Exact solutions for longitudinal vibration of rods coupled by translational springs (Q1577864)

The objective is to present exact analytical solutions for longitudinal vibration of non-uniform rods with conservated 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 to an ordinary differential equation with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for area variation. The authors derive exact solutions for free longitudinal vibration of rods with varying cross-section. The initial parameter method and the transfer matrix method are proposed to derive the frequency equation for longitudinal vibrations 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 second-order determinant for any number of translational springs and concentrated masses. The proposed methods can be used to investigate the axial stiffness and mass distribution among the rods to obtain 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 a good agreement with the full-scale measured data. Thus the proposed methods are applicable to engineering practices.