A Transfer-Matrix-Perturbation Approach to the Dynamics of Chains of Nonlinear Sliding Beams

Chains of nonlinear shear indeformable beams with distributed mass, resting on movable supports, are considered. To determine the dynamic response of the system, the transfer-matrix approach is merged with the harmonic balance method and a perturbation method, thereby transforming the original space-temporal continuous problem into a discrete one-dimensional map $xk+1=F(xk)$ expressed in terms of the state variables $xk$ at the interface between adjacent beams. Such transformation does not imply any discretization, because it is obtained by integrating the single-element field equations. The state variables consist of both first-order variables, namely, transversal displacement and couples, and second-order variables, which are longitudinal displacement and axial forces. Therefore, while the linear problem is monocoupled, the nonlinear one becomes multicoupled. The procedure is applied to determine frequency-response relationship under free and forced vibrations.