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A spur pinion of pitch diameter 50 mm rotates at 200 rad/s and transmits 3 kW power. The pressure angle of the tooth of the pinion is 20°. Assuming that only one pair of the teeth is in contact, the total force (in newton) exerted by a tooth of the pinion on the tooth on a mating gear is _______

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Concept:

Spur gear:

  • Spur gears are those gears in which teeth are straight and parallel to the axis of the shaft.
  • These types of gear are noisy, more fatigue and have less life.
  • Spur gears are used for very low power transmission.

Force analysis in Spur gear:

  • Total force (F) acts on the common normal of the mating gears.
  • Total force can be divided into two components.

1.Tangential component (Ft) = Fcosϕ

2. Radial component (Fr) = Fsinϕ

   where ϕ is pressure angle

  • Ftis used to produce the power required and Fr provides thrust.
  • Torque produced can be calculated as:

          T = Ft × R    (where R is pitch radius)

  • The power produced can be calculated as:

           P = T × ω   (where ω is the angular velocity of gear)

Calculation:

Given:

ω = 200 rad/s,

\(D = 50{\rm{}}mm = 0.05{\rm{}}m \Rightarrow R = \frac{D}{2} = 0.025m\)

P = 3 kW = 3000 W

ϕ = 20°

\(\begin{array}{l} P = Tw \Rightarrow T = \frac{P}{w} = \frac{{3000}}{{200}} = 15N.m\\ T = {F_t} \times R \Rightarrow {F_t} = \frac{T}{R} = \frac{{15}}{{0.025}} = 600N\\ F = \frac{{{F_t}}}{{\cos \phi }} = \frac{{600}}{{\cos 20^\circ }} = 638.5N \end{array}\)

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