The free transverse vibration of circular plates with elastic boundary conditions was carried out using the Spectro-Geometric method. First, the displacement admissible function was invariantly expressed in spectral form as an improved trigonometric series expansion. Elastic boundary supports were physically realized by setting restraining springs along the edge with the current method. Then, the unknown series expansion coefficients were treated as the generalized coordinates, and the Rayleigh-Ritz method was employed to derive the characteristic equation for the circular plate with elastic boundary support. Finally, the corresponding experimental setup was constructed, and the experimental study was performed. The numerical results and the comparison between the reference results, finite element software (FEA) results, and test results demonstrated the correctness of the current method.