Design criteria of bistable nonlinear energy sink in steady-state dynamics of beams and plates

A bistable nonlinear energy sink (BNES) conceived for the passive vibration control of beam and plate structures under harmonic excitation is investigated. By applying an Incremental Harmonic Balance (IHB) method together with an adjusted arc-length continuation technique, the frequency and amplitude responses are obtained, and their respective trends are discussed in detail from three aspects. The simplest single-mode dynamics is first considered with a special focus on the coupled effect of the cubic nonlinear stiffness and the negative linear stiffness, where an analytical treatment using complex-averaging method is also applied to obtain the slow invariant manifold for understanding the underlying dynamics. Then the multi-mode dynamics of the beam are discussed in variation of each parameter. As a result, a simple step-by-step design rule for the BNES is summarized. Finally, the obtained results and design criteria of the BNES in the beam case are extended to a 2D plate, realizing a broadband control for multi-mode plate vibration. It is found that compared to a traditional cubic one, a BNES can have a better performance both on the frequency and amplitude point of view.