Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

6.8 power Transmission
161
POwER TRANSMISSION
One of the most common uses for a circular shaft is the transmission of power from motors
or engines to other devices and components. Power is defined as the work performed in a
unit of time. The work W done by a torque T of constant magnitude is equal to the product
of the torque T and the angle φ through which the torque rotates:
W
= Tφ
(6.15)
Power is the rate at which the work is done. Therefore, Equation (6.15) can be differentiated
with respect to time t to give an expression for the power P transmitted by a shaft
subjected to a constant torque T:
dW
P = T d φ
dt
= dt
(6.16)
The rate of change of the angular displacement dφ /dt is the rotational speed or angular
velocity ω. Therefore, the power P transmitted by a shaft is a function of the magnitude of
the torque T in the shaft and its rotational speed ω; that is
where ω is measured in radians per second.
P
= Tω
(6.17)
power Units
In SI, an appropriate unit for torque is the newton-meter (N· m). The corresponding SI unit
for power is termed a watt:
Nm ⋅
P = T ω = (N⋅ m)(rad/s) = = 1watt = 1W
s
In U.S. customary units, torque is often measured in lb · ft; thus, the corresponding power
unit is
lb⋅ft
P = T ω = (lb⋅ ft)(rad/s) =
s
In U.S. practice, power is typically expressed in terms of horsepower (hp), which has the
following conversion factor:
lb⋅ft
1hp = 550 (6.18)
s
Rotational Speed Units
The rotational speed ω of a shaft is commonly expressed either as frequency f or as revolutions
per minute (rpm). Frequency f is the number of revolutions per unit of time. The
standard unit of frequency is the hertz (Hz), which is equal to one revolution per second
(s −1 ). Since a shaft turns through an angle of 2π radians in one revolution (rev), rotational
speed ω can be expressed in terms of frequency f measured in Hz:
f rev 2π
rad
ω = ⎛ 2π
f rad/s
⎝ ⎜ ⎞
s ⎠
⎟ ⎛ ⎝ ⎜ ⎞
rev ⎠
⎟ =
Accordingly, Equation (6.17) can be written in terms of frequency f (measured in Hz) as
P = Tω
= 2π
fT
(6.19)