Let us take an example of an automobile suspension (a second order system).

How does your automobile keep its tires on a bumpy road?

From Newton's first law, there is a balance between the
forces acting on a body and its mass x acceleration.

This is rearranged to give:

Where:

- W is the suspended weight
- g is Gravity
- K is Hooke's constant
- C is a viscous damping constant

If we take t2 =
W/gcK and 2zt = C/K and** F** = F/K
then:

Taking the Laplace transform (realizing that X(0) and dX/dt]x=0_0)

For a transfer function of:

To understand how these dynamics respond to a "bump" lets take a forcing function of a step response, i.e., F(s) = 1/s then

Which has the form:

It is apparent that nature of the solution to this problem depends on z.

Note the inversions of the Laplace Transforms on pages 48-49, examples 21-23.

Case 1 : zis <1

use ex. 21

or eq 22:

This solution is obviously oscillatory over a period of time.

Case 2: z= 1

The roots are equal and real and

Case 3: z>1 roots are real and unequal: we have solved this before.

Which has the form:

which has a response to a step function as equation 20

The decay ratio is = exp (-2¹z/ (1 - z2)1/2)

The Overshoot is = exp (-¹z/ (1 - z2)1/2)

The Period of oscillations is (1/2¹) [(1 - z2)1/2/t]

What does this mean with respect to the original variables ?

The decay ratio is = exp (-2¹z/ (1 - z2)1/2)

The Overshoot is = exp (-¹z/ (1 - z2)1/2)

The Period of oscillations is (1/2¹) [(1 - z2)1/2/t]

What happens if W( the car weight), C(the damping constant) or K (the spring constant) increase or decrease ?

What are the generic problems that this solution represents
?