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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 15.01.2018 Proff. Braghin, Collina, Sabbioni Consider the mechanical system represented in the figure above. A mass m moves on a flat ground without friction and it is connected, on the left side, to a moving plane y by means of a spring with stiffness k. On the other side it is connected to a belt transmission, infinitely rigid, driven by a DC motor whose stator is fixed in its centre while two springs of stiffness k1 limit its rotation. An external force F is applied on the mass . The force is proportional to mass speed according to the following equation 𝐹 = −𝑏𝑥̇ m Considering the DC motor as ideal (La→ 0), stator stiffness as completely rigid (k1 → ∞), and considering the constraint displacement y as a disturbance: 1. Write the equation of motion of the electro-mechanical system. Compute the required motor voltage Va0 in steady-state condition (x=0, y=0). Linearize the equation of motion around the equilibrium position and analyse the system stability in the time domain according to the value of parameter b. 2. Apply a proportional and derivative control on the mass displacement x acting on the motor voltage. 3. Calculate the frequency response function between the disturbance y and the mass displacement x. Draw the corresponding amplitude and phase diagrams, discussing the effect of the control gains. Considering now the actuator dynamics (La ≠ 0), stator stiffness as completely rigid ( k1 → ∞), and considering the constraint displacement y as a disturbance: 4. Write the equation of motion of the system , including an internal current loop : apply a Proportional-Derivative control on the mass displacement and a proportional controller on the internal current loop of the DC motor. Analyse the stability of the…