# 14 Gear Design 1 2019 Spring ```Gear Design – Part 1
These slides are prepared by Suat Kadıoğlu
and modified by Sezer &Ouml;zerin&ccedil;
Design of Helical and Spur Gears
• Gear tooth fails due to bending (yielding or
fatigue) or due to wear.
• Wear is used as a general term for surface
failure, which is usually caused by pitting.
• Pitting: Formation of pits due to repetitive
contact stresses (surface fatigue).
• Abrasion (wear due to foreign particles) and
scoring (due to lubrication failure) can also result
in failure.
Photo-elastic analysis of two gear teeth in contact.
Note that due to bending of the tooth two types of stress
occurs at the tooth flank:
• On the right side tensile stresses
• On the left side compressive stresses.
These stress states repeat themselves every revolution, and
may result in fatigue failure.
Moreover, contact stresses occur at the contact point, which
may result in surface fatigue.
Design Method
• A method developed by American Gear
Manufacturers Association (AGMA) will be
followed. It is called AGMA method.
• Underlying principles of AGMA method, which
historically provided the background for this
method, will be reviewed first.
• These principles can be used for a rough
estimate of gear dimensions, power rating, and
similar.
Lewis Bending Equation
• Provides the basis for gear design against
bending failure.
• Proposed by Wilfred Lewis in 1892.
• In calculating the bending stress at the
root of the teeth, the tooth form is taken
into account.
Face width, 8mnF16mn
 max
Ft
Section modulus:
I/c=(Ft3/12)/(t/2)=Ft2/6
 max
a
t/2

=
x
t/2
b
a
b
M
=
I /c
6Wt 
=
Ft 2
t2
x=
4
tangent to fillet
 max
6Wt 
Wt 4
Wt
=
=
=
2
2
Ft
4F / 6 t
2 3 Fx
 max
Wt p
=
2 3 Fxp
2x
let y =
3p
then
multiply both numerator and
denominator by circular pitch, p.
 max
Wt
=
Fpy
Original form of
Lewis equation
Lewis Form factor
• Lewis form factor, y, can be obtained by graphical layout
of the gear tooth or by numerical computation.
• Noting that p = p/P = pm (where P is the diametral pitch)
Y = py can be defined and bending stress can be
expressed as:
Wt
Wt P
 max =
or  max =
FmY
FY
Assumptions in Using Lewis Equation
• Only the bending stress due to load Wt is considered;
hence, the compression due to Wr is neglected. In
reality, the radial force causes compressive stresses
over the base cross-section.
• The teeth do not share the load and the highest force is
exerted at the tip of the tooth. In reality, the tip condition
is not the worst case, as another tooth will be in contact
(contact ratio &gt; 1), and the load will be shared.
Assumptions in Using Lewis
Equation
• When the contact proceeds
towards the middle of the
tooth due to the rotation of the
gear, only one tooth will start
occurs near the middle of the
tooth. But in this case the
moment arm will be smaller.
More detailed analysis shows
that maximum stress tends to
occur when a single pair of
teeth is in contact.
• Stress concentration is not
taken into account.
effects are not considered.
Dynamic Effects
[MPa]
• Dynamic effects
increases the bending
stress. They are taken
into account by the
dynamic (velocity)
factor, Kv.
• In the 19th century, Carl
G. Barth first introduced
the velocity factor.
• Kv expressions are
empirically derived
(based on
experiments).
 max
K vWt
=
FmY
[N]
[mm]
3.05 + V
Kv =
cast iron, cast profile
3.05
6 .1 + V
Kv =
cut or milled profile
6 .1
3.56 + V
Kv =
3.56
5.56 + V
Kv =
5.56
hobbed or shaped
profile
shaved or ground
profile
V is the pitch line velocity in [m/s]
Surface Durability
• Types of surface failure for gear tooth:
– pitting (formation of pits due to repetitive high contact
stresses)
– scoring (lubrication failure)
– abrasion (due to presence of foreign materials)
• Scoring and abrasion should be prevented by maintaining
the lubrication well and keeping the gears free from hard
abrasive particles.
• But pitting is a type of failure that goes into the design of
gears.
• Hertz theory for contact stresses is utilized for analysis.
Hertz Contact Stresses
Surface Compressive stress
 c = pmax
2F
=
pb
1/ 2
 2F

1 +  2
b=

(
)
(
)
p

1
/
d
+
1
/
d
1
2 

For gears
F = Wt / cos 
 = Face width, F
d1, 2 = 2r1, 2
1 − 2
=
E
Instantaneous radii of curvature at the
contact point of the tooth profile
c
2
Wt  (1 / r1 ) + (1 / r2 )
=


pF cos   1 +  2 
c can be calculated at any
point on the tooth profile by
effect on Wt.
Along pressure line, only pure rolling exists. But since contact
point has a finite width, sliding will also take place. The Hertz
contact approach does no take friction into account.
Experimentally it is observed that wear occurs first near the
pitch line, and at pitch point P. The radius of curvature there:
r1 =
d p sin 
2
r2 =
d g sin 
2
Define an elastic
coefficient Cp as
follows (by AGMA):
1
Cp =
p ( p +  g )
Adding dynamic factor Kv as a multiplier for Wt, and using Cp:
 c = −C p
K vWt  1 1 
 + 
F cos   r1 r2 
negative sign means
compressive stress
Factor of Safety
• Factor of safety for bending is defined as:
St
loss - of - function - load
nt =
=
 max
St is the gear
bending strength
• Note that contact stress is not linearly dependent
to applied transmitted force Wt, but rather
proportional to 𝑊𝑡 .
• The factor of safety for contact, nc, is defined as;
2
2
 loss - of - function - load 
Sc
 = 2
nc = 

 c
Sc is the gear
surface strength
AGMA Stress Equations
• There are two stress equations, one for
bending and the other for contact (pitting
resistance).
• In AGMA terminology, the stresses are
referred to as stress numbers.
• The stress equations are based on the
previously explained equations but there
are many correction factors.
AGMA Bending Equation
Lewis equation with dynamic correction:
1 KH KB
 = Wt K o K v K s
Fm YJ
 max
K vWt
=
FmY
2𝜋𝑥
𝑌=
3𝑝
• Wt: tangential transmitted load [N]
• Ko: Overload Factor (depends on application, driving vs. driven machinery)
• Kv: Dynamic Factor (dynamic effects in load depending on pitch line velocity
and tooth form)
• Ks: Size Factor
The equation given here
• F: Face width (here we use the US letter
is in SI units. The values
in parentheses are U.S.
to be consistent with previous slides)
units, that is why there are
• m: transverse module [mm]
two different letters for
• KH or (Km): Load distribution factor
some of these
coefficients.
• KB : Rim thickness factor
• YJ (J): Geometry factor for bending, which includes root fillet stress
concentration factor, Kf (that is, an extended Lewis form factor).
AGMA Contact Stress Equation
Original form of
contact stress
equation
 c = ZE
 c = −C p
K vWt  1 1 
 + 
F cos   r1 r2 
KH ZR
Wt K o K v K s
d pF ZI
𝐶𝑝 = 𝑍𝐸 =
1
𝜋 𝜅𝑝 + 𝜅𝑔
The equation given here is in SI units. The
values in parentheses below are U.S.
units, that is why there are two different
letters for some of these coefficients.
• ZE (Cp): Previously defined elastic coefficient [N/mm2]1/2
• ZR (Cf): Surface condition factor
• dp: Pitch diameter of the pinion [mm] width (here we use the US
letter to be consistent with previous slides)
• ZI (I): Geometry factor for pitting resistance
• Other factors are the same as those in bending equation.
• In AGMA method, the stresses obtained from these equations are
compared with corresponding “AGMA gear strengths” which are
also referred to as &quot;allowable stress numbers&quot;.
AGMA Strength Equations
• In general gear strengths (St for bending,
Sc for contact) are given in terms of the
hardness of the material for a certain
number of stress cycles.
• St and Sc are also modified by various
factors to take into account the different
service conditions and requirements.
Bending and Contact Strength Equations
 all
St YN
=
S F Y YZ
• St : Bending strength [N/mm2]
• YN: Stress cycle (life) factor for
bending
• YQ (KT): Temperature factor
• YZ (KR): Reliability factor
• SF: AGMA factor of safety for
bending.
 c,all
S c Z N ZW
=
S H Y YZ
• Sc : Contact strength [N/mm2]
• ZN: Stress cycle (life) factor for
contact
• CH or ZW (): Hardness ratio factor
for pitting resistance
• YQ (KT): Temperature factor
• YZ (KR): Reliability factor
• SH: AGMA factor of safety for
pitting.
1 ksi = 6.89 MPa
 all
St YN
=
S F Y YZ
 all
St YN
=
S F Y YZ
 all
St YN
=
S F Y YZ
Sc Curves and Equations
 c,all
S c Z N ZW
=
S H Y YZ
Remarks on Gear Strengths
• The use of St and Sc are restricted to gear
problems.
• They are for
in a single direction)
– 10 million stress cycles
– 99% reliability
as with idler gears, AGMA recommends
using 70% of St values.
Geometry Factors
I (ZI) and J (YJ)
 = Wt K o K v K s
1 KH KB
Fm YJ
 c = Z E Wt K o K v K s
KH ZR
d pF ZI
• These are used to account for tooth form and load sharing.
• I and J depend on face contact ratio mF.
• mF = F/px (face width / axial pitch). We will consider spur
gears (mF = 0) and helical gears (mF &gt; 1).
• Helical gears with mF &lt; 1 are &quot;low contact ratio&quot; gears and
they are not much different than spur gears as far as noise is
concerned.
Y
J=
K f mN
obtained from tooth
layout for highest point
of single-tooth contact
m N = 1 .0
for spur gears
normal base pitch
pN
mN 
0.95Z
for helical gears
with mF &gt; 2.0
Lab length of action
mN is the ratio of face width to minimum total length of lines
of contact.
Note that mN 
pN
0.95Z
is an approximation.
p N = pn cos n
Lab = (r2 + a2 ) 2 − (r2 cos  ) + (r1 + a1 ) 2 − (r1 cos  ) − (r1 + r2 ) sin 
2
2
• The book gives ready-to-use graphs for J, which makes the
process easier.
• For spur gears with F = 20 degrees and FDI teeth, use Fig.
14-6.
• For helical gears with Fn=20 degrees and mF  2, use Fig.
14-7 and 14-8. (the latter is used to correct the values
obtained from the former.)
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