exam 20210615

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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 15.06.2021 Consider the mechanical system represented in the figure placed in the vertical plane. A mass m is moved on an inclined plane with slope equal to α through one pulley (mass M, inertia J and radius R). The mass m is connected to a cable that is fixed at the opposite end to the piston of a double acting hydraulic actuator driven by a servo- valve. Consider the cable between the pulley and the piston as flexible (stiffness k and damping r). There is no sliding between the cable and the pulley. The mass m is subjected to dynamic friction. The behaviour of the friction coefficient µd as a function of the relative velocity is represented in the figure. All the moments of inertia are computed with respect to the centre of mass. Consider the actuator as ideal (β → ∞) and the cable as completely rigid (k → ∞): 1. Write the equation of motion of the mechanical -hydraulic system; linearize the equation of motion around the steady state velocity and analyse the system stability in the time domain. 2. Apply a Proportional control on the velocity of the mass m acting on the spool displacement of the servo- valve. Given a step reference, compute the system time response and plot it discussing the effect of the control gain. Consider now the hydraulic actuator dynamics (β ≠ 0) and still consider the cable as completely rigid (k → ∞): 1. Write the equation of motion of the system in the Laplace domain, introducing a Proportional- Integral control on the velocity of the mass m acting on the spool displacement of the servo- valve. Analyse the stability of the control system in the time domain. 2. Draw the block diagram of the controlled system. 3. Compute the open and closed loop transfer functions between the mass velocity…