On the Use of Momentum Balance and the Assumed Modes Method in Transverse Impact Problems

The objective of this investigation is to examine the validity of applying the assumed modes method and the generalized impulse momentum approach that involves the coefficient of restitution in the analysis of transverse impact in constrained elastic systems. A simple impact model that consists of a rotating beam which is subjected to a transverse impact by a mass (impact hammer) moving with a constant velocity is used. For the purpose of comparison and in order to check the validity of using the proposed technique, the transverse deformation of the beam with respect to the beam coordinate system is described using three different assumed sets of orthogonal functions. The different sets of modes are the clamped-free modes, pin-free modes, and a set of assumed harmonic functions. The system mass matrix that accounts for the coupling between the rigid body motion and the elastic deformation is identified and used with the Jacobian matrix of the kinematic constraints and the coefficient of restitution to define the algebraic generalized impulse momentum equations that describe the transverse impact. The series solution obtained using the generalized impulse momentum equations is used to define the generalized impulse, the jump discontinuity in the system reference and modal velocities, and the jump discontinuities in the generalized joint reaction forces. It is shown in this investigation, that by increasing the number of elastic degrees of freedom, the jump discontinuity in the angular velocity of the rod as well as the generalized impulse converge to zero regardless of the assumed complete set of modes used. The effect of the coefficient of restitution and the mass ratio on the jump in the system velocities and the generalized reaction forces is also examined.