Gears were, in the early days, like bearings: very magical things. According to the scientific understanding of the time, they could not possibly work. The faces of two gears touching can be approximately represented as a single line contact of infinite ‘thin-ness’ along the faces of the gears. When you calculated the forces acting here, they were of course infinite (the line having no area, and the equation for stress being sigma=force/area). This was very upsetting for early gear pioneers, who were pleased when German physicist Heinrich Hertz published Über die Berührung fester elastischer Körper (On the contact of solid elastic bodies).1
Hertz’s work accurately determined that two circles forced together actually ‘squidge’, resulting in a much larger contact area than might be expected. This became known as the Hertz Contact Theory, and is still used today.
Engineers were happy that the stresses in their gears now seemed to be below the level at which the gears ought to break. However, they were most vexed by the fact that their calculations showed that the gears which had worked for decades could not possibly work for any sustained length of time. This was because gears need a lubricant to stop the teeth rubbing together, and the calculations still stubbornly showed that even with the now much larger contact zone as predicted by Hertz, the level of the oil film was far below the rough surfaces of the teeth. This meant that the oil couldn’t keep the surfaces apart – they ought to tear themselves to shreds in hours.
In around 1893, Carl Barus came to the rescue with the theory now known as the Barus Law. This stated that – to everyone’s great surprise – the factor in an oil that stops it getting flattened by pressure (its viscosity) doesn’t just go up the more you press it between two contacts, it actually skyrockets. This meant that if you added Hertz’s realization that two cylinders touching don’t really contact as a ‘line’ to the fact that the oil viscosity in this wider contact was considerably higher than expected from simple lab tests on oils, suddenly gears and bearings made sense to those trying to calculate their behavior.
However, the engineering world took a while to start putting these pieces together. This was possibly because without the Barus Law being added to it, Hertz’s work didn’t make sense for gear contacts. Without the exponential increase in viscosity inside the contact zone, the oil film was not predicted to survive the pressures needed to deform the steel contacts elastically.
Then, at Moscow Central Institute for Mechanical Engineering, Alexander Ertel-Morenstein became the first to combine the work of Hertz (elastic deformation) with hydrodynamics and tribology (existing fluids laws added to the Barus Law). Thus, the field of elastohydrodynamics was born, where the correct contact width was known, due to Hertz; the correct fluid viscosity under pressure was known (thanks to Barus); and equations showed how all this combined in a gear contact, revealing that the real oil film thickness was higher than the roughness of the metal underneath.
The problem was, Ertel had been doing much of this work in Russia during World War II. By now he was desperately sick with tuberculosis, and no one outside the country knew of his research. In 1945 he wrote The Calculation of Hydrodynamic Lubrication of Moving Curved Surfaces under High Loading, about which Dr A N Grubin, the head of the institute’s bearings department, stated, “The level of this work approaches that of the classical works of the lubrication theories by Petrov, Reynolds, Hertz.” Anyone who knows anything about Russians and mathematics understands that this meant it was very difficult work indeed.
By the time the war ended, Ertel was incredibly ill. Taking advantage of a posting to East Berlin, he reportedly faked his own death in a swimming accident and defected to the West, where he was welcomed and restored to health.
To keep his existence a secret from the Russian authorities, he assumed the alias Alexander von Mohrenstein (adding ‘von’ for a more convincing Germanic feel) and began work under Prof. Gustav Niemann at TH Braunschweig. His groundbreaking paper was published in 1949 under the name of his former head of department, Dr Grubin.
In 1966, Prof. M M Khrushchev at Moscow Central Institute for Mechanical Engineering received a letter from Ertel, which said, “In Grubin’s paper, there is no mention that the theory and calculations were conducted by me. I find this very indecent of him. However, should you see Grubin, tell him that I have forgiven him. With kind regards, your A Ertel (Dr A Morenstein).”
In my own undergraduate thesis I used the ‘Grubin’ method to calculate gear oil film thickness. It was only in the last few years that I discovered that it was not really Grubin who worked out how (and why) gears and bearings work.
Ertel lived a long life and died in 2001. Most people who have investigated the story believe that Grubin probably did not seek to plagiarize him. Aware of the importance of the work, he knew it had to be disseminated in the West. However, it could not be published under Ertel’s name for fear of bringing him to the attention of the NKVD (Russian security services), who would not be best pleased that one of the country’s top scientists had run away and been given a nice job working for the capitalists. Thus, Grubin published it under his own name to protect Ertel. The whole story was finally revealed after exhaustive research by the English tribologist Prof. A Cameron, published in the article Righting a 40-year-old wrong: A. M. Ertel – the true author of ‘Grubin’s EHL’ Solution, Tribology Int. 18, 92 (1985).
1) https://home.uni-leipzig.de/pwm/web/download/Hertz1881.pdf
