The dynamic instability of plates is investigated with geometric nonlinearities being included in the model, which allows one to determine the amplitude of the parametric vibrations. A modal analysis allowing one spatial mode is performed on the nonlinear equations of motion and the resulting nonlinear Mathieu equation is solved by the incremental harmonic balance method, which takes several temporal harmonics into account. When viscous damping is included, a new algorithm is proposed to solve the equation system obtained by the incremental method. For this purpose, a new characterization of the parametric vibration by its total amplitude - or Euclidean norm - is introduced. This algorithm is particularly simple and convenient for computer implementation. The instability regions are obtained with a high degree of accuracy.