# Blog: Middle Temperature Error

Date: 2 February 2017

Copyright © David Boettcher 2006 - 2017 all rights reserved.I make additions and corrections to this web site frequently, but because they are buried somewhere on one of the pages the changes are not very noticeable, so I decided to create this blog section to highlight new material. Here below you will find part of one of the pages that I have recently either changed or added to significantly.

The section reproduced here is from my page about Temperature effects in watches.

If you have any comments or questions, please don't hesitate to contact me via my Contact me page.

### Middle Temperature Error

HSN article about Middle Temperature Error. A313S152

Download article: MTE.pdf

Download spreadsheet: MTE.xlsx

In the late eighteenth century a phenomenon was observed in marine chronometers with temperature compensation. It was found that if the device was brought to time at a certain temperature it would lose at higher and lower temperatures. This effect was not observed in watches without temperature compensation because it was dwarfed by other effects, principally the change in rate with temperature due to variation in the elasticity of the balance spring. The effect became observable in marine chronometers with compensation for this major, primary source, of temperature caused error.

To minimise the total error over the range of temperatures a marine chronometer was expected to encounter, the timing was adjusted so that it was fast at a "middle" temperature and correct at two temperatures either side of this. This gaining rate at the middle temperature was called the "middle temperature error".

Because the effect only became noticeable once the primary source of error, the change in the elasticity of the balance spring with temperature, had been compensated, it was also called "secondary error".

The cause of middle temperature error is briefly outlined further down on this page. If you are not already familiar with middle temperature error you might want to read that short summary first. The phenomenon of middle temperature or secondary error is discussed more fully in an article published in the February 2017 edition of the Horological Science Newsletter (HSN). The HSN newsletter is published by NAWCC Chapter #161. The interest of Chapter #161 is the study and distribution of information about the science of horology. Chapter membership is available to members of the NAWCC. The editor of HSN, Bob Holmström has kindly agreed that my article and spreadsheet can also be downloaded from this web page.

The spreadsheet that accompanies the article allows you to interactively explore the effects of MTE. I strongly recommend that you download and try it. You don't need to do any spreadsheet programming, it is already set up. You just alter the values of the thermal coefficients of expansion and elasticity, and charts built into the spreadsheet immediately show you the effect on timekeeping. It's really simple so give it a go, and if you have any problems just drop me an email. If you don't have the Microsoft Office Excel spreadsheet application, then Libre Office contains an excellent alternative and is available absolutely free from Download Libre Office.

The spreadsheet can also be used to investigate the effects of the individual components. For example, to see the effect of thermal expansion of the balance alone, then set all the coefficients apart from the "Thermal expansion/ºC – balance:" to zero. As a check, a brass balance should show a loss of 1.64 seconds per day for a temperature increase of one degree C, a steel balance 0.95 seconds per day per degree C.

The article and spreadsheet can be downloaded from these links: article: MTE.pdf, spreadsheet: MTE.xlsx.

If you have any comments or questions, please don't hesitate to contact me via my Contact me page.

#### Explanations of Middle Temperature Error

An explanation for middle temperature error was developed by the Reverend George Fisher and published in The Nautical Magazine of 1842 under the name of E. J. Dent, of the chronometer makers Frodsham & Dent. Fisher's name is not mentioned in Dent's article, probably because Fisher had incurred the displeasure of the Astronomer Royal by publishing a paper that suggested that the going of chronometers could be affected by magnetism in iron ships. The Astronomer Royal refuted this in polite but strong terms and Fisher probably decided to keep a low profile as a result.

The explanation published by Dent was based on the observation that the equation for the period of a balance controlled timepiece can be written as:

$$ T = {\pi} \sqrt \frac { m k^2}{S} $$where mk^{2} is the moment of inertia of the balance, m being the mass and k the radius of gyration, and S represents the force of the balance spring. Observations by Dent had indicated that the elasticity of the balance spring varied linearly with temperature.

Fisher surmised that it was the ratio within the square root of k^{2} to a linear S that was the cause of middle temperature error. Clearly a curve produced by plotting the k^{2} term against temperature on a chart could only intersect with a straight line representing a linear S term at either one or, at most, two points. This is the explanation related by Commander Gould in "The Marine Chronometer".

Fisher's explanation considers the relationship of inertia to spring force within the encompassing square root sign of the equation for period. An alternative way of visualising the effect is by recognising that the period is proportional to the square root of the square of the radius of gyration k and rewriting the equation as:

$$ T \propto \frac { k} {\sqrt S} $$If both k and S vary linearly with temperature, as was supposed, then a plot of k against temperature on a chart would be a straight line which could only intersect with a curve representing the square root of the linear S term at either one or, at most, two points. This is the explanation related by A. L. Rawlings in "The Science of Clocks and Watches".

The two different versions use the same geometric or mechanical phenomenon to explain middle temperature error, the only difference is that the Fisher / Dent version considers the ratio k^{2} to S inside the square root sign, whereas Rawlings looks at the whole equation for period including the square root and considers the ratio of k to √ S .

Middle temperature error or secondary error: seconds per day.

1: Blue line; square root effect alone.

2: Orange line; square root plus non-linear elasticity of balance spring.

This "square root" explanation is rational, logical and correct, and it held sway as the only explanation for middle temperature error for many years. However, although the reasoning is correct, the magnitude of the middle temperature error produced by this effect is smaller than observed in practice. This is not a criticism of Fisher and Dent and others, it is just that until there was an easy way of calculating the effect and plotting it, there was no reason to seek an alternative explanation.

The shortcoming in the square root explanation was noticed by Peter Baxandall during the updating of Rawlings' classic work by the BHI, and subsequently investigated by Philip Woodward in a paper published in the Horological Journal of April 2011. Woodward showed that the middle temperature error produced by the square root effect alone was around one tenth of a second per day, only a small fraction of observed values.

There is no doubt that the square root effect does exist, but a cause for the remaining majority of the error must be sought.

In the early twentieth century Dr Guillaume explained that middle temperature error, which he called "secondary error" or "Dent's error", is due to the fact that the elasticity of the balance spring does not vary linearly with temperature but has a curvature. It is not clear that Guillaume ascribed any of the middle temperature error to the square root effect, he doesn't mention it in his writings. From his work on changes in the dimensions and elastic moduli of nickel steels with changes in temperature he would have known that any such effect would be small, it appears that he probably thought it would be negligible. Dr Guillaume's explanation of the effect can be found at The Guillaume "integral" balance.

The chart here shows the two effects on the timekeeping of a machine brought to time at 5°C and 35°C, with a middle temperature of 20°C. The blue line shows the middle temperature error that results from the square root explanation alone. It is very small, of the order of one tenth of a second per day. The orange line shows the error that results when the curvature in the temperature response of the elasticity of the balance spring is also taken into account. This results in an error of more than two seconds per day, which is in accordance with observed values.

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Copyright © David Boettcher 2006 - 2017 all rights reserved. This page updated February 2017. W3CMVS.