Analytical cinematic model

We developed an analytic bidimensional model of Eyson CVT to verify numerical model results comparing it to theoretical ones. This analytical model deals with revolution multiplier kinematics and is useful to preview gearbox transmission ratio trend in time for a chosen input-output shafts offset.

Planar simplified mechanism model is constituted by the principal gear (with primitive radius Rm), which rotate with ωm about a fixed axis, and 4 lateral gears (with primitive radius Rl), which can rotate with ωl about their own axis. Lateral wheels axis can slide on the right disk slopes resting at the constant distance Rm+Rl from the  principal gear axis because they are mounted on supports that rotate around the principal axis with ωs. Each wheel moves with a different distance Rc from the disk axis. The disk rotates with ωc around a fixed axis which rests at a distance Ts from the principal gear axis.

Figure A: Adopted nomenclature

For the presence of unilateral bearings between supports and lateral shafts, which sustain lateral wheels, one can set for every lateral wheel:

where supports angular velocity is:

 δ and Δ are defined in the figure above. They are correlated by:

Rc can be expressed as:

So a sliding coefficient can be defined:

Setting velocities equilibrium in any of the 4 gear contact point:

ωm is fixed imposing input shaft boundary conditions, ωc assumes a unique value for everyone of 4 gear contacts, ωl varies from wheel to wheel with the term Rc /(cos(δ -Δ)) of each wheel.

During an output shaft revolution every wheel freely rotates about its axis with ωl  > ωs when its axis is far from output shaft axis. When it enters into the output shaft angular range -45° ≤ δ ≤ +45°, lateral shaft rotation is locked by unilateral bearing so principal gear moves lateral gears around input shaft axis transferring motion and torque to the disk.

Therefore the slowest of the 4 wheels covers the output shaft angular range -45° ≤ δ ≤ +45° and it rotates about its axis with the minimum value of ωl that ideally respects the condition ωl  – ωs ≥0, i.e. ωl  = ωs.

Gearbox transmission ratio could be defined considering locked wheel motion. In fact only for this wheel SL can be ideally set to zero. In this hypothesis ideal velocity ratio can be written as:

With Δ and δ obviously referred to the nearest to disk axis wheel.

In the numerical model the unilateral bearings permit a small angular sliding even in the sense impeded by ideal working of the bearing. In analytic model this sliding effect can be represented artificially setting the sliding coefficient SL < 0. In this case:

Numerical results can be compared with analytic results. In the diagram below numerical transmission ratio is calculated by simulation on the CVT choosing the proper angular interval when only one wheel is locked. Data regard a proof where constant Tin = 100 Nm, C = 3 m2 Kg/(rad s) and Ts= 20 mm is set. Analytic results are been obtained with previous formulas leaving the same numerical model geometrical data. In the ideal case SL = 0, in the other case SL is fixed constant and equal to -2%.

Numerical results fit quite well the analytic model transmission ratio trend. In the initial locking wheel transitory the unilateral bearing has a wide sliding in the forbidden verse which bring to power losses and to an unwanted maximum point of the transmission ratio. Then the bearing is locking gradually.

Figure B: Transmission ratio comparison assuming null sliding coefficient

 

Figure C: Transmission ratio comparison with sliding coefficient of -2%

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