Chebyshev Formulation for In-Plane Vibration Analysis of Arbitrary Laminated Polygonal Plates

This paper proposes a unified solution for in-plane vibration analysis of laminated arbitrary polygonal plates based on the Chebyshev polynomial method. As an innovative point of present work, the arbitrarily shaped polygonal plate is composed of two arbitrarily shaped quadrilateral plates rigidly coupled for simulation. The quadrilateral plate is uniformly mapped into a unit square plate through the four-node coordinate transformation. In the present approach, whatever the shape of the plate and the type of boundary conditions, the in-plane displacement components of the plate after transformation are expanded as Chebyshev polynomial of the first kind. The artificial spring technique is adopted here to simulate the general boundary restraints and the rigid coupling of coupled plates. Based on the Rayleigh–Ritz method, the accurate solution can be obtained through the energy function of the laminated polygonal plate. The in-plane vibration characteristics including the natural frequency and the corresponding mode shapes of the polygonal plate can be easily obtained by solving the eigenvalue problem. A large number of numerical results of the laminated polygonal plate with different shapes and boundary conditions are presented, it is found the present method shows rapid convergence, high computing efficiency, and high accuracy by comparing with those solutions calculated from the finite element method and the results in the experimentation. Besides, comprehensive studies on the effects of geometrical properties and material parameters of rhombic plates, pentagonal plates, and hexagonal plates on their in-plane vibration characteristics are also presented.