Research Papers

Modeling Inelastic Collisions With the Hunt–Crossley Model Using the Energetic Coefficient of Restitution

[+] Author and Article Information
Daniel A. Jacobs

Department of Mechanical Engineering,
Stanford University,
Stanford, CA 94305
e-mail: dajacobs@stanford.edu

Kenneth J. Waldron

Department of Mechanical Engineering,
Stanford University,
Stanford, CA 94305
e-mail: kwaldron@stanford.edu

1Corresponding author.

Manuscript received December 6, 2011; final manuscript received August 29, 2014; published online January 12, 2015. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 10(2), 021001 (Mar 01, 2015) (10 pages) Paper No: CND-12-1003; doi: 10.1115/1.4028473 History: Received December 06, 2011; Revised August 29, 2014; Online January 12, 2015

Modeling collision and contact accurately is essential to simulating many multibody systems. The three parameter Hunt–Crossley model is a continuous collision model for representing the contact dynamics of viscoelastic systems. By augmenting Hertz's elastic theory with a nonlinear damper, Hunt and Crossley captured part of the viscoelastic and velocity dependent behavior found in many systems. In the Hunt–Crossley model, the power parameter and the elastic coefficient can be related to the physical properties through Hertz's elastic theory but the damping coefficient cannot. Generally, the damping coefficient is related to an empirical measurement, the coefficient of restitution. Over the past few decades, several authors have posed relationships between the coefficient of restitution and the damping constant but key challenges remain. In the first portion of the paper, we derive an approximate expression for Stronge's (energetic) coefficient of restitution that has better accuracy for high velocities and low coefficient of restitution values than the published solutions based on Taylor series approximations. We present one method for selecting the model parameters from five empirical measurements using a genetic optimization routine. In the second portion of the paper, we investigate the application of the Hunt–Crossley model to an inhomogeneous system of a rubber covered aluminum sphere on a plate. Although this system does not fit the inclusion criteria for the Hunt–Crossley, it is representative of many systems of interest where authors have chosen the Hunt–Crossley model to represent the contact dynamics. The results show that a fitted model well predicts collision behavior at low values of the coefficient of restitution.

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Ruina, A., Bertram, J. E., and Srinivasan, M., 2005, “A Collisional Model of the Energetic Cost of Support Work Qualitatively Explains Leg Sequencing in Walking and Galloping, Pseudo-Elastic Leg Behavior in Running and the Walk-to-Run Transition,” J. Theor. Biol., 237(2), pp. 170–192. [CrossRef] [PubMed]
Kuo, A. D., Donelan, J. M., and Ruina, A., 2005, “Energetic Consequences of Walking Like an Inverted Pendulum: Step-to-Step Transitions,” Exercise Sport Sci. Rev., 33(2), pp. 88–97. [CrossRef]
Nichol, J. G., Singh, S. P. N., Waldron, K. J., Palmer, L. R., and Orin, D. E., 2004, “System Design of a Quadrupedal Galloping Machine,” Int. J. Rob. Res., 23(10–11), pp. 1013–1027. [CrossRef]
Alexander, R. M., Bennett, M. B., and Ker, R. F., 1986, “Mechanical Properties and Function of the Paw Pads of Some Mammals,” J. Zool., 209(3), pp. 405–419. [CrossRef]
Raman, C. V., 1918, “The Photographic Study of Impact at Minimal Velocities,” Phys. Rev., 12(6), pp. 442–447. [CrossRef]
Stoianovici, D., and Hurmuzlu, Y., 1996, “A Critical Study of the Applicability of Rigid-Body Collision Theory,” ASME J. Appl. Mech., 63(2), pp. 307–316. [CrossRef]
Chatterjee, A., and Ruina, A., 1997, “A New Algebraic Rigid Body Collision Model With Some Useful Properties,” ASME J. Appl. Mech., 65(4), pp. 939–951. [CrossRef]
Stronge, W. J., 1991, “Unraveling Paradoxical Theories for Rigid Body Collisions,” ASME J. Appl. Mech., 58(4), pp. 1049–1055. [CrossRef]
Wang, Y., and Mason, M. T., 1992, “Two-Dimensional Rigid-Body Collisions With Friction,” ASME J. Appl. Mech., 59(3), pp. 635–642. [CrossRef]
Hunt, K. H., and Crossley, F. R. E., 1975, “Coefficient of Restitution Interpreted as Damping in Vibroimpact,” ASME J. Appl. Mech., 42(2), pp. 440–445. [CrossRef]
Yigit, A. S., Christoforou, A. P., and Majeed, M. A., 2011, “A Nonlinear Visco-Elastoplastic Impact Model and the Coefficient of Restitution,” Nonlinear Dyn., 66(4), pp. 509–521. [CrossRef]
Ismail, K., and Stronge, W., 2012, “Viscoplastic Analysis for Direct Impact of Sports Balls,” Int. J. Nonlinear Mech., 47(4), pp. 16–21. [CrossRef]
Aryaei, A., Hashemnia, K., and Jafarpur, K., 2010, “Experimental and Numerical Study of Ball Size Effect on Restitution Coefficient in Low Velocity Impacts,” Int. J. Impact Eng., 37(10), pp. 1037–1044. [CrossRef]
Jackson, R., Green, I., and Marghitu, D., 2010, “Predicting the Coefficient of Restitution of Impacting Elastic-Perfectly Plastic Spheres,” Nonlinear Dyn., 60(3), pp. 217–229. [CrossRef]
Jackson, R. L., and Green, I., 2005, “A Finite Element Study of Elasto-Plastic Hemispherical Contact Against a Rigid Flat,” ASME J. Tribol., 127(2), pp. 343–354. [CrossRef]
Seifried, R., Minamoto, H., and Eberhard, P., 2010, “Viscoplastic Effects Occurring in Impacts of Aluminum and Steel Bodies and Their Influence on the Coefficient of Restitution,” ASME J. Appl. Mech., 77(4), p. 041008. [CrossRef]
Zhang, X., and Vu-Quoc, L., 2002, “Modeling the Dependence of the Coefficient of Restitution on the Impact Velocity in Elasto-Plastic Collisions,” Int. J. Impact Eng., 27(3), pp. 317–341. [CrossRef]
Wu, C.-Y., Li, L.-Y., and Thornton, C., 2005, “Energy Dissipation During Normal Impact of Elastic and Elastic–Plastic Spheres,” Int. J. Impact Eng., 32(1–4), pp. 593–604. [CrossRef]
Zhang, Y., and Sharf, I., 2009, “Validation of Nonlinear Viscoelastic Contact Force Models for Low Speed Impact,” ASME J. Appl. Mech., 76(5), p. 051002. [CrossRef]
Diolaiti, N., Melchiorri, C., and Stramigioli, S., 2005, “Contact Impedance Estimation for Robotic Systems,” IEEE Trans. Rob., 21(5), pp. 925–935. [CrossRef]
Haddadi, A., and Hashtrudi-Zaad, K., 2008, “A New Method for Online Parameter Estimation of Hunt–Crossley Environment Dynamic Models,” IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, Sept. 22–26, pp. 981–986.
Yamamoto, T., Vagvolgyi, B., Balaji, K., Whitcomb, L. L., and Okamura, A. M., 2009, “Tissue Property Estimation and Graphical Display for Teleoperated Robot-Assisted Surgery,” IEEE International Conference on Robotics and Automation, Kobe, Japan, May 12–17, pp. 4239–4245.
Minamoto, H., and Kawamura, S., 2011, “Moderately High Speed Impact of Two Identical Spheres,” Int. J. Impact Eng., 38(2–3), pp. 123–129. [CrossRef]
Tabor, D., 1948, “A Simple Theory of Static and Dynamic Hardness,” Proc. R. Soc. Lond. Ser. A, 192(1029), pp. 247–274. [CrossRef]
Kuwabara, G., and Kono, K., 1987, “Restitution Coefficient in a Collision Between Two Spheres,” Jpn. J. Appl. Phys., 26(Part 1, No. 8), pp. 1230–1233. [CrossRef]
Gugan, D., 2000, “Inelastic Collision and the Hertz Theory of Impact,” Am. J. Phys., 68(10), pp. 920–924. [CrossRef]
Zhang, Y., and Sharf, I., 2011, “Force Reconstruction for Low Velocity Impacts Using Force and Acceleration Measurements,” J. Vib. Control, 17(3), pp. 407–420. [CrossRef]
Herbert, R. G., and McWhannell, D. C., 1977, “Shape and Frequency Composition of Pulses From an Impact Pair,” ASME J. Manuf. Eng. Sci., 99(8), pp. 513–518. [CrossRef]
Lankarani, H. M., and Nikravesh, P. E., 1990, “A Contact Force Model With Hysteresis Damping for Impact Analysis of Multibody Systems,” ASME J. Mech. Des., 112(3), pp. 369–376. [CrossRef]
Marhefka, D., and Orin, D., 1999, “A Compliant Contact Model With Nonlinear Damping for Simulation of Robotic Systems,” IEEE Trans. Syst., Man Cybern., Part A, 29(6), pp. 566–572. [CrossRef]
Gonthier, Y., McPhee, J., Lange, C., and Piedbœuf, J.-C., 2004, “A Regularized Contact Model With Asymmetric Damping and Dwell-Time Dependent Friction,” Multibody Syst. Dyn., 11(3), pp. 209–233. [CrossRef]
Zhang, Y., and Sharf, I., 2004, “Compliant Force Modelling for Impact Analysis,” ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Salt Lake City, UT, Sept. 28–Oct. 2, pp. 595–601.
Jacobs, D. A., and Waldron, K. J., 2008, “A Nonlinear Model for Simulating Contact and Collision,” Advances in Mobile Robotics—Proceedings of the Eleventh International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, Coimbra, Portugal, Sept. 8–10, pp. 930–936.
Stronge, W. J., 2004, Impact Mechanics, Cambridge University, Cambridge, UK.
Holland, J. H., 1992, “Adaptation in Natural and Artificial Systems: An Introductory Analysis With Applications to Biology, Control, and Artificial Intelligence,” Complex Adaptive Systems, MIT Press, Cambridge, MA.
Carlos, C., and Enrique, A., 2006, “Evolutionary Algorithms,” Handbook of Bioinspired Algorithms and Applications, Chapman & Hall/CRC Computer & Information Science Series, Chapman and Hall/CRC, Boca Raton, FL.
Cross, R., 1999, “The Bounce of a Ball,” Am. J. Phys., 67(3), pp. 222–227. [CrossRef]


Grahic Jump Location
Fig. 1

Comparison of the error in the approximation of the natural log function using Taylor's and Pade's method

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Fig. 2

Comparison of simulated coefficient of restitution using the Hunt–Crossley and proposed model for α values of 0.25 (dashed-dotted), 0.50 (dashed), and 0.75 (dotted)

Grahic Jump Location
Fig. 3

Comparison of the approximation error in the simulated coefficient of restitution using the Hunt–Crossley and proposed model for α values of 0.25 (dashed-dotted), 0.50 (dashed), and 0.75 (dotted)

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Fig. 4

Illustration of the experimental setup

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Fig. 5

Experimental collision measurements as a function of initial collision velocity

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Fig. 6

Hysteresis loops for several collision velocities (a) 0.6655 m/s, (b) 0.9448 m/s, (c) 1.1263 m/s, (d) 1.5098 m/s, (e) 1.8491 m/s, and (f) 2.1556 m/s

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Fig. 7

Result comparison for collision velocity 0.6655 m/s

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Fig. 8

Result comparison for collision velocity 2.1556 m/s

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Fig. 9

Estimated stiffness values of the GA as a function of the initial collision velocity using the standard Hunt–Crossley model power (n = 3/2)

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Fig. 10

Comparison of GA estimation for collision velocity of 0.6655 m/s between free selection and constrained selection of the power parameter

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Fig. 11

Estimated model parameters and the trend lines as a function of collision velocity

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Fig. 12

Comparison of the experimental data to the simulation results using the trend line estimations for the model parameters

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Fig. 13

Force and position versus time for 0.6655 m/s

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Fig. 14

Force and position versus time for 1.689 m/s



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