Let us consider a device that consists of a ball rolling on the inside rim of a

Question:

Let us consider a device that consists of a ball rolling on the inside rim of a hoop [11]. This model is similar to the problem of liquid fuel sloshing in a rocket. The hoop is free to rotate about its horizontal principal axis as shown in Figure E7.1. The angular position of the hoop may be controlled via the torque T applied to the hoop from a torque motor attached to the hoop drive shaft. If negative feedback is used, the system characteristic equation is

(a) Sketch the root locus.
(b) Find the gain when the roots are both equal.
(c) Find these two equal roots.
(d) Find the settling time of the system when the roots are equal.