相邻两边固支其余两边自由矩形正交各向异性薄板弯曲的辛叠加解

Symplectic superposition solution for bending problem of orthotropic rectangular plate with two adjacent edges free and the others clamped

应用辛叠加方法,求出相邻两边固支其余两边自由矩形正交各向异性薄板在均匀荷载下弯曲问题的解析解。先将原方程转换成Hamilton正则方程;用辛方法计算出一边滑支对边简支条件下对应Hamilton算子的辛本征值和辛本征函数系,证明该辛本征函数系的辛正交性,进而在Cauchy主值意义下证明了它的完备性;根据辛本征函数系的完备性,得到了对应Hamilton正则方程的通解,再应用叠加方法计算出原弯曲问题的解析解;最后通过两个具体矩形薄板的数值算例,验证了所得辛叠加解的正确性。

The bending problem of orthotropic rectangular thin plate under a uniform load with two adjacent edges free and the others clamped is solved by the symplectic superposition method. First, the eigenvalues and eigenfunction system of Hamiltonian operator with one side simply supported and the opposite side slidinglysupported are calculated by the symplectic method, and the symplectic orthogonality of the eigenfunction system and its completeness in the sense of Cauchy’s principal value are proved. According to the completeness of the eigenfunction system, the general solution of the corresponding Hamiltonian canonical equations is obtained. Then, the analytical solution of the bending equation of orthotropic rectangular thin plate under a uniform load with two adjacent edges free and the others clamped is derived by the superposition method. Finally, the validity of the obtained analytical solution is verified by two specific numerical examples.