Last week, we discussed the one-dimensional problem of two identical masses connected to each other by a spring of constant *k*_{2}, and to fixed walls by springs of constant *k*_{1}, with the springs all at resting length at equilibrium. Note that the resting length of the springs was not a factor in the computations or the solution. Suppose we extend the line to three masses instead of two, giving us two springs of constant *k*_{2}.

What is the behavior of these masses for small displacements along the line?

In analogy to the previous problem, let the position of the masses be given by *x*_{1}, *x*_{2}, and *x*_{2}, with each being zero at the initial positions, and positive values for rightward displacement. Using *ΣF*=*ma*, we have equations of motion:

,

,

and

.

We again let and , and use matrices to combine the equations. Doing this, we have:

.

Here, we again attempt a solution of the form

.

Substituting, we obtain:

So again we have that is an eigenvector of the matrix with eigenvalue *r*^{2}.

Using: , we find the equation for the eigenvalues *x* to be:

This gives us eigenvalues:

,

,

,,

all of which are negative (as ). Thus we have oscillatory behavior, with angular frequencies , , and (in increasing order).

Finding the corresponding eigenvectors, we see they are , , and .

We note that for *v*_{1}, all three terms are positive; for *v*_{2}, the middle mass is stationary and the other two move in opposite directions; and for *v*_{3}, the middle mass moves opposite the other two.

We can express this as changes of phase as we move from left to right: the lowest mode has zero changes in phase; the next mode has one change in phase, at the middle mass; the highest mode has two changes of phase, between each mass. Compare this to last week’s result, when we had two modes: the lower-frequency mode had no phase changes, the higher one had one phase change.

### Like this:

Like Loading...

*Related*

Tags: Coupled Oscillator, Friday Physics, physics, Vibrational Modes

This entry was posted on July 4, 2008 at 12:01 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

July 11, 2008 at 12:01 am |

[…] Friday 28 In the previous two posts (here and here), we looked at a collection of masses (2 and 3,respectively) connected in a line by springs of […]

July 18, 2008 at 5:49 am |

[…] ith mode has i-1 zeroes (nodes) in the interval 0<v<n+1, and thus i-1 changes of phase, as we observed previously (noting the nodes via changes in relative phase). Note that our horizontal displacement of the […]