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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 analysis of the frequency of vibration in spiral springs, specifically in relation to valve spring breakages.

Identifier  ExFiles\Box 56\2\  Scan096
Date  28th August 1930 guessed
  
-2-

where E is the elasticity or rate of the whole
spring, namely the load in poundals per foot of compression
and M is the mass of the whole spring in pounds. The more
immediately useful rearrangement of this expression for the
frequency of vibration of a spiral spring is

n = (d / 8πkD²) * √(2Cg / w) (approximately)

As there appears good ground for supposing that
valve spring breakages occur as a result of dynamic stresses
caused by oscillations of the spring coils, it is generally
agreed to be desirable that the natural frequency of vibration
of the spring shall be as high as possible consistent with
the usual static quantities being as desired. It is of
interest to examine what direction is indicated by the use of
the above results in which to vary the design of the spring
to keep the natural frequency of vibration as high as possible.

Omitting the invariable portion of the expression
for the frequency above given (i.e. C shear modulus,
g gravity and w density of wire) it is seen that the
frequency number of vibration of a spiral spring varies as

d / kD²

It is also seen from the statical results
given that the ratio of the stress to the deflection
  
  


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