非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

Free vibration analysis for rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation by DTM

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法(DTM),研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。

For free vibration problem of rectangular plates with variable thickness resting on a non-uniform foundation and by an effective solving numerical method called differential transformation method(DTM),the dimensionless natural frequency characteristics are investigated.Two opposite edges of plates are assumed to be simply supported and other two edges can be changed into simply supported,camped or free boundary conditions arbitrarily.By using DTM,dimensionless normalized governing differential equation of rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation and boundary conditions are transformed to the equivalent algebraic equations,which can derive equations of dimensionless natural frequency.The example results are back to cases for uniform Winkler rectangular plates and rectangular plates with variable thickness,which are compared with different methods in present literature.The result shows that DTM have very higher accuracy and stronger applicability.Finally,the influence of the varied foundation parameter,the varied thickness parameter and the aspect ratio on dimensionless natural frequencies are analyzed for different boundary conditions and deriving the first six mode shapes for CSCS plate with variable thickness resting on a non-uniform Winkler elastic foundations.