It's about a vibration model call Hoffmann with the equations of motion: [[a,b],[c,d]] (ddotx/ddoty)+[[2,1-triangle],[1,2]](x/y)=0.

Ilnaus5

Ilnaus5

Answered question

2022-09-22

Problem with complex eigenvalue in these Researches?
It's about a vibration model call Hoffmann with the equations of motion:
[ 1 0 0 1 ] ( x ¨ y ¨ ) + [ 2 1 Δ 1 2 ] ( x y ) = 0
And then they calculated the complex eigenvalue:
s 1 , 2 = ± [ 2 ± 1 Δ ] 1 2
I really don't undersatand how did they calculate this formula. Because i think the complex eigenvalue in this situation must be looked like this:
s 1 , 2 = 2 ± 1 Δ

Answer & Explanation

Julianne Mccoy

Julianne Mccoy

Beginner2022-09-23Added 10 answers

Step 1
Define u = d x / d t and v = d y / d t, so that you can write the differential equations as
d x d t = u d y d t = v d u d t = 2 x ( 1 Δ ) y d v d t = x 2 y
Step 2
Or in matrix form d X d t = A X ,, with X = ( x , y , u , v ) T and A = ( 0 0 1 0 0 0 0 1 2 Δ 1 0 0 1 2 0 0 ) .
The formal solution to this problem is X ( t ) = e A t X ( 0 ) for which you need to find the eigenvalues of A: ± [ 2 ± ( 1 d ) 1 / 2 ] 1 / 2

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Research Methodology

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?