# Gear ration

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1. Jul 24, 2016

### Siddiqui

Hello
I am using two motor which are coupled with the mechanical coupling. The coupling has moment of inertia. The gear ratio of both motors is same 26:1.
I want to know i also need to divide the coupling moment of inertia with the gear ratio?
Thany

2. Jul 24, 2016

### jack action

I'm not sure what you mean by «coupling», so I will show you how to find the equivalent moment of inertia of a simple system with gear ratio.

Imagine you have a gear set with gear 1 and 2. They have a gear ratio $GR$, and each gear has inertia $I$ and angular acceleration $\alpha$. We know the input torque $T_{1\ in}$ and angular acceleration $\alpha_1$ of gear 1. Doing the sum of moments on each gear:
$$T_{1\ in} - T_{1\ out} = I_1\alpha_1$$
$$T_{2\ in} - T_{2\ out} = I_2\alpha_2$$
And we also know that:
$$T_{1\ out} = GR T_{2\ in}$$
$$\alpha_2 = GR \alpha_1$$
We now have 4 equations, 4 unknowns ($T_{1\ out}, T_{2\ in}, T_{2\ out}, \alpha_2$). Finding $T_{2\ out}$ starting with the first equation:
$$T_{1\ in} - T_{1\ out} = I_1\alpha_1$$
$$T_{1\ in} - GR T_{2\ in} = I_1\alpha_1$$
$$T_{1\ in} - GR \left(T_{2\ out} + I_2\alpha_2\right) = I_1\alpha_1$$
$$T_{1\ in} - GR \left(T_{2\ out} + I_2 GR \alpha_1\right) = I_1\alpha_1$$
$$T_{1\ in} - GR T_{2\ out} = \left(I_1 + GR^2 I_2\right)\alpha_1$$
We now have an equation of the form $T_{in} - T_{out} = I\alpha$ (sum of moments), but for the complete gear set, based on the input torque and acceleration. Note that the inertia of the second gear is multiplied by the square of the gear ratio.

3. Nov 6, 2016

### Murli

Please sketch what you mean by "coupling two motors which are coupled with mechanical coupling".