Normal modes / Eigenfrequencies

In summary, the conversation discusses a dynamical system represented by coupled ordinary differential equations. The parameters A, B, and C are positive and nonzero, and for certain values of these parameters, the system can exhibit oscillatory behavior. The person is attempting to solve for the normal modes of this system and is on the right track by setting up a matrix and solving for the eigenfrequencies. They also mention the importance of using functions that represent damped oscillations in their solution.
  • #1
xago
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1.
Homework Statement

Someone studying a dynamical system in another field of science tells you that when they
attempt to model the experiment they’ve been examining they obtain the following set of
coupled ordinary differential equations.

[tex]\dot{x}[/tex]= -Ax + By
[tex]\dot{y}[/tex]= -Cx

In what follows you should assume that the material parameters A, B, C are all positive and
non zero. They also tell you that for certain material parameters (that is, for certain A, B, C)
they can sometimes obtain oscillatory behaviour, albeit damped, but sometimes they do not.
Note that x and y are restricted to be real.

(a) Show that this is indeed possible by solving for the normal modes of this system. That is,
find the eigenfrequencies for this system.

The Attempt at a Solution



So basically my idea of eigenfrequencies are the frequencies at which the system oscillates and all motion of the system is the superposition of these two frequencies/motions. The first thing I'm doing is putting the constants into a matrix:

| -A B | |x| = |0|
| C 0 | |y|= |0|

Then i solve for determinant and get the eigenfrequencies. Basically I just wanted to know if I'm on the right track and this will help me prove that this is solving for the normal modes.
 
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  • #2
Normal modes imply:

[tex]x(t) = D e^{i\omega t}[/tex]

So you need to plug in similar functions for x and y. Then you solve for the eigenfrequences.
 
  • #3
Shouldn't the function you plug in be of the form of a solution for a damped oscillation?
 
  • #4
If [tex]\omega[/tex] is imaginary then it will be damped.
 
  • #5


Yes, you are on the right track. Solving for the eigenfrequencies of a system is a common approach to understanding its normal modes. In this case, the eigenfrequencies will give you the frequencies at which the system will oscillate and the corresponding eigenmodes (eigenvectors) will give you the motion of the system at those frequencies. So by solving for the eigenfrequencies of this system, you will be able to determine whether it exhibits oscillatory behavior or not, depending on the values of A, B, and C. Keep in mind that the eigenfrequencies may be complex numbers, so they will have both a real and imaginary component. This means that the system may exhibit both oscillations and damping. Good luck with your calculations!
 

1. What are normal modes/eigenfrequencies?

Normal modes, also known as eigenfrequencies, are the characteristic vibrations or oscillations of a physical system. They represent the natural frequencies at which a system will vibrate after being disturbed.

2. How are normal modes/eigenfrequencies calculated?

Normal modes/eigenfrequencies are calculated by solving the eigenvalue problem of the system, which involves finding the eigenvalues and eigenvectors of the system's governing equations. This can often be done analytically, but in more complex systems, numerical methods may be necessary.

3. What is the significance of normal modes/eigenfrequencies?

The normal modes/eigenfrequencies of a system provide important information about its behavior and stability. They can reveal the natural resonances of a system and help identify potential modes of failure or instability.

4. How do normal modes/eigenfrequencies relate to the physical properties of a system?

The normal modes/eigenfrequencies of a system are directly related to its physical properties, such as mass, stiffness, and damping. Changes in these properties will affect the system's eigenfrequencies and can be used to tune or control its behavior.

5. Can normal modes/eigenfrequencies be observed in real-world systems?

Yes, normal modes/eigenfrequencies can be observed in various physical systems, such as musical instruments, buildings, and bridges. They can also be observed in microscopic systems, such as atoms and molecules, through techniques such as Raman spectroscopy.

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