Vibration control through the robust nonlinear absorber with negative stiffness and internal resonance creation

In this study, a nonlinear absorber that works with a negative stiffness mechanism is suggested to mitigate vibration, and its effect on the reduction of vibration is investigated. The negative stiffness, which is inherently nonlinear, creates internal resonance; therefore, the vibration energy can be transmitted from low-frequency to high-frequency vibrating modes, causing vibration suppression. The nonlinear absorber is added to the primary nonlinear system, and when the main system is subjected to external resonance due to harmonic excitation, the negative stiffness parameter of absorber is so adjusted that autoparametric resonance occurs and vibration is reduced. First, the mathematical model of the system is presented and the governing differential equations of the motion are derived, and then, using the multiple scale method, the equations are solved for the case without, and with the 1:3 internal resonance. The responses and their stability are inspected, discussed, and compared. After that, the effect of negative stiffness and damping parameters on vibration amplitude reduction is investigated and the adequacy of the proposed absorber will be demonstrated by numerical analysis. Finally, the energy exchange between the primary system and the absorber will be demonstrated by plotting the responses in the state space and the displacement response Fourier spectrum.

Study of an electromechanical nonlinear vibration absorber: Design via analytical approach

This article deals with the behavior and design of a homogeneous beam linked to an electrical absorber, via a piezoelectric material patched on the beam. The beam presents a cubic nonlinearity, which is consistent with geometrical nonlinearities for clamped-free beams. A cubic nonlinearity is added to the electrical circuit for similarity purposes. The coupled electromechanical equations are reduced to a two degrees of freedom system. The equation representing the mechanical response of the system is seen as the main system, whereas the equation coming from the electrical circuit would represent a nonlinear absorber. An analytical treatment at different time scales is endowed to identify electrical coefficients allowing the design of the electromechanical vibration absorber behavior. The equilibrium and singular points are identified, allowing the definition of ranges of forcing amplitudes leading to periodic and modulated regimes. The analytical results are compared with those obtained from direct numerical integration of the two degree of freedom system.