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A new model of the contact force for the collision between two solid bodies

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Abstract

A collision between two bodies is a usual phenomenon in many engineering applications. The most important problem with the collision analysis is determining the hysteresis damping factor or the hysteresis damping ratio. The hysteresis damping ratio is related to the coefficient of restitution. In this paper, an explicit expression is determined for this relation. For this reason, a parametric expression is considered for the relation between the deformation and its velocity of the contact process. This expression consists of two unknown constants. Using the energy balance, a new explicit parametric expression between the hysteresis damping factor and the coefficient of restitution is derived. For determining the unknown constants, the root mean square (RMS) of the hysteresis damping ratio of this new expression with respect to the numerical model is minimized. This new model is completely suitable for the whole range of the coefficient of restitution. So, the new model can be used in the hard and soft impact problems. Finally, three numerical examples of two colliding bodies, the classic bouncing ball problem, the resilient impact damper, and a planar slider–crank mechanism, are presented and analyzed.

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Abbreviations

RMS:

Root of mean square

\(F\) :

Contact force

\(K\) :

Generalized stiffness parameter

\(\Delta\) :

Relative normal deformation

\(R_{1}\) and \(R_{2}\) :

Radii of spheres

\(\sigma _{1}\) and \(\sigma _{2}\) :

Material parameters

\(v_{1}\) and \(v_{2}\) :

Poisson’s ratios

\(E_{1}\) and \(E_{2}\) :

Young moduli

\(D\) :

Damping coefficient

\(C\) :

Hysteresis damping factor

\(N\) :

Exponent

\(\dot{\delta }\) :

Relative normal velocity of two contacting bodies

\(c_{r}\) :

Coefficient of restitution

\(x_{1}\) and \(x_{2}\) :

Displacements of the center of mass

\(\delta _{1}\) and \(\delta _{2}\) :

Deformations of two spheres

\(m_{1}\) and \(m_{2}\) :

Masses of spheres

\(\ddot{x}_{1}\) and \(\ddot{x}_{2}\) :

Acceleration of the centers of mass of two spheres

\(m\) :

Equivalent mass

\(\ddot{\delta }\) :

Relative normal acceleration of two contacting bodies

\(V_{1}^{-}\) and \(V_{2}^{-}\) :

Velocities of the two spheres at the initial instant of contact

\(V_{12}\) :

Common velocity of both spheres in the maximum deformation instant

\(\dot{\delta }^{-}\) and \(\dot{\delta }^{+}\) :

Impacting and the separating velocities

\(\Delta T_{\mathrm{com}}\) :

Change in the kinetic energy in the compression period

\(\Delta T_{\mathrm{res}}\) :

Change in the kinetic energy in the restitution period

\(\Delta T\) :

Total change in the kinetic energy in the contact process

\(\Delta W_{\mathrm{com}}\) :

Work done by the Hertz contact force in the compression period

\(\Delta W_{\mathrm{res}}\) :

Work done by the Hertz contact force in the restitution period

\(\Delta W\) :

Total work done by the Hertz contact force in the contact process

\(\delta _{\max } \) :

Maximum deformation

ln:

Natural logarithm

\(\Delta E_{\mathrm{com}}\) :

Energy loss due to the damping force in the compression period

\(\Delta E_{\mathrm{res}}\) :

Energy loss due to the damping force in the restitution period

\(\Delta E\) :

Total energy loss due to the damping force in the contact process

\(h_{r}\) :

Hysteresis damping ratio

\(H_{0}\) :

Initial height of bouncing ball

\(g\) :

Gravity acceleration

\(V_{0}\) :

Initial velocity of bouncing ball

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  1. Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran

    Hossein Safaeifar & Anooshirvan Farshidianfar

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  1. Hossein Safaeifar
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Correspondence to Anooshirvan Farshidianfar.

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Safaeifar, H., Farshidianfar, A. A new model of the contact force for the collision between two solid bodies. Multibody Syst Dyn 50, 233–257 (2020). https://doi.org/10.1007/s11044-020-09732-2

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