Definitions

# Balance spring

A balance spring, or hairspring, is a fine spiral or helical spring used in mechanical watches, marine chronometers, and other timekeeping mechanisms to control the rate of vibration of the balance wheel. A balance spring usually has a regulator lever to adjust the rate of the timepiece.

## History

There is some dispute as to whether it was invented around 1660 by British physicist Robert Hooke or Dutch scientist Christian Huygens, with the likelihood being that Hooke first had the idea, but Huygens built the first functioning watch that used a balance spring. Before that time, balance wheels or foliots without springs were used in clocks and watches, but they were very sensitive to fluctuations in the driving force, causing the timepiece to slow down as the mainspring unwound. The introduction of the balance spring effected an enormous increase in the accuracy of pocketwatches, from perhaps several hours per day to 10 minutes per day, making them useful timekeepers for the first time. The first balance springs had only a few turns.

A few early watches had a Barrow regulator, which used a worm drive, but the first widely used regulator was invented by Thomas Tompion around 1680. In the Tompion regulator the curb pins were mounted on a semicircular toothed rack, which was adjusted by fitting a key to a cog and turning it. The modern regulator, a lever pivoted concentrically with the balance wheel, was patented by Joseph Bosley in 1755, but it didn't replace the Tompion regulator until the early 1800s.

## Purpose

The balance spring is an essential part of the balance wheel; together they form a harmonic oscillator. The balance spring provides the linear restoring force that reverses the motion of the wheel so it oscillates back and forth. The motion of the balance wheel is approximately simple harmonic motion, i.e., a sinusoidal motion of constant period. Its resonant period is resistant to changes from perturbing forces, which is what makes it a good timekeeping device. The stiffness of the spring, its spring coefficient, $kappa,$ in N-m/radian, along with the balance wheel's moment of inertia, $I,$ in kg-m2, determines the wheel's oscillation period $T,$ in seconds:
$T = 2pisqrt\left\{I/kappa\right\},$
This period controls the rate of the timepiece.

## Regulator

In order to adjust the rate, the balance spring usually has a regulator, a moveable lever with a narrow slit on the end through which the last turn of the spring passes. The portion of the spring after the slit is held stationary, so the slit controls the useable length of the spring. Moving the regulator slides the slit up or down the spring, changing its effective length, and thus $kappa,$. Moving it away from the spring's attachment point (stud) shortens the spring, making it stiffer, increasing the balance's oscillation rate, and making the timepiece gain time.

The regulator interferes slightly with the motion of the spring, causing inaccuracy, so precision timepieces like marine chronometers and some high end watches are free sprung, meaning they don't have a regulator. Their rate is adjusted by weight screws on the balance instead.

## Material

A number of materials have been used for balance springs. Early on, steel was used, but without any hardening or tempering process applied; as a result, these springs would gradually weaken and the watch would start losing time. Some watchmakers, for example John Arnold, used gold, which avoids the problem of corrosion, but retains the problem of gradual weakening. Hardened and tempered steel was first used by John Harrison and subsequently remained the material of choice until the 20th century.

## Effect of temperature

The modulus of elasticity of materials is dependent on temperature. For most materials, this temperature coefficient is large enough that variations in temperature significantly affect the timekeeping of a balance wheel and balance spring. The earliest makers of watches with balance springs, such as Robert Hooke and Christian Huygens observed this effect without finding a solution to it.

John Harrison, in the course of his development of the marine chronometer, solved the problem by a "compensation curb" -- essentially a bimetallic thermometer which adjusted the effective length of the balance spring as a function of temperature. While this scheme worked well enough to allow Harrison to meet the standards set by the Longitude Act, it was not widely adopted.

Around 1765, Pierre Le Roy (son of Julien Le Roy) invented the compensation balance, which became the standard approach for temperature compensation in watches and chronometers. In this approach, the shape of the balance is altered, or adjusting weights are moved on the spokes or rim of the balance, by a temperature-sensitive mechanism. This changes the moment of inertia of the balance wheel, and the change is adjusted such that it compensates for the change in modulus of elasticity of the balance spring. The compensating balance design of Thomas Earnshaw, which consists simply of a balance wheel with bimetallic rim, became the standard solution for temperature compensation.

### Elinvar

While the compensating balance was effective as a way to compensate for the effect of temperature on the balance spring, it could not provide a complete solution. The basic design suffers from "middle temperature error": if the compensation is adjusted to be exact at extremes of temperature, then it will be slightly off at temperatures between those extremes. Various "auxiliary compensation" mechanisms were designed to avoid this, but they all suffer from being complex and hard to adjust.

Around 1900, a fundamentally different solution was created by Charles Édouard Guillaume, inventor of elinvar. This is a nickel-steel alloy with the property that the modulus of elasticity is essentially unaffected by temperature. A watch fitted with an elinvar balance spring requires either no temperature compensation at all, or very little. This simplifies the mechanism, and it also means that middle temperature error is eliminated as well, or at a minimum is drastically reduced.

## Isochronism

A balance spring obeys Hooke's Law: the restoring torque is proportional to the angular displacement. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. This is an essential property for accurate timekeeping, because no mechanical drive train can provide absolutely constant driving force. This is particularly true in watches and portable clocks which are powered by a mainspring, which provides a diminishing drive force as it unwinds. Another cause of varying driving force is friction, which varies as the lubricating oil ages.

Early watchmakers empirically found approaches to make their balance springs isochronous. For example, John Arnold in 1776 patented a helical (cylindrical) form of the balance spring, in which the ends of the spring were coiled inwards. In 1861 M. Phillips published a theoretical treatment of the problem. He demonstrated that a balance spring whose center of gravity coincides with the axis of the balance wheel is isochronous.