A weak-form quadrature element method for modal analysis of electric motor stators

Abstract

The modal analysis of electric stators based on energy principles has been extensively studied. However, existing methods often involve complexities in constructing axial mode shape functions to accommodate various boundary conditions (BCs). This paper presents a weak-form quadrature element method (QEM) for the modal analysis of motor stators, where Lagrange interpolation polynomials are used to construct axial mode shape functions, and Gauss–Lobatto-Legendre (GLL) quadrature is employed to perform the differentiations and integrations in the energy formulation. This approach significantly simplifies the derivation of characteristic equation coefficients and allows for the easy application of different BCs. The effectiveness of the proposed method is validated through modal experiments on both cylindrical shells and real motor stators under various BCs. Due to the invariance of basis in linear space, QEM achieves the same level of accuracy as other series expansion methods, provided that the polynomial order is consistent. This has been demonstrated through comparisons with both the power series expansion method and Chebyshev polynomial expansion method reported in recent literatures. Furthermore, this study identifies a notable computational error in (2,1) mode frequency of cylindrical shells and motor stators under free boundaries. Mutual validations with the transfer matrix method and exact solutions demonstrate that this error does not arise from the solution method itself but is instead inherent to the underlying shell theory.