Finding the Angles of Coupled Pendulums with Eigenvalue Analysis

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In summary, the conversation discusses a problem involving three coupled pendulums with identical masses and springs. The goal is to find the angles of the pendulums as a function of time. The conversation suggests using a Lagrangian and Euler-Lagrange equations, as well as solving an eigenvalue problem to find the frequencies and amplitudes. The solution should include an extra term to account for initial conditions.
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Homework Statement



I have three coupled pendulums: each of identical mass, and hung fromt the ceiling with identical massless rod. They are connected by identical massless springs. [itex]\phi[/itex][itex]_{1}[/itex], [itex]\phi[/itex][itex]_{2}[/itex], and [itex]\phi[/itex][itex]_{3}[/itex] represent their angles from the vertical, hanging position. Find these three angles as a function of time.

Homework Equations





The Attempt at a Solution


I started by setting up the Lagrangian, but I'm really not sure where to go from here. Any help?
 
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  • #2
Have you drawn a picture? That should always be your first step with mechanics problems. Use the picture to label all the interactions and your generalized coordinates. Don't forget to use the interactions to find the relationship between the coordinates.
Taylor's "Classical Mechanics" goes through this problem in detail in the chapter on coupled oscillations.
 
  • #3
I think he already has the Lagrangian. Assuming you didn't make any mistake, you'd have to write Euler-Lagrange equations using your Lagrangian. This will give you the equations of motion for each mass.
 
  • #4
Since you have an oscillatory system, your Lagrangian should be for the form [itex]L = \frac{1}{2}\Sigma m_{ik}\dot{\phi}_{i}\dot{\phi}_{k} - \frac{1}{2}\Sigma k_{ik}\phi_{i}\phi_{k} [/itex]. So your equations of motion would be able to be written in matrix form. Is this what you have?
 
  • #5
Here is the caveat: at t=0, ϕ1=0, ϕ2=, and ϕ3=C.

I have the Lagrangian correct and I have the matrix form of the equation.

From there, how do I figure out the angles as a function of time, given those initial conditions?
 
  • #6
Solve the eigenvalue problem to find the frequencies (eigenvalues) and the amplitudes (eigenvectors). You can account for initial conditions by including an extra term in your general solution. Your solution should be something like [itex] \phi_{i} = A_{i}cos(\omega*t - \delta) [/itex] one of those terms can be used to account for the IC's (i indexes the coordinates 1,2,3; in other words [itex]\phi[/itex] is a "vector" you will have three such solutions, one for each eigenvalue [itex]\omega_{\alpha}[/itex]). Which one and how would you solve for it?
 

1. What are three coupled pendulums?

Three coupled pendulums refer to a system of three pendulums that are connected to each other by a common support. This arrangement allows the pendulums to swing and interact with each other, causing interesting and complex motions.

2. What factors affect the motion of three coupled pendulums?

The motion of three coupled pendulums is affected by several factors, such as the length and mass of each individual pendulum, the distance between the pendulums, and the initial conditions of the system. These factors can influence the amplitude, frequency, and phase of the pendulums' motion.

3. How is the motion of three coupled pendulums described mathematically?

The motion of three coupled pendulums can be described using mathematical equations and principles, such as Newton's laws of motion, conservation of energy, and harmonic motion equations. These equations can help predict and analyze the behavior of the pendulums in the system.

4. What is the significance of studying three coupled pendulums?

Studying three coupled pendulums can provide insights into complex systems and phenomena, such as synchronization, chaos, and resonance. It can also have practical applications in fields such as engineering, physics, and mathematics.

5. Can three coupled pendulums exhibit chaotic behavior?

Yes, three coupled pendulums can exhibit chaotic behavior, especially when the pendulums have different lengths and initial conditions. Chaotic behavior is characterized by unpredictable and sensitive responses to small changes in the system, making it a complex and interesting phenomenon to study.

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