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Research Papers

# A Contact Force Model With Nonlinear Compliance and Residual Indentation1

[+] Author and Article Information
Xiaogang Xiong

Ryo Kikuuwe

e-mail: kikuuwe@mech.kyushu-u.ac.jp

Motoji Yamamoto

e-mail: yama@mech.kyushu-u.ac.jp
Department of Mechanical Engineering,
Kyushu University,
Fukuoka, 819-0395, Japan

This paper is an extended version of the authors’ conference paper [26]. Simulation results regarding the parameter $γ$ are newly included here. More detailed results and discussions are also presented.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 4, 2012; final manuscript received March 22, 2013; accepted manuscript posted May 7, 2013; published online September 16, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 81(2), 021003 (Sep 16, 2013) (8 pages) Paper No: JAM-12-1370; doi: 10.1115/1.4024403 History: Received August 04, 2012; Revised March 22, 2013; Accepted May 07, 2013

## Abstract

The contact between two bodies is a complicated phenomenon in which the force and the relative position have nonlinear relations. Empirical results in the literature show that, in some mechanical systems such as biological tissues, the relation between the contact force and the indentation is characterized by the following three features: (i) continuity of the force at the time of collision, (ii) a Hertz-like nonlinear force-indentation curve, and (iii) nonzero indentation at the time of loss of contact force. The conventional Hunt–Crossley (HC) model does not capture the feature (iii) as the model makes the contact force and the indentation reach zero simultaneously. This paper proposes a compliant contact model based on a differential-algebraic equation that satisfies all three features. The behaviors of the model and the effect of the parameters in the model are investigated through numerical simulations.

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## Figures

Fig. 1

The graphs of dio(x) and max(0,-x)

Fig. 2

Force-indentation curves: (a) typical empirical result adopted from, e.g., Fig. 4 in Ref. [1], Fig. 2.6 in Ref. [2], and Fig. 2 in Ref. [5], (b) the KV model (5), (c) the HC model (6) without any external force (solid line) and with an external pulling force (dashed line), and (d) the authors’ previous contact model (7)

Fig. 3

The force-indentation curves of the new model (10) and the HC model (6) integrated by the RK4 with a time step size of 0.001 s. The parameters are chosen as: fe = 0 N, λ = 1.5, K = 104 N/m1.5, γ = 2×103s-1, β1 = 3×10-3 s, β2 = 0.1 s/m1.5, and b2 = 0.35 s/m. The initial conditions are set as: p(0) = -0.1 m, p·(0) = 2 m/s, and a(0) = 0m1.5.

Fig. 4

Simulation of the bouncing motion by using the new model (10) and the HC model (6) integrated by the RK4 with a time step size of 0.001 s. The parameters are set as: fe = Mg, λ = 1.5, K = 104 N/m1.5, γ = 2×103s-1, β1 = 1.45×10-3 s, β2 = 0.2 s/m1.5, and b2 = 0.2 s/m. The initial conditions are set as: p(0) = -0.5 m, p·(0) = 0 m/s, and a(0) = 0m1.5.

Fig. 5

Influence of β1 on the behaviors of the new model (10) integrated by the RK4 with a time step size of 0.001 s. The parameters are set as: fe = 0 N, λ = 1.5, and K = 104 N/m1.5. The initial conditions are set as: p(0) = -0.1 m, p·(0) = 2 m/s, and a(0) = 0m1.5.

Fig. 6

Influence of β2 on the behaviors of the new model (10) integrated by the RK4 with a time step size of 0.001 s. The parameters are set as: fe = 0 N, λ = 1.5, and K = 104 N/m1.5. The initial conditions are set as: p(0) = -0.1 m, p·(0) = 2 m/s, and a(0) = 0m1.5.

Fig. 7

Influence of γ on the behaviors of the new model (10) integrated by the RK4 with a time step size of 0.001 s. The parameters are set as: fe = 0 N, λ = 1.5, and K = 104 N/m1.5. The initial conditions are set as: p(0) = -0.1 m and a(0) = 0m1.5.

Fig. 8

(a), (b) The COR as a function of β1, β2 and the impact velocity obtained from the new model (10). (c), (d) The COR as a function of the impact velocity. The parameters are set as: fe = 0 N, λ = 1.5, and K = 104 N/m1.5. The initial conditions are set as: p(0) = -0.1 m and a(0) = 0m1.5.

Fig. 9

(a) The COR as a function of γ and the impact velocity obtained from the new model (10). (b) The COR as a function of the impact velocity. The parameters are set as: fe = 0 N, λ = 1.5, and K = 104 N/m1.5. The initial conditions are set as: p(0) =-0.1 m and a(0) = 0m1.5.

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