Inelastic collision of ball with rotating beam (juggling robot)

[madsci]

Homework Statement

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 $\theta$ and rotational rate $\dot{\theta}$. After collision, since the ball's position is constrained to the surface, it is parameterized by where $\phi$, 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 $\dot{\theta}$.

Homework Equations

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

The Attempt at a Solution

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

 

[anathema]

Thank you for sharing your project with us! Based on the information you have provided, it seems like you are on the right track in terms of understanding the dynamics of the system. In order to solve for the post-collision state, you will need to use the principle of conservation of momentum and the equations for angular momentum.

First, let's consider the conservation of momentum in the tangential direction. As you mentioned, the component of the ball's pre-collision velocity that is tangential to the beam's surface will become the post-collision rolling velocity. This means that the tangential component of the ball's momentum will be transferred to the beam. So, using the equation you provided, we have:

m_b v_{b,t} + I_b \omega_b = (m_b + m_{beam})v_{beam,t}

Where m_b is the mass of the ball, v_{b,t} is the tangential component of the ball's pre-collision velocity, I_b is the moment of inertia of the ball, \omega_b is the angular velocity of the ball, m_{beam} is the mass of the beam, and v_{beam,t} is the tangential component of the beam's post-collision velocity.

Next, let's consider the conservation of angular momentum. This means that the total angular momentum of the system before and after the collision will remain the same. So, we have:

I_b \omega_b + I_{beam} \omega_{beam} = I_b \omega_{b,final} + I_{beam} \omega_{beam,final}

Where I_{beam} is the moment of inertia of the beam, and \omega_{beam} is the angular velocity of the beam before and after the collision.

Now, using these two equations, you can solve for the post-collision velocities and angular velocities of the ball and beam. Keep in mind that the post-collision velocities of the ball will depend on the angle \phi and the post-collision angular velocity of the beam will depend on the angle \theta, so you will need to use the equations you provided for x and y to solve for these angles.

I hope this helps you solve for the post-collision state of your juggling robot. Good luck with your project!