Determining Eigen Vector

[engineertobe]

Hi all,

First I will copy/paste the given question and then I will explain what I have done and where I am stuck.



Consider the double mass spring system shown in the figure below.

​

The positions x

1 ​

and x

2 ​

of the two masses are given by the system

| m1 0 | * | x1" | = | -(k1+k2) k2 | * | x1 | +F
| 0 m2 | | x2" | | k2 -(k2+k3) | | x2 |

​

​

Let m

1=1​

m

2=1​

k

1=1​

k

2=4 ​

and k

3=1​

. If the forcing imparts a force F

1=1cos(8​

t

) ​

on the first mass and a force F

2=0​

on the second and the masses start from rest (x

1(0)=​

x

2(0)=0​

) and at their equilibrium positions (x

1(0)=​

x

2(0)=0​

), find the resulting motion of the system.


I set up filled in the equation ( | denotes side of matrix).

| 1 0 | * | x1" | = | -5 4 | * | x1 | + | cos(8t) |
| 0 1 | | x2" | = | 4 -5 | | x2 | | 0 |

solved | -5- E 4 |
| 4 -5-E |

for the eigen values which turned out to be 1, 9. Now as I try to solve for the eigen vectors I am left with

| -6 4 | for eigen value 1
| 4 -6 |

and

| -14 4 | for eigen value 9
| 4 -14 |


So my first question is, is it right up to this point? And secondly, how can I find an eigen vector that satisfies both equations if in the first, the first number is bigger, and in the second the second number is bigger?

 

[smoky]

Hmm, calc 4 at UM? Me too.

Well, I see that I started to do what you did and saw that lambda^2+10lambda+9=0 which given eigen values of -1, -9. Not positive. I am stuck from here tho. I tried to do what we did in lecture today, but we ran out of time with forcing terms. Ill let you know if i figure it out!

 

[smoky]

http://kaharris.org/teaching/216/Lectures/lec26/lec26.pdf

Hooray for randomly finding things online.

 

[DrPhil]

I set up filled in the equation ( | denotes side of matrix).

[FONT=courier new]| 1  0 | * | x1" | = | -5  4 | * | x1 | + | cos(8t) |
| 0  1 |   | x2" | = | 4  -5 |   | x2 |   |   0     |

solved | -5-E    4  |
       |  4    -5-E | [/FONT]

for the eigenvalues which turned out to be - 1, - 9. Now as I try to solve for the eigenvectors I am left with
. . .

almost OK up to finding the eigenvalues, but you haven't captured the idea of the eigenvectors yet. When the original matrix multiplies an eigenvector, the result is the same as multiplying the eigenvector by its eigenvalue.

\(\displaystyle \begin {pmatrix} -5 & 4 \\ 4 & -5 \end {pmatrix} ×\begin {pmatrix} \alpha_1 \\ \alpha_2 \end {pmatrix}
=\ -1 × \begin {pmatrix} \alpha_1 \\ \alpha_2 \end {pmatrix}\)

\(\displaystyle \begin {pmatrix} -5 & 4 \\ 4 & -5 \end {pmatrix} ×\begin {pmatrix} \beta_1 \\ \beta_2 \end {pmatrix}
=\ -9 × \begin {pmatrix} \beta_1 \\ \beta_2 \end {pmatrix}\)

where alpha and beta are the eigenvectors.

Have you gotten past this point yet?