- #1
noamriemer
- 50
- 0
Hello!
I have a question regarding a method I saw every now and then:
Say I have a system containing of two masses, attached to one another by a spring. Each is attached to a wall by another spring.
Now I wish to know the eigen vectors and eigen values( [itex]\omega[/itex]) of movement.
I get:
[itex]\ddot x_1= \frac {-2k} {m} x_1+\frac {k} {m} x_2[/itex]
[itex]\ddot x_2= \frac {-2k} {m} x_2+\frac {k} {m} x_1[/itex]
That is because the sysem is completely symmetric. Now, I get
[itex]{\omega_1}^{2}= \frac {3k} {m}[/itex] and [itex] {\omega_2}^{2}= \frac {k} {m} [/itex]
But if I try to find the eigen vectors using a matrix,
|5 -1 |
|-1 5 |
As you can see, the only solution is the trivial one.
So, what I saw was done in this case, was defining new coordinates:
[itex]y_1=x_1+x_2[/itex]
[itex]y_2=x_1-x_2[/itex]
And now it works, and everything is fine.
What I don't understand is why.
How is it that when I move to y-s it is ok?
Shouldn't I get the same result?
Thank you!
I have a question regarding a method I saw every now and then:
Say I have a system containing of two masses, attached to one another by a spring. Each is attached to a wall by another spring.
Now I wish to know the eigen vectors and eigen values( [itex]\omega[/itex]) of movement.
I get:
[itex]\ddot x_1= \frac {-2k} {m} x_1+\frac {k} {m} x_2[/itex]
[itex]\ddot x_2= \frac {-2k} {m} x_2+\frac {k} {m} x_1[/itex]
That is because the sysem is completely symmetric. Now, I get
[itex]{\omega_1}^{2}= \frac {3k} {m}[/itex] and [itex] {\omega_2}^{2}= \frac {k} {m} [/itex]
But if I try to find the eigen vectors using a matrix,
|5 -1 |
|-1 5 |
As you can see, the only solution is the trivial one.
So, what I saw was done in this case, was defining new coordinates:
[itex]y_1=x_1+x_2[/itex]
[itex]y_2=x_1-x_2[/itex]
And now it works, and everything is fine.
What I don't understand is why.
How is it that when I move to y-s it is ok?
Shouldn't I get the same result?
Thank you!