Normal mode

We are all quite familiar with a simple harmonic oscillator, such as a frictionless pendulum, or a mass-on-a-spring system. Since there is no friction (or energy dissipation), given a non-zero initial condition, it will swing back-and-forth, with the same amplitude, and the same frequency. Mathematically speaking, its dynamic can be described with a simple second order Ordinary Differential Equation (ODE), x'' + \omega^2 x = 0, where \omega is the resonant frequency. For the sake of simplicity, I will let \omega = 1. Apparently, x = A sin(\omega t) is a solution, and the value A is determined by the initial conditions.

However, things will be different there is another oscillator coming into play, and the second oscillator interacts with the first one. Below is a schematic picture of such condition (courtesy of wikipedia).

Let's call the motion of the left oscillator x_1, and that of the right oscillator x_2, both are functions of time (t). The leftmost and rightmost spring dictates the spring constant of the left and right oscillator, respectively. Together with their masses, m_1 and m_2, we can have the uncoupled resonant frequencies of \omega_1 = sqrt(k_1/m_1), and \omega_2 = sqrt(k_2/m_2). For the sake of the simplicity, again I will let m_1 = m_2 = 1, and k_1 = k_2 = 1. Therefore, both \omega_1 and \omega_2 are 1.

Now, let's say the middle spring has a spring constant of k, and k = 0.5. I am being sloppy with units here, but let's just assume them are all in standard SI units. Now I can write down the equation of motion that describe the motions of x_1 and x_2 as:

Since we already know all the values of the parameters, I can just mindlessly solve for x_1(t) and x_2(t) numerically, say, with Mathematica. With the initial conditions of: x_1(0) = x_2(0) = 0, x_1'(0) = 1, and x_2'(0) = 0.1, below is what I get:

Neither x_1 nor x_2 looks anything like a regular sinusoidal function, as in the uncoupled case! Although if I squeeze my eyes, I can convince myself there are still some periodicities in those figures... It turns out that, instead of looking at x_1 and x_2 individually, we I should be looking at is (x_1 + x_2) and (x_1 - x_2):

Voila! We now see the nice clean waves again, although with different frequencies.

The new quantities, y_1 = (x_1 + x_2) and y_2 = (x_1 - x_2), are called normal modes, and the corresponding frequencies are the eigen-frequencies of the system. Instead of looking at the seemingly arbitrary x's, one can get a much better grip of the system by looking at the y's. The wikipedia page actually do a pretty good job on describing what transformation should one do to go from x's to y's, but here I just want to have a visual confirmation, to reveal the value of the normal mode.

Update: found a terrific site with good lecture notes on this subject!