**Task 1**

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.

- 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

- 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).
- Use the derived transfer function to model the system and plot the step response for the system by Matlab or Simulink.

**Task 2**

A common example of base excitation is caused by a vehicle moving along a bumpy road surface as shown in Figure 2. This motion produces a displacement input to the suspension system via the wheels. The second task is to calculate and draw a displacement transmissibility ratio diagram for a quarter car with 250 kg, the spring constant is 10000 N/m, but varying damping constant to be 1000, 2000, 3000, 5000, and 10000 N.s/m. If the vehicle driver wishes to reduce the vehicle’s body displacement, what suggestion you could make for the driver and why?

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