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# NHA2414: Draw a free body diagram (FBD) and derive the equation of motion of m with y(t) as the input, and obtain the transfer function: Dynamic Analysis and Control Assignment, UOH, UK

The quarter-car model of a vehicle suspension and its free body diagram is shown in Figure 1. In this simpliﬁed model, the masses of the wheel, tire, and axle are neglected, and the mass m represents one-fourth of the vehicle mass. The spring constant k models the elasticity of both the tire and the suspension spring. The damping constant c models the shock absorber. The equilibrium position of m when y=0 is x=0. The road surface displacement y(t) can be derived from the road surface profile and the car’s speed.

1. Draw a free body diagram (FBD) and derive the equation of motion of m with y(t) as the input, and obtain the transfer function.

If assume:

m=250 kg

k=10000, 30000, 50000 N/m

c=1000, 2000, 3000 N.s/m

1. Plot magnification ratio vs frequency ratio (r=0-4) diagrams for the parameters given above (you can draw the three curves in one diagram for three different k values and do the same for the three c values as well).
2. Use the derived transfer function to model the system and plot the step response for the system by Matlab or Simulink.