Dynamics of a Rolling Wheel on a Sliding Platform

[john87]

Hi, I have a problem that I am working on and I am running into some questions.

The question states (the image is attached):
This system consists of the disk D and the bar B that is welded at point Q.
The other end of this bar is fixed at point O. Assume that the disk rolls on the
thin horizontal plate P supporting it. This plate in turn slides over the plane.
Determine the constraints, discuss the nature of them, and also determine the
number of degrees of freedom of the system.

This is what I have so far for constraints:

1) omega of disk dotted with b2 is zero (1-constraint)
2) velocity of point O to O' is zero: therefore, velocities of x0 = 0, y0=0, z0=0 (3-constraints)
3) for the plate: Velocity of plate dotted with b3 is 0 (1-constraint)

Then I also said the velocity of the plate at C' is equal to the velocity of the disk at C. So, velocity from Q to C is R*thetadot in b2 direction. I know this relationship, but I am not sure how to go about the kinematics of the plate and then set that equal to the velocity of the disk at C to get another constraint. I know that the plate has 2 degrees of freedom, and the disc has 1 degree of freedom.

Thanks for the help.

 

[Navin A S]

To answer your question, the last constraint is obtained by considering the kinematics of the plate. Since the plate is sliding on a plane, its motion can be described in terms of its linear velocity components vx and vy along the x- and y-axes, respectively. The velocity of the disk at point C can then be related to these two velocities as follows: velocity of disk at C = vx b1 + vy b2. Therefore, the last constraint can be written as: velocity of disk at C = velocity of plate at C'. This gives you a total of 6 constraints and the system has 3 degrees of freedom.