Automobile manufacturers are contemplating building active suspension systems. The simplest change is to make shock absorbers with a changeable damping, b(u1 ). It is also possible to make a device to be placed in parallel with the springs that has the ability to supply an equal force, u2 , in opposite directions on the wheel axle and the car body.

(a) Modify the equations of motion in Example 2.2 to include such control inputs.

(b) Is the resulting system linear?

(c) Is it possible to use the forcer u2 to completely replace the springs and shock absorber? Is this a good idea?

Example 2.2

A Two-Mass System: Suspension Model

Figure 2.4 shows an automobile suspension system. Write the equations of motion for the automobile and wheel motion assuming one-dimensional vertical motion of one quarter of the car mass above one wheel. A system consisting of one of the four wheel suspensions is usually referred to as a quarter-car model. Assume that the model is for a car with a mass of 1580 kg, including the four wheels, which have a mass of 20 kg each. By placing a known weight (an author) directly over a wheel and measuring the car's deflection, we find that ks = 130,000 N/m. Measuring the wheel's deflection for the same applied weight, we find that kw 1,000,000 N/m. By using the results in Section 3.3, Fig. 3.18(b), and qualitatively observing that the car's response as the author jumps off matches the t, = 0.7 curve, we conclude that b = 9800 N-sec/m. Solution. The system can be approximated by the simplified system shown in Fig. 2.5. The coordinates of the two masses, x and y, with the reference directions as shown, are the displacements of the masses from their equilibrium
conditions. The equilibrium positions are offset from the springs' unstretched positions because of the force of gravity. The shock absorber is represented in the schematic diagram by a dashpot symbol with friction constant b. The magnitude of the force from the shock absorber is assumed to be proportional to the rate of change of the relative displacement of the two masses-that is, the force = b(y - x). The force of gravity could be included in the free-body diagram; however, its effect is to produce a constant offset of x and y. By defining x and y to be the distance from the equilibrium position, the need to include the gravity forces is eliminated.