A solution method for free vibration of intact and cracked polygonal thin plates using the Ritz method and Jacobi polynomials. (17th February 2022)

• Record Type: Journal Article
• Authors: Song, Yuyu; Xue, Kai; Li, Qiuhong
• Is Part Of: Journal of sound and vibration. Volume 519(2022)
• Publication Date: 2022-02-17
• Subjects: Polygonal plate -- Cracked polygonal plate -- Kirchhoff theory -- Jacobi polynomials -- Ritz method -- Free vibration; Sound -- Periodicals; Vibration -- Periodicals; Son -- Périodiques; Vibration -- Périodiques; Sound; Vibration; Periodicals; Electronic journals; 620.205
• DOI: 10.1016/j.jsv.2021.116578
• Languages: English
• ISSNs: 0022-460X

Abstract

Highlights: A unified solution method for intact and cracked polygonal plates is presented. Polygonal domain of integration is divided into triangular and trapezoidal domains. The existence of crack is determined by the stiffness of the connecting springs. The proposed method is appropriate for thin plates with arbitrary constraints. The proposed method is also appropriate for concave polygonal plates. Abstract: A method for solving for the free vibration of intact and straight through-cracked polygonal thin plates with arbitrary boundary conditions is proposed. To obtain the expressions for the energy integral, the polygonal domain of integration is divided into triangular and trapezoidal domains by a set of parallel lines passing through the vertexes. The energy integral can then be calculated analytically in closed form using the properties of Jacobi orthogonal polynomials, which are the admissible functions of the Ritz method in the framework of the Kirchhoff plate theory. The boundary conditions are modeled using linear springs to restrain the plate edges. The model for solving polygonal thin plate vibration for arbitrary boundary conditions is thus obtained. Accordingly, the cracked polygonal plate is decomposed into several polygonal subdomains based on the crack by applying the technique of domain decomposition. Each subdomain is solved as an intact plate. The subdomains are then connected by a set of linear springs with variable stiffness, and the crack or rigid connection is determined by setting the spring stiffness to zero or infinity, respectively. The convergence and accuracy of the proposed method are verified by comparing the numerical examples with results available in the literature. The proposed method is also applied to determine the natural frequencies of a concave pentagonal plate and cracked regular hexagonal plates. The effect of crack length and inclination on the natural frequencies and mode shapes of cracked regular hexagonal plates with several boundary conditions are studied. The natural frequencies of cracked regular hexagonal plates tabulated are the first published vibration data, and can be used as a reference for future studies.