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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 03.02.2020 Proff. Bucca, Braghin, Sabbioni The vehicle represented in figure (overall mass M, comprehensive of car, motor and wheels), is driven by a DC motor (inertia 𝑱𝒎), which actuates the rear wheels through a transmission with unitary efficiency and transmission ratio 𝝉. The wheels (inertia J and radius R) rotate without sliding on the road surface. The torsional stiffness of the motor shaft is 𝒌𝑻. An aerodynamic force 𝐹𝑤 = 𝑏(𝑣 − 𝑈)2 opposes the vehicle motion (U is the wind speed, to be considered much smaller than v). Considering the DC motor as ideal ( La→ 0), the shaft as completely rigid ( 𝒌𝑻 → ∞), 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. Discuss possible advantages of using a Proportional-Derivative or a Proportional-Integral control over a purely Proportional controller on the speed of the vehicle. Choose and apply the best solution among the three. 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 shaft as completely rigid (𝒌𝑻→ ∞): 4. Write the equation of motion of the system, including an internal current loop: apply a Proportional-Integral control on the velocity of the v ehicle and a proportional controller on the internal current loop of the DC motor. Analyse the stability of the…