Longitudinal Vibrations of a Lightly Damped Viscoelastic Rod Growing in Both Lateral and Axial Directions

Abstract

Free longitudinal vibrations of a viscoelastic growing rod in both lateral and axial directions are considered in terms of the classical model. Exact and approximate WKB solutions are obtained for the single-mode case for both undamped and damped vibrations of the rod. The coupled multi-mode system is investigated numerically. It is shown that because of the rod’s growth in the axial direction, the amplitude of vibration tends to increase, and vice versa, because of the rod’s growth in the lateral direction its amplitude tends to decrease. The notion of ”critical damping” of a growing rod is introduced. At critical damping, the vibration amplitude remains constant. It is found that because of initial excitation of the lower-order vibration modes, the coupled higher-order mode amplitudes decrease with the growth of the mode number. However, with the growth of the mode number, the initial excitation of higher-order modes in the rod’s vibrations induces an effective amplitude increase in the lower-order modes.