Modeling and simulation of nonlinear motion of ship heave and pitch in head sea
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摘要: 为了提高航海模拟器的行为真实感，使用混合Green函数法构建船舶运动数学模型，对船舶垂荡运动和纵摇运动进行求解。基于已有Rankine源数学模型，通过人为引入虚拟控制面将计算域分为内外域两部分，内域使用Rankine源法，外域使用Green函数法，构建混合Green函数三维时域线性数学模型并对船舶运动的波浪力进行求解，对Wigley Ⅰ型船不同方法的仿真结果进行分析；为进一步考虑非线性因素对船舶航行的影响，基于四叉树法对船体网格进行动态生成，对船舶运动的傅汝德-克雷洛夫(F-K)力和静恢复力进行求解，对Wigley Ⅰ型船不同的波长船长比和不同方法的仿真结果进行了可行性分析。研究结果表明：提出的线性数学模型计算效率远高于Rankine源法，垂荡仿真结果与试验结果误差为10.86%，纵摇仿真结果与试验结果误差为14.28%；而当非线性数学模型波长船长比为1.25时，计算所得的垂荡F-K力幅值结果、纵摇F-K力幅值结果与Green函数非线性时域计算所得的计算结果相差均较小，误差都在5.00%以内，与三维线性时域相比误差较大，误差在30.00%以内；当非线性数学模型波长船长比为2时，计算所得的垂荡F-K力幅值结果、纵摇F-K力幅值结果与Green函数非线性时域计算所得的计算结果相差均较小，其误差都在3.00%以内，与三维线性时域相比误差较大，误差在20.00%以内；由于非线性方法需要在瞬时湿表面上计算，而线性方法在平均湿表面上计算，导致垂荡F-K力计算结果相差较大；与试验结果对比，计算所得的垂荡幅值响应因子和三维线性时域方法均和试验结果误差不大，误差均在20.00%以内；相对于纵摇幅值响应因子，波长船长比为1.75时存在共振现象导致2种方法误差均较大，波长船长比不等于1.75时，提出的方法误差明显小于三维时域方法误差。建立的三维时域非线性数学模型可以应用在航海模拟器上，可用于航海动态仿真数值分析。Abstract: The hybrid Green function method was adopted to build a mathematical model of ship motion, and the heave motion and pitch motion of the ships were solved to improve the behavioral realism of navigation simulators. Based on the existing Rankine source mathematical models, the computational domain was divided into an inner and outer domain by artificially introducing a virtual control surface. The inner domain employs the Rankine source method, and the outer domain utilizes the Green function method. A 3D time-domain linear mathematical model of the hybrid Green function was built and the wave force of ship motion was solved. The simulation results of the Wigley Ⅰ hull with different methods were analyzed. The hull grid was dynamically generated based on the quadtree method, the Froude-Krylov (F-K) force and hydrostatic restoring force of ship motion were solved, and feasibility analysis was conducted on the simulation results of the Wigley Ⅰ hull with different wave length-to-ship length ratios and methods to further consider the influence of nonlinear factors on ship navigation. The results show that the computational efficiency of the proposed linear mathematical model is much higher than that of the Rankine source method. The error between the heave simulation results and experimental results is 10.86%, and the error between the pitch simulation results and experimental results is 14.28%. When the wave length-to-ship length ratio in the nonlinear mathematical model is 1.25, the heave F-K force amplitude results and pitch F-K force amplitude results calculated by this paper are slightly different from those calculated by Green function nonlinear time-domain calculation, and the errors are all within 5.00%. Compared with the 3D linear time domain, the errors are large and within 30.00%. When the wave length-to-ship length ratio in the nonlinear mathematical model is 2, the heave F-K force amplitude results and pitch F-K force amplitude results calculated by this paper are slightly different from those calculated by the Green function nonlinear time-domain calculation, and the error is within 3.00%. Compared with the 3D linear time domain, the error is large and within 20.00%. The nonlinear method needs to be calculated on the instantaneous wet surface, while the linear method is calculated on the average wet surface, thereby resulting in large errors in the calculation results of the heave F-K force. Compared with the experimental results, there is little error between heave amplitude response factor and 3D linear time-domain method in this paper and the experimental results, and the errors are all within 20.00%. Compared with the pitch amplitude response factor, under the wave length-to-ship length ratio of 1.75, resonance occurs, resulting in large errors in both methods. When the wave length-to-ship length ratio is not equal to 1.75, the error of the proposed method is significantly smaller than that of the 3D time-domain method. The built 3D time-domain nonlinear mathematical model can be applied to navigation simulators and can be adopted for the numerical analysis of navigation dynamic simulation.
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Key words:
- ship engineering /
- navigation simulator /
- hybrid Green function /
- ship seakeeping /
- Wigley Ⅰ hull /
- F-K force /
- quadtree
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图 4 基于非线性时域法船舶运动数值求解
Figure 4. Numerical solution of ship motion based on nonlinear time-domain method
图 10 四叉树法船体湿表面网格示意
Figure 10. Schematic of the wet surface grid of the hull using the quadtree scheme
图 21 不同船舶长宽比的垂荡运动时历
Figure 21. Time history of heave motion when ships with different length-to-width ratios
图 22 不同船舶长宽比的纵摇运动时历
Figure 22. Time history of pitch motion when of ships with different length-to-width ratios
表 1 Wigley Ⅰ型船主尺度
Table 1. Main particulars of Wigley Ⅰ hull
参数名称 值 参数名称 值 垂线间长/m 3 纵摇惯性半径/m 0.75 船宽/m 0.3 重心距离基线距离/m 0.17 吃水/m 0.187 5 干舷吃水比 1 排水体积/m3 0.094 6 船中剖面系数 0.909 
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表 2 Fn=0.2, Wigley Ⅰ型船运动时历求解时间对比
Table 2. Comparison of time history solution times for Wigley Ⅰ hull ship motion at Fn=0.2
方法 网格划分
时间/s速度势求解
时间/s波浪力计算
时间/s本文方法 1.53 1 191.99 0.54 Rankine源法 5.53 148 766.04 1.77
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表 3 当Fn=0.2，Wigley Ⅰ型船垂荡运动RAO与试验值相对误差
Table 3. Relative error of Wigley Ⅰ hull's heave motion RAO compared with experimental value at Fn=0.2
λ/L 1.25 1.40 1.50 1.60 1.75 2.00 相对误差/% 本文方法 6.40 2.25 2.08 1.25 2.58 3.03 Green函数法 6.98 0.11 13.65 0.31 1.34 0.71 三维线性时域 15.70 9.55 9.06 7.81 6.80 6.67
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表 4 当Fn=0.2，Wigley Ⅰ型船纵摇运动RAO与试验值相对误差
Table 4. Relative error of Wigley Ⅰ hull's pitch motion RAO compared with experimental value at Fn=0.2
λ/L 1.25 1.40 1.50 1.60 1.75 2.00 相对误差/% 本文方法 4.24 12.90 5.31 0.21 28.44 10.29 Green函数法 3.21 4.11 10.88 4.32 27.71 9.41 三维线性时域 16.97 17.74 21.95 16.11 35.87 13.53
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