Materiale universitario di Control and Actuating Devices for Mechanical Systems per il corso di Mechanical Engineering presso Politecnico di Milano. Il documento tratta: Master of Science Course in Mechanical Engineering Politecnico di Milano - Bovisa Campus CONTROL OF MECHANICAL SYSTEMS - 059180 Prof. Edoardo Sabbioni Academic Year 2025-2026 – 1st Semester EXAM SIMULATION (total time 90 min.) Name: Surname: Personal code: 1 𝑀 = 0.5 [𝑘𝑔] 𝐽 =…
Materiale universitario di Control and Actuating Devices for Mechanical Systems per il corso di Mechanical Engineering presso Politecnico di Milano. Il documento tratta: Master of Science Course in Mechanical Engineering Politecnico di Milano - Bovisa Campus CONTROL OF MECHANICAL SYSTEMS - 059180 Prof. Edoardo Sabbioni Academic Year 2025-2026 – 1st Semester EXAM SIMULATION (total time 90 min.) Name: Surname: Personal code: 1 𝑀 = 0.5 [𝑘𝑔] 𝐽 =…
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Master of Science Course in Mechanical Engineering Politecnico di Milano - Bovisa Campus CONTROL OF MECHANICAL SYSTEMS - 059180 Prof. Edoardo Sabbioni Academic Year 2025-2026 – 1st Semester EXAM SIMULATION (total time 90 min.) Name: Surname: Personal code: 1
𝑀 = 0.5 [𝑘𝑔] 𝐽 = 0.25 [𝑘𝑔𝑚2 ] 𝑵𝒔 𝒓 = 𝟏[ ] 𝒎 𝑵𝒔 𝛼= 𝟐[ ] 𝑊ℎ𝑒𝑟𝑒 𝒎 𝑅 = 1 [𝑚] 𝑁𝑚 𝑘𝜙 = 0.001 [ ] 𝐴 𝐿𝑎 = 0.166 [𝐻] 𝑅𝑎 = 1 [𝑜ℎ𝑚] { 𝑇𝑖 = 0.33 Consider the system represented in the figure, assuming that the disturbance d is null (d=0). Part 1 1. Write the linearized equation of motion of the system (placed in the vertical plane) considering that a force 𝐹 = 𝛼𝑥̇ acts on the mass M (according to the direction indicated in the figure). Highlight the generalized mass and damping of the system. 2. Calculate the poles of the passive system. 2
Let’s control the speed 𝒙̇ of the mass M by means of the torque T, supplied by a proportional (P) regulator. 3. Analyze the stability of the feedback control system and calculate the number of unstable poles (if any). 4. Let’s now consider d≠0. Calculate the transfer function Ld between the speed 𝑥̇ of the mass M and the disturbance d and draw the Bode diagrams of Ld, highlighting the effect of the control gain Kp (assume a stable controlled system). Consider that T is provided by a permanent magnet DC motor rigidly connected to disk 1 and assume that the disturbance d is null (d=0). 5. Write the equation of motion of the electro-mechanical system. 6. Write the equation of the transfer funcion between input voltage of the motor Va and the speed of the mass M, 𝑥̇ . Part 2 Let’s control the speed 𝒙̇ of the mass M by means of a Proportional-Integral (PI) regulator acting on the armature voltage of the motor Va. 7. Draw the block diagram of the feedback control system. 8. Calculate the equation of the open-loop transfer funcion. 9. Analyse the stability of the system applying the Bode criterium (if possible), and the Nyquist criterion. Assume that the motor introduces a real negative pole at high frequency (P=-6), which only slightly modifies the position of the poles of the passive system. 10. Plot a qualitative root locus of the feedback control system. Assume that the motor introduces a real negative pole at high frequency (P=-6), which only slightly modifies the position of the poles of the passive system. 11. Write the eqation of motion of the feedback control system using the state- space form. 3
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