Coupled driven and damped oscillators

In summary, the conversation is about analyzing a system of elastically coupled oscillators using Fourier expansion. The differential equation being examined is of the form m\frac{d^2\hat{y}_k}{dt^2} + \gamma\frac{d\hat{y}_k}{dt} - \kappa\Delta^2\hat{y}_k = \hat{f}_k(t), and the goal is to show that a particular solution is given by \hat{y}_k(t) = A(\nu(k), \omega(k, \gamma))\sin(\nu(k)t + \phi). The conversation also mentions trying different methods and struggling to find the correct solution.
  • #1
Runei
193
17
Hello,

I'm trying to analyze a system of elastically coupled oscillators, whose masses are all the same, using Fourier expansion. So the differential equation I am looking at right now is of the form

[itex]m\frac{d^2\hat{y}_k}{dt^2} + \gamma\frac{d\hat{y}_k}{dt} - \kappa\Delta^2\hat{y}_k = \hat{f}_k(t)[/itex]

Here the [itex]\hat{y}_k[/itex] is a particular Fourier coefficient of a position vector with all of its components being the position function y(t) of each of the particles. The [itex]\hat{f}_k[/itex] is the corresponding Fourier coefficient for the driving force. So basically

[itex]\vec{y} = <y_0(t), y_1(t), y_2(t), ..., y_{M-1}(t)>[/itex]

and

[itex]\vec{f} = <f_0(t), f_1(t), f_2(t), ..., f_{M-1}(t)> [/itex]

And these two vectors have then been decomposed into Fourier representation. After taking the inner product it was possible to create M differential equations of the form above, which I am now working with.

Now -- I am trying to show that a particular solution to the differential equation is given by

[itex]\hat{y}_k(t) = A(\nu(k), \omega(k, \gamma))\sin(\nu(k)t + \phi)[/itex]

When the Fourier coefficients for the force are oscillating like

[itex]\hat{f}_k(t) = \hat{f}_k(0)sin(\nu(k)t) = \frac{e^{i\nu(k)t} - e^{-i\nu(k)t}}{2i}\hat{f}_k(0)[/itex]

It might be me whose blind (or tired after hours of work), but when I insert the particular solution (or what should be), I get to the point where I have the following:

[itex]A(\nu(k), \omega(\gamma, k))\left[-m\nu(k)^2sin(\nu(k)t+\phi) + \gamma\nu(k)\cos(\nu(k)t+\phi) - \kappa\lambda(k)\sin(\nu(k)t+\phi)\right] = \frac{e^{i\nu(k)t} - e^{-i\nu(k)t}}{2i}[/itex]

I've tried several things such as taking out the sine, and converting it using the addition trigonometric identity. I also tried converting the sines and cosines to complex exponentials, but I just can't seem to see where I should go.

So basically I'm out of creativity and a almost out of coffee. So any help, hints, tips or guidance would be very much appreciated.

Thank you.
 
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  • #2
Error:

The first equation is supposed to be:

[itex]m\frac{d^2y_k}{dt^2}+\gamma\frac{dy_k}{dt}−\kappa\lambda_ky_k=f_k(t)[/itex]

Without the lattice laplacian. The lambda is an eigenvalue to the lattice laplacian.
 

What is a coupled driven oscillator?

A coupled driven oscillator is a system consisting of two or more oscillators that are connected and driven by an external force or energy source. The oscillators affect each other's motion and their frequencies can become synchronized.

What is a damped oscillator?

A damped oscillator is a system where the oscillations decrease in amplitude over time due to the dissipation of energy. This can be caused by external factors such as friction or internal factors such as resistance within the system.

How does coupling affect the motion of oscillators?

Coupling affects the motion of oscillators by altering their frequencies and causing them to become synchronized. If the oscillators have the same frequency, they will reinforce each other's motion. If they have different frequencies, they will interfere with each other's motion.

What is the difference between a coupled driven oscillator and a coupled damped oscillator?

A coupled driven oscillator is a system driven by an external force, while a coupled damped oscillator has an additional factor of energy dissipation. This energy dissipation can cause the oscillators to eventually come to rest.

What are some real-life examples of coupled driven and damped oscillators?

Coupled driven and damped oscillators can be found in many systems in nature, such as the motion of molecules in a chemical reaction, the swinging of a pendulum with air resistance, and the synchronization of fireflies flashing in unison.

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