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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 18.06.2018 Proff. Braghin, Collina, Sabbioni Consider the mechanical system represented in the figure above. A DC motor drives a disc (mass M, radius R, inertia J) rolling without sliding on a slope through a double pulley system (all having the same mass M, inertia J and radius R). Between the motor and the first pulley there is a gearbox (ideal efficiency and gear ratio ), while the cable between the two pulleys can be modelled as a parallel spring-damper system (k, r). A torque TR acts on the rolling disc as shown in the figure. This torque can be modelled a quadratic function of the angular speed of the disc. a Motor M, J, R Front view Lateral view M, J, R k, rTR(w) M, J, R TR(w) = a + bw + cw2 Consider the DC motor as ideal (La→ 0) and the spring k as completely rigid (k → ∞): 1. Write the equation of motion of the electro-mechanical system; find the steady-state velocity of the system, then linearize the equation of motion around the steady state velocity and analyse the system stability in the time domain. 2. Apply a Proportional control on the velocity of the centre of the rolling disc acting on the motor voltage. Given a step referen ce, compute the time response of the system and plot it discussing the effect of the control gain. 3. Calculate the frequency response function between the reference and the mass velocity. Draw the corresponding amplitude and phase diagrams, discussing the effect of the control gain. Consider now the motor dynamics (La ≠ 0) and still consider the spring k as completely rigid (k → ∞): 4. Write the equation of motion of the system , including an internal current loop : apply a Proportional-Integral control on the velocity of the centre of the rolling disc and a proportional…