# What are the modes of vibration on an oscillating spring?

I am looking at the problem of a coupled oscillator, whereby we have 3 springs connected between two walls in the following way: wall, then spring (k), then mass (m), then spring (2k), then mass (m), then spring (k), then wall.

I have calculated the characteristic frequencies (I think) by using the fact that we will have

$$m \dfrac{d^2x}{dt^2}=-kx+2k(y-x)=-3kx+2ky$$ and $$m\dfrac{d^2y}{dt^2}=-ky-2k(y-x)=-3ky+2kx.$$

Thus we have in a new coordinate system

$$m\pmatrix{\ddot{X}\\\ddot{Y}}=k\pmatrix{-3&-2\\2&-3}\pmatrix{x\\y}.$$

Evaluating the matrix in the previous expression I obtain Eigenvalues $\mu_1=-5$ and $\mu_2=-1$. The the unit Eigenvectors will be $$\hat{e}_1=\dfrac{1}{\sqrt{2}}\pmatrix{1\\1}$$ and $$\hat{e}_2=\dfrac{1}{\sqrt{2}}\pmatrix{1\\-1}.$$

So in the new coordinate system I have $$\pmatrix{\ddot{X}\\\ddot{Y}}=\dfrac{k}{m}\pmatrix{\mu_1&0\\0&\mu_2}\pmatrix{X\\Y}.$$ This implies that $$\ddot{X}=-\dfrac{5k}{m}X$$ and $$\ddot{Y}=-\dfrac{k}{m}Y.$$

Solving these equations we obtain $$X=Asin(\omega_1t+\alpha)$$ and $$Y=Bsin(\omega_2t+\beta),$$ where $\omega_1=\sqrt{\dfrac{5k}{m}}$ and $\omega_2=\sqrt{\dfrac{k}{m}}$.

Now I need to find the characteristic frequency and the mode of vibration. (Please do not give any answers as this is a homework assignment that I would like to solve myself.)

Q1. Am I correct in thinking that the characteristic frequency is simply $f=\omega/2\pi$, or am I misunderstanding what the characteristic frequency is?

Q2. What is the mode of vibration? I understand what this is for a longitudinal wave oscillating between two fixed points, but I think this system is a transverse one, and I am struggling to think what the modes of vibration could be.

Please can someone verify whether my understanding of what the characteristic frequency is is correct, and if not could you explain what it is. Also, can someone explain what the mode of vibration is in a transverse system like this one.

• fyi, In your Latex, you need to escape the "sin" using "\" so it looks right. – Nasser Jun 2 '16 at 3:14
• Thank you, however you are about half a year too late, realised this not too long ago ;) – ODP Jun 18 '16 at 20:28

The eigenvalues are frequencies of two normal modes of oscillations. Now, there may be different conventions, and one may define frequency as $\omega$ or $\frac{\omega}{2\pi}$, but in my opinion, $\omega$ will do.