Equations of Motion for Three Coupled Pendula with Low Spring Constant k

[ekkilop]

Homework Statement

Derive the equations of motion for three identical pendula A, B and C, of mass m and length L coupled together (A to B and B to C) with two identical springs of low spring constant k.


Can't quite appreciate the forces acting on these pendula as they all should be dependent of each other. Any help would be appreciated.

 

[timman_24]

I would tackle the problem by first stating the kinetic energy and potential energy of the pendula.
$$T=\frac{1}{2}m\left(b\dot\theta_{1} \right)^2 + \frac{1}{2}m\left(b\dot\theta_{2} \right)^2 + \cdot\cdot\cdot$$
Assuming small oscillations with the usual cosine/sin substitutions:
$$U=\frac{mgb}{2}\left(\theta^2_{1}+\theta^2_{2} \right) + \frac{b^2\kappa}{2}\left(\theta_{1}-\theta_{2} \right)^2$$

U is for a two pendula system, you should be able to figure out how to add in another pendulum.

From here you can use the Lagrangian which should be rather straight forward since we don't have any degenerating forces.

At least that would probably be the way I would tackle it.

 

[ekkilop]

Thanks a bunch. That did the trick =)
Of pure curiosity, what would happen if we in fact had a degenerating force?