Modal damping variations in nonlinear dynamical systems

For any linear dynamical system coupled with one or more nonlinear dynamical attachments, the effect of nonlinear energy content on modal damping variations of the entire system has not been clearly addressed in the literature. Accordingly, a novel method is employed here to formulate the amplitude-dependent modal damping matrix for such nonlinear dynamical systems using an amplitude-dependent stiffness approach. The proposed method is directly applied into the equations of motion where numerical and analytical solutions are not required to be known a priori. This advantage is highly desirable to study the dynamical behavior of nonlinear dynamical systems by direct application of methods into equations of motion. Accordingly, the modal damping content variations under the effect of amplitude-dependent stiffness are investigated here. The method is based on linearizing the nonlinear coupling stiffness where a scaled amplitude-dependent stiffness has been obtained to replace the original nonlinear coupling stiffness in the system. Accordingly, the amplitude-dependent modal damping matrix in modal coordinates is obtained and investigated. Consequently, new significant findings regarding modal damping content variations under the effect of the change in nonlinear energy during oscillation are achieved through this study. Furthermore, the nonlinear amplitude-dependent modal damping matrix of the equivalent system is found to be satisfying all matrix similarity conditions with the linear amplitude-independent modal damping matrix of the original system. These findings are expected to be of significant impact on passive nonlinear targeted energy transfer for shock mitigation and energy-harvesting fields.