Eigenvalue Problem -- Justification

[Trying2Learn]

TL;DR

Justify common frequency, yet different amplitudes

Hello!

Suppose you have two masses, that are connected by a spring.
Each mass is, in turn, connected by a spring to a wall
So there is a straight line: left wall to first mass, first mass to second mass, second mass to right wall

This problem can be analyzed as an eigenvalue problem.

We assume a solution for the displacement of each mass as:

x1(t) = X1 exp(i*omega*time)
x2(t) = X2 exp(i*omega*time)

Do you see that?

We assume a solution with:

• an unspecified common frequency
• two different amplitudes

So my question is: can someone justify this assumption?

I know the process works and we must solve for an eigenvalue problem.

But could someone provide (in words), a justification for this assumption?

 

[hilbert2]

The solution will look like that only after changing the positions of the masses to normal mode coordinates, and even then the frequencies are usually not equal. The mass coordinates evolve independently only if the central spring has vanishing spring constant.

However, I may be interpreting this wrong if the capital coordinates X_1 and X_2 are also functions of time.

 

[Trying2Learn]

The solution will look like that only after changing the positions of the masses to normal mode coordinates, and even then the frequencies are usually not equal. The mass coordinates evolve independently only if the central spring has vanishing spring constant.

However, I may be interpreting this wrong if the capital coordinates X_1 and X_2 are also functions of time.

I am not so sure...

If the masses are the same and all three spring constants are different, and there is no damping and no forcing functions...

Almost all textbooks make the assumption I did:
They assume the SAME angular frequency response
They assume different amplitude response

They NORMALIZE the modes, AFTER the solution -- I get that.

But to start the solutoin, they do make the assumptions, enumerated above.

But I have never read a justification for this assumption IN ADVANCE of the eigenvalue solution implementation.

 

[DrClaude]

It comes from harmonic analysis. You make the assumption that the motion can be described by a superposition of modes where all masses move at the same frequency (with the same phase), and in the end you get that you can indeed reproduce any harmonic motion in terms of normal modes.

 

[Trying2Learn]

It comes from harmonic analysis. You make the assumption that the motion can be described by a superposition of modes where all masses move at the same frequency (with the same phase), and in the end you get that you can indeed reproduce any harmonic motion in terms of normal modes.

Yes, I agree... I understand... but that is EXACTLY my point...

You make the assumption.
You see that you can reproduce any harmonic motion.

But I want someone to justify this assumption BEFORE you do the work.

Somebody thought this up. Someone must have had some intuition to justify that assumption.

I mean: I have two different masses here. Different spring constants. What gives me the right to dare to assume the frequency response is the same?

That is what I am hoping to learn.

 

[anorlunda]

Somebody thought this up. Someone must have had some intuition to justify that assumption.

That is what I am hoping to learn.

Are you asking for the identity of that someone? Perhaps Joseph Fourier.

What research have you done on the history of harmonic analysis?

 

[Trying2Learn]

Are you asking for the identity of that someone? Perhaps Joseph Fourier.

What research have you done on the history of harmonic analysis?

I assume not enough because this is the second time someone has assumed that by saying "harmonic" it should be clear. (I am NOT being disrespectful, I am trying to get to the bottom of this.)

There are two masses here. And, say, if it was the eigenvaule of a 10 story building, 10 masses.

Are you saying that the solution just assumes all masses move with the same frequency?

I do not understand how you can justify that IN ADVANCE of the work of solving the problem.

I am hoping for: "Intuitively, it makes sense to assume all masses move with the same frequency because... X"

 

[DrClaude]

The thing is that you have to study the mathematics of harmonic analysis to understand how it works. It is the mathematics that will show you that any motion of the spring-connected masses can be understood as the superposition of different vibration modes where all the masses move with the same frequency.

When you have bunch of masses connected by linear springs, you get a set of coupled differential equations that all look the same, $ddot{\mathbf{x}} = \mathbf{x}/m$, where the $\mathbf{x}$ are displacements from equilibrium. It took the genius of Fourier to figure out that such equations could be solved as sums of sines and cosines.

I do not understand how you can justify that IN ADVANCE of the work of solving the problem.

By standing on the shoulders of giants. Once it has been shown that harmonic series can be use to solve such problems, we "simply" need to apply it.

It is not much different than what is done when analyzing vibrating strings. How knowledgeable are you about that?

 

[Trying2Learn]

The thing is that you have to study the mathematics of harmonic analysis to understand how it works. It is the mathematics that will show you that any motion of the spring-connected masses can be understood as the superposition of different vibration modes where all the masses move with the same frequency.

When you have bunch of masses connected by linear springs, you get a set of coupled differential equations that all look the same, $ddot{\mathbf{x}} = \mathbf{x}/m$, where the $\mathbf{x}$ are displacements from equilibrium. It took the genius of Fourier to figure out that such equations could be solved as sums of sines and cosines.By standing on the shoulders of giants. Once it has been shown that harmonic series can be use to solve such problems, we "simply" need to apply it.

It is not much different than what is done when analyzing vibrating strings. How knowledgeable are you about that?

I am so sorry...

I do understand Fourier series.
I do understand how the problem comes down to differential equations with the same structure.
I do understand how any response is the sum of many cos/sin, etc.

But I do NOT understand how, when cast as an Eigenvalue problem, you can say, in words, SIMPLY
"We assume all masses respond with the same frequency and we are justified because... hell... because the entire structure of a ten story building will oscillate with the same frequency... something like that...

 

[hutchphd]

The proof of the pudding is in the eating.
The result can always be substituted into the original set of equations and shown to be a solution...does that offer any satisfaction to your question ?

 

[Dr.D]

The answer is that the assumption is justified by the fact that it works. As said by @hutchphd, the proof of the pudding is in the eating. If it did not work, nobody would be interested in this assumption. It is the fact that it leads to an acceptable solution that justifies making the assumption.

 

[Trying2Learn]

Well, thank you to everyone for your patience.

I sort of expected it would come down to that.

But I just needed to hear it said explicitly.