Theory of Machine
In under damped vibrating system, if x₁ and x₂ are the successive values of the amplitude on the same side of the mean position, then the logarithmic decrement is equal to

log(x₁.x₂)
log(x₁/x₂)
loge(x₁/x₂)
x₁/x₂

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Theory of Machine
For an isochronous Hartnell governor (where r₁ and r₂ = Maximum and minimum radius of rotation of balls respectively, S₁ and S₂ = Maximum and minimum force exerted on the sleeve respectively, and M = Mass on the sleeve)

S₁/S₂ = r₁/r₂
(m.g - S₁)/(m.g - S₂) = r₂/r₁
(m.g + S₁)/(m.g + S₂) = r₁/r₂
S₂/S₁ = r₁/r₂

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Theory of Machine
The frictional torque transmitted in a truncated conical pivot bearing, considering uniform pressure, is

(1/2). μ W cosecα (r₁ + r₂)
(2/3). μ W cosecα (r₁ + r₂)
(2/3). μ W cosecα [(r₁³ - r₂³)/(r₁² - r₂²)]
(1/2). μ W cosecα [(r₁³ - r₂³)/(r₁² - r₂²)]

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