Movement of connected masses under external force

[nickmacias]

Hello,


First time post here, so if I violate some rules of protocol etc. please forgive and let me know! Anyway, I've been asking profs and other physicists this question, and the more-knowledgable the person I ask, the more complex the answer sounds!

Two basic setups:
(1) two masses, connected by a massless rod, resting on a plane, no friction, etc. etc. apply a force off the center-of-mass. I know it will translate and rotate...but what are the equations for describing the rotation? (translation is just F=ma, right?); and
(2) say you have three masses M1 M2 M3; M1 and M2 joined by a rod; M2 and M3 joined by a rod; but the rods can pivot (in a plane) where they're attached to the masses. Think atoms maybe: M1-r1-M2-r2-M3 and M1, M2 and M3 can move around, so the angle between the rods can be whatever. Start off with M1 M2 M3 all alligned. Now I apply a force to M3, perp. to the line of masses. What is the motion?

My question is general: I'm heading towards a 2-D or 3-D lattice of such masses and rods, where some pivots are free, some have resistance, some are fixed; and I want to apply a force somewhere and model the motion. I don't expect a closed form solution! I just want to know if there's a reasonably-simple way to model this, say, across a small time interval, maybe study the effect at the application point and the immediately-connected masses, then propagate that to the adjacent masses, and so on.

Sorry if this is a ridiculous question (either too simple or too complex). I'm pretty stuck, but really need to model this. Any help would be greatly appreciated! Thanks...

 

[vanesch]

The difficult part will be if there is friction in some joints. If all the joints are frictionless, then there is a general technique which you can easily apply: Lagrangian dynamics. First, you have to write down all the independent degrees of freedom of your system (angles of the moving joints, position of the COM of the whole system,...). Next, you express the position (the x,y,z coords) of each of the masses as a function of the degrees of freedom you just wrote down (this will need some geometry). Once you have that, you can express the velocity of each of the mass points and hence their kinetic energy T.
Concerning your force(s?), you have to write it as a potential energy as a function of the position of its point of attack, also expressed as a function of the degrees of freedom. This will give you the potential energy V of the system.
Finally, you define the lagrangian L = T - V and you apply the Euler-Lagrange equations, which will give you a system of differential equations whose solutions are the dependence on time of the values of the degrees of freedom.

The problem is that this technique only works for frictionless systems (there are extensions for specific cases of friction).

 

[nickmacias]

Thanks vanesch, this looks extremely helpful - I'll see what I can do with it. The initial analysis assumes everything is frictionless, so your info may help me with the first-order work. I'll deal with friction later ;)


-Nick