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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS (Proff. Braghin, Collina) 23.02.2015 k, r Js, R Jr, r JD , MD, RD Cr(wD) a Consider the mechanical system represented in the figure placed in the vertical plane. The stator of a DC motor (light grey), having external radius R and inertia Js, is connected to the ground through a spring-damper system (k,r). The rotor (dark grey), having external radius r and inertia Jr, moves a disc (mass MD, inertia JD and radius RD)through a cable unwrapping from the disc itself on a plane inclined by an angle a. The disc is subjected to a torque Cr, function of the angular speed of the disc as shown in the picture. The cable and the plane are parallel each other. Consider the motor as ideal (L→0) and the spring k as completely rigid (k→∞): 1. Write the equation of motion of the electro-mechanical system and compute the steady state condition of the system according to a given applied voltage V0. Then study the stability of the system about its steady state speed in the time domain. 2. Considering a velocity control of the disc , choose the most prope r control type (motivating the choice). 3. Draw the system response to a step reference and discuss the effect of the control gain (or gains) on the response. Consider now the motor dynamics (L ≠ 0) and still consider the spring k as completely rigid (k→∞): 4. Write the equation of motion of the system, introducing a Proportional -Integral control on the disc speed acting on the motor voltage. Analyse the stability of the controlled system in the time domain. 5. Analyse the stability of the controll ed system in the Laplace domain both for a stable and unstable uncontrolled system. 6. For each case analysed at the previous point, draw the corresponding root locus. Consider the actuator…