Effect of elasticity and manufacturing tolerances on the kinematic and dynamic performances of a Cardan Joint

Cardan joint is a very widely used assembly in many mechanical fields [1]. It is comprised of an input and output shafts with two forks and a cross. Each fork hole is connected to a cross pin by means of a revolute joint. In transmission simulations, it is often included using a “black box” approach, by adding coupling relationships between the relative velocities of the connected shafts. On the other side, in many applications, it is also necessary to model and simulate the joint in details in order to assess the internal reaction forces between pins and holes. The most severe complication is that the Cardan joint is an overconstrained system. In standard multibody simulations in which the kinematic restraints are modeled introducing constraint equations, this may lead to the indeterminacy of the reaction forces. Moreover, in an overconstrained system, it is impossible to take into account the effect of manufacturing and assembling errors that are always present in the actual parts. In fact, the presence of these errors is somewhat managed by the clearances and elasticity of the components which are in general neglected in a rigid body model with kinematic constraint equations. Scientific literature often reports study on universal joints by simplifying the mechanism deleting the overabundant constraints [2-6]. Other studies in multibody dynamics introduce the presence of clearances in revolute and cylindrical for overcoming the redundancies [7]. In order to avoid all these problems and to produce a multibody model able to take into account the effects of elasticity and manufacturing errors, a specific modeling technique has been used. All the kinematic pin-hole constraints have been replaced by penalty systems of two intelligent nonlinear piecewise springs by using the approach proposed by Brutti et al. [8] According to this approach, the springs have variable stiffness and damping characteristics in order to take into account different types of contact (line contact, single point contact, two-point contact). Moreover, the joint forks and cross pins have been modeled using the discrete flexible multibody techniques by splitting the rigid beam-shaped bodies into several smaller bodies connected by matrix spring elements that simulate the elastic compliance of the structure. Several design scenarios have been simulated including geometrical and dimensional errors of components. In particular, four errors have been included: position error on the alignment of the holes of one of the fork, angular misalignment between the holes of one of the fork, angular misalignment on the alignment of two perpendicular pins of the cross, position error of one of the pin of the cross. The influence of these errors on the performance of the joint have been evaluated in terms of kinematic parameters (velocity and acceleration of the output shaft with respect to the input one) and in terms of dynamic parameters (forces between pin and holes). The results are presented in graphs and tables using dimensionless influence parameters. They can be used for the optimization of the allocation of tolerances and for improving the design of Cardan Joint in specific field such as high-speed mechanisms, precision devices, high-efficiency driveline, etc.