimensional dynamic formulations are necessary in understanding the utility or limitations of simpler techniques, which may not account for rotational inertia and coupling effects[40]. Wittrick [53] has provided a middle-ground approach to this problem by proposing a multi- dimensional model assuming the propagation of extensional and rotational waves described by respective coupled wave equations. This is shown in Figure 7.11. Figure 7.11: Governing equations and diagramatic representation of Wittrick’s coupled wave model. Taken from [53]. This model was compared with a more complex formulation that assumed the coil to behave as a Timoshenko beam. The complex formulation requires twelve simultaneous gov- erning equations relating forces and moments with displacements and rotations. This is shown in Figure 7.12. 91 Figure7.12: Governing equations and diagramatic representation of the Timoshenko beam model used to validate Wittrick’s coupled wave model. Taken from [53]. It was found that for small helix angles, large spring index, and large-wavelength forcing, the simpler theory had negligible error. The assumption of large-wavelength forcing may prohibittheuseofthistechniqueforcasesofshockloadingforlonesprings. Inthemechanism presented in this work, it is possible that the inertia of the hammer component results in a flattening of the exitation to greatly lower the frequency content of excitation due to impact. Others have developed sophisticated multi-dimensional techniques to deal with heli- cal compression springs subjected to shock loading. Examples includes specialized finite elements[39], and numerical solutions to nonlinear PDEs [2]. 7.4.5 Finite Element Model of Spring and Hammer In this work, a nonlinear finite element model was used as a benchmark to simulate the true multi-dimensional response of the compression spring subjected to an impact load. This method should tend to give more accurate results over analytical models near boundaries, which happen to be a region of particular interest in this study, as it does not 92 rely on Saint-Venant’s principle. In addition, it is capable of handling nonuniformity at the transitional regions of the spring coil, which may affect the reflection of waves near spring boundaries. A distributed spring-mass system was developed using FEMAP to provide a reference point on the accuracy of the DTFM model, and determine the extent to which dynamic loading may play a role in spring failure. The transient response was numerically computed using the ADINA 601 Advanced Nonlinear solver. The assembly was geometrically simplified to contain a squared, ground helical spring, with a cylindrical mass at the other end with a chosen density that accounted for the entire mass of the hammer (.626 kg). This model is shown in Figure 7.13. Figure 7.13: Basic depiction of the model with constraints. It is inherently assumed that the transfer of energy to the spring occurs due to rigid body motion of the hammer rather than wave propagation within the hammer, which is believed to be the case due to the hammers complex geometry and relatively short length. A comparison of the physical hammer assembly and the analysis model is shown in Figure 7.14. 93 Figure 7.14: Physical hammer assembly and its finite element model counterpart. Note that the shaft going through the middle of the spring is hollow. The system was fixed at the rear end of the spring, and the hammer component was cylindrically constrained to prevent lateral motion, which is analogous to the linear spline that cylindrically constrains the hammer in the physical assembly. A contact region was placed between each of the coils but was not activated over the duration of the simulation. Forcing was chosen identically to that described in Eq. (7.6) applied across the face of the cylindrical mass. Solid hex elements were used for the uniform main body of the spring, while tetrahedral elements were preferred towards the nonuniform ends. This model had 29774 elements, and simulations were run with a time step of 5μs. Typical run times were approximately 90 minutes on a 2016 ASUS Zenbook with a 2.59GHz 4-core i7 processor. 94 7.5 Experimentation 7.5.1 Test Setup Two separate testbeds were used to observe the functional life of the spring across various combinations of spring and spacer materials. The location of the spacer is shown in Figure 7.15. It is expected that if damping spacers provide a significant increase in cycle life, dynamic behavior may play a role in producing high stresses. Figure 7.15: Spacer location with respect to the spring and hammer. The first testbed was an instrumented, isolated percussion subassembly identical to that within the prototype TRIDENT drill, described in Chapter 3. The second testbed was a full prototype TRIDENT drill to confirm the fidelity of the percussion testbed. In both cases, the intent was to determine a functional design solution within limited time constraints, resulting in a relatively small sample size. In both testbeds, percussion was activated for 100-200k cycles at a time, with partial disassembly and inspection between sequential sessions to inspect for damage. The ham- mering rate was set between 750-850 Blows Per Minute (BPM), which was fast enough to prevent damage to the cam, but allowed the motor to operate continuously for multiple hours 95 without overheating. In real-world drilling operations, percussion is intended to be activated with a rate approximately 1000BPM but typically only for a few seconds at a time. The initial spring that failed was made of 17-7 precipitation hardened stainless steel (17-7PH per AMS 5678) and was not shot-peened. The resulting failure investigation was conducted using a batch of unpeened 302 series stainless steel springs to observe the effect of spacers with various shock attenuation capacities. The 302 series material was chosen due to its well documented characteristics as well as the expectation that it might fail sooner due to its lower yield stress, allowing for shorter life tests and more variation of spacer materials. In all cases, the prototype drill was preloaded against a basalt block, while the percussion subassembly was fixed to impact an analogous 1.6m replacement drill rod, resting on a steel plate supported by rubber dampers. It should be noted that while the percussion subassembly hammered onto this uniform cylindrical steel drill rod, the drill string on the prototype drill was composed of both steel and titanium with complex geometry, such as hollow segments and auger flutes. Figure 7.16: Standard deviation and average of spring actuator current over the lifetime of the spring. Spring failure can be retroactively determined using motor telemetry, which appears as three abrupt drops in current standard deviation. 1st, 2nd, and 3rd spring fractures denoted at roughly 300, 600, and 650 thousand cycles. 96 The number of cycles until failure was determined by observation of the motor current, which was sampled at 100Hz and 46Hz in the percussion subassembly and prototype drill respectively. It was assumed that a sudden drop in the standard deviation of the motor cur- rent signified a sudden decrease in spring resistance, which was confirmed by visual inspec- tion. An example of a compression spring undergoing three subsequent drops in motor current standard deviation is shown in Figure 7.16. A photo of this same spring after test- ing is shown in Figure 7.17 where it can be seen that it has fractured 3 times (resulting in four coil fragments). This technique in itself has value in that it can be used to investigate potential mechanism failure during flight operations, so long as motor telemetry is available to operators. Figure 7.17: Fragments of the initial spring used in this study which failed in three sequential fractures. An in-tact spring from the same batch is shown for comparison. Lastly, uponsuspectinghigherstressesthananticipated, anOil-TemperedChromeSilicon (OTCS) spring wire was used as an alternative to the stainless varieties. OTCS springs are preferredinhighstressenvironmentsonearth, andhavemuchhigheryieldandshearstrength than stainless varieties. Each spring configuration for the test campaign is given in Table 7.2. 97 Table 7.2: Spring configurations used in the percussion subassembly testbed. 17-7 PH and 302 SS wire was .148" thick, while OTCS wire was .156" thick. Test ID Spring Configuration 1.0 17-7 PH wire, steel spacer, 2J compression 1.1 302 SS wire, steel spacer, 2J compression 1.2 302 SS wire, neoprene rubber spacer, 2J compression 1.3 302 SS wire, Vespel SP-1 spacer, 2J compression 1.4 OTCS wire, bronze spacer, 2J compression 2.0 17-7 PH wire, steel spacer, 2J compression 2.1 302 SS wire, steel spacer, 2J compression 2.2 302 SS wire, steel spacer, 2J compression 2.3 OTCS wire, bronze spacer, 2.75J compression and 2J compression The 17-7 PH and 302 series springs had equal wire diameters of .148" and a mean wire diameter of .822", but the 17-7PH spring had 4.60 active coils while the 302SS had had the OTCS spring had 4.33 in order to maintain 2J spring energy. Additionally, a 954 bronze spacer was used in OTCS trials due to the larger diameter of the commercially available OTCS springs, which required the spacer to act as a low friction bushing as well. Regarding dynamic attenuation, was assumed that the bronze spacer would only negligibly deviate from the other metallic spacers in other trials. This OTCS spring also contained several other geometric differences, including a mean coil diameter of 1.04", a thicker wire diameter of .156", and 2.7 active coils in order to achieve a 2J compression. 7.5.2 Experimental Results This experiment was intended for the purpose of identifying a functional design solution within limited time constraints and its resulting small sample size does not permit rigorous statistical analysis. Results are shown in Table 7.3. 98 Table 7.3: Operational life of various spring and spacer combinations. Both the neoprene spacer and the stronger OTCS material resulted in significant increases in cycle life. Test ID Spring Configuration Cycles Until Failure x10 3 1.0 17-7 PH wire, steel spacer <400 1.1 302 SS wire, steel spacer <400 1.2 302 SS wire, neoprene rubber spacer >1000 1.3 302 SS wire, Vespel SP-1 spacer <450 1.4 OTCS wire, bronze spacer >1000 2.0 17-7 PH wire, steel spacer <200 2.1 302 SS wire, steel spacer <200 2.2 302 SS wire, steel spacer <200 2.3 OTCS wire, bronze spacer >2000 First, it can be seen that the percussion testbed did not provide a more severe loading environment than the prototype drill as all springs failed earlier when tested within the prototype TRIDENT drill. It can be seen that in the case with the neoprene rubber spacer, the 302SS spring survived 1M cycles without failure. This is more than twice the maximum experimental service life of the 17-7 PH spring, despite 302 series steel having 30% lower yield strength than the 17-7 PH. It is assumed that this is a result of the shock attenuation resulting from the neoprene damper. Hence, a significant aspect of failure in the presented stainless springs could be explained by dynamic behavior rather than quasi-static fatigue failure or unsatisfactory manufacturing techniques. As previously mentioned, the OTCS material satisfied the life criteria for this mechanism, although it is not ideal for spaceflight as it is not a stainless variety. 99 7.6 Results and Discussion The wave-speed-based DTFM model and natural-frequency-based DTFM model were compared with the numerical FEA solution to determine which might be more suited for dynamic analysis of this system. An example of this comparison is shown in Figure 7.18 for a .5J impact with shear stress measured in the inside of the middle coil. Figure 7.18: Maximum shear stress in the middle coil during a .5J rebound after a hammer impact. It can be seen that the DTFM model based on predicted wave speed significantly deviates from the FEA results, whereas the natural frequency basis resulted in general agreement. Additionally, the above forcing energy and coil position showed the most benign stress envi- ronment, yet the wave speed basis resulted in shear stresses as high as 185ksi, which is beyond the shear strength of most varieties of steel, making this method nonphysical. For this reason, the following dynamic analysis was conducted using the natural frequency basis only. 100 Results using FEA and DTFM with a natural frequency basis are presented in Figures 7.20-7.23 and are compared to the discrete-element oscillator model. The maximum shear stress over time is shown for .5J and 1.5J impact events, in the middle coil and fixed-end coil as described in Figure 7.19. These locations were chosen in order to determine if dynamic behavior results in stress concentrations in particular regions of the spring. Figure 7.19: Two locations used to measure shear stress over time in the following simulations. Figure 7.20: Maximum shear stress in the middle coil after a .5J pulse applied to the face of the hammer. 101 Figure 7.21: Maximum shear stress in the end coil after a .5J pulse applied to the face of the hammer. Figure 7.22: Maximum shear stress in the middle coil after a 1.5J pulse applied to the face of the hammer. 102 Figure 7.23: Maximum shear stress in the middle coil after a 1.5J pulse applied to the face of the hammer. In FEA and DTFM with impacts of both energy levels, the fixed coil experienced higher peak shear stresses than the middle coil, which explains the tendency for the compression spring in TRIDENT to fail toward the end. For example, in the .5J impact case computed in FEA, the fixed-end coil experienced a maximum stress of 69.9ksi, whereas the middle coil only reached 57.6ksi, a difference of more than 20%. This is due to reflection at the fixed boundary, which further suggests that extensional wave propagation may play a dominant role in the failure of the spring. This also explains the tendency for springs to fail towards this end in the mechanism, as shown in Figure 7.24. 103 Figure 7.24: Repeated failure in the fixed-end coil was observed. The discrete system underestimates the peak shear stress, and does not explain the dynamics of the rebound event because it does not capture wave behavior. For example, in the .5J impact case, the discrete-element model underestimates peak shear stress by roughly 30% in the fixed-end coil compared to FEA. This suggests that the inertia of the spring coils plays a significant role in the generation of stresses and it cannot be considered massless in this event. The transient response resulting from the DTFM model agrees well with the response as determined by FEA, particularly in the fixed-end coil. This is likely due to its ability to incorporate constructive interference of reflected extensional waves, and is clearly advanta- geous over the discrete-element impact model. It consistently bounds the computed FEA response, and overestimates peak shear stresses in the spring by no more than 12%. Even though it overestimates peak shear stress in the middle coil by as much as 25%, this coil generally experiences lower stresses overall and was never found to be the first coil to fail over the duration of the experiment. 104 In the more benign impact case (.5J) the FEA dynamic response shows a peak shear stress of 69.9ksi, which is nearly 90% of the peak shear stress that occurs during the primary quasi-static loading cycle. This means that a rebound energy as low as .5J could result in effectively twice as many stress cycles, thereby reducing the percieved fatigue life of the spring by almost 50%. In the 1.5J case, midline and amplitude shear stress resulting from impact recoil are higher than the quasi-static cycling at 64.8ksi and 36.0ksi respectively. Although more sophisticated fatigue analysis techniques are required for highly complicated stress cycles, this combination of midline and amplitude fails the Gooman criterion for 10M cyclesin 17-7 PH stainless steel, as shown in Figure 7.25. Figure7.25: Goodman diagram showing permissible combinations of midline and amplitude shear stress for 17-7PH material to survive 10M cycles. Quasi-static techniques permit 10M cycles, but a 1.5J dynamic recoil can dramatically decrease the expected life. 105 Thus, not only can this recoil effectively double the number of stress cycles put on the spring, this increase in stress can dramatically decrease the total allowable cycles before failure. These result suggest that dynamic effects can be a significant factor in reducing the life of the spring. Combined with experimental results showing the potential for stainless springs with an appropriate damper, this suggests that stainless springs may be capable of attaining a sufficient service life when combined with appropriate shock-attenuating elements. 7.7 Conclusions In this work, the repeated failure of a compression spring in a hammering mechanism was analyzed using limited experimental results combined with a series of dynamic analyses. The results of this work suggest that springs in hammering systems of this type are likely to experience additional stresses due to dynamic effects which can be modeled using methods such as FEA, or DTFM. Such springs should not be considered massless stiffness elements. The one-dimensional DTFM model appeared to bound the shear stress as calculated by FEA, and appears to be a significantly better description of an impact at the hammer face compared to a discrete-element model. The discrete-element model greatly underestimates the shear stress in both the middle coil and the end coil, but the precise effect that addi- tional shear stress has on the service life of a given spring will require further analysis and experimentation to accurately quantify. The drawbacks of this analysis are mainly in the very small sample size, and for this reason the results should be considered a case study of a particular specialized mechanism, and not a thorough analysis on the general behavior of compression springs. It is common to ascribe spring failure to manufacturing techniques (or lack thereof), such as peening, heat treatment, and surface conditions. However in this particular case, dynamic effects may play just as large of a role, if not actually the predominant life limiting factor. That being said, while failure occurring near end coils can be explained by dynamic 106 effects, it does confound with the tendency for end coils to fail due to stress concentrations, inability to fully peen tight locations, and wear between the transitional region between the end and active coils [57, 42, 4]. It is possible that the recurrence of end coil failure is due to a combination of dynamic effects and manufacturing practices. WhiletheOTCSmaterialwassuccessfulinachievingthedesiredmechanismlife, stainless steels are greatly preferred in space mechanisms primarily due to their excellent corrosion resistance. This work suggests that stainless varieties can likely attain sufficiently long service life when dynamic effects can be mitigated via incorporation of dampers or other wave attenuation techniques. This is a preliminary work resulting from the recent discovery of this failure. Future work should include increasing the number of tested springs to better quantify endurance limits, modeling various dampers in finite element analysis, as well as high-speed video or another technique to determine to what degree a rebound occurs. 107 Chapter 8 Application of DTFM to a Manual Soil Penetrometer Device 8.1 Motivation Dynamic Cone Penetrometers (DCPs) are a commonly used field testing device in a wide variety of geotechnical investigations, including construction, environmental sciences, and extraterrestrial planetary science. These devices utilize repeated impacts from a dropped mass on a long slender steel rod, which results in a longitudinal elastic wave that allows the device to penetrate the soil or pavement below. Based on the specified energy and depth of penetration, geotechnical professionals can estimate specific parameters of interest, such as bearing capacity, and subsurface stratigraphy. These parameters are used to inform the design of roads and foundations, as well as provide useful data for environmental science and geology research. 108 Figure 8.1: U.S. Air Force Air Field Assesment team using a DCP to measure soil strength at a military airfield in Afghanistan. Photo by U.S. Air Force Master Sgt. Tracy L. DeMarco (Dec. 2, 2009). Public Domain. Between different DCP manufacturers, device performance varies and typically requires empirical calibration for various cone tips, soil types, and energy delivered in order to esti- mate parameters of interest, such as California Bearing Ratio (CBR). Due to the extensive manual effort required to operate these devices for long periods, as well as the challenges of operation in low-gravity extraterrestrial applications, there has been interest in developing automatic DCPs (ADCP) that rely on robotic hammering mechanisms. These devices need to mimic the performance of manually operated systems, despite significant differences in the mechanism of percussion generation. It is known that the geometry of impactors directly affects the shape of their resulting force wave [Lundberg]. However, in the design and analysis of DCPs and ADCPs, there is rarely consideration for how the shape of its resulting impulse may play a role in the ability of the device to penetrate various types of soil. This is a concern as it neglects the dynamics principles by which the device operates and is important to understand when designing alternative methods of percussion. For example, the fact that a given DCP might use 50J impacts does not necessarily provide information on the peak forces and wavelengths of the 109 incident percussive waves. This can provide misleading results when encountering hard, brittle rock such as basalt, which requires very high magnitude, short duration pulses to fracture. It is possible that hard, brittle rocks reflect a significant portion of the delivered energy [Lundberg], which may end up being converted to vibration in the rod. For this reason, two devices utilizing the same impact energy, with different pulse shapes, may behave differently depending on the properties of the penetrated medium. Modeling of DCP systems is an important step in gaining insight into their dynamic behavior, and is critical in development of such devices for more advanced applications, such as extraterrestrial geology. In this work, DTFM was used to construct a 1D model of a commercial DCP, and experimental data was used to provide information on impact characteristics. 8.2 Proposed Model The DCP used in this work is manufactured by Kessler Soils Engineering, and is shown in its deconstructed state in Figure 8.2. Figure8.2: DisassembledDCPthatisbeingusedinthiswork. PhotobyKesslerSoilsEngineering. 110 This device uses an 8.1kg mass which is lifted to a gravitational energy of 50J, then droppedontoasteelanvil, whichgeneratesapercussivewavethatallowstherodtopenetrate the soil below. The drill rod itself is 95cm in length and 1.59cm (.625”) diameter and composedofhardenedstainlesssteel. Thedevicealsocontainsanupperrodofequaldiameter and 60cm length which functions as a handle for the user as well as a guide for the dropped mass. Due to the length to diameter ratio of the drill rod (L/d≈ 60), it is assumed that this behaves as a theoretical slender rod, which permits the use of 1D wave theory. DTFM was used to construct two 1D models of this DCP. The first model (full model) contains two uniform segments that are separated by a lumped mass (M = 1.22kg), rigidly constrained on the left, with a mass on the right end to represent the handle piece (m = 1kg). The second model is simplified (simplified model) and only contains a uniform bar that is rigidly fixed on the left. These two models are shown in Figure 8.3. Figure 8.3: Proposed 1D models of the Kessler DCP. The upper model is the simplified model containing only a rigid boundary on the left, the lower model is the full model which takes into account the upper segment as well as the masses of the anvil and the handle piece. The simplified model can be directly constructed using the techniques found in Section 2.2, where l 1 = .95m, E = 193.14GPa, A = 1.979x10 −3 m 2 , and ρ L = 1.57kg/m. 111 The full model can be derived starting with the methods described in Section 2.3, except with the addition of lumped masses. In this specific case we can make modifications to the fundamental matrices, impedance transition matrices, and boundary matrices to describe the problem at hand. Inthecaseoftwouniformsegmentsofequalstiffnessanddensity, thefundamentalmatrix can be written as U(x,s) =          e F 1 (s)(x) , for 0<x<l 1 . e F 2 (s)(x−l 1 ) T 1 e F 1 (s)l 1 , for l 1 <x<l 1 +l 2 . (8.1) where F 1 (s) =F 2 (s) =    0 1 s 2 r 2 0    (8.2) with r 2 = EA ρ L . (8.3) The addition of a lumped mass between the two segments is taken into account by T 1 =    1 0 Ms 2 EA 1   . (8.4) As before, the boundary impedance matrix is computed as Z(s) =det(M b +N b U(l 1 +l 2 ,s)). (8.5) Assuming a fixed boundary on the left, and a free boundary with a lumped mass on the right, M b =    1 0 0 0   , N b =    0 0 m 1 s 2 EA    (8.6) 112 Naturalfrequenciesarecomputedastherootsofthecharacteristicequationaftersubstituting jω for s, Δ(ω) =det(M b +N b W (ω)) (8.7) where W (ω) =U(l 1 +l 2 ,jω) =e F 2 (jω)(l 2 ) T 1 e F 1 (jω)l 1 . (8.8) This is a transcendental equation and will result in an infinite number of natural frequencies serialized by k =1,2...∞. As described in Section 2.2, assuming no external distributed loads and an initially qui- escent system, the transient response can be computed as    u(x,t) ∂u(x,t) ∂x    = 2Σ ∞ k=1 U(x,jω k )Q k I b,k (t) (8.9) Q k = adjZ(jω k ) jZ D (ω k ) . (8.10) In this case, different from Section 2.3, the addition of lumped masses is considered, and thus Z D (ω k ) =det(M k +N k d W (ω k )) +det(d M,k +N k W (ω k )) +det(M k +d N,k W (ω k )). (8.11) In this case, d W =E 2 (ω k )T 1 (ω k )e F 1 (jω k )l 1 +e F 2 (jω k )l 2 δT 1 (ω k )e F 1 (jω k )l 1 +e F 2 (jω k )l 2 T 1 E 1 (ω k ) (8.12) where δT 1 =    0 0 2Mω k EA 0    (8.13) 113 M k =    0 EA 0 0   , N k =    0 0 −mω 2 k EA    (8.14) E i =     l i r S ik −l i ω k C ik + r ω 2 k S ik −l i ω k r 2 C ik + 1 r S ik l i r S ik     (8.15) with S ik = sin( ω k l i r ) and C ik = cos( ω k l i r ). In order to apply forcing directly to mass M, I b,k (t) can be written using the general form provided in [54] as I b,k (t) = Z t 0 sin(ω k (t−τ))    γ L (τ) γ R (τ)   dτ−M b n X i=1 −1 1 EA i U −1 (x + i ,jω k ) Z t 0 sin(ω k (t−τ))(q i (τ)+K i z i (τ))dτ    0 1    (8.16) where q i is the forcing applied to the lumped mass at location x i , K i is the stiffness of a discrete spring at this location, andz i is the base excitation of this spring. In the case of the DCP, we assume that there is no spring at this location, and also that γ R =γ L = 0 because there is no forcing assumed at the boundaries of the system. Thus, given these assumptions and applying this to this system with only two segments, I b,k can be more simply written as I b,k (t) =−M b 1 EA U −1 (l 1 ,jω k ) Z t 0 sin(ω k (t−τ))q i (τ)dτ    0 1    (8.17) In both cases, the forcing q i (t) was assumed to be in the form of an exponential decay pulse, which is the theoretical result of the impact of a mass on a slender rod, as described in Section 2.4.4. This was also compared with the response of a rectangular pulse applied to the full model. In both cases, experimental data was used to tune the forcing to match both the magnitude and duration of the incident pulse. These two models were implemented in MATLAB, and the transient response was ana- lytically computed using the first 75 modes for each model at 5 μs time increments, with a .005m spatial increment. 114 8.3 Experiment AKesslerSoilsEngineeringDCPwasinstrumentedusing3mm350Ωbiaxialstraingauges in a full bridge configuration just below the anvil as shown in Figure 8.4. This location was chosen as it maximizes the time until reflection, which allows for more clear observation of the incident pulse. A lower boundary was simulated using a hardened carbon steel plate on top of a concrete floor. The 8.12kg impactor was dropped from heights of 5, 10, and 15cm, which have associated potential energies of 3.98, 7.97, and 11.95J. Larger energies were not attempted in order to prevent damage to sensing equipment, and no penetrometer cone was used. Figure 8.4: DCP used in this experiment. Strain gauges are placed a small distance below the anvil in order to delay reflection time for a clear measurement of the incident pulse. Transient strain was measured at 200kS/s on an oscilloscope after being amplified with a strain gauge signal conditioner that provided 10V excitation, set to 100X gain and 600kHz bandwidth output. Strain was computed as = −2V/V e K GF ((ν + 1)−V/V e (ν− 1)) (8.18) where V is the measured signal, V e is the excitation voltage, ν is Poisson’s ratio, and K GF is the gauge factor. 8.4 Results and Discussion 8.4.1 Experimental Results Raw force data for the three impact energies are shown in Figure 8.5. 115 Figure 8.5: Raw time-history force of a DCP impactor (8.12kg) being dropped from three heights (5cm, 10cm, and 15cm). Strain was sampled at 200 kS/s and converted to force using the known area and elastic modulus of the drill rod. Time-history force was filtered using a 6th-order Butterworth filter with a cutoff fre- quency of 30kHz. Cutoff frequency was initially chosen at 50kHz from inspection of the frequency content of the 15cm drop, shown in Figure 8.6 but was iterated lower until suffi- cient smoothness was achieved in the quiescent region of the time-history data. This filter resulted in negligible distortion peaks and was used for all trials. Performance of this filter is shown applied to the 15cm drop trial in Figure 8.7. 116 Figure8.6: Frequency content of the raw 15cm drop response. An increase in amplitude is present around 50kHz and may indicate a band of noise. Figure 8.7: Filtered and unfiltered time-history force resulting from a 15cm drop with the DCP. A 6th order Butterworth filter was used with a cutoff frequency of 30kHz. This filter was chosen as it shows significant smoothing of the quiescent region, with minimal loss of peak magnitudes. 117 This allows us to determine certain key features of the impact event which may be useful in modeling, which are annotated in Figure 8.8. The peak-to-peak time between percieved incident and reflected pulses was measured at around 370μs. Given that strain gauges were placed .90m from the rigid boundary, and given a wave speed of 17-4 stainless steel of approximately 4935 m/s, a 1.8m round trip should take 364μs, which agrees with the measured reflection. An effective full pulse appears to exist with a duration of 380μs and magnitude of 16.6kN, which is slightly longer than the theoretical round trip time, and is sensitive to measurement technique. Expectations for a rigid mass impacting a slender rod were established in Section 2.4.4 which allows us to assume that this pulse continues to decay, and thus the incident pulse captured in Figure 8.9 describes the dominant behavior. Figure 8.8: Annotated incident pulse of a 15cm (11.97J) drop with the DCP used in this work. 118 Figure 8.9: Illustration of the section of time-history force that will be used as the incident pulse to compare with modeling. 8.4.2 Application of DTFM model DTFM was used to analytically compute a transient response for various cases of interest. Two cases involved the full model with a rectangular pulse and exponential pulse applied respectively, which was compared to an exponential pulse applied to the simplified model. In the simplified model, the exponential pulse applied had a peak of 16.6kN and a time constant of 200μs. In the case of the full model, due to the including of the mass, forcing was tuned to roughly match the 16.6kN peak force of the experimental incident wave. In the simplified model, the exponential pulse had a time constant In the full model the exponential pulse had a time constant of 100μs and the rectangular pulse had a duration of 200μs. Results for all cases are shown in Figure 8.10 compared with experimental data. Computation of the transient strain and displacement of the full model with 75 modes, at 350 times, and 310 different locations had a computation time of 663 seconds. 119 Figure 8.10: Incident pulse used in modeling efforts. This pulse resembles exponential decay, which is the theoretical result of a rigid mass impacting a slender rod. Figure 8.11: Incident pulse used in modeling efforts. This pulse resembles exponential decay, which is the theoretical result of a rigid mass impacting a slender rod. 120 The exponential decay pulse applied to the full model agreed well with the experimental data, and was used to represent the 5cm and 10cm impacts as well. Pulse tuning resulted in a force magnitude of 80kN applied to mass M. Magnitudes for these cases were chosen by comparing the proportional velocities of each impact. By assuming the gravitational potential energy of the impactor was converted to kinetic energy before impact, it is assumed that the 5, 10, and 15cm drop cases resulted in impact velocities of .99, 1.40, and 1.72m/s. The 5cm and 10cm cases had 58% and 81% lower velocity, so peak force applied to the model was scaled down in this way, resulting in 46.4kN and 64.8kN respectively. Results comparing the model response with experimental data are shown in Figure 8.12. Figure 8.12: DTFM transient response for the 5cm and 10cm case compared with experimen- tal data. An exponential pulse was applied to the full model in both cases, with a magnitude proportional to the relative velocity at impact. 8.4.3 Discussion and Conclusion It is clear that the simple model does not align with experimental results, and thus is not an adequate description of this system. This is particularly shown by the large tensile forces that occur as a result of this model. This implies that the masses of the anvil, as well as the 121 upper rod segment, may be important in the transfer of force, and thus affect the way this device can penetrate various media. The exponential pulse applied to mass M showed good agreement with experimental data with the 15cm drop case. When extrapolated to the other drop cases using the propotional velocity technique, it agreed well with the lower energy cases as well. As expected, the responses of the impacts of various energies had peak forces roughly proportional to the velocity at impact. It should be noted that the experimental results implied that there was only compression present during this test. In a system with discrete masses, and spring-supported or free boundaries, this is unlikely unless there is nonlinear behavior present, such as temporary separation of rod segments (gapping). This model may be improved using methods that can handle nonlinear behavior such as separation of components or impactor rebound. It is likely that geotechnical tools such as DCPs will be necessary for upcoming scien- tific programs on the Moon, where subsurface conditions may be unknown until they are encountered. For this reason, it is important to understand the nuances in percussive wave generation and propagation in order to make precise measurements in uncertain environ- ments. 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Creator Goldman, Samuel David (author)

Core Title An approach to dynamic modeling of percussive mechanisms

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2021-08

Publication Date 07/17/2021

Defense Date 03/17/2021

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

Tag 1D,distributed transfer function method,drilling,dynamic cone penetrometer,dynamic modeling,dynamics,hammer,hammer drill,impact,ISRU,Lunar,nonuniform structure,OAI-PMH Harvest,percussion,rock drilling,space,space mechanisms,wave,wave propagation

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Advisor Flashner, Henryk (committee chair), Gruntman, Mike (committee member), Oberai, Asad (committee member), Shiflett, Geoffrey (committee member), Yang, Bingen (committee member)

Creator Email samdgoldman@gmail.com,sdgoldma@usc.edu

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