Abstract
In this study, we develop an analytical formula to approximate the damping coefficient as a function of the coefficient of restitution for a class of continuous contact models. The contact force is generated by a logical point-to-point force element consisting of a linear damper connected in parallel to a spring with Hertz force–penetration characteristic, while the exponent of deformation of the Hertz spring can vary between one and two. In this nonlinear model, it is assumed that the bodies start to separate when the contact force becomes zero. After separation, either the restitution continues or a permanent penetration is achieved. Therefore, this model is capable of addressing a wide range of impact problems. Herein, we apply an optimization strategy on the solution of the equations governing the dynamics of the penetration, ensuring that the desired restitution is reproduced at the time of separation. Furthermore, based on the results of the optimization process along with analytical investigations, the resulting optimal damping coefficient is analytically expressed at the time of impact in terms of system properties such as the effective mass, penetration velocity just before the impact, coefficient of restitution, and the characteristics of the Hertz spring model.
Highlights
In some applications, multibody systems may experience intermittent motions due to the collisions between different components, existence of the joint clearance, or joint lockingM
To determine the damping coefficient, we apply an optimization process to ensure that the desired restitution is reproduced at the time of separation
The contact force model consists of a spring–damper force element in which the spring obeys force–penetration characteristics of the Hertz model
Summary
Multibody systems may experience intermittent motions due to the collisions between different components, existence of the joint clearance, or joint locking. Since applied loads (forces and moments) are bounded, their impulses during the very short period of contact are ignored in the momentum balance equations These equations along with the elastic characteristic of the impact, which is reflected in the coefficient of restitution, are used to compute the jumps in the velocities of the system [22, 23]. These estimations have been expressed in terms of the coefficient of restitution and of the geometrical and mechanical properties of the contacting bodies, which are reflected in the parameters of the Hertz contact model Since all of these methods use approximations, in some situations the continuous model with the estimated damping factor may not necessarily recover the desired restitution based on which the estimation has been generated.
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