On the Use of the Energetic Coefficient of Restitution in Flexible Multibody Dynamics

The generalized impulse-momentum balance equations are the result of applying impulsive dynamics to the collision of bodies in multibody systems. The impulse-momentum balance equations have been applied to rigid body multibody systems [1], and to multibody systems including rigid and flexible bodies [2]. Flexibility was considered via the approach of floating frame of reference, the component mode synthesis technique was used to reduce the number of flexible co-ordinates and Newton's rule was used to define the coefficient of restitution. It was shown that when an infinite set of mode shapes is used, the generalized impulse-momentum balance equations result in a velocity jump only at the contact area between the colliding bodies, and the other parts of the system remaining unaltered [3].

The basic assumption in impulsive dynamics is that the period of contact between the colliding bodies is so short that the coordinates remain unaltered during the period. In the approach of floating frame of reference, the coordinates are divided into the reference and flexible coordinates. The reference coordinates describe the rigid body motion in the frame of reference attached to the bodies, while the flexible coordinates account for the flexible degrees of freedom used to describe elastic deformation in the bodies. The assumption of constant coordinates contrasts with the analysis of impact-induced vibrations, which arise from the elastic motion of the flexible bodies during the contact period. It has been shown [4] that the equations are still valid because the collision process is solved a step approach (i.e., simulating the impact process requires the use of several balances). The system's configuration changes among balances, thus allowing the flexible-coordinates to change as well. The number of times that the generalized impulse-momentum balance equations need to be solved to finish the impact process depends on the number of coordinates used to describe flexibility, as well as on the time step employed. The coefficient of restitution, included in the formulation to account for energy losses in the vicinity of the contact area is difficult to interpret under these conditions. This is easily understood because continuous contact is simulated as a virtual succession of instantaneous impacts. At any such impact part of the energy is assumed to be locally lost. But the number and severity of the fictitious instantaneous impacts cannot be controlled in advance and, therefore, different losses of energy can be got for the same coefficient of restitution depending on the number of coordinates or on the time step employed.