The model of an automobile suspension

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 with   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 ?