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Link to thread of next article. Mass-spring system represented by inductor-capacitor system. Date: 1996/11/11
Consider an infinite linear system of identical masses separated by identical springs. A system of inductors and capacitors if connected in the right way will have the same differential equation of motion as the above linear mass spring system (see page 512 "Theoretical Mechanics" by T.C. Bradbury for example)
Question, what is the circuit diagram (if one exists) of a system of inductors and capacitors that would have the same differential equation of motion of a 2-dimensional (and 3-dimensional) system of masses and springs ?
Thanks for any references or ideas!
Re: Mass-spring system represented by inductor-capacitor. Date: 1996/11/12
In article <567e56$2m7e@r02n01.cac.psu.edu> ale2@psu.edu (ale2) writes:
> > Question, what is the circuit diagram (if one exists) of a system of > inductors and capacitors that would have the same differential equation > of motion of a 2-dimensional (and 3-dimensional) system of masses and > springs ?
I think i have an answer for you. For the 2-dimensional case consider the set of points in the infinite plane R^2 with integer coordinates. For each point in that set connect one end of the capacitor to "ground" and the other end of the capacitor to the point. Now connect each point in the above set to its four nearest neighbors with an inductor. Current can flow from ground "through" the capacitor and then have a choice of flowing through four inductors. Now cut out a finite piece of the above infinite system, cut only inductors.
The N-dimensional case should now be straightforward, the current can flow from ground "through" the capacitor and then have a choice of flowing through 2N inductors.
If the above is correct it is interesting that in the mechanical case in 2-D there are twice as many springs as masses but in the electrical analog there twice as many inductors ("masses") as capacitors ("springs").
The anchored string revisited, but now in 3D. Date: 1996/11/13
If i plug Iain Mains book twenty more times he said he might get me a free copy (not).
In Iain Main's book "Vibration and Waves in Physics" he considers the physics of the anchored string. Now what gives me goosebumps about this system is that it has the same frequency wavelength relationship as a massive quanta in one dimension!
(children and mental midgets are easily impressed)
A while back i asked the readers of sci.physics to come up with a physical system which has the same frequency wavelength relationship as a massive quanta in 3 dimensions. Well either no one cared about my question or they did not see it (I've been kilefiled?) ?
I think i have something now that works ?
Consider an infinite 3 dimensional system of masses and springs such that each mass has six springs attached to it in a symmetric fashion, and all the masses are hooked together by the springs and form a cubic array. This is the system one considers as a simple model of vibrations in solids?
Now transform the above system:
1) replace each spring with an inductor and capacitor in series,
2) each mass is replaced with one end of a capacitor and the other end of the capacitor is grounded.
Now perturb the system at some small region (apply an oscillating voltage at a point where one of the masses once was) for a long time with less than some critical frequency and energy is not absorbed after steady state is reached, but increase the frequency above the critical value and energy propagates out of the small region?
Homework, come up with a physical system which models (some, not all aspects of the):
Re: The anchored string revisited, but now in 3D ? Date: 1996/11/15
In article <56hrmr$1jj6@r02n01.cac.psu.edu> ale2@psu.edu (ale2) writes:
> > > > Consider an infinite 3 dimensional system of masses and springs such > > that each mass has six springs attached to it in a symmetric fashion, > > and all the masses are hooked together by the springs and form a cubic > > array. This is the system one considers as a simple model of vibrations > > in solids? > > > > Now transform the above system: > > > > 1) replace each spring with an inductor and capacitor in series, > > > > 2) each mass is replaced with one end of a capacitor and the other end > > of the capacitor is grounded. > >
For more fun make both the inductor and capacitor in 1) above, variable and variable independently of all others (let it change with time?). Changing the capacitance in a region has the effect of changing the "potential" (but with a twist?) and changing the inductance in a region has the effect of changing the effective mass (but with a twist?).
Summer reruns :^( the anchored string and massive quanta. Date: 1996/09/13
I came across an interesting section in the book "Vibrations and Waves in Physics" by Iain Main. In it Main considers the physics of the anchored string, a string under tension with an additional sideways restoring force (This can be accomplished by attaching many little springs to the string and anchoring them properly.
One interesting property of this system is that the anchored string has the same frequency-wavelength characteristics of any massive quanta, that is, for very long wavelengths the frequency approaches a non-zero minimum w_o. If one forces this system with a frequency less then w_o the net work done on the system averaged over one cycle will be zero. Kind of makes me think of how a low energy photon can turn into a virtual electron-positron pair for a short times. This system can also be modified to display the tunnelling of waves. Suppose that for some large distance L along the anchored string the additional restoring force becomes much larger. Then waves encountering this region with greater then some critical wavelength will be completely reflected. If the distance L is not so long, then part of the wave will be transmitted having tunnelled through the "barrier". Also interesting is the fact that the anchored string has "massless" modes. If the string moves the plane perpendicular to the plane which contains the springs then the string will have a dispersion curve similar to massless quanta. Another interesting aspect of this system is that there is a small coupling between the massive modes and the massless modes. If either mode is populated then the string will be curved and therefore be under slightly greater tension. I believe that this is a clue in trying to figure out quanta, imagine, the electron and photon fields on a single string (we have not yet figured out the quantization part ): I am surprised that I have not yet seen this example in any quantum mechanics textbooks. Think of electrons and photons and think of the anchored string, but properly, we must think of the 3-dimensional version of the anchored string, what ever that might be?
A field of in phase oscillators in a moving frame. Date: 1996/11/27
Let each point, (rho,phi,z), in the plane z=0 have associated with it an oscillator whose phase is a function of time and the cylindrical coordinate phi:
where w is some constant, t is time, and phi is the angle of a point (rho,phi,0) in the plane z=0.
Let us represent the phase of this system at each point with a color. Consider the following function which maps the phase angle to the colors of an Artist's Color Wheel:
for all other angles make the "natural" continuous map to the other colors of the rainbow. Notice we could have gone around the Artist's Color Wheel in the other direction.
Now "paint" the plane z=0 by giving each point the color that corresponds to its phase, [wt + phi]. Consider the "paint-job" as time increases.
For more fun consider the paint-job as it looks in a relativistically moving frame that passes near the point (0,phi,0) on the left and on the right.
Consider also a frame that moves relativistically around the point (0,phi,0) or some other point for all possible orientations, now what does the paint-job look like in all cases.
Now is a job for Mathematica! If you can figure out the proper transformation (should be undergraduate stuff) you will see a very pretty picture (I'm guessing here). The basics of what the paint-job looks in the moving frame should require no calculations and only an understanding of the basics of relativity. For the circular paths I'm a little uncertain of the transformation, any help?
Well lets simplify the last part. Let the phase now only be a function of time and repeat all of the above.
$ 0.25 system for solving specific differential equations. Date: 1996/11/11
Suppose you have a system consisting of pair of coupled pendulums with masses M and M'. Let each pendulum have a variable length L and L' . Suppose the coupling is also variable. Also suppose one wants to determine how the above system "moves" for some given initial conditions? Here are two ways to answer this question:
1) solve the damn equations dummy! %^)
2) build the above system for about 25 cents and 30 minutes time.
For best results choose 1). For a change of pace choose 2). The following are the materials required:
It should not take too much imagination to put the above together?
Hints: Suspend clothes hanger with two pieces of fishing line, by suspending the hanger with just one thread the system goes crazy, too much freedom. Use short lengths for smaller coupling. Use the clothespins for clamping the fishing line to some stationary object (from which hangs the clothes hanger).
Fun for the whole family, satisfaction (not) guaranteed!
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