Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

In this study, a multi-parameter perturbation method is used for the solution of a functionally-graded, thin, circular piezoelectric plate. First, by assuming that elastic, piezoelectric, and dielectric coefficients of the functionally-graded materials vary in the form of the same exponential function, the basic equation expressed in terms of two stress functions and one electrical potential function are established in cylindrical coordinate system. Three piezoelectric coefficients are selected as perturbation parameters, and the established equations are solved by the multi-parameter perturbation method, thus obtaining up to first-order perturbation solutions. The validity of the perturbation solution obtained is verified by numerical simulations, based on layer-wise theory. The perturbation process indicates that adopting three piezoelectric coefficients as perturbation parameters follows the basic idea of perturbation theory—i.e., if the piezoelectricity may be regarded as a kind of introduced disturbance, the zero-order solution of the disturbance system corresponds exactly to the solution of functionally-graded plates without piezoelectricity. The result also indicates that the deformation magnitude of piezoelectric plates is smaller than that of plates without piezoelectricity, due to the well-known piezoelectric stiffening effect.

Vibration stability analysis of dual motor harmonic synchronous excitation nonlinear vibration conveyer

To enhance the stability of a harmonic vibration synchronous conveyer, this paper establishes a nonlinear dynamical model for this kind of vibration machine, and the effects and compensation function on the stability produced by the nonlinearity of a master vibration spring have been analyzed. A small parameter perturbation method has been used to analyze the effects of a nonlinear force on the conveyer when a fluctuating impact was loaded onto the machine. The reaction between motion stability of the vibration conveyer and the synchronization of the two motors was also investigated. The results of experiments and practical applications demonstrated the correctness of the motion stability analysis of this nonlinear vibration conveyer and its application validity. In conclusion, significant reference values for design, dynamic analysis, testing, and application of the nonlinear vibration conveyer, with harmonic synchronous vibration, actuated by two motors have been achieved.

A Multi-Parameter Perturbation Solution for Functionally Graded Piezoelectric Cantilever Beams under Combined Loads

In this study, we use a multi-parameter perturbation method to solve the problem of a functionally graded piezoelectric cantilever beam under combined loads, in which three piezoelectric coefficients are selected as the perturbation parameters. First, we derive the two basic equations concerning the Airy stress function and electric potential function. By expanding the unknown Airy stress function and electric potential function with respect to three perturbation parameters, the two basic equations were decoupled, thus obtaining the corresponding multi-parameter perturbation solution under boundary conditions. From the solution obtained, we can see clearly how the piezoelectric effects influence the behavior of the functionally graded piezoelectric cantilever beam. Based on a numerical example, the variations of the elastic stresses and displacements as well as the electric displacements of the cantilever beam under different gradient exponents were shown. The results indicate that if the pure functionally graded cantilever beam without a piezoelectric effect is regarded as an unperturbed system, the functionally graded piezoelectric cantilever beam can be looked upon as a perturbed system, thus opening the possibilities for perturbation solving. Besides, the proposed multi-parameter perturbation method provides a new idea for solving similar nonlinear differential equations.

Controlled synchronization of two nonidentical homodromy coupling exciters driven by inductor motors in a vibratory system

In this work, the controlled synchronization of two nonidentical homodromy coupling exciters driven by inductor motors in a vibratory system is investigated. According to the previous works, using small parameter perturbation method deduces the conditions of implementing self-synchronous motion of two exciters. The shortage of self-synchronization method in design of vibratory system is found. The controlled synchronization method is proposed by employing sliding mode control and proportional–integral method on two inductor motors based on the master-slave control strategy to replace the self-synchronization. The stability of the controllers is proved by Lyapunov theorem. The performances of the control system are demonstrated by numerical simulation, which shows the controlled synchronization method is feasible. Additionally, the effects of various uncertainties including internal parameter perturbations and external disturbances on the control system are discussed, which indicate the proposed controllers have a good robustness.