Abstract
The paper presents an analytical solution for the centric viscoelastic impact of two smooth balls. The contact period has two phases, compression and restitution, delimited by the moment corresponding to maximum deformation. The motion of the system is described by a nonlinear Hunt–Crossley equation that, when compared to the linear model, presents the advantage of a hysteresis loop closing in origin. There is only a single available equation obtained from the theorem of momentum. In order to solve the problem, in the literature, there are accepted different supplementary hypotheses based on energy considerations. In the present paper, the differential equation is written under a convenient form; it is shown that it can be integrated and a first integral is found—this being the main asset of the work. Then, all impact parameters can be calculated. The effect of coefficient of restitution upon all collision characteristics is emphasized, presenting importance for the compliant materials, in the domain of small coefficients of restitution. The results (variations of approach, velocity, force vs. time and hysteresis loop) are compared to two models due to Lankarani and Flores. For quasi-elastic collisions, the results are practically the same for the three models. For smaller values of the coefficient of restitution, the results of the present paper are in good agreement only to the Flores model. The simplified algorithm for the calculus of viscoelastic impact parameters is also presented. This algorithm avoids the large calculus volume required by solving the transcendental equations and definite integrals present in the mathematical model. The method proposed, based on the viscoelastic model given by Hunt and Crossley, can be extended to the elasto–visco–plastic nonlinear impact model.
Highlights
The behavior of a mechanical system is described by using ordinary differential equations or differential equations which contain partial derivatives
The sudden variation in time of the geometrical and kinematical parameters of a mechanical system leads to the incidence of forces of considerable intensities
Allowing for the entire kinetical energy variation to be retrieved as heat generated by internal friction, Hunt and Crossley [38] show that it is inadequate to describe the impact by using a second-order linear homogenous differential equation, since it leads to a hysteresis loop open in origin, and foreseeing that, at the end of collision, the bodies attract each other instead of rejecting
Summary
The behavior of a mechanical system is described by using ordinary differential equations or differential equations which contain partial derivatives. Allowing for the entire kinetical energy variation to be retrieved as heat generated by internal friction, Hunt and Crossley [38] show that it is inadequate to describe the impact by using a second-order linear homogenous differential equation (the internal friction force being proportional to the deformation rate and to the elastic force), since it leads to a hysteresis loop open in origin, and foreseeing that, at the end of collision, the bodies attract each other instead of rejecting They prove that, in order to get a hysteresis loop closing in origin when the elastic force is of Hertzian type, it is necessary to adopt a nonlinear viscoelastic model, which ensures zero values of the damping force for the initial and final moments [38], and for which the coefficient of deformation rate should be proportional to deformation.
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