Effect of Nonlinearities on the Vibration Reduction Performance of a System With Multiple Nonlinear Tuned Mass Dampers

Abstract

Abstract In this study, a single-degree-of-freedom structure with multiple nonlinear tuned mass dampers is investigated. A sinusoidal excitation is assumed to be applied to the structure. In order to obtain the results, a nonlinear mathematical model of the general system is obtained in the form of a set of nonlinear ordinary differential equations. Dry friction and cubic stiffness nonlinearities are considered for the nonlinear terms. To obtain steady-state frequency responses of the system, these nonlinear ordinary differential equations are transformed into a set of nonlinear algebraic equations by using Harmonic Balance Method. Fourier transformation is used in the calculation of the Fourier coefficients of the nonlinear forcing terms. Then, numerical solutions to the resulting nonlinear algebraic equations are obtained using Newton’s Method with Arc Length Continuation. The effect of the number of tuned mass dampers and their arrangement, i.e. in series or in parallel, on vibration reduction is observed by performing optimization on several cases and comparing them. In order to observe the separate and combined effects of the nonlinearities, and the number of tuned mass dampers used, several case studies are carried out using optimized case parameters with different combinations of the nonlinearities. The results of these case studies and the linear model are compared with each other to see the effect of the nonlinearities. Additionally, results of the system with optimum nonlinear tuned mass dampers are compared with that of the optimum linear tuned mass dampers and the effect of nonlinearities are discussed.