To study the free in-plane vibration for the orthotropic circular, annular and sector plates with general boundary conditions, a modified Fourier-Ritz approach is developed. In this approach, several auxiliary closed-form functions are added to the standard Fourier cosine series to obtain a robust function. The introduction of these auxiliary functions can eliminate all the potential discontinuities of the original displacement function and its derivatives in whole domain and then effectively improve the convergence of the results. All the displacements are expressed with the modified Fourier series expansion and the arbitrary boundary conditions and the appropriate continuity conditions along the radial edges are realized by introducing the artificial boundary spring technique and artificial coupling spring technique. In addition, the Ritz procedure based on the energy functions of the plates is adopted to obtain the accurate solution since the constructed displacement field is adequately smooth in the whole solution domain. By numerical examples involving the plates of various shapes and with different boundary conditions, the reliability, accuracy and versatility of the current method get fully demonstrated. On this basis, some new results for the free in-plane vibration problem of orthotropic circular, annular and sector plates with

The vibrations of circular, annular, and sector plates are traditionally considered as different boundary value problems and often treated using different solution algorithms and procedures. This problem is further compounded by the fact that the solution for each type of plate typically needs to be adapted to different boundary conditions. In this paper, a simple solution method is proposed for a unified vibration analysis of annular, circular and sector plates with arbitrary boundary conditions. Regardless of the shapes of the plates and the types of boundary conditions, the displacement solutions are invariably expressed as a new and simple form of trigonometric series expansion with an accelerated convergence rate. The unification of seemingly different boundary value problems for the circular and annular plates and their sector counterparts is physically accomplished by applying a set of coupling springs to ensure appropriate continuity conditions along the radial edges. The accuracy, reliability and versatility of the current method are fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions. It should be noted that the current method can be easily applied to sector plates with an arbitrary inclusion angle up to 2 π.

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