Describing the motion of a linked body?

[greswd]

A general example of a linked body is an object that is made up of rigid bodies joined together by pivots, such as hinges and ball-socket joints.

If the pivots are frictionless, and I apply a force to one rigid body, how can I go about describing the subsequent angular acceleration of the individual rigid bodies, and also the entire linked body about its barycenter?

 

[Dale]

I would use the Lagrangian approach. Even for something as simple as a double pendulum it winds up being easier.

 

[K^2]

Which, with exception of a few simple cases, will usually produce something that can only be solved numerically. So if you hope for a closed-form expression, forget about it. But if you need it for a simulation, do what DaleSpam suggested. Write down the Lagrangian with an undetermined Lagrange Multiplier for every joint or other constraint you have. That will give you a system of equations for each [itex]\ddot{q_i}[/tex] to be solved for each time step, and then you can use Verlet or Runge-Kutta methods to integrate these.

 

[greswd]

Which, with exception of a few simple cases, will usually produce something that can only be solved numerically. So if you hope for a closed-form expression, forget about it. But if you need it for a simulation, do what DaleSpam suggested. Write down the Lagrangian with an undetermined Lagrange Multiplier for every joint or other constraint you have. That will give you a system of equations for each [itex]\ddot{q_i}[/tex] to be solved for each time step, and then you can use Verlet or Runge-Kutta methods to integrate these.

So it's possible to get a numerical solution to any degree of accuracy? I've always wondered how mathematicians definitively prove that the system is inherently chaotic and that no closed form solutions exist.

But even if a pattern exists it's difficult to find because the subsequent motion is so highly dependent on initial conditions.

 

[greswd]

what about a very simple compound body?

Just one mass, floating in deep space, with a rod pivoted to it.

 

[K^2]

So it's possible to get a numerical solution to any degree of accuracy?

Not in general. There are going to be special cases where you can, but in general, these things tend towards chaos.

If you have just two masses connected by a single joint, you can use conservation of momentum and angular momentum to greatly reduce degrees of freedom, giving you a closed form solution. Everything past that would require some approximations, I believe.

 

[greswd]

Not in general. There are going to be special cases where you can, but in general, these things tend towards chaos.

If you have just two masses connected by a single joint, you can use conservation of momentum and angular momentum to greatly reduce degrees of freedom, giving you a closed form solution. Everything past that would require some approximations, I believe.

Thanks. So in certain cases things are non-chaotic, but it's generally difficult to prove whether system is definitely chaotic and/or lacks a closed form solution.

 

[Dale]

If you are interested in chaos, I would start with a double pendulum. It is relatively easy to solve, and it is well studied. Here is a good start:
http://en.wikipedia.org/wiki/Double_pendulum

 

[greswd]

Hmmm, I am interested in finding out how mathematicians prove that a system is chaotic and that no symbolic solutions exist.

 

[greswd]

For instance, the three body problem of Sun-Moon-Earth might have a definite closed form solution that we just cannot find.

 

[Dale]

Hmmm, I am interested in finding out how mathematicians prove that a system is chaotic and that no symbolic solutions exist.

These are two somewhat different properties. A system can be non-chaotic and still not have a symbolic solution.

However, I don't think that "no symbolic solutions" is something that is actually mathematically proven. It is simply that we don't know any such solutions. It also depends critically on what functions are admissible in your set of symbolic colutions.

 

[greswd]

A system can be non-chaotic and still not have a symbolic solution.

What kind of system is described as such?

However, I don't think that "no symbolic solutions" is something that is actually mathematically proven. It is simply that we don't know any such solutions. It also depends critically on what functions are admissible in your set of symbolic colutions.

Interesting, it sort of opens the possibility that someday someone might solve the Sun-Earth-Moon problem.

Like the CMI problems.

 

[TipTop314]

Hey DaleSpam,

Do you mind giving me a little direction on how to set up the problem described here using the Lagrangian approach? I just recently learned about and applied the Lagrangian approach to describe the motion of a double pendulum (in this case the set up is given by many sources online). What I want to do now is describe the motion of the double pendulum if the support pivot disappears and the connected rigid bodies are now flying through the air.

Thanks,
TipTop