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Inelastic collision of ball with rotating beam (juggling robot)

  1. May 13, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to build a "juggling" robot but I'm getting stuck on the dynamics of catching. The robot follows the dynamics and terminology of a similar one presented in the attached paper (equation 1). Basically, there is a circular ball flying through the air which lands (and rolls) on a pivoting beam. The beam has some angle [itex]\theta[/itex] and rotational rate [itex]\dot{\theta}[/itex]. After collision, since the ball's position is constrained to the surface, it is parameterized by where [itex]\phi[/itex], the angle to the ball w.r.t. the beam's local coordinate frame. The surfaces are frictionless, so rolling dynamics are ignored, but the rotational inertia of the beam is considered. My question is, exactly how is the momentum transferred in the (perfectly inelastic) collision? I want the component of the ball's pre-collision velocity that is tangential to the beam's surface to become the post-collision rolling velocity, and I want the component of the ball's pre-collision velocity that is normal to the beam's surface to transfer momentum to the beam, changing the beam's [itex]\dot{\theta}[/itex].


    2. Relevant equations

    For a ball, momentum is [itex]p=m v[/itex], and for the beam, angular momentum is [itex]L=r\times p = I \omega[/itex].
    I understand that with two free-moving balls, the momentum transfer associated with an inelastic collision is [itex]m_1 v_1 + m_2 v_2 = (m_1 + m_2)v_{12}[/itex], but I don't understand how to apply that to a situation in which one of the masses is rotating about a fixed axis.

    3. The attempt at a solution

    I have the full state pre-collision [itex]\left[\theta,x,y,\dot{\theta}_-,\dot{x}_-,\dot{y}_-\right][/itex], as well as the equations relating the ball's post-collision world coordinates to parameterized coordinates:
    [itex]x=x(\theta,\phi)[/itex]
    [itex]y=y(\theta,\phi)[/itex]
    [itex]\dot{x}_+=\dot{x}_+(\theta+\phi,\dot{\theta}_++ \dot{\phi}_+)[/itex]
    [itex]\dot{y}_+=\dot{y}_+(\theta+\phi,\dot{\theta}_++ \dot{\phi}_+)[/itex].
    How do I use the dynamics, namely momentum, to solve for system's post-collision state?
     

    Attached Files:

    Last edited: May 13, 2012
  2. jcsd
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