Objectives  This paper aims to study the free vibration characteristics of complicated shape plate coupled with fluid under elastic boundary conditions.

  Methods  To this end, the rectangular domain enveloping the complicated shape plate domain is selected, and the plate displacement is expressed by the improved Fourier series in the rectangular domain. Combined with the Rayleigh integral, the relationship between the plate displacement and the surface sound pressure is established, and the integral formula is transformed into polar coordinates to avoid singularity. Aiming at the problem that it is difficult to obtain the explicit expression of the boundary curve in the variable limit integral in local polar coordinates, the structural boundary curves are treated by "replacing curve with straight" to simplify the Rayleigh integral. Based on the energy principle, a semi analytical method for analyzing the free vibration characteristics of complicated shape plate contacting with water on one side is established.

  Results  The numerical examples of rectangular plate, circular plate and some complicated shape plates are given. Compared with the finite element method and literature results, the convergence and accuracy of the method are verified, and the influence of elastic boundary conditions on the added virtual mass incremental (AVMI) factor of plate is discussed. The AVMI factor of each mode reaches the maximum near the dimensionless displacement spring stiffness of 10³. At this time, the structure is relatively most affected by the fluid.

  Conclusions  This method has strong adaptability and high calculation efficiency. It reveals the free vibration law of complicated shape plate coupled with fluid, and has certain engineering guiding significance.