Vibration of straight-curved-straight hollow shafts

This work presents the governing equations for both the in-plane motion and out-of-plane motion of a curved hollow shaft. Two types of shaft structures, which are a curved hollow shaft and a straight-curved-straight hollow shaft, are considered. An analytical method is also presented to examine the free vibration of the two types of shaft structures. The orthogonality of any two distinct sets of mode shape functions for both the in-plane motion and out-of-plane motion of the structures is also derived. The first in-plane modal frequency of a structure is greater than the first out-of-plane modal frequency of the same structure.