变刚度Winkler地基上受压非均质矩形板的自由振动与屈曲特性

基于经典薄板理论,利用Hamilton原理建立变刚度Winkler地基上受压非均质矩形板自由振动与屈曲问题的控制微分方程并进行无量纲化。通过一种半解析方法-微分变换法(DTM)研究其无量纲固有频率和屈曲临界载荷特性。采用DTM将其无量纲控制微分方程及边界条件变换为等价的代数方程,得到含有频率和屈曲载荷的特征方程。将该问题退化为面内变刚度矩形板情形,其DTM解与精确解进行对比,结果表明DTM具有非常高的精度和很强的适用性。计算出在不同边界条件下屈曲临界载荷并分析地基刚度变化参数、弹性模量变化参数、密度变化参数、面内载荷和长宽比对矩形板无量纲固有频率的影响,给出了不同边界条件下变刚度Winkler地基上受压非均质矩形板的前三阶振型。

Free vibration and buckling characteristics of compressed non-homogeneous rectangular plates on Winkler foundation with variable stiffness

Based on the classical thin plate theory,the governing differential equation of free vibration and buckling of a compressed non-homogeneous rectangular plate on Winkler foundation with variable stiffness was established by using Hamilton’s principle,and then its dimensionless form was obtained.The characteristics of the plate’s dimensionless natural frequencies and buckling critical loads were studied with a semi-analytical method called the differential transformation method(DTM).DTM was used to convert dimensionless governing differential equation and boundary conditions into equivalent algebraic equations,and derive characteristic equations of frequencies and buckling loads.Then,the problem was degenerated into the case of an in-plane variable stiffness rectangular plate,and its DTM solution was compared with the analytical solution.The results showed that DTM have very higher accuracy and stronger applicability.Finally,the buckling critical loads were calculated under different boundary conditions,and the effects of foundation stiffness parameters,elastic modulus parameters,density parameter,in-plane loads and length-width ratio on the plate’s dimensionless natural frequencies were analyzed.The first three modal shapes of the compressed non-homogeneous rectangular plate on Winkler foundations with variable stiffness were deduced under different boundary conditions.