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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

## Abstract

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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## Figures

Fig. 1

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

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)

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)

Fig. 4

Illustration of the experimental setup

Fig. 5

Experimental collision measurements as a function of initial collision velocity

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

Fig. 7

Result comparison for collision velocity 0.6655 m/s

Fig. 8

Result comparison for collision velocity 2.1556 m/s

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)

Fig. 10

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

Fig. 12

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

Fig. 13

Force and position versus time for 0.6655 m/s

Fig. 14

Force and position versus time for 1.689 m/s

Fig. 11

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

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