Effect of Finite Rotation on the Propagation of Elastic Waves in Constrained Mechanical Systems

In structural systems, impact-induced longitudinal elastic waves travel with finite speeds that depend on the material properties. Using Fourier method of analysis, the exact wave motion can be described as the sum of infinite number of harmonic waves which have the same phase velocity. In this case the medium is said to be nondispersive, since the phase velocities of the harmonic waves are equal and equal to the group velocity of the resulting wave motion. In mechanism systems with intermittent motion, on the other hand, elastic members undergo finite rotations. In this investigation, the effect of the finite rotation, coefficient of restitution, and impact conditions on the propagation of the impact-induced waves in costrained elastic systems is examined. The system equations of motion are developed using the principle of virtual work in dynamics. The jump discontinuities in the system variables as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution. It is shown that the phase velocities of different harmonic waves are no longer equal, that is, dispersion occurs in perfectly elastic mechanism members as the result of the finite rotation. The analysis presented in this paper shows that the finite rotation has more significant effect on the phase velocity of the low frequency harmonics as compared to the high frequency harmonics. A rotation-wave number that depends on the material properties and the wave length is defined for each harmonic wave. It is shown that if the angular velocity of the elastic member becomes large such that the rotation-wave number of a mode exceeds one, the associated modal displacement is no longer oscillatory.