A particle is connected to a massive wheel by a rigid rod. The wheel can roll without slipping on a horizontal surface. The particle is free to rotate around the centre of the wheel.
I believe the system has two degrees of freedom: The centre of the wheel and the particle each have x- and y- positions, and the wheel has an angle of rotation. The constraints are:
- The wheel is on a horizontal surface, fixing its y-coord
- The wheel cannot slip, so its x-coord is directly proportional to its angle of rotation
- The particle is a fixed distance from the centre of the wheel
Is this correct?
This leaves two generalised coordinates, which I have taken to be the angle of rotation of the wheel and the angle between the rod and the y-axis.
After struggling (and failing) with a Newtonian approach I constructed a Lagrangian for the system and applied the Euler-Lagrange equation(s), using the angles as generalised coordinates. After much algebra out popped two second-order non-linear differential equations.
To my surprise, I could eliminate the wheel rotation completely from one equation, leaving it concerning only the rod angle and its derivatives. What, if at all, is the significance of this?
And finally, I would like to simulate the situation computationally. Is there a general way of simulating rigid constraints acting on rigid bodies/particles, or must one find and solve (numerically) the differential equations governing the system?