Analytical solutions for the free vibration of rectangular laminated thin plates

Fiber-reinforced composite laminated plates are widely used in engineering fields such as aerospace, vehicles and ships. Accurately and rapidly solving natural modes of laminates is significant for their designs and dynamic property evaluation. Considering both symmetric and antisymmetric cross-ply and angle-ply laminates, this paper presents an analytical solution method for the natural modes of rectangular thin laminates based on the iterative separation-of-variable (iSOV) method. For symmetric cross-ply laminates, high-precision closed-form analytical natural modes are obtained by the iSOV method. For symmetric angle-ply laminates with bending-torsion coupling, a semi-analytical solution method based on the Rayleigh-Ritz principle is proposed. In this method, the closed-form mode functions obtained by the iSOV method serve as basis functions, and the frequency equation and mode functions are then achieved according to the Rayleigh quotient. For the free vibration of antisymmetric laminates, this paper establishes a theoretical model that involves only one independent displacement, i.e., the deflection, by assuming in-plane resultant forces to be zero, then uses the iSOV method to solve for closed-form natural modes for some boundary conditions and semi-analytical natural modes for other boundary conditions, greatly simplifying the solution complexity. Numerical results indicate that the proposed methods can effectively produce natural modes with high accuracy.