# 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.

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### Middle Temperature Error

HSN article about Middle Temperature Error. A103S77

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.

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 correct at two temperatures either side of a "middle" temperature. This resulted in a gaining rate at the middle temperature, which 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 reason for the 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 is discussed more fully in an article published in the February 2017 edition of the Horological Science Newsletter (HSN). A spreadsheet that allows you to interactively explore the effects of MTE accompanies the article. 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 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 based on the observation that the equation for the period of a balance controlled timepiece could be written as:

$$ T = {\pi} \sqrt \frac { m k^2}{S} \qquad or \qquad T \propto \frac { k} {\sqrt S} $$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 accelerating effect of the balance spring. Observations by E. J. Dent of the chronometer makers Frodsham & 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 explanation was published in The Nautical Magazine of 1842 under Dent's name. This is the explanation related by Commander Gould in "The Marine Chronometer".

An alternative way of visualising the effect is using the second version of the equation, where T is proportional to k and inversely proportional to the square root of S. If k and S both vary linearly with temperature, a line produced by plotting k against temperature on a chart would be straight and 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 Dent version considers the ratio k^{2} to S inside the square root sign, whereas Rawlings looks at the whole equation and considers the ratio of k to √ S .

Middle temperature / secondary error.

Blue line; square root effect alone.

Orange line; square root effect plus non-linear elasticity.

This 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 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. However, although there is no doubt that the effect exists, a cause for the remaining majority of the error must be sought.

In the early twentieth century it was noted by Dr Guillaume that the secondary error, which Guillaume referred to as "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 insignificant.

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.