Geometric Problems On The Motion Of Rigid Bodies In Plane

This paper focuses on the instantaneous center and instantaneous center line on plane moving rigid body. During the study, we firstly use the differential geometry method to analyze the trajectory of rigid body and study the position of instantaneous center on relative motion of a pair of rigid body in plane, and obtain the representation formula of instantaneous center. Secondly, by discussing about instantaneous center of four-bar mechanism, we verify the correctness of this formula and expound the significance of it. Based on the instantaneous center formula, we calculate the instantaneous velocity and instantaneous acceleration.This research changes the study methods on instantaneous center before. By using the formula, it completely gets rid of drawing instantaneous center graph to study the instantaneous center problems. Moreover, in the process of solving the instantaneous center, it don’t have to consider forms about rigid body motions and the possible situations of instantaneous center producing. Under some kind of moving forms, given the trajectory of a rigid body and the instantaneous origin coordinate of moving frame, we can successfully get the exact instantaneous center coordinate about this movement by the formula. This approach can simplify the solving process of instantaneous center on complex mechanisms and solves the entire instantaneous centers concisely, intuitively and accurately.The paper also studies the Euler-Savary formula and provides its proving process based on vector and its geometric significance. With the instantaneous center formula, the paper derives a very important physical concept-inflection circle. On this basis, an identical equation about curvature of instantaneous center line is proved. Finally, the article illustrates the concept of Burmester point, discusses it preliminarily and derives an equation which must be satisfied under some conditions.