AXOIDS OF HOOKE’S JOINT

Abstract

The analysis of literary sources devoted to the research and construction of cardan transmissions using the Hooke's joint showed that the peculiarities of the spherical movement of the cardan joint crosspiece are insufficiently covered. This is explained by the relatively small weight of the crosspiece, due to which it almost does not affect the dynamics of the cardan transmission. The need to take into account the mass-geometric parameters of the crosspiece and to analyze comprehensively its movement may arise during accurate calculations of high-speed cardan transmissions. In this case, it is convenient to consider the movement of the crosspiece as rolling without sliding of the moving axoid on the stationary axoid. This approach makes it possible to more fully investigate the peculiarities of the spherical movement of the crosspiece and gives a visual representation of the orientation of its instantaneous axis of rotation in fixed and moving coordinate systems. Based on this, this work is devoted to the construction of a kinematic model of the Hooke's joint crosspiece, the movement of which is considered as rolling without sliding of a moving axoid on a stationary one.
 As a result of the kinematic analysis of the Hooke's joint, the equations of the hodographs of the vector of the absolute angular velocity of the crosspiece in the fixed and moving coordinate systems were obtained, and the parameters of the fixed and moving axoids of the crosspiece of the cardan joint were determined. It is proved that the stationary axoid of the crosspiece of the Hooke's joint is an oblique elliptic cone with the apex in the center of the crosspiece, and its opposite generatrices, crossing the major axis of the guide ellipse, coincide with the axes of the drive and driven shafts of the joint. The moving axoid of the crosspiece is a closed conical surface with a variable opening angle, the maximum value of which is equal to twice the value of the interaxial transmission angle. With small values of the interaxial angle, which occur in the practice of using a single Hooke’s joint, a moving axoid can be approximately considered as a straight circular cone with a vertex in the center of the crosspiece and an opening angle equal to twice the value of the interaxial transmission angle. The error of this assumption increases rapidly with the increase of the interaxial transmission angle. The correctness of the obtained mathematical model was confirmed by computer modeling in the Autodesk Inventor environment.
 The obtained results can be used in the study of dynamic processes that occur in cardan transmissions using Hooke's joint.