Use Simulink to simulate the control system of Problem 11.19, and plot the controlled response of the system. A reference voltage vr is applied to the piezoelectric block, which increases linearly from 0 to 110V in 5 s then remains constant for the following 25 s. The cantilever has a length l = 15 mm, its rectangular cross-section is defined by w = 3 mm, h = 0.5 mm, and the material elasticity modulus is E = 210 . The piezoelectric actuator is defined by la = 35 mm, A = 65 mm2 , t = 6300 kg/m3 , Em = 55 GPa, and d = 4.5 × 10-10 m/V. The strain gauges have a sensitivity Kg = 2, and the input voltage to the Wheatstone bridge is vi = 18 V. Problem 11.19 A piezoelectric block, which is fixed at one end and free at the other, is supplied a reference voltage vr and, consequently, deforms axially. Using a lumped-parameter model of the piezoelectric block with inertia, stiffness and viscous-type losses (damping) taken into account, design a unity-feedback control system, where the sensing of the block’s free end is performed by a cantilever provided with four strain gauges connected in a Wheatstone bridge. Propose physical components for the summing point and proportional controller and determine all transfer functions involved in this control system.
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