Infinite Grid of Springs

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Kosak
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Infinite Grid of Springs

How would a spring behave if it were in an infinite grid of springs? Like the nerd sniping problem where there is an infinite grid of resistors. If a force is added to point 1, what is the resulting behavior of point 2.

BlackSails
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Re: Infinite Grid of Springs

Its easy, you just find the eigenvalues of an infinitely large matrix.

danpilon54
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Re: Infinite Grid of Springs

Locally Im not so sure, but if the size of the gap between springs was very small compared to your area of interest, wouldnt it act much like the 2D wave equation? You would have an initial position and velocity at every point and have outward propagating longitudinal and transverse waves. The speed of such waves would average to a value determined by the gap of the springs and the spring constants. This of course assumes that all the springs are they same spring constant and distance apart.

Other than that not sure how to find the exact solution; probably isnt possible.
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BlackSails
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Re: Infinite Grid of Springs

It would behave much like vibrations in a crystal I think. In fact, I think the lots of springs model is one of the most commonly used ones.

Tass
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Re: Infinite Grid of Springs

You, sir, name?
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Re: Infinite Grid of Springs

Try going continuous (i.e. looking at the problem from so far away that the distance between the springs is vanishingly small), that is, making the sum a differential equation. I think you basically end up with something similar to the wave equation applied to a sheath living in three dimensions, to which finding a steady-state solution for some boundary conditions is relatively easy to find.

When you have such a solution, you could try to hook it into the edges of a finite discrete area. At this point, you're in numeric solution land, but it's just crazy enough to work™.
I edit my posts a lot and sometimes the words wrong order words appear in sentences get messed up.

mattdude
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Re: Infinite Grid of Springs

i should think the force required to move any point would be infinite. when adding springs in series, the k's add inversely (like capacitors in series or resistors in parallel). but there's also all the springs that are in parallel to any point, which simply add. so you'd have a simple expression like lim (n->infinity) nk = infinity. although i'm only a second year physics major, so i may be wrong. any ideas?

Tass
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Re: Infinite Grid of Springs

mattdude wrote:i should think the force required to move any point would be infinite. when adding springs in series, the k's add inversely (like capacitors in series or resistors in parallel). but there's also all the springs that are in parallel to any point, which simply add. so you'd have a simple expression like lim (n->infinity) nk = infinity. although i'm only a second year physics major, so i may be wrong. any ideas?

It is most certainly not infinite, and it can't be treated as simple as that.

danpilon54
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Re: Infinite Grid of Springs

The force would not be infinite because there will be a characteristic speed (much like the speed of light) in the problem, making only a finite number of springs affect a given point in time.
Mighty Jalapeno wrote:Well, I killed a homeless man. We can't all be good people.

_Big_Mac_
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Re: Infinite Grid of Springs

danpilon54 wrote:The force would not be infinite because there will be a characteristic speed (much like the speed of light) in the problem

I'd say speed of sound is a better analogy.

mattdude
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Re: Infinite Grid of Springs

danpilon54 wrote:The force would not be infinite because there will be a characteristic speed (much like the speed of light) in the problem, making only a finite number of springs affect a given point in time.

ahhh of course. classically i think my solution is ok but i think you're definitely correct in introducing relativity. i figured there was a flaw in my position but i couldn't figure it out. thanks

Seraph
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Re: Infinite Grid of Springs

mattdude wrote:
danpilon54 wrote:The force would not be infinite because there will be a characteristic speed (much like the speed of light) in the problem, making only a finite number of springs affect a given point in time.

ahhh of course. classically i think my solution is ok but i think you're definitely correct in introducing relativity. i figured there was a flaw in my position but i couldn't figure it out. thanks

Clasically I still think your solution doesn't work. Lets define 5 junctions in the infinate field of springs. Point C will be the Center, while A, B, E, and F are the four junctions connected to C by spings.
Now lets say that A, B, E and F are fixed (which is the same as saying it would take an infinite force to move them). Point C is now the junction of four springs (which have their far ends anchored), given that those springs have a finite spring constant I can't see it would take an infinate amount of force to make the point C move.

mattdude
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Re: Infinite Grid of Springs

Seraph wrote:
mattdude wrote:
danpilon54 wrote:The force would not be infinite because there will be a characteristic speed (much like the speed of light) in the problem, making only a finite number of springs affect a given point in time.

ahhh of course. classically i think my solution is ok but i think you're definitely correct in introducing relativity. i figured there was a flaw in my position but i couldn't figure it out. thanks

Clasically I still think your solution doesn't work. Lets define 5 junctions in the infinate field of springs. Point C will be the Center, while A, B, E, and F are the four junctions connected to C by spings.
Now lets say that A, B, E and F are fixed (which is the same as saying it would take an infinite force to move them). Point C is now the junction of four springs (which have their far ends anchored), given that those springs have a finite spring constant I can't see it would take an infinate amount of force to make the point C move.

you make an interesting argument. indeed it would definitely seem that my solution is wrong. in that case, i'm assuming this is far beyond my second year knowledge, but i'll continue to think about it.

Rentsy
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Re: Infinite Grid of Springs

Sorry.

LaserGuy
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Re: Infinite Grid of Springs

There are two approaches that you can take to solve this problem:

1) Go continuous... on a large enough scale, you can treat this as vibrations in a 2-d infinite plane. This can be solved using the wave equation and fourier transforms. Or probabably conformal mapping, if you prefer complex analysis.

2) Stay discrete... in the easiest case, you assume that each spring only supplies a force to its nearest neighbours. This makes the problem quite tractable, and the solution can be expressed in terms of the normal modes of the spring system. Any elementary solid state physics text will cover this (at least in 1d) in a section on lattice vibrations or phonons, and the generalization to higher dimensions is not that tricky. If you want to go passed nearest neighbours to a higher order solution, getting a sensible solution will probably require a fairly computation-intensive numerical scheme. There's probably a paper or two on the subject floating around in one of the academic journals on solid-state physics that looks at this.

 If you aren't familiar with solving these types of problems, I'd suggest starting with the 1d case. It will be a lot easier to figure out what's going on, and generalizing to higher dimensions won't enhance your understanding of the problem significantly relative to the added effort needed to solve it.