果实振动响应时的空中运动数学模型

现有的高速摄影技术主要用于研究果实运动时以果实表面某一特征点作为果实质心的运动状态,该方法实际上只能反映出果实表面特征点的运动轨迹及瞬时位移、速度及加速度,并不能反映果实质心在空间瞬时的平移、摆动及旋转姿态。该文提出了一种将果实空间运动分解为对应果实空间运动瞬时姿态的平移、摆动及旋转的计算方法。通过制作实体单位连体基坐标系并确定其初始静态位置,建立果实在绝对坐标系中的表面特征点坐标与连体基坐标系的转换关系,确定在运动过程中果实上连体基坐标的动态绝对坐标,基于相邻两时刻点连体基坐标中的位置变化关系计算果实瞬时动态位移、速度及加速度,以及果实摆动与旋转的瞬时角度、角速度及角加速度运动参数。应用ADAMS计算软件,通过设定特定的平移、摆动及旋转的复合运动关系进行运动仿真,应用该文构建的计算公式进行计算,将计算结果与理论仿真值进行对比,确定计算公式的计算精度。位移最大单向平均绝对误差只有5.9×10~(-8) mm,且位移、速度及加速度的绝对误差存在10~3数量级的逐步放大,位移与速度的相对误差完全一致,加速度相对误差则大于位移与速度,最大加速度平均绝对误差与平均相对误差分别为6.5×10~(-2) mm/s~2及4.13×10~(-2)%,摆动与旋转的最大平均绝对误差分别为5.43×10~(-2)与9.51×10~(-2)°/s~2。结果表明,该文构建的计算方法应用于求解果实的瞬时运动姿态是可行的。

Mathematical model of fruit's aerial movement in vibration response

High-speed photography which can take a lot of pictures catching the movement state of the fruit in rapid motionin during a very short time has become the mainstream method to study fruit motion. The mean limitation of the existing researchs is that tracking movement of one feature point on fruit surface can not reflect the exact movement of the fruit,especially swing and rotation. In this paper, We divided the spatial motion of the fruit into the translation, swing and rotation of the instantaneous attitude corresponding fruit spatial motion and proposed a method to calculate 3 kinds of spatial motion of fruit by 3 feature points on fruit surface. First, we constructed a coordinate system called characteristic point coordinates with 2 space vectors from 3 surface feature points. Second, we assumed that the fruit was a rotating body. Then, the coordinate system which we called the conjoined base coordinate system was established with the origin of the joint of the fruit stalk ,in which the Z-axis coincides with the rotation axis of the fruit. The movement of the fruit was represented by the movement of the conjoined base coordinate origin. The swing of the fruit was represented by the swing of the Z-axis of the conjoined base coordinate system. The rotation of the fruit was represented by the rotating around Z-axis of the Y-axis or X-axis in the conjoined base coordinate system. The third, we got the transformation relationship between 2 coordinate systems through the space vector relationship. Final, we used the spatial coordinates of the feature points and coordinate relationship gotten above to calculate the position and attitude of conjoined base coordinate frame by frame. Conjoined base coordinate of 2 adjacent time could calculate the instantaneous dynamic displacement, speed and acceleration of the fruit, as well as the motion parameters such as instantaneous angle, angular speed and angular acceleration for fruit swing and rotation.The ADAMS computing software was used to simulate the motion by setting a specific complex motion relationship of translation, swing and rotation. The calculation formula constructed in this paper was used for calculation and the calculation results were compared with the theoretical simulation values to determine the calculation accuracy of the calculation formula. The maximum one-way average absolute error of displacement was only 5.9×10-8 mm, and the absolute errors of displacement, speed and acceleration were enlarged gradually by the order of magnitude of 103. The relative errors of displacement and speed were identical, while the relative error of acceleration was larger than that of displacement and speed. The average absolute error and average relative error of maximum acceleration were 6.5×10-2 mm/s2 and 4.13×10-2%, respectively. The maximum average absolute errors of swing and rotation were 5.43×10-2 and 9.51×10-2 °/s2, respectively. The results show that the method constructed in this paper is feasible to solve the instantaneous motion attitude of the fruit.