A new type of hierarchical control is proposed for a four-wheel-steering (4WS) vehicle, in which both the sideslip angle and yaw rate feedback are used, and the saturation of the control variables (i.e., the front and rear steering angles) is considered. The nonlinear three degrees of freedom (3DOF) 4WS vehicle model is employed to describe the uncertainties originating from the operating situations. Further, a normal front-wheel-steering (2WS) vehicle with a drop filter of the sideslip angle is selected as the reference model. The inputs for the rear and front steering angles of the linear 2DOF 4WS, required to achieve the performances described by the reference model, are obtained and controlled by the upper controller. Further, the lower controller is designed to eliminate the state error between the linear 2DOF and nonlinear 3DOF 4WS vehicle models. The simulation results of several vehicle models with/without the controller are presented, and the robustness of the hierarchical control system is analyzed. The simulation results indicate that using the proposed hierarchical controller yields the same performance between the nonlinear 4WS vehicle and the reference model, in addition to exhibiting good robustness.

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 a critical issue for designing very high quality factor microresonators. This paper derives the entropy generation, associated with the irreversibility in heat conduction, that is used for ring resonators in in-plane vibration and presents an analytical model of thermoelastic damping according to heat increments calculated by entropy theory. We consider the heat flow only in radial thickness of the ring and obtain a complex temperature field that is out of phase with the mechanical stress. The thermoelastic dissipation is calculated in the perspective of heat increments that appear due to entropy generation. The analytical model is validated by comparing with an LR (Lifshitz and Roukes) model, finite-element method and measurement. The accuracy of the present model is found to be very high for different ambient temperatures and structures. The effects of structure dimensions and vibration frequencies on entropy generation and thermoelastic damping is investigated for ring resonators under in-plane vibration.

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.

The structure of a microresonator will affect the vibration characteristics and the performance of the system. Inspired by the structural characteristics of the human tympanic membrane, this paper proposed a microresonator with the bionic structure of a tympanic membrane. The structure of a tympanic membrane was simplified to a regular shape with three structural parameters: diameter, height, and thickness. To imitate the tympanic membrane, the contour surface of the bionic structure was modeled based on the formula of transverse vibration mode of a circular thin plate. The geometric model of the bionic structure was established by using the three structural parameters and the contour surface equation. The dynamic properties of the bionic model were studied by the finite element method (FEM). We discuss the modal characteristics of the bionic structure and study the effect of structural parameters and scale on the dynamic properties. The advantages of the bionic structure were investigated by a comparison with circular plate microresonators. The results illustrate that the bionic structure can significantly improve the resonant frequency and have a much larger effective area of vibration displacement.

Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.