Damped Harmonc Oscillation with No Driving Force
This note discusses the damped simple harmonic oscillator with and without a driving force. I ask you to refer to Chapter 16 of the text - go to the differential equation which follows eq 16-39 in the text. That differential equation reads:
We will solve this differential equation by first turning it into an alegrabic equation. This will be done by assuming a solution of the form:
You will recall that:
So we see that equation (2) resembles equation (16-40) in the text. Now we will see the power of using exponentials:
Dividing (1) by m and substituting using (3) we find:
You might do this for yourself and then look at the details of the derivation.
Notice, that if b = 0, there is no damping and the frequency is just the natural frequency of the system. More examples and plots of damped harmonic motion are available. Forced harmonic motion is also discussed.