Constraint equations for a sphere rolling on a fixed sphere

[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

 

[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 the simpler case of circle rolling on circle (1 free DOF).

M

 

[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

 

[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

 

[hmoein]

thank you very much

 

[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?

 

[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)?