The system represented in the picture is placed in the vertical plane. The mass m is...

  

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The system represented in the picture is placed in the vertical plane. The mass m is moved horizontally by the rotor of a DC motor (radius Rr, inertia Jr, motor resistance Ra, motor inductance La, motor constant K'T') through two pulleys. The radius of each one of the two pulleys is equal to R, while their inertia is equal to J. Consider the friction force between the mass m and the ground; the behavior of the friction coefficient fd with respect to the mass velocity is represented in the plot. The stator of the motor (inertia Js) is connected to the ground through two spring-damper systems. The stiffness of the two springs is equal to k, while the damping coefficient of the dampers is equal tor. The distance between the two spring-damper systems is equal to 2L. Consider La = 0 and the stiffness of the springs k as infinite: 1. Develop the mathematical model of the electro-mechanical system; linearize it around a steady state velocity v. Consider now La f.. 0 and the stiffness of the springs k as infinite: 2. Develop the linearized equation of motion of the system, considering a Proportional-Integral control on the velocity of the mass m, acting on the motor voltage. 3. Draw the block diagram of the system described at point 2 and compute the open-loop and closed-loop transfer functions. 4. Analyze the system stability in the Laplace domain, considering both the cases of stable and unstable uncontrolled system. Develop the Bode plot for analyzed cases. 5. Develop the root locus for the analyzed cases k,r J,R Js Complex Engineering Problem m, fd 2L V Jr, Rr J,R k,r fd fdo Viim V The system represented in the picture is placed in the vertical plane. The mass m is moved horizontally by the rotor of a DC motor (radius Rr, inertia Jr, motor resistance Ra, motor inductance La, motor constant K'T') through two pulleys. The radius of each one of the two pulleys is equal to R, while their inertia is equal to J. Consider the friction force between the mass m and the ground; the behavior of the friction coefficient fd with respect to the mass velocity is represented in the plot. The stator of the motor (inertia Js) is connected to the ground through two spring-damper systems. The stiffness of the two springs is equal to k, while the damping coefficient of the dampers is equal tor. The distance between the two spring-damper systems is equal to 2L. Consider La = 0 and the stiffness of the springs k as infinite: 1. Develop the mathematical model of the electro-mechanical system; linearize it around a steady state velocity v. Consider now La f.. 0 and the stiffness of the springs k as infinite: 2. Develop the linearized equation of motion of the system, considering a Proportional-Integral control on the velocity of the mass m, acting on the motor voltage. 3. Draw the block diagram of the system described at point 2 and compute the open-loop and closed-loop transfer functions. 4. Analyze the system stability in the Laplace domain, considering both the cases of stable and unstable uncontrolled system. Develop the Bode plot for analyzed cases. 5. Develop the root locus for the analyzed cases k,r J,R Js Complex Engineering Problem m, fd 2L V Jr, Rr J,R k,r fd fdo Viim V