classical mechanics - Dynamics of moment of inertia

I'd like to be able to determine the angular acceleration of a system of two rotating masses, which are connected so as to have a variable mechanical advantage between the two. My background with mechanics went as far as a course in statics, so I'm not sure how to proceed with this.

If I have a single mass of some shape and apply a torque to it, I know that the angular acceleration depends on the moment of inertia of that object. But suppose I have a system of two objects, e.g., gears, and apply torque to one of them, and want to know the angular acceleration. I'd assuming that the effective moment of inertia, at the point where I apply the torque, is the moment of the directly driven mass, plus the moment of the secondary mass multiplied by the mechanical advantage between the gears, and that using this 'effective moment of inertia' with the input torque would tell me how fast the input mass accelerates. (The acceleration of the 2nd mass being implied, as there's only 1 dof here) Not sure if this general approach is even correct, and then there's the real problem .

Introducing the variable mechanical advantage is what's giving me trouble here. If I take the contribution to the effective moment of inertia from the 2nd mass as just its 'intrinsic' moment times the mechanical advantage at any given instant, am I missing something? Calculus instincts tell me that there ought to be a contributing term from the rate-of-change of the mechanical advantage, too.

Answer

There is indeed a term involving the time derivative of the changing coupling between the masses.

First, let's derive the equation for a single mass.

$$L = \frac{1}{2} I\, \dot{\theta}^2 - V(\theta)$$

$$\frac{\partial L}{\partial \dot{\theta}} = I\, \dot{\theta}$$

$$\frac{\partial L}{\partial \theta} = -\frac{dV}{d\theta} = \tau$$

$$\tau = \frac{d}{dt} \left( I\, \dot{\theta} \right) = I\, \ddot{\theta}$$

This shows you that the angular acceleration is proportional to the torque.

Now, assume we have two masses. The driven mass has moment of inertia $I_1$ and angular velocity $\dot{\theta}$. The secondary mass has moment of inertia $I_2$ and angular velocity $\dot{\theta_2} = a(t)\, \dot{\theta}$, where $a(t)$ is the changing coupling (for example a changing belt position in a continuously variable transmission).

$$L = \frac{1}{2} I_1\, \dot{\theta}^2 + \frac{1}{2} I_2\, a(t)^2\, \dot{\theta}^2 - V(\theta)$$

$$\frac{\partial L}{\partial \dot{\theta}} = \left( I_1 + a(t)^2 I_2 \right) \dot{\theta}$$

$$\frac{\partial L}{\partial \theta} = -\frac{dV}{d\theta} = \tau$$

$$\tau = \frac{d}{dt} \left( \left( I_1 + a(t)^2 I_2 \right) \dot{\theta} \right)$$

$$\tau = \left( I_1 + a(t)^2 I_2 \right) \ddot{\theta} + 2 I_2\, a(t) \frac{da}{dt} \dot{\theta}$$

The last term, proportional to $a \dot{a} \dot{\theta}$, is the funny term you're looking for. It says that when the coupling is changing, you need to apply some torque just to keep the angular velocity $\dot{\theta}$ constant. Another way to think of it is that in the absence of external torque, $\dot{\theta}$ is no longer constant (as it was for the single mass), but instead $(I_1 + a(t)^2 I_2) \dot{\theta}$ is constant, because that's the real angular momentum.