Kinematics of Hooke’s Joint

Abstract

Problem. An analysis of the literature has shown that some issues of the kinematics and dynamics of the Hooke’s joint are not fully covered. In particular, this applies to the determination of the kinematic characteristics of the relative movements of the links in the movable joints of the crosspieces and cardan yokes, as well as the acceleration of the crosspiece in the study of the hinge. These kinematic characteristics are the basis for dynamic analysis of the cardan transmission; they are used at research of operation of the bearings of a cardan joint cross; they are necessary to determine the energy expended to overcome the forces of friction in these kinematic pairs; they form the basis for evaluating transmission efficiency. Goal. The goal of this work is to determine the kinematic characteristics of the cross in its absolute and relative motions. Methodology. The research is based on the graphical and analytical method of kinematic analysis of spherical motion of a rigid body. At the initial stage, the calculated dependences for determining the angular velocity of the cross in its absolute and relative motions were obtained. The relative angles of rotation of the links in the movable joints of the crosspiece and cardan yokes were determined by integrating the corresponding differential equations. To determine the angular acceleration of the crosspiece, the theorem on the addition of angular accelerations of a rigid body rotating around two non-parallel axes was used. Results. The study of the kinematics of the cardan joint made it possible to identify and eliminate ambiguities in determining the relative angular velocities of the elements of the kinematic pairs of the cross and cardan yokes. Functional dependences for determination of angular acceleration of a crosspiece in absolute and relative movements, and also relative angles of rotation of links in kinematic pairs of a crosspiece and cardan yokes were received. It was shown that the angular acceleration of the crosspiece is a periodic function of the angle of rotation of the input link. During the operation of the hinge, the crosspiece creates insignificant periodic dynamic loads, which should be taken into account in accurate calculations of high-speed cardan transmissions. Practical value. The work is devoted to the study of the kinematics of the asynchronous hinge, which is the foundation for its dynamic analysis and ultimately aimed at improving the output performance of the cardan transmission. The research method combines mathematical calculations with simplicity and clarity of graphic constructions, which contributes to a deeper understanding of the kinematic features of a hinge of uneven angular velocities. The obtained results can be used in the design and study of motor-transmission units, which include the Hooke’s joint.