[spring system] 3 degree of freedom system and its properties

[renvox]

Hello, I was given the attached 3 degree of freedom spring system with the purpose of analyzing it.

I came up with the following equation of motion

and then I ran Matlab to calculate the corresponding natural frequencies and mode shapes using eigenvalues and eigenvectors; I was asked to see what happens when value of stiffness k_12 is changed. This is the plot of the value k₁₂ against the natural frequencies.

The problem is that I do not know WHY values of natural frequencies are insensitive at low values of k₁₂ and why both the 1st and 2nd natural frequencies are insensitive to changes in k₁₂ when values are large (first two level off and the third one seems to go to infinity).

I assume it has something to do with the equation for force due to a spring between 2 masses but I cannot figure it out. That is why I ask for your help - thanks in advance.

 

[tiny-tim]

hello renvox!

… I do not know WHY values of natural frequencies are insensitive at low values of k₁₂

if k₁₂ is a weedy little thing, why should changing it make any noticeable difference?

… and why both the 1st and 2nd natural frequencies are insensitive to changes in k₁₂ when values are large (first two level off and the third one seems to go to infinity).

what are the three eigenvectors (the three modes of oscillation)?