Bao Siyuan, Wang Shuodao
School of Civil Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, China.
School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078, USA.
R Soc Open Sci. 2017 Aug 16;4(8):170484. doi: 10.1098/rsos.170484. eCollection 2017 Aug.
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.
针对具有一般边界条件的矩形和环形扇形板的面内自由振动，开发了一种通用的求解方法。对于环形扇形板，引入对数径向变量简化了基本理论和总能量的表达式。将两种不同形状的坐标、几何参数和势能组织在一个统一的框架中，使得通用的求解方法变得可行。通过使用改进的傅里叶 - 里兹方法，允许函数以三角函数形式表示，这使得能够明确组装矩形和环形扇形板的全局质量矩阵和刚度矩阵，从而使该方法在计算上有效，特别是在分析环形扇形板时。此外，改进的傅里叶展开消除了原始法向和切向位移函数及其导数在整个域中的潜在不连续性，并加速了收敛。针对这两种形状的广义傅里叶 - 里兹方法具有通用性、准确性和效率的特点。通过一些数值例子证明了这些特性。
