Nonlinear Analytical Design of Nonlinear Tuned Mass Dampers and Nonlinear Primary Structures Based on Complex Variable Averaging and Multiscale Methods

With the development of modern structures in the direction of higher and more complexity, the existence of multiple factors in the design and requirements of structures can lead to structures prone to nonlinear properties. The tuned mass damper (TMD), a widely implemented passive control mechanism, plays a crucial role in the engineering field by effectively reducing vibrations within primary structures. Nevertheless, its deployment frequently induces nonlinear dynamics due to the substantial displacements resulting from TMD operation or the integration of limiting devices. This research delineates a computational framework for a single-degree-of-freedom nonlinear primary system regulated by a nonlinear tuned mass damper (NTMD), designed to emulate near-fault seismic phenomena via a sinusoidal load. The study concentrates on the nonlinear attributes of both the NTMD and the primary system. Utilizing the complex variable averaging method in conjunction with the multiscale technique, complex variable equations and slow invariant manifolds are formulated for the system under 1:1 resonance conditions, with their accuracy and validity substantiated through numerical simulations. Expanding upon the derived complex variable equations and slow invariant manifolds, this study examines the impact of nonlinear coefficients within the NTMD and the primary system on both the damping performance of the NTMD and the stability of the primary system. Furthermore, this research delves into the effects of mass ratio fluctuations on the damping effectiveness of the TMD and the control efficiency of the primary system, as well as the emergence of jump phenomena in the presence of significant nonlinear coefficients. The analytical outcomes underscore the critical need to account for the inherent nonlinearities in both the TMD and the primary system, which can have detrimental effects. By considering the mass ratio as a key design parameter, optimizing it can enhance the TMD’s vibration suppression capabilities and the primary system’s control behavior, while also reducing the likelihood of jump phenomena and improving overall structural stability.