Control of nonlinear vibration of beams subjected to moving loads using tuned mass dampers

Using von Kármán nonlinear beam theory, Hamilton’s principle and D’Alembert principle, the coupled nonlinear dynamic equations of the beam, moving vehicle and tuned mass damper (TMD) are established. The partial differential equation of the beam is transformed into ordinary differential equations of multimode using the Galerkin method. The viscous damping and internal damping of the beam are considered according to the proportional damping and Kelvin–Voigt model, respectively. Based on the Newmark conjunction with Newton–Raphson method, the optimum stiffness, damping and location parameters of the TMD are given for linear and nonlinear beams under the fixed load and moving vehicle. The classical control schemes (Den Hartog, Warburton) and numerical methods are compared. Also, taking the energy dissipation ratio as the control objective, the present results are verified by those of the previous literature.