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Research Papers

# Validation of Nonlinear Viscoelastic Contact Force Models for Low Speed Impact

[+] Author and Article Information
Yuning Zhang

Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 2K6, Canadayuning.zhang@mail.mcgill.ca

Inna Sharf

Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 2K6, Canadainna.sharf@mcgill.ca

This term is a misnomer since, in fact, energy dissipation does occur in these impacts. However, since it is the standard terminology in the field, it is adhered to in this paper.

Equation (6.10) from Johnson (21) yields a slightly lower coefficient of 21.167 in Eq. 4.

The dilatational wave speed $c1$ may be more appropriate for the geometries considered here; however, the corresponding value is higher than $c0$ resulting in a lower value of $Twave$. Therefore, the result obtained here provides a conservative estimate.

The pendulum test rig cannot be used to accurately measure the postimpact velocity because of the vibration of the pendulum string and the ball after impact.

We estimate the frequencies corresponding to the longitudinal and dilatational waves to be 51 kHz and 57 kHz, respectively.

The compression time is measured as the time to peak force. Results in Fig. 1 confirm that this is a very accurate approximation of the compression time (time to maximum deformation).

J. Appl. Mech 76(5), 051002 (Jun 16, 2009) (12 pages) doi:10.1115/1.3112739 History: Received February 22, 2008; Revised January 11, 2009; Published June 16, 2009

## Abstract

Compliant contact force modeling has become a popular approach for contact and impact dynamics simulation of multibody systems. In this area, the nonlinear viscoelastic contact force model developed by Hunt and Crossley (1975, “Coefficient of Restitution Interpreted as Damping in Vibroimpact,” ASME J. Appl. Mech., 42, pp. 440–445) over 2 decades ago has become a trademark with applications of the model ranging from intermittent dynamics of mechanisms to engagement dynamics of helicopter rotors and implementations in commercial multibody dynamics simulators. The distinguishing feature of this model is that it employs a nonlinear damping term to model the energy dissipation during contact, where the damping coefficient is related to the coefficient of restitution. Since its conception, the model prompted several investigations on how to evaluate the damping coefficient, in turn resulting in several variations on the original Hunt–Crossley model. In this paper, the authors aim to experimentally validate the Hunt–Crossley type of contact force models and furthermore to compare the experimental results to the model predictions obtained with different values of the damping coefficient. This paper reports our findings from the sphere to flat impact experiments, conducted for a range of initial impacting velocities using a pendulum test rig. The unique features of this investigation are that the impact forces are deduced from the acceleration measurements of the impacting body, and the experiments are conducted with specimens of different yield strengths. The experimental forces are compared with those predicted from the contact dynamics simulation of the experimental scenario. The experiments, in addition to generating novel impact measurements, provide a number of insights into both the study of impact and the impact response.

###### FIGURES IN THIS ARTICLE
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Copyright © 2009 by American Society of Mechanical Engineers
Topics: Force , Pendulums
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## Figures

Figure 4

Geometry of angle scale

Figure 5

e as a function of vi for the two specimens

Figure 6

Approximations of e as a function of vi: (a) specimen C1 and (b) specimen C2

Figure 7

Acceleration waveforms: (a) specimen C1: vi=0.0938 m/s, (b) specimen C1: vi=0.5 m/s, (c) specimen C2: vi=0.0938 m/s, and (d) specimen C2: vi=0.5 m/s

Figure 8

Frequency spectrum of acceleration signal for specimen C1 at vi=0.5 m/s: (a) during impact and (b) after impact

Figure 9

All impact force profiles with vi=0.5 m/s: (a) specimen C1 and (b) specimen C2

Figure 10

Model of pendulum impact system

Figure 11

Simulated impact force profiles for specimen C1: (a) vi=0.0938 m/s (original), (b) vi=0.0938 m/s (zoomed in), (c) vi=0.5 m/s (original), and (d) vi=0.5 m/s (zoomed in)

Figure 12

Simulated impact force profiles for specimen C2: (a) vi=0.0938 m/s (original), (b) vi=0.0938 m/s (zoomed in), (c) vi=0.5 m/s (original), and (d) vi=0.5 m/s (zoomed in)

Figure 13

Comparison of impact force profiles for specimen C1 between simulation (dash line) and experimental results (solid line): (a) vi=0.0938 m/s, (b) vi=0.15 m/s, (c) vi=0.2060 m/s, (d) vi=0.2989 m/s, (e) vi=0.3910 m/s, and (f) vi=0.50 m/s.

Figure 14

Comparison of impact force profiles for specimen C2 between simulation (ZS: dash line; Hertz: dash-dot line), and experimental (solid line) results: (a) vi=0.0938 m/s, (b) vi=0.15 m/s, (c) vi=0.2060 m/s, (d) vi=0.2989 m/s, (e) vi=0.3910 m/s, and (f) vi=0.50 m/s

Figure 1

Experimental setup for force measurement using pendulum test rig (not to scale)

Figure 2

Pendulum mechanism

Figure 3

Experimental setup for velocity measurement (not to scale)

Figure 17

Experimental velocity versus penetration diagrams: (a) for vi=0.0938 m/s and (b) for vi=0.5 m/s

Figure 15

Comparison of impact force profiles between specimens C1 (solid line) and C2 (dotted line): (a) vi=0.0938 m/s, (b) vi=0.15 m/s, (c) vi=0.2060 m/s, (d) vi=0.2989 m/s, (e) vi=0.3910 m/s, and (f) vi=0.50 m/s

Figure 16

Experimental impact force versus penetration diagrams: (a) for vi=0.0938 m/s and (b) for vi=0.5 m/s

## Errata

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