GEAR CONTACT STRESS DESIGN

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GEAR CONTACT STRESS DESIGN

Description
Limit State Function
Random Variables
Analysis
Results

Description:

Evaluate the integrity of the given gear using the following limit-state function and random variables. Make recommendations for potential design improvement.

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Limit-State Function:

( 1)

( 2)

where

m_g = Gear ratio = 3.78

f: Face width (in.)

T_p: Torque (lb/in.)

P_d: Diametrical pitch (1/in.)

N₁: Number of teeth of the pinion

f: Pressure angle (deg.)

E: Modulus of elasticity (psi)

n: Poisson’s ratio

s_N: Allowable surface pressure (ksi)

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Random Variables:

Case 1:

X1 = f ~ Normal (m=0.5, s=0.025)

X2 = T_p ~ Normal (m=108, s=5.4)

X3 = P_d ~ Normal (m=9, s=0.45)

X4 = N₁ ~ Normal (m=18, s=0.9)

X5 = f ~ Normal (m=20, s=1.0)

X6 = E ~ Normal (m=30*10⁶, s=1500000)

X7 = n ~ Normal (m=0.25, s=0.0125)

X8 = s_N ~ Normal (m=88, s=4.4)

Case 2:

X1 = f ~ Lognormal (m=0.5, s=0.025)

X2 = T_p ~ Lognormal (m=108, s=5.4)

X3 = P_d ~ Lognormal (m=9, s=0.45)

X4 = N₁ ~ Lognormal (m=18, s=0.9)

X5 = f ~ Lognormal (m=20, s=1.0)

X6 = E ~ Lognormal (m=30*10⁶, s=1500000)

X7 = n ~ Lognormal (m=0.25, s=0.0125)

X8 = s_N ~ Lognormal (m=88, s=4.4)

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Analysis:

Anal1: Calculate the failure probability of the limit-state function in Eq. 1 using different methods.

Case1: For Random Variables - Case 1

Case2: For Random Variables - Case 2

Anal2: CDF/PDF Analysis for limit-state function defined in Eq. 2 for Pf = 0.000001~0.999999.

Case1: For Random Variables - Case 1

Case2: For Random Variables - Case 2

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Results:

Analysis 1: Failure Probability Analysis using SORM – Point- Fitting

CASE	MPP Method	Pf1 Value	Pf2 Value	No. of g	No. of Ñg
Case 1	U Based/X Linearized	3.37187E-02	3.47789E-02	104	4
Case 2	U Based/X Linearized	3.31927E-02	3.34408E-02	104	4

Analysis 2: CDF/PDF analysis using SORM-Point-Fit

Case 1

***** SUMMARY OF CDF ANALYSIS *****

First/Second-Order CDF of Limit-State Function 26

Number of points in each CDF curve: 7

1st-order b	1st-order Pf	2nd-order b	2nd-order Pf	G-Value
3.0902332E+00	9.9999710E-04	3.0866970E+00	1.0119717E-03	3.0215517E+09
1.7932989E+00	3.6462573E-02	1.7914846E+00	3.6607818E-02	3.8157124E+09
7.4344663E-01	2.2860566E-01	7.4297396E-01	2.2874880E-01	4.6098732E+09
-1.3415200E-01	5.5335881E-01	-1.3350743E-01	5.5310394E-01	5.4040339E+09
-8.8480174E-01	8.1186822E-01	-8.8322066E-01	8.1144140E-01	6.1981947E+09
-2.1142374E+00	9.8275250E-01	-2.1111694E+00	9.8262109E-01	7.7865162E+09
-3.0902332E+00	9.9900000E-01	-3.0860030E+00	9.9898566E-01	9.3748377E+09

CDF of Limit State Function G26:

***** SUMMARY OF PDF ANALYSIS *****

First/Second-Order PDF of Limit-State Function 26

1st-order b	1st-order Pf	2nd-order b	2nd-order Pf	G-Value
-1.8359020E-09	6.1816310E-12	-1.3167226E-08	4.4821966E-11	3.0215517E+09
-1.4575210E-09	1.1646530E-10	-1.7885491E-09	1.4338207E-10	3.8157124E+09
-1.2018496E-09	3.6369731E-10	-1.0742023E-09	3.2518361E-10	4.6098732E+09
-1.0176374E-09	4.0234183E-10	-9.2781525E-10	3.6686061E-10	5.4040339E+09
-8.7870719E-10	2.3700379E-10	-6.6747149E-10	1.8028137E-10	6.1981947E+09
-6.8365231E-10	2.9180805E-11	-1.3742267E-09	5.9038508E-11	7.7865162E+09
-5.5326595E-10	1.8628913E-12	-3.0202259E-09	1.0303061E-11	9.3748377E+09

PDF of Limit State Function G26:

No. of g-functions called = 672

No. of derivatives of g called = 31

Case 2:

***** SUMMARY OF CDF ANALYSIS *****

First/Second-Order CDF of Limit-State Function 26

Number of points in each CDF curve: 14

1st-order b	1st-order Pf	2nd-order b	2nd-order Pf	G-Value
4.7534259E+00	9.9999191E-07	4.7562939E+00	9.8590477E-07	2.2451890E+09
3.8645828E+00	5.5639621E-05	3.8674554E+00	5.4988778E-05	2.6308152E+09
3.0997344E+00	9.6847128E-04	3.1026171E+00	9.5909089E-04	3.0164415E+09
2.4288267E+00	7.5738848E-03	2.4316990E+00	7.5141076E-03	3.4020677E+09
1.5558188E+00	5.9875587E-02	1.5586980E+00	5.9534004E-02	3.9805071E+09
8.0411087E-01	2.1066647E-01	8.0701606E-01	2.0982869E-01	4.5589465E+09
1.4448525E-01	4.4255865E-01	1.4742477E-01	4.4139840E-01	5.1373859E+09
-4.4286621E-01	6.7106874E-01	-4.3987258E-01	6.6998525E-01	5.7158252E+09
-1.4531718E+00	9.2691197E-01	-1.4500754E+00	9.2648119E-01	6.8727040E+09
-2.3012525E+00	9.8931132E-01	-2.2980361E+00	9.8922011E-01	8.0295827E+09
-3.0312132E+00	9.9878213E-01	-3.0278676E+00	9.9876857E-01	9.1864615E+09
-3.6713385E+00	9.9987936E-01	-3.6678917E+00	9.9987772E-01	1.0343340E+10
-4.2408471E+00	9.9998887E-01	-4.2372984E+00	9.9998869E-01	1.1500219E+10
-4.7534259E+00	9.9999900E-01	-4.7497673E+00	9.9999898E-01	1.2657098E+10

CDF analysis of Limit State Function G26:

***** SUMMARY OF PDF ANALYSIS *****

First/Second-Order PDF of Limit-State Function 26

1st-order b	1st-order Pf	2nd-order b	2nd-order Pf	G-Value
-2.5012476E-09	1.2376909E-14	-2.8689111E-08	1.4003941E-13	2.2451890E+09
-2.1283545E-09	4.8513205E-13	-5.5109135E-09	1.2422715E-12	2.6308152E+09
-1.8511615E-09	6.0523910E-12	-2.9846322E-09	9.6714356E-12	3.0164415E+09
-1.6371113E-09	3.4197425E-11	-2.9290138E-09	6.0758222E-11	3.4020677E+09
-1.3941574E-09	1.6580640E-10	-1.4770563E-09	1.7487968E-10	3.9805071E+09
-1.2131692E-09	3.5028801E-10	-1.1458659E-09	3.3008161E-10	4.5589465E+09
-1.0731606E-09	4.2368359E-10	-1.0079226E-09	3.9775695E-10	5.1373859E+09
-9.6163702E-10	3.4780198E-10	-7.7186628E-10	2.7953537E-10	5.7158252E+09
-7.9523030E-10	1.1037008E-10	-9.8965076E-10	1.3797248E-10	6.8727040E+09
-6.7705746E-10	1.9123847E-11	-1.0979501E-09	3.1242415E-11	8.0295827E+09
-5.8883717E-10	2.3752068E-12	-1.1304043E-09	4.6061900E-12	9.1864615E+09
-5.2048366E-10	2.4572474E-13	-1.1029367E-09	5.2733374E-13	1.0343340E+10
-4.6601639E-10	2.3118785E-14	-1.0406598E-09	5.2408997E-14	1.1500219E+10
-4.2159002E-10	2.0861515E-15	-1.7670630E-09	8.8972852E-15	1.2657098E+10

PDF analysis of Limit State Function G26:

No. of g-functions called = 2100

No. of derivatives of g called = 138

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Last Updated 02/08/10