Objectives  A large number of structures with openings are used in engineering projects, which is of great significance for studying the vibration characteristics of plates with openings.

  Methods  By introducing an improved Fourier series, the permissible displacement function of rectangular plates with openings is simulated. By using displacement springs and angle springs, arbitrary boundary conditions are simulated. While obtaining the overall energy functional, the kinetic energy and strain energy of the opening part are subtracted. Based on the Rayleigh-Ritz method, the Lagrange energy functional of the structure is constructed and the variational extremum of unknown coefficients in the Fourier series obtained, transforming the original vibration problem into a problem of eigenvalue equations. The paper studies the influence on the free vibration of rectangular plates with different shaped and sized openings.

  Results  A comparison with the results of finite element software ANSYS shows that the current method is accurate and reliable,

  Conclusions  providing a reference for practical engineering.