Quantitative Analysis of Uncertainty at the End of the Towed Cable in Underwater Towing Systems
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摘要: 在多变的海洋环境中, 水下拖曳系统拖缆优化设计与拖体精确控制的关键在于拖缆末端不确定性的量化。针对传统不确定性量化方法蒙特卡罗(MC)法计算成本高、精度低的问题, 提出一种基于多项式混沌(PC)的拖缆末端不确定性量化方法。利用拉丁超立方采样获取拖缆参数的样本集, 并代入集中质量法模型求得拖缆末端位置坐标。通过PC方法生成拖缆末端响应的代理模型, 根据正交多项式的特点量化拖缆末端的不确定性, 同时与MC方法进行对比。结果表明: 相比于MC方法, PC方法的计算结果关于样本数量的收敛速度更快, 精度更高; 运动响应不确定性与拖缆轴向长度近似正比例关系; 缆长增大将导致末端的不确定性增大, 且增大趋势逐渐平缓; 拖缆参数不确定性一定时, 增大母船航速有助于提高拖体在高度上的稳定性。PC方法的准确性和高效性得到验证。同时, 拖缆末端不确定性量化分析结果可为相关工程问题提供指导。Abstract: In the ever-changing marine environment, the key to the optimal design of the towed cable and the precise control of the towed body in the underwater towing system is the quantification of uncertainty at the end of the towed cable. The Monte Carlo(MC) method, a traditional uncertainty quantification method, has high computation costs and low accuracy. In view of this, a method of uncertainty quantization at the end of a towed cable based on polynomial chaos(PC) was proposed. Latin hypercube sampling was used to obtain sample sets of the towed cable parameters, and the sample sets were substituted into the lumped-mass method model to obtain the coordinate of the end position of the towed cable. A proxy model of the end response of the towed cable was generated by the PC method, and the uncertainty of the end was quantified according to the characteristics of the orthogonal polynomials. At the same time, the results of the PC method were compared with those of the MC method. The results show that compared with the MC method, the PC method has a faster convergence speed in terms of sample size and higher accuracy. The uncertainty of motion response is approximately proportional to the axial length of the towed cable; the increase in cable length leads to the increase in uncertainty at the end, and the increasing trend is gradually flattened. When the uncertainty of the towed cable parameters is constant, increasing the speed of the mother ship helps to improve the stability of the towed body at height. The accuracy and efficiency of the PC method have been verified. Meanwhile, the quantitative analysis results of the uncertainty at the end of the towed cable guide engineering problems.
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图 2 末端质点坐标分量标准差计算结果
Figure 2. Calculation results of standard deviation of end particle coordinate components
图 3 误差指标随基函数阶数和样本数量的变化
Figure 3. Variation of error with order of basis function and number of samples
图 4 PC方法与MC方法末端质点坐标分量标准差计算结果
Figure 4. Comparison of calculation results of standard deviation of end particle coordinate components between PC method and MC method
图 5 末端质点标准差随轴向长度变化曲线
Figure 5. The standard deviation of the end particle changes with axial lengths
图 6 末端质点标准差随缆长变化曲线
Figure 6. Standard deviation of end particle changes with cable lengths
图 7 末端质点标准差随母船航速变化曲线
Figure 7. The standard deviation of the end particle changes with the speeds of the mother ship
表 1 不确定性量化考虑的8个主要输入参数
Table 1. Eight main input parameters considered for uncertainty quantification
变量 拖缆参数 数值 x1 缆长/m 3.66 x2 直径/m 0.003 05 x3 切向阻尼系数 0.051 1 x4 法向阻尼系数 1.2 x5 线密度/(kg/m) 0.4 x6 水流速率/(m/s) 1.542 x7 液体密度/(kg/m3) 1 024 x8 重力加速度/(m/s2) 9.8 
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表 2 拖缆参数分布
Table 2. Distributions of towed cable parameters
变量 分布 x1/m x1~N(3.66, 5.36×10−3) x2/m x2~N(3.05×10−3, 8.37×10−9) x3 x3~N (5.11×10−2, 2.35×10−6) x4 x4~N (1.2, 1.30×10−3) x5/(kg/m) x5~N (4×10−1, 1.44×10−4) x6/(m/s) x6~N (1.542, 2.14×10−3) x7/(kg/m3) x7~N (1.024×103, 9.44×102) x8/(m/s2) x8~N (9.8, 8.64×10−4)
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