Tuned mass damper with nonlinear viscous damping

Abstract

This paper investigates the effect of tuned mass dampers with nonlinear viscous damping elements. The tuned mass damper is assumed to be attached to a single-degree-of-freedom system excited by white noise. Statistical linearization is used to analyze the system and the accuracy of this procedure is verified by stochastic simulation. The optimal linear tuned mass damper is defined in terms of minimizing the standard deviation of the structural displacement. The structural damping is shown to have little influence on the optimal parameter values for the linear tuned mass damper. The (approximate) optimal nonlinear tuned mass damper is defined as the system, which by statistical linearization identifies an optimal equivalent linear tuned mass damper. This is shown to lead to explicit expressions for the optimal parameter values of the nonlinear tuned mass damper. The fact that statistical linearization leads to very accurate results, implies that the optimal nonlinear tuned mass damper is practically as effective as a linear tuned mass damper. However, the nonlinear tuned mass damper must be tuned to a specific amplitude and excitation intensity, in contrast to a conventional linear tuned mass damper. The theory is demonstrated for a tuned mass damper with viscous power-law damping and for a tuned mass damper with Bingham-type damping. The probability distribution of the displacement of the structure seems to be a close approximation to a Gaussian distribution despite the nonlinearity. Furthermore, it is shown by stochastic simulation, that the approximate optimal nonlinear tuned mass damper (identified via statistical linearization) is in fact very close to the true optimum.