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From the Rolls-Royce experimental archive: a quarter of a million communications from Rolls-Royce, 1906 to 1960's. Documents from the Sir Henry Royce Memorial Foundation (SHRMF).
Technical bulletin discussing the causes and solutions for high speed wobbles, focusing on tyre pressure and size.

Identifier  ExFiles\Box 45\2\  Scan004
Date  1st January 1927 guessed
  
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-2-

(b) High speed wobbles.

(1) Tyre pressure. The main factor within the control of the customer in this connection is the tyre pressure. The higher the tyre pressure, the less the likelihood of high speed wobbles developing and the less the amplitude of the wobble when it does occur. It should be noted that the pressure of balloon tyres is much more likely to vary in service/than that of high pressure tyres (See - Variation of pressure with atmospheric temperature).
Care should be taken when recommending a customer to inflate tyres to a high pressure, as unless the steering can be provided with a little friction somewhere it may suffer from 'joggles' at high speeds.

(11) Tyre size. Until recently we have been under the impression that a larger section tyre will always be worse for high speed wobbles than a small section tyre, provided that the casings of both contain the same number of plies. However, our recent tests have shewn that the 6.75" Dunlop balloon tyre is without doubt less prone to engender high speed wobbles than the 6" tyre of the same make, if both are run anywhere near the same inflation pressure. Our views with regard to the advantage of the large tyre are that this tyre has a greater rotational moment of inertia than the smaller tyre. Our experiments shew that anything which increases the flywheel effect of the wheel and tyre without markedly increasing the transverse moment of inertia of the axle tends to reduce wobbles i.e., the gyroscopic couple of the wheel resists a high speed wobble. The matter being somewhat complex, it is safe to say that an owner complaining of 'shimmy' with any tyre other than Dunlop or Michelin, should be recommended to try the former at least on his front wheels.

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