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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 15.01.2016 Consider the mechanical system represented in the figure placed in the vertical plane. The stator of a DC motor (light grey), having inertia Js, is connected to the ground through two equal spring-damper systems (k2, r2). The rotor (dark grey), having external radius r and inertia Jm, moves a disc (inertia JD and radius RD) through a rigid cable. The disc moves a massless vertical beam, hinged to a horizontal beam (mass M, inertia J, length L) in B. This beam is connected to another massless vertical beam in C. A torsional spring-damper element (r1, k1) connects the beam CD to the ground. Consider the motor as ideal (La→0) and the spring K2 as completely rigid (K2→∞): 1. Write the linearized equation of motion of the electro-mechanical system and study its stability in the time domain. 2. Apply a Proportional-Derivative control on the horizontal displacement of M acting on the motor voltage. Compute the time response of the controlled system to a step reference input. 3. Draw the time history of the response and discuss how it is affected by the variation of the control gains. Consider now the motor dynamics (La ≠ 0) and still consider the spring K2 as completely rigid (K2→∞): 4. Write the equation of motion of the system, introducing a Proportional-Integral control on the horizontal displacement of M and a Proportional current feedback loop. Analyse the stability of the controlled system in time domain. 5. Draw the block diagram of the controlled system and compute the open loop and closed loop transfer functions. 6. Analyse the stability of the controlled system in Laplace domain under the assumption of unstable uncontrolled system, studying the effect of control parameters. For each case analysed, draw…