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

Furthermore, since gears have teeth, shafts connected by gears are always synchronized
exactly with one another.
A basic gear assembly is shown in Figure 6.14. In this assembly, torque is transmitted
from shaft (1) to shaft (2) by means of gears A and B, which have radii of R A and R B , respectively.
The number of teeth on each gear is denoted by N A and N B . Positive internal
torques T 1 and T 2 are assumed in shafts (1) and (2). For clarity, bearings necessary to support
the two shafts have been omitted. This configuration will be used to illustrate basic
relationships involving torque, rotation angle, and rotation speed in torsion assemblies
with gears.
Torque
To illustrate the relationship between the internal torques in shafts (1) and (2), free-body
diagrams of each gear are shown in Figure 6.15. If the system is to be in equilibrium, then
each gear must be in equilibrium. Consider the free-body diagram of gear A. The internal
torque T 1 acting in shaft (1) is transmitted directly to gear A. This torque causes gear A to
rotate counterclockwise. As gears A and B rotate, the teeth of gear B exert a force on gear
A that acts tangential to both gears. This force, which opposes the rotation of gear A, is
denoted by F. A moment equilibrium equation about the x axis gives the relationship
between T 1 and F for gear A:
T
Σ Mx
= T1
− F × RA
= 0 ∴ F =
R
1
A
(a)
R B
R A
NA
T 2
(2)
NB
B
= Number of teeth
on gear A
= Number of teeth
on gear B
(1)
FIGURE 6.14 Basic gear
assembly.
R B
R A
T 2
(2)
F
F
B
A
A
x′
x′
T 1
x
Next, consider the free-body diagram of gear B. If the teeth of gear B exert a force F on gear A,
then the teeth of gear A must exert a force of equal magnitude on gear B, but that acts in the
opposite direction. This force causes gear B to rotate clockwise. A moment equilibrium
equation about the x′ axis then gives
Σ Mx′ = − F × RB − T2
= 0
(b)
If the expression for F determined in Equation (a) is substituted into Equation (b), then the
torque T 2 required to satisfy equilibrium can be expressed in terms of the torque T 1 :
T1
− − = ∴ =−
R R T 0 T T R B
R
A
2 2 1
B
(c)
A
The magnitude of T 2 is related to T 1 by the ratio of the gear radii. Since the two gears rotate
in opposite directions, however, the sign of T 2 is opposite from the sign of T 1 .
Gear Ratio. The ratio R B /R A in Equation (c) is called the gear ratio, and this ratio is
the key parameter that dictates relationships between shafts connected by gears. The gear
ratio in Equation (c) is expressed in terms of the gear radii; however, it can also be expressed
in terms of gear diameters or gear teeth.
The diameter D of a gear is simply two times its radius R. Accordingly, the gear ratio
in Equation (c) could also be expressed as D B /D A , where D A and D B are the diameters of
gears A and B, respectively.
For two gears to interlock properly, the teeth on both gears must be the same size. In other
words, the arclength of a single tooth—a quantity termed the pitch p—must be the same for both
gears. The circumference C of gears A and B can be expressed either in terms of the gear radius,
(1)
FIGURE 6.15 Free-body
diagrams of gears A and B.
T 1
mecmovies 6.9 presents an
animation that illustrates basic
gear relationships for torque,
rotation angle, rotation speed,
and power transmission.
x
C = 2πR C = 2πR
A A B B
155