A unified semi-analytical method for free in-plane and out-of-plane vibrations of arbitrarily shaped plates with clamped edges

Abstract This paper proposes a unified semi-analytical method for the free in-plane and out-of-plane vibrations of arbitrarily shaped plates with clamped edges. The method is derived using boundary conformal mapping (BCM). The BCM is first proposed here to overcome the mathematical difficulty in describing the clamped boundary conditions of complex or/and irregular shaped plates. Elliptical, rectangular and horseshoe plates with clamped edges are studied to demonstrate the proposed method. After that, another two groups of arbitrarily shaped plates including three polygonal plates and three curve-edged plates are studied. The efficiency of the proposed method is checked through a convergence study. Results from the proposed method agree well with the available results in the literature and with finite element results. All the case studies demonstrate that the proposed method is versatile and gives good accuracy for both the in-plane and out-of-plane vibrations of arbitrarily shaped plates with clamped edges. New results from the proposed method can serve as benchmarks for further study on the vibration of plates, especially of irregular shaped plates.