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. An electrical actuator provides

the input displacement, x, to the rig, based on a control input, z. The corresponding displacement of the mass is y. 0.1 9 Electrical actuator 10 11 00 W Figure Q2.1 (a) Demonstrate that the transfer function for the plant G₂(s) = for k=10000 N/m and m = 200kg, can be written as G₂ (s) = +100 Y(a) X(s) M Calculate the magnitude and the phase of G₂, (s) at the frequencies given in the table (magnitude in linear scale, phase in degrees) and sketch a polar plot using the calculated points. rad m F ܠܠܠܠܠܦܝ in ¡Y (@j)| IX (@j)] Y(@j) ²X(wj) As a control engineer you must decide on a suitable controller for this plant and on what should be the minimum sampling frequency of any sensors. Note the frequency of interest is usually between 0.5-50Hz and the surroundings are noisy. Justify your answers./n(b) Practical experimentation on the system has indicated that the electrical actuator has its own transfer function, G. (s): where a is the actuator parameter. Write down the closed loop characteristic equation for the complete system, that is 1 + G₂ (s)G, (s)G(s)H(s) = 0, when using the controller expression you defined as suitable in (a). Rewrite the CLCE in its standard form, ready to sketch the roots loci for the complete system when studying varying values of a, the actuator parameter. (c) Figure Q2.2 shows the roots loci for the complete system when studying varying values of a, the actuator parameter: Imaginary Axis (seconds) 50 10 гда -10 -50 0.94 0.975 0.994 120 0.994 x(s) G₁(s) = u(s) s+a 0.975 0.94 100 0.88 80 0.88 -140 -120 -100 Root Locus 0.8 0.68 0.5 0.25 60 40 20 0.8 0.68 0.5 0.25 -77 -60 -40 -25 Real Axis (seconds) 0 20 Figure Q2.2 Calculate the minimum value of a that will result in all roots being real and estimate the associated settling time. Sketch the time response of the system to a unit input step for the calculated a.