Justification (or philosophy) of modal analysis

[Trying2Learn]

TL;DR Summary

justify a modal analysis

Hello

I have tried asking this different ways, here. I do not know exactly what I am asking. So I am hoping to try again.

Say we have a differential equation for a multi-body system.

We cannot solve it because of the coupling (too difficult). So, we turn to a modal analysis (eigen value problems).

I am fine with how to conduct one, and I am fine with what the modes mean. However, I remain unsatisifed as to a justification for such an approach.

Why do we do a modal analysis? Who FIRST thought this up? What justified them to even consider this? The whole idea of searching for a solution such that the matrix describing the differential equatoin must be singular to ensure that there exist non-null solutions -- who justified this?

At the END of a modal analysis, with no damping (I wish to focus only on this), we have a solution that converts KE to PE and back. What is it, about that statement that motivates a modal analysis (or is it the other way: does a modal analysis show us the KE and PE switch back and forth?)

I get that say, in a 2-mass system connected by springs, that there are natural shapes and natural frequencies. But why? What is it about a particular physical problem that even makes us assume there is such a thing as a natural frequency?

I get the physical intution, say of a swing in a park (for one body), but what is it about the physical world that justifies the possibility of a solution based on natural frequencies?

I do not really know what I am asking, but I am unhappy with existing explanations on how we can justify a modal analysis.

Anyone?

(Something else that irritates me. In a differential equation (for, say a harmonic oscillator), if the forcing functoin matches the natural frequency, resonance occurs. How is THAT resonance related to the resonance that occurs when you take two mass and put them in one of the natural modes and relase them. Is there a common denominator here?)

 

[Ssnow]

Hi @Trying2Learn, in my opinion these frequencies are intimately related to the mechanical structures of the physical system considered. They are also called "natural frequencies" or "resonant frequencies". Frequencies can tell you "how waves of the vibrating system can be added" so the problem is connected to the problem detemine the energy. Every molecule can be view as an harmonic oscillator "very small". Everything is composed by molecules so everything is an harmonic oscillator, this is sufficiently to justify modal analysis ...
Ssnow

 

[Trying2Learn]

Hi @Trying2Learn, in my opinion these frequencies are intimately related to the mechanical structures of the physical system considered. They are also called "natural frequencies" or "resonant frequencies". Frequencies can tell you "how waves of the vibrating system can be added" so the problem is connected to the problem detemine the energy. Everything is composed by molecules so everything is an harmonic oscillator, this is sufficiently to justify modal analysis ...
Ssnow

I understand your explanation, but I am looking for an explanation that reveals a close connection between what we observe and the math we intend to use.

You wrote: "Everything is composed by molecules so everything is an harmonic oscillator."

I agree. But I cannot see how that statement "beckons" a modal analysis. (I can wave my hands and talk about fundamental frequencies, but I just do not see the "motivating" connection.

For example, knowing that the system has, say, a multitude of masses, each connected by a spring, as below...
(IGNORE DAMPING and applied forces FOR NOW)

... how do you KNOW that modal analysis will provide the foundation for all solutions?

All I see is that if I use Hamilton's principle, I will get three coupled differential equations. I try to solve it,
and find that if I use the standard approach, I obtain the NULL solution. So, then I am MOTIVATED to
seek a matrix that is rank deficient, (determinant = 0) to get the characteristic equations.

How do I KNOW this will work?

It is such a strange thing to want to do: "Create a matrix based statement of the three coupled differential equations and assume each mass moves with the same frequency (justify?) and that we see a null solution (justify) to get the characteristic equations and then normalized eigenvectors (Justify?) It works, I get it, but how do you "feel/know/intuit" it, in advance?

 

[Ssnow]

Ok, let us consider the problem from the point of view of "chaos theory". Sometimes in the practical events these resonant frequencies can prevent from a "chaotic disaster". If you study the system from the point of view of frequencies you are able to understand what is the accurate level of chaos that infect you mechanical system. In this sense the study of resonant frequencies can be an index also of the chaotic component of your mechanical system ...
Ssnow