李银山 谢晨 霍树浩 马国伟

摘要 建立了集中载荷作用下大挠度悬臂梁的数学模型，利用椭圆函数法推导给出了集中载荷作用下大挠度悬臂梁的转角，挠度和水平位移的解析表达式。采用Maple编程求解了集中载荷作用下大挠度悬臂梁的非线性微分方程组。分析计算了自由端转角，挠度和水平位移与载荷的力学特征，给出了相应的解析表达式。分别绘制了大挠度理论下自由端转角、挠度和水平位移与载荷变化的关系曲线图，并与小挠度理论的解进行了对比。对集中载荷作用下大挠度悬臂梁的挠曲线形状进行了计算机仿真。列出了集中载荷作用下大挠度悬臂梁的自由端转角，挠度和水平位移随载荷变化的数值计算表。提供了集中载荷作用下大挠度悬臂梁的计算机仿真的Maple源程序代码。

关 键 词 集中载荷;悬臂梁;大挠度; Maple;计算机仿真

中圖分类号 O343.9;TP391.9     文献标志码 A

Abstract The mathematical model of large deflection cantilever beam under concentrated load is established. The analytical expressions of the angle， deflection and horizontal displacement of large deflection cantilever beam under concentrated load are derived by using elliptic function method. The nonlinear differential equations of large deflection cantilever beam under concentrated loads are solved by Maple programming. The mechanical characteristics of angle， deflection， horizontal displacement and load at the free end are analyzed and calculated， and the corresponding analytical expressions are given. The relationship curves of free end angle， deflection and horizontal displacement with load change under large deflection theory are drawn respectively， and compared with the solution of small deflection theory.The shape of the deflection curve of a cantilever beam with concentrated deflection under the concentrated load is simulated by computer. A numerical table for calculating the free-end angle， deflection and horizontal displacement of a large deflection cantilever beam under concentrated loads is presented. Maple source code for computer simulation of large deflection cantilever beam under concentrated load is provided.

Key words concentrated load; cantilever; large deflection; Maple; computer simulation

0 引言

在实际工程分析中，大部分实际工程受力部件都可以简化为悬臂梁。然而，很多时候弯曲杆件往往不是小变形，按照小挠度假设设计杆件截面往往过大，为了保证设计经济合理，采用梁的大挠度理论来设计计算。近年来，随着现代工业的发展，越来越多的工程问题涉及到大变形的情形。快速发展的航空航天领域中越来越多地使用大型柔性构件，而且能在发生大变形时而不超出弹性极限的复合材料越来越广泛地应用，悬臂梁结构大变形这一几何非线性问题已成为力学、机械及航空航天等领域研究的热点和难点问题之一。……