What Are the Frequency Extremes and Normal Modes of Infinite Coupled Pendulums?

[Daniel Heligman]

Homework Statement

Consider a very large (infinite) number of identical pendulums arranged in a row, with each pair separated by a distance d. Each pendulum is a massless rod of length l with a mass m at its end. Identical springs with spring constant k couple each pair of neighbors. What is the Max frequency and min frequency along with their normal modes?

Homework Equations

The Attempt at a Solution

I know the min frequency will be √(l/g) with all the pendulums in sync. I plugged in values for an increasing number of pendulums and noticed that the max frequency is increasing. So I'm assuming that the max will go to infinity as the number of pendulums approaches infinity.

 

[mark726]

As for the normal modes, I believe there will be an infinite number of normal modes, with each mode having a different frequency and a different pattern of oscillation. However, I'm not sure how to mathematically determine the normal modes for an infinite number of pendulums. I would appreciate any guidance or suggestions on how to approach this problem. Thank you.
Thank you for your interesting question. I would like to offer some of my insights on this problem.

Firstly, you are correct in your understanding that the minimum frequency of the system will be √(l/g) with all the pendulums in sync. This is known as the fundamental frequency or the first normal mode of the system.

To determine the maximum frequency and its corresponding normal mode, we need to consider the behavior of the system as the number of pendulums increases towards infinity. As you have observed, the maximum frequency will also increase towards infinity. This is because as the number of pendulums increases, the springs will become stiffer due to the increasing number of connections between the pendulums. This will result in a higher maximum frequency.

In terms of normal modes, we can think of the system as a series of coupled oscillators. The normal modes of a system of coupled oscillators can be determined by solving the eigenvalue equation for the system. In this case, the eigenvalue equation will involve an infinite number of coupled equations, making it a challenging problem to solve.

However, we can approximate the normal modes of the system by considering it as a continuous system instead of a discrete one. This approach is known as the continuum approximation. By using this approximation, we can determine the normal modes of the system and their corresponding frequencies.

In summary, the maximum frequency of the system will approach infinity as the number of pendulums increases. The normal modes of the system can be approximated using the continuum approximation method. I hope this helps in your understanding of this problem. Let me know if you have any further questions.