Analysis on nonlinear stiffness isolators revealing damping thresholds

Nonlinear vibration isolators with stiffness nonlinearity show promise for a broadband isolation performance without degrading the load capacity. An analytical method is proposed to predict all the possible frequency responses of nonlinear stiffness isolators under different damping. The vibration equation is transformed into an algebraic equation through harmonic balance method. The equation is regarded as a function of the frequency instead of the amplitude, leading to a general analytical method with its complexity independent with the order of the nonlinear stiffness. A damping threshold function is proposed based on the root conditions of the frequency, and the whole damping region is divided into large damping, medium damping and small damping sub-regions accordingly. The root conditions and the frequency responses in each subregion are analytically analyzed. All the possible responses are classified into five categories based on the damping thresholds and the existence of a jump phenomenon. Nonlinear phenomena of full-band isolation, bounded response and unbounded response are revealed according to multivalued roots of the frequency, and their sufficient and necessary conditions are presented. Simulations are performed on nonlinear isolators with cubic stiffness, high-order stiffness, and non-polynomial stiffness, respectively. The damping thresholds, frequency response categories, and the nonlinear characteristics of each category are demonstrated for the studied cases. A highorder harmonic balance method and the Runge-Kutta method are adopted to verify the effectiveness of the proposed method. This work presents a general analytical method for the prediction of frequency responses, the determination of damping thresholds, and the theoretical illustrations of unique behaviors for nonlinear stiffness isolators.