In this investigation, an analytical method is presented for the free vibration analysis of T coupling plates with general boundary conditions. The in-plane vibration and out-of-plane vibration have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses. Regardless of boundary conditions, the transverse and in-plane vibration displacement functions are sought as a new form of trigonometric expansion with accelerated convergence. The displacement functions can overcome all the relevant discontinuities of the elastic boundaries. The expansion coefficients are considered as the generalized coordinates, and determined using the Rayleigh-Ritz method. The free vibration analysis of T coupling plate with various boundary and coupling conditions are carried out using the current method. The reliability and accuracy of the current method are validated with FEM results. This study provides a reliable and theoretical basis for the vibration control of coupling plate structure.