Pendulum with non-rigid string

[agostino981]

Hello guys, this is my first post so if I have something done wrong, please tell me. :P

When we come across with pendula, they are all with rigid rod, whether with mass or not. I wonder if the pendulum has a non-rigid string which can estimate the motion of string when the pendulum rises above $\pi$.

My plan is like this:

Assume the string is a complex pendulum which are composed of numerous rigid rod which that the accuracy of the model will increase (hopefully). Air resistance and resistance due to friction between the string and the hook or whatever is negligible.

As string can't be folded to much, there must be a curvature and the resistance against the bending will be shown as $k \theta$, where $k$ is the constant depending on the material and $\theta$ is the angle between normal and the rod.

How to input the resistance force due to angle of bending into Lagrangian mechanics? And, any suggestions?

-Agos

 

[LightHero]

One way to incorporate the resistance force of the non-rigid string into Lagrangian mechanics is to treat it as a potential energy term in the Lagrangian. You can do this by adding a term to the Lagrangian of the form U(θ) = kθ^2, where θ is the angle between the normal and the rod, and k is a constant depending on the material. This term represents the potential energy due to bending of the string. The other terms in the Lagrangian would remain unchanged. Then the Euler-Lagrange equations can be used to solve for the motion of the pendulum.