Fx
Fy
a
T
w
f
Wt W
US System Force SI System US System
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198,000H T n
H power, hp T torque, lbf - in
W r W t tan f
Wr
• Wt
t
T 2T r d
60(103 ) H W nd
• Bending stress at root Failure by bending will occur when the significant tooth stress equals or exceeds either the yield strength or the bending endurance strength. • Contact stress on tooth surface A surface failure occurs when the significant contact stress equals or exceeds the surface endurance strength.
• The AGMA standards are revised from time to time. Make sure to check with AGMA for the newest revision (ANSI/AGMA 2001-C95 was used here). • Make sure to use the correct unit systems (U.S. system or metric system)
Fb3 F43 F23 0.546 0.199 0.347 kN
02 - 7
Question
What if Gear 3 is not an idler (i.e., it transmits/consumes power)?
Tooth Failure Modes
Tooth Failure Modes
3
r3
Fbx b x Fby
t F23
Forces from Gear 2
F23
t
60(103 ) P2 60(103 )(2.5) r n2d 2 (1750)(50) F43 0.546 kN
Fb
r t F23 F23 tan f (0.546) tan 20

F23
Note that gear 3 is an idler and it transmits no power to its shaft.
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r r t t t t M b 0 F23 F23 0.199 kN r3 F43 r3 0 F43 F23 0.546 kN F43 t r X 0 Fbx3 F23 F43 0.546 0.199 0.347 kN Y 0 y t r
Standards to Use
• AGMA (American Gear Manufacturers Association) Stress calculation Bending stress Contact Stress Strength calculation Bending strength Contact strength
AGMA bending stress
WtP Ko Kv K s K m K B FJ
P: Diametral pitch J: Geometry factor for bending strength (including stress concentration factor) Ko: Overload factor (Application factor) Kv: Dynamic factor Ks: Size factor Km: Load distribution factor KB: Rim-thickness factor
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Spur Gear Force Analysis: Free Body Diagram
Fa2
f f
2 Ta 2 a
f
W
w
W
W
f
• The two rotating gears in mesh can be viewed as two rotating pitch cylinders without slippery, with a contact force applied at the pitch point.
Lewis Form Factor
Lewis Form Factor
The Lewis form factor is independent of tooth size.
The Lewis Form Factor for the 20º full-depth involute teeth.
Number of Teeth 12 13 14 15 16 17 18 19 20 21 22 24 26
rf
6W t WtP , or F (4 x) F (2 / 3) xP P : Diametral pitch
Lewis Equation
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Y
WtP FY
2 xP , Y 3
Lewis form factor
02 - 9
Y is defined as the Lewis form factor, which accounts for the effect of the tooth geometry on the bending stress level.
y
c b 50T
4
3
30T
20T
a 2
x
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02 - 5
Spur Gear Force Analysis Example: Solution I
y
Free Body Diagram
y c
4
50T
t F43 F43 20
3
Fbx b 3
r F43
b x Fby
t F23
30T
20T a 2
Fb
x
Notes
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02 - 8
Spur Gear: Tooth Bending
Bending Stress at Root
Mc (W t l )t / 2 6W t l 3 I Ft / 12 Ft 2 F : Face width
Wr
Wt
W
Lewis Equation
The Lewis Equation relates the bending stress with the variable x in the figure. By similar triangles, t/2 l t2 l 1 or x or x t/2 4l t 2 4x
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Pitch Diameters
02 - 6
d 2 N 2m 20 2.5 50 mm d 4 N 4m 30 2.5 75 mm
Spur Gear Force Analysis Example: Solution II
y
Free Body Diagram
t F43 F43 20
t W
d : mm Wt : kN d : in Wt : lbf
396,000H nd
02 - 4
Using equilibrium equations to determine Fx and Fy
Spur Gear Force Analysis: Example
Pinion 2 runs at 1750 rev/min and transmits 2.5 kW to idler gear 3. The teeth are cut on the 20ºfull-depth system and have a module of m = 2.5 mm. Draw a free-body diagram of gear 3 and show all the forces that act upon it.
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• The contact force is always along line ab, passing through the pitch point P. • The force F, remains constant as long as the transmitted torque is constant. • For tooth analysis, the force F is applied at different location with a different angles from the initial to final contacts.
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3 Tb 3 b Fb3
02 - 3
Spur Gear Force Analysis: Force Calculation
Power and Torque
SI System • T
H
w
H
3
30(10 ) H T w n
H power, kW T torque, kN - mm w rad/s, n rpm
School of Engineering Mechanical Engineering
TOPIC
ME 462
02
© Tulong Zhu, All rights reserved.
Design of Mechanical Systems
Gears II Spur Gears Analysis