Forced vibration analysis of a system equipped with the nonlinear displacement-dependent (NDD) damper

In this paper, forced vibration analysis of a mass-spring system equipped with a nonlinear displacement-dependent (NDD) damper is elaborated. To this end, the nonlinear governing differential equation of the system is derived for two types of soft- and hard- periodic excitations. In order to obtain the displacement of the excited system, the approximate analytical solution of the governing equation is developed using the multiple scales method (MSM). The proposed analytical formulations are performed for several cases of hard- and soft-excitation and are also verified by the numerical fourth-order Runge-Kutta method. Moreover, the performance of the NDD damper is analyzed and compared with the traditional linear damper used in the both hard- and soft-excited vibration analyses. For a same external periodic force, a comparison has also been carried out between the responses of the hard- and soft-excitations. It is found that the organizing the external force based on its amplitude into two types as soft- and hard- excitation, leads to a better estimated response in the forced vibration analysis. Moreover, the NDD damper has a superior performance in reducing the vibration amplitude thorough tending to the steady-state solution compared to the traditional linear damper.