b1 k1 b2 kg [13 k3 M It" locomotive | 1 cars Flt} Shown in the gure above is a simplied model for the alongtrack dynamics of a train pulling three...

b1 k1 b2 kg [13 k3 M It“locomotive | 1 cars Flt} Shown in the figure above is a simplified model for the along—track dynamics of atrain pulling three cars along a level section of track. The mass mmg models theengin eflocomotive unit which provides the hauling or braking force for thesystem, and the masses maul, mam-2 and man-3 model the un—powered cars that arepulled long the tracks. The coupling and buffers that connects the locomotive tothe first car and subsequent cars to one an other can be modelled using springsand dampers between each connection point as shown. In this question you will use numeric methods to determine the response of thesystem [i.e. the horizontal positions and velocities of the locomotive and threecars vs. time], based on the dynamic model shown above, to an input braking force profile which is applied when the train is travelling at a steady forwardspeed of 50 mfs. The braking force is applied in the opposite direction of motion in two pulses, such that the magnitude of force is: t = 0 to t = 1.0 seconds, F = 400kNt = 1.0 to t = 2.0 seconds, F = 0 t a 2.0 seconds, F = 300104 i. Develop the equations of motion for the train and cars model shown(10 Marlm] a. Draw four freebody diagrams, one for each of meng, mom-1, mm: andmm. Use at coordinates for each of the four masses that representtheir horizontal displacements with respect to equilibrium. b. Derive the equations of motion for the train model [differentialequations for x513, xgl, IEZ,I;3,IF;IH,I;.1, xéz, 1;.3 as a function ofFEE], mug, mam, mm, mm, k1,. k2,. k3,. b1, b2 and b3].