Dynamic vibration absorbers (DVAs) are widely used in engineering practice because of their good vibration control performance. Structural design or parameter optimization could improve its control efficiency. In this paper, the viscoelastic Maxwell-type DVA model with an inerter and multiple stiffness springs is investigated with the combination of the traditional theory and an intelligent algorithm. Firstly, the expressions and approximate optimal values of the system parameters are obtained using the fixed-point theory to deal with the $H_{\infty }$ optimization problem, which can provide help with the range of parameters in the algorithm. Secondly, we innovatively introduce the particle swarm optimization (PSO) algorithm to prove that the algorithm could adjust the value of the approximate solution to minimize the maximum amplitude by analyzing and optimizing the single variable and four variables. Furthermore, the validity of the parameters is further verified by simulation between the numerical solution and the analytical solution using the fourth-order Runge–Kutta method. Finally, the DVA demonstrated in this paper is compared with typical DVAs under random excitation. The timing sequence and variances, as well as the decreased ratios of the displacements, show that the presented DVA has a more satisfactory control performance. The inerter and negative stiffness spring can indeed bring beneficial effects to the vibration absorber. Remarkably, the intelligent algorithm can make the resonance peaks equal in the parameter optimization of the vibration absorber, which is quite difficult to achieve with theoretical methods at present. The results may provide a theoretical and computational basis for the optimization design of DVA.

Dynamic vibration absorbers (DVAs) are extensively used in the prevention of building and bridge vibrations, as well as in vehicle suspension and other fields, due to their excellent damping performance. The reliable optimization of DVA parameters is key to improve their performance. In this paper, an H${}_{\infty }$ optimization problem of a novel three-element-type DVA model including an inerter device and a grounded negative stiffness spring is studied by combining a traditional theory and an intelligent algorithm. Firstly, to ensure the system’s stability, the specific analytical expressions of the optimal tuning frequency ratio, stiffness ratio, and approximate damping ratio with regard to the mass ratio and inerter–mass ratio are determined through fixed-point theory, which provides an iterative range for algorithm optimization. Secondly, the particle swarm optimization (PSO) algorithm is used to further optimize the four parameters of DVA simultaneously. The effects of the traditional fixed-point theory and the intelligent PSO algorithm are comprehensively compared and analyzed. The results verify that the effect of the coupling of the traditional theory and the intelligent algorithm is better than that of fixed-point theory alone and can make the two resonance peaks on the amplitude–frequency response curves almost equal, which is difficult to achieve using fixed-point theory alone. Finally, we compare the proposed model with other DVA models under harmonic and random excitation. By comparing the amplitude–frequency curves, stroke lengths, mean square responses, time history diagrams, variances and decrease ratios, it is clear that the established DVA has a good vibration absorption effect. The research results provide theoretical and algorithm support for designing more effective DVA models of the same type in engineering applications.

This paper presents a grounded nonlinear energy sink (NES) with negative stiffness attached to a damped primary system, thereby forming a 2-DOF coupled vibration system. The dynamic characteristics of the proposed NES are thoroughly investigated. Using the complexification averaging method, the governing equations for saddle-node bifurcation, codimension-2 bifurcation and Hopf bifurcation are derived. Moreover, the slow invariant manifold of the system is gained, and the necessary conditions for the strongly modulated response (SMR) induced by bifurcations are identified based on the multi-scale method. The vibration mechanism of the system is validated by the analysis of slow invariant manifolds, phase trajectories, time histories and Poincaré maps. A comparison with the grounded damping NES reveals that the introduction of negative stiffness influences the system’s stability and the occurrence of SMR. Furthermore, neglecting primary system damping can lead to misjudgment of system stability. The evaluation of damping dissipation ratio and energy spectrum on vibration reduction effect indicates that negative stiffness facilitates strong energy transfer between the primary system and the NES, thereby enhancing the damping efficiency of the system. These results are of great significance for improving the vibration reduction performance of systems in engineering applications, and provide a theoretical basis for the design of other types of NES.