# Coupled Oscillations

I have a burning question,

I was trying to find the solutions for a double mass coupled oscillation. So I found out the eigenvectors and then I arrived at this step

$$\left( \begin{array}{c} \ddot{x_1} \\ \ddot{x_2} \end{array} \right)=\lambda \left( \begin{array}{c} x_1 \\ \ x_2 \end{array} \right)$$
(the second matrix is without the accents, I think the latex code will take a while to refresh)

ok so my question is, why is one of the solutions displayed as

$$x_{1}+x_{2}=A_{1}\cos{(\omega t+\phi)}$$

when from the first equation, it is evident that

$$\ddot{x_1}=\lambda{x_1}$$

so

$$x_1=A_1\cos{(\omega t+\phi)}$$

I simply don't understand why the above is not acceptable. Also, I am having trouble in relating the addition of the equations (equation 2) to the solution for the eigenvectors. By the way, I also know the solution for the eigenvectors.

Last edited:

Related Other Physics Topics News on Phys.org
Isn't X_1 + X_2 a mode coordinate?

The 2 mode coordinates being the sum and difference of X_1 and X_2. They are ways of looking at the motion of the system as a whole, not X_1 and X_2 individually.

isn't the matrix constructed from x_1 and x_2 individually?

I also do not understand what is a mode coordinate, could you explain this to me if this is important?

tiny-tim
Homework Helper
I was trying to find the solutions for a double mass coupled oscillation.

I simply don't understand why the above is not acceptable.
Hi Oerg!

It is acceptable, but it's not as simple nor as conceptually deep as using normal modes such as x1 ± x2

From http://en.wikipedia.org/wiki/Coupled_oscillation#Coupled_oscillations
The apparent motions of the individual oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into normal modes.

Hi Oerg!

It is acceptable, but it's not as simple nor as conceptually deep as using normal modes such as x1 ± x2

From http://en.wikipedia.org/wiki/Coupled_oscillation#Coupled_oscillations

there was another solution that is

$$x_{1}+x_{2}=A_{2}\cos{(\omega t+\phi _{1})}$$

and then with the first equation in the original post, x1 and x2 is then given as

$$x_1=\frac{1}{2}(A_{1}\cos{(\omega _{0}t-\phi)} +A_{2}\cos{(\sqrt{3}\omega _{0}t-\phi _{1})})$$

by the way,

$$\lambda =1$$

and

$$\lambda =3$$

are the eigenvalues for the problem. So there seems to be a discrepancy for the equations for x_1 and x_2.Where have I gone wrong

help im drowning arghhhhhhh

tiny-tim
Homework Helper
So there seems to be a discrepancy for the equations for x_1 and x_2.Where have I gone wrong
i don't follow

what discrepancy are you referring to?

why is the last equation from my last post different from the correct solution to the de?

Also, how do I obtain the correct solutions from the eigenvector

I think i understand a little now, the eigenvalues that I found was when all the masses displayed the same frequency of oscillation.

But how do I prove that the equations for the positions of the masses are a superposition of normal modes with the eigenvectors?

tiny-tim
Homework Helper
I think i understand a little now, the eigenvalues that I found was when all the masses displayed the same frequency of oscillation.

But how do I prove that the equations for the positions of the masses are a superposition of normal modes with the eigenvectors?
Hi Oerg!

Stop using all these technical words

x1 + x2 = A cosBt, x1 - x2 = C cosDt,

obviously x1 = (AcosBt + CcosDt)/2 … that's year-1 arithmetic!

We solve it that way round because you only have to look at the formula for x1 on its own to see that it's much more difficult to solve than x1 + x2

but there's no magic of "superposition" or "normal modes" to understand

Hi Oerg!

Stop using all these technical words

x1 + x2 = A cosBt, x1 - x2 = C cosDt,

obviously x1 = (AcosBt + CcosDt)/2 … that's year-1 arithmetic!

We solve it that way round because you only have to look at the formula for x1 on its own to see that it's much more difficult to solve than x1 + x2

but there's no magic of "superposition" or "normal modes" to understand

hmm, i know about the equations "x1 + x2 = A cosBt, x1 - x2 = C cosDt" for a two mass system, it is just the addition and subtraction and then the acceleration and the position are common terms.

But what about a system with a higher number of masses and springs? How do i know
x?+x?+x?+..=Acoswt+phi

So I was trying to see how by finding out the eigenvectors for a two mass system for simplicity, that x1+x2=Acoswt+phi. Im still at a loss though.

tiny-tim
Homework Helper
But what about a system with a higher number of masses and springs? How do i know
x?+x?+x?+..=Acoswt+phi

So I was trying to see how by finding out the eigenvectors for a two mass system for simplicity, that x1+x2=Acoswt+phi. Im still at a loss though.

every matrix has eigenvectors, and each eigenvector is a combination of "basis" vectors, and by definition of eigenvector that combination is going to satisfy the shm equation ∑'' = -w2∑, so ∑ = Acoswt+phi

so we have a Ax=b and b can be expressed as a linear combination of the eigenvalues multiplied by the respective eigenvectors? This is because eigenvectors are orthogonal. so in this spirit we have the solutions for a 3 mass system as

$$\left( \begin{array}{cc} \ddot{x_1} \\ \ddot{x_2} \\ \ddot{x_3} \end{array} \right)=\lambda _{1}v_{1}x+\lambda _{2}v_{2}x+\lambda _{3}v_3x$$

where

$$x=\left( \begin{array}{cc} x_1 \\ x_2 \\ x_3 \end{array}\right)$$

is this correct?

ahh i think i understand now, I read up on a chapter on diagonalization and now I understand how it can be applied to solve the system of differential equations. Thanks for your help tiny tim.