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).