# Diagonalize a coupled damped driven oscillator

• jamie.j1989
In summary, the conversation discusses the attempt to follow a paper on diagonalizing equations 5 and 6, and the struggle with understanding eq 3, which was initially written incorrectly. The solution of using a driving frequency and rearranging into an eigenvalue problem is suggested. The inclusion of damping in the equation is also questioned, and the use of a transformation matrix U for a normal mode analysis is mentioned.
jamie.j1989

## Homework Statement

I am trying to follow a paper, https://arxiv.org/pdf/1410.0710v1.pdf, I want to get the results obtained in equations 5 and 6 but can't quite work out how eq 3 has been diagonalized.

eq 3

## The Attempt at a Solution

As the system is driven i thought I'd try a solution first and then try to rearrange into an eigenvalue problem. I tried the solution ##x=Ae^{i\omega t}## where ##\omega## is the driving frequency and A some constant, I tried this solution as the system will oscillate at the driving frequency?

I think that eq 3 (in the [ ] at the beginning) should be $\frac{d^2}{dt^2} + \gamma \frac{d}{dt} + \Omega^2 _0$

I didn't even notice that mistake and have been reading it as you have corrected. Would you suggest trying a solution and rearranging is the best method?

on further review, maybe they don't treat the damping initially, $\gamma$ doesn't appear in the diagonalized matrix...

Yes you're right, if i don't consider damping I get the correct expressions for the eigenvalues, why would they not include the damping term? Also I'm struggling to understand the transformation matrix U, Is this a rotating frame transformation?

$U$ allows for a normal mode analysis..., thus decoupling the equations of motion.

## 1. What is a coupled damped driven oscillator?

A coupled damped driven oscillator is a physical system consisting of two or more oscillators that are connected and influenced by external forces, such as a driving force and damping forces.

## 2. Why is diagonalization important for coupled damped driven oscillators?

Diagonalization is important for coupled damped driven oscillators because it allows us to find the normal modes of the system, which are the independent oscillations of each oscillator. This helps us understand the behavior of the system and make predictions about its future motion.

## 3. How is diagonalization achieved for coupled damped driven oscillators?

Diagonalization for coupled damped driven oscillators involves solving a system of differential equations using matrix methods. This results in finding the eigenvalues and eigenvectors of the system, which can then be used to construct the normal modes.

## 4. What is the significance of the eigenvalues in diagonalization of coupled damped driven oscillators?

The eigenvalues obtained through diagonalization represent the frequencies of the normal modes of the system. These frequencies can be used to analyze the behavior of the system and make predictions about its future motion.

## 5. How does damping affect the diagonalization of coupled damped driven oscillators?

Damping affects the diagonalization of coupled damped driven oscillators by introducing complex eigenvalues, which represent the decay rates of the normal modes. This means that the oscillations of the system will decrease over time due to the dissipation of energy.

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