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DISCUSSION THREAD
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DISCUSSION and HINTS
In this problem, we desire to relate the rotation rates of the slotted wheel and the disk. With the two rigid bodies connected by a pin-in-slot joint, we are not able to use the rigid body kinematics equations by themselves. Let's discuss that below.
Before we do, however, can you think of a good application for such a mechanism design? Take a look at this short video.
Velocity analysis
Here, we can use the rigid body velocity equation to relate the motions of P and C:
vP = vC + ωdisk x rP/C
However, we cannot use a rigid body velocity equation to relate the motion of points O and P (the reason for this is that O and P are not connected by a rigid body). In its place, we can use the moving reference frame velocity equation with an observer attached to the slotted wheel:
vP = vO + (vP/O)rel + ω x rP/O
where ω is the angular velocity of the observer, and (vP/O)rel is the velocity of P as seen by our observer on the disk. Note that with the observer being attached to the slotted wheel, this observer sees motion of P only along the slot.
Combine these two equations to produce two scalar equations.
Acceleration analysis
We will use the same procedure for acceleration as we did for velocity - use a rigid body equation for the disk and a moving reference frame equation relating O and P:
aP = aC + αdisk x rP/C + ωdisk x (ωdisk x rP/C)
aP = aO + (aP/O)rel + α x rP/O + 2ω x (vP/O)rel + ω x (ω x rP/O)
where α is the angular acceleration of the observer, and (aP/O)rel is the acceleration of P as seen by our observer on the slotted wheel. Again, note that with the observer being attached to the slotted wheel, this observer sees motion of P only along the slot.
Combine these two equations to produce two scalar equations.
Since the observer only sees motion of P along the slot, does that mean that any acceleration not in line with the slot will not be part of a p/o rel?
You are correct - the acceleration seen by an observer on the Geneva wheel is aligned with the slot. The observed acceleration does not have a component that is perpendicular to the slot.
How does switching phi from 120 degrees to 150 degrees affect the system? Because we have axes attached to the slotted wheel, shouldn't r(P/C) and r(P/O) remain the same? If so, what are the other differences to keep in mind?
the change will be the velocity of point P is in a different direction. The magnitudes of each r is the same, but the I and J components are not. Using i and j connected to the angle theta, you can see that at 120 degrees P moves solely in the direction of the slot (-i) but at 150 degrees the motion will be different, although the observed motion is still within the slot. This alters both omega and alpha for the disk
Wouldn't the angle theta change with the angle phi as the axes we select are attached to the wheel not theta which will, I assume, rotate?
How can we find what R2 is? R2 is not the same as distance d correct?
I was able to use the law of sines to find R2 with a triangle formed by OPC. One angle of the triangle is theta and another can be found by doing 180 - phi. From there you know the side across from theta (R1) and the side across from the angle you just found (R2) and can just plug into the law of sines formula. I'm not sure if there is a better approach, but this seemed to work for me.
I'm assuming our answers are in terms of w1 where w1 =w disk?
We are given w1! It is 6 rad/s
How should we approach part b? Should we solve new values for the lengths and such to solve for the new angular velocity and angular acceleration? I’m not sure how to approach it to solve it.
You will need to find the new value for theta and the distance from O to P.
For part B, is it possible to reestablish an i and j coordinate system once the wheel has spun, therefore the position vectors stay as Rj or Ri and do not have to be broken up into components using trig. This way the math for Wpo is the same and is still 0 k?