A boundary perturbation method is developed to determine the fundamental frequency of vibrating plates. The method is then applied to wavy, star shape and polygonal plates with clamped boundary conditions. Approximate analytical solutions of the fundamental frequency are obtained with an accuracy of O(e^4), where e is the deviation from the unit circle.
Yüce, H., & Wang, C. Y. (2007). Fundamental frequency of clamped plates with circularly periodic boundaries. Journal of sound and vibration, 299(1-2), 355-362.