Abstarct: Similar to any flexible structure, plates undergo dynamic and vibrational loads when in operation, Such loads result in Defects in the plate. Cracks are the most important Defect that can be formed in different parts of a flexible structure and change its physical properties such as stiffness or mass, consequently changing dynamic and vibrational properties such as natural frequencies and the shape of modes which in turn reduce functionality and strength of the structure. Free in-plane vibrations were examined in this study for intact and cracked plates. Fourier series expansions are often employed for solving vibrational problems but these expansions could be problematic for convergence along boundary edges. That is why in this study, the Displacement components of the plate are first defined by modified Fourier series expansions; then, in order to examine free in-plane vibrations in an intact plate, modified Fourier series expansion and artificial spring boundary techniques are employed. In the following sections, in order to investigate free in-plane vibrations of cracked plates, modified Rayleigh-Ritz method is used. The cracked plate is simulated using the artificial spring boundary technique and then, substituting the modified displacement components in energy functions and using Lagrangian function and Rayleigh-Ritz method, dynamic and vibrational properties of the cracked plate are extracted in cantilever boundary conditions. Examining and evaluating the accuracy and convergence of the modified Fourier series and modified Raleigh-Ritz, it is safe to say that using these methods along with an adequate number of displacement component expansion sentences provides accurate results. In the following sections, the effect of geometrical and physical parameters on changes in natural frequencies and shapes of the modes are analyzed using the modified Rayleigh-Ritz method for intact plates. Moreover, the effect of relative depth of cracks, and geometrical and physical parameters on dynamic and vibrational properties of the cracked plate, such as the ratio of intact plate natural frequency to that of the cracked plate, and the shapes of modes free in-plane vibration in cantilever boundary conditions, are analyzed using the modified Rayleigh-Ritz method.