Reducing nonlinear vibration of locally resonant plates via multi-frequency resonators

The nonlinear vibration reduction in locally resonant cantilever thin plate with multi-frequency periodically distributed resonators is investigated, considering the geometric nonlinearity of the plate. Based on the von Kármán nonlinear plate theory, the strain and kinetic energy of the plated are constructed. The resonators are modeled by discrete mass–spring subsystems. The plate and resonators are combined by implementing their interactions at the connection point. The natural frequencies and modes of the cantilever plate are obtained via the Ritz method with modified characteristic functions in a linear analysis. The problem of nonlinear vibration for the locally resonant plate is discretized into a multi-degree-of-freedom system using the natural modes and Lagrange approach. The effects of multi-frequency periodically distributed resonators on the plate dynamics are demonstrated by studying the amplitude–frequency responses and the bifurcations in conjunction with the maximum Lyapunov exponents of the multi-frequency locally resonant plate. It is demonstrated that the complex nonlinear dynamic behavior of the plate in a specific frequency band can be suppressed by the periodically distributed linear resonators, and the vibration reduction over a wide frequency range can be achieved by appropriately setting the frequencies of the multi-frequency resonators.