Mechanics, 3 coupled masses. don't worry, no springs

[elegysix]

this is a system of three masses attached to a string, hanging on two fixed pulleys.

I want to find the equation of motion of the middle mass, given the initial condition P(0)=X
Assuming all friction, stretching of the strings, and momentum of the pulleys are negligible.

The string has a fixed length overall, the length of string between masses is equal, and the masses are also equal.


I get confused because I'm trying to keep track of too many variables, there are 4 angles, 3 tensions, 3 positions... I know there's a better way to do this, any help would be appreciated.


here's a diagram I made... thanks for any help!

[PLAIN]http://img828.imageshack.us/img828/1610/3mass2pulleysystem.jpg

 

[jrlaguna]

It's not going to be easy anyway. But I think the best way to go, with difference, is lagrangian mechanics. Have you tried that?

 

[Bill_K]

Tell us this is not a homework problem, elegysix.

 

[elegysix]

don't worry its not a homework problem. I like to think up mechanics problems and try to solve them in my spare time. This particular one gave me trouble.

I'd try langrange's method, but what should I choose as my generalized coordinates?

I've been trying to relate the tensions to each other as a function of P, but this has been proving to be an algebraic nightmare itself...

 

[Vanadium 50]

The picture doesn't match the description. At equilibrium, all three masses are at the bottom and the string is triangular.

 

[elegysix]

Indeed, at equilibrium the masses will sit at the bottom similar to how I've shown in the first diagram. the lower diagram is an illustration of the system at the initial condition P(0)=X.

The string is only triangular due to the symmetry of the system at equilibrium. the masses move due to the tensions of the connected strings, they are not in any fixed position, only fixed to a string of constant length.

 

[my_wan]

I believe the situation described here would have significant chaotic behavior. Much like when a pair of pendulums are coupled. In such situations you cannot predict the behavior to any degree accuracy. Coupled pendulums are far simpler yet their patterns cannot explicitly be predicted either.

 

[elegysix]

the reason I thought up this problem was because I wanted to create a system of masses that are self-leveling no matter the initial conditions. I believe that the orientation of this system at equilibrium will be such that a line drawn between the two outer masses, will be perfectly level. But the question is, how do I show that?

 

[Bill_K]

You need six generalized coordinates, the Cartesian coordinates for each particle: (x_i, y_i). Three constraints: the constant length of each of the three segments of the string. So, the system has 6 - 3 = 3 degrees of freedom.

T = ½ ∑ m_i(x_i^·2 + y_i^·2)
V = g ∑ m_iy_i

Maintaining the constraints requires three Lagrange multipliers, representing the tensions in the three string segments. Each of the constraints follow this pattern: take L² = (x₁-x₂)² + (y₁-y₂)² and differentiate it: 0 = (x₁-x₂)(x₁-x₂)^· + (y₁-y₂)(y₁-y₂)^·. These give you three more equations, nine in all to be solved for nine variables. Also, the coefficients that go with the Lagrange multipliers can be picked out of the differentiated constraints. For example, the equations of motion for the mass m₂ in the center will be
m₂x₂^·· + λ₁ (x₁-x₂)/L + λ₂ (x₂-x₃)/L = 0
m₂y₂^·· + m₂ g + λ₁ (y₁-y₂)/L + λ₂ (y₂-y₃)/L = 0

Solution of the nine equations left as an exercise.

 

[elegysix]

Thanks so much Bill! that is just the help I needed to start working this out.

 

[meldraft]

Actually, your system only has 2 degrees of freedom (as long as we consider that the string cannot be lax). One of the three is dependent on the others. I would pick the angles at the joints as the generalized coordinates. If you can express one of the angles as a function of the other two (and the lengths of course) you only have 2 equations. This will make solving the system much more manageable.

It's, of course, impossible to solve by hand

 

[Bill_K]

Actually, your system only has 2 degrees of freedom

1) Center mass moves up and down
2) Three masses swing together side to side
3) String moves along the pulleys

If you can express one of the angles as a function of the other two (and the lengths of course) you only have 2 equations.

If you think that two equations, or even three, can determine the positions of all three masses along with three unknown tensions, please write them down.