Higher‐order perturbation solution of a free vibration problem of coupled piezoelectricity by Kato's theory

Kato's theory, which relies on the analytical properties of the resolvent of a differential operator, is used in this paper to obtain a closed‐form perturbation solution of an electromechanical free‐vibration problem of coupled piezoelectricity. Making use of the fact that piezoelectric coupling is not too strong, the natural frequencies and the mode shapes of the through‐thickness vibration of an infinite piezoelectric plate are calculated using a perturbation method. Explicit perturbation formulae have been derived up to the third‐order terms for the natural frequencies, whereas the closed‐form perturbation expressions for the mode shapes have been obtained up to the first‐order terms through the use of a projection operator. Numerical results are discussed for a plate with fixed‐free and free‐free faces. Since exact solutions exist for the problem in hand, the validity of the proposed method can be proved. The method differs from the perturbation approach usually used in quantum mechanics, since it does not rely on the expansion of the perturbation solution in series of the unperturbed eigenmodes; since the solutions are obtained in closed form, no issues of the convergence of infinite series arise. The perturbation approach used allows to obtain an approximate solution of a coupled problem by solving only the corresponding uncoupled elastic problem. The paper is a good illustration of the application of Kato's analytical method to solving a physical problem, which may be of interest for other vibration problems, too.