I have a problem which reduces to a simplified 1D spring-mass system, in which I do not care about spring constants nor masses (they can assumed to be the same everywhere). I'm pretty sure it boils to solving a system of linear equations, but I can't quite figure out how to formulate it. Pardon the lack of formalism of the description below:
Given a series of
n nodes along a 1D axis and
m springs connecting any two nodes (adjacent or not) with rest lengths r1 to rm and spring constant
k (can be assumed to be 1 for simplicity). Solve for the distances d1 to dn-1 between every adjacent node when the spring system comes to rest.
I used a physics engine to simulate an example with 4 nodes and the following springs, where the notation
x-y:r means a spring connecting nodes
y with rest length
1-2:100, 2-3:100, 3-4:100, 1-2:100, 2-4:167, 1-3:167, 3-4:100. The result is as illustrated below, where the two central nodes have been pulled slightly closer together due to the action of the longer springs:
How do I go about solving a system like this? If it is possible to do so with a series of linear equations, what would they look like in my example?