This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

中文翻译：

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具有表面或内部长裂纹的各向同性板的自由振动的广义微分积分法分析

本文基于基尔霍夫理论，提出了一种分析具有各种边界条件的裂纹各向同性板的自由振动的新方法。假设各向同性板具有部分贯通表面或内部裂纹。裂纹被认为平行于板边缘之一。裂纹的存在修改了基于线弹簧模型制定的控制微分方程。广义微分求积法将得到的控制微分方程离散化，转化为代数方程组。然后，使用特征值分析来确定裂纹板的固有频率。给出了一些数值结果来证明所得结果的准确性和收敛性。为了证明该方法的有效性，结果与有限元解决方案和现有文献进行了比较。此外，还研究了裂纹深度、裂纹沿厚度位置、裂纹长度和不同边界条件对固有频率的影响。