30 01 17

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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 30.01.2017 Proff. Braghin, Collina, Sabbioni Consider the mechanical system represented in the figure placed in the vertical plane. A DC motor is connected to the driving pulley, with radius R 1, of a conveyor belt. The motor rotor inertia is Jm while the stator inertia is J s. Two springs with stiffness k connect the stator to the ground. The conveyor belt is connected to a disk with rolling radius R2 while the outer radius is 2R2. A bar with mass M is moved by the disk with a velocity v. A force F, opposite to the bar motion, is applied to the bar with modulus dependent on velocity v as shown in the graph. Consider the actuator as ideal (La → 0) and the springs 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 bar and analyse the system stability in the time domain. 2. Apply a Proportional-Integral control on the velocity of the bar 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. Consider now the actuator dynamics (La ≠ 0) and still consider the springs as completely rigid (k → ∞): 3. Write the equation of motion of the system, introducing a Proportional -Integral control on the velocity of the bar. Analyse the stability of the controlled system in time domain. 4. Draw the block diagram of the control system and compute the open and closed loop transfer functions between the velocity of the bar and the reference. 5. Analyse the stability of the control system in Laplace domain considering both cases (stable/unstable uncontrolled system) , studying the effect of control parameter s. For each case…