Instantaneous center of rotation

[Bauxiet]

Homework Statement

Where are the instantaneous centers of rotation?
Instantaneous center of rotation =
1. Velocity is 0 in this point.
2. This can be also a point where 2 parts "touch" each other and have the same velocity in this point.

Homework Equations

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The Attempt at a Solution

Is this correct? Are those the points of instantaneous centers of rotation?

Thanks guys

 

[BvU]

Hi,

Is that the full problem statement ? I see an $\omega_p$, and $\omega_Z$ and an $\omega R$. Is there a fixed axis through $O$ ?

 

[Bauxiet]

Hi,

Is that the full problem statement ? I see an $\omega_p$, and $\omega_Z$ and an $\omega R$. Is there a fixed axis through $O$ ?

That is indeed not the full problem statement. But my problem statement is which are the instantaneous centers of velocity?
There is a fixed axis through O. If you need the angular speeds they are:

wz = 5 rad/s (given)
wa = 15 rad/s
wR = 3 rad/s

Everything else is in the picture.
Thanks

 

[haruspex]

Is this correct? Are those the points of instantaneous centers of rotation

If you mean the points marked red on the periphery of the small gear, no.
If I understand the diagram, both gears centred on O are rotating about O, so their instantaneous centres are obvious. It remains to find those points on the small gear where the velocity is instantaneously zero.
Consider a point Q on the small gear distance x from its centre, P, angle QPO=θ.
Find the radial and tangential velocities of Q relative to the centre O.

 

[Bauxiet]

If you mean the points marked red on the periphery of the small gear, no.
If I understand the diagram, both gears centred on O are rotating about O, so their instantaneous centres are obvious. It remains to find those points on the small gear where the velocity is instantaneously zero.
Consider a point Q on the small gear distance x from its centre, P, angle QPO=θ.
Find the radial and tangential velocities of Q relative to the centre O.

Are you sure? For example on the picture underneath, the red point was de center of velocity for the red gear (small and big are one solid gear).

I am a little bit confused. How can i find the velocity center. It is not because a disk is fixed because of a axis that the axis is the velocity point. A bicycle wheel also has an axis but the velocity center is where the wheel touches the ground. But to stay on the topic, how can I find the velocity centers in this gears? It is really confusing.

 

[haruspex]

In the diagram "Today 00:03:57", the outer gear appears stationary, so the point of contact with that will be instantaneously at rest. In the problem inpost #1, the outer gear is rotating.
First, find the velocity of the centre of gear P. It will be easiest to deal in terms of velocity components along and perpendicular to OP.
Then find the velocity components of Q relative to P centre.

Edit: it occurs to me that following what I wrote above might lead you into much more work than necessary. Start by considering just the case where Q lies on the line through the gear centres.

 

[Bauxiet]

In the diagram "Today 00:03:57", the outer gear appears stationary, so the point of contact with that will be instantaneously at rest. In the problem inpost #1, the outer gear is rotating.
First, find the velocity of the centre of gear P. It will be easiest to deal in terms of velocity components along and perpendicular to OP.
Then find the velocity components of Q relative to P centre.

Edit: it occurs to me that following what I wrote above might lead you into much more work than necessary. Start by considering just the case where Q lies on the line through the gear centres.

Hi,

What do you mean with "Q"? I already calculated all the velocities and angular speeds. I had to know all the instantaneous centers of velocity for this. I am really confused about this. I think the two instantaneous centers are at the center points of the gears (this is what my intuition would say). Although it is difficult to imagine for me. How can you calculate or know if for sure?

1. To start from the beginning. They say the middle gear has a angular speed of 5 rad/s. Because the middle gear is fixed on the axis and it can't move in any direction, it should be rotating around the middle point? Is this correct?

2. Now we can calculate the velocity at the edge of this gear. It will be 0,75 m/s. So where gear Z en P touch each other, the speed must be the same. Because they say that the lever OA doesn't have any rotating speed (w=0). So this lever is fixed and can also NOT move. The gear P is in this case forced to rotate around the point A? Is this correct?

3. Last stage, the speed at the edge of gear P where it touches gear R will be the same. Here you can calculate the angular speed of the biggest gear R.

Because of the fixed lever and the fixed gear O. De gears were forced to rotate around their middle points? Is this all above correct? Thanks guys!

 

[haruspex]

What do you mean with "Q"?

As I defined it in post #4.

I had to know all the instantaneous centers of velocity for this

I've never heard of instantaneous centre of velocity. I assume it is the same as centre of rotation.

I think the two instantaneous centers are at the center points of the gears

they say that the lever OA doesn't have any rotating speed (

You did not mention that previously.
But, now I plug in the rotation rates and radii, I see that it is indeed the case. The centre of the small gear will not move. This makes it trivial... the centres of rotation are simply the centres of the gears. This leaves me wondering why you were asked to find them.
The question would have been much more interesting if the small gear were moving around point O.

 

[haruspex]

now I plug in the rotation rates and radii, I see that it is indeed the case.

.. or did you calculate those rates from the premiss that rod OA is fixed?

 

[Bauxiet]

As I defined it in post #4.

I've never heard of instantaneous centre of velocity. I assume it is the same as centre of rotation.You did not mention that previously.
But, now I plug in the rotation rates and radii, I see that it is indeed the case. The centre of the small gear will not move. This makes it trivial... the centres of rotation are simply the centres of the gears. This leaves me wondering why you were asked to find them.
The question would have been much more interesting if the small gear were moving around point O.

Thanks guys. Yes indeed, the detail of the fixed lever did not get my attention. Thats why I was confused.
Another question loose from this exercise.

What with the acceleration? Image you have a wheel, the instantaneous center of rotation is at the ground. This wheel also has an acceleration a. Is this the acceleration from the view of the OR or from the middlepoint of the wheel (circle)?

Thanks

 

[haruspex]

What with the acceleration? Image you have a wheel, the instantaneous center of rotation is at the ground. This wheel also has an acceleration a. Is this the acceleration from the view of the OR or from the middlepoint of the wheel (circle)?

What does OR stand for?
If you are told the acceleration of the wheel is a then I would take that to refer to the centre of the wheel.