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Internal Resonance of a Nonlinear Vibration Absorber

An approach for implementing an active nonlinear vibration absorber is presented. The strategy uses the saturation phenomenon that is exhibited by multi-degree-of-freedom systems with cubic nonlinearities possessing one-to-one internal resonance. The proposed technique consists of introducing a second-order controller and coupling it to the plant through a sensor and an actuator, where both the feedback and control signals are cubic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. We use the method of multiple scales to obtain an approximate solution of these equations and investigate the vibration stability. There are two cases of fixed points. In the first case, the response amplitude is symmetric about the origin and divided into two branches with increasing magnitudes for decreasing and increasing the natural frequency ω and the coefficient of external excitation f respectively. In the second case, the response amplitudes are symmetric about the origin for variation of all parameters but the symmetry disappeared for increasing detuning parameter σ.