Differential Equations/Homogenous 2

From Wikibooks, the open-content textbooks collection

[edit] Mechanical Vibrations

One place homogenous equations of constant coefficients are used is in mechanical vibrations. Lets imagine a mechanical system of a spring, a dampener, and a mass. The force on the string at any point is F = - kx where k is the spring constant. The force on the dampener is F = − cv where c is the damping constant. And of course, the net force is F = ma. That gives us a system where

ma = − cv − kx

Remember that v = x' and a = x''. This gives us a differential equation of

mx'' = − cx' − kx

mx'' + cx' + kx = 0

In the case where c=0, we have just a mass on a spring. In this case, we have . Since k and m are both positive (by the laws of phsyics), the result is always a . This makes sense from a physical perspective- a spring moving back and forth forms a periodic wave of frequency