# Finding velocity components of point in the middle of a tricycle

1. Oct 18, 2014

### Krus

1. The problem statement, all variables and given/known data
I have a tricycle with a following data (see attachment):
• Distance (d) between center of front wheel (F) and center between rear wheels (C) = 325 mm
• Velocity of front wheel (VF) = 50 mm/s
• Angle of front wheel (α) = changing between -90;90
• Distance between point C and P = 129 mm
Out of this data, I calculated the angular velocity by using the following formula:
ω = (sin(α) * vF) / d

What I need, is the component velocity of a point between the front wheel and the center between rear wheels (P) in its local coordinate system. This local coordinate system is for sure turning with the vehicle.

2. Relevant equations

3. The attempt at a solution
To calculate the velocity in point P i used the angular velocity and the distance between C and P:

vP = ω * 129 mm

This is for sure right, if α = ±90. But If α = 0, my vP is also 0, which can't be.
So somehow the velocity connected with the front wheel angle α must have an influence as well.

Finally, I need to split the velocity of point P into its X- and Y-component.

Krus

#### Attached Files:

• ###### Untitled.png
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2. Oct 18, 2014

### BvU

Hello Krus, and welcome to PF.

surprises me. In the local coordinate system of the vehicle, P isn't moving at all. :)

3. Oct 30, 2014

### Krus

Hi :)

OK, I'm sorry.. The description was maybe a bit wrong.
I try to explain what I need in some examples:

If vF = 50 mm/s and α = 0, then the y-component of the velocity would also be 50 mm/s, x-component and ω is then 0.

If vF = 50 mm/s and α = 90, then the vehicle is turning around its point C -> x-component and ω can't be 0 in point P.

If vF = 50 mm/s and α = 45, then the vehicle drives in a circle around a center point, which is outside of the vehicle. Then there is a smaller ω than when α = 90, but I have a higher y-component velocity.

I mean in point C, can only exist a y-velocity and an angular velocity. It can't happen that there is a velocity in x-direction. But in point P it is different, because it's not the turning centre.

So there must be a formula, how to calculate the speed in components (x-,y- and angular) for point P.

Sorry, It's kind of difficult to describe the problem :(

4. Oct 31, 2014

### haruspex

I suggest figuring out from the geometry where the centre of rotation is. P then describes an arc about that point.
I guess you mean the co-ordinate system fixed on the ground at the point where P happens to be at this instant, with the y axis along CP.

5. Nov 1, 2014

### Krus

Yes, you're right, that's what I mean.:)

Do you have any Idea how to get these velocities?
For sure I can get the center of rotation out of the geometry, but how to use these data to get the velocities?

Thanks

6. Nov 2, 2014

### haruspex

Can you figure out the angular velocity?
If you draw a radius from P to the centre of rotation, how long is that radius and what is the direction of movement of P in relation to that?

7. Nov 2, 2014

### Krus

Yes...

The angular velocity and all other values (see image) are the following:

ω = (VF * sin(α)) / d
r = d / tan(α)

q
= √(r2 + CP2)

direction of movement of P:
β
= atan(CP / r)

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8. Nov 2, 2014

### haruspex

Yes to all those. I was rather hoping you'd see how to finish it from there.
What is the speed in the direction of movement? What are the x and y components of that?

9. Nov 2, 2014

### Krus

Well, i guess the following:

Speed of movement in P:
vP = q * ω

Components:
vPX = VP * sin(β)
vPY = VP * cos(β)

Right? ;)

10. Nov 3, 2014

### haruspex

Yes, that all looks right.