Shock absorbers are a type of damped oscillator.
They must be tuned to the weight of the car and the type of driving.
We are solving the harmonic oscillator equation:
We have general solutions of the form:
We seek complex solutions of the form:
We denote a complex number in "Cartesian form" as:
Here, is the real part and is the imaginary part; both are real numbers.
The complex conjugate of is:
We can also write a complex number in "polar form" as:
where is the magnitude of the complex number and is the phase. Both are real numbers.
The product of a complex number and its complex conjugate is:
The sum of a complex number and its complex conjugate is:
The difference of a complex number and its complex conjugate is:
A complex number is plotted in the complex plane such that lies in the second quadrant.
Where does the complex conjugate lie?
Ignore the marked point in the figure.
We constructed a solution of the form:
We can plot it in the complex plane and see the real and imaginary parts, and how they change in time.
We can plot the solution on the complex plane. For this, , and the amplitude is .
The solution rotates counterclockwise in the complex plane, following the rainbow from violet to red.
The real part is just the projection of the complex solution onto the real axis. Just how far along the real axis is the solution at any given time.
That looks like a time trace, but not quite, it's the real projection. The colors scheme is the same as before.
We just flip the axes to produce the time trace that you are used to seeing. The color scheme is the same as before.
We constructed a solution for the weakly damped harmonic oscillator:
where .
Click when you and your table are done.