The fluid inerter described by the fractional derivative model is introduced into the traditional nonlinear energy sink (NES), which is called fractional-order NES in this paper. The slowly varying dynamic equation (SVDE) of the system coupled with fractional-order NES is obtained by the complex averaging method, in which the fractional derivative term is treated using the fractional Leibniz theorem. Then, the discriminants (Δ, Δ₁, and Δ₂) of the number of equilibrium points are derived. By using the variable substitution method, the characteristic equation for judging the stability is established. The results show: (1) the approximate SVDE is sufficient to reflect the slowly varying characteristics of the system, which shows that the mathematical treatment of the fractional derivative term is reliable; (2) the discriminant conditions (Δ₁, Δ₂) can accurately reflect the number of equilibrium points, and the corresponding range of nonlinear parameter κ can be calculated when the system has three equilibrium points. The expressions of Δ₁, Δ₂ are simpler than Δ, which is suitable for analysis and design parameters; (3) the stability discrimination methods of schemes 1 and 2 are accurate. Compared with scheme 2, scheme 1 is more prone to various responses, especially various strongly and weakly modulated responses. In scheme 2, the inertia effect of mass can be completely replaced by integer order inerter. Compared with integer order inerter, the introduction of fractional order inerter, whether in series or in parallel, means that the amplitude of the equilibrium point on the NES vibrator is smaller, but it is also for this reason that it is not easy to produce a modulated response with scheme 2, and the vibration suppression effect of the main structure is not good.

This study investigates the critical role of resonance capture dynamics in determining the energy dissipation performance of nonlinear energy sinks (NES). A fluid inerter combining mass amplification and damping characteristics is proposed as a core component, based on which two configurations of fractional-order NES (configured in series and parallel) are systematically constructed. The applicability of the complex averaging method in fractional-order systems has been addressed through fractional calculus (such as Leibniz properties), enabling it to be analyzed like integer-order systems. Employing the multi-scale perturbation method, the energy transfer mechanism between the primary oscillator and the NES is derived, leading to the analytical determination of optimal cubic stiffness and maximum energy transfer efficiency. Comparative simulation shows that the parameters of the inerter directly affect the magnitude of critical damping. The optimal cubic stiffness design method is more reliable than traditional methods and can ensure effective target energy transfer triggering. Further analysis of dissipation time shows that the performance of fractional-order NES is superior to integer-order NES; notably, the dissipation time of series fractional-order NES is significantly shorter than that of parallel and traditional NES. In summary, this study provides theoretical guidance for the design of lightweight and high-performance NES and will also promote the application of fractional calculus theory in the field of engineering vibration reduction.

The understanding and application of fluid-type inerters by scholars have been on the rise. However, due to their intricate multiphase mechanical properties, existing models still have considerable room for improvement. This study presents two fractional-order models and conducts parameter identification by integrating them with classical experimental data. The first model is an independent fractional-order model. In comparison with traditional models, it demonstrates significantly higher fitting accuracy in frequency regions beyond the ultra-low frequency range. The second model is a segmented fractional-order model, which determines segments according to critical frequencies. Although this model enhances the overall fitting accuracy, it also leads to increased complexity. To tackle this complexity issue, a rough design strategy is proposed to minimize the critical frequency. Research indicates that under such a strategy, the inertial effect dominates the behavior of the fluid inerter. Even when the independent fractional-order model is used, a high fitting accuracy can be achieved. Consequently, by designing the structural parameters and fluid medium of the fluid inerter based on the rough design strategy, the model can be simplified. Moreover, compared with traditional nonlinear inerter models, the transfer function and eigenvalue analysis methods can be effectively applied. This enables the acquisition of more comprehensive theoretical research results, thereby greatly facilitating theoretical analysis.

This article focuses on the study of elastic beams with fractional-order inertial damping structures at both ends, with the aim of exploring their dynamic characteristics, damping effects, and parameter selection rules in depth, providing theoretical and practical support for engineering applications. Firstly, using the generalized Hamilton principle, two dynamic models of an elastic beam are established for two different boundary conditions. Next, using the complex modal analysis method, a design method for the critical damping of the first and second modes of an elastic beam was proposed for the first time, and the accuracy of the critical damping calculation formula was verified. Simulation analysis shows that the higher the derivative order and inertance, the lower the main resonance frequency, and the greater the critical damping. Then, using the main resonance amplitude and frequency attenuation rate (R_A and R_Ω) as indicators, an analysis was conducted on the impact of damper parameters on vibration suppression effects. The results indicate that the introduction of fractional-order inerter can reduce the main resonance amplitude and frequency, and critical damping plays a significant role in the vibration suppression process. Based on the optimal average R_A range (95–98%) and higher cost-effectiveness, selecting a damping value of 0.05~0.6 times the critical damping ensures better overall vibration suppression performance, providing an important reference for the vibration suppression design of elastic beams in practical engineering.