Hartnell Governor:

The Hartnell governor is illustrated in Figure. The two bell crank leners have been provided that can have rotating motion around fulcrums O and O′. One ending of each bell crank lever carries a ball at one ending of one arm and a roller at the ending of other arm. The rollers make contact along the sleeve. The frame is associated to the spindle. A helical spring is mounted around the spindle among frame and sleeve. With the rotation of the spindle, all of these parts rotate.

Along the increase of speed, the radius of rotation of the balls enhanced and the rollers lift the sleeve against the spring force. Along the decrease in speed, the sleeve moves downwards. The movement of the sleeve are transferred to the throttle of the engine through linkages.

  Hartnell Governor

 Assume r₁ = from spindle axis minimum radius of rotation of ball centre, in m,

             r₂ = from spindle axis maximum radius of rotation of ball centre, in m,

 S₁ = At minimum radius spring force exerted on sleeve, in N,

S₂ = At maximum radius spring force exerted on sleeve, in N,

m = Mass of each ball ( kg)

M = Mass of sleeve ( kg)

N₁ = Minimum speed of governor at minimum radius (in rpm)

N₂ = Maximum speed of governor at maximum radius( in rpm)

ω₁ and ω₂ = Corresponding minimum and maximum angular velocities( in r/s)

 (F_C)₁ = Centrifugal force equivalent to minimum speed = m × ω₂ × r ,

(F_C)₂ = Centrifugal force equivalent to maximum speed = m × ω₂ × r  ,

 s = Stiffness of spring or the force needed to compress the spring by one m,

r = Distance of fulcrum O from the governor axis or radius of rotation,

a = Length of ball arm of bell-crank lever, that is. distance OA, and

b = Length of sleeve arm of bell-crank lever, that is distance OC.

Letting the position of the ball at radius 'r₁', as illustrated in Figure (a) and taking moments of all of the forces around O

(a)                                                                     (b)

Letting the position of the ball at radius 'r₂' as illustrate in Figure (b) and taking the moments of all of the forces around O′

If θ₁ and θ₂ are extremely small and mass of the ball is negligible in compare to the spring force, the terms m_g tan θ₁ and m_g tan θ₂ might be ignored.

For ball radius 'r'