Q2 A schematic representation of a vibration test rig is shown in Figure Q2.1, where m is a mass positioned between two springs, each of stiffness k. The connection to the ground has two dampers of damping coefficient c. An electrical actuator provides the input displacement, x, to the rig, based on a control input, u. The corresponding displacement of the mass is y. C 171 C Electrical actuator Figure Q2.1 v(s) (a) Figure Q2.2 shows the Bode plot of the plant transfer function G(s) = y(s) x(s) If m=150kg, using the Bode plot, estimate the natural frequency and the damping ratio of the plant, and derive the values of both k and c. Would modifying the mass, m, be a good strategy to obtain a larger low frequency gain or a bigger damping ratio? Give arguments to support your answer. Phase (deg) Magnitude (dB) 40 20 0 -20 -40 -60 Bode Diagram System: G1 Frequency (rad/s): 4 Magnitude (dB): 28 0 45 System: G1 Frequency (rad/s): 4. Phase (deg): -89.4 -90 -135 -180 10:1 100 10 102 Frequency (rad/s) Figure Q2.2 (7 marks) (b) Assuming initially that x= u and that the damping c in the plant is negligible. determine the gains of a proportional + derivative controller, Ge(s), that yields a critically damped closed loop system and a step settling time of 0.2s. Hint: you can use Gp(s) = $2+16 (4 marks) (c) Practical experimentation on the system has indicated that the electrical actuator has its own transfer function, Ga(s): x(s) Ga(s) = u(s) sta a where a is the actuator parameter. Sketch the roots loci for varying values of a when including both the controller calculated in point (b) and the actuator transfer function and assuming again than the plant has negligible damping, i.e. CLCE = 1+ Ga(s)G(s)G(s) = 0 with G(s) and Gp (s) as defined in (b). Hint: you can use the following: (s+60) (s+20)(s-0.18 -2.3)(s-0.18+ 2.3j) (s2+40s+400)2 (5 marks) (d) Use your roots loci sketch to determine a value of a than ensures the non- dominant CLCE roots are critically damped. Sketch the response of the controlled system to a unit step input. (4 marks)