A 17-tooth 20° pressure angle spur pinion rotates at 1800 rev/min and transmits 4 hp to a 52-tooth disk gear. The diametral pitch is 10 teeth/in, the face width 1.5 in, and the quality standard is No. 6. The gears are straddle-mounted with bearings immediately adjacent. The pinion is a grade 1

Question

A 17-tooth 20° pressure angle spur pinion rotates at 1800 rev/min and transmits 4 hp to a 52-tooth disk gear. The diametral pitch is 10 teeth/in, the face width 1.5 in, and the quality standard is No. 6. The gears are straddle-mounted with bearings immediately adjacent. The pinion is a grade 1 steel with a hardness of 240 Brinell tooth surface and through-hardened core. The gear is steel, through-hardened also, grade 1 material, with a Brinell hardness of 200, tooth surface and core. Poisson’s ratio is 0.30, $J_{P}=0.30$ , $J_{G}=0.40$ , and Young’s modulus is $30\left(10^{6}\right)$ psi. The loading is smooth because of motor and load. Assume a pinion life of $10^{8}$ cycles and a reliability of 0.90, and use $Y_{N}=1.3558 N^{-0.0178}, Z_{N}=1.4488 N^{-0.023}$ . The tooth profile is uncrowned. This is a commercial enclosed gear unit.

(a) Find the factor of safety of the gears in bending.

(b) Find the factor of safety of the gears in wear.

(c) By examining the factors of safety, identify the threat to each gear and to the mesh.

(a) Find the factor of safety of the gears in bending.

(b) Find the factor of safety of the gears in wear.

(c) By examining the factors of safety, identify the threat to each gear and to the mesh.

Accepted Answer

There will be many terms to obtain so use Figs. 14–17 and 14–18 as guides to what is needed.

[latex]d_{P}=N_{P} / P_{d}=17 / 10=1.7 ext { in } \quad d_{G}=52 / 10=5.2 ext { in }[/latex]

[latex]V=\frac{\pi d_{P} n_{P}}{12}=\frac{\pi(1.7) 1800}{12}=801.1 ft / min[/latex]

[latex]W^{t}=\frac{33000 H}{V}=\frac{33000(4)}{801.1}=164.8 lbf[/latex]

Assuming uniform loading, [latex]K_{o}=1[/latex]. To evaluate [latex]K_{v}[/latex], from Eq. (14–28) with a quality number [latex]O_{n}=6[/latex],

[latex]\begin{array}{l}A=50+56(1-B) \B=0.25\left(12-Q_{v} ight)^{2 / 3}\end{array}[/latex] (14–28)

[latex]\begin{array}{l}B=0.25(12-6)^{2 / 3}=0.8255 \A=50+56(1-0.8255)=59.77\end{array}[/latex]

Then from Eq. (14–27) the dynamic factor is

[latex]K_{v}=\left\{\begin{array}{ll}\left(\frac{A+\sqrt{V}}{A} ight)^{B} & V ext { in } ft / min \\left(\frac{A+\sqrt{200 V}}{A} ight)^{B} & V ext { in } m / s\end{array} ight.[/latex] (14–27)

[latex]K_{v}=\left(\frac{59.77+\sqrt{801.1}}{59.77} ight)^{0.8255}=1.377[/latex]

To determine the size factor, Ks , the Lewis form factor is needed. From Table 14–2, with [latex]N_{P}=17 ext { teeth }[/latex] teeth, [latex]Y_{P}=0.303[/latex]. Interpolation for the gear with [latex]N_{G}=52[/latex] teeth yields [latex]Y_{G}=0.412[/latex]. Thus from Eq. (a) of Sec. 14–10, with F = 1.5 in,

Table 14–2 Values of the Lewis Form Factor Y (TheseValues Are for a Normal Pressure Angle of 20°, Full-Depth Teeth, and a Diametral Pitch of Unity in the Plane of Rotation)
Number of Teeth	Y	Number of Teeth	Y
12	0.245	28	0.353
13	0.261	30	0.359
14	0.277	34	0.371
15	0.29	38	0.384
16	0.296	43	0.397
17	0.303	50	0.409
18	0.309	60	0.422
19	0.314	75	0.435
20	0.322	100	0.447
21	0.328	150	0.46
22	0.331	300	0.472
24	0.337	400	0.48
26	0.346	Rack	0.485

[latex]\sigma=\frac{M}{I / c}=\frac{6 W^{t} l}{F t^{2}}[/latex] (a)

[latex]\left(K_{s} ight)_{P}=1.192\left(\frac{1.5 \sqrt{0.303}}{10} ight)^{0.0535}=1.043[/latex]

[latex]\left(K_{s} ight)_{G}=1.192\left(\frac{1.5 \sqrt{0.412}}{10} ight)^{0.0535}=1.052[/latex]

The load distribution factor [latex]K_{m}[/latex] is determined from Eq. (14–30), where five terms are needed. They are, where F = 1.5 in when needed:

[latex]K_{m}=C_{m f}=1+C_{m c}\left(C_{p f} C_{p m}+C_{m a} C_{e} ight)[/latex] (14–30)

Uncrowned, Eq. (14–30): [latex]C_{m c}=1[/latex],

[latex]K_{m}=C_{m f}=1+C_{m c}\left(C_{p f} C_{p m}+C_{m a} C_{e} ight)[/latex] (14–30)

Eq. (14–32): [latex]C_{p f}=1.5 /[10(1.7)]-0.0375+0.0125(1.5)=0.0695[/latex]

[latex]C_{p f}=\left\{\begin{array}{ll}\frac{F}{10 d}-0.025 & F \leq 1 ext { in } \\frac{F}{10 d}-0.0375+0.0125 F & 1

Bearings immediately adjacent, Eq. (14–33): [latex]C_{p m}=1[/latex]

[latex]C_{p m}=\left\{\begin{array}{ll}1 & ext { for straddle-mounted pinion with } S_{1} / S<0.175 \1.1 & ext { for straddle-mounted pinion with } S_{1} / S \geq 0.175\end{array} ight.[/latex] (14–33)

Commercial enclosed gear units (Fig. 14–11): [latex]C_{m a}=0.15[/latex]

Eq. (14–35): [latex]C_{e}=1[/latex]

[latex]C_{e}=\left\{\begin{array}{cl}0.8 & ext { for gearing adjusted at assembly, or compatibility } \1 & ext { is improved by lapping, or both }\end{array} ight.[/latex] (14–35)

Thus,

[latex]K_{m}=1+C_{m c}\left(C_{p f} C_{p m}+C_{m a} C_{e} ight)=1+(1)[0.0695(1)+0.15(1)]=1.22[/latex]

Assuming constant thickness gears, the rim-thickness factor [latex]K_{B}=1[/latex]. The speed ratio is [latex]m_{G}=N_{G} / N_{P}=52 / 17=3.059[/latex]. The load cycle factors given in the problem statement, with [latex]N ext { (pinion) }=10^{8}[/latex] cycles and [latex]N( ext { gear })=10^{8} / m_{G}=10^{8} / 3.059[/latex] cycles, are

[latex]\left(Y_{N} ight)_{P}=1.3558\left(10^{8} ight)^{-0.0178}=0.977[/latex]

[latex]\left(Y_{N} ight)_{G}=1.3558\left(10^{8} / 3.059 ight)^{-0.0178}=0.996[/latex]

From Table 14.10, with a reliability of 0.9, [latex]K_{R}=0.85[/latex]. From Fig. 14–18, the temperature and surface condition factors are [latex]K_{T}=1 ext { and } C_{f}=1[/latex]. From Eq. (14–23), with [latex]m_{N}=1[/latex] for spur gears,

Table 14–10 Reliability Factors [latex]K_{R}\left(Y_{Z} ight)[/latex] Source: ANSI/AGMA 2001-D04.
Reliability	[latex]K_{R}\left(Y_{Z} ight)[/latex]
0.9999	1.50
0.999	1.25
0.99	1.00
0.90	0.85
0.50	0.70

[latex]I=\left\{\begin{array}{ll}\frac{\cos \phi_{t} \sin \phi_{t}}{2 m_{N}} \frac{m_{G}}{m_{G}+1} & ext { external gears } \\frac{\cos \phi_{t} \sin \phi_{t}}{2 m_{N}} \frac{m_{G}}{m_{G}-1} & ext { internal gears }\end{array} ight.[/latex] (14–23)

[latex]I=\frac{\cos 20^{\circ} \sin 20^{\circ}}{2} \frac{3.059}{3.059+1}=0.121[/latex]

From Table 14–8, [latex]C_{p}=2300 \sqrt{ ps }[/latex].

Table 14–8 Elastic Coefficient [latex]C_{p}\left(Z_{E} ight), \sqrt{ psi }(\sqrt{ MPa })[/latex] Source: AGMA 218.01
Gear Material and Modulus of Elasticity [latex]E_{G}, lbf / in ^{2}( MPa )^{*}[/latex]
Pinion Material	Pinion Modulus of [latex]\begin{array}{l} ext { Elasticity Ep }\ ext { psi (MPa) * }\end{array}[/latex]	Steel [latex]\begin{array}{l}30 imes 10^{6} \\left(2 imes 10^{5} ight)\end{array}[/latex]	Malleable Iron [latex]\begin{array}{c}25 imes 10^{6} \\left(1.7 imes 10^{5} ight)\end{array}[/latex]	Nodular Iron [latex]\begin{array}{c}24 imes 10^{6} \\left(1.7 imes 10^{5} ight)\end{array}[/latex]	Cast Iron [latex]\begin{array}{r}22 imes 10^{6} \\left(1.5 imes 10^{5} ight)\end{array}[/latex]	Aluminum Bronze [latex]\begin{array}{r}17.5 imes 10^{6} \\left(1.2 imes 10^{5} ight)\end{array}[/latex]	Tin Bronze [latex]\begin{array}{c}16 imes 10^{6} \\left(1.1 imes 10^{5} ight)\end{array}[/latex]
Steel	[latex]30 imes 10^{6}[/latex]	2300	2180	2160	2100	1950	1900
	[latex]\left(2 imes 10^{5} ight)[/latex]	(191)	(181)	(179)	(174)	(162)	(158)
Malleable iron	[latex]25 imes 10^{6}[/latex]	2180	2090	2070	2020	1900	1850
	[latex]\left(1.7 imes 10^{5} ight)[/latex]	(181)	(174)	(172)	(168)	(158)	(154)
Nodular iron	[latex]24 imes 10^{6}[/latex]	2160	2070	2050	2000	1880	1830
	[latex]\left(1.7 imes 10^{5} ight)[/latex]	(179)	(172)	(170)	(166)	(156)	(152)
Cast iron	[latex]22 imes 10^{6}[/latex]	2100	2020	2000	1960	1850	1800
	[latex]\left(1.5 imes 10^{5} ight)[/latex]	(174)	(168)	(166)	(163)	(154)	(149)
Aluminum bronze	[latex]17.5 imes 10^{6}[/latex]	1950	1900	1880	1850	1750	1700
	[latex]\left(1.2 imes 10^{5} ight)[/latex]	(162)	(158)	(156)	(154)	(145)	(141)
Tin bronze	[latex]16 imes 10^{6}[/latex]	1900	1850	1830	1800	1700	1650
	[latex]\left(1.1 imes 10^{5} ight)[/latex]	(158)	(154)	(152)	(149)	(141)	(137)

Next, we need the terms for the gear endurance strength equations. From Table 14–3, for grade 1 steel with [latex]H_{B P}=240 ext { and } H_{B G}=200[/latex], we use Fig. 14–2, which gives

Table 14–3 Repeatedly Applied Bending Strength [latex]S_{t} ext { at } 10^{7}[/latex] Cycles and 0.99 Reliability for Steel Gears Source: ANSI/AGMA 2001-D04.
Material Designation	Heat Treatment	Minimum Surface [latex] ext { Hardness } 1[/latex]	Allowable Bending Stress Number [latex]S _{ t ,}^{2}[/latex] psi
Material Designation	Heat Treatment	Minimum Surface [latex] ext { Hardness } 1[/latex]	Grade 1v	Grade 2	Grade 3
[latex] ext { Steel }^{3}[/latex]	Through-hardened	See Fig. 14–2	See Fig. 14–2	See Fig. 14–2	—
	[latex] ext { Flame }^{4}[/latex] or induction	See Table 8*	45 000	55 000	—
	[latex] ext { hardened }{ }^{4}[/latex] with type
	[latex] ext { A pattern }{ }^{5}[/latex]
	[latex] ext { Flame }^{4}[/latex] or induction	See Table 8*	22 000	22 000	—
	[latex] ext { hardened }{ }^{4}[/latex] with type
	[latex] ext { B pattern }{ }^{5}[/latex]
	Carburized and hardened	See Table 9*	55 000	[latex]\begin{array}{l}65000 ext { or } \70000^{6}\end{array}[/latex]	75 000
	[latex] ext { Nitrided }^{4,7}[/latex] (through- hardened steels)	83.5 HR15N	See Fig. 14–3	See Fig. 14–3	—
Nitralloy 135M, Nitralloy N, and 2.5% chrome (no aluminum)	[latex] ext { Nitrided }^{4,7}[/latex]	87.5 HR15N	See Fig. 14–4	See Fig. 14–4	See Fig. 14–4

Notes: See ANSI/AGMA 2001-D04 for references cited in notes 1–7.

[latex]{ }^{1} ext { Hardness }[/latex] to be equivalent to that at the root diameter in the center of the tooth space and face width.

[latex]{ }^{2} ext { See }[/latex] tables 7 through 10 for major metallurgical factors for each stress grade of steel gears.

[latex]{ }^{3} ext { The }[/latex] steel selected must be compatible with the heat treatment process selected and hardness required.

[latex]{ }^{3} ext { The }[/latex] allowable stress numbers indicated may be used with the case depths prescribed in 16.1.

[latex]{ }^{5} See[/latex] figure 12 for type A and type B hardness patterns.

[latex]{ }^{6} ext { If }[/latex] bainite and microcracks are limited to grade 3 levels, 70 000 psi may be used.

[latex]{ }^{3} ext { The }[/latex] overload capacity of nitrided gears is low. Since the shape of the effective S-N curve is flat, the sensitivity to shock should be investigated before proceeding with the design. [7]

*Tables 8 and 9 of ANSI/AGMA 2001-D04 are comprehensive tabulations of the major metallurgical factors affecting [latex]S_{t} ext { and } S_{c}[/latex] of flame-hardened and induction-hardened (Table 8) and carburized and hardened (Table 9) steel gears.4

Similarly, from Table 14–6, we use Fig. 14–5, which gives

Table 14–6 Repeatedly Applied Contact Strength [latex]S_{c} ext { at } 10^{7}[/latex] Cycles and 0.99 Reliability for Steel Gears Source: ANSI/AGMA 2001-D04.
Material Designation	Heat Treatment	Minimum Surface [latex] ext { Hardness } 1[/latex]	Allowable Contact Stress [latex] ext { Number, }^{2}{ }^{\prime} S_{c}, ext { psi }[/latex]
Material Designation	Heat Treatment	Minimum Surface [latex] ext { Hardness } 1[/latex]	Grade 1	Grade 2	Grade 3
[latex] ext { Steel }^{3}[/latex]	Through [latex] ext { hardened }{ }^{4}[/latex]	See Fig. 14–5	See Fig. 14–5	See Fig. 14–5	—
	[latex] ext { Flame }^{5}[/latex] or induction	50 HRC	170 000	190 000	—
	[latex] ext { hardened }^{5}[/latex]	54 HRC	175 000	195 000	—
	Carburized and [latex] ext { hardened }^{5}[/latex]	[latex] ext { See Table } 9^{*}[/latex]	180 000	225 000	275 000
	[latex] ext { Nitrided }^{5}[/latex] (through	83.5 HR15N	150 000	163 000	175 000
	hardened steels)	84.5 HR15N	155 000	168 000	180 000
2.5% chrome (no aluminum)	[latex] ext { Nitrided }^{5}[/latex]	87.5 HR15N	155 000	172 000	189 000
Nitralloy 135M	[latex] ext { Nitrided }^{5}[/latex]	90.0 HR15N	170 000	183 000	195 000
Nitralloy N	[latex] ext { Nitrided }^{5}[/latex]	90.0 HR15N	172 000	188 000	205 000
2.5% chrome (no aluminum)	[latex] ext { Nitrided }^{5}[/latex]	90.0 HR15N	176 000	196 000	216 000

Notes: See ANSI/AGMA 2001-D04 for references cited in notes 1–5.

[latex]{ }^{1} ext { Hardness }[/latex] to be equivalent to that at the start of active profile in the center of the face width.

[latex]{ }^{2} ext { See }[/latex] Tables 7 through 10 for major metallurgical factors for each stress grade of steel gears.

[latex]{ }^{3} ext { The }[/latex] steel selected must be compatible with the heat treatment process selected and hardness required.

[latex]{ }^{4} ext { These }[/latex] materials must be annealed or normalized as a minimum.

[latex]{ }^{5} ext { The }[/latex] allowable stress numbers indicated may be used with the case depths prescribed in 16.1.

*Table 9 of ANSI/AGMA 2001-D04 is a comprehensive tabulation of the major metallurgical factors affecting [latex]S_{t} ext { and } S_{c}[/latex] of carburized and hardened steel gears.

[latex]\left(S_{c} ight)_{P}=322(240)+29100=106400 psi[/latex]

[latex]\left(S_{c} ight)_{G}=322(200)+29100=93500 psi[/latex]

From Fig. 14–15,

[latex]\left(Z_{N} ight)_{P}=1.4488\left(10^{8} ight)^{-0.023}=0.948[/latex]

[latex]\left(Z_{N} ight)_{G}=1.4488\left(10^{8} / 3.059 ight)^{-0.023}=0.973[/latex]

For the hardness ratio factor [latex]C_{H}[/latex], the hardness ratio is [latex]H_{B P} / H_{B G}=240 / 200=1.2[/latex]. Then, from Sec. 14–12,

[latex]\begin{aligned}A^{\prime} &=8.98\left(10^{-3} ight)\left(H_{B P} / H_{B G} ight)-8.29\left(10^{-3} ight) \&=8.98\left(10^{-3} ight)(1.2)-8.29\left(10^{-3} ight)=0.00249\end{aligned}[/latex]

Thus, from Eq. (14–36),

[latex]C_{H}=1.0+A^{\prime}\left(m_{G}-1.0 ight)[/latex] (14–36)

[latex]C_{H}=1+0.00249(3.059-1)=1.005[/latex]

(a) Pinion tooth bending. Substituting the appropriate terms for the pinion into Eq. (14–15) gives

[latex]\sigma=\left\{\begin{array}{ll}W^{t} K_{o} K_{v} K_{s} \frac{P_{d}}{F} \frac{K_{m} K_{B}}{J} & ext { (U.S. customary units) } \W^{t} K_{o} K_{v} K_{s} \frac{1}{b m_{t}} \frac{K_{H} K_{B}}{Y_{J}} & ext { (SI units) }\end{array} ight.[/latex] (14–15)

[latex]\begin{aligned}(\sigma)_{P} &=\left(W^{t} K_{o} K_{v} K_{s} \frac{P_{d}}{F} \frac{K_{m} K_{B}}{J} ight)_{P}=164.8(1) 1.377(1.043) \frac{10}{1.5} \frac{1.22(1)}{0.30} \&=6417 psi\end{aligned}[/latex]

Substituting the appropriate terms for the pinion into Eq. (14–41) gives

[latex]S_{F}=\frac{S_{t} Y_{N} /\left(K_{T} K_{R} ight)}{\sigma}=\frac{ ext { fully corrected bending strength }}{ ext { bending stress }}[/latex] (14–41)

[latex]\left(S_{F} ight)_{P}=\left(\frac{S_{t} Y_{N} /\left(K_{T} K_{R} ight)}{\sigma} ight)_{P}=\frac{31350(0.977) /[1(0.85)]}{6417}=5.62[/latex]

Gear tooth bending. Substituting the appropriate terms for the gear into Eq. (14–15) gives

[latex](\sigma)_{G}=164.8(1) 1.377(1.052) \frac{10}{1.5} \frac{1.22(1)}{0.40}=4854 psi[/latex]

Substituting the appropriate terms for the gear into Eq. (14–41) gives

[latex]\left(S_{F} ight)_{G}=\frac{28260(0.996) /[1(0.85)]}{4854}=6.82[/latex]

(b) Pinion tooth wear. Substituting the appropriate terms for the pinion into Eq. (14–16) gives

[latex]\sigma_{c}=\left\{\begin{array}{ll}C_{p} \sqrt{W^{t} K_{o} K_{v} K_{s} \frac{K_{m}}{d_{P} F} \frac{C_{f}}{I}} & ext { (U.S. customary units) } \Z_{E} \sqrt{W^{t} K_{o} K_{v} K_{s} \frac{K_{H}}{d_{w 1} b} \frac{Z_{R}}{Z_{I}}} & ext { (SI units) }\end{array} ight.[/latex] (14–16)

[latex]\begin{aligned}\left(\sigma_{c} ight)_{p} &=C_{p}\left(W^{t} K_{o} K_{v} K_{s} \frac{K_{m}}{d_{p} F} \frac{C_{f}}{I} ight)_{p}^{1 / 2} \&=2300\left[164.8(1) 1.377(1.043) \frac{1.22}{1.7(1.5)} \frac{1}{0.121} ight]^{1 / 2}=70360 psi\end{aligned}[/latex]

Substituting the appropriate terms for the pinion into Eq. (14–42) gives

[latex]S_{H}=\frac{S_{c} Z_{N} C_{H} /\left(K_{T} K_{R} ight)}{\sigma_{c}}=\frac{ ext { fully corrected contact strength }}{ ext { contact stress }}[/latex] (14–42)

[latex]\left(S_{H} ight)_{P}=\left[\frac{S_{c} Z_{N} /\left(K_{T} K_{R} ight)}{\sigma_{c}} ight]_{P}=\frac{106400(0.948) /[1(0.85)]}{70360}=1.69[/latex]

Gear tooth wear. The only term in Eq. (14–16) that changes for the gear is [latex]K_{s}[/latex]. Thus,

[latex]\left(\sigma_{c} ight)_{G}=\left[\frac{\left(K_{s} ight)_{G}}{\left(K_{s} ight)_{P}} ight]^{1 / 2}\left(\sigma_{c} ight)_{P}=\left(\frac{1.052}{1.043} ight)^{1 / 2} 70360=70660 psi[/latex]

Substituting the appropriate terms for the gear into Eq. (14–42) with [latex]C_{H}=1.005[/latex] gives

[latex]\left(S_{H} ight)_{G}=\frac{93500(0.973) 1.005 /[1(0.85)]}{70660}=1.52[/latex]

(c) For the pinion, we compare [latex]\left(S_{F} ight) P ext { with }\left(S_{H} ight)_{P}^{2}[/latex] , or 5.73 with [latex]1.69^{2}=2.86[/latex], so the threat in the pinion is from wear. For the gear, we compare [latex]\left(S_{F} ight)_{G} ext { with }\left(S_{H} ight)_{G}^{2}[/latex], or 6.96 with [latex]1.52^{2}=2.31[/latex], so the threat in the gear is also from wear.

Question 14.4: A 17-tooth 20° pressure angle spur pinion rotates at 1800 re...

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