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

 

[PJC01]

Hi Ofer,

Calculating the equilibrium point of a network of springs can be achieved through a combination of mathematical equations and computer simulations. The first step would be to define the physical properties of the system, such as the number of springs, the k constant, and the resting lengths of each spring, as well as the fixed positions of the springs in 3D space. This information can then be used to create a mathematical model of the system, which can be solved using techniques such as Hooke's law and the principle of superposition.

Additionally, computer simulations can be used to visualize and analyze the behavior of the network of springs. By inputting the physical properties and fixed positions of the springs, the simulation can calculate the equilibrium point of the system and display it graphically. This can also be useful in predicting the behavior of the system when the fixed positions of the springs are moved.

In summary, calculating the equilibrium point of a network of springs involves a combination of mathematical modeling and computer simulations. By accurately defining the physical properties and fixed positions of the springs, the resting position of the point where all springs are connected can be determined. I hope this helps in your research. Best of luck!

Sincerely,