2. Magnetic levitation trains provide a high-speed, very low friction alternative to steel wheels on steel...

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2. Magnetic levitation trains provide a high-speed, very low friction alternative to steel wheels on steel rails. The train floats on an air gap, z, as shown in the figure below. The levitation force F is controlled by the coil current i in the levitation coils and may be approximated by FLK i Assume the train has mass M and denote gravity by g. The vertical displacement of the train is y; the vertical displacement of the guideway is r. Note, z = y-r. Also, this system has two inputs, i(t) and r(t). (a) (10%) Draw a free body diagram of this system. Hint, show the lift force represented by a nonlinear spring between the guideway conducting plate and the base of the vehicle. y L Train car Rotor coil Levitation" coil Rotor coil Stator coils Conducting plate Levitation coil b. (5%) This system has two inputs, guideway elevation changes r and the current i in the coil. Assuming the desired equilibrium gap is ze, determine the required equilibrium current i in terms of ze. (c) (15%) Write the component equations for the vertical dynamics of this suspension system. (d) (10%) Linearize your equation for F for small perturbations about equilibrium, i.e. FL FLe +ai + bz, a=? b=? (e) (15%) Using your linear equation found in part d, solve for the transfer function relating Y(s) to R(s) and for the transfer function relating Y(s) to I(s). 2. Magnetic levitation trains provide a high-speed, very low friction alternative to steel wheels on steel rails. The train floats on an air gap, z, as shown in the figure below. The levitation force F is controlled by the coil current i in the levitation coils and may be approximated by FLK i Assume the train has mass M and denote gravity by g. The vertical displacement of the train is y; the vertical displacement of the guideway is r. Note, z = y-r. Also, this system has two inputs, i(t) and r(t). (a) (10%) Draw a free body diagram of this system. Hint, show the lift force represented by a nonlinear spring between the guideway conducting plate and the base of the vehicle. y L Train car Rotor coil Levitation" coil Rotor coil Stator coils Conducting plate Levitation coil b. (5%) This system has two inputs, guideway elevation changes r and the current i in the coil. Assuming the desired equilibrium gap is ze, determine the required equilibrium current i in terms of ze. (c) (15%) Write the component equations for the vertical dynamics of this suspension system. (d) (10%) Linearize your equation for F for small perturbations about equilibrium, i.e. FL FLe +ai + bz, a=? b=? (e) (15%) Using your linear equation found in part d, solve for the transfer function relating Y(s) to R(s) and for the transfer function relating Y(s) to I(s).