Abstract Based on general shell theory and the first order shear deformation theory, an accurate relationship between strains and displacements of a twisted plate is derived by the Green strain tensor. An equation of equilibrium for free vibration is given by the principle of virtual work and the governing equation is solved by using the Rayleigh–Ritz method with sets of orthonormal polynomials in which only the first polynomials are defined according to the geometric boundary conditions of a plate and the others are generated by the Gram–Schmidt process. The numerical verification is carried out by comparing with previous results of cantilever plates. Vibration characteristics of cantilever twisted plates such as frequency parameters and corresponding mode shapes are obtained by the present numerical method, and the effects of the twist angle, the aspect ratio and the thickness ratio on them are studied.