Modal Analysis for Localization in Multiple Nonlinear Tuned Mass Dampers Installed on a Structure

Abstract

Localization may occur in systems where multiple nonlinear units are coupled with each other. Additionally, localization may occur in multiple nonlinear tuned mass dampers (TMDs) installed on a structure subjected to harmonic excitation. This paper clarifies the cause of localization via modal analysis. First, the equations of motion for a structure with multiple nonlinear TMDs, which are expressed in physical coordinates, are transformed to equations in modal coordinates. When identical TMDs are used, the lowest and highest modes are decoupled in the linear terms to the other modes; however, all these modes are nonlinearly coupled with each other. In addition, the lowest and highest modes are directly excited by an external force, but the other modes are not directly excited. Therefore, the modal equations of motion form an autoparametric system. Van der Pol's method is employed to determine the frequency response curves (FRCs) for harmonic vibrations in modal coordinates. The FRCs and time histories in modal coordinates are calculated when two or three TMDs are installed on the structure; subsequently, those in physical coordinates are determined via coordinate transformation. Numerical results indicate that localization may occur when more than two modes appear simultaneously. Furthermore, the excitation frequency range where localization occurs is theoretically clarified based on backbone curves.