Calculating equilibrium point of a network of springs

[o_z]

Hi,

I'm trying to find a way to calculate the resting position of a network of springs that is built as follows: n number of springs with identical k constant, but with different resting lengths are connected together at one end of each spring. The other end of each spring is fixed to some point in 3d space - meaning, that position cannot change by the spring, only the end that is connected to all the other springs can move.
Now, if I move the fixed positions of all/some of the springs, how can I calculate the resting (equilibrium) position in space of the point in which all springs are connected?


Thanks in advance,
Ofer

 

[aracharya]

Consider the "floating point". Each spring yields a force with an x component , a y component, and a z component. If we sum all the x components, we get the net force in the x direction. Likewise for y & z.

For the "floating point" to be in equilibrium, the net force in each direction (x,y,z) should be zero. Therefore we have three equations that we have to http://en.wikipedia.org/wiki/Simultaneous_equations" :

Net force in x direction = 0
Net force in y direction = 0
Net force in z direction = 0

We have three equations and three unknowns (the coordinates of the floating point). Therefore we can solve these equations to get the equilibrium position of the floating point.

 

[aracharya]

Note however that not all solutions are valid equilibrium points.

For example consider the case where the fixed points are distributed on a circle with equal angular spacings. Then if the springs are under tension, the centre of the circle is an equilibrium point. However if the springs are in compression, then the centre point, though a solution to the equations above, is not a valid equilibrium point since it is unstable. What determines whether a solution to the above is a true equilibrium point is whether the point corresponds to a minima of the total potential energy of the system (http://en.wikipedia.org/wiki/Minimum_total_potential_energy_principle" )

 

[o_z]

Thanks for the replies!

The basic problem I'm having is: how do I know the force that each spring yields if I don't know the equilibrium point? What I mean is, I know the force's strength, but not it's vector. So how can I solve those equations without knowing the different x,y and z elements of the force?

In the mean time, I've written a simplified spring network solver for this specific case, that seems to solve the issue for my needs. But I'm still interested in knowing if there's a way to calculate that without simulation.

Thanks,
o