A refined nonlinear vibration absorber

Presented here is a theoretical study on how to use saturation phenomena to design nonlinear vibration absorbers and how to improve their stability and effective frequency bandwidth. The so-called original saturation control method uses 2 : 1 internal resonances and saturation phenomena to suppress steady-state vibrations of a dynamical system by connecting it to a second-order controller using quadratic position coupling terms, which do not really suppress vibration to zero as a linear vibration absorber does. However, a linear vibration absorber uses direct position feedbacks and splits one natural frequency of the original system into two and hence spill-over effects exist when the system is subjected to broad-band and/or transient excitations. Although a saturation controller does not split one natural frequency into two, one large-amplitude nonlinear solution coexists with a small-amplitude linear solution outside of the resonance area. Hence, the existence of spillover effects depends on initial conditions. A refined nonlinear vibration absorber is designed by using a quadratic velocity coupling term in the controller and adding a negative velocity feedback to the system. It is shown that the quadratic velocity coupling term enables a saturation controller to suppress system vibrations to zero. Moreover, the linear velocity feedback enhances the capability of suppressing transient vibrations and prevents the system from having the large-amplitude nonlinear response. Two equations describing the first-mode vibration of a stainless-steel beam and a saturation controller from the authors’ previous experimental work are used in this theoretical study. Both perturbation and direct numerical integration solutions are presented. Guidelines for designing nonlinear vibration absorbers are derived.