This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations and the distributed time delay. First, we consider the pitchfork bifurcation of the system driven by both a high-frequency parametric excitation and a low-frequency external excitation. Utilizing the method of direct partition of motion, the original system is transformed into an effective integer-order slow system, and the supercritical and subcritical pitchfork bifurcations are observed in this case. Then, we study the chaotic behavior of the system when the two excitation frequencies are equal. The necessary condition for the existence of the horseshoe chaos from the homoclinic bifurcation is obtained based on the Melnikov method. Besides, the parameters effects on the routes to chaos of the system are detected by bifurcation diagrams, largest Lyapunov exponents, phase portraits, and Poincaré maps. It has been confirmed that the theoretical predictions achieve a high coincidence with the numerical results. The techniques in this paper can be applied to explore the underlying bifurcation and chaotic dynamics of fractional-order models.

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.

In this paper, we propose a novel Maxwell dynamic vibration absorber (DVA) with lever, inerter, and grounded stiffness. Firstly, the governing equation of the coupled system is established. The analytical formula of the amplitude amplification factor of the primary system and the natural frequencies of the coupled system are derived. There are three fixed points in the amplitude–frequency response curve of the primary system, which are independent of damping. Then, based on H_∞ optimization criterion, two possible optimal parameter designs of the proposed model are obtained. Considering the practical engineering application and ensuring the stability of the system, the optimal grounded stiffness ratio is selected, and six working ranges of inerter–mass ratio are calculated. Furthermore, the performance of the vibration reduction is compared for six cases. It is found that when the values of the mass ratio, lever amplification ratio, and inerter–mass ratio change in different intervals, and the optimal grounded stiffness ratio has different cases of negative, zero, and positive results. Especially when the stiffness coefficient of the viscoelastic Maxwell model and another grounded stiffness are positive at the same time, the vibration absorption effect is better theoretically. Finally, comparing with the traditional DVAs, the performance of the novel DVA is better under harmonic excitation and random excitation. The results could provide theoretical guidance for the design of inerter-based Maxwell-type DVA with a lever component.

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.