# Thread: Harmonic Oscillator ODE Questions

1. ## Harmonic Oscillator ODE Questions

Hi all - first time poster, hoping I might get a little guidance.

I am working through applied engineering math homework and I have a problem concerning harmonic oscillation. Based on a given diagram with a mass attached to a spring, with a dashpot attached to the mass, I am given a second order linear ODE based on the stated forces acting on the system:

my" = -ky - cy', where the position of the mass, y, is a function of time, t.

I have arranged the ODE to:

my" + cy' + ky = 0.

I am to find the general solution in terms of m, c, k, and any other arbitrary constants.

So far, I have established that the characteristic equation is:

mr^2 + cr + k = 0

In the problem statement, I am told that c and k are negligible compared to m...

Now, I am trying to solve the ODE by means of establishing the characteristic equation and solving for the characteristic roots (eigenvalues) via the quadratic formula. As you may know, there are three cases which dictate what your general solution may be:

1) b^2-4ac = 0
2) b^2-4ac > 0
3) b^2-4ac < 0

This is where I am stuck--- I cannot determine which case the characteristic roots may satisfy based on the problem statement. The fact that it is stated that "c and k are negligible compared to m" makes me feel this is the key to finding my roots -- does anyone have any pointers that might help me solve this ODE?

THANKS!!!!!

2. Originally Posted by mrwiseguy85
Hi all - first time poster, hoping I might get a little guidance.

I am working through applied engineering math homework and I have a problem concerning harmonic oscillation. Based on a given diagram with a mass attached to a spring, with a dashpot attached to the mass, I am given a second order linear ODE based on the stated forces acting on the system:

my" = -ky - cy', where the position of the mass, y, is a function of time, t.

I have arranged the ODE to:

my" + cy' + ky = 0.

I am to find the general solution in terms of m, c, k, and any other arbitrary constants.

So far, I have established that the characteristic equation is:

mr^2 + cr + k = 0

In the problem statement, I am told that c and k are negligible compared to m...

Now, I am trying to solve the ODE by means of establishing the characteristic equation and solving for the characteristic roots (eigenvalues) via the quadratic formula. As you may know, there are three cases which dictate what your general solution may be:

1) b^2-4ac = 0
2) b^2-4ac > 0
3) b^2-4ac < 0

This is where I am stuck--- I cannot determine which case the characteristic roots may satisfy based on the problem statement. The fact that it is stated that "c and k are negligible compared to m" makes me feel this is the key to finding my roots -- does anyone have any pointers that might help me solve this ODE?

THANKS!!!!!
Let's examine this discriminant:

$b^2-4ac\implies c^2-4mk$ (from your quadratic equation)

The fact that it is stated that "c and k are negligible compared to m" makes me feel this is the key to finding my roots
Given this, I would think that $c^2-4mk\longrightarrow 0$

So (in my opinion) there would only be one solution to your characteristic equation.