- #1
KLoux
- 176
- 1
Hello,
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
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