Solving for equations of motion in a system of three coupled oscillators

In summary, the conversation is about a problem with writing the equations of motion for a system with three masses and four springs. The use of the Lagrangian method is discussed, as well as the potential energy spring extensions and the definition of coupling. The possibility of finding special solutions for an infinite chain is also mentioned. There is some confusion about the presence of a middle spring in the system.
  • #1
quasar_4
290
0
Hello all.

I am having a substantially difficult time with what should be, actually, a very simple problem.

I have three masses, each with a spring on each side (so three masses and four springs total in the system). My problem is writing down the equations of motion. I can do it when there are two masses and three springs, but I'm not sure what's happening with this many. I tried writing down the Lagrangian to compare to my Newtonian equations of motion, but they weren't the same, so somethings definitely wrong with one (or both) of them.

Basically, the problem is - assuming mass one is displaced to the right, we have a term with k2(x2-x1) where k2 is the second spring constant, and x2-x1 is the separation distance between mass 1 and 2. But now we also have another spring on the other side of mass 2, so do I need another term for k3(x3-x2)?

What happens with the middle spring?

Also, I prefer the Lagrangian method.. it just may not be right because of the same problem. Would the potential energy spring extensions be something like x1, x1-x2,x2-x3 and -x3? That's what I used for constructing my potential energies, but not sure these separations are correct...

Thanks!
 
Physics news on Phys.org
  • #2
I'm not familiar with the Lagrangian representation, but yes, the force on each mass depends on the displacement of its neighbours. This is the definition of coupling.
Essentially the acceleration and displacement are related by a matrix that is tri?diagonal.

If I remember correctly it is reasonably easy to find some special solutions for an infinite chain, eg harmonic waves. A terminated chain will be slightly more complicated.

Not sure what you mean by middle spring... for a 3mass/4spring system there is a middle mass but not middle spring. The middle mass has force k(x1-2x2+x3).
 
  • #3


Hello there,

It seems that you are trying to analyze a system of three coupled oscillators. This can definitely be a challenging problem, as it involves multiple masses and springs interacting with each other. It is important to note that the equations of motion for this system will depend on the specific setup and arrangement of the masses and springs. However, I can provide some general guidance that may help you in solving this problem.

Firstly, it is important to understand that each mass and spring will have its own equation of motion, which will then be coupled to the other equations through the interactions between the masses and springs. This means that you will have three individual equations of motion for each mass, and they will all be connected through the springs.

In terms of setting up the equations, you are correct in considering the displacement between each mass and its adjacent masses. This means that for mass 1, you will have terms for k2(x2-x1) and k1(x1-x0), where x0 is the equilibrium position of mass 1. Similarly, for mass 2, you will have terms for k2(x2-x1) and k3(x3-x2), where x3 is the equilibrium position of mass 2. The potential energy terms will also follow this pattern, with potential energies associated with the displacement between each mass and its adjacent masses.

I am not sure what you mean by the "middle spring," but I assume you are referring to the spring between mass 1 and mass 2. In that case, you will need to consider the displacement between these two masses as well, leading to the term k2(x1-x2).

As for using the Lagrangian method, it can definitely be a useful approach for solving this type of problem. However, it is important to make sure that your Lagrangian is set up correctly and includes all relevant terms. I suggest double-checking your Lagrangian and comparing it to your Newtonian equations of motion to ensure they are consistent.

I hope this helps and good luck with your analysis!
 

1. What are three coupled oscillators?

Three coupled oscillators refer to a system of three harmonic oscillators that are connected to each other through some form of coupling, such as a spring or a pendulum. This system exhibits complex behavior due to the interactions between the oscillators.

2. How do three coupled oscillators behave?

The behavior of three coupled oscillators can vary depending on the strength of the coupling and the initial conditions. Generally, the oscillators will exhibit a combination of synchronous and asynchronous motion, and can also display patterns such as beats, amplitude death, or chaos.

3. What is the significance of three coupled oscillators?

Three coupled oscillators are often used as a simplified model for more complex systems in physics, biology, and engineering. They can help us understand how interactions between multiple components can influence the overall behavior of a system.

4. How are three coupled oscillators mathematically modeled?

Three coupled oscillators can be mathematically modeled using a set of differential equations that describe the motion of each oscillator and their interactions. These equations can be solved numerically to predict the behavior of the system over time.

5. What are some real-world applications of three coupled oscillators?

Three coupled oscillators have been used to model phenomena such as the synchronization of fireflies, the behavior of populations in ecology, and the motion of atoms in a crystal lattice. They can also be applied in engineering to study the vibrations of structures or the synchronization of power grids.

Similar threads

Replies
13
Views
895
  • Classical Physics
Replies
1
Views
575
Replies
5
Views
310
  • Classical Physics
Replies
7
Views
1K
  • Classical Physics
Replies
1
Views
675
Replies
3
Views
959
  • Classical Physics
4
Replies
131
Views
4K
  • Classical Physics
Replies
3
Views
651
  • Introductory Physics Homework Help
Replies
19
Views
2K
  • Classical Physics
Replies
21
Views
1K
Back
Top