Herein, the fractional derivative is firstly proposed to describe the constitutive relationship of a fluid inerter, and a dynamic model of a vibration isolation system (VIS) with both a fractional-order inerter and damper is established. The effects of the inerter parameters on the displacement transmissibility are analyzed; the results indicate that the fractional constitutive model can express the inertia and damping effects simultaneously, and the ratio between them can be changed by adjusting the derivative order. Furthermore, semi-analytical calculation methods of fractional critical damping, and optimal fractional critical damping and corresponding derivative order are proposed and verified through three examples.

In recent years, the fractional order model has been employed to state of charge (SOC) estimation. The non integer differentiation order being expressed as a function of recursive factors defining the fractality of charge distribution on porous electrodes. The battery SOC affects the fractal dimension of charge distribution, therefore the order of the fractional order model varies with the SOC at the same condition. This paper proposes a new method to estimate the SOC. A fractional continuous variable order model is used to characterize the fractal morphology of charge distribution. The order identification results showed that there is a stable monotonic relationship between the fractional order and the SOC after the battery inner electrochemical reaction reaches balanced. This feature makes the proposed model particularly suitable for SOC estimation when the battery is in the resting state. Moreover, a fast iterative method based on the proposed model is introduced for SOC estimation. The experimental results showed that the proposed iterative method can quickly estimate the SOC by several iterations while maintaining high estimation accuracy.

In this study, a dynamic model of a sandwich beam with viscoelastic coating based on a fractional constitutive equation was developed. The fractional model is a generalized model of sandwich beams and can be modified into integer models. The wave propagation method was extended for vibration analysis of fractional sandwich beams. Further, the influence of fractional order and size parameters on the dynamic characteristics of the sandwich beam was investigated, and the steady-state frequency response of the laminated beam was calculated. The results can provide a theoretical basis for the size design of laminated beams with damping requirements.

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.

Thermoelastic damping is an important dissipation mechanism in microresonators. This paper presents an analytical model of thermoelastic damping for out-of-plane vibration of microrings with circular cross-section by considering two-dimensional heat conduction. The temperature field of the circular cross-section is calculated by using the Bessel function and free boundary conditions. The coupled motion of bending and torsion in out-of-plane mode has been considered to calculate the mechanical energy. The derivation shows that the analytical expression of thermoelastic damping can be considered as a product of Zener model and the energy ratio of pure bending energy stored to total elastic energy stored. The present model is verified by comparing with the finite-element method. The convergence of the analytical expression has been examined and the characteristics of the expression have been studied by using a normalized form. The effect of geometry on thermoelastic damping has been studied. The results show that thermoelastic damping in microrings of circular cross-section under out-of-plane mode depends on geometry, scales, and vibration frequencies.