Science > Physics > coupled oscillators to correct a clock - how to mathematically model?
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Science > Physics |
| User: |
"Dave" |
| Date: |
02 Apr 2005 03:08:13 PM |
| Object: |
coupled oscillators to correct a clock - how to mathematically model? |
I'm hoping for some physical/mathematical insight into coupled
oscillators. Is there any physicists or mathematician that can help? The
crux of the problem concerns whether you can force a pendulum to
oscillate at a frequency close, but not exactly equal to its natural
resonate frequency, and how to model it.
In particular I'd like to know what happens if a non-idealised pendulum
running at a frequency of f1 (where f1 is not too stable) is supplied
(or removed) energy at a frequency *exactly* f2.
f1 and f2 are very close in frequency (say fractional difference of less
than 1 part in 1000).
For a simple (idealised) pendulum in a clock, its resonate frequency f1
will
f1 = 1/( 2 Pi sqrt(length/acceleration_due_to_gravity))
but it will depend on temperature (expansivity of the metal forming the
pendulum) and other factors, whereas f2 can be assumed to be extremely
stable.
Can the pendulum be forced to oscillate at *exactly* f2, despite small
changes in its natural resonate frequency f1?
If you can only *add* energy to the pendulum, could this only work if
the frequency of oscillation f1 was < f2 ?
There was an article, which I have not yet got a copy of, which appeared
in the Amateur Scientist of Scientific America (I think the reference is
Scientific America "Clock. pendulum-type equipped with quartz-crystal
oscillator", 1974 Sep, pg 192) but if anyone has it, I'd like a copy. If
not, I can dig it out from the university library.
The author made a pendulum clock very accurate by putting a magnet on
the pendulum, then using an electromagnet applied energy at a stable
frequency f2, derived from a stable crystal. The crystal frequency being
chosen such that the clock run perfect time if the pendulum was forced
to oscillate with a frequency f2, rather than its natural frequency f1.
But this method seems to suffer from some problems.
1) It required that the clock first be set to run slow (f1 < f2). This
suggests to me that he was applying more energy than actually needed, as
he was constantly having to supply energy to speed the pendulum up.
2) There is some discussion about whether this with work with all
pendulum clocks.
Any ideas how I might model this, to at least determine if
a) If it is essential to supply energy to the pendulum
b) If the resonate frequency f1 must be less than f2.
I suspect it it much easier to supply energy (apply power to the
electromagnet), but in principle you could extract energy be having the
electromagnet develop power into a load. If that could be used to charge
the battery powering the circuit, the power consumption could be reduced.
If anyone has any Mathematica (or Matlab for that matter) code for this
sort of thing, let me know, as I have access to both, although I know
Mathematica a lot better.
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| User: "" |
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| Title: Re: coupled oscillators to correct a clock - how to mathematically model? |
02 Apr 2005 08:16:14 PM |
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Dave wrote:
<<snip>>
There was an article, which I have not yet got a copy of, which
appeared
in the Amateur Scientist of Scientific America (I think the reference
is
Scientific America "Clock. pendulum-type equipped with quartz-crystal
oscillator", 1974 Sep, pg 192) but if anyone has it, I'd like a copy.
If
not, I can dig it out from the university library.
The author made a pendulum clock very accurate by putting a magnet on
the pendulum, then using an electromagnet applied energy at a stable
frequency f2, derived from a stable crystal. The crystal frequency
being
chosen such that the clock run perfect time if the pendulum was
forced
to oscillate with a frequency f2, rather than its natural frequency
f1.
But this method seems to suffer from some problems.
1) It required that the clock first be set to run slow (f1 < f2).
This
suggests to me that he was applying more energy than actually needed,
as
he was constantly having to supply energy to speed the pendulum up.
2) There is some discussion about whether this with work with all
pendulum clocks.
Any ideas how I might model this, to at least determine if
a) If it is essential to supply energy to the pendulum
b) If the resonate frequency f1 must be less than f2.
I suspect it it much easier to supply energy (apply power to the
electromagnet), but in principle you could extract energy be having
the
electromagnet develop power into a load. If that could be used to
charge
the battery powering the circuit, the power consumption could be
reduced.
If anyone has any Mathematica (or Matlab for that matter) code for
this
sort of thing, let me know, as I have access to both, although I know
Mathematica a lot better.
Hey, I REMEMBER that article, but I don't have a copy.. (:
IIRC, there is a slight adeendum to your model... The correction
system can handle excursions on either side of the desired frequency.
The driving magnetic coil (behind the back of the pendulum compartment)
is switched on, as you say, by a quartz crystal timer. It operates to
pull the small magnet on the pendulum towards the equilibrium position,
for about 1/4 of a pendulum cycle. (The longer this duty cycle, the
greater the frequency mismatch tha can be overcome.)
If the pendulum is dead on time, it arrives at equilibrium in the
middle of this electromagnet duty cycle, and it is pulled forward for a
period of time, and BACKWARDS for the SAME period of time, No energy
added...
If the pendulum is running a little slow, it experiences an pull in the
direction of motion for a longer part of the duty cycle, and a
retarding force for a smaller part of the duty cycle. Energy input, and
faster motion for that part of the swing, so the pendulum won't be as
late next time around.
If the pendulum is running fast, it misses some of the accelerating
pull, and experiences more of the retarding pull...Energy extracted,
and slower motion for that part of the swing, so the pendulum won't be
as early next time.
It was a neat idea, and the only thing visible from the front of the
clock was a small magnet on the pendulum...
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| User: "Julian V. Noble" |
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| Title: Re: coupled oscillators to correct a clock - how to mathematicallymodel? |
03 Apr 2005 08:55:08 PM |
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Dave wrote:
I'm hoping for some physical/mathematical insight into coupled
oscillators. Is there any physicists or mathematician that can help? The
crux of the problem concerns whether you can force a pendulum to
oscillate at a frequency close, but not exactly equal to its natural
resonate frequency, and how to model it.
In particular I'd like to know what happens if a non-idealised pendulum
running at a frequency of f1 (where f1 is not too stable) is supplied
(or removed) energy at a frequency *exactly* f2.
f1 and f2 are very close in frequency (say fractional difference of less
than 1 part in 1000).
For a simple (idealised) pendulum in a clock, its resonate frequency f1
will
f1 = 1/( 2 Pi sqrt(length/acceleration_due_to_gravity))
but it will depend on temperature (expansivity of the metal forming the
pendulum) and other factors, whereas f2 can be assumed to be extremely
stable.
Can the pendulum be forced to oscillate at *exactly* f2, despite small
changes in its natural resonate frequency f1?
If you can only *add* energy to the pendulum, could this only work if
the frequency of oscillation f1 was < f2 ?
There was an article, which I have not yet got a copy of, which appeared
in the Amateur Scientist of Scientific America (I think the reference is
Scientific America "Clock. pendulum-type equipped with quartz-crystal
oscillator", 1974 Sep, pg 192) but if anyone has it, I'd like a copy. If
not, I can dig it out from the university library.
The author made a pendulum clock very accurate by putting a magnet on
the pendulum, then using an electromagnet applied energy at a stable
frequency f2, derived from a stable crystal. The crystal frequency being
chosen such that the clock run perfect time if the pendulum was forced
to oscillate with a frequency f2, rather than its natural frequency f1.
But this method seems to suffer from some problems.
1) It required that the clock first be set to run slow (f1 < f2). This
suggests to me that he was applying more energy than actually needed, as
he was constantly having to supply energy to speed the pendulum up.
2) There is some discussion about whether this with work with all
pendulum clocks.
Any ideas how I might model this, to at least determine if
a) If it is essential to supply energy to the pendulum
b) If the resonate frequency f1 must be less than f2.
I suspect it it much easier to supply energy (apply power to the
electromagnet), but in principle you could extract energy be having the
electromagnet develop power into a load. If that could be used to charge
the battery powering the circuit, the power consumption could be reduced.
If anyone has any Mathematica (or Matlab for that matter) code for this
sort of thing, let me know, as I have access to both, although I know
Mathematica a lot better.
You need to look at the theory of nonlinear oscillations. What you
are talking about is "entrainment" or synchronization, something EE's
use all the time, not to mention biological nonlinear oscillators.
Unfortunately this is rarely discussed in books on nonlinear systems.
I discuss it in sickening detail in the chapter on nonlinear dynamics
in my upcoming book on mathematical methods of physics. This is not yet
available, so I will e-mail you the relevant pages in *.pdf format,
if you send me an e-mail with your address on it. Obviously, the
one you gave here doesn't quite cut it.
I realize the theory is difficult, but it is a hard subject. At least
you will have the references to look at.
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
"For there was never yet philosopher that could endure the
toothache patiently."
-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
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| User: "Sam Wormley" |
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| Title: Re: coupled oscillators to correct a clock - how to mathematicallymodel? |
02 Apr 2005 04:08:54 PM |
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Dave wrote:
I'm hoping for some physical/mathematical insight into coupled
oscillators. Is there any physicists or mathematician that can help? The
crux of the problem concerns whether you can force a pendulum to
oscillate at a frequency close, but not exactly equal to its natural
resonate frequency, and how to model it.
Zee: http://scienceworld.wolfram.com/physics/CoupledPendula.html
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| User: "Dave" |
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| Title: Re: coupled oscillators to correct a clock - how to mathematicallymodel? |
02 Apr 2005 06:45:06 PM |
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Sam Wormley wrote:
Dave wrote:
I'm hoping for some physical/mathematical insight into coupled
oscillators. Is there any physicists or mathematician that can help?
The crux of the problem concerns whether you can force a pendulum to
oscillate at a frequency close, but not exactly equal to its natural
resonate frequency, and how to model it.
Zee: http://scienceworld.wolfram.com/physics/CoupledPendula.html
But the very first line of that says "Given two pendula with the same
mass m and length l attached with a spring." This is equivalent to
saying "giving two oscillators with the same resonate frequency f",
which is clearly not the case here.
I could replace l by l1 and l2, but it is clearly going to be a lot more
complicated result. Still, nothing Mathematica can't crunch through I
guess.
It's late here now (well actually it's 1:45 am and time I went to bed,
so that is for later.
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| User: "Dieter Michel" |
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| Title: Re: coupled oscillators to correct a clock - how to mathematicallymodel? |
02 Apr 2005 04:59:58 PM |
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Hi Dave,
The crux of the problem concerns whether you can
force a pendulum to oscillate at a frequency close,
but not exactly equal to its natural resonate
frequency, and how to model it.
the model will be a harmonic oscillator performing
forced oscillations imposed by an external (periodic)
force (e.g. an electromagnet).
In particular I'd like to know what happens if a
non-idealised pendulum running at a frequency of f1
(where f1 is not too stable) is supplied (or removed)
energy at a frequency *exactly* f2.
Depends a little an the kind of energy supply.
In general, the motion of a forced harmonic
oscillation is a combination of the time-course of
the external force (or it's peridicity respectively)
and the eigenfrequency (natuaral resonance frequency)
of the harmonic oscillator (here: the pendulum) itself.
Example:
Consider a pendulum with a period of 1 second.
If the external force is your finger pushing
it sidewards exactly at the point of it's rest
position (i.e. the lowest point of it's oscillation)
each time it comes back, you will see the pendulum
perform a half period exactly as if it were freely
oscillating.
However, you will also see the periodicity of the
pushing force (here: 2Hz), because by pushing it
you prevent it from performing a full oscillation
period. You will rather see only movement to one
side of the resting position. This oscillation
will have the period of the driving force (2Hz),
but will also exhibit characteristics of the
free oscillation (between the pushes).
If you apply the driving force throughout the
whole oscillation period, for example with
some kind of linear motor acting upon the
end of the pendulum, you can force the pendulum
to exactly follow the driving force and it's
frequency.
This is, for example, what a loadspeaker does.
It's eigenfrequency may be something between
40Hz and 60Hz, but it's diaphragm will move
according to the driving force (here: music
frequencies).
f1 and f2 are very close in frequency (say
fractional difference of less than 1 part in 1000).
The eigenfrequency of the oscillator and the
frequency of the driving force do not need to
be similar (see loudspeaker example).
However, the amplitude of the oscillation will
decrease with increasing difference.
Also, the less dampened the oscillator, the
greater the amplitude loss with increasing
frequency difference.
Can the pendulum be forced to oscillate at
*exactly* f2, despite small changes in its
natural resonate frequency f1?
Yes, by driving the pendulum with a periodic
force of exactly f2.
If you can only *add* energy to the pendulum,
could this only work if the frequency of
oscillation f1 was < f2 ?
No, that doesn't matter with respect to the
resulting frequency. Amplitude (and phase)
will be different, but in case of only
slight corrections (like in a pendulum clock)
that will not be a problem.
1) It required that the clock first be set
to run slow (f1 < f2). This suggests to me that
he was applying more energy than actually needed,
as he was constantly having to supply energy to
speed the pendulum up.
I presume that in that project the driving force
was not a constant sine function with frequency f2
but rather a periodic push like in my first example.
In that case, the pendulum exhibits an oscillation
with it's eigenfrequency between the pushes.
Then, a pendulum with a frequency f1 < f2 may be
more useful for a prectical implementation, because
you can give it a push with an electronic pulse
applied to the electromagnet.
Otherwise, you would need an electronic brake to
decelerate the pendulum a little bit in each cycle.
2) There is some discussion about whether this
with work with all pendulum clocks.
What you will probably need is some kind of a linear
motor (like in a loudspeaker) that is able to control
the motion of the pendulum throughout the whole
oscillation cycle. If you feed that motor with a
sinusoidal driving signal derived from a quartz
oscillator, you should get a very precise pendulum
clock.
Of course, you could remove the pendulum and use
the quartz directly, but that's not what we are
talking about here ;-)
Best regards,
Dieter Michel
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| User: "Dave" |
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| Title: Re: coupled oscillators to correct a clock - how to mathematicallymodel? |
02 Apr 2005 08:15:48 PM |
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Dieter Michel wrote:
Hi Dave,
The crux of the problem concerns whether you can
force a pendulum to oscillate at a frequency close,
but not exactly equal to its natural resonate
frequency, and how to model it.
the model will be a harmonic oscillator performing
forced oscillations imposed by an external (periodic)
force (e.g. an electromagnet).
Cheers.
In particular I'd like to know what happens if a
non-idealised pendulum running at a frequency of f1
(where f1 is not too stable) is supplied (or removed)
energy at a frequency *exactly* f2.
Depends a little an the kind of energy supply.
In general, the motion of a forced harmonic
oscillation is a combination of the time-course of
the external force (or it's peridicity respectively)
and the eigenfrequency (natuaral resonance frequency)
of the harmonic oscillator (here: the pendulum) itself.
OK, that is bad news, but since you said in general, and not
specifically always, that is not too worrying.
Example:
Consider a pendulum with a period of 1 second.
If the external force is your finger pushing
it sidewards exactly at the point of it's rest
position (i.e. the lowest point of it's oscillation)
each time it comes back, you will see the pendulum
perform a half period exactly as if it were freely
oscillating.
However, you will also see the periodicity of the
pushing force (here: 2Hz), because by pushing it
you prevent it from performing a full oscillation
period. You will rather see only movement to one
side of the resting position. This oscillation
will have the period of the driving force (2Hz),
but will also exhibit characteristics of the
free oscillation (between the pushes).
That makes sense. However, since it is not essential that the pendulum
exhibit characteristics of the forcing frequency (f2) over any part of
the oscillation, that does not matter. So if it moves faster on one
half-cycle than on the other, that is not an issue. I guess it means
there will be a small error, as the movements of the minute hand will be
small, large, small, large ... but since the movement will be too small
to discern with the eye, that will not be a problem.
If you apply the driving force throughout the
whole oscillation period, for example with
some kind of linear motor acting upon the
end of the pendulum, you can force the pendulum
to exactly follow the driving force and it's
frequency.
From a practical point of view, it would not be possible to move it
throughout the who cycle.
This is, for example, what a loadspeaker does.
It's eigenfrequency may be something between
40Hz and 60Hz, but it's diaphragm will move
according to the driving force (here: music
frequencies).
OK, that is making a lot of sense now.
Does the following make any sense to you? If so, I think I understand
the principal a *lot* better now, and also how it *might* be practical
to improve it.
I guess if you push it in two places hard enough (each side) towards the
equilibrium position (centre) you can force it to any frequency you
choose *faster* than it's resonate frequency, (which is basically what
you said when you mentioned forcing it to twice its natural frequency by
pushing it in the centre.) You could not however force it to move slower
than it's resonate frequency, which would explain why the original
design called for the clock to be set to run slow.
However, if you hang onto it for a while at some point in the cycle, you
should be able to slow it below its resonate frequency. In which case
you would need to set the clock fast, not slow.
Obviously whether you need to speed the pendulum up or slow it down is
determined by whether you set the clock fast or slow. However it is
possible to speed it up or slow it down by deciding if you want to give
it an extra shove toward the centre, or hang onto it.
In order to push towards the centre you *MUST* have a magnet attached to
the pendulum,as otherwise there is no method to push it towards the
centre. However, in order to retard it at each end, you would *not* need
a magnet on the pendulum, although it is probably better to have one as
the attractive force will be greater.
Hence assuming there is a magnet on the pendulum, one could speed it up,
*or* slow the pendulum down (whatever you choose), simply by deciding on
the direction the current is going to flow in the electromagnet.
Then how about this for a modification, which might not be practical, as
it might cause the clock to stop, and would certainly reduce the time
the clock could run for without needing to be rewound. It would however
have the advantage you might be able to avoid having any power source to
power the regulator.
You set the clock fast, but retard the pendulum in the middle, where it
is moving with maximum velocity. You retard it by turning the clock into
a dynamo, extracting power from it.
So in summary my thoughts are
1) You can set the clock fast, then slow it by retarding it somewhere in
its travel, and still get the overall frequency right.
2) You could just as easily set the clock slow, and give it a push at
the ends of the travel - as the author of the article did.
3) By running the clock fast, one could avoid having a power source by
just extracting energy on each cycle to slow the pendulum. The issue
would be that unless the clock was very accurate to start with, you
would need to extract a lot of energy to slow the pendulum, and it might
cause the clock to stop.
Any of that make sense? If (3) has some chance of working, I might try
that!!! The idea of a self contained unit, with no need for a battery is
attractive, but I suspect it is impractical.
.
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