A rather interesting type of coupled oscillator.

In summary, the conversation discusses a problem with a helical rope and the attempt to solve part 3 of it. The equations of motion and the Lagrangian of the system are provided, and the person is attempting to solve for x(t) and theta(t). Different methods are tried, including assuming solutions of the form x=Ae^(iwt+phi) and theta=Be^(iwt+phi), but the person is unsuccessful and requests help.
  • #1
dHannibal
10
0

Homework Statement


The problem can be found here. http://wopho.org/dl.php?id=17&dirfile=selection-problem/helical_rope.pdf" I am attempting to solve part 3.

Homework Equations


The Lagrangian of the system is: [tex]L= \frac{m\dot{x}^2}{2}+\frac{mr^2\dot{\theta}^2}{2}-k \left( x^2+(r\theta)^2-2x_0\sqrt{x^2+(r\theta)^2} \right) [/tex]
Also,the equations of motion are:
[tex]\ddot{x}=\frac{2k}{m} \left(\frac{x_0}{\sqrt{x^2+(r\theta)^2}} -1 \right)x [/tex]
[tex]\ddot{\theta}=\frac{2k}{m} \left(\frac{x_0}{\sqrt{x^2+(r\theta)^2}} -1 \right) \theta [/tex]

The Attempt at a Solution


I need to solve the equations of motion for [tex]x(t)[/tex] and [tex] \theta (t) [/tex]. First I tried assuming [tex] r \theta << x [/tex] but it leads to equations of motion of the form

[tex]
\ddot{x}=\frac{2k}{m} \left(\frac{x_0}{x+\frac{(r \theta)^2}{2x}} -1 \right)x
[/tex]
which is not particulaly useful. Pointing out that [tex] \frac{\ddot{x}}{x} = \frac{\ddot{\theta}}{\theta} [/tex] I tried assuming solutions of the form [tex] x= Ae^{(iwt + \phi)} [/tex] and [tex] \theta= Be^{(iwt + \phi)} [/tex] but I was again unsuccessful. I was stuck at this point.
Thank you.
 
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  • #2
hi dHannibal! :smile:

try putting x'' = v dv/dx and integrating :wink:
 
  • #3
Hi tinytim,
but how do i get rid of [tex]\theta[/tex]'s in the square root when integrating?
 

1. What is a coupled oscillator?

A coupled oscillator is a system of two or more oscillators that are connected and interact with each other, causing them to synchronize their movements.

2. How does a coupled oscillator work?

A coupled oscillator works by exchanging energy between the oscillators through their coupling. This energy exchange causes the oscillators to adjust their movements and eventually synchronize.

3. What are the applications of coupled oscillators?

Coupled oscillators have many applications in various fields such as physics, biology, and engineering. They can be used to explain phenomena like heartbeats, synchronized fireflies, and electronic circuits.

4. What are the different types of coupling in coupled oscillators?

There are various types of coupling in coupled oscillators, including linear, nonlinear, and time-delayed coupling. Each type of coupling affects the synchronization behavior of the oscillators differently.

5. How is a rather interesting type of coupled oscillator different from other types?

A rather interesting type of coupled oscillator refers to a specific type of coupling that produces unique and complex synchronization patterns. This type of coupling often involves non-identical oscillators and nonlinear interactions.

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