Dynamical analysis of a general mass-spring arrangement in beam systems

The free vibration analysis of general structural arrangements consisting of beams with discrete spring-mass oscillators is presented. The dynamical model considered by Nicholson and Bergman, in which the driving force exerted by the discrete oscillator at a node depends on the displacement at that node, is extended to account for the more general case where the driving forces exerted by the discrete oscillators also depend on the displacements of the adjacent nodes. A detailed derivation of the governing differential equation of motion for such systems is presented. The self-adjoint property of the resulting differential operator is shown which assures that the eigenvalues of the dynamical system are both real and positive. The orthogonality conditions for the eigenfunctions are also derived. Two methods, namely, the Green’s function method and the particular integral method, are developed to treat the resulting differential equation of motion. Detailed parametric studies are performed to study the vibrational characteristics of general dynamical systems. A Galerkin method solution is also derived for comparison and for validation of the formulations presented.