Long Addendum Gears 1946

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Volume 2, Issue 1, 2011Compact Design for Non-Standard Spur Gears

Tuan Nguyen, Graduate Student, Department of Mechanical Engineering, The University of Memphis,[email protected] H. Lin, Professor, Department of Mechanical Engineering, The University of Memphis,[email protected]

Abstract

This paper presents procedures for compact design of non-standard spur gear sets with the objective ofminimizing the gear size. The procedures are for long and short addendum system gear pairs that have anequal but opposite amount of hob cutter offset applied to the pinion and gear in order to maintain thestandard center distance. Hob offset has shown to be an effective way to balance the dynamic tooth stressof the pinion and the gear to increase load capacity. The allowable tooth stress and dynamic response areincorporated in the process to obtain a feasible design region. Various dynamic rating factors wereinvestigated 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 spaceinvolving the gear size, diametral pitch, and operating speed was developed to illustrate the optimal designof the non-standard spur gear pairs.

Operating gears over a range of speeds creates variations in the dynamic response and changes therequired gear size in a trend that parallels the dynamic factor. The dynamic factors are strongly affected bythe system natural frequencies. The peak values of the dynamic factor within the operating speed rangesignificantly influence the optimal gear designs. The refined dynamic factor introduced in this study yieldsmore compact designs than those produced by AGMA dynamic factors.

Introduction

Designing gear transmission systems involves selecting combinations of gears to produce a desired speedratio. 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 speedratio and fit in the available space. This can lead to undercut of the teeth in the pinion which further reducesstrength.

One solution to the pinion design problem is to specify nonstandard gears in which the tooth profile of thepinion is shifted outward slightly, thus increasing its strength, while the gear tooth profile may be decreasedby an equal amount. These changes in tooth proportions may be accomplished without changing operatingcenter distance and with standard cutting tools by withdrawing the cutting tool slightly as the pinion blank iscut and advancing the cutter the same distance into the gear blank. This practice is called the long andshort addendum system. When gears are cut by hobs, this adjustment to the tooth proportions is called hobcutter offset.

Several studies of non-standard gears have been performed to determine the proper hob cutter offset toproduce 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 pinionteeth was approximately equal to that in the gear teeth. They developed hob offset charts for various velocityratios and for several changes in center distance. Mabie, Walsh, and Bateman [2] performed similar butmore detailed work about tooth stress equalization and compared the results with those found using eitherthe Lewis form factor Y or the AGMA formulation. Mabie, Rogers, and Reinholtz [3] developed a numericalprocedure to determine the individual hob offsets for a pair of gears to obtain maximum ratio of recess toapproach action, to balance tooth strength of the pinion and gear, to maintain the desired contact ratio, andto avoid undercutting. Liou [4] tried to balance dynamic tooth stresses on both pinion and gear by adjustinghob 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,

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smaller housings, and smaller inertial loads. Gear designs must satisfy constraints, such as bending strengthlimits, pitting resistance, and scoring. Many approaches for improved gear design have been proposed inprevious literature of references [5] to [16]. Previous research presented different approaches for optimalgear design. Savage et al. [12] considered involute interference, contact stresses, and bending fatigue. Theyconcluded that the optimal design usually occurs at the intersection point of curves relating the toothnumbers and diametral pitch required to avoid pitting and scoring. Wang et al. [13] expanded the model toinclude 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 contactstress constraints at the initial point of contact.

More recently, the optimal design of gear sets has been expanded to include a wider range ofconsiderations. Andrews et al. [14] approached the optimal strength design for nonstandard gears bycalculating the hob offsets to equalize the maximum bending stress and contact stress between the pinionand gear. Savage et al. [15] treated the entire transmission as a complete system. In addition to the gearmesh 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 multi-objective 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. Thesestudies use the Lewis formula assuming that the static load is applied at the tip of the tooth. Someconsidered 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, whichincreases as a simple function of pitch line velocity, a gear dynamics code by Lin et al. [5, 6, and 17] wasused 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 spurgear sets of the long and short addendum system incorporating dynamic considerations. Constraint criteriaemployed for this investigation include the involute interference limits combined with the tooth bendingstrength 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 wherethe solid line indicates a pinion with fewer than the minimum number of teeth required to preventinterference.

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 thepinion teeth are undercut. To avoid undercutting, the hob can be withdrawn a distance e, so that the

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addendum line of the hob passes through the interference point E. This condition is shown as dotted inFigure 1 and results in the hob cutting a pinion with a wider tooth. As the hob is withdrawn, the outsideradius of the pinion must also be increased (by starting with a larger blank) to maintain the same clearancebetween the tip of the pinion tooth and the root of the hob tooth. To show the change in the pinion toothmore clearly, the withdrawn hob in Figure 1 was moved to the right to keep the left side of the tooth profilethe same in both cases.

The width of the enlarged pinion tooth on its cutting pitch circle can be determined from the tooth space ofthe hob on its cutting pitch line. From Figure 2, this thickness can be expressed by the following equation:

2

p+2etan=t ( 1 )

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 ahob offset an amount e; e will be negative if the hob is advanced into the gear blank. In Figure 1, the hobwas 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 notbecome undercut or pointed. The equation that describes the relationship between e and other toothgeometry is

R-R+P

k=

OP-OA+AB=e

pb

d

cos

( 2 )

Therefore,

e =k

P-R (1- )

d

p2

cos ( 3 )

e =1

P( k -

N

2)

d

2sin ( 4 )

Two equations that were developed from involutometry find particular application in this study:

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cos cos=R

R

B

A

B

A ( 5 )

B BA

A

At = 2R (t

2B

R+inv -inv ) ( 6 )

By means of these equations, the pressure angle and tooth thickness at any radius RB can be found if thepressure angle and tooth thickness are known at a reference radius RA. This reference radius is the cuttingpitch 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 e1 and e2, respectively, aremeshed together, they operate on pitch circles of radii R

'1 and R

'2 and at pressure angle . 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 t1 and t2 onthe cutting pitch circles of radii R1 and R2.

Figure 3 Non-standard gears cut by offset hob cutters and operated at new pressure angle .

To determine the pressure angle at which these gears will operate, we found

'

2

'

1

2

1

1

2

R

R

N

N==

( 7 )

and

2

'

2

1

'

1'

2

'

1

22

N

R

N

Rtt

==+ ( 8 )

Substituting Eq. (6) into Eq. (8) and dividing by 2R'1,

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1

'

2

2

'

1

'

2'

1

1 ([]([N

)inv-inv+R2

t

R

R)inv-inv+

R2

t =+ ( 9 )

Rearranging this equation gives,

))(1(22'

'

1

'

2

12

2'

1

'

2

1

1

invinvR

R

NR

t

R

R

R

t

++=+ ( 10 )

By substituting Eq. (7) and 2R = N/Pd into the above equation and multiplying by N1/Pd ,

1 2

d

1 2

d

t + t =P

+N +N

P(inv - inv )

(11)

By substituting Eq. (1) for t1 and t2,

)inv-(invP

N+N+

P=

2

p+e2+

2

p+e2

d

21

d

21

tantan ( 12 )

Simplifying the equation above gives,

2 ( e + e ) + p =P

+N +N

P(inv -inv )1 2

d

1 2

d

tan

( 13 )

By substituting p = /Pd and solving for inv ',

inv = inv +2 P (e + e )

N +N

d 1 2

1 2

tan

( 14 )

or

P2

)inv-)(invN+N(=e+e

d

2121

tan

( 15 )

Expressing the above equation in metric unit with m as module,

1 21 2

e + e =m (N +N )( inv - inv )

2

tan

( 16 )

In this study, we limit our investigation to the case that the hob cutter is advanced into the gear blank thesame amount that it is withdrawn from the pinion, therefore, e2 = -e1 and, from Eq. (15) or (16),

= .

Because there is no change in the pressure angle, R'1 = R1 and R

'2 = R2, and the gears operate at the

standard center distance. If the offsets are unequal (e2 -e1), then the center distance must change. Thisproblem 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 addendumsystem satisfying design requirements that include loads and power level, gear ratio and materialparameters. 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 beoptimized.

(17)C R Rp= +1 p2where

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Rp1

pitch radius of gear 1

Rp2

pitch radius of gear 2

Design Parameters and Variables

The following table lists the parameters and variables used in this study:

Table 1 Basic gear design parameters and variables

Gear Parameters Design Variables

Bending strength and contact strengthlimitsOperating torqueGear ratioFace widthPressure angleHob cutter offset

Number of pinion teethDiametral pitchOperating speed

Design Constraints

Involute Interference Involute interference is defined as a condition in which there is an obstruction on thetooth surface that prevents proper tooth contact (Townsend et al. [18]); or contact between portions of toothprofiles that are not conjugate (South et al. [19]). Interference occurs when the driven gear contacts a non-involute portion (below the base circle) of the driving gear. Undercutting occurs during tooth generation if thecutting tool removes the interference portion of the gear being cut. An undercut tooth is weaker, lessresistant to bending stress, and prone to premature tooth failure. DANST has a built-in routine to check forinterference.

Bending Stress Tooth bending failure at the root is a major concern in gear design. If the bending stressexceeds the fatigue strength, the gear tooth has a high probability of failure. The AGMA bending stressequation can be found in the paper by Savage et al. [13] and in other gear literature. In this study, a modifiedHeywood formula for tooth root stress was used. This formula correlates well with the experimental data andfinite element analysis results (Cornell et al. [20]):

jj j f

f

f

f f f

L

f

jj

f

WF

hR

lh h l

hh h

= + + cos [ . ( ) ] [ . ( tan ) tan ].1 0 262

6 0 72 10 72

(18)

where

j root bending stress at loading position j.

Wj transmitted load at loading position j.

j load angle, degree

F face width of gear tooth, inch

approximately 1/4, according to Heywood [21].Rf fillet radius, inch

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],

j allt L

T R

S K

K K = (19)

where

all allowable bending stress

St AGMA bending strength

KL life factor

KT temperature factor

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KR 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 failuremodes include pitting, scoring, and wear. Pitting is a gear tooth failure in the form of tooth surface cavities asa result of repeated stress applications. Scoring is another surface failure that usually results from high loadsor lubrication problems. It is defined as the rapid removal of metal from the tooth surface caused by metalcontact due to high overload or lubricant failure. The surface is characterized by a ragged appearance withfurrows in the direction of tooth sliding (Shigley et al. [23]). Wear is a fairly uniform removal of material fromthe 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

Hjj jW

F

E E

=+

+

cos

cos

1 1

1 1

1 2

1

2

1

2

2

2

(20)

where

Hj contact stress at loading position j

Wj transmitted load at loading position j.

j load angle, degree

F face width of gear tooth, inch

pressure angle, degree

1,2 radius of curvature of gear 1,2 at the point of contact, inch

1,2 Poissons ratio of gear 1,2

E1,2

modulus of elasticity of gear 1,2, psi

The AGMA recommends that this contact stress should also be considered in a similar manner as thebending endurance limit (Shigley et al. [22]). The equation is

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Hj c all CL H

T R

SC C

C C =, (21)

where

c,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 developpits and is evaluated at the lowest point of single tooth contact rather than at the less critical pitch point asrecommended by AGMA. Gear tip scoring failure is highly temperature dependent (Shigley et al. [23]) andthe 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 stressevaluated at the initial point of contact. A more rigorous method not used here is the PVT equation or theBlok 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 bythe authors. DANST has been validated with experimental data for high-accuracy gears at NASA LewisResearch Center (Oswald et al. [24]). DANST considers the influence of gear mass, meshing stiffness, toothprofile 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 toothpairs. After the gear dynamic load is found, the dynamic load factor can be determined by the ratio of themaximum gear dynamic load during mesh to the applied load. The applied load equals the torque divided bythe base circle radius. This ratio indicates the relative instantaneous gear tooth load. Compact gearsdesigned 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 Algor ithm

An algorithm was developed to perform the analyses and find the optimum design of non-standard gears ofthe long and short addendum system (LASA). The computer algorithm of the dynamic optimal design canstart with the pinion tooth number loop. Within each pinion tooth number loop, all of the basic parameters areneeded in dynamic analysis, including the diametral pitch, addendum ratio, dedendum ratio, gear materialproperties, transmitted power (torque), face width, and hob offset value. The DANST code also calculatesadditional 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 naturalfrequencies (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 checkfor involute interference and to calculate the meshing stiffness variations and static transmission errors of thegear pair. If there was a possibility of interference, the number of pinion teeth was increased by one and thestatic process was repeated. The number of gear teeth was also increased according to the specified gearratio. Results from the static analyses were incorporated in the equations of motion of the gear set to obtainthe dynamic motions of the system. Instantaneous dynamic load at each contact point along the tooth profilewas determined from these motions. The varying contact stresses and root bending stresses during the gearmesh 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 thisset were added to a candidate group. At each value of the diametral pitch, the most compact gear set in thecandidate group would have the smallest center distance. These different candidate designs can becompared 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, theanalyses can be repeated at different speeds. The program will stop when all the stresses are below the

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design constraints. Thus, a feasible optimal gear set operating at the system natural frequency can beobtained. As the diametral pitch varies, the optimal gear sets determined from each diametral pitch value arecollected to form the optimal design space. This design space, when transferred into the center distancespace, which is our merit function of optimal design, will indicate all the feasible optimal gear sets to beselected 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 agear design problem by Shigley and Mitchell [19], and later used by Carroll and Johnson [13] as an examplefor optimal design of compact gear sets. The sample gear set transmits 100 horsepower at an input speed of1120 rpm. The gear set has standard full depth teeth and a speed reduction ratio of 4. In this study, the facewidth of the gear is always chosen to be one-half the pinion pitch diameter. In other words, the length to

diameter ratio () is 0.5.

Table 2 Basic design parameters of sample gear sets

Pressure Angle, (degree) 20Gear Ratio, Mg 4.0

Length to Diameter Ratio, 0.5Transmitted Power (hp) 100.0

Applied Torque (lb-in) 5627.264

Input Speed (rpm) 1120.0Modulus of Elasticity (Mpsi) 30

Poissons Ratio, 0.3Scoring and Pitting Stress Limits, SS and SP (Kpsi) 79.23

Bending Stress Limit, Sb (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 andgear. It reduces the dynamic stress in the pinion and increases stress in the gear to achieve balance. Ingeneral, increasing the offset improves the balance in dynamic tooth strength. However, the best hob offsetvaries with the transmission speed and is limited by the maximum allowable offset that renders the piniontooth pointed. The hob offset amount is varied from 0.03 inch to 0.09 inch with an increment of 0.02 inch inthis investigation.

Table 3 displays Carrolls [13] optimal design results for the sample gears of standard system without hobcutter offset. The optimal design is indicated in bold type and by an arrow. In the table, Pd is the diametral

pitch, NT1 and NT2 represent the number of teeth of pinion and gear, respectively, CD is the centerdistance, FW is the face width, CR is the contact ratio, and Sb, SS,and SP are the stress limit for bending,scoring, and pitting, respectively.

Table 3 Carrolls opt imization results of sample standard gear set [13]

(Using Lewis too th stress formula)

Pd NT1 NT2 CD FW CR Sb Ss Sp2.00 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 78.28512.00 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 stressconstraint curves at the lowest point of single tooth contact. This creates a gear set that has NT1 = 64 andPd = 9.8 for a theoretical center distance of 16.333 in. The minimum practical center distance (16.50 in.) is

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obtained when NT1 = 66 and Pd = 10.0.

For comparison with the above results, we used the same AGMA dynamic factorKv (Equation 6) but with the

modified Heywood tooth bending stress formula (Eq. 2) in the calculations. Table 4 lists the optimizationresults obtained.

Table 4 Optimization results of sample standard gears

(Using modi fied Heywood tooth stress formula)

Pd NT1 NT2 CD FW CR Sb Ss Sp2.00 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 78.13712.00 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 Pd = 10.0. This is very close to Carrolls 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. Thedifferences between Carrolls results and those reported here are likely due to the use of different formulasfor bending stress calculations.

Figure 5 shows graphically the design space for the results presented in Table 4, depicting the stressconstraint curves of bending, scoring, and pitting. The region above each constraint curve indicates feasibledesign space for that particular constraint. In the figure, the theoretical optimum is located at the intersectionpoint of the scoring stress and the bending stress constraint.

Figure 5 Design space for determination of optimized results of sample standard gears using modifiedHeywood tooth stress formula.

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Table 5 displays the data for the optimum compact design of the sample non-standard gears (long and shortaddendum system) with a hob offset of 0.03 inch. The practical minimum center distance is equal to 15.000inches and occurs at two values of Pd = 10.00 and 12.00, with NT1 = 60 and 72. The practical minimumcenter distance decreases as the diametral pitch Pd increases from 2.00 to 10.00; it then increases as thediametral pitch Pd 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 Ss Sp

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 57.99020.00 151 604 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 Pd = 10.0with a practical minimum center distance of 15.750 inch and NT1 = 63,. As can be seen from the table, theminimum center distance decreases as the diametral pitch increases from 2.00 to 10.00. When the diametralpitch is greater than 10, the minimum center distance increases with the diametral pitch value.

Table 6 Optimization resul ts for non-standard gears with 0.05 inch hob of fset

Pd NT1 NT2 CD FW CR Sb Ss Sp

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 practicalminimum center distance is equal to 17.500 and occurs at Pd = 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 thenincreases as diametral pitch increases from 8.00. At Pd = 6.00 and 8.00, the minimum center distance forboth cases is unchanged. Some higher diametral pitch values do not produce optimization results due to thepointed 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 Pd value increases from 2.00 to 4.00 for non-standard gears and from 2.00 to 6.00 for standard gears.

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210

190

Feasible170

PinionToothNu

mber

Design150

130

110

9

7

5

3

1

2.00 3.00 20.004.00 6.00 8.00 10.00 12.00 16.00

Diametral Pitch

0.09 in0.03 in 0.05 in 0.07 inHob Offset

(a)

29

27

25Feasible

CenterDist

ance(in.)

23Design21

1917

15

13

11

9

7

5

2.00 3.00 20.004.00 6.00 8.00 10.00 12.00 16.00

Diametral

Hob Offset 0.09 in0.03 in 0.05 in 0.07 in

(b)

Figure 6 Design space and optimal gear sets for standard and non-standard gears, Gear ratio Mg = 4,pressure angle = 20

0.

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At diametral pitch values less than or equal to 3.00, non-standard gears with hob offset values of 0.07 inchesand 0.09 inches appear to be more compact than standard gears while non-standard gears with hob offsetvalues of 0.03 inches and 0.05 inches have the same center distance as standard gears (except non-standard gears pair at Pd = 2.00 with hob offset 0.05 inch that has slightly smaller center distance thanstandard 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. Comparisonsbetween 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 designthan hob offsets of 0.03 inches and 0.05 inches when diametral pitch is less than or equal to 4.00. Withdiametral pitches that are greater than 4.00, non-standard gears with hob offset values of 0.07 inches and0.09 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 morecompact design. When the diametral pitch is greater than 4.00, increasing the hob offset value will result in alarger gears center distance or a less compact design. When the diametral pitch value is high (greater thanor equal to 10.00), non-standard gears at higher hob offsets (0.07 inches and 0.09 inches) seem verysensitive to subject load which result in a much larger gear pair. Thus, the degree of sensitivity increaseswith 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 Pd =20.00 and 0.09 inches at Pd = 16.00 and 20.00.

Conclusions

This paper presents a method for optimal design of standard spur gears for minimum dynamic response. Astudy was performed using a sample gear set from the gear literature. Optimal gear sets were compared fordesigns based on the AGMA dynamic factor and a refined dynamic factor calculated using the DANST geardynamics code. The operating speed was varied over a broad range to evaluate its effect on the requiredgear size. A design space for designing optimal compact gear sets can be developed using the currentapproach.

The required size of an optimal gear set is significantly influenced by the dynamic factor. The peak dynamicfactor at system natural frequencies can dominate the design of optimal gear sets that operate over a widerange of speeds. In the current investigation, refined dynamic factors calculated by the dynamic gear codedeveloped by the authors allow a more compact gear design than that obtained by using the AGMA dynamicfactor values.

Compact gears designed using the modified Heywood tooth stress formula are similar to those designedusing the simpler Lewis formula for the example cases studied here. Design charts such as those shownhere can be used for a single speed or over a range of speeds. For the sample gears in the study, adiametral pitch of 10.0 was found to provide a compact gear set over the speed range considered.

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References

1. Walsh, E. J. and Mabie, H. H., A Simplified Method for Determining Hob Offset Value in the Design ofNonstandard Spur Gears, 2nd OSU Applied Mechanism Conference, Stillwater, Oklahoma, (Oct. 1971).

2. Mabie, H. H., Walsh, E. J., and Bateman, Determination of Hob Offset Required to GenerateNonstandard Spur Gears with Teeth of Equal Strength, Mechanism and Machine Theory, Vol. 18, No. 3, pp.

181-192, (1983).

3. Mabie, H. H., Roggers, C. A., and Reinholtz, C. F., Design of Nonstandard Spur Gears Cut by a Hob,Mechanism and Machine Theory, Vol. 25, No. 6, pp. 635-644, (1990).

4. Liou, C. H., Design of Nonstandard Spur Gears for Balanced Dynamic Strength, Ph.D. Dissertation,University of Memphis, (1994).

5. Lin, H.H., Townsend, D.P., and Oswald, F.B., 1989, Profile Modification to Minimize Spur GearDynamic Loading, Proc. of ASME 5th Int. Power Trans. and Gearing Conf., Chicago, IL, Vol. 1,pp. 455465.

6. Liou, C.H., Lin, H.H., and Oswald, F.B., 1992, Effect of Contact Ratio on Spur Gear Dynamic Load,Proc. of ASME 6th Int. Power Trans. and Gearing Conf., Phoenix, AZ, Vol. 1, pp. 2933.

7. Bowen, C.W., 1978, The Practical Significance of Designing to Gear Pitting Fatigue Life Criteria,ASME Journal of Mechanical Design, Vol. 100, pp. 4653.

8. Gay, C.E., 1970, How to Design to Minimize Wear in Gears, Machine Design, Vol. 42, pp. 9297.

9. Coy, J.J., Townsend, D.P., and Zaretsky, E.V., 1979, Dynamic Capacity and Surface Fatigue Life forSpur and Helical Gears,ASME Journal of Lubrication Technology, Vol. 98, No. 2, pp. 267276.

10. Anon, 1965, Surface Durability (Pitting) of Spur Gear Teeth,AGMA Standard 210.02.

11. Rozeanu, L. and Godet, M., 1977, Model for Gear Scoring, ASME Paper 77-DET-60.

12. Savage, M., Coy, J.J., and Townsend, D.P., 1982, Optimal Tooth Numbers for Compact StandardSpur Gear Sets,ASME Journal of Mechanical Design, Vol. 104, pp. 749758.

13. Carroll, R.K. and Johnson, G.E., 1984, Optimal Design of Compact Spur Gear Sets, ASME Journalof Mechanisms, Transmissions, and Automation in Design, Vol. 106, pp. 95101.

14. Andrews, G.C. and Argent, J.D., 1992, Computer Aided Optimal Gear Design, Proc. of ASME 6thInt. Power Trans. and Gearing Conf., Phoenix, AZ, Vol. 1, pp. 391396.

15. Savage, M., Lattime, S.B., Kimmel, J.A., and Coe, H.H., 1992, Optimal Design of Compact SpurGear Reductions, Proc. of ASME 6th Int. Power Trans. and Gearing Conf., Phoenix, AZ, Vol. 1,pp. 383390.

16. Wang, H.L. and Wang, H.P., 1994, Optimal Engineering Design of Spur Gear Sets, Mechanism andMachine Theory, Vol. 29, No. 7, pp. 10711080.

17. Lin, H.H., Wang, J., Oswald, F.B., and Coy, J.J., 1993, Effect of Extended Tooth Contact on the

Modeling of Spur Gear Transmissions, AIAA-932148.

18. Townsend, D.P., 1992, Dudleys Gear Handbook, 2nd edition, McGraw-Hill Inc.

19. South, D.W. and Ewert, R.H., 1992, Encyclopedic Dictionary of Gears and Gearing, McGraw-Hill.

20. Cornell, R.W., 1981, Compliance and Stress Sensitivity of Spur Gear Teeth, ASME Journal ofMechanical Design, Vol. 103, pp. 447459.

21. Heywood, R.B., 1952, Designing by Photoelasticity, Chapman and Hall, Ltd.

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22. Shigley, J.E. and Mitchell, L.D., 1983, Mechanical Engineering Design, 4th Ed., McGraw-Hill.

23. Shigley, J.E. and Mischke, C.R., 1989, Mechanical Engineering Design, 5th Ed., McGraw-Hill.

24. Oswald, F.B., Townsend, D.P., Rebbechi, B., and Lin, H.H., 1996, Dynamic Forces in Spur Gears -Measurement, Prediction, and Code Validation, Proc. of ASME 7th Int. Power Trans. and Gearing Conf.,San Diego, CA, pp. 915

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Documents

Long Addendum Gears 1946