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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 01.09.2017 Proff. Braghin, Collina, Sabbioni Consider the vehicle represented in the figure above. A DC motor drives the vehicle, with overall mass m (vehicle + wheels + motor), on a slope through a belt transmission, with stiffness k, which connects the motor pulley (radius r m, inertia J m) to the rear wheels pulley (radius r ). The wheels (inertia J and radius R) rotate without sliding on the road surface. An aerodynamic force 𝐹𝐹= 𝑏𝑏(𝑣𝑣− 𝑢𝑢)2 opposes to the vehicle motion (u is the wind speed). m Considering the DC motor as ideal ( La → 0), the belt as completely rigid (k → ∞), and considering the wind speed u as a disturbance: 1. Write the equation of motion of the electro -mechanical system. Compute the required motor voltage Va0 in steady-state condition (v=v0, u=u0). Linearize the equation of motion around the steady state velocity of the vehicle and analyse the system stability in the time domain. 2. Choose and apply a suitable control on the velocity of the vehicle acting on the motor voltage. Justify the choice. 3. calculate the frequency response function between the disturbance u and the vehicle velocity v. Draw the corresponding amplitude and phase diagrams, discussing the effect of the control gain (or gains). Considering now the actuator dynamics (La ≠ 0), still consider the belt as completely rigid (k → ∞) and not considering the wind speed (u=0): 4. Write the equation of motion of the system , including an internal current loop : apply a Proportional-Integral control on the velocity of the vehicle and a proportional controller on the internal current loop of the DC motor. Analyse the stability of the controlled system in time domain. 5. Draw the block diagram of the control led system and compute the…