The modelling problem

The multibody virtual model of Edyson CVT is been developed with LMS Virtual.Lab software. Simulations made on this virtual model permit to analyze the gearbox working.

The modelling purpose was to recreate all product features resumed in previous paragraphs. We chose to represent an external spur gear based CVT, thought for operating with torque of about 100รท150 Nm.

Figure A: Single mechanism

The entering power is given to an input shaft, which is linked with a revolute joint to a base. The base is fixed to ground. The input shaft is integral to a principal gear that is linked to other 4 wheels with gear joints. Every lateral gear is embedded on a lateral shaft. These shafts lay on the principal axis of the unilateral bearings that are fixed to 4 supports which can rotate about the input shaft. On the other side lateral shafts are constrained to slide along disk slopes. The disk is welded to the output shaft and is fixed to the slide with a revolute joint. The slide position can be controlled along a direction which is orthogonal to the rotation axis of shafts. A slide translation causes a variation in the output shaft offset respect to the input shaft.

If, instead of the unilateral bearings, one uses simple cylindrical joints to fix lateral shafts to supports, the mechanism could not transmit power to output shaft, for any sort of entering power given to the input shaft. In this case the gearbox has null mechanical efficiency and input power is dissipated making rotate lateral shafts around their axis. For right working of the transmission, unilateral rotation of lateral shafts about their axis respect to support motion has to be imposed. LMS Virtual.Lab does not permit to set such ideal unilateral constraint, so it was necessary to introduce a submodel of unilateral bearing.

The unilateral bearing submechanism is based on a realistic model of cylindrical bearing which permits motion only in one sense. In particular the submechanism we are speaking about is inspired by SDP/SI ROLLER CLUTCHES, which description is available here below (taken from www.SDP-SI.com).

 

The simplified Virtual.Lab unilateral bearing model is represented below.

Figure B: Virtual model of the unilateral bearing

As already mentioned, this bearing model permits free rotation of lateral shaft, fixed to inner ring, respect to support, fixed to outer ring, only in one sense of revolution. This behavior was realized developing a proper forces system through which outer ring can transmit torque and power to inner ring passing through one roller. We chose to refer all forces to only one roller, represented by a sphere, for simplicity and faster calculation. Active roller is constrained to move on a linear path contained between inner ring and wedge-shaped cavity of outer ring. Contact forces act between the sphere and two rings. The contact forces are characterized by a friction coefficient, that quantifies contact tangential forces, by touching material stiffness properties, which quantify contact normal forces with respect to penetration deep, and by a restitution coefficient, which quantifies elasticity or inelasticity of shocks. A preloaded spring is positioned between the roller and the outer ring (as shown in the real model) to force the sphere against the wedge-shaped cavity.

Contact forces parameters, spring features so as model shapes are been calibrated finding a good compromise between locking rings relative rotation in forbidden sense and ensuring negligible mechanical power losses due to friction during rotation about allowed direction. In the operating conditions set for simulations of CVT, actual calibration of bearing model variables gives, as result, good mechanical efficiency of unilateral bearing during its revolution around the allowed verse but also little relative angular displacements of the rings when forced to rotate in the forbidden sense. In particular, the highest relative angular velocity in the forbidden sense is developed in the transitory time interval that represents the passage of the lateral wheel from its free motion to its torque transmission from the principal gear to the disk. During angular sliding of rings in the forbidden verse, high torques passes through bearings and so even if relative rotational velocities are light, mechanical losses are relevant.

Most of mechanical power losses of the CVT model are localized inside unilateral bearing submechanisms but other friction losses are also been introduced to give a more realistic behavior to gearbox operating. All these power dissipations are been modeled as friction losses concentrated in some idealized joints through which the model is been built. These friction losses considerate:

  • rotation of the input shaft about its principal axis supported by a simple cylindrical bearing fixed to the base;
  • rotation of the output shaft supported by a simple cylindrical bearing fixed to the slide;
  • sliding of one end of the lateral shafts through disk slopes;
  • rotation of supports around the input shaft;
  • sliding contact between gear teeth.

For modeling losses, friction coefficients are been chosen taking example from technical literature.

All results reported in this technical relation are referred to CVT models where is been set a position of the slide in a normal direction respect to the shafts rotation one, an input velocity or an input torque imposed to the input shaft and a value of damping coefficient that limits output shaft motion. In fact resistance torque assigned to output shaft is been chosen to be proportional to the angular velocity of output shaft through a constant damping coefficient.

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