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# 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)* eax cos bx + k eax sin bx

where a and b are coefficients of the solution r = a + bi for the quadratic equation 24.2r2 + zr + 2400 = 0  letting z=400

and k = (-1/12 * a)/bNote 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