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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).
Mathematical procedure and equations for a ring governor.

Identifier  ExFiles\Box 61b\4\  scan0004
Date  13th January 1920
  
Procedure :- Ring Governor.
K + PA{Mr Paterson}/R = S² sin 2(Φ + A/R)k (4)
Unknown :- K, P, and R
K = initial tension of spring
A = deg. advance
P = torque of spring per degree movement of ring
R = reduction factor
S = engine R.P.M. (hundreds)
Φ = initial angle of ring
k = constant for ring
Substitute values for A=0, A=.5 max, and A=max, and corresponding values of S₀, S.₅ and S₁.₀
Then K = S₀² k sin 2 Φ (1), give value to Φ, K is then fixed
(a) K + PA.{Mr Paterson}₅/R = k S.₅² sin 2(Φ + A.{Mr Adams}₅/R) } (3)
(b) K + PA{Mr Paterson}₁.₀/R = k S₁.₀² sin 2(Φ + A₁.₀/R) } now from ③(a)
2K + PA{Mr Paterson}₁.₀/R = 2k S.₅² sin 2(Φ + A.{Mr Adams}₅/R) and combining with ③(b)
K = k[2 S.₅² sin 2(Φ + A.{Mr Adams}₅/R) - S₁.₀² sin 2(Φ + A₁.₀/R)] or
y=0 = K/k - 2S.₅² sin 2(Φ + A.{Mr Adams}₅/R) + S₁.₀² sin 2(Φ + A₁.₀/R) (2)
Substitute values of R to give y=0, solve for P in ③, substitute in ④ to check original curve.
  
  


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