786 Mechanical Engineering Design

Standard gears may not be the most economical design that meets the functional
requirements, because no application is standard in all respects.10 Methods of design-
ing custom gears are well understood and frequently used in mobile equipment to
provide good weight-to-performance index. The required calculations including opti-
mizations are within the capability of a personal computer.

PROBLEMS
Problems marked with an asterisk (*) are linked to problems in other chapters, as summarized
in Table 1–2 of Section 1–17.
Because the results will vary depending on the method used, the problems are presented
by section.

Section 14–1
14–1 A steel spur pinion has a pitch of 6 teeth/in, 22 full-depth teeth, and a 20° pressure
angle. The pinion runs at a speed of 1200 rev/min and transmits 15 hp to a 60-tooth
gear. If the face width is 2 in, estimate the bending stress.
14–2 A steel spur pinion has a diametral pitch of 10 teeth/in, 18 teeth cut full-depth with a
20° pressure angle, and a face width of 1 in. This pinion is expected to transmit 2 hp
at a speed of 600 rev/min. Determine the bending stress.
14–3 A steel spur pinion has a module of 1.25 mm, 18 teeth cut on the 20° full-depth
system, and a face width of 12 mm. At a speed of 1800 rev/min, this pinion is expected
to carry a steady load of 0.5 kW. Determine the bending stress.
14–4 A steel spur pinion has 16 teeth cut on the 20° full-depth system with a module of
8 mm and a face width of 90 mm. The pinion rotates at 150 rev/min and transmits
6 kW to the mating steel gear. What is the bending stress?
14–5 A steel spur pinion has a module of 1 mm and 16 teeth cut on the 20° full-depth
system and is to carry 0.15 kW at 400 rev/min. Determine a suitable face width based
on an allowable bending stress of 150 MPa.
14–6 A 20° full-depth steel spur pinion has 20 teeth and a module of 2 mm and is to trans-
mit 0.5 kW at a speed of 200 rev/min. Find an appropriate face width if the bending
stress is not to exceed 75 MPa.
14–7 A 20° full-depth steel spur pinion has a diametral pitch of 5 teeth/in and 24 teeth and
transmits 6 hp at a speed of 50 rev/min. Find an appropriate face width if the allow-
able bending stress is 20 kpsi.
14–8 A steel spur pinion is to transmit 20 hp at a speed of 400 rev/min. The pinion is cut
on the 20° full-depth system and has a diametral pitch of 4 teeth/in and 16 teeth. Find
a suitable face width based on an allowable stress of 12 kpsi.
14–9 A 20° full-depth steel spur pinion with 18 teeth is to transmit 2.5 hp at a speed of
600 rev/min. Determine appropriate values for the face width and diametral pitch based
on an allowable bending stress of 10 kpsi.
14–10 A 20° full-depth steel spur pinion is to transmit 1.5 kW hp at a speed of 900 rev/min.
If the pinion has 18 teeth, determine suitable values for the module and face width.
The bending stress should not exceed 75 MPa.

10
See H. W. Van Gerpen, C. K. Reece, and J. K. Jensen, Computer Aided Design of Custom Gears,
Van Gerpen–Reece Engineering, Cedar Falls, Iowa, 1996.
 Spur and Helical Gears 787

Section 14–2
14–11 The steel pinion of Problem 14–4 is to mesh with a steel gear with a gear ratio of 4:1.
The Brinell hardness of the teeth is 200, and the tangential load transmitted by the
gears is 6 kN. If the contact fatigue strength of the steel can be estimated from the
AGMA formula of Sc = 2.22 HB + 200 MPa, estimate the factor of safety of the drive
based on a surface fatigue failure.
14–12 A speed reducer has 20° full-depth teeth and consists of a 20-tooth steel spur pinion
driving a 50-tooth cast-iron gear. The horsepower transmitted is 12 at a pinion speed
of 1200 rev/min. For a diametral pitch of 8 teeth/in and a face width of 1.5 in, find
the contact stress.
14–13 A gear drive consists of a 16-tooth 20° steel spur pinion and a 48-tooth cast-iron
gear having a pitch of 12 teeth/in. For a power input of 1.5 hp at a pinion speed of
700 rev/min, select a face width based on an allowable contact stress of 100 kpsi.
14–14 A gearset has a module of 5 mm, a 20° pressure angle, and a 24-tooth cast-iron spur
pinion driving a 48-tooth cast-iron gear. The pinion is to rotate at 50 rev/min. What
horsepower input can be used with this gearset if the contact stress is limited to 690 MPa
and F = 60 mm?
14–15 A 20° 20-tooth cast-iron spur pinion having a module of 4 mm drives a 32-tooth cast-
iron gear. Find the contact stress if the pinion speed is 1000 rev/min, the face width
is 50 mm, and 10 kW of power is transmitted.
14–16 A steel spur pinion and gear have a diametral pitch of 12 teeth/in, milled teeth, 17 and
30 teeth, respectively, a 20° pressure angle, a face width of 78 in, and a pinion speed
of 525 rev/min. The tooth properties are Sut = 76 kpsi, Sy = 42 kpsi and the Brinell
hardness is 149. Use the Gerber criteria to compensate for one-way bending. For a
design factor of 2.25, what is the power rating of the gearset?
14–17 A milled-teeth steel pinion and gear pair have Sut = 113 kpsi, Sy = 86 kpsi and a hard-
ness at the involute surface of 262 Brinell. The diametral pitch is 3 teeth/in, the face
width is 2.5 in, and the pinion speed is 870 rev/min. The tooth counts are 20 and 100.
Use the Gerber criteria to compensate for one-way bending. For a design factor of 1.5,
rate the gearset for power considering both bending and wear.
14–18 A 20° full-depth steel spur pinion rotates at 1145 rev/min. It has a module of 6 mm,
a face width of 75 mm, and 16 milled teeth. The ultimate tensile strength at the invo-
lute is 900 MPa exhibiting a Brinell hardness of 260. The gear is steel with 30 teeth
and has identical material strengths. Use the Gerber criteria to compensate for one-way
bending. For a design factor of 1.3 find the power rating of the gearset based on the
pinion and the gear resisting bending and wear fatigue.
14–19 A steel spur pinion has a pitch of 6 teeth/in, 17 full-depth milled teeth, and a pres-
sure angle of 20°. The pinion has an ultimate tensile strength at the involute surface
of 116 kpsi, a Brinell hardness of 232, and a yield strength of 90 kpsi. Its shaft
speed is 1120 rev/min, its face width is 2 in, and its mating gear has 51 teeth. Use
a design factor of 2.
(a) Pinion bending fatigue imposes what power limitation? Use the Gerber criteria to
compensate for one-way bending.
(b) Pinion surface fatigue imposes what power limitation? The gear has identical
strengths to the pinion with regard to material properties.
(c) Determine power limitations due to gear bending and wear.
(d) Specify the power rating for the gearset.
788 Mechanical Engineering Design

Section 14–3 to 14–19

14–20 A commercial enclosed gear drive is to be designed. Initially, the following is specified.
The drive is to transmit 5 hp with an input pinion speed of 300 rev/min. The drive is to
consist of two spur gears, with a 20° pressure angle, and the pinion is to have 16 teeth
driving a 48-tooth gear. The gears are to be grade 1 steel, through-hardened at 200 Brinell,
made to No. 6 quality standards. The gears will be uncrowned, centered on their shafts
between bearings. A pinion life of 108 cycles is desired, with a 90 percent reliability.
(a) Use a trial value of diametral pitch of 6 teeth/in and a face width of 2 in. For the pinion,
(i) determine the bending stress, allowable bending stress, and bending factor
of safety.
(ii) determine the contact stress, allowable contact stress, and contact factor
of safety.
(b) Assume gears are readily available in face-width increments of 0.5 in, with face
widths in the range of three to five times the circular pitch. Specify a diametral pitch
and face width such that the minimum factor of safety for the pinion is equal to 2.
14–21 A 20° spur pinion with 20 teeth and a module of 2.5 mm transmits 120 W to a 36-tooth
gear. The pinion speed is 100 rev/min, and the gears are grade 1, 18-mm face width,
through-hardened steel at 200 Brinell, uncrowned, manufactured to a No. 6 quality
standard, and considered to be of open gearing quality installation. Find the AGMA
bending and contact stresses and the corresponding factors of safety for a pinion life
of 108 cycles and a reliability of 0.95.
14–22 Repeat Problem 14–20, part (a), using helical gears each with a 20° normal pitch angle
and a helix angle of 30° and a normal diametral pitch of 6 teeth/in.
14–23 A spur gearset has 17 teeth on the pinion and 51 teeth on the gear. The pressure angle
is 20° and the overload factor Ko = 1. The diametral pitch is 6 teeth/in and the face
width is 2 in. The pinion speed is 1120 rev/min and its cycle life is to be 108 revolu-
tions at a reliability R = 0.99. The quality number is 5. The material is a through-
hardened steel, grade 1, with Brinell hardnesses of 232 core and case of both gears.
For a design factor of 2, rate the gearset for these conditions using the AGMA method.
14–24 In Section 14–10, Equation (a) is given for Ks based on the procedure in Example 14–2.
Derive this equation.
14–25 A speed-reducer has 20° full-depth teeth, and the single-reduction spur-gear gearset
has 22 and 60 teeth. The diametral pitch is 4 teeth/in and the face width is 314 in. The
pinion shaft speed is 1145 rev/min. The life goal of 5-year 24-hour-per-day service is
about 3(109) pinion revolutions. The absolute value of the pitch variation is such that
the quality number is 6. The materials are 4340 through-hardened grade 1 steels, heat-
treated to 250 Brinell, core and case, both gears. The load is moderate shock and the
power is smooth. For a reliability of 0.99, rate the speed reducer for power.
14–26 The speed reducer of Problem 14–25 is to be used for an application requiring 40 hp
at 1145 rev/min. For the gear and the pinion, estimate the AGMA factors of safety for
bending and wear, that is, (SF)P, (SF)G, (SH)P, and (SH)G. By examining the factors of
safety, identify the threat to each gear and to the mesh.
14–27 The gearset of Problem 14–25 needs improvement of wear capacity. Toward this end
the gears are nitrided so that the grade 1 materials have hardnesses as follows: The
pinion core is 250 and the pinion case hardness is 390 Brinell, and the gear core
hardness is 250 core and 390 case. Estimate the power rating for the new gearset.
 Spur and Helical Gears 789

14–28 The gearset of Problem 14–25 has had its gear specification changed to 9310 for
carburizing and surface hardening with the result that the pinion Brinell hardnesses
are 285 core and 580–600 case, and the gear hardnesses are 285 core and 580–600
case. Estimate the power rating for the new gearset.

14–29 The gearset of Problem 14–28 is going to be upgraded in material to a quality of

grade 2 (9310) steel. Estimate the power rating for the new gearset.

14–30 Matters of scale always improve insight and perspective. Reduce the physical size of
the gearset in Problem 14–25 by one-half and note the result on the estimates of
transmitted load W t and power.

14–31 AGMA procedures with cast-iron gear pairs differ from those with steels because life
predictions are difficult; consequently (YN)P, (YN)G, (ZN)P, and (ZN)G are set to unity. The
consequence of this is that the fatigue strengths of the pinion and gear materials are the
same. The reliability is 0.99 and the life is 107 revolution of the pinion (KR = 1).
For longer lives the reducer is derated in power. For the pinion and gearset of
Problem 14–25, use grade 40 cast iron for both gears (HB = 201 Brinell). Rate the
reducer for power with SF and SH equal to unity.

14–32 Spur-gear teeth have rolling and slipping contact (often about 8 percent slip). Spur
gears tested to wear failure are reported at 108 cycles as Buckingham’s surface fatigue
load-stress factor K. This factor is related to Hertzian contact strength SC by

1.4K
SC = √
(1∕E1 + 1∕E2 ) sin ϕ

where ϕ is the normal pressure angle. Cast iron grade 20 gears with ϕ = 14 12° and
20° pressure angle exhibit a minimum K of 81 and 112 psi, respectively. How does
this compare with SC = 0.32HB kpsi?

14–33 You’ve probably noticed that although the AGMA method is based on two equa-
tions, the details of assembling all the factors is computationally intensive. To
reduce error and omissions, a computer program would be useful. Write a program
to perform a power rating of an existing gearset, then use Problem 14–25, 14–27,
14–28, 14–29, and 14–30 to test your program by comparing the results to your
longhand solutions.

14–34 In Example 14–5 use nitrided grade 1 steel (4140) which produces Brinell hard-
nesses of 250 core and 500 at the surface (case). Use the upper fatigue curves on
Figures 14–14 and 14–15. Estimate the power capacity of the mesh with factors of
safety of SF = SH = 1.

14–35 In Example 14–5 use carburized and case-hardened gears of grade 1. Carburizing and
case-hardening can produce a 550 Brinell case. The core hardnesses are 200 Brinell.
Estimate the power capacity of the mesh with factors of safety of SF = SH = 1, using
the lower fatigue curves in Figures 14–14 and 14–15.

14–36 In Example 14–5, use carburized and case-hardened gears of grade 2 steel. The core
hardnesses are 200, and surface hardnesses are 600 Brinell. Use the lower fatigue
curves of Figures 14–14 and 14–15. Estimate the power capacity of the mesh using
SF = SH = 1. Compare the power capacity with the results of Problem 14–35.

14–37* The countershaft in Problem 3–83 is part of a speed reducing compound gear train
using 20° spur gears. A gear on the input shaft drives gear A. Gear B drives a gear
790 Mechanical Engineering Design

on the output shaft. The input shaft runs at 2400 rev/min. Each gear reduces the speed
(and thus increases the torque) by a 2 to 1 ratio. All gears are to be of the same mate-
rial. Since gear B is the smallest gear, transmitting the largest load, it will likely be
critical, so a preliminary analysis is to be performed on it. Use a diametral pitch of
2 teeth/in, a face width of 4 times the circular pitch, a Grade 2 steel through-hardened
to a Brinell hardness of 300, and a desired life of 15 kh with a 95 percent reliability.
Determine factors of safety for bending and wear.

14–38* The countershaft in Problem 3–84 is part of a speed reducing compound gear train
using 20° spur gears. A gear on the input shaft drives gear A with a 2 to 1 speed
reduction. Gear B drives a gear on the output shaft with a 5 to 1 speed reduction. The
input shaft runs at 1800 rev/min. All gears are to be of the same material. Since gear
B is the smallest gear, transmitting the largest load, it will likely be critical, so a
preliminary analysis is to be performed on it. Use a module of 18.75 mm/tooth, a face
width of 4 times the circular pitch, a Grade 2 steel through-hardened to a Brinell
hardness of 300, and a desired life of 12 kh with a 98 percent reliability. Determine
factors of safety for bending and wear.

14–39* Build on the results of Problem 13–46 to find factors of safety for bending and wear
for gear F. Both gears are made from Grade 2 carburized and hardened steel. Use a
face width of 4 times the circular pitch. The desired life is 12 kh with a 95 percent
reliability.

14–40* Build on the results of Problem 13–47 to find factors of safety for bending and wear
for gear C. Both gears are made from Grade 2 carburized and hardened steel. Use a
face width of 4 times the circular pitch. The desired life is 14 kh with a 98 percent
reliability.