Re: [Phys-l] question about coupled oscillators

[Jack Uretsky <jlu@hep.anl.gov>]

Hi all-
	I'm not sure what Brian means by "common mounting board".  Two 
pendula with coupled mounting points can be chaotic with an attractor at 
pi phase difference (I claim no expertise on this topic), but if the 
mounting points are each rigid the statement would not be true.
	As far as uncoupled women are concerned, better we not go there on 
this list.
				Regards,
					Jack



On Sat, 4 Nov 2006, Brian Whatcott wrote:

> Which reminds me: two pendula on a common mounting board will tend to
> beat at pi phase difference - like a tuning fork.
> The more general observation is that oscillators at near frequencies
> tend to be entrained with very little coupling indeed - this observation
>  has been applied to women at college I see.  Which is to say, eigenmodes
> and coupled oscillations are not the same....
>
> Brian Whatcott    Altus OK
>
> At 09:51 PM 11/3/2006, you wrote:
> > Hi Carl-
> >         The general statement goes something like this (in the 2 normal
> > mode case):
> >         There are 2 coupled amplitudes, call them X1 and X2.  Your system
> > is such that you can find 2 normal modes, with frequencies fA and fB.
> > Corresponding to each frequency there is an eigenvector that is a linear
> > combination of X1 and X2.  The eigenvectors, call them XA and XB are
> > orthogonal to each other in the usual case.  The eigenmodes can be excited
> > independently of each othe (that's what makes them "eigen"), so there
> > should be no relationship between the phases of the excitations.
> >         The simplest case to work out, that I know of, is two pendulums
> > connected by a spring.
> >                                         Regards,
> >                                                 Jack
> >
> >
> >
> >
> > On Fri, 3 Nov 2006, Carl Mungan wrote:
> >
> > > In writing down the assumed form for the displacement of each
> > > oscillator i in a coupled system, in general one includes a phase
> > > constant d_i in the argument of the cosine:
> > >
> > > x_i (t) = A_i cos (w*t + d_i)
> > >
> > > However, in all the examples I can think of, the relative phases of
> > > the oscillators *in a single normal mode* always turn out to be
> > > either 0 or pi. Why can't we get other values?
> > >
> > > In particular, the "weak coupling" problem (where you pull just one
> > > oscillator aside and then get beats) looks like an example of pi/2
> > > relative phase difference between the two oscillators. Except I don't
> > > have a single normal mode; I have two normal modes equally excited.
> > > But there's a part of me that wants to call this something like
> > > another normal mode since each particle is oscillating in a nicely
> > > repetitive manner. (Okay, maybe I need to require the two normal mode
> > > frequencies to be commensurate for this statement to be exactly
> > > correct.) That is, I'm wanting to define "normal mode" as any
> > > smoothly repetitive pattern, but apparently that's too loose a
> > > definition to be right.
> > >
> > > Please help clarify my thinking. -Carl
>
>
> Brian Whatcott    Altus OK    Eureka!
>
>
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