Free vibrations of two rods connected by multi-spring–mass systems

This study deals with the longitudinal free vibrations of a system in which two rods are coupled by multi-spring–mass devices. The dynamics of this system are coupled through the motion of the masses. By using the transfer matrix method and considering the compatibility requirements across each spring connection position, the eigensolutions (natural frequencies and mode shapes) of this system can be obtained easily for different boundary conditions. The characteristic equation encompasses a function of the eigenvalues, the location of the spring connection positions, the ratio of the stiffness of the springs, the ratio of the lengths of the rods, the ratio of the sectional properties of the rods and the suspended masses. Some numerical results are presented to demonstrate the method proposed in this article.