# Conservation of Energy - Gear Box vs. CVT

## Main Question or Discussion Point

I am working on a mechanical model for a Snowmobile and trying to figure out what the differential equations becomes when you have a CVT (Continuous Variable Transmission) instead of a Gear Box.

Assume that you have two shafts connected with each another through a (lossless) and stiff gear box, consisting of two gears only. If I understand it correctly, the differential equations describing this system are

J1*d2(Theta1)/dt2= M1 + g*M
J2*d2(Theta2)/dt2= M2 - 1/g*M

where Ji, i ={1,2}, are the Moment of Inertia's of the shafts (including gears) respectively and Mi, i={1,2}, are torques acting on the corresponding shafts (friction, external torques, a.s.o.). M is an auxiliary Torque describing the coupling between the two axes, and of course g is the gear ratio. Thetai, i={1,2} of course are the corresponding angles. We also assume that dg/dt is constant (except when changing gear instantly and re-initialize the system).

By using the fact that Theta2 = g*Theta1 (+ Constant), M can be eliminated and the equations above be reduced to a single equation, not shown here.

Questions:
1. Is everything above correct?

2. How will the approach change when the shafts are connected through a CVT and therefore dg/dt is NOT equal to zero? I understand conservation of energy must be used, but how?

3. If you have a fixed gear box and change the gear (instantly), will there be any transient effects, like Dirac pulses? This is mostly neglected in the literature, so I do not know if there will be any transients or not.

4. How do you extend the models for fixed gear box and CVT in order to include losses inside the gear box?

5. Are there any good books for modeling gear boxes and CVTs?