Specify suitable materials for the pinion and the gear from Example Problem 9-1. Design for a reliability of fewer than one failure in 10 000. The application is an industrial saw that will be fully utilized on a normal, one-shift, five-day-per-week operation.

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The results of Example Problem 9-1 include the expected bending stress number for both the pinion and the gear as follows:

[latex]s_{tP}= 29 700[/latex] psi [latex] s_{tG} = 23 700[/latex] psi

We should consider the number of stress cycles, the reliability, and the safety factor to complete the calculation indicated in Equation (9-20).

[latex]\frac{K_R(SF)}{Y_N} s_t\lt s_{at}[/latex] (9-20)

Stress Cycle Factor, [latex] Y_N[/latex] : From the problem statement

[latex]n_p = 1750[/latex] rpm, [latex]N_p = 20[/latex] teeth, and [latex]N_G = 70[/latex] teeth. Let's use these data to determine the expected number of cycles of stress that the pinion and gear teeth will experience. The application conforms to common industry practice, calling for a design life of approximately 20 000 h as suggested in Table 9-7. The number of stress cycles for the pinion is

TABLE 9-7 Recommended design life
Application	Design life (h)
Domestic appliances	1000-2000
Aircraft engines	1000-4000
Automotive	1500-5000
Agricultural equipment	3000-6000
Elevators, industrial fans, multipurpose gearing	8000-15 000
Electric motors, industrial blowers, general industrial machines	20 000-30 000
Pumps and compressors	40 000-60 000
Critical equipment in continuous 24-h operation	100 000-200 000
Source: Eugene A. Avallone and Theodore Baumeister 111, eds. Marks' Standard Handbook for Mechanical Engineers. 9th ed. New York: McGraw-Hill, 1986.

[latex]N_{cP} = (60)(L)(n_p)(q) = (60)(20 000)(1750)(1) = 2.10×10^9[/latex] cycles

The gear rotates more slowly because of the speed reduction. Then

[latex]n_G = n_P(N_P/N_G) = (1750 \ rpm)(20/70) = 500[/latex] rpm

We can now compute the number of stress cycles for each gear tooth:

[latex]N_{cG} = (60)(L)(n_G)(q) - (60)(20 000)(500)( 1) = 6.00 ×10^8[/latex] cycles

Because both values are above the nominal value of [latex]10^7 [/latex] cycles, a value of [latex]Y_N[/latex] must be determined from Figure 9-22 for both the pinion and the gear:

[latex]Y_{Np} = 0.92[/latex] [latex]Y_{NG} = 0.96[/latex]

Reliability Factor, [latex]K_R[/latex]; For the design goal of fewer than one failure in 10 000, Table 9-8 recommends [latex]K_R = 1.50[/latex]

TABLE 9-8 Reliability factor, [latex]K_p[/latex]
Reliability	[latex]K_R[/latex]
0.90, one failure in 10	0.85
0.99, one failure in 100	1
0.999. one failure in 1000	1.25
0.9999, one failure in 10 000	1.5

Factor of Safety; This is a design decision. Reviewing the discussion of factors throughout Example Problem 9-1 and this problem, we see that virtually all factors typically considered in adjusting stress on the teeth and strength of the material have been taken into account.
Furthermore, when we select a material, it will likely have strength and hardness values somewhat above the minimum acceptable values. Therefore, as a design decision, let's use SF= 1.00.

Adjusted Value of [latex]s_{at}[/latex]; We can now complete Equation (9-20) and use it for material selection. For the pinion.

[latex]\frac{K_R(SF)}{Y_{NP}} s_t=\frac{(1.50)(1.00)}{0.92}(29 700 \ psi) = 48 450 \ psi < s_{at} [/latex]

For the gear.

[latex]\frac{K_R(SF)}{Y_{NG}} s_t=\frac{(1.50)(1.00)}{0.96}(23 700 \ psi) = 37 050 \ psi < s_{at} [/latex]

Now. referring to Figure 9-10, and deciding to use Grade 1 steel, we find that the required allowable bending stress number for the pinion is greater than permitted for a through hardened steel. But Table 9-3

TABLE 9-3 Allowable stress numbers for case-hardened steel gear materials
	Allowable bending stress number, [latex]s_{ac}[/latex](ksi)			Allowable contact stress number, [latex]s_{ac}[/latex] (ksi)
Hardness at surface	Grade 1	Grade 2	Grade 3	Grade 1	Grade 2	Grade 3
Flame- or induction-hardened:
50 HRC	45	55		170	190
54 HRC	45	55		175	195
Carburized and case-hardened
55-64 HRC	55			180
58-64 HRC	55	65	75	180	225	275
Nitrided, through-hardened steels:
83.5 HR 15N		See Figure 9-14.		150	163	175
84.5 HR 15N		See Figure 9-14.		155	168	180
Nitrided. nitralloy 135 [latex]M^a[/latex]
87.5 HR 15N		See Figure 9-15.
90.0 HR 15N		See Figure 9-15.		170	183	195
Nitrided, nitralloy [latex]N^a[/latex]:
87.5 HR 15N		See Figure 9-15.
90.0 HR 15N		See Figure 9-15.		172	188	205
Nitrided. 2.5% chrome (no aluminum):
87.5 HR 15N		See Figure 9-15.		155	172	189
90.0 HR 15N		See Figure 9-15.		176	196	216
Source: Extracted from AGMA Standard 2001 -C95. Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, with the permission of the publisher. American Gear Manufacturers Association, 1500 King Street. Suite 201. Alexandria. VA 22314. Nitralloy is a proprietary family of steels containing approximately 1 .0% aluminum which enhances the formation of hard nitrides

indicates that a carburized, case-hardened steel with a case
hardness of 55 to 64 HRC would be satisfactory, having a value of [latex]s_{at} = 55[/latex] ksi = 55 000 psi.
Referring to Appendix 5, we see that virtually any of the listed carburized materials could be used. Let's specify AISI 4320 SOQT 300, having a core tensile strength of 218 ksi, 13% elongation, and a case hardness of 62 HRC.
For the gear. Figure 9-10 indicates that a through-hardened steel with a hardness of 320 HB would be satisfactory. From Appendix 3, let's specify AISI 4340 OQT 1000 having a hardness of 363 HB, a tensile strength of 171 ksi, and 16% elongation.

Comments
These materials should provide satisfactory .service, considering bending strength. The following section considers the other major failure mode: pitting resistance. It is possible, perhaps likely, that the requirements to meet that condition will be more .severe than for bending.

Question 9.2: Specify suitable materials for the pinion and the gear from ...

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