This paper presents the modelling and free vibration analysis of two elastically coupled plates in with general boundary conditions. Both out of plane and in-plane vibration are considered in in the present analysis thus making the methodology suitable for analysis of high frequency in-plane vibration along with low frequency bending characteristics. The plate displacement function has been derived using combination of trial beam functions , expressed as modified Fourier cosine series, in x and y direction. The modification of Fourier series is achieved by adding supplementary function to the basic Fourier series which has resulted in faster convergence rate of the series expansion. Finally, Rayleigh-Ritz method is employed to the Lagrangian function to derive the frequency matrix. The accuracy and effectiveness of the proposed method are validated through numerical examples and comparison with analytical and FEM results presented in literature.