Forced Vibration of Coupled Extensional‐Torsional Systems

Abstract

The dynamic response of coupled extensional‐torsional systems caused by variable forces and torques Is discussed. The general solution to the equations of motion is found, and closed‐form solutions are obtained for various harmonic excitations. Based on these solutions, other forced vibration problems can be solved using the Fourier expansions and the method of superposition. The solutions show that due to the coupled behavior, two types of periodic waves will propagate through the structure, one characterizing the extensional‐compressive deformation and the other the torsional deformation. Each component wave is composed of two parts: one follows the frequency of the excitation and the other vibrates according to its own natural frequency. The transient and steady‐state motions of the vibration are studied in detail in the analysis. The natural frequency of the coupled system is the same as that of the free vibration, but the shape of the transient motion is different from that of the free vibration and varies depending on the excitations. The forced vibration of the helical spring is used as examples to illustrate the theory developed.