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

Next, consider a free-body diagram that cuts through shafts (1) and (2) and
includes gear B. Once again, a positive internal torque will be assumed in shafts
(1) and (2). The teeth of gear E exert a force F on the teeth of gear B. If the
radius of gear B is denoted by R B , a moment equilibrium equation about the x
axis can be written as
T 1
R B
(1)
B
NB
= 40 teeth
(2)
T 2
Σ M = T − T − F × R = 0
(b)
x
2 1
B
F
Next, consider a free-body diagram that cuts through shaft (3) and includes
gear E as shown. A positive internal torque T 3 is assumed to act in shaft (3).
Since the teeth of gear E exert a force F on the teeth of gear B, equilibrium
requires that the teeth of gear B exert a force of equal magnitude in the opposite
direction on the teeth of gear E. With the radius of gear E denoted by R E , a
moment equilibrium equation about the x′ axis can be written as
T3
Σ Mx′ = −T3
− F × RE
= 0 ∴ F = −
(c)
R
E
T 3
(3)
F
NE=
60 teeth
E
R E
x′
The results of Equations (a) and (c) can be substituted into Equation (b) to give
= − × = ⋅ − ⎛ T3
⎞
T T F R
−
⎝
⎜
⎠
⎟ R = ⋅ + T R 1 2
B 18 kip in. B 18 kip in. 3
R
R
E
B
E
The gear radii R B and R E are not known. However, the ratio R B /R E is simply the gear
ratio between gears B and E. Since the teeth on both gears must be the same size in
order for the gears to mesh properly, the ratio of the teeth on each gear is equivalent
to the ratio of the gear radii. Consequently, the torque in shaft (1) can be expressed in
terms of N B and N E , the number of teeth on gears B and E, respectively:
T
B
= 18 kip⋅ in. + T N (d)
N
1 3
Equation (d) summarizes the results of the equilibrium considerations, but there are still
two unknowns in it: T 1 and T 3 . Consequently, this problem is statically indeterminate.
To solve the problem, an additional equation must be developed. This second equation
will be derived from the relationship between the angles of twist in shafts (1) and (3).
E
Step 2 — Geometry of Deformation: The rotation of gear B is equal to the angle of
twist in shaft (1):
φ
= φ
B 1
Similarly, the rotation of gear E is equal to the angle of twist in shaft (3):
φ
= φ
E 3
However, since the gear teeth mesh, the rotation angles for gears B and E are not
independent. The arclengths associated with the respective rotations must be equal,
but the gears turn in opposite directions. Thus, the relationship between the gear
rotations can be stated as
R φ
=−R
φ
B B E E
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