A numerical approach is proposed for estimating the mode shape and natural frequency of an orthotropic plate. By applying an iterative method similar to Stodola's method to the governing equation not of a free vibration but of a forced one, a series solution is derived. By using this solution as a trial function, the application of Galerkin's technique to the governing equation of a free vibration yields approximate equations for estimating the natural frequency and the mode shape. It is a merit of this approach that one can estimate directly the arbitrary order natural frequency and mode shape. For the discussion of the convergency and accuracy of the solution, a simply supported orthotropic rectangular plate with an intermediate support is demonstrated. From these results, it follows that one can estimate values with sufficient accuracy. As numerical examples, natural frequencies are computed for various flexural rigidities and aspect ratios.