Influence of Reference Conditions on the Analysis of Impact-Induced Elastic Waves

Abstract

Impact problems involving flexible bodies that undergo largerotations and translations were solved by using the floating frame ofreference approach. Analytically defined modes were selected to describedeformation. Different mode shapes can be used depending on the type ofattachment between the body frames of reference and the flexible bodies.Evaluating the kinematic coefficient of restitution during impactentails using the equivalent rigid body velocities of the flexiblebodies. In general, however, such velocities cannot be identified withreference velocities. When the colliding bodies can undergo large translations, freeattachments allow the use of mode shapes that possess zero totalmomentum. In this case, reference motion coincides with the equivalentrigid body motion of the flexible bodies and is not coupled to elasticmotion. When the colliding solids can rotate freely, the use of modeshapes with a zero angular momentum does not uncouple reference andelastic motion. However, if small deformation occurs during freerotation of the flexible body, the derivative of the reference angleremains roughly constant. Such a derivative can be interpreted as theequivalent angular velocity of the flexible body. When using a rigidattachment between the body frames of reference and the flexible bodies,reference motion does not coincide with the equivalent rigid bodyvelocity of the flexible bodies, which leads to non-constant referencevelocities when the bodies move freely. As a result, the equivalentrigid body velocities must be evaluated as linear combinations ofreference and elastic velocities. As shown in this paper, the use of free or rigid attachments asreference conditions, and their corresponding mode shapes allows theaccurate description of impact-induced elastic waves. However, theevaluation of elastic efforts with rigid attachments provides spuriouselastic forces at the attachment point. Such forces decrease as thenumber of modes increases.