Most mechanical systems or structures are subject to parametric or self excitations. In the present work, simultaneous principal parametric resonance of two-degree-of-freedom systems with quadratic and cubic non-linearities subject to multi-frequency parametric excitations in the presence of two-to-one internal resonance is investigated. Two approximate methods are applied to construct a set of first order, non-linear ordinary differential equations governing the modulation of the amplitudes and phases of oscillations. The applied methods are; the method of multiple time scale perturbation and the generalized synchronization methods. Steady state solutions and their stability are studied for selected values of the different parameters. The obtained results from both methods are in excellent agreement.
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.
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