经典悬链线理论精确解与近似解的非线性数值计算

Nonlinear numerical computation of exact and approximate solutions of classical catenary theory

针对经典悬链线数学解中存在两个未知参数,即水平张力h和广义倾角α迄今尚未妥善解决的问题,进行了深入细致的分析。利用悬链线两点边值约束条件和不可拉伸假设,推导出求解隐含独立未知量水平张力的超越方程。引进互逆的无量纲参数求解超越方程中的水平张力,使得水平张力形式上具有最简单的参数依赖关系。探讨了广义倾角β,α和θ与几何参数的相互关系,得出广义倾角α不是独立未知参数的结论。提出了水平距离趋于0和趋于极限距离的各种近似解、在真小数全局计算范围内的近似解以及这些近似解关于精确解的误差程度,其结果在工程上具有应用价值。

There are two unknown parameters in the mathematic solution of classical catenary,namely horizontal tension h and generalized angleα.They are analysed in detail.By using the constraint condition of the two-point boundary value problem and the assumption of non-extension,a transcendental equation to solve the implicit and independent unknown horizontal tension is deduced.A group of reciprocal dimensionless parameters are adopted,which leads to the simplest expression the horizontal tension as a function of relevant parameters.The interrelation of the generalized angleβandαas well asθwith the geometric parameters is discussed,and it is concluded that generalizedαis not an independent unknown parameter.The authors put forward a number of approximate solutions which can simulate the situations when horizontal distances tend to zero,or to the limit distance or vary within the scope of the global calculation of true decimal,also have discussed the degree of accuracy of approximate solutions in relation to the exact solution.These mathematic solutions of classical catenary are of great significance in engineering.