带裂纹矩形板自由振动解析解

选取带有补充项的双重正弦傅里叶级数作为振型函数通解,来解析研究带裂纹矩形板的自由振动特性。先将带裂纹矩形板分割成若干小矩形板,利用各小矩形板的边界条件,并结合振型函数中待定常数的物理意义,简化得到各小矩形板的振型函数,再结合各板的控制方程、未使用的边界条件、公共边协调条件及本文提出公共自由角点的协调条件,建立求解频率的代数方程组,然后将其转化为广义特征值问题来求解带裂纹矩形板的无量纲频率;最后选取具体参数进行计算并与文献结果对比,吻合良好,证明了本文采用的研究方法以及所提出公共角点协调条件的正确性和合理性。由于该振型函数能满足矩形板的任意边界约束,且其中的待定常数具有明确的物理意义,所以可使矩形板问题的求解统一化、简单化和规律化。

Analytical study of free vibration for cracked rectangular plate

A double sine Fourier series with supplementary functions is adopted to investigate a cracked rectangular plate.It is decomposed into several sub-domains and a simplified function for each sub-domain is derived.For each specific function,the boundary conditions and the physical meanings for constants in the mode functions are considered.The equation system,which contains the governing equations,the remaining boundary conditions,compatibility conditions for shared boundaries and the proposed equilibrium condition for shared free corners in this paper,is converted to a generalized eigenvalue problem to compute the non-dimensional frequencies.An example is calculated and the results in this paper are close to published data,which verifies the correctness of derivaition.Moreover,the validity of the proposed equilibrium condition for shared free corners is proved.The adopted function,whose canstant contain certain physical meanings,could conveniently investigates layered rectangular plates with aribitrary boundary conditions.Therefore,the method in this paper could unify the solution and simplify the approach for the study of rectangular plates.