# Ball and Hoop

S

#### Sobhath Ramesh Ekanayake

I have used Lagrangian to model the Hoop-Ball relationship, then used Tellegen's theorem to get Motor-Hoop relationship.

I have simplified the Motor-Hoop Transfer Func. (TF) to

M(s) = K*wn = Theta(s)
------- --------
s(s+wn) R(s)

Where wn = 2, K = 1.01, Theta(s) is hoop position in radians, R(s) is input reference voltage to motor.

I don't seem to have got the correct Hoop-Ball model, thought. I get a TF that looks like,

H(s) = sKb = Phi(s)
----------------------- ------
2 Theta(s)
Jb*s + Kb*s + mg(rh - rb)

obtained using Lagrangian Equations, where Phi(s) is ball position with respect to vertical in radians.

If I put the two systems together to model the entire Ball 'n Hoop, I don't get a realistic step response. With a unit-step the ball seems to maintain a non-zero angle with constant hoop velocity. Instead it should oscillate then damp to zero.

I hypothesised using sH(s) would make the simulation more realistic, and it did.

But I'm not sure why my Lagrangian didn't give me a correct solution.

Can u please help me model this system, and get a proper step response. I have included a schematic diagram.