Complex Numbers Applied To a 1986 Toyota Suspension System
1. The equation y=(1/12) * e^{8.264x} cos(5.536x)
 0.1244 * e^{8.264x} sin(5.536x) describes
vertical motion in feet (y) versus time in seconds (x) for a mass of
my old 2600 lb 1986 Toyota that is (was) supported by springs
and a dampener (shock absorbers). The spring constant and damping
coefficients were determined experimentally (in my garage and on my
computer). The higher the spring constant value is, the stiffer
the spring, the higher the dampening coefficient b value is, the
stiffer the shock absorber.
Here is the graph of position y vs. time x.
2. The general equation for this model when the dampening
coefficient z (stiffness of the shock) is 400 is
y = (1/12)* e^{ax }cos bx
+ k e^{ax} sin bx
where a and b are coefficients of
the solution r = a + bi for the
quadratic equation 24.2r^{2} + zr + 2400
= 0 letting z=400
and k = (1/12 * a)/b.
Note that "b" is the coefficient of the
imaginary part of our complex solution!
The equation derived is y=(1/12) * e^{8.264x} cos(5.536x)
 0.1244 * e^{8.264x} sin(5.536x)
NOTE: You can now reduce stiffness of your shocks (z) and derive
and graph an equation to predict the effect. You are able to
model system behavior for worn out shocks!
For a PowerPoint Presentation showing an animation of the motion,
go to Mathematics
Applications PowerPoint Presentation. For a detailed project
you can use in a Differential Equations course, go to
Differential Equations Car
Project.
