Semi-analytical solutions for free vibration analysis of arbitrarily shaped plates

Using the closed-form natural mode functions of rectangular plates as Ritz basis functions, this work presents a semi-analytical solution method for the free vibration analysis of arbitrarily shaped plates with stiffeners and cutouts. In the present method, the plates are approximated by the Kirchhoff plate theory, the stiffeners are modeled by a curvilinear beam theory, and the cutouts are considered as removing energy stored in cutout domain. The arbitrarily shaped plate (physical plate) is built in a rectangular plate (background plate) that covers the physical plate, retaining the energy of the physical plate and removing the residual energy of the background plate. The closed-form natural mode functions of the background plate are used as basis functions. The closed-form natural modes are obtained by the iterative Separation-Of-Variable (iSOV) method, and an effective procedure to make use of the iSOV method is presented to obtain the natural modes as many as acquired. The present method decomposes the arbitrarily shaped plate into several curvilinear triangular and quadrilateral patches, and transforms each patches into a standard square domain using the blending function method, where the integration of energy functional is performed by Gauss quadrature. To validate the present method, comprehensive comparisons of the present results with those of commercial software and available literature are presented, revealing that the present method using fewer basis functions can achieve accurate natural modes, even higher modes, attributing to the superiority of the present novel basis functions. The present basis function derived from rectangular plates can be viewed as a natural extension and improvement of the beam functions widely used in Ritz method.