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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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