Abstract
A generalized solution procedure is developed for in-plane free vibration of rectangular and annular sectorial plates with general boundary conditions. For the annular sectorial plate, the introduction of a logarithmic radial variable simplifies the basic theory and the expression of the total energy. The coordinates, geometric parameters and potential energy for the two different shapes are organized in a unified framework such that a generalized solving procedure becomes feasible. By using the improved Fourier–Ritz approach, the admissible functions are formulated in trigonometric form, which allows the explicit assembly of global mass and stiffness matrices for both rectangular and annular sectorial plates, thereby making the method computationally effective, especially when analysing annular sectorial plates. Moreover, the improved Fourier expansion eliminates the potential discontinuity of the original normal and tangential displacement functions and their derivatives in the entire domain, and accelerates the convergence. The generalized Fourier–Ritz approach for both shapes has the characteristics of generality, accuracy and efficiency. These features are demonstrated via a few numerical examples.
Highlights
The in-plane vibration of built-up structures is found to have a significant effect on the sound radiation and transmission of vibration energies [1,2]
In-plane vibration analysis is important when inspecting the hulls of ships under the impacts
For the in-plane vibration of plate structures, several analytical solutions are developed, e.g. the variational method by Kantorovich–Krylov in [4], the superposition method by Gorman [5], the direct separation of variables and eigenvalue-problem approach by Xing and Liu [3,6], the strong form of the governing equation solved via a two-dimensional improved Fourier series by Du et al [7] and the Ritz method based on a set of trigonometric functions by Dozio [8], just to name a few
Summary
The in-plane vibration of built-up structures is found to have a significant effect on the sound radiation and transmission of vibration energies [1,2]. Wang et al [25] used a modified Fourier–Ritz approach [26,27] to solve the free in-plane vibration of orthotropic circular, annular and sectorial plates subjected to general boundary conditions. The modified Fourier series technique has been extended to study the in-plane vibration of plate and shell structures with general boundary conditions by the modified Ritz method [25,28,29,30,31]. In this large volume of literature, plates of different shapes are always treated separately and solved by different approaches. A few numerical examples are presented to demonstrate the versatility of the generalized approach
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