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

R B
NB
= 40 teeth
φ B
B
where R B and R E are the radii of gears B and E, respectively. Since the gear rotation
angles are related to the shaft angles of twist, the preceding equation can be
expressed as
R φ =− R φ
(e)
B 1 E 3
Step 3 — Torque–Twist Relationships: The angles of twist in shafts (1) and (3) can
be expressed as
R E
φ E
φ
1
TL 1 1
= and φ3
=
JG
1 1
TL
JG
3 3
3 3
(f)
E
Step 4 — compatibility Equation: Substitute the torque–twist relationships
[Equation (f)] into the geometry-of-deformation equation [Equation (e)] to
obtain
NE
= 60 teeth
1 1
R TL
B
JG
1 1
=−R
E
TL
JG
3 3
3 3
which can be rearranged and expressed in terms of the gear ratio N B /N E :
N
N
B
E
TL
JG
1 1
1 1
TL 3 3
=− (g)
JG
3 3
Note: The compatibility equation has two unknowns: T 1 and T 3 . This equation can be
solved simultaneously with the equilibrium equation [Equation (d)] to calculate the
internal torques in shafts (1) and (3).
Step 5 — Solve the Equations: Solve for the internal torque T 3 in Equation (g):
T
=−T N N
3 1
B
E
⎛
⎝
⎜
L
L
Then substitute the result into Equation (d):
1
3
⎞ ⎛ J
⎠
⎟
⎝
⎜
J
⎞ G3
⎠
⎟ ⎛ ⎝ ⎜
⎞
G ⎠
⎟
3
1
1
T
= 18 kip⋅ in. + T N N
1 3
⎡
= 18 kip⋅ in. + ⎢−T N 1
⎣ N
⎛
= 18 kip⋅in.
− T
N 1
⎝
⎜
N
Now group the T 1 terms to obtain
B
E
B
E
B
E
⎞
⎠
⎟
⎛
⎝
⎜
2
L
L
⎛
⎝
⎜
1
3
L
L
⎞ ⎛ J
⎠
⎟
⎝
⎜
J
1
3
3
1
⎞ ⎛ J
⎠
⎟
⎝
⎜
J
⎞ G
⎠
⎟ ⎛ ⎝ ⎜ G
3
1
3
1
⎞ ⎤ N
⎠
⎟ ⎥
⎦ N
⎞ G3
⎠
⎟ ⎛ ⎝ ⎜
⎞
G ⎠
⎟
2
⎡ NB
L1
J3
G3
T1
1+ ⎛ 18kip in.
⎝ ⎜ ⎞ ⎛ ⎞ ⎛ ⎞
NE
⎠
⎟
⎝
⎜
L3
⎠
⎟
⎝
⎜
J1
⎠
⎟ ⎛ ⎝ ⎜
⎞ ⎤
⎢
G1
⎠
⎟ ⎥
⎣⎢
⎦⎥ = ⋅ (h)
1
B
E
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