轴对称圆薄板大挠度微分方程的数值分析方法  

A numerical method to the nonlinear differential equations of thin circular plates with large deflections

作　　者：曹天捷[1] 

机构地区：[1]中国民航大学机场学院,天津300300

出　　处：《应用力学学报》2016年第3期427-433,546,共7页Chinese Journal of Applied Mechanics

摘　　要：根据轴对称问题的特点,利用级数展开和求极限法则,证明了轴对称大挠度圆薄板在圆心处应满足的边界条件,并以圆薄板轴对称大挠度弯曲变形微分方程为基础,建立了圆心处非奇异的轴对称大挠度圆板弯曲微分方程,从而可以方便地利用现有的常微分方程数值求解方法(如变步长龙格-库塔法)对实心圆板的轴对称问题进行数值求解,又不必像摄动法那样推导复杂的公式。在数值求解轴对称圆板大挠度弯曲变形微分方程时,将非线性微分方程的求解主要归结为迭代求解圆心处三个未知边界条件的问题,即圆心处的径向膜力、圆心处的挠度、圆心处挠度的二阶导数,并提出了相应的求解方法。实例中,对于圆薄板受均布横向荷载的问题,分析了周边固支边界条件下的非线性弯曲问题,给出了中心挠度参数大范围变化时的荷载和部分边界值变化曲线,并与经典摄动解进行了对比。对比结果可见,本文方法和摄动法的解非常接近,在量纲归一化中心挠度不超过4.0时,两种方法解的相对误差均小于5.0%。另外,本文还分析了与挠度有关的液体压力作用下和集中荷载作用下周边固支圆板的非线性弯曲问题。通过算例可见:本文方法可以灵活处理不同的荷载问题;对于不同的问题,计算过程相似,不必推导复杂的计算公式,计算精度容易控制。The boundary conditions at the center for large deflected thin circular plates are firstly presented by using Taylor's series expansion and the limit process. Based on the von Kármán equations, the nonlinear differential equations for the thin circular plates without singularity are then presented. Under the boundary conditions and the modified differential equations presented in the paper, the numerical solution to ordinary differential equations, such as Runge-Kutta method with variable step size, can be easily used to solve the thin circular plates with large deflections, and it is not necessary for one to derive the complicated formulas like the perturbation method. During solving the equations for large deflected thin circular plates, the key process is to solve for the three unknown boundary values at the center-the membrane force, the deflection and the second derivative of the deflection at the center, and a three-fold method of bisection is presented to solve for the three unknowns. In an example, a nonlinear thin circular plate clamped along the outer edge under uniform load is analyzed. Some results about dimensionless loads and boundary values are given and compared with those by classical perturbation method. It can be seen from the comparison that the solutions by the presented method and by classical perturbation method are quite close, and the relative errors between the results by the two methods are all less than 5.0% when the dimensionless deflection at the center is less than or equal to 4.0. In the other two examples, nonlinear thin circular plates clamped along the outer edge under the liquid pressure depending on the deflection of the plate and under a concentrated load at the center are also analyzed. It can be seen from the examples that the presented method can deal with problems with different loads easily, in which the procedures are quite similar, it needs not to derive complicated formulas and the precision can be controlled easily.

关 键 词：von Kármán方程 大挠度圆板 数值解 三重二分法 膜力 

分 类 号：O343.5[理学—固体力学]