# Sphere rolling

1. Aug 16, 2010

### hmoein

hi , every one!
I have a problem with a sphere rolling on a fixed sphere. My problem is to find relationship between coordinate of center of sphere (X,Y,Z) and orientations (alpha, beta, gamma) or Euler angles of sphere. as we know a sphere has 6 DOF in space (3 coordiantes and 3 rotation) when a sphere rolling on surface we expect that it have 3 dof beacuse of relation beween coordinate and rotation.
for example when a circle roll on a surface the x coordinate of its center is:
X=R*teta (R = radius of circle) and it has one DOF.
Like the circle rolling i want to find the relations for sphere.
thanks
hossein

2. Aug 20, 2010

### drMS

Hi
for a sphere of center C and radius R rolling on a fixed sphere centered at origin with radius 1 you have (using polar reference)

- a relation for contact: $$C = (1+R) e_{r}$$
- relation for rolling without sliding: $$\dot{C} = R * \phi \times e_{r}$$,

where $$\dot{C} = (1+R) \omega \times e_{r}$$ (the latter is the time derivative of the first eq.),

and where $$e_{r}$$ describe the versor pointing the moving ball center, $$\phi$$ is the moving ball angular velocity (or displacement) and $$\omega$$ the angular velocity (or displacement) related to $$e_{r}$$ through the relation $$\dot{e_{r}} = \omega \times e_{r}$$.

Then the ball has 3 free DOF, $$\omega$$ and $$\phi_{//}=\phi \cdot e_{r}$$, with

$$d \phi_{\bot}=d \omega (1+R)/R$$.

Look to the attached mathematica file for teh simpler case of circle rolling on circle (1 free DOF).

M

#### Attached Files:

• ###### circle.nb
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3. Aug 21, 2010

### hmoein

hi drMs
thanks for your answer. i don't understand about \\time e_{r}. and what is difference between \\phi and \\omega?
could you expalin more?
best regard
hossein

4. Aug 21, 2010

### drMS

Hi-
$$\times$$ means vector product. $$\phi$$ is the (free) angular velocity vector describing the rotation of the ball. $$\omega$$ is the (free) angular velocity vector describing the rotation of the versor $$e_{r}$$ (which I used for the lagrangian parameters of the moving ball center).

M

5. Aug 21, 2010

### hmoein

thank you very much

6. Aug 21, 2010

### hmoein

Hi drMs
suppose that the moving sphere is in contact with th efixed one at one contact point.
is the rotattion about the z axis (axis that is perpendicular to the contact surface and pass through center of sphere ) rolling?

7. Aug 22, 2010

### drMS

Hi
It is not really clear to me the question. You mean the spin motion (rotation of the ball with rotation vector parallel to the segment connecting the two centers)?