Question 1 [10 marks] [60 minutes] [LO. 2] m High-speed electric trains use a mechanical arm called a pantograph (Figure

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Question 1 10 Marks 60 Minutes Lo 2 M High Speed Electric Trains Use A Mechanical Arm Called A Pantograph Figure 1 (151.55 KiB) Viewed 26 times

Question 1 [10 marks] [60 minutes] [LO. 2] m High-speed electric trains use a mechanical arm called a pantograph (Figure 1) to transfer electric current from an overhead wire to the train. The pantograph typically consists of a two- arm frame linkage that provides an upward force to maintain contact between a small pan-head and the catenary wire. Figure 2 shows a two-mass lumped mechanical model of the pantograph where m₁ = 12 kg is the head mass, m₂ = 5 kg is the frame mass, and k, = 0.1 is the stiffness of the "shoe" contact between the head and catenary wire. The head suspension is modeled by lumped stiffness k₂ = 2.7 and lumped friction coefficient b₁ = 3.8 while the frame suspension only involves a lumped friction coefficient b₂ = 4.4 - A pneumatic piston provides the force fa(t) that pushes up on the frame so that the shoe remains in contact with the wire. Displacements z, and z, are measured from the static equilibrium positions and zw(t) is the displacement of the overhead wire. m/s N m/s a) Derive the mathematical model of the pantograph system assuming z; > Zw, Z₂ > z; (i.e., both springs are in compression) and 2, >2,,and 2₂ > 0. Note that the stiffness element k1 can only be in compression; that is, the wire cannot "pull" in tension on the head mass. Note that for this system, you should not consider the gravity forces. [3 marks] b) Obtain a complete SSR where the two inputs are overhead wire displacement z₁, (t) and piston force fa(t) and the two measured outputs are the (compressive) contact force between the wire and head mass and the relative displacement between the head and frame masses (i.e., Z₁ — Z₂). You may assume that the spring k, is always in compression. Note that for state variables, you need to consider positions and velocities of the system inertias. You are required to show all the steps involved in finding your final answer, even the smallest details. Make sure in your final answer, all numerical values are substituted and equations are simplified to the simplest form. [7 marks] Upper arm of frame Pan-head Lower arm of frame Base Figure 1 schematic of the pantograph Overhead Wire Head suspension { Frame Head Frame suspension k₂ X₂ www {† www 3 IN Contact shoe C 15 B₂ 140 D Base Figure 2 schematic of the equivalent system for pantograph