4.2 Resistive Force Theory Based Locomotion
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Development of a reduced-order modeling technique for granular locomotion by Shashank Agarwal B.Tech., Indian Institute of Technology Gandhinagar (2014) Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2019 @ Massachusetts Institute of Technology 2019. All rights reserved. Signature redacted Author....... Department of Mechanical Engineering A!/ A / December 19, 2018 Signature redacted Certified by... Kenneth Kamrin Associate Professor Thesis Supervisor Signature redacted Accepted by............. MASSACHUSETTS INSTITUTE Prof. Nicolas Hadjiconstantinou OF TECHNOLOGY Chair, Graduate Program Committee FEB '2 5 2019 UiBiARIES ARCHIVES 2 Development of a reduced-order modeling technique for granular locomotion by Shashank Agarwal Submitted to the Department of Mechanical Engineering on December 19, 2018, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract The work is aimed towards the development and expansion of a reduced-order mod- eling technique called granular Resistive Force Theory(RFT) for modeling wheeled locomotion on granular beds. A combination of various modeling techniques, namely plasticity-based continuum modeling, discrete element method (DEM) modeling, along with RFT and collaborative experimentation are used for evaluation and ex- panding upon the capabilities of granular RFT. A dimensionless formulation cor- responding to the onset of catastrophic rise in slipping of wheels during granular locomotion is proposed. This limit also corresponds to the limits of the existing form of RFT in modeling wheel-granular media interaction accurately. Along with granular locomotion, general problems of granular intrusion have also been studied extensively using continuum modeling to characterize the general force response of different granular media based on various system parameters. Thesis Supervisor: Kenneth Kamrin Title: Associate Professor 3 4 :; .a---.-.raM- --. au ..-t-3:. m-3 -;--=r: --a-.09.w -l--M---:- 5-? --... 6-3-- ----:-, -. !2---.<--.--':.-- n--' -:':.-r ag.p~ ieqe:p g ru g -|-u:.:.gg~..-P~r P..'- m -:-:- .:.:..- ..4-:,..:.-.-.. ..-.---.....+;..: ,-. -,..-ky -.: 73 -;-:. -.-.- -.-- -.--.-.: i;-g.( .- - Acknowledgments The completion of this work would not have been possible without the guidance and support of my research and academic advisor Prof Kenneth Kamrin. Ken has continually encouraged me to be a better student and researcher providing me with his invaluable tutoring and guiding me how to look into the details of simple yet challenging problems from a perspective of a researcher. I thank my parents and elder sister for their encouragement and constant support from overseas. Special thanks to our collaborator Andy Karsai and Prof Daniel Goldman at Georgia Institute of Technology whose expertise in experimentation has been vital for this work. I am thankful for all of my peers MechE, and the members of the Kamrin Group for making this entire experience a positive one. I am also indebted to an innumerable number of friends who have been there for me throughout my time at MIT. In particular, though, I want to thank Maytee Chantharayukhonthorn, my lab-mate and roommate, whose friendship I could not do without. 5 6 Contents 1 Introduction: Granular media and its triple nature 13 2 Locomotion and wheels 15 2.1 History of development of wheels over time .. ......... ... 15 2.2 Differences of granular locomotion from rigid surface locomotion . .. 16 2.3 Granular intrusion: A mechanics viewpoint ... ......... ... 17 2.3.1 M icro-Inertia .......... ........... ...... 18 2.3.2 M acro-Inertia . ........... ............ ... 19 2.3.3 Elastic wave fluidization ............ ......... 20 3 Traditional methods of modeling wheel locomotion 21 3.1 Bekker's pressure-sinkage relation ...... ............. 21 3.2 Modified Bekker model: Wong and Rees model . ........... 22 4 Resistive Force Theory (RFT): A new approach to model granular intrusion 25 4.1 Background .... ........ ........ ......... ... 25 4.2 Resistive Force Theory based locomotion ...... ........ .. 26 4.3 Using RFT for force calculations ..... ........ ........ 27 4.4 Modeling wheel locomotion using implicit RFT code ...... .... 29 5 Computationally intensive methods of simulating granular media 33 5.1 Background ..... ....... ...... ...... ...... .. 33 5.2 Discrete Element Method (DEM I MD) .... ...... ...... 33 5.3 Continuum modeling ........... ............ .... 35 5.3.1 Constitutive laws .... ....... ........ ...... 35 5.3.2 Im plem entation ...... ........ ........ .... 37 6 Low speed Locomotion 39 6.1 Circular wheel locomotion ........ ............ .... 39 6.1.1 Experimental details ......... .............. 39 6.1.2 Continuum modeling using MPM ....... ......... 43 7 6.1.3 RFT based modeling ....................... 44 6.1.4 R esults ......... ............ .......... 45 6.1.5 Conclusion .. .............. ............. 54 6.2 Non-Circular wheel locomotion ... ............ ...... 56 6.2.1 Experimental details .............. ......... 56 6.2.2 RFT based modeling ................... .... 58 6.2.3 R esults ........ ....................... 58 6.2.4 Conclusion ........... .................. 58 7 Locomotion optimization with RFT 61 7.1 Introduction ................. ............... 61 7.2 Experimental setup ....................... ..... 62 7.2.1 W heel ............................... 62 7.2.2 Test rig .. ........................ .... 62 7.3 D iscussion .......... ............ ........... 63 8 High speed locomotion 65 8.1 Experimental setup ....... ............ ......... 65 8.2 RFT based Modeling ......... ............ ...... 66 8.3 Continuum modeling (MPM) .... ............ ...... 66 8.4 Observations and analysis ...... ............ ...... 68 8.5 Effects of initial boundary conditions on equilibrium state ...... 69 8.5.1 Effect of initial translation velocity . .............. 70 8.5.2 Effect of angular velocity ramp rate ...... ......... 71 8.6 Evaluation of critical velocity for grousered wheel locomotion . .... 73 9 Other general intrusions in sand 77 9.1 Background .... ............ ............ .... 77 9.2 Plane-strain granular intrusion ..... ............ .... 79 9.2.1 Projected area defines the resistive force magnitude ... ... 79 9.2.2 Superposition of forces .......... ........... 80 9.3 Force & flow transition in plowed granular media ........... 81 9.4 Depth based lift and drag variation on cylindrical objects ....... 83 8 List of Figures 1-1 A real life example of sand flow .... ... .... .... .... 14 2-1 Evolution of wheel over time. .. ... .... .... .... ... 15 3-1 Schematic of Bekker's Bevameter intrusment .... ..... ... 22 3-2 Stress field representation on rigid wheel as per TM model . ... 23 4-1 Comparison of RFT and Plasticity theory results in plate intrusion 27 4-2 Fitted form of RFT force functions . ..... .... ..... .. 29 4-3 Sample RFT simulations for various wheel shapes ... .... .. 31 5-1 Schematic of two particle interaction in DEM ... ...... .. 34 5-2 Schematic representation of Non-cohesive graular model .... 35 6-1 Schematic and lab setup for force-slip experiments .. ..... ... 40 6-2 Wheels nomenclature for slow locomotion study ..... ..... .. 42 6-3 A sample continuum simulation of wheel Type C .. ...... ... 43 6-4 A sample RFT modeling of wheel type B ..... ..... ..... 44 6-5 Wheel C on loose and compacted Poppy Seeds ... ..... ..... 47 6-6 Wheel A and B on compacted Poppy Seeds . .... .... .... .. 50 6-7 Wheel C on MS and MMS ...... ...... ...... ...... 51 6-8 Comparison of RFT and TM model for wheel C on MS ... ..... 54 6-9 Flapped wheel used in non-circular wheel locomotion study .... .. 57 6-10 Observations made in non-circular wheel locomotion studies . .... 59 7-1 Representation of system parameters associated with the flapped wheel 62 7-2 Lab setup for flapped wheel experiments ... ...... ..... 63 7-3 Normalized velocity variation with flapped wheel ..... ..... 64 8-1 Grousered wheel test setup for high-speed locomotion studies . ... 66 8-2 Sample continuum simulation of grousered wheel ..... ..... 67 8-3 High speed locomotion performance of grousered wheel: ... ... 69 8-4 Effect on initial velocity on equilibrium state for grousered wheels . 70 8-5 Variation of system parameters for different initial Velocity ... .. 71 9 8-6 Effect on the ramp rates on equilibrium state for grousered wheels . 72 8-7 Dependence of critical angular velocity on various system parameters 74 9-1 Sokolovskii analytic form of Slip lines with characteristic log spiral .. 77 9-2 Continuum modeling: Flat plate intrusion in granular media . .... 78 9-3 Continuum modeling: Flat plate Intrusion in unconsolidated media 78 9-4 Continuum modeling: Intrusion of triangular intruder ....... .. 79 9-5 Continuum modeling: Unsymmetric Triangle intrusion flow pattern 80 9-6 Continuum modeling: Unsymmetric Triangle intrusion ...... .. 81 9-7 Force and flow transition in plowed granular media .. ...... .. 82 9-8 Lift and drag forces in granular media .... ............ 84 9-9 Lift and drag on cylinder: Variation of state variables ........ 85 10 ~"- ~ ~- List of Tables 2.1 Evolution of wheel over time ...... ...... ....... .... 16 4.1 Generic values of fitting parameters in analytic form of RFT ..... 28 6.1 Granular material properties for slow locomotion studies ....... 41 6.2 Specifications of the wheels used in slow locomotion studies .