Origin and Solution of Pre-crash Speed Calculation Error
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摘要: 为准确计算汽车碰撞事故碰撞前车速，基于动量守恒定理的汽车碰撞事故模型，在事故分析中采用反推法求解. 根据事故现场勘查信息,应用运动学公式计算碰撞后车速，再将碰撞后车速代入模型计算碰撞前车速. 以实车碰撞试验对模型计算结果进行检验后，发现用模型计算的碰撞前车速存在误差. 为此以实车碰撞试验为对象，根据模型的求解过程和误差的传递过程，研究了碰撞前车速误差的成因和处理方法，以提高交通事故分析的准确性. 首先，应用矩阵理论研究了碰撞前车速误差的形成原因；其次，应用反推迭代算法建立了碰撞前车速误差的处理方法；最后，通过实例应用验证了该方法的可行性. 研究结果和实例应用表明：用运动学公式计算碰撞后车速所产生的误差是造成碰撞前车速误差的决定性原因，只需对碰撞后车速误差进行简单的1次处理就能使应用该模型计算碰撞前车速所产生的误差归0.Abstract: In accident analysis vehicle collision model based on the momentum conservation is solved by the inverse algorithm to calculate pre-crash speed accurately. Then post-crash speed is calculated by a kinematics formula with survey data from accident scene, and is used to calculate the pre-crash speed by the model. In this work, the pre-crash speed error emerges when examining the model calculation results by a car crash test. From the model solving and error propagation processes, the origin and solution of the pre-crash speed error was studied by the car crash test to improve the accuracy of traffic accident analysis. Firstly, the matrix theory was applied to study the origin of the pre-crash speed error. Secondly, the inverse iterative algorithm was applied to establish a solution of the pre-crash speed error. Finally, an example was used to verify the solution. Study results and example applications show that the error in post-crash speed calculation with the kinematics formula is the decisive to the pre-crash speed error. Thus, the pre-crash speed error can be eliminated completely by simply correcting the post-crash speed error.
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Key words:
- pre-crash speed /
- error /
- post-crash speed /
- accident scene
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表 1 试验数据
Table 1. Trial data
试验参数 车辆1 车辆2 mj /kg 977 976 ρj /m 1.220 1.230 ljτ（aj）/m −0.530 1（−0.530 1） −0.075 2（−0.075 2） ljn（bj）/m −0.984 4（0.984 4） 0.739 4（−0.739 4） vjn /（m•s−1） −16.581 3（−16.581 3） −4.378 7（−4.378 7） vjτ /（m•s−1） −2.284 8（2.284 8） 17.790 0（−17.790 0） ωj /（rad•s−1） −5.347 9（5.347 9） −8.021 1（8.021 1） vj0n /（m•s−1） −19.598 4（−19.598 4） 0（0） vj0τ /（m•s−1） −12.246 4（12.246 4） 22.780 0（−22.780 0） ωj0 /（rad•s−1） 0（0） 0（0） 注：括弧中的数值对应模型2. 
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表 3 碰撞前车速误差的处理过程
Table 3. Correction process of pre -crash speed
模型 力学参数 算法 碰撞后车速 碰撞后车速误差 碰撞前车速 试验结果 碰撞前车速误差 1 μ=−3.651 293
k=−0.736 506反
推
法${ {{v} }_1} = \left[ {\begin{array}{*{20}{c} } { - 16.581\;3} \\ { - 2.284\;8} \\ { - 4.378\;7} \\ {17.790\;0} \\ { - 5.347\;9} \\ { - 8.021\;1} \end{array} } \right]$ $\begin{aligned}& { {{v} }_1} - { {{v} }_{11} } = \\& \left[ {\begin{array}{*{20}{c} } {1.470\;7} \\ {15.608\;0} \\ { - 2.830\;7} \\ { - 10.642\;2} \\ { - 9.633\;1} \\ { - 10.706\;6} \end{array} } \right]\end{aligned}$ ${ {{v} }_{01} } = \left[ {\begin{array}{*{20}{c} } { - 18.127\;7} \\ {3.361\;6} \\ { - 2.830\;7} \\ {12.137\;8} \\ { - 9.633\;1} \\ { - 10.706\;6} \end{array} } \right]$ $\begin{aligned}& { {{v} }_{01{\rm{T} } } } = \\& \left[ {\begin{array}{*{20}{c} } { - 19.598\;4} \\ { - 12.246\;4} \\ 0 \\ {22.780\;0} \\ 0 \\ 0 \end{array} } \right]\end{aligned}$ ${ {{v} }_{01} } - { {{v} }_{01{\rm{T} } } } = \left[ {\begin{array}{*{20}{c} } {1.470\;7} \\ {15.608\;0} \\ { - 2.830\;7} \\ { - 10.642\;2} \\ { - 9.633\;1} \\ { - 10.706\;6} \end{array} } \right]$ 处
理${ {{v} }_{11} } = \left[ {\begin{array}{*{20}{c} } { - 18.052\;0} \\ { - 17.892\;8} \\ { - 1.548\;0} \\ {28.432\;2} \\ {4.285\;2} \\ {2.685\;5} \end{array} } \right]$ ${ {{v} }_{011} } = \left[ {\begin{array}{*{20}{c} } { - 19.598\;4} \\ { - 12.246\;4} \\ 0 \\ {22.780\;0} \\ 0 \\ 0 \end{array} } \right]$ ${{{v}}_{011}} - {{{v}}_{01{\rm{T}}}} = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}} \right]$ 2 εn=
−0.508 759
ετ=
−0.892 757反
推
法${ {{v} }_2} = \left[ {\begin{array}{*{20}{c} } { - 16.581\;3} \\ {2.284\;8} \\ { - 4.378\;7} \\ { - 17.790\;0} \\ {5.347\;9} \\ {8.021\;1} \end{array} } \right]$ $\begin{aligned}& { {{v} }_2} - { {{v} }_{21} } = \\& \left[ {\begin{array}{*{20}{c} } { - 1.619\;3} \\ { - 8.232\;5} \\ {0.262\;4} \\ {3.259\;1} \\ {4.840\;2} \\ {9.097\;7} \end{array} } \right]\end{aligned}$ ${ {{v} }_{02} } = \left[ {\begin{array}{*{20}{c} } { - 21.217\;7} \\ {4.013\;9} \\ {0.262\;4} \\ { - 19.520\;9} \\ {4.840\;2} \\ {9.097\;7} \end{array} } \right]$ $\begin{aligned}& { {{v} }_{02{\rm{T} } } } = \\& \left[ {\begin{array}{*{20}{c} } { - 19.598\;4} \\ {12.246\;4} \\ 0 \\ { - 22.780\;0} \\ 0 \\ 0 \end{array} } \right]\end{aligned}$ ${ {{v} }_{02} } - { {{v} }_{21{\rm{T} } } } = \left[ {\begin{array}{*{20}{c} } { - 1.619\;3} \\ { - 8.232\;5} \\ {0.262\;4} \\ {3.259\;1} \\ {4.840\;2} \\ {9.097\;7} \end{array} } \right]$ 处
理
1${ {{v} }_{21} } = \left[ {\begin{array}{*{20}{c} } { - 14.962\;0} \\ {10.517\;3} \\ { - 4.641\;1} \\ { - 21.049\;1} \\ {0.507\;7} \\ { - 1.076\;6} \end{array} } \right]$ ${ {{v} }_{021} } = \left[ {\begin{array}{*{20}{c} } { - 19.598\;3} \\ {12.246\;4} \\ 0 \\ { - 22.780\;0} \\ {0.000\;1} \\ 0 \end{array} } \right]$ ${ {{v} }_{021} } - { {{v} }_{02{\rm{T} } } } = \left[ {\begin{array}{*{20}{c} } {0.000\;1} \\ 0 \\ 0 \\ 0 \\ 0.000\;1 \\ 0 \end{array} } \right]$ εn1=
−0.508 756
ετ1=
−0.892 759处
理
2${ {{v} }_{21} } = \left[ {\begin{array}{*{20}{c} } { - 14.962\;0} \\ {10.517\;3} \\ { - 4.641\;1} \\ { - 21.049\;1} \\ {0.507\;7} \\ { - 1.076\;6} \end{array} } \right]$ ${ {{v} }_{022} } = \left[ {\begin{array}{*{20}{c} } { - 19.598\;4} \\ {12.246\;4} \\ 0 \\ { - 22.780\;0} \\ 0 \\ 0 \end{array} } \right]$ ${{{v}}_{022}} - {{{v}}_{02{\rm{T}}}} = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}} \right]$
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表 4 力学参数误差
Table 4. Mechanical parameter error
力学参数 原始值 准确值 相对误差/% μ −3.651 293 −3.651 293 0 k −0.736 506 −0.736 506 0 εn −0.508 759 −0.508 756 0.000 511 ετ −0.892 757 −0.892 759 0.000 224
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