Talk:Balance wheel

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This article seems to focus on one narrow part of the subject: compensated, bimetallic balance wheels. When and where were balance wheels invented? Why was the spring added? Were balance wheels used in anything besides watches? What do modern balance wheels look like? How accurate are they? --Chetvorno 00:22, 15 June 2007 (UTC)

Good points. One issue is that the balance wheel is a component of a system (the mechanical watch). Some of what we'd want to discuss relates to the system, not the component. For example, you can't speak of accuracy of a balance. A watch has accuracy, and the way the balance is constructed is a factor, but it also depends on the balance spring, the escapement and so on.
Mechanical watch might be a good spot for this. Paul Koning 10:59, 15 June 2007 (UTC)
I've taken off some of the tags since those issues have now been cured. That was a nice rewrite, Chetvorno. Paul Koning 14:51, 19 June 2007 (UTC)

[edit] Modern balance wheels

I have a photo of a chronometer (in my father's collection), built during world war 2 by the Hamilton Watch Company. It clearly has an elinvar balance spring because the balance wheel has no cuts in it. But apart from that, the balance wheel looks like the classic kind, with screws at various points around the circumference.

I was told (but don't have a reference at hand) that elinvar still has a (very small) temperature coefficient, so it is desirable for the balance wheel to compensate for that. And supposedly that's done by using a bimetallic rim whichs is not cut, or perhaps a rim of one material with spokes of another. Supposedly, the result is that the wheel changes shape slightly (from circular to elliptical) when temperature changes, a smaller effect than the Earnshaw pattern classic compensation balance. And if that's correct, then the screws would indeed be compensation adjustment screws.

I'll see if I can get more data, and perhaps permission to include the photo. Paul Koning 20:32, 22 June 2007 (UTC)

That's fascinating! I've heard the same thing, about uncut alloy balance wheels being temperature compensated, but I was unable to find anything about it. I have a Bulova 23j watch from 1956 with an uncut balance with weights, and it says 'adjusted to heat and cold' on the movement, so I always wondered. If you could find a reference, that would be a good thing to include.
I have an image of an alloy balance on Wikipedia Commons, Benrus_Watch_Balance_Wheel.jpg that I was going to include in the article, but your photo sounds better. --Chetvorno 02:09, 23 June 2007 (UTC)
One key difference is that I don't have permission yet to make that photo available.... so use yours. Even if I do get permission it will take a while (months not days). Paul Koning 14:50, 25 June 2007 (UTC)

[edit] Mathematical Model

I was thinking of adding a section on the math of balance wheels at the end, similar to below. Comments? Too boring? --Chetvorno 00:39, 23 June 2007 (UTC)

(See definition of terms at end)

A balance wheel is a lightly damped impulse driven harmonic oscillator. The equation of motion is:

I \ddot \theta + c \dot \theta + k\theta = \tau(t)

A 'perfect' balance, with no friction c\, or drive torque \tau\, has the solution:

\theta = A\cos{(\omega_0t+\phi)} \,       where \omega_0 = \sqrt {k/I} \,

So it oscillates at a constant amplitude with the frequency f_0 = 2\pi\omega_0 = 2\pi\sqrt{k/I}\,, called the natural resonant frequency. The goal of balance wheel design is to approach the behavior of this 'perfect' balance.

The general unforced (homogeneous) solution is:

\theta = A e^{-\alpha t} \cos {( \omega t + \phi )} \,
Where:
\alpha = c / 2I \,
\omega = \sqrt { k/I - (c/2I)^2 } = \sqrt { \omega_0^2 - (c/2I)^2 }

The friction c\,  of real balances is low enough that they are very underdamped, sharply resonant systems. Under these conditions, where c << (kI)^{1/2} \,, a common dimensionless parameter used to characterize the timekeeping ability of balance wheels and other resonators is the Q, or quality factor:

Q = (2\pi) \frac {\mbox{energy stored in balance}}{\mbox{energy lost during one period}} = (2\pi) \frac {\mbox{energy stored in balance}}{\mbox{energy provided by escapement during one period}}\,
= \frac {\sqrt{4kI}}{c} = \frac{2I\omega_0}{c}\,

The higher the Q, the less energy is required from the escapement per period to replace the energy lost to friction. Since the escapement is the main source of disturbances to the motion of the balance, a balance's possible precision as a timekeeper is roughly proportional to it's Q.

It can be seen that to get higher Q, other things being equal, requires a larger, heavier balance (I), a faster beat (ω0), or lower friction (c). This is the direction balance design has taken over the years. The Q of modern watch balances is around 100 to 300. This is why they are not as good timekeepers as pendulums (Q ~ 10^4), or quartz crystals (Q ~ 10^4 – 10^6).

==Definition of terms==
\theta = \, Angle of rotation of balance wheel from rest position (radians)
I = \, Moment of inertia of balance wheel (Kg-m)
c = \, Coefficient of angular friction from all sources (Kg-m^2/radian-s)
k = \, Spring coefficient of angular elasticity (Newton-m/radian)
\tau = \, Drive torque on balance from escapement (Newton-m)
A = \, Maximum amplitude of vibration (radians)
\omega = \, Angular frequency of vibration (radians/sec)
f =\, Beat/2 = Frequency of vibration (Hz)


Interesting. The difference in Q seems to correspond reasonably well to the difference in achievable accuracy (order of a second a year for a high quality quartz clock vs. maybe 10x that for a good mechanical chronometer). But there are other aspects that I don't see here. For example, it's a rule of escapement design that you want to apply the impulse at the zero crossing. Why (mathematically) is that? Paul Koning 14:49, 25 June 2007 (UTC)