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Describing the motion of a linked body? 
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#1
Nov2412, 04:59 AM

P: 147

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 ballsocket 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? 


#2
Nov2412, 05:31 AM

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P: 16,963

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



#3
Nov2412, 06:09 AM

Sci Advisor
P: 2,470

Which, with exception of a few simple cases, will usually produce something that can only be solved numerically. So if you hope for a closedform 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 RungeKutta methods to integrate these.



#4
Nov2412, 08:19 AM

P: 147

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


#5
Nov2412, 08:32 AM

P: 147

what about a very simple compound body?
Just one mass, floating in deep space, with a rod pivoted to it. 


#6
Nov2412, 08:48 AM

Sci Advisor
P: 2,470

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. 


#7
Nov2412, 10:58 AM

P: 147




#8
Nov2412, 01:43 PM

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P: 16,963

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 


#9
Nov3012, 09:05 AM

P: 147

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



#10
Dec212, 07:55 AM

P: 147

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



#11
Dec312, 10:47 AM

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P: 16,963

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. 


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