Optimal damping coefficient for a class of continuous contact models
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Optimal damping coefficient for a class of continuous contact models Mohammad Poursina1
· Parviz E. Nikravesh2
Received: 1 July 2019 / Accepted: 21 May 2020 © The Author(s) 2020
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. Keywords Impact · Continuous contact model · Hertz spring · Damping coefficient · Coefficient of restitution · Multibody systems
1 Introduction In some applications, multibody systems may experience intermittent motions due to the collisions between different components, existence of the joint clearance, or joint locking
B M. Poursina
[email protected] P.E. Nikravesh [email protected]
1
Department of Engineering Sciences, University of Agder, Grimstad 4879, Norway
2
Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
M. Poursina, P.E. Nikravesh
[4, 5, 14, 18, 24, 26]. Different methods can be used to model the impact in a multibody system. These approaches have been reviewed in detail in [7, 19]. Piecewise approach has largely been used to model the impact. This technique is mainly applicable when the duration of contact is significantly short. Based on this assumption, the configuration of the system including translational and angular coordinates are considered to remain unchanged during the period of contact. However, the system experiences a discontinuous change at the velocity level. In the piecewise method, the integration of the equations of motion is halted just before the impact occurs. Then impulse–momentum equations are formed. These equations relate the change in both linear and angular momenta of the system to impulsive forces and moments. Since applied loads (forces and moments) are bounded, their impulses during the very short period of contact are ignored in the momentum balance equations.
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