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CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 09.07.2018 Proff. Braghin, Collina, Sabbioni M,J,2Lmp 2L 2L r1,k1 r2,k2 A B Z=Z0eiWt C D P Consider the mechanical system represented in the figure above and placed in a vertical plane. The figure shows the static equilibrium position of the sy stem. The system is made by three uniform beams (AB, BC and CD), connected as shown in the figure and whose length is 2L. Point P represents the centre of gravity of beam BC. Consider the mass and inertia of beam BC equal to M and J respectively, while the other two beams are massless. A spring -damper system (k 2, r 2) connects the middle of the beam CD to the ground, which is moving following an imposed harmonic motion Z. The system is driven by a hydraulic double -effect actuator, whose piston is connected t o the structure in B through a spring -damper system (k 1, r 1) and whose external cylinder is fixed to the ground. The actuator is driven by a servovalve. Consider the area of both the chambers equal to S, and the internal leakage coefficient equal to Ci. Considering the connection between the actuator and the system as rigid (k 1→∞) and the actuator dynamics faster than that of the mechanical system (→∞): 1. Write the equation of motion of the system, linearizing around the equilibrium position represented in the figure. Study the stability of the system in time domain. 2. Suppose to use the actuator as damper (the servovalve is not attached to the actuator). Compute the value of the internal leakage coefficient Ci so that the non -dimensional damping of the system is greater than an assigned value h0. 3. Introduce a Proportional-Derivative control on the horizontal posi tion of point P and acting on the spool displacement of the servovalve. Compute the frequency…