函数传递法求解悬链线问题

马艳红; 时成龙; 洪杰; 李超

doi:10.13700/j.bh.1001-5965.2020.0367

函数传递法求解悬链线问题

doi: 10.13700/j.bh.1001-5965.2020.0367

马艳红^{1, 2, ,},
时成龙¹,
洪杰^{1, 2},
李超¹

1.
北京航空航天大学能源与动力工程学院, 北京 100083
2.
先进航空发动机协同创新中心, 北京 100083

详细信息

通讯作者:
马艳红, E-mail: mayanhong@buaa.edu.cn

中图分类号: V239
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- 被引次数: 0
出版历程
- 收稿日期: 2020-07-29
- 录用日期: 2020-09-11
- 网络出版日期: 2021-10-20

Solving catenary problem using function transfer method

MA Yanhong^{1, 2
, ,},
SHI Chenglong¹,
HONG Jie^{1, 2},
LI Chao¹

1.
School of Energy and Power Engineering, Beihang University, Beijing 100083, China
2.
Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100083, China

More Information

Corresponding author: MA Yanhong, E-mail: mayanhong@buaa.edu.cn

摘要

摘要:
悬链线问题是一类经典又多变的力学问题，其曲线构形指导着桥梁等工程应用的结构设计。为了求得结构特征或载荷特征不同的悬链线的变形，提出了一种基于传递矩阵法思想的、通用性较好的求解方法。首先，提炼出悬链力学模型，将悬链顺序划分成若干简单单元，结合单元力平衡和本构-几何关系解得单元特征函数方程组，顺序嵌套单元特征函数方程组获得整体特征函数方程组；然后，使用离散Newton迭代法对该非线性方程组进行求解，获得悬链的受力和变形；最后，算例验证了结果与解析解的一致性。函数传递法对具有复杂结构特征和载荷特征的悬链线问题有很好的适用性，对求解其他可划分为若干首尾相接结构单元的结构系统的广义变形也适用。
- 函数传递法 /
- 传递矩阵法 /
- 悬链线 /
- Newton迭代法 /
- 广义变形
Abstract:
Catenary problem is a kind of classical and changeable mechanical problem, whose curve configuration guides the structural design of engineering applications such as bridge. In order to obtain the deformation of catenary with different structural or load characteristics, a general method based on transfer matrix method is proposed. The catenary mechanics model is extracted, and the catenary is divided sequentially into several simple elements. The characteristic function group of element state parameters is obtained by combining element force balance and constitutive-geometry relationship. The whole characteristic function group is obtained by nesting element characteristic function group sequentially. Then the discrete Newton iterative method is used to solve the equations of nonlinear whole characteristic function group, and finally the forces and deformation of catenary are obtained. An example showed that the results were consistent with the analytical solution. The function transfer method is applicable to the catenary problems with complex structural characteristics and load characteristics, and also applicable to solving the generalized deformation of other structural systems which can be divided into several head-to-tail structural elements.
- function transfer method /
- transfer matrix method /
- catenary /
- Newton iterative method /
- generalized deformation

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图 1 悬链结构系统力学模型

Figure 1. Mechanical model of cable chain structure system

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图 2 单元受力分析

Figure 2. Load analysis of element

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图 3 结构变形求解流程

Figure 3. Flowchart for solving structure deformation

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图 4 刚体杆件单元受力分析

Figure 4. Load analysis of rigid rod element

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图 5 悬链线解析解与数值解曲线

Figure 5. Analysis and numerical solution curves of catenary

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图 6 两组悬链线解

Figure 6. Two groups of catenary solutions

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图 7 不同跨长比的悬链线

Figure 7. Catenary with different span/length ratios

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图 8 悬链线构形参数间的变化关系

Figure 8. Relationship between configuration parameters of catenary

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图 9 经典悬链线解与附加重物悬链线解

Figure 9. Classical catenary solution and catenary with additional weight

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图 10 经典悬链线解与带弹性悬链线解

Figure 10. Classical catenary solution and catenary with elasticity

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图 11 经典悬链线与悬索桥曲线

Figure 11. Classical catenary and suspension bridge curve

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图 12 经典悬链线与跳绳曲线

Figure 12. Classical catenary and rope skipping curve

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表 1 常见边界条件参数

Table 1. Common boundary condition parameters

边界条件	挠度	挠角	弯矩	剪力
简支	0	θ_x	0	F_x
自由	y_x	θ_x	0	0
弹性支承	y_x	θ_x	0	K_springy_x

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表 2 悬链算例参数

Table 2. Example parameters of catenary

参数	数值
跨度L_span/m	1.5
索链总长L_line/m	2.0
横截面圆半径R_line/m	1.0×10^-3
材料密度ρ/(kg·m^-3)	7 800

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表 3 悬链算例结果

Table 3. Example results of catenary

解析/数值解	高低差Δy/m	索端水平拉力F_x-end/N	索端角θ_end/(°)
解析解	0.588 6	0.0137 8	60.96
数值解	0.589 5	0.0135 9	59.72
相对误差/%	0.15	-1.4	-2.0

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