On the symplectic superposition method for analytic free vibration solutions of right triangular plates
- PDF / 856,839 Bytes
- 17 Pages / 595.276 x 790.866 pts Page_size
- 84 Downloads / 191 Views
O R I G I NA L
Yushi Yang · Dongqi An · Houlin Xu · Peng Li · Bo Wang · Rui Li
On the symplectic superposition method for analytic free vibration solutions of right triangular plates
Received: 5 March 2020 / Accepted: 24 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The analytic free vibration solutions of triangular plates are important for both rapid analyses and preliminary designs of similar structures. Due to the difficulty in solving the complex boundary value problems of the governing high-order partial differential equations, the current knowledge about the analytic solutions is limited. This study presents a first attempt to explore an up-to-date symplectic superposition method for analytic free vibration solutions of right triangular plates. Specifically, an original problem is regarded as the superposition of three fundamental subproblems of the corresponding rectangular plates that are solved by the symplectic eigenexpansion within the Hamiltonian-system framework, involving the coordinate transformation. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition. By comparison with the numerical results for the right triangular plates under six different combinations of clamped and simply supported boundary constraints, the fast convergence and high accuracy of the present approach are well confirmed. Within the current solution framework, the extension to the problems of more polygonal plates is possible. Keywords Hamiltonian system · Symplectic superposition · Triangular plate · Free vibration · Analytic solution 1 Introduction Triangular plates play an essential role in the field of mechanical and civil engineering, with prominent applications to, e.g., submarine elevators, aircraft wings, and building curtain walls. Since free vibration is one of the most vital mechanical behaviors of such structures, understanding the vibration characteristics such as the natural frequencies and associated mode shapes becomes necessary for both the analysis and design. Although various classic analytic approaches have been developed for plate problems, they are generally valid for some special cases, e.g., the circular plates and rectangular plates with Lévy-type boundary conditions [1]. The complexity in shape would inevitably limit the scope of a classic analytic method. Taking the free vibration of a triangular plate as an example, it has been difficult to conduct the accurate analysis for such a plate with different boundary constraints in a unified analytic way. Accordingly, various numerical methods are adopted to obtain the approximate solutions with acceptable accuracy for engineering applications. While many previous studies on the free vibration of triangular plates and similar problems have been summarized in a renowned treatise by Leissa [2], the subject has received continuous attention until recently. Y. Yang · D. An · H. Xu · B. Wang · R. Li (B) State Key Laboratory
Data Loading...
