Kinematic And Contact Analysis Of Complex Joints In Multibody Systems

In traditional dynamics analysis of multibody systems, joint functions are replaced by a set of algebraic kinematic constraint equations. At the same time, joint masses are assigned to the inboard and outboard bodies of the joint. However, in order to reduce the friction effect, improve the accuracy of constraints and achieve certain special purposes, the joints of actual mechanical systems usually have complicated structures, such as rolling bearings in rotating machinery, constant velocity universal joints in vehicle drive systems, etc. All of these joints have some components which are not fixed on the inboard and outboard bodies, including steel balls, cage and ball cage, etc. The existence of these additional parts increases the difficulty of kinematic and dynamic contact analysis of these complex joints. In traditional contact analysis methods, the details of relative motion in clearance joints are necessary to locate contact points. The kinematic constraint must be realeased and continuous contact force models are used for calculating impact forces, no matter how small the joint clearance is. In practice, the mechanical joint clearance is uaually in the order of micrometer magnitude. Therefore, it is not optimal that a very small time step size is required. In addition, the contact stiff and damping parameters are difficult to choose without a doubtless theory. This paper proposes the contact analysis method for small clearance joints in terms of joint reaction forces, and investigates dynamic contact analysis of rolling bearing revolute joint. In consideration of assembly with no clearance in Rzeppa constant velocity universal joints, kinematic analysis and simulation of the joints are studied based on kinematics of multibody systems.Since the joint clearances of many actual mechanical systems are relatively small, the kinematic constraints of the joints can be achieved with good accuracy. Therefore, the impact effects on the kinematical behavior of mechanical systems can be ignored. Joint reaction forces are the resultants of joint contact forces with respect to joint definition points. Based on this physical fact, a set of equivalent equations of joint contact forces and joint reaction forces are supplemented for solving contact positions and contact forces in joints during dynamic analysis of multibody systems. Firstly, gap functions for describing contacts in the joints are obtained through a relative motion analysis; Secondly, because the gap functions are not independent with each other, their relationship can be given; Thirdly, based on the complementary relationship between the gap functions and contact forces, as well the equivalent relationship between the reaction forces and contact forces in the joints, the relationship between the joint reaction forces and contact forces is supplemented; Finally, the contact positions and forces in the joints can be obtained through solving the supplementary equations. Taking a planar prismatic joint and a spatial revolute joint for example, the simulation results of the method in this paper and the continuous contact force model are compared with each other. It can be shown by the comparison that the joint contact force curve calculated by the method in this paper is relatively smoother than that of the continuous contact force model, which contains a lot of high frequency oscillations. However, the tendencies and magnitudes of the joint contact force curves obtained by these two methods are generally consistent with each other. In the method of this paper, non-penetration assumption is adopted in clearance joints and it is not necessary to release joint constraints. Contact positions and contact forces in joints can be obtained through solving a set of equivalent equations. Therefore, the method of this paper has higher calculation efficiency.The hypothesis of the contact method for clearance joints in terms of the equivalent relationship between joint reaction forces and contact forces is that, the impact effects on the kinematical behavior of mechanical systems can be ignored. Taking a planar revolute clearance joint for example, a non-penetration contact analysis is given to demonstrate this hypothesis. The journal motion modes are divided into three categories, namely, the free motion, impact and permanent contact modes. Firstly, the kinematic constraint of the revolute clearance joint is removed in the free motion. Then, the impact process is simulated by the differential analysis of impulse-based, where Stronge's improved model for restitution is employed to determine the relative velocity after impact. Finally, in the continuous contact mode, the equality constraint between the journal and bearing is activated and a modified Coulomb law is adopted. The switch between different contact modes is identified by the state of the journal and bearing, including the gap and normal relative velocity. Due to that the continuous contact mode maintains for most of the time and a large integration step size can be adopted, the proposed method has a high efficiency. Numerical examples prove the infinitesimal effect of impact in joints with small clearances.Traditional dynamic contact analysis of rolling bearings considers all contacts among the bearing components. Therefore, it has a large computational complexity and cannot be used for contact analysis in joints of multibody systems. Based on the equivalent relation of joint reaction forces and contact forces in joints, dynamic contact analysis in a spatial revolute joint composed of a pair of rolling bearings is studied. A rolling bearing has many components, such as several steel balls, a cage, inner and outer rings, etc. There are tiny clearances between the steel balls and the cage pockets, as well as between the steel balls and the inner and outer raceways. Through clearance assembly analysis, the following conclusion can be obtained that the steel balls are not in contact with the cage pockets in the axial and radial directions of rolling bearings, and the cage is driven by the steel balls carrying loads stably in the tangential direction of the raceways. Through pure rolling analysis of a steel ball, the rotational motion of the cage can be obtained. The relative motion between the shaft fixed with the two inner rings and the bearing housing fixed with the two outer rings is thus defined. Considering the conditions for a steel ball in contact with its raceways simultaneously and carrying loads stably, the following essential conclusions can be obtained:1) there are two steel balls carrying loads stably at most in each rolling bearing; 2) these two balls must be adjacent. By the definitions of the resultant of contact forces between the two adjacent steel balls and the outer raceway in each rolling bearing and the orientation angle of the resultant, the total unknown variables can be decreased to five parameters, including the two resultants of contact forces, their two orientation angles and the axial displacement of the shaft. Finally, the contact positions and contact forces in rolling bearing joints are obtained by solving the equivalent relation of joint reaction forces and contact forces in joints.Rzeppa constant velocity universal joint is a new driving device to transmit the constant velocity rotation between two cross shafts. The steel balls and the inner and outer orbits are assembled with no clearance in a Rzeppa constant velocity universal joint. The motion of the steel balls plays a key role in the function realization and parameter design of the joint. This paper carries out the kinematic analysis of BJ type and VL type Rzeppa constant velocity universal joints and analyzes the motion of the steel balls under the constraints of the inner and outer orbits and the cage. The rigorous proof of constant velocity characteristic is presented, and the kinematic differential equations of the steel balls are formulated. On the basis of the above investigations, the software for kinematic analysis and simulation of Rzeppa constant velocity universal joint is written. The software can model BJ type and VL type Rzeppa constant velocity universal joints parametrically and simulate the motion of the joints accurately. By solving the motion of the steel balls, some important parameters are obtained, such as the displacements of the steel balls with respect to the inner race, the outer race and the cage, etc. These results are helpful for the parameter design of Rzeppa constant velocity universal joint.