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.