The present study deals with nonlinear vibration analysis of the plate having at least one free edge. the plate has a part-through central crack with arbitrary limit length which is parallel to one side of the plate. Due to complexity of the governing equation of motion, the Galerkin method is used for solving the problem. Therefore, the appropriate admissible functions satisfying free edge conditions in the cracked plate, must be employed. The beam functions can not satisfy free edge conditions in the plate, nevertheless these functions used in many researches in the literature, which lead to high numerical errors in computing the frequency and mode shapes. Therefore, in this research, a new admissible functions is proposed which can obviate incapability of the beam functions to accurately estimate natural frequency of the intact and cracked plate and reduce the related computational errors. The effect of changes in thickness and non-dimensional crack length on natural frequency of the cracked plate are investigated using proposed functions, and frequency response curves indicating the dependence of frequency on amplitude is derived for different boundary conditions. Also, the influence of crack length on the changes in the nonlinear behavior of the cracked plate are investigated. The results, are compared with those of the available results in the literature.