Re: [Phys-l] question about coupled oscillators

[Brian Whatcott <betwys1@sbcglobal.net>]

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!