Torque losses through gears due to inertia

[ZachGriffin]

I'm trying to work out a formula to calculate the amount of torque lost in a two gear system due to the acceleration the second gear. I'm assuming there is nothing lost to friction etc

For example, If I have two gears, G1 and G2, with respective inertia's of 2 and 4. G1 has 10 teeth and G2 has 80 teeth for a ratio of 8.0 or 0.125 depending on which way you are going. When looking on the G1 side, the effective inertia of the geared system can be found as:

G1's inertia + (1/(GearRatio)²) * G2's inertia

which equates to 2 + (1/(8)²) * 4 = 2 + 0.015625 * 4

So the effective inertia is 2.0625, as the high gear ratio of 8 reduces the influence of G2's inertia.

Applying 100Nm of torque to the first gear, the acceleration now becomes 100/2.0625 = 48.48

On the other side the effective inertia is equal to 4 + (1/(0.125)²) * 2
= 4 + 64 * 2
= 132

As the torque is now multiplied by the gear ratio, we have 800Nm of torque so the acceleration now becomes 800/132 = 6.06, exactly 8 times less than G1's acceleration, so the formula is correct.

What I'd like to know is how to calculate the effective torque, so for example G1 has an inertia of 2, and accelerates at 48.48 which means the effective torque was 96.96Nm. The overall loss from accelerating G2's inertia was 100 - 96.96 = 3.04Nm. On the other side, G2 has an inertia of 4 and accelerates at 6.06 which means the effective torque was 24.24Nm. The overall loss from accelerating G1's inertia was 800 - 24.24 = 775.76Nm.

I've been trying to work out a way to calculate the effective torque using the gear ratio and inertia in a formula, rather than using the effective moment of inertia calcs as above. I'm hoping somebody here knows, fingers crossed. Any help is much appreciated

 

[K^2]

You either have to balance torques and solve for angular acceleration, or use effective moment of inertia. There isn't really a third option.