B Can a Mass Suspended by Springs in a Box Exhibit Chaotic Behavior?

[houlahound]

I have never seen a problem as such;

A mass in the isocentre of a box. The springs are identical unstretched length, they are attached to the walls maximally distant from each other.

The mass is suspended from springs attached to the walls of the box.What is the least number of springs to have a chaotic response of the mass?

No reason why here, just thinking of building something like this to see what happens. I think it would be cool to watch and then use odd numbers of different springs, attachment points etc.

 

[zwierz]

.Ffirst one should define what "chaotic response" is. I mean mathematical definition, not bla-bla-bla

 

[Nidum]

Ffirst one should define what "chaotic response" is. I mean mathematical definition, not bla-bla-bla

So tell us what you think the mathematical definition is .

 

[hilbert2]

I don't think the motion will be anything you'd call chaotic if the springs are Hookean.

 

[zwierz]

So tell us what you think the mathematical definition is .

for which purpose? in dynamical systems the standards of mathematical rigor are very well known by mathematical community. If such a question arises then it is early to study dynamical chaos and one should take a course of math.analysis first

 

[houlahound]

I just mean aperiodic motion of the mass. May not be technical chaos.

 

[hilbert2]

I just mean aperiodic motion of the mass. May not be technical chaos.

Even a mass that oscillates independently in the x and y directions (i.e. the potential energy is of the form $\frac{1}{2}k_x x^2 + \frac{1}{2}k_y y^2$ with no cross terms) moves only quasiperiodically if the two angular frequencies $\sqrt{\frac{k_i}{m}}$ are incommensurable real numbers. But this is not what chaos means.

 

[houlahound]

K, then repeating same starting conditions would lead to different trajectories?

ie; the system would be so sensitive to starting conditions repeatability would be lost - chaotic?

 

[hilbert2]

K, then repeating same starting conditions would lead to different trajectories?

ie; the system would be so sensitive to starting conditions repeatability would be lost - chaotic?

That doesn't have anything to do with periodicity or lack thereof. A system that consists of coupled Hookean oscillators is never chaotic in that sense as far as I know, but it still doesn't have to move periodically.

 

[Khashishi]

If a single spring has a fixed resting point, then the x motion and y motion are completely separable
$H = kd^2 = k(x-x_0)^2 + k(y-y_0)^2$
But, if the spring pivot can rotate, it couples x and y in a complicated way, because there isn't a resting point $(x_0, y_0)$ but rather a resting circle $(x-x_p)^2 + (y-y_p)^2 = r^2$

It looks fairly chaotic in a simulation.
https://www.myphysicslab.com/springs/2d-spring/2d-spring-en.html

 

[Khashishi]

Oh, the sim also includes gravity. I guess without gravity, it isn't chaotic.

 

[alan2]

Here is the simplest case, its behavior is well known. https://www.google.com/search?q=can+a+mass+and+springs+be+chaotic&ie=utf-8&oe=utf-8