Dynamics of a string coupled pendulum

[physics_cosmos]

Homework Statement

My problem/task is to explain in elementary terms the dynamics of a string coupled pendulum, the same as in this diagram:

Is it simple to make a free body diagram for the pendulums? Is it possible to understand the motion as being caused by SHM oscillation of the top horizontal spring from which the weighted pendulum are suspended?

Thanks in advance for any help with this problem.

 

[haruspex]

Is this the full statement of the problem? E.g. do we have to consider arbitrary directions of oscillation of the pendula? In arbitrary phases?

 

[physics_cosmos]

Is this the full statement of the problem? E.g. do we have to consider arbitrary directions of oscillation of the pendula? In arbitrary phases?

Assume that initially one of the pendula is displaced at an angle perpendicular to the plane defined by the fixed posts indicated in the diagram, and that the other pendula is initially at equilibrium.

It is a self-imposed problem that I have set myself, to understand the dynamics of this system. Unlike a spring coupled pendulum, the forces involved are not so obvious and I am not sure where to start.

 

[Fred Wright]

This problem can be solved by the technique of Lagrangian dynamics. The solution has a closed form for small angular displacements. It would be fun and easy to set up an experiment to observe the oscillatory transfer of angular momentum from one pendulum to the other.

 

[physics_cosmos]

Hi, thanks for the reply. I am looking more for some qualitative analysis. Exactly forces are involved? The tension in the long suspending string, as in the situation for SHM. In addition the tension from the horizontal string must come into account, I guess? Just a bit lost as to how to put all of this together.

 

[haruspex]

Hi, thanks for the reply. I am looking more for some qualitative analysis. Exactly forces are involved? The tension in the long suspending string, as in the situation for SHM. In addition the tension from the horizontal string must come into account, I guess? Just a bit lost as to how to put all of this together.

You have chosen a very complicated problem.
There are five different tensions.
The two junctions between the strings lie on spheres centred at the tops of the posts, but the distance between them adds a constraint, making three degrees of freedom.
At any instant, the three strings at a junction lie in a plane (why?).
The positions of the bobs relative to the junctions have one more degree of freedom each, and their velocities add two more each.
That's nine interacting variables.

 

[EconBabe]

The problem has long been solved in mathematically closed-form:
https://www.youtube.com/channel/UCtS9bRhPwVjq2i4MyKU72sA?view_as=subscriber