Gear stress lewis_formula

Lewis Gear Formula

1. W
t
A
B
C
The above photo-elastic analysis of two gear teeth in contact shows that there are
two types of high stress on the teeth. At A and B we see the tensile and compressive
stresses due to bending of the tooth. Note that the compressive stress has a greater
magnitude due to the radially inward component of the tooth force W
t
. The bending
stress is cyclic as it occurs once per revolution of the gear and will, thus, lead to a
potential fatigue failure.
At C we have a contact stress situation as the two, approximately cylindrical
surfaces roll and slide on each other during tooth contact. Thisstress may lead to a
surface fatigue of the tooth.

2. Lewis Equation for Gear Strength.
Historically, the first equation used for the bending stress wasthe Lewis equation.
This is derived by treating the tooth as a simple cantilever andwith tooth contact
occurring at the tip as shown above. Only the tangent component (W
t
) is
considered. It is also assumed that only one pair of teeth is incontact. Stress
concentrations at the tooth root fillet are ignored. It can be shown that the
maximum bending stress occurs at the tangent points on the parabola shown
above. Use of the standard equation for bending stress (σ= Mc/I) leads to:
and, letting y = 2x/3p we get the original Lewis
equation: where p = circular pitch
Replacing p with the diametral pitch P ( = π/p) gives the more usual form:
3/2Fxp
pW
t
=σ
Fpy
W
t
=σ
σ= W
t
P/FY, where
W
t
= tangential tooth load
P = diametral pitch
F = face width of tooth
σ= bending stress in gear tooth
Y = Lewis form factor

3. The Lewis form factor is a function of the number of teeth N andthe pressure angle
φand is given in the following chart. Note that the old 14.5 degree form factor is not
generally used today.
Dynamic effects:
It is found that, if a pair of gears is run under load, the safeload on them
decreases as the running speed increases. Specifically, it is a function of the
pitch line velocity V = πdN/12 ft/min where d is the pitch diameter in
inches and N is the rotation speed in rpm.
This is accounted for by the use of a velocity factor K
v
.
Values of K
v
derived by Barthare used with the Lewis equation:
K
v
= (600 + V)/600 for crude gears (typically cast metal)
K
v
= (1200 +V)/1200 for cut or milled teeth
Hence we get the gear stress as
We will not use the Lewis equation or the Barthvelocity factors in this
course as they have been superseded by the AGMA equations.
FY
PWK
tv
=σ

4. AGMA Gear Stress Equation
This equation modifies the Lewis equation to take into account:
1. the effect of the radial forceW
r
2. the root stress concentrations
3. the effects of having multiple pairs of teeth in contact
σ= W
t
PK
v
K
o
K
m
K
s
K
B
/FJ
J = AGMA gear geometry factor
K
v
= velocity factor
K
o
= overload factor
K
m
= mounting factor
K
s
= size factor
K
B
= rim thickness factor
One important difference is that the AGMA geometry factor J is afunction of the numbers
of teeth on both gears.
All of the equations and data in this document applies to designof spur gears using standard
US units (inches, pounds, etc.)

6. Dynamic factor K
v
:
AGMA has defined quality control numbers Q
v
. Classes 3 to 7 cover most
commercial quality gears and classes 8 to 12 cover high precision gears. Values
of K
v
as a function of Q
v
and pitch line velocity (in ft/min) are given in the chart
above. These are derived from the following equations:
where
The maximum velocity for any given Q
v
value is:
The following chart and equations give older values for K
v.
These are used in the
gear.exeprogram discussed elsewhere in the ME356 web site notes..
B
v
A
VA
K







+
= ()( )
3/2
1225.015650
v
QBandBA −=−+=
( )[ ] min/3
2
max
ftQAV
v
−+=

8. The overload and load distribution factors are highly qualitative in nature as you
will see from sections 14-8 and 14-11 in your text. For the purposes of this
course you should use the values given in the following two tables.
Overload factor K
O
Load distribution factor K
m
Size factor K
s
: use the following table
1.010121.1005
1.029101.1244
1.05281.1563
1.06571.1762.5
1.08161.2022
Factor K
s
Pitch PFactor K
s
Pitch P
The rim thickness factor is only needed for large gears with a spokedhub –see secton14-
16 in your test.

9. Gear surface stresses:
Surface failure of gear teeth has two causes –(a) fatigue failure leading to pitting or
spalling and (b) wear due to the sliding contact that occurs between involute tooth
surface during tooth contact (except at the pitch point). These failures are related to
the high, very local contact stresses that occur. The compressive stress at the contact
point between two cylinders is given by:
where F = force pressing the cylinders together
l = length of the cylinders
b = half width of the contact surface
Where ν
1
, E
1
, ν
2
, E
2
, d
1
and d
2
are the elastic constants and diameters of the cylinders.
For two gear teeth we can replace F by W
t
/cosφ, p
max
by σ
C,
d by 2r and l by the face
width F. This gives the surface contact compressive stress as:
bl
F
p
π
2
max=
()[] ( )[ ]
() ()
2/1
21
2
2
21
2
1
/1/1
/1/12






+
−+−
=
dd
EE
l
F
b
νν
π
()()
()[] ()[]
2
2
21
2
1
212
/1/1
/1/1
cos EE
rr
F
W
t
C
ννφπ
σ
−+−
+
=
Where r
1
and r
2
are the radii of curvature of the tooth surfaces at the contactpoint.
As the first evidence of tooth wear is seen at the pitch point, the curvatures at the
pitch point are used:
Where d
P
and d
G
are the pitch diameters of the pinion and gear respectively.
Note that the denominator in the above contains only the elasticconstants for the
materials of the gear and the pinion. We thus define the elasticcoefficient C
p
:
2
sin
2
sin
21
φφ
GP
d
rand
d
r ==
()[] ()[]
2/1
22
/1/1
1






−+−
=
PPGG
p
EE
C
ννπ
This is tabulated below for various common combinations of gear and pinion
materials.

10. The expression in the contact stress equation that contains onlyvalues that are
dependent on the geometry of the gear and pinion can be written as:
We get the surface geometry factor for spur gears only.
All of this leads to:
P
G
P
G
G
GP
d
d
N
N
mratiospeedtheifand
ddrr
I ==








+=+= ,,
11
sin
211
21
φ
12
sincos
+
=
G
G
m
m
I
φφ
AGMA Surface Stress Equation:
W
t
, K
o
, K
v
, K
s
, K
m
and F are the same quantities that were defined above for the
bending stress equation. The elastic coefficient C
p
and the geometry factor I are
defined above. d
P
is the piniondiameter.
The surface condition factor C
f
has not yet been evaluated by AGMA so its
value is always 1.0.
I
C
Fd
K
KKKWC
f
P
m
svotpC=σ

11. Strength of gear materials:
AGMA uses its own data for strengths. These values should only be used in
gear tooth strength calculations.
The allowable bendingstrength numbers (S
t
) are given in your text in
Figures 14-2, 14-3, 14-4 and Tables 14-3, 14-4 (pages 735-737).
These values are modified by a number of factors to arrive at the allowable
bending stress σ
all
.
Where: Y
N
is the stress cycle factor (Figure 14-14, page 751)
K
T
is the temperature factor =1.0 for temp ≤250 F.
K
R
is the reliability factor (Table 14-10, page 752)
S
F
is the AGMA factor of safety
RT
N
F
t
all
KK
Y
S
S
=σ
The allowable contact strength numbers are given in Figure 14-5 (page
738) and Tables 16-6, 14-7 (pages 739-740) in your text.
These values are modified by a number of factors to arrive at the
allowable bending stress σ
c,all
.
where K
T
, K
R
are defined above and:
Z
N
is the stress cycle life factor (Figure 14-15, page 751)
S
H
is the AGMA factor of safety
C
H
is the hardness ratio factor. This is discussed in section 14-12 in your
text. For the purposes of this course, use C
H
= 1.0.
RTHKKS
HNC
allc
CZS
=
,
σ

12. Design equations for spur gears:
These are simply two separate equations that must both be used in a
gear set design.
1.Bending strength -σ= σ
all
2.Surface strength -σ
C
= σ
c,all
There are two variables to be found in the design of a gear set
consisting of a gear and a pinion, the face width F and the diametral
pitch P. The pitch P must be one of the preferred standard values (see
Table 13-2 in your text). Note that you can choose any of these pitch
values and then solve for F. You should aim for tooth proportions such
that the face width lies in the range 3p to 5p where p is the circular
pitch p = π/P inch. If you must violate this rule, do so with values less
than 3p.
You should use the bending strength equation separately for the gear
and pinion if they are made of different materials or have different heat
treatment. Otherwise, you only have to design for the pinion as the
larger gear will have a larger geometry factor J.
The surface strength calculation is a completely different case and
must always be done in addition to the bending strength
calculation. You only have to do this for the gear set member with the
lower contact strength as the calculation already incorporates the
elastic properties and geometry of both gears.