Helical Gears, Bevel Gears, and Wormgearing
395
E
A
z
B
z
A
y
B
y
B
x
=
W
rP
108 lb
43.5 lb
43.5
36 lb
z
y
2.22
b
a
E
W
xP
1.03
36 lb
36 lb
0
H11002
64.5
(lb)
V
x
64.5 lb
B
x
H11005
99
99 lb
b
2.22
W
tP
313 lb
a
1.03
214 lb
W
xP
H11005
36 lb
V
(lb)
0
H11002
214
0
M
(lb
·
in)
H11002
66.4
H11002
96.6
H11005
M
E
y
M
E
y
2
H11005
M
E
z
2
H11001
220
2
H11001
96.6
2
Resultant
H11005
H11005
H11005
240 lb
·
in
(
b
) Vertical plane (
xy
)
Maximum bending moment
H11002
220
H11005
M
E
z
(
a
) Horizontal plane (
xz
)
M
(lb
·
in)
0
0.84
FIGURE 10–11
Pinion shaft bending moments
Summing moments about
B
yields
0
=
W
rP
(
b
)
+
W
xP
(
r
m
)

A
y
(
L
P
)
0
=
108(2.22)
+
36(0.84)

A
y
(3.25)
A
y
=
64.5 lb
Step 3.
To find
B
x
: Summing forces in the
x
direction yields
B
x
=
W
xP
=
36 lb
This is the thrust force on bearing
B.
Step 4.
To find the total radial force on each bearing: Compute the resultant of the
y
 and
z
components.
A
=
4
A
y
2
+
A
z
2
=
2
64.5
2
+
214
2
=
224 lb
B
=
4
B
y
2
+
B
z
2
=
2
43.5
2
+
99.2
2
=
108 lb
Bearing Reactions, Gear Shaft: Bearings C and D
Using similar methods, we can find the forces in Figure 10–12.
C
z
=
142 lb
C
x
=
41.1 lb
f
C
=
148 lb (radial force on C)
D
z
=
171 lb
D
x
=
77.1 lb
f
D
=
188 lb
(radial force on
D
)
D
y
=
W
xG
=
108 lb
(thrust force on
D)
Summary
In selection of the bearings for these shafts, the following capacities are required:
Bearing
A:
224lb radial
Bearing
B:
108lb radial; 36lb thrust
Bearing
C:
148lb radial
Bearing
D:
188lb radial; 108lb thrust
396
PART TWO
Design of a Mechanical Drive
10–9
STRESSES IN STRAIGHT
BEVEL GEAR TEETH
This section generally follows AGMA Standard 2003
C10,
Rating the Pitting Resistance and Bending Strength of
Generated Straight Bevel, Zerol Bevel and Spiral Bevel Gear
Teeth
, considered to be the primary standard in the United
States (Reference 10). However, only straight bevel gears are
treated here. The standard presents design analysis in both
the U.S. unit system based on diametral pitch,
P
d
, and the SI
Metric unit system based on metric module,
m.
This book will maintain similar notations and sym
bols for gear tooth features, allowable stresses, and modi
fying factors that were initially presented in Chapter 9 for
spur gear design to facilitate the comparison of design ap
proaches. The reader should note that Standard 2003C10
presents design analysis in SI units using terminology from
ISO standards that employ radically different symbol sets. In
this book, we maintain similar notations for factors in both
systems except for the basic terms, diametral pitch,
P
d
, and
metric module,
m.
10–8
BENDING MOMENTS ON
SHAFTS CARRYING BEVEL
GEARS
Because there are forces acting in two planes on bevel gears,
as discussed in the preceding section, there is also bending in
two planes. The analysis of the shearing force and bending
moment diagrams for the shafts must take this into account.
Figures 10–11 and 10–12 show the resulting diagrams
for the pinion and the gear shafts, respectively, for the gear
pair used for Example Problems 8–3, 10–4, and 10–5. Notice
that the axial thrust load on each gear provides a concen
trated moment to the shaft equal to the axial load times the
distance that it is offset from the axis of the shaft. Also no
tice that the maximum bending moment for each shaft is the
resultant of the moments in the two planes. On the pinion