# Static solutions of a series of coupled pendulums

• Robin04
In summary, the equations of motion for a system of pendulums coupled by a torsion spring can be described by the equation ¨Φi=−kml2(2Φi−Φi−1−Φi+1)−glsin(Φi). In order to avoid the indices going out of range at the boundaries, we define Φ0 = Φ1 and ΦN+1 = ΦN. Looking for static solutions, we obtain a necessary condition of ∑isin(Φi)=0, which can be used to narrow the domain of solutions. With a simplified case of N=3 and all constants equal to 1, we can determine that
Robin04

## Homework Statement

The equation of motions of a series of pendulums coupled by a torsion spring is this:
##\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})##, where k is the torsion spring constant, m is the mass of a single pendulum, and l is the length of a single pendulum. We have N pendulums and we define ##\Phi_0 = \Phi_1##, and ##\Phi_{N+1}=\Phi_N##, in order to avoid the indices to go out of range at the boundaries. I'm looking for static solutions of this system and I'm expecting to get something like a static soliton.

## The Attempt at a Solution

First, for all ##i##, ##\ddot{\Phi_i} = 0##
Then, to make it a bit simpler in the beginning I tried with N = 3, and all constants are 1.
The equations are:
##\Phi_2-\Phi_1-\sin(\Phi_1)=0##
##\Phi_1+\Phi_3-2\Phi_2-\sin(\Phi_2) = 0##
##\Phi_2-\Phi_3-\sin(\Phi_3)=0##
If i add these equations, I get ##\sum_i \sin(\Phi_i)=0##, which is a nicer necessary condition.

I'm struggling to see through this problem. There has to be infinite solutions. I can't give any random value to any ##\Phi## because if I put it into the equations and calculate the rest, my condition for the sums of the sines doesn't work. What tools/tricks can I use to narrow the domain?
I also tried ##\Phi_3=\arcsin(-\sin(\Phi_1)-\sin(\Phi_1+\sin(\Phi_1))## which is a very messy function.

Robin04 said:
The equation of motions of a series of pendulums coupled by a torsion spring is this:
¨Φi=−kml2(2Φi−Φi−1−Φi+1)
Ops, I made a mistake there. I left out the term from the gravitational potential. The correct equation of motion is:
##\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})-\frac{g}{l}\sin(\Phi_i)##

Robin04 said:
The equations are:
##\Phi_2-\Phi_1-\sin(\Phi_1)=0##
##\Phi_1+\Phi_3-2\Phi_2-\sin(\Phi_2) = 0##
##\Phi_2-\Phi_3-\sin(\Phi_3)=0##
The first eqn implies Φ1 and Φ2 have the same sign. Likewise, 2 and 3, so...

## 1. What is a static solution of a series of coupled pendulums?

A static solution of a series of coupled pendulums is a configuration in which all the pendulums are in equilibrium and do not move. This means that the forces acting on each pendulum are balanced, resulting in a stable and stationary system.

## 2. How many pendulums are typically included in a series of coupled pendulums?

The number of pendulums in a series of coupled pendulums can vary, but it is usually more than two. This allows for more complex and interesting dynamics to occur, as the motion of one pendulum can affect the motion of the others.

## 3. Can the pendulums in a series of coupled pendulums have different lengths?

Yes, the pendulums in a series of coupled pendulums can have different lengths. This can lead to interesting and unpredictable behaviors, as the different lengths will affect the natural frequencies of each pendulum.

## 4. How are the pendulums coupled in a series of coupled pendulums?

The pendulums in a series of coupled pendulums are typically connected by a rigid rod or string. This allows for the transfer of energy and motion between the pendulums, creating complex and interconnected dynamics.

## 5. What factors affect the static solutions of a series of coupled pendulums?

The static solutions of a series of coupled pendulums can be affected by various factors such as the number of pendulums, their lengths, the strength of the coupling between them, and the initial conditions of the system. These factors can lead to a wide range of behaviors and solutions.

Replies
7
Views
2K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
1
Views
5K
Replies
16
Views
2K
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
34
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
266
• High Energy, Nuclear, Particle Physics
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
884