As one of the most representative passive control devices, tuned mass dampers (TMDs) are widely used in civil engineering, aerospace, machinery, and other fields, after years of research and improvement. However, due to their large displacement and the use of a limiting device, they inevitably exhibit some nonlinear characteristics in practical engineering applications. This nonlinearity is often ignored; however, neglecting it in the design process can adversely affect the control performance. Therefore, considering the nonlinearity of a TMD while designing TMD parameters can make the calculation results closer to reality and benefit the structural design. In this paper, we derived the approximate analytical solution of TMD amplitude using the complex variable averaging method by considering the nonlinearity generated by a TMD in the vibration process. Theoretical optimal design parameters were obtained by analytical comparison, and we compared the computational time consumption of this method and the numerical method. The results showed that the optimized parameters of the TMD obtained by nonlinear design possessed a good vibration reduction effect both before and after the TMD generated nonlinear characteristics. Additionally, the complex averaging method generated frequency response curves tens of times faster than the numerical method.

The tuned mass damper (TMD), as a representative passive control device, has been widely used in various fields such as mechanical vibration and civil engineering. In practical engineering applications, nonlinear characteristics of the TMD inevitably occur due to the installation of limiting devices or large displacements. Neglecting such nonlinearity may adversely affect the control performance of the TMD. Based on this situation, the parameters of the nonlinear tuned mass damper (NTMD) were analyzed, considering the linear and cubic nonlinear stiffness of the TMD, in order to improve its control performance. In this paper, a numerical model of a single-degree-of-freedom controlled structure using an NTMD was established, and the complex variable averaging method was employed to derive the response expression of the structure under the 1:1 resonance condition. Subsequently, the nonlinear design equation for the NTMD frequency was obtained. Based on this nonlinear design formula, the design accuracy was analyzed under 1:1 resonance, undamped and softened stiffness conditions. The results indicate that the NTMD based on a nonlinear design exhibits good control performance.

Tuned mass dampers (TMDs) are widely used as vibration damping devices in engineering practice. However, during use, TMDs inevitably exhibit some nonlinear characteristics that may negatively impact engineering applications. To improve the practical performance of TMDs, the cubic nonlinear stiffness of the TMD is considered, and a nonlinear design is implemented. A numerical model of a single-degree-of-freedom main structure controlled by an NTMD is developed, and the steady-state amplitude solution of the system is obtained using the complex variable averaging method. The results show that a jump phenomenon may occur in the structure. To address this, a multivalued solution discrimination formula based on the complex variable averaging method is proposed. The discriminant formula for the jump phenomenon obtains the frequency ratio and nonlinear coefficient curves of the critical jump state, and four different system response areas are obtained. This helps the structure avoid the jump phenomenon while ensuring stability of the main structure and improving the control performance of the NTMD.