How Do Forces Transmit Through a Chain When One Link is Disturbed?

[gogza]

Folks,

Imagine a chain, floating in a frictionless, weightless environment.

Just like this:

--------------------

where every "-" is link.

If a force acts on just one link of the chain "Link0", causing it to buckle like this (and the force continues to act) :

-------------\_/---
(apoligies for the crudeness of my diagram)

What forces will act on the links either side of Link0, causing them to move? Do the first nearest neighbours (to Link0) have the same magnitude of force?
If they don't, is it because the mass on either side is different?

Help! Its been a few years since I last forayed into classical mechanics!

Thanks,
Gogza

 

[rayjohn01]

To Gogza
This problem stumped mathematicians for eons and they were still wrong thinking that the chain formed a parabola.
The trouble with thinking about links one at a time is that you set yourself up for a dynamic solution and without the inclusion of friction you will end up with an oscillating chain -- which never stops.
Clearly AT equilibrium all forces at all points are in balance , the forces acting are 1) gravity acting vertically at each point ie. at every link.
At any point in the chain the mass above that point is supported by the suspension
and the mass below that point is supported by the link , the forces cause tension in the chain which is NOT fixed but varies along it's length. But they do balance since if they did not the link would move.
The upshot which I cannot show is a Cosh form of curve ( i.e exponential in form )
something I have often tried to prove but always failed.
Ray.

 

[Loren Booda]

Is this not the "catenary" problem solved with the help of a Lagrangian or Hamiltonian?

 

[rayjohn01]

Well it is certainly the catenary problem and the final result is simple -- but I do not know HOW to solve it. As opposed to the usual problem this is one where you must find the formula which fits, not an answer from a given formula, it is beyond my maths.
Ray

 

[gogza]

Some Clarification

Thanks rayjohn01/Lore Booda for your replies,

Having looked into "cantenary" problem, I don't think it covers what I'm looking for.

Perhaps I've not explained myself properly. I'll have another go.

OK we've got our frictionless chain again and this time its lying out straight on a frictionless table, both ends are free.

If I take my finger, place it on just one link and push this link forward the rest of the links will follow.

The force applied by my finger will be used to:
a) accelerate the link I'm touching.
b) accelerate the links to the left of this link.
c) accelerate the links to the right of this link.

If I push the middle link forward, I'll get a symmetrical curve. The force applied, after a), will be split equally for b) and c).

However, what happens if I push a link closer towards the left end of the chain? I'm thinking its one of these answers, of course the correct answer would be best!

1) the forces for b) and c) are not equal because the mass on either side is different. The curve remain symmetrical.

or

2) the forces for b) and c) are equal(?) but the curve is not longer symmetrical.

I guess what I'm looking for really, is equations that give the forces acting on each link, depending on its distance from the point of applied force (if possible!)

Thanks for your help so far,
Gogza

 

[HallsofIvy]

To Gogza
This problem stumped mathematicians for eons and they were still wrong thinking that the chain formed a parabola.

But only those mathematicians who, like you, could not READ. The original post said "weightless environment".

 

[rayjohn01]

To Gogza
The chain curves in general will not be symmetrical due to the difference in inertial mass on either side of the chosen link ( there is NO friction ) , curves will be straight lines
(IF the force is very light wrt the masses ) their angles will be such that the vectors
sum to zero if you consider the total link mass on either side of the link.
However there is a nasty trick here ( due to your removal of friction ) any applied force will cause the links to accelerate and then NOT to stop .
Second since the chain is clearly not stiff -- if you consider a small motion of the chosen link it will be normal to the chain , the tension it causes is directed down the chain on either side according to mass distribution , as regards the last links they see a force ~ parallel to the chain their mass resists this but they travel INWARDS
The amount of resistance is dependant on how many links you consider , that is the tension is different down the chain therefore the inwards accelleration is also different -- That is I believe the chain forms two different cusp like shapes which continually change if the force is maintained until they form straight lines behind the leading link .
I suggest this is a very nasty excercise in calculus which is DYNAMIC until the force is removed. Even then the implied shapes will mean complex analyisis as there are two not one -- unless you choose the central link.
It 'might' be simpler to simulate this.
Unlike 'Ivy' I AM assuming mass (not weight) -- if there's no mass then any force creates infinite velocity of a straight chain. And Mr Ivy can direct his answers to the questioner.
Ray

 

[gogza]

Thanks rayjohn01,

It 'might' be simpler to simulate this.

Exactly what I want to do! (Oops, perhaps I should have mentioned that, sorry!)

The question is how do I simulate this?

I'm trying to work out how a single link will move after considering the external force applied to it (this only affects one link) and the internal forces applied by the other links in the chain.

For each link, I can model the external forces applied to the chain (straightforward, enough). But I've a huge problem modelling the internal forces in the chain.

If I consider an end link then the only force causing this to move (inward, as mentioned), it exterted by its neighbour. But what is this force? I think its as simple as summing up all the forces acting on the other links. BUT (and as you can see it its a big but) I don't know how inertia of the links affect these forces.

I guess my problem lies with inertia of the other links and how this affects the internal forces.

I will not be defeated! :tongue2:

Thanks again,
Gogza

 

[rayjohn01]

Gogza
I have done a chain simulation in the past with elastic links but I lost the program in a computer crash --- but I will try to remember what exactly I did -- it was written in QBasic 4.0 available as a free download and can calc and give you a dynamic visual output. First thing is to recognize that choosing anyone link splits the chain in two ( i.e. they are independant so you only need to deal with and end chain situation) For your purpose.
If I recall correctly I used elastic connections so that the movements of the connected masses could be considered one at a time in a repeating series-- but I will have to dwell on it for a while , obviously you want to start off simple like two masses only -- then extend to whatever.
The mass to mass link funcion can be any function you can dream up i.e. linear non-linear time varying etc. To simulate a real link just make it stiff but not infinitely so , you must use small steps like all normal time simulations and allow cause and effect to propogate thru' the structure -- otherwise you get infinities , and if you think about it this probably is real anyway.
In my program I had several small chains lying on a 3D table which were mutually repulsive with highly elastic links. I would start the simulation by kicking one mass and the allowing the whole thing to evolve --
A point of consideration is to put a boundary of some sort so they do not wander off screen to infinity -- I used 4 reflection walls like a snooker table
and I could rotate the whole view so as to be looking down on the table or from one side like looking down an Ice rink.
In this type of simulation the point masses do not have to be the same , and in QB you can graph them as circles or spheres which are then considered as the mass boundary so they do not overlap i.e. the link function is between mass boundaries not their centers or alternately the mass boundaries become reflection walls ( that gets tricky because you then deal with spherical or circular reflections ).
Again if you wish to be really adventureous you can do the whole thing
in 3dimensions not two , which I did but with single point masses only something like a miniature solar system bound by a cube.
I do not know what type of programming you may be used to but I personally find that QB 4.0 ( not earlier and not later ) is very powerful for this type of thing and relatively simple to use . Currently I have about 100 programs written including a complete 3D drawing package -- these together with the exec and libraries is still less than 2Mbytes.
The only draw back is aside from the fact that it is no longer supported by Microsoft is that the best screen ( for visual output ) known as 'screen 12'
is only 640x480 pixels . However anything you draw can be captured by a 'print screen' and imported to 'paint' as a bitmap ( at least in windows 98 -- not sure if XP allows this ) .
Simulations speeds are very dependant on your particular machine -- QB is a DOS program and windows takes priority -- but when I wrote that specific program I was using a 50 Mhz machine so today there is little problem.
Also I do not believe that there are any longer any tutorial books but there is an excellent QB site on the internet with very experienced programmers
. Any way have fun and if I think of any specific problem solutions I used I will post to Gogza either in this thread or in general physics.

Ps Do Not Be Defeated if you can do this you will learn a great deal and will be able to apply it to many other things.
Ray.