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I am trying to derive the equations of motion for a system that is essentially an inverted pendulum on a wheel. The pendulum is connected to the wheel by a motor. I have a motor torque

*M*and a disturbance torque

*D*. The radius of the wheel is

*r*.

My generalized coordinates are the angle of the pendulum, [tex]\theta[/tex], relative to vertical and the horizontal position of the center of the wheel,

*x*.

I'm having trouble wrapping my head around the generalized forces (which here, I think, are only the motor torque and disturbance moment). Do I need to consider both the motor torque and the reaction force at the wheel center, or only the motor torque? In other words, is the motor torque doing work in

*x*, or only in [tex]\theta[/tex]. Should my virtual work equation look like this:

[tex]\delta W = \frac{M + D}{r} \delta x + \left( M + D \left) \delta \theta[/tex]

or this:

[tex]\delta W = \left( M + D \left) \delta \theta[/tex]

For a while I was considering using Lagrange multipliers (after looking at the example of the cylinder rolling down an incline), but I've convinced myself that since

*x*and [tex]\theta[/tex] really are independent in my problem, I don't need to use Lagrange multipliers. Or, I need to add a generalized coordinate for the angle of the wheel and use Lagrange multipliers.

So, I'm pretty confused... what am I missing? What is it that I don't understand?

Thanks,

Kerry