Free Vibration of Combined Dynamical Systems

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Abstract
A method for analyzing the free vibration of combined linear undamped dynamical systems attached at discrete points is shown. The method uses separation of variables to exhibit the harmonic motion of the system and to derive a generalized differential equation for the normal modes. Green's functions for the vibrating component systems are used to solve the generalized differential equation and derive the characteristic equation for the natural frequencies of the system. The characteristic equation can then be solved for the exact natural frequencies and exact normal modes. The method is demonstrated for two types of dynamical systems involving beams and oscillators. For two particular systems, approximate natural frequencies determined through a Galerkin's method and the finite element method are compared to the exact natural frequencies. The generalized orthogonality relation for each system is derived.