Bevel and Worm Gears
AGMA Equation Factors
Chapter 15
Week four-Lecture three
Presented by Team 5:
Luis Aguilar
Javier Cervantes
Jhonson Jaimes
Robert Meeker
MECHANICAL SYSTEM DESIGN 3309-032
HOUSTON ENGINEERING CENTER
THE UNIVERSITY OF TEXAS AT TYLER
February 26, 2022
AGMA Equation Factors
Outline 15-3
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𝐾𝑜 (𝐾𝐴 ):
𝑆𝐻 𝑎𝑛𝑑 𝑆𝐹 :
𝐾𝑉 :
𝐶𝑠 (𝑍𝑥 ):
𝐾𝑠 (𝑌𝑥 ):
𝐾𝑚 (𝐾𝐻𝛽 ):
𝐶𝑋𝐶 (𝑍𝑋𝐶 ):
𝐾𝑋 (𝑌𝛽 ):
• 𝐼(𝑍𝐼 ):
• 𝐽(𝑌𝐽 ):
Overload Factor
Safety Factor
Dynamic Factor
Size Factor for Pitting Resistance
Size Factor for Bending
Load Distribution Factor
Crowning Factor for Pitting
Lengthwise Curvature Factor for
Bending Strength
Pitting Resistance Geometry
Factor
Bending Strength Geometry
Factor
• 𝐶𝐿 (𝑍𝑁𝑇 ):
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Stress-Cycle Factor for Pitting
Resistance
𝐾𝐿 (𝑌𝑁𝑇 ):
Stress-Cycle Factor for Bending
Strength
𝐶𝐻 (𝑍𝑊 ):
Hardness Ratio Factor
𝐾𝑇 (𝐾𝜃 ):
Temperature Factor
𝐶𝑃 (𝑍𝐸 ):
Coefficient for Pitting Resistance
𝑆𝑎𝑐 (𝜎𝐻 ):
Allowable Contact Stress
Number
𝑆𝑎𝑡 (𝜎𝐹𝑙𝑖𝑚 ): Allowable Bending Stress
Number
Reverse Loading
𝐶𝑅 (𝑍𝑍 ) 𝑎𝑛𝑑𝐾𝑅 (𝑌𝑍 ): Reliability Factor Elastic
[1][3]
1
Straight-Bevel Gear Wear and Bending
Geometry
𝑁𝑃
𝑑𝑃 =
𝑃𝑑
𝑁𝑃
𝛾=
𝑁𝐺
𝑁
−1 𝑃
Γ = 𝑡𝑎𝑛
𝑁𝐺
𝑑𝑎𝑣 = 𝑑𝑃 − 𝐹𝑐𝑜𝑠Γ
𝑡𝑎𝑛−1
•
•
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•
𝑑𝑝 : 𝑂𝑢𝑡𝑒𝑟 𝑝𝑖𝑡𝑐ℎ 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
𝑜𝑓 𝑔𝑒𝑎𝑟 & 𝑝𝑖𝑛𝑖𝑜𝑛
𝛾, Γ: 𝑃𝑖𝑡𝑐ℎ 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑝𝑖𝑛𝑖𝑜𝑛 𝑎𝑛𝑑 𝑔𝑒𝑎𝑟
𝑁𝑃 : 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑜𝑜𝑡ℎ 𝑜𝑛 𝑝𝑖𝑛𝑖𝑜𝑛
𝑃𝑑 : 𝐷𝑖𝑎𝑚𝑒𝑡𝑟𝑎𝑙 𝑃𝑖𝑡𝑐ℎ
Force Analysis
Strength analysis
2𝑇
𝑡
𝑊 =
𝑑𝑎𝑣
𝑊𝑡 =
𝑊𝑟
𝑊𝑎
=
𝑊 𝑡 𝑡𝑎𝑛𝜙𝑐𝑜𝑠𝛾
=
𝑊 𝑡 𝑡𝑎𝑛𝜙𝑠𝑖𝑛𝛾
2𝑇
𝑑𝑃
𝑊 𝑟 = 𝑊 𝑡 𝑡𝑎𝑛𝜙𝑐𝑜𝑠𝛾
𝑊 𝑎 = 𝑊 𝑡 𝑡𝑎𝑛𝜙𝑠𝑖𝑛𝛾
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𝑊 𝑡 : 𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
𝑊 𝑟 : 𝑅𝑎𝑑𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
𝑊 𝑎 : 𝐴𝑥𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 or thrust load
𝑑𝑎𝑣 : 𝐼𝑓 𝑎𝑛𝑦𝑜𝑛𝑒 𝑘𝑛𝑜𝑤𝑠 𝑤ℎ𝑎𝑡 𝑡ℎ𝑖𝑠 𝑠𝑡𝑎𝑛𝑑𝑠 𝑓𝑜𝑟 𝑝𝑙𝑒𝑎𝑠𝑒 𝑡𝑦𝑝𝑒 𝑖𝑡 𝑏𝑐
𝐼 𝑑𝑜𝑛𝑡 𝑘𝑛𝑜𝑤 𝑦𝑒𝑡 𝑜𝑟 𝑤𝑒 𝑐𝑎𝑛 𝑙𝑒𝑎𝑣𝑒 𝑖𝑡 𝑖𝑛 𝑏𝑙𝑎𝑛𝑘.
[1]
2
Straight-Bevel Gear Wear
Gear Contact Stress
𝑺𝒄 = 𝝈𝒄 =
𝟏
𝑾𝒕
𝑪𝒑 (
𝑲 𝑲 𝑲 𝑪 𝑪 )𝟐
𝑭𝒅𝒑 𝑰 𝒐 𝒗 𝒎 𝒔 𝒙𝒄
(15-1)
• Elastic Coefficient for Pittin Resistance
𝑪𝑷 (𝒁𝑬 ):
𝐶𝑃
1
2 )/𝐸 +1−𝑣 2 )/𝐸 ))
𝜋((1−𝑣𝑃
𝑃
𝐺
𝐺
(15-21)
 Or can be found on Table 14-8 @ pdf
attached.
• Overload Factor 𝑲𝒐 (𝑲𝑨 ):
It is used to account for any externally applied
loads exceeding the normal tangential load 𝑊 𝑡
Table 15-2
[2]
• Dynamic Factor 𝑲𝑽 :
It is used to account for deviations from the
uniform angular speed due to inaccuracies in
manufacturing and meshing of gears
[1][2]
𝐴+ 𝑣𝑡 𝐵
)
𝐴
𝐾𝑣 = (
(15-5)
 𝐾𝑣 Is found from figure 15-5 as a function of
𝑄𝑣 where:
𝐴 = 50 + 56 1 − 𝐵
2
𝐵 = 0.25(12 − 𝑄𝑣 ) 3
(15-6)
 And pitch-line velocity 𝑣𝑡 where:
𝜋𝑑𝑝 𝑛𝑝
𝑣𝑡 =
12
𝑣𝑡𝑚𝑎𝑥 = [𝐴 + (𝑄𝑣 −3)]2
(15-7)
(15-8)
[1][3]
3
Straight-Bevel Gear Wear
• Crowning Factor for Pitting 𝑪𝑿𝑪 (𝒁𝑿𝑪 ):
• Load Distribution Factor 𝑲𝒎 (𝑲𝑯𝜷 ):
𝐾𝑚 = 𝐾𝑚𝑏 + 0.0036𝐹 2
(15-11)
The teeth of most bevel gears are crowned to
accommodate the shaft deflections
[1]
 Where:
𝐾𝑚𝑏
1.00 𝑏𝑜𝑡ℎ 𝑚𝑒𝑚𝑏𝑒𝑟𝑠 𝑠𝑡𝑟𝑎𝑑𝑑𝑙𝑒 − 𝑚𝑜𝑢𝑛𝑡𝑒𝑑
=
1.10 𝑂𝑛𝑒 𝑚𝑒𝑏𝑒𝑟 𝑠𝑡𝑟𝑎𝑑𝑑𝑒𝑙 − 𝑚𝑜𝑢𝑛𝑡𝑒𝑑
1.25 𝑛𝑒𝑖𝑡ℎ𝑒𝑟 𝑚𝑒𝑚𝑏𝑒𝑟 𝑠𝑡𝑟𝑎𝑑𝑑𝑙𝑒 − 𝑚𝑜𝑢𝑛𝑡𝑒𝑑
𝐶𝑋𝐶 (𝑍𝑋𝐶 ) =
1.5 𝐶𝑟𝑜𝑤𝑛𝑒𝑑 𝑡𝑒𝑒𝑡ℎ
2 𝑛𝑜𝑛 − 𝑐𝑟𝑜𝑤𝑛𝑒𝑑 𝑡𝑒𝑒𝑡ℎ
(15-12)
[1][3]
• Pitting Resistance Geometry Factor 𝑰(𝒁𝑰 ):
• Size Factor for Contact Stress 𝑪𝒔 (𝒁𝒙 ):
0.5
𝐹 < 0.5 𝑖𝑛
𝐶𝑠 = 0.125𝐹 + 0.4375 0.5 ≤ 𝐹 ≤ 4.5 𝑖𝑛
1
𝐹 > 4.5 𝑖𝑛
(15-9)
Figure 15-7: Enter the figure with the number of
pinion, move to the number of gear teeth
contour, and read from the abscissa
[1]
[1]
4
Straight-Bevel Gear Wear
Gear Wear Strength
𝑺𝒘𝒄 = (𝝈𝒄 )𝒂𝒍𝒍 =
𝑺𝒂𝒄 𝑪𝑳 𝑪𝑯
𝑺𝑯 𝑲𝑻 𝑪𝑹
• Stress-Cycle Factor for Pitting Resistance
𝑪𝑳 (𝒁𝑵𝑻 ):
(15-2)
• Allowable Contact Stress 𝑺𝒂𝒄 𝝈𝑯 :
Material property under compressive cyclic loading
conditions.
 The Tables 15-4, 15-5 provide values for steel
gears and iron gears. Figure 15-12 displays
allowable contact stress for grade 1 and 2
materials
[1][2]
It account for lives other than 109 cycles. The
value is found on Table 15-8
2
𝐶𝐿 =
3.4822𝑁𝐿−0.0602
103 ≤ 𝑁𝐿 < 104
104 ≤ 𝑁𝐿 ≤ 1010
15 − 14
•
Reliability Factor 𝑪𝑹 (𝒁𝒁 ) 𝒂𝒏𝒅𝑲𝑹 (𝒀𝒁 ):
Used to account for reliabilities other than 0.99
 𝐾𝑅 : Reliability factor for bending strength
 𝐶𝑅 : Reliability factor for contact strength [3]
5
Straight-Bevel Gear Wear
• Reliability Factor 𝑪𝑹 (𝒁𝒁 ) 𝒂𝒏𝒅𝑲𝑹 (𝒀𝒁 ): (cont.)
 Where: 𝐶𝑅 =
Table 15-3
𝑌𝑍 = 𝐾𝑅 =
𝐶𝐻 = 1 + 𝐵1 𝑁 𝑛 − 1
 𝐵1 = 0.00898 𝐻𝐵𝑃
𝐾𝑅 , and values are found on
0.50 − 0.25𝑙𝑜𝑔 1 − 𝑅 0.99 ≤ 𝑅 ≤ 0.999
0.70 − 0.15𝑙𝑜𝑔 1 − 𝑅 0.90 ≤ 𝑅 ≤ 0.99
(15-19) , (15-20)
• Hardness Ratio Factor 𝑪𝑯 (𝒁𝑾 ):
It is used to account for the difference of the
hardness between gear and pinon.
[1][2]
𝐻𝐵𝐺
− 0.00829 (15-16)
𝐶𝐻 = 1 + 𝐵2 450 − 𝐻𝐵𝐺
 𝐵2 = 0.00075 exp(−0.52𝑅1 )
(15-17)
• Temperature Factor 𝑲𝑻 (𝑲𝜽 ):
It accounts for the change in material strength at
increased temperature. Figure 15-9
[2]
𝐾𝑇 =
1
32℉ ≤ 𝑇 ≤ 250℉
460+𝑇
𝑇 > 250℉
(15-18)
710
6
Straight-Bevel Gear Bending
Gear Bending Stress
𝑺𝒕 = 𝝈 =
𝑾𝒕
𝑲 𝑲
𝑷𝒅 𝑲𝒐 𝑲𝒗 𝒔 𝒎
𝑭
𝑲𝒙 𝑱
• Lengthwise Curvature Factor for Bending
Strength 𝑲𝑿 (𝒀𝜷 ):
(15-3)
𝐾𝑥 = 𝑌𝛽 = 1
• Size Factor for Bending 𝑲𝒔 (𝒀𝒔 ):
𝐾𝑠 =
0.4867 +
0.2132
𝑃𝑑
0.5
0.5 ≤ 𝑃𝑑 ≤ 16 𝑡𝑒𝑒𝑡ℎ/𝑖𝑛
𝑃𝑑 > 16 𝑡𝑒𝑒𝑡ℎ/𝑖𝑛
For straight bevel gears:
(15-10)
(15-13)
• Bending Strength Geometry Factor 𝑱(𝒀𝑱 ):
It is used to account for geometry of the tooth and
location of the load 𝑊 𝑡 .
 The value of J is found from Figure 15-7
[1][2]
7
Straight-Bevel Gear Bending
• Allowable Bending Stress Number 𝑺𝒂𝒕 (𝝈𝑭𝒍𝒊𝒎 ):
Gear Bending Strength
𝑺𝒘𝒕 = 𝝈𝒂𝒍𝒍 =
𝑺𝒂𝒕 𝑲𝑳
𝑺𝑭 𝑲𝑻 𝑲𝑹
(15-4)
• Stress-Cycle Factor for Bending Strength
𝑲𝑳 (𝒀𝑵𝑻 ):
It accounts for lives other than 107 cycles
2.7
102 ≤ 𝑁𝐿 < 103
6.1514𝑁𝐿−0.1182 103 ≤ 𝑁𝐿 < 3(10)6
𝐾𝐿 =
1.6831𝑁𝐿−0.0323 3(10)6 ≤ 𝑁𝐿 ≤ (10)10
1.3558𝑁𝐿−0.0178 3(10)6 ≤ 𝑁𝐿 ≤ (10)10
Material property under tensile cyclic loading
conditions (tensile fatigue strength)
 𝑆𝑎𝑡 values are found from Tables 15-6, 15-7
and Figure 15-13
 Values are based on 107 cycles and 0.99
reliability
 For reverse loading such as in idler gears,
AGMA recommends using 70% of value
Critical
General
[1][2]
8
Wear Safety Factor
Bending Safety Factor
• Safety factors 𝐒𝐇 𝐚𝐧𝐝 𝐒𝐅 :
Use to account for unquantifiable elements affecting the stresses.
𝝈𝒂𝒍𝒍
𝝈
Based on strenght
(𝝈𝑪 )𝒂𝒍𝒍
𝑺𝑯 =
𝝈𝑪
Based on strenght
𝑺𝑭 =
(𝝈𝑪 )𝒂𝒍𝒍 𝟐
)
𝝈𝑪
(Based on W t , can be compared
directly with SF )
𝒏𝒘 =
𝒏𝒘 = (
𝝈𝒂𝒍𝒍
𝝈
(Based on W t , same as SF )
 When designing a safety factor become a design factor
 When analyzing, the safety factor is the ratio of strength to stress
 When comparing bending and contact factors of safety, we compare 𝑆𝐹 𝑤𝑖𝑡ℎ 𝑆𝐻
2
[1][2]
9
WEEK FIVE - LECTURE FOUR
Straight- Bevel Gear Analysis
In the following lecture, we are going to
solve example 15-1, on page 808 of the
book, where some of the equations
used in Bevel-Gears Stress and
Strengths and equation factors are
applied
Due date: February 5
Thank you!
10
References
[1] Richard BUDYNAS and J. Keith NISBETT, "15-1 Bevel Gear-General," Shigley's Mechanical Engineering
Design, 11th edition. MCGRAW-HILL EDUCATION
[2] The Hashemite University, ACUploads, Wikipedia·https://eis.hu.edu.jo/ACUploads/10526/
[Online source accesses 02/24/2022]
[3] STYDYLIB, Shigley's Mechanical Engineering Design, Chapter 15: Bevel and Worm gears, Kuei-Yuan
Chan. https://studylib.net/doc/5705167/shigley-9e-si-chap15 [Online source accessed 02/24/2022]