Lagrangian of 3 masses connected by springs, non-parallel.

[Daley192303]

Writing the Lagrangian for 3 masses and 2 springs in a line is easy.

KE=1/2(m*v^2)

L=KE(m1)+k/2(l1-(x2-x1))^2+KE(m2)+k2/2[L2-(x3-x2)]^2+KE(m3)

However, I wish to model non-linear linkages of the above 3 masses and 2 springs.

Suppose that the second spring (m2-m3) is angle θ away from the axis of the first spring (m1-m2).

I am quite daunted by the flexing or bending of the springs.

 

[haruspex]

Writing the Lagrangian for 3 masses and 2 springs in a line is easy.

KE=1/2(m*v^2)

L=KE(m1)+k/2(l1-(x2-x1))^2+KE(m2)+k2/2[L2-(x3-x2)]^2+KE(m3)

However, I wish to model non-linear linkages of the above 3 masses and 2 springs.

Suppose that the second spring (m2-m3) is angle θ away from the axis of the first spring (m1-m2).

I am quite daunted by the flexing or bending of the springs.

If you simply replace the lengths by vectors, that should work. The angle will not be constant, right?

 

[Daley192303]

@haruspex Good question about the angle being constant. The angle of intersection should be constant for a given simulation, however the springs themselves may bend.

Using vectors sounds promising. I am not sure if the vectors (alone) would contain the necessary information given Young's modulus, which is a variation of Hook's Law that applies to the flexing that would occur given the constant angle of intersection, which I should have mentioned to begin with.

The tricky thing about Young's modulus is that while the angles between the springs may be constant at the central mass, the relative angles between the three masses is variable due to flexing of the springs.