How to Model a Two-Mass Pendulum System Using ODEs?

[mathgirl]

Can anyone help me figure out how to model this pendulum system using ODE's? It is a two-mass system in which the two masses are placed at opposite ends of a massless rod, with a fulcrum somewhere in the middle. The smaller mass is length k away from the fulcrum and the larger mass is length L away from the fulcrum.

 

[dextercioby]

Could u attach a drawing,please and explain what sort of oscillations are u looking for...?

The physics involved would be only a wise application of Newton's second law for nonlinear movement (extended bodies).

Daniel.

 

[M98Ranger]

Sure, I can try to help you with modeling this pendulum system using ODEs. First, let's define some variables:

- Let m1 be the mass of the smaller mass and m2 be the mass of the larger mass.
- Let k be the distance of the smaller mass from the fulcrum and L be the distance of the larger mass from the fulcrum.
- Let x1 and x2 be the horizontal positions of the smaller and larger masses respectively, with x1 = -k and x2 = L.
- Let θ be the angle made by the rod with the vertical direction.

Now, we can use Newton's second law to write down the equations of motion for the two masses:

m1x1'' = T1sinθ - m1gcosθ

m2x2'' = T2sinθ - m2gcosθ

where T1 and T2 are the tensions in the rod acting on the smaller and larger masses respectively.

Next, we need to relate the horizontal positions x1 and x2 to the angle θ. Since the rod is massless, the distance between the two masses remains constant, and we can use the Pythagorean theorem to write:

(x2 - x1)^2 + (L - k)^2 = (L + k)^2

Expanding and simplifying, we get:

x2^2 - 2x1x2 + x1^2 + L^2 - 2Lk + k^2 = L^2 + 2Lk + k^2

Simplifying further, we get:

x2^2 - 2x1x2 + x1^2 = 4Lk

Now, we can use the small angle approximation sinθ ≈ θ and cosθ ≈ 1 to rewrite the equations of motion as:

m1x1'' = T1θ - m1g

m2x2'' = T2θ - m2g

We also need to express the tensions T1 and T2 in terms of θ. From the diagram, we can see that:

T1sinθ = m1x1'' + m1gcosθ

T2sinθ = m2x2'' + m2gcosθ

Substituting these into the equations of motion, we get: