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Sphere rolling

  1. Aug 16, 2010 #1
    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. jcsd
  3. Aug 20, 2010 #2
    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: [tex] C = (1+R) e_{r}[/tex]
    - relation for rolling without sliding: [tex] \dot{C} = R * \phi \times e_{r} [/tex],

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

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

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

    [tex]d \phi_{\bot}=d \omega (1+R)/R[/tex].

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

    M
     

    Attached Files:

  4. Aug 21, 2010 #3
    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
     
  5. Aug 21, 2010 #4
    Hi-
    [tex]\times[/tex] means vector product. [tex]\phi[/tex] is the (free) angular velocity vector describing the rotation of the ball. [tex]\omega[/tex] is the (free) angular velocity vector describing the rotation of the versor [tex]e_{r}[/tex] (which I used for the lagrangian parameters of the moving ball center).

    M
     
  6. Aug 21, 2010 #5
    thank you very much
     
  7. Aug 21, 2010 #6
    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?
     
  8. Aug 22, 2010 #7
    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)?
     
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