Homework (Due on 9/11/25, 23:59)

Instructions 

A. Show all of your workings and drawings clearly.

B. Submit your homework as a pdf. The homework can be typed, written digitally or via pen and paper. Do ensure that the orientation of your submitted homework is upright.

C. You may use numerical software such as MATLAB for this homework. However, you have to sketch the root locus manually and not generate it via MATLAB or other software. This root locus sketch can be done via pen and paper, or digitally.

1. (a) Figure 1 shows gear 1 driving gear 2 and a cable drum, where gear 2 and the cable drum share the same rotating shaft. Gear 1, gear 2 and the cable drum have a radius of 3r, 5r and 2r, respectively. The moment of inertia of gear 1 is J while the combined moment of inertia of gear 2 and the cable drum is 4J. As the cable drum rotates, it will also hoist up a weight of mass M. All the angular and linear displacements of the components are shown in Figure 1. The rotary damping coefficient of gear 1 is c. The combined rotary damping coefficient of gear 2 and the drum is 2c. The linear damping coefficient of the mass is c. To actuate this system, we have mounted a motor at gear 1 and the torque supplied by the motor is given as. Consider the weight of the mass and assume zero initial conditions to derive the block diagram of the system where the displacement x is the output. Express ℒ(τ) = T(s).  (6 marks)
   
   (b) Determine and explain if the system is stable for output x. (3 marks)

    

   Fig. 1. A hoisting system, where a motor torque of  is generated at gear 1 to lift the mass. 

2. Draw the root locus for the unity feedback system shown in Fig. 2 by answering all the questions below.

Fig. 2. A unity feedback system.