11 01 21 1

Pagina 1

CONTROL AND ACTUATING DEVICES FOR MECHANICAL SYSTEMS 11.01.2021 Consider the mechanical system represented in the figure placed in the vertical plane. A bar of length 3L, with mass 𝑀1 and moment of inertia 𝐽1, slides without friction on a horizontal plane and is connected to the rod of a hydraulic actuator by a spring-damper group characterized by stiffne ss 𝑘1 and damping coefficient 𝑟1. The piston of the hydraulic actuator has a mass equal to 𝑚𝑝. A disc with mass 𝑚𝑑, moment of inertia 𝐽𝑑 and radius R is pinned to the ground is its centre and is in contact with the bar. The contact is such that slippage is prevented. A bar with length 2L, mass 𝑀2 and moment of inertia 𝐽2 is hinged to the disc at one end, while is connected to a massless beam on the other end. Such beam is connected to the ground both with a pin and with a torsional spring-damper group with torsional stiffness 𝑘𝑇 and damping coefficient 𝑟𝑇. All the moments of inertia are computed with respect to the centre of mass. Consider the elongations of the springs to be null in the configuration depicted in figure. Consider the actuator as ideal (β→∞) and the spring K1 as completely rigid (k1→∞): Collect only the required information (in red) into a single document and upload it before the end of time defined for the first block. Write the equation of motion of the system and linearize it around the static equilibrium position depicted in the figure ( = 0). 1. Linearized equation of motion, indicating the equation of the generalized mass, damping, stiffness and external force. 2. Stability analysis Apply a Proportional-Derivative control on the rotation 𝜃 of the beam acting on the displacement of the spool valve. 3. Linearized equation of motion, indicating the equation of the generalized mass, damping,…