Dynamic asymmetry of piezoelectric shell structures

Abstract

The non-symmetry and asymmetric dynamic characteristics of piezoelectric shell structures are studied analytically and numerically. The basic formulations of piezoelectric structures are firstly given in the tensor form. The differential equations for displacements and electric potential are derived from the formulations, which validate that the description in curvilinear coordinates with the electrical and mechanical coupling induces the non-symmetry in generalized stiffness and then the dynamic asymmetry of piezoelectric structures. Also expanding the structural displacements and electric potential with various spatial characteristics can induce the non-symmetry in generalized stiffness matrix of the converted ordinary differential equations. Then, for the multi-degree-of-freedom system with asymmetric stiffness matrix, the conventional right modes or eigenvectors are proved to have not the auto-orthogonality relations with the mass or stiffness weight. However, the left and right eigenvectors have the cross-orthogonality relations with the mass or stiffness weight, which can be used for uncoupling the asymmetric systems. Furthermore, the non-symmetric stiffness matrix is divided into the corresponding symmetric and anti-symmetric stiffness matrices. The algebraic equations for eigenvalues and singular values of the asymmetric, symmetric and anti-symmetric systems are given. It is obtained that the eigenvalues are equal to the corresponding singular values for the symmetric system, the singular values are equal to the products of the unit imaginary number and corresponding eigenvalues for the anti-symmetric system, and the differences of the eigenvalues and corresponding singular values for the asymmetric system depend on the anti-symmetric stiffness. Also the upper limits of the absolute and relative differences of the singular values of the asymmetric system to the corresponding eigenvalues of the symmetric system, and the upper and lower limits of the singular values of the asymmetric system are obtained, respectively. Finally, the spherically symmetric piezoelectric shell described in spherical coordinates is studied in detail to show the asymmetric dynamic characteristics. The differential equations for the radial displacement and electric potential of the shell structure are obtained to illustrate the non-symmetric generalized stiffness involving elastic and piezoelectric constants. Eliminating the electric potential, expanding the displacement in space, and using the Galerkin method yield the ordinary differential equations, which represent a multi-degree-of-freedom dynamic system with the asymmetric generalized stiffness matrix. Numerical results are given to illustrate the eigenvalues and modes of the asymmetric system different from those of the corresponding symmetric system, the relative differences of the eigenvalues of the asymmetric to symmetric system for different piezoelectric constants and geometric parameters, and the non-orthogonality of the left or right eigenvectors for the asymmetric system. The analytical and numerical results on the non-symmetric dynamics of piezoelectric shell structures are useful for accurate analysis and design.