Compact Design for Non-Standard Spur Gears 1 - Scientific ...

Volume 2, Issue 1, 2011
Compact Design for Non-Standard Spur Gears
Tuan Nguyen, Graduate Student, Department of Mechanical Engineering, The University of Memphis,
tnguyen1@memphis.edu
Hsiang H. Lin, Professor, Department of Mechanical Engineering, The University of Memphis,
hlin1@memphis.edu
Abstract
This paper presents procedures for compact design of non-standard spur gear sets with the objective of
minimizing the gear size. The procedures are for long and short addendum system gear pairs that have an
equal but opposite amount of hob cutter offset applied to the pinion and gear in order to maintain the
standard center distance. Hob offset has shown to be an effective way to balance the dynamic tooth stress
of the pinion and the gear to increase load capacity. The allowable tooth stress and dynamic response are
incorporated in the process to obtain a feasible design region. Various dynamic rating factors were
investigated and evaluated. The constraints of contact stress limits and involute interference combined with
the tooth bending strength provide the main criteria for this investigation. A three-dimensional design space
involving the gear size, diametral pitch, and operating speed was developed to illustrate the optimal design
of the non-standard spur gear pairs.
Operating gears over a range of speeds creates variations in the dynamic response and changes the
required gear size in a trend that parallels the dynamic factor. The dynamic factors are strongly affected by
the system natural frequencies. The peak values of the dynamic factor within the operating speed range
significantly influence the optimal gear designs. The refined dynamic factor introduced in this study yields
more compact designs than those produced by AGMA dynamic factors.
Introduction
Designing gear transmission systems involves selecting combinations of gears to produce a desired speed
ratio. If the ratio is not unity, the gears will have different diameters. The tooth strength of the smaller gear
(the pinion) is generally weaker than that of the larger one (the gear) if both are made of the same material.
In some designs, pinions with very small tooth numbers must be used in order to satisfy the required speed
ratio and fit in the available space. This can lead to undercut of the teeth in the pinion which further reduces
strength.
One solution to the pinion design problem is to specify nonstandard gears in which the tooth profile of the
pinion is shifted outward slightly, thus increasing its strength, while the gear tooth profile may be decreased
by an equal amount. These changes in tooth proportions may be accomplished without changing operating
center distance and with standard cutting tools by withdrawing the cutting tool slightly as the pinion blank is
cut and advancing the cutter the same distance into the gear blank. This practice is called the long and
short addendum system. When gears are cut by hobs, this adjustment to the tooth proportions is called hob
cutter offset.
Several studies of non-standard gears have been performed to determine the proper hob cutter offset to
produce gear pairs of different sizes and contact ratios with similar tooth strength. Walsh and Mabie [1]
investigated the method to adjust the hob offset values for pinion and gear so that the stress in the pinion
teeth was approximately equal to that in the gear teeth. They developed hob offset charts for various velocity
ratios and for several changes in center distance. Mabie, Walsh, and Bateman [2] performed similar but
more detailed work about tooth stress equalization and compared the results with those found using either
the Lewis form factor Y or the AGMA formulation. Mabie, Rogers, and Reinholtz [3] developed a numerical
procedure to determine the individual hob offsets for a pair of gears to obtain maximum ratio of recess to
approach action, to balance tooth strength of the pinion and gear, to maintain the desired contact ratio, and
to avoid undercutting. Liou [4] tried to balance dynamic tooth stresses on both pinion and gear by adjusting
hob offset values when cutting non-standard gears of long and short addendum design.
Designing compact (minimum size) gear sets provides benefits such as minimal weight, lower material cost,
1

smaller housings, and smaller inertial loads. Gear designs must satisfy constraints, such as bending strength
limits, pitting resistance, and scoring. Many approaches for improved gear design have been proposed in
previous literature of references [5] to [16]. Previous research presented different approaches for optimal
gear design. Savage et al. [12] considered involute interference, contact stresses, and bending fatigue. They
concluded that the optimal design usually occurs at the intersection point of curves relating the tooth
numbers and diametral pitch required to avoid pitting and scoring. Wang et al. [13] expanded the model to
include the AGMA geometry factor and AGMA dynamic factor in the tooth strength formulas. Their analysis
found that the theoretical optimal gear set occurred at the intersection of the bending stress and contact
stress constraints at the initial point of contact.
More recently, the optimal design of gear sets has been expanded to include a wider range of
considerations. Andrews et al. [14] approached the optimal strength design for nonstandard gears by
calculating the hob offsets to equalize the maximum bending stress and contact stress between the pinion
and gear. Savage et al. [15] treated the entire transmission as a complete system. In addition to the gear
mesh parameters, the selection of bearing and shaft proportions were included in the design configuration.
The mathematical formulation and an algorithm are introduced by Wang et al. [16] to solve the multiobjective
gear design problem, where feasible solutions can be found in a three-dimensional solution space.
However, none of them was for compact design study.
Most of the foregoing literature dealt primarily with static tooth strength of standard gear geometries. These
studies use the Lewis formula assuming that the static load is applied at the tip of the tooth. Some
considered stress concentration and the AGMA geometry and dynamic factors. However, the operating
speed must be considered for dynamic effects. Rather than using the AGMA dynamic factor, which
increases as a simple function of pitch line velocity, a gear dynamics code by Lin et al. [5, 6, and 17] was
used here to calculate a dynamic load factor.
The purpose of the present work is to develop a design procedure for a compact size of non-standard spur
gear sets of the long and short addendum system incorporating dynamic considerations. Constraint criteria
employed for this investigation include the involute interference limits combined with the tooth bending
strength and contact stress limits.
Model Formulation
Spur Gears Cut by a Hob Cutter
The following analysis is based on the study of Mabie et al. [3]. Figure 1 shows a hob cutting a pinion where
the solid line indicates a pinion with fewer than the minimum number of teeth required to prevent
interference.
Figure 1 Spur gear tooth cut by a standard hob or a withdrawn hob.
The addendum line of the hob falls above the interference point E of the pinion, so that the flanks of the
pinion teeth are undercut. To avoid undercutting, the hob can be withdrawn a distance e, so that the
2

addendum line of the hob passes through the interference point E. This condition is shown as dotted in
Figure 1 and results in the hob cutting a pinion with a wider tooth. As the hob is withdrawn, the outside
radius of the pinion must also be increased (by starting with a larger blank) to maintain the same clearance
between the tip of the pinion tooth and the root of the hob tooth. To show the change in the pinion tooth
more clearly, the withdrawn hob in Figure 1 was moved to the right to keep the left side of the tooth profile
the same in both cases.
The width of the enlarged pinion tooth on its cutting pitch circle can be determined from the tooth space of
the hob on its cutting pitch line. From Figure 2, this thickness can be expressed by the following equation:
p
t = 2etanφ
+
( 1 )
2
Figure 2 A hob cutter with offset e for enlarged pinion tooth thickness
Equation (1) can be used to calculate the tooth thickness on the cutting pitch circle of a gear generated by a
hob offset an amount e; e will be negative if the hob is advanced into the gear blank. In Figure 1, the hob
was withdrawn just enough so that the addendum line passed through the interference point of the pinion.
The withdrawn amount can be increased or decreased as desired as long as the pinion tooth does not
become undercut or pointed. The equation that describes the relationship between e and other tooth
geometry is
Therefore,
e = AB + OA - OP
k
= + Rb
cosφ
- R p
Pd
e= k
P - 2
R p(1-
)
d
e= 1
P
d
(k- N
2
( 2 )
cos φ ( 3 )
)
2
sin φ ( 4 )
Two equations that were developed from involutometry find particular application in this study:
3

cos = cos
R A
φBφ A
( 5 )
RB
t B=2R B(
t A
2 R
A
+inv -inv )
φA φ B
( 6 )
By means of these equations, the pressure angle and tooth thickness at any radius RB can be found if the
pressure angle and tooth thickness are known at a reference radius R A. This reference radius is the cutting
pitch radius, and the tooth thickness on this cutting pitch circle can be easily calculated for any cutter offset.
The reference pressure angle is the pressure angle of the hob cutter.
When two gears, gear 1 and gear 2, which have been cut with a hob offset e 1 and e 2, respectively, are
meshed together, they operate on pitch circles of radii R ' 1 and R ' 2 and at pressure angle φ. The thickness of
the teeth on the operating pitch circles can be expressed as t ' 1 and t ' 2, which can be calculated from
Equation (6). These dimensions are shown in Figure 3 together with the thickness of the teeth t 1 and t 2 on
the cutting pitch circles of radii R 1 and R 2.
Figure 3 Non-standard gears cut by offset hob cutters and operated at new pressure angle φ’.
To determine the pressure angle φ at which these gears will operate, we found
and
ω2
=
ω
t
'
1
1
t
'
2
N 1 =
N
2
2πR
=
N
R
R
1
'
1
'
1
'
2
2πR
=
N
2
'
2
+ ( 8 )
Substituting Eq. (6) into Eq. (8) and dividing by 2R ' 1,
4
( 7 )

1
[
t
+ ( inv
2 R1
Rearranging this equation gives,
t
2R
1
R
'
' 2 2
'
φ - invφ
) ] [
t
+ ( inv φ - invφ
) =
'
R1
2 R2
π
N
+ ( 9 )
'
'
R2
t2
π R2
'
+ = + ( 1 + )( invφ
− inv )
( 10 )
'
'
R 2R
N R
1 φ
1
2
1
By substituting Eq. (7) and 2R = N/Pd into the above equation and multiplying by N 1/P d ,
π
1 2
t 1+ t 2=
+
P d d
N + N
By substituting Eq. (1) for t1 and t 2,
2 e
P
1
(inv -inv )
φ′ φ (11)
p
p π N 1+
N 2
φ + + 2 e2
tan φ + = + (invφ
′ - inv φ )
2
2 Pd
Pd
1 tan ( 12 )
Simplifying the equation above gives,
2 (e + e )+ p= +
P N π + N
tan φ 1 2
d Pd
By substituting p = π /Pd and solving for inv φ ' ,
or
1 2
inv = inv + 2 P d( e 1+ e 2)
tan φ
φ′ φ
N 1+ N 2
( N1
+ N 2 )(invφ
′ - inv φ )
e1+
e2
=
2 Pd
tan φ
(inv -inv )
Expressing the above equation in metric unit with m as module,
1 2
e 1+ e 2=
m ( N + N )(inv φ′ -inv φ )
2
tan φ
1
φ′ φ ( 13 )
In this study, we limit our investigation to the case that the hob cutter is advanced into the gear blank the
same amount that it is withdrawn from the pinion, therefore, e2 = -e 1 and, from Eq. (15) or (16), φ ‘ = φ.
Because there is no change in the pressure angle, R ' 1 = R 1 and R ' 2 = R 2, and the gears operate at the
standard center distance. If the offsets are unequal (e2 ≠ -e 1), then the center distance must change. This
problem is not considered in the present investigation.
Objective Function
The design objective of this study is to obtain the most compact gear set of the long and short addendum
system satisfying design requirements that include loads and power level, gear ratio and material
parameters. The gears designed must satisfy operational constraints such as avoiding interference, pitting,
scoring distress and tooth breakage. The required gear center distance C is the chosen parameter to be
optimized.
where
C = R + R
p1
p2
5
( 14 )
( 15 )
( 16 )
(17)

R p1 pitch radius of gear 1
R p2 pitch radius of gear 2
Design Parameters and Variables
The following table lists the parameters and variables used in this study:
Design Constraints
Table 1 Basic gear design parameters and variables
Gear Parameters Design Variables
Bending strength and contact strength Number of pinion teeth
limits
Diametral pitch
Operating torque
Operating speed
Gear ratio
Face width
Pressure angle
Hob cutter offset
Involute Interference Involute interference is defined as a condition in which there is an obstruction on the
tooth surface that prevents proper tooth contact (Townsend et al. [18]); or contact between portions of tooth
profiles that are not conjugate (South et al. [19]). Interference occurs when the driven gear contacts a noninvolute
portion (below the base circle) of the driving gear. Undercutting occurs during tooth generation if the
cutting tool removes the interference portion of the gear being cut. An undercut tooth is weaker, less
resistant to bending stress, and prone to premature tooth failure. DANST has a built-in routine to check for
interference.
Bending Stress Tooth bending failure at the root is a major concern in gear design. If the bending stress
exceeds the fatigue strength, the gear tooth has a high probability of failure. The AGMA bending stress
equation can be found in the paper by Savage et al. [13] and in other gear literature. In this study, a modified
Heywood formula for tooth root stress was used. This formula correlates well with the experimental data and
finite element analysis results (Cornell et al. [20]):
Wj
cos β j hf
6l f . h
tan β
07 .
072 L
j
σ j = [ 1+ 026 . ( ) ][ + ( 1 − νtan β j ) − ]
2
F
2R
h h l h h
where
σ root bending stress at loading position j.
j
Wj transmitted load at loading position j.
β load angle, degree
j
F face width of gear tooth, inch
ν approximately 1/4, according to Heywood [21].
Rf fillet radius, inch
f
f f f
other nomenclature is defined in Figure 4 and References [20] and [21]. To avoid tooth failure, the bending
stress should be limited to the allowable bending strength of the material as suggested by AGMA [22],
St K L
σ j ≤ σall
= (19)
K K
T R
where
σ all allowable bending stress
S t AGMA bending strength
K L life factor
K T temperature factor
6
f
f
(18)

K R reliability factor
Kv dynamic factor
Figure 4 Tooth geometry for stress calculations using modified Heywood formula.
Surface Stress The surface failure of gear teeth is an important concern in gear design. Surface failure
modes include pitting, scoring, and wear. Pitting is a gear tooth failure in the form of tooth surface cavities as
a result of repeated stress applications. Scoring is another surface failure that usually results from high loads
or lubrication problems. It is defined as the rapid removal of metal from the tooth surface caused by metal
contact due to high overload or lubricant failure. The surface is characterized by a ragged appearance with
furrows in the direction of tooth sliding (Shigley et al. [23]). Wear is a fairly uniform removal of material from
the tooth surface.
The stresses on the surface of gear teeth are determined by formulas derived from the work of Hertz [18].
The Hertzian contact stress between meshing teeth can be expressed as
σ
Hj
=
+
Wj
β j ρ ρ
π F φ − ν ν
+
E E
−
⎡ 1 1 ⎤
cos
⎢
⎥
⎢ 1 2 ⎥
2
2
cos ⎢ 1 1 1 2 ⎥
⎣⎢
⎦⎥
where
σ contact stress at loading position j
Hj
Wj transmitted load at loading position j.
βj load angle, degree
F face width of gear tooth, inch
φ pressure angle, degree
ρ radius of curvature of gear 1,2 at the point of contact, inch
1,2
ν Poisson’s ratio of gear 1,2
1,2
E modulus of elasticity of gear 1,2, psi
1,2
1
2
The AGMA recommends that this contact stress should also be considered in a similar manner as the
bending endurance limit (Shigley et al. [22]). The equation is
7
(20)

σ ≤ σ = S
C C
Hj c, all C
L H
CTCR (21)
where
σc,all allowable contact stress
Sc AGMA surface fatigue strength
CL life factor
CH hardness-ratio factor
CT temperature factor
CR reliability factor
According to Savage et al. [12], Hertzian stress is a measure of the tendency of the tooth surface to develop
pits and is evaluated at the lowest point of single tooth contact rather than at the less critical pitch point as
recommended by AGMA. Gear tip scoring failure is highly temperature dependent (Shigley et al. [23]) and
the temperature rise is a direct result of the Hertz contact stress and relative sliding speed at the gear tip.
Therefore, the possibility of scoring failure can be determined by Equation (20) with the contact stress
evaluated at the initial point of contact. A more rigorous method not used here is the PVT equation or the
Blok scoring equation (South et al. [18]).
Dynamic Load Effect One of the major goals of this work is to study the effect of dynamic load on optimal
gear design. The dynamic load calculation is based on the NASA gear dynamics code DANST developed by
the authors. DANST has been validated with experimental data for high-accuracy gears at NASA Lewis
Research Center (Oswald et al. [24]). DANST considers the influence of gear mass, meshing stiffness, tooth
profile modification, and system natural frequencies in its dynamic calculations.
The dynamic tooth load depends on the value of relative dynamic position and backlash of meshing tooth
pairs. After the gear dynamic load is found, the dynamic load factor can be determined by the ratio of the
maximum gear dynamic load during mesh to the applied load. The applied load equals the torque divided by
the base circle radius. This ratio indicates the relative instantaneous gear tooth load. Compact gears
designed using the dynamic load calculated by DANST will be compared with gears designed using the
AGMA suggested dynamic factor, which is a simple function of the pitch line velocity.
Gear Design Application
Design Algorithm
An algorithm was developed to perform the analyses and find the optimum design of non-standard gears of
the long and short addendum system (LASA). The computer algorithm of the dynamic optimal design can
start with the pinion tooth number loop. Within each pinion tooth number loop, all of the basic parameters are
needed in dynamic analysis, including the diametral pitch, addendum ratio, dedendum ratio, gear material
properties, transmitted power (torque), face width, and hob offset value. The DANST code also calculates
additional parameters that are needed in dynamic analysis, such as base circle radius, pitch circle radius,
addendum circle radius, stiffness, and the system natural frequencies, etc. One of the system natural
frequencies (the one that results in the highest dynamic stresses) will be used as pinion operating speed.
For this study, the diametral pitch was varied from two to twenty. Static analysis was first performed to check
for involute interference and to calculate the meshing stiffness variations and static transmission errors of the
gear pair. If there was a possibility of interference, the number of pinion teeth was increased by one and the
static process was repeated. The number of gear teeth was also increased according to the specified gear
ratio. Results from the static analyses were incorporated in the equations of motion of the gear set to obtain
the dynamic motions of the system. Instantaneous dynamic load at each contact point along the tooth profile
was determined from these motions. The varying contact stresses and root bending stresses during the gear
mesh were calculated from these dynamic loads.
If all the calculated stresses were less than the design stress limits for a possible gear set, the data for this
set were added to a candidate group. At each value of the diametral pitch, the most compact gear set in the
candidate group would have the smallest center distance. These different candidate designs can be
compared in a table or graph to show the optimum design from all the sets studied.
The analyses above are for gears operating at a single speed. To examine the effect of varying speed, the
analyses can be repeated at different speeds. The program will stop when all the stresses are below the
8

design constraints. Thus, a feasible optimal gear set operating at the system natural frequency can be
obtained. As the diametral pitch varies, the optimal gear sets determined from each diametral pitch value are
collected to form the optimal design space. This design space, when transferred into the center distance
space, which is our merit function of optimal design, will indicate all the feasible optimal gear sets to be
selected from with given operating parameters.
Design Applications
Table 2 shows the basic gear parameters for the sample gear sets to be studied. They were first used in a
gear design problem by Shigley and Mitchell [19], and later used by Carroll and Johnson [13] as an example
for optimal design of compact gear sets. The sample gear set transmits 100 horsepower at an input speed of
1120 rpm. The gear set has standard full depth teeth and a speed reduction ratio of 4. In this study, the face
width of the gear is always chosen to be one-half the pinion pitch diameter. In other words, the length to
diameter ratio (λ) is 0.5.
Table 2 Basic design parameters of sample gear sets
Pressure Angle, φ (degree) 20
Gear Ratio, M g 4.0
Length to Diameter Ratio, λ 0.5
Transmitted Power (hp) 100.0
Applied Torque (lb-in) 5627.264
Input Speed (rpm) 1120.0
Modulus of Elasticity (Mpsi) 30
Poisson’s Ratio, ν 0.3
Scoring and Pitting Stress Limits, S S and S P (Kpsi) 79.23
Bending Stress Limit, S b (Kpsi) 19.81
Hob Cutter Offset (in) 0.03, 0.05, 0.07, 0.09
Cutting gears with offset hobs is an effective way to balance the dynamic tooth strength of the pinion and
gear. It reduces the dynamic stress in the pinion and increases stress in the gear to achieve balance. In
general, increasing the offset improves the balance in dynamic tooth strength. However, the best hob offset
varies with the transmission speed and is limited by the maximum allowable offset that renders the pinion
tooth pointed. The hob offset amount is varied from 0.03 inch to 0.09 inch with an increment of 0.02 inch in
this investigation.
Table 3 displays Carroll’s [13] optimal design results for the sample gears of standard system without hob
cutter offset. The optimal design is indicated in bold type and by an arrow. In the table, P d is the diametral
pitch, NT1 and NT2 represent the number of teeth of pinion and gear, respectively, CD is the center
distance, FW is the face width, CR is the contact ratio, and S b, S S, and S P are the stress limit for bending,
scoring, and pitting, respectively.
Table 3 Carroll’s optimization results of sample standard gear set [13]
(Using Lewis tooth stress formula)
Pd NT1 NT2 CD FW CR Sb S s S p
2.00 19 76 23.750 4.750 1.681 2.872 72.497 51.396
2.25 20 80 22.222 4.444 1.691 3.553 72.162 55.853
2.50 21 84 21.000 4.200 1.701 4.275 72.532 59.950
3.00 23 92 19.167 3.833 1.717 5.820 74.100 67.213
4.00 27 108 16.875 3.375 1.745 9.202 77.876 78.860
6.00 40 160 16.667 3.333 1.805 12.703 66.972 78.309
8.00 53 212 16.563 3.313 1.840 16.191 63.302 78.247
10.00 66 264 16.500 3.300 1.863 19.670 61.463 78.285 ←
12.00 86 344 17.917 3.583 1.887 19.727 53.219 69.603
16.00 132 528 20.625 4.125 1.917 19.726 42.459 57.137
20.00 185 740 23.125 4.625 1.934 19.742 35.708 48.760
The theoretical optimum for this example occurs at the intersection of bending stress and contact stress
constraint curves at the lowest point of single tooth contact. This creates a gear set that has NT1 = 64 and
P d = 9.8 for a theoretical center distance of 16.333 in. The minimum practical center distance (16.50 in.) is
9

obtained when NT1 = 66 and P d = 10.0.
For comparison with the above results, we used the same AGMA dynamic factor K v (Equation 6) but with the
modified Heywood tooth bending stress formula (Eq. 2) in the calculations. Table 4 lists the optimization
results obtained.
Table 4 Optimization results of sample standard gears
(Using modified Heywood tooth stress formula)
Pd NT1 NT2 CD FW CR Sb S s S p
2.00 19 76 23.750 4.750 1.681 2.847 66.873 52.390
2.25 19 76 21.111 4.222 1.681 3.935 78.539 61.589
2.50 20 80 20.000 4.000 1.691 4.718 76.864 65.812
3.00 22 88 18.333 3.667 1.709 6.384 76.054 73.234
4.00 28 112 17.500 3.500 1.751 8.466 66.656 76.216
6.00 41 164 17.083 3.417 1.808 12.215 58.956 77.066
8.00 54 216 16.875 3.375 1.842 15.785 56.336 77.689
10.00 67 268 16.750 3.350 1.865 19.369 55.015 78.137 ←
12.00 87 348 18.125 3.625 1.888 19.637 47.915 69.817
16.00 134 536 20.938 4.188 1.918 19.638 38.076 57.045
20.00 188 752 23.500 4.700 1.935 19.718 32.006 48.616
As can be seen from the table, the minimum practical center distance (16.750 in.) is obtained when NT1 =
67 and P d = 10.0. This is very close to Carroll’s design but his optimal gear set will exceed the design limit of
19.81 Kpsi (from Table 2) for maximum bending stress on the pinion according to our calculations. The
differences between Carroll’s results and those reported here are likely due to the use of different formulas
for bending stress calculations.
Figure 5 shows graphically the design space for the results presented in Table 4, depicting the stress
constraint curves of bending, scoring, and pitting. The region above each constraint curve indicates feasible
design space for that particular constraint. In the figure, the theoretical optimum is located at the intersection
point of the scoring stress and the bending stress constraint.
Figure 5 Design space for determination of optimized results of sample standard gears using modified
Heywood tooth stress formula.
10

Table 5 displays the data for the optimum compact design of the sample non-standard gears (long and short
addendum system) with a hob offset of 0.03 inch. The practical minimum center distance is equal to 15.000
inches and occurs at two values of P d = 10.00 and 12.00, with NT1 = 60 and 72. The practical minimum
center distance decreases as the diametral pitch P d increases from 2.00 to 10.00; it then increases as the
diametral pitch P d increases from 12.00 to 20.00.
Table 5 Optimization results for non-standard gears (LASA) with 0.03 inch hob offset
Pd NT1 NT2 CD FW CR Sb S s S p
2.00 20 80 25.000 5.000 1.691 2.684 71.166 23.861
3.00 25 100 20.833 4.167 1.732 5.325 74.722 27.375
4.00 30 120 18.750 3.750 1.762 8.534 79.217 25.057
6.00 39 156 16.250 3.250 1.801 12.526 78.846 60.440
8.00 50 200 15.625 3.125 1.833 16.074 76.410 63.863
10.00 60 240 15.000 3.000 1.854 21.053 78.169 67.826←
12.00 72 288 15.000 3.000 1.872 24.637 76.021 68.245←
16.00 107 428 16.719 3.344 1.904 25.312 61.583 57.990
20.00 151 604 18.875 3.775 1.924 25.647 51.281 49.269
In Table 6, when the hob offset is increased to 0.05 inch, the optimum compact design occurs at P d = 10.0
with a practical minimum center distance of 15.750 inch and NT1 = 63,. As can be seen from the table, the
minimum center distance decreases as the diametral pitch increases from 2.00 to 10.00. When the diametral
pitch is greater than 10, the minimum center distance increases with the diametral pitch value.
Table 6 Optimization results for non-standard gears with 0.05 inch hob offset
P d NT1 NT2 CD FW CR S b S s S p
2.00 19 76 23.750 4.750 1.681 3.005 78.361 26.984
3.00 25 100 20.833 4.167 1.732 5.296 73.095 29.197
4.00 30 120 18.750 3.750 1.762 8.611 77.748 30.469
6.00 42 168 17.500 3.500 1.811 14.076 78.573 30.307
8.00 52 208 16.250 3.250 1.838 17.460 75.663 61.022
10.00 63 252 15.750 3.150 1.859 23.580 78.122 65.023←
12.00 79 316 16.458 3.292 1.880 25.628 71.270 62.258
16.00 127 508 19.844 3.969 1.915 25.293 54.128 49.583
20.00 196 784 24.500 4.900 1.936 25.309 41.570 38.476
The optimization results for hob offsets of 0.07 and 0.09 inches are very similar. For both cases the practical
minimum center distance is equal to 17.500 and occurs at P d = 6.00 and 8.00, with NT1 = 42 and 56,
respectively. Minimum center distance decreases as the diametral pitch increases from 2.00 to 6.00; it then
increases as diametral pitch increases from 8.00. At P d = 6.00 and 8.00, the minimum center distance for
both cases is unchanged. Some higher diametral pitch values do not produce optimization results due to the
pointed teeth.
Comparisons between non-standard gear designs at different hob offset values are shown in Figures 6(a)
and 6(b). The compact design improves significantly as P d value increases from 2.00 to 4.00 for nonstandard
gears and from 2.00 to 6.00 for standard gears.
11

Pinion Tooth Number
Center Distance (in.)
210
190
170
150
130
110
9
7
5
3
1
29
27
25
23
21
19
17
15
13
11
9
7
5
Feasible
Design
2.00 3.00 4.00 6.00 8.00 10.00 12.00 16.00 20.00
Hob Offset
Diametral Pitch
0.03 in 0.05 in 0.07 in
(a)
Feasible
Design
0.09 in
2.00 3.00 4.00 6.00 8.00 10.00 12.00 16.00 20.00
Diametral
Hob Offset 0.03 in 0.05 in 0.07 in 0.09 in
Figure 6 Design space and optimal gear sets for standard and non-standard gears, Gear ratio M g = 4,�
pressure angle φ = 20 0 .
(b)
12

At diametral pitch values less than or equal to 3.00, non-standard gears with hob offset values of 0.07 inches
and 0.09 inches appear to be more compact than standard gears while non-standard gears with hob offset
values of 0.03 inches and 0.05 inches have the same center distance as standard gears (except nonstandard
gears pair at P d = 2.00 with hob offset 0.05 inch that has slightly smaller center distance than
standard gears pair).
At a diametral pitch value of 4.00, non-standard gears and standard gears have the same compact design.
At diametral pitch that is greater than 6.00, standard gears appear more compact than non-standard gears.
The difference is extremely large at high diametral pitch values and high hob offset values. Comparisons
between non-standard gears indicate that the curves in this study are almost identical to the one in Study 8.
Therefore, similar conclusions are derived in this study.
From Figure 6(b), hob offset values of 0.07 inches and 0.09 inches seem to result in better compact design
than hob offsets of 0.03 inches and 0.05 inches when diametral pitch is less than or equal to 4.00. With
diametral pitches that are greater than 4.00, non-standard gears with hob offset values of 0.07 inches and
0.09 inches become less compact when compared to non-standard gears at smaller hob offset values.
Generally, at a diametral pitch value of 4.00 or less, increasing the hob offset value will result in a more
compact design. When the diametral pitch is greater than 4.00, increasing the hob offset value will result in a
larger gears center distance or a less compact design. When the diametral pitch value is high (greater than
or equal to 10.00), non-standard gears at higher hob offsets (0.07 inches and 0.09 inches) seem very
sensitive to subject load which result in a much larger gear pair. Thus, the degree of sensitivity increases
with hob offset value at high diametral pitch values. As can be seen in the plots, because of the sensitivity in
geometry, we could not obtain results for non-standard gears with hob offset values of 0.07 inches at P d =
20.00 and 0.09 inches at P d = 16.00 and 20.00.
Conclusions
This paper presents a method for optimal design of standard spur gears for minimum dynamic response. A
study was performed using a sample gear set from the gear literature. Optimal gear sets were compared for
designs based on the AGMA dynamic factor and a refined dynamic factor calculated using the DANST gear
dynamics code. The operating speed was varied over a broad range to evaluate its effect on the required
gear size. A design space for designing optimal compact gear sets can be developed using the current
approach.
The required size of an optimal gear set is significantly influenced by the dynamic factor. The peak dynamic
factor at system natural frequencies can dominate the design of optimal gear sets that operate over a wide
range of speeds. In the current investigation, refined dynamic factors calculated by the dynamic gear code
developed by the authors allow a more compact gear design than that obtained by using the AGMA dynamic
factor values.
Compact gears designed using the modified Heywood tooth stress formula are similar to those designed
using the simpler Lewis formula for the example cases studied here. Design charts such as those shown
here can be used for a single speed or over a range of speeds. For the sample gears in the study, a
diametral pitch of 10.0 was found to provide a compact gear set over the speed range considered.
13

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15