High-precision approximate analytical solutions for free bending vibrations of thin plates and an improvement

For free bending vibrations of thin plates, based on the variable separation method, the analytical solutions of deflection functions were obtained in the Hamiltonian system, and the eigenvalue equations about the 2 coordinate axes were established. Then the vibration frequencies were solved as parametrical variables, and the mode shapes of different orders were got. The analytical form of the deflection function was in fact a high-precision approximate solution satisfying the displacement boundary conditions. From the approximate frequency values calculated with the Rayleigh-Ritz method, it is found that the previous analytical method had great precision, thus the effectiveness was demonstrated. In addition, for clamped or simply supported boundary conditions, different points were properly selected as the coordinate origins. The unified forms of the frequency equations were given. These forms were used to discuss the rectangular plates with 4 edges clamped, or 4 edges simply supported, or some clamped and the other(s) simply supported, and so on. The correctness of the frequency equation forms was verified with the symmetry of the plates deformations. The linkage and transformation between the frequency equations under different boundary conditions were obtained.