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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 21.07.2017 Proff. Braghin, Collina, Sabbioni Consider the mechanical system represented in the figure above. A DC motor drives a mass m moving on a slope through a double pulley system (mass M, inertia J and radius R). Between the motor and the first pulley there is a gearbox ( unitary efficiency and gear ratio τ), while the cable between the two pulleys can be modelled as a parallel spring-damper system (k, r). The friction between the mass m and the slope can be modelled considering a dynamic friction coefficient µd. The relationship between the friction coefficient and the mass velocity is represented in the figure. α m µd Motorτ M, J, R µd v Front view Lateral view M, J, R k, r v0 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; linearize the equation of motion around the steady state velocity of the mass m and analyse the system stability in the time domain. 2. Apply a Proportional control on the velocity of the mass m 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 actuator 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 mass m and a proportional controller on the internal current loop of the DC motor. Analyse the stability of the…