Exact solutions for longitudinal vibration of rods coupled by translational springs

Abstract

The objective of this paper is to present exact analytical solutions for longitudinal vibration of non-uniform rods with concentrated masses coupled by translational springs. Using appropriate transformation, the governing differential equation for longitudinal vibration of a rod with varying cross section is reduced to Bessel's equation or an ordinary differential equation with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the area variation. The exact solutions for free longitudinal vibration of rods with varying cross-section are derived. The initial parameter method and the transfer matrix method are proposed to establish the frequency equation for the longitudinal vibration of two rods coupled by translational springs. The advantage of the proposed methods is that the frequency equation for two rods coupled by translational springs can be established in terms of a determinant of 2-order for any number of translational springs and concentrated masses. The proposed methods can be used to solve the problem of free longitudinal vibration of uniform and non-uniform rods with concentrated masses coupled by various translational springs, and thus to investigate the axial stiffness and mass distribution among the rods to alter the system's dynamic characteristics. A numerical example shows that the fundamental longitudinal natural frequency of two reaction towers coupled by a pipe calculated by the proposed methods is in good agreement with the full scale measured data, suggesting the proposed methods are applicable to engineering practices.