NASA

Technical Memorandum 107222

/

/

Army Research Laboratory

Technical Report ARL-TR-1087

Balancing Dynamic Strength of Spur Gears

Operated at Extended Center Distance

Hsiang Hsi Lin and Chuen-Huei Liou

The University of Memphis

Memphis, Tennessee

Fred B. Oswald and Dennis P. Townsend

Lewis Research Center

Cleveland, Ohio

Prepared for the

Seventh International Power Transmission and Gearing Conference

sponsored by theAmerican Society of Mechanical Engineers

San Diego, California, October 6-9, 1996

National Aeronautics and

Space Administration

U.S. ARMY

RESEARCH LABORATORY

BALANCINGDYNAMICSTRENGTHOFSPURGEARSOPERATEDAT

EXTENDEDCENTER DISTANCE

Hsiang Hsi Lin and Chuen-Huei Liou*

The University of Memphis

Memphis, Tennessee

Fred B. Oswald, and Dennis P. Townsend

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio

ABSTRACT

This paper presents an analytical study on using hob offset to balance the dynamic tooth strength of spur gears operated at a center distance greater than the standard value. This is an extension of a static

study by Mabie and others. The study was limited to the offset values

that assure the pinion and gear teeth will neither be undercut nor become

pointed. The analysis presented in this paper was performed using

DANST-PC, a new version of the NASA gear dynamics code.

The operating speed of the transmission influences the amount of

hob offset required to equalize the dynamic stresses in the pinion and

gear. The optimum hob offset for the pinion was found to vary within a

small range as the speed changes. The optimum value is generally greater

than the optimum value found by static procedures. For gears that must

operate over a wide range of speeds, an average offset value may be

used.

NOMENCLATURE

C center distance, mm (in)

c tooth clearance, mm (in)

e hob offset, mm (in)

k hob addendum parameter

m module, mm

N number of teeth

Pd diametral pitch, in d

R o outside radius, mm On)

R Pitch radius, mm (in)

pressure angle, degrees

Subscripts

1 driving gear (pinion)

2 driven gear (gear)

INTRODUCTION

Gears may be designed to operate at a non-standard center distance

to introduce backlash, to accommodate space constraints (Townsend,

1991), and to adjust for anticipated deflections under load as well as

geometry changes due to thermal effects. In some gear designs it is necessary to connect shafts at a predetermined center distance and to produce

a desired velocity ratio. However, when standard gears cannot be used,

the design is much more complicated. In these cases, the cutting tool can

be offset from the standard setting to produce nonstandard gears which

will mesh at the desired center distance.

In designs with large module (small pitch) teeth, the number of teeth

is small on both the pinion and gear. If the pinion teeth are enlarged to

provide proper strength then the pinion teeth may reach below the base

circle of the gear and cause undercut. In such cases, the center distance

must be increased. This design is called the extended center distance

system. For gear,s cut by hobs, the adjustment of the cutting tool is called

hob cutter offset.

In previous work, Walsh and Mabie (1971), Mabie et al. (1983),

and Mabie et al. (1990), procedures and design charts were developed to

determine the hob offset for the pinion and gear so that the stress in the

pinion and gear teeth were approximately equal. They developed charts

relating hob offset at various velocity ratios and changes in center

*Presently at Central Police University, Taoyuan, Taiwan.
distance. However, their analyses dealt with the static tooth strength.

They used the Lewis formula (Shigley and Mitchell, 1983) to calculate

the tooth root stress and based calculations assuming the maximum static

load was applied at the tip of the tooth.

Under dynamic conditions, especially at high speed, the tooth load

can be significantly greater than the static load, and the maximum

dynamic tooth load may occur at a location other than the tooth tip CLin

et al., 1989). Furthermore, non-standard gears have transmission errors

different from gears with standard tooth proportions. The dynamic response of a gear system, excited by the transmission error, can be considerably different for nonstandard gears. Dynamic effects should be

considered for the design of high speed gears.

THEORY AND ANALYSIS

Extended Center Distance Gears

The analysis and formulation for nonstandard gears operating at an

extended center distance can be found from the previous work of Mabie

and Reinholtz (1987). The relationship between the amount of hob offset and gear parameters was found to be:

(N 1+ N2)(inv¢'- inv¢)

e 1 + e 2 = (1)

2 Pd tan O

The above equation can be expressed in metric units with m as module,

m( N 1 + N2Xinv_'-invO)

e ! + e 2 = (2)

2tan0

where e I is the hob offset for the pinion and e 2 is the offset for the gear,

is the standard pressure angle and ¢" is the operating pressure angle.

The involute function inv 0 is defined as (tan ¢ - ¢).

When a pair of gears (the pinion and the gear) are moved apart a

distance AC from the standard center distance C, the extended center

distance is

C'= C + AC = Rf + R_ (3)

where R{ and R_ are the operating pitch radii of the pinion and the

gear, respectively. From basic gear geometry, the center distance extension is

AC = c(C°s* Lc°s ¢' 1) (4) -

and the operating pitch radii are

(5)

R,_ = C'( Nl N23+N2 (6)

From Fig. 1, the outside radii of the gear pair, expressed in U.S. customary units, are

k

Rol = C' - R 2 - e 2 + _ (7) Pd

k

Re2 = C" - R l - e I + Pd (8)

where k is the addendum coefficient value and is equal to 1.00 for a

standard full-depth tooth. The above equations can be expressed in metric units as

Ci

R°I "_RI. 01 i

............. 6_;ti_o/h.......... ...

/

I

02

(a) Pinion

I

!

....::..::..::..:.. / °::"<:ii;ii;ii;iii.i;ii;ii;ii;°°°°:°°°°

O' 2

(b) Gear

Figure 1 .---Gear pair cut by offset hobs to mesh at an extended center distance.
Rol= C'- R2 -e2 + m (9)

Ro2 = C" - R 1 -e I + m (10)

Note that the sum of the hob offsets, e I and e 2 , does not necessarily

equal the increase in the center distance (C) over the standard value.

There is no easy way of determining the hob offsets independently of

each other. The values are usually selected by assuming one of them and

calculating the other from either Eqs. 1 or 2 depending on the units used

(U.S. or metric).

Gear Dynamic Model

The computer program DANST-PC was used for the dynamic analysis (Oswald et al., 1993). The DANST model employs four torsional

degrees of freedom to represent a typical gear transmission. The model

includes driving pinion and driven gear, connecting shafts, motor, and

load. DANST predicts the dynamic response of a transmission for several parameters including dynamic load and tooth bending stress. In two

validation studies, the predictions of DANST compared very well with

experimental measurements (Oswald et al., 1991 and Oswald et al., 1996).

A more detailed discussion of the dynamic model and tooth root

stress calculation can be found in the literature (Comell, 1981, and Lin

et al., 1993).

RESULTS AND DISCUSSION

The following section describes a parameter study in which the

dynamic analysis was applied to a sample gear set to compare results

with the static results obtained by Mabie, et al. (1990). An example problem from their paper was used for the investigation. Parameters for the

sample gears used are given in Table 1. The gear set has a standard center

distance of 5.00 inches (127 ram) but the operating center distance was

extended by 0.25 inches (6.35 mm). In the study, the hob offset on the

pinion was treated as a variable and the corresponding offset on the gear

was calculated from Eq. (1).

The sample gears for this study have a perfect involute tooth profiles with no modifications. From previous work (Lin, et al., 1993 and

1989) we know that modifying the tooth profile can significantly reduce

gear dynamics. However, in this study, no modifications were considered.

Figures 2 and 3 illustrate the effect of pinion hob offset (e l) as

well as rotation speed on the dynamic stress of the pinion. The most

prominent feature in both figures is the resonant response which occurs

at 13500 rpm. The "notches" which appear along the resonant speed

contours result from numerical error. Minimum values of dynamic tooth

stress appear as a valleys in the three dimensional figure (Fig. 2) and can

be located in the contour diagram (Fig. 3). The system resonant speed

was minimally affected by the pinion hob offset. In this and the cases to

follow, we maintained the center distance at 5.25" (133.35 mm). The

hob offset for the pinion was varied as shown in the figure and the offset

for the gear was calculated from Eq. (1).

The pinion hob offset was varied between 0.015 and 0.213 inch

(0.38 and 5.41 mm). These limits were required to avoid having undercut or pointed teeth in either pinion or gear. The optimal pinion hob

offset from the static procedure is 0.159 inch (4.04 mm). This is shown

as a dashed line in the figure. At lower speeds (1000 to 5000 rpm) the

pinion dynamic stress reaches its minimum when the pinion hob offset is

less than the optimal static value. (Two areas marked with the letter "A"

in Fig. 3.) However, at higher speeds (6000 to 13000 rpm), the minimum dynamic pinion stress occurs at hob offsets greater than the static

optimal value (letter "B" in Fig. 3).

Figures 4 and 5 show the effect of pinion hob offset and rotation

speed on dynamic stress of the gear. Unlike Figs. 2 and 3, the minimum

TABLE 1.---Gear Parameters

Gear type ................................................................. Full depth, involute tooth

Number of teeth (Pinion/Gear) ..........................................................20, 30

Module M, mm (diametral pitch P, l/in.) ................................. 5.08 (5)

Pressure angle, deg ................................................................................................ 20

Face width, mm (in.) ....................................................................... 635 (0.25)

Applied torque. N-m (lb-in.) ......................................................54.25 (480)

Static tooth load, N/m (lb/in.) .............................................178 850 (1022)
¢-

=;

p.

C

0

"_E

Q.

0.24

0.20

0.16

B

0.12

0.08

0.04

o.oo I I I I I

1000 3000 5000 7000 9000 11 000 13 000

Rotating speed, RPM

,/

t

I

1500O

Figure 3.--Contour diagram showing the effect of pinion hob offset and rotating speed on pinion

dynamic stress. (The optimal static case hob offset is shown by the dashed line.)

Figure 4._Effect of pinion hob offset (el) and rotating speed on gear

dynamic stress. (The hob offset for the gear is determined from eq. 1.)

4
0.24

0.20

f IX'll

=I'll

At"

Ill

• 0"16 _l_J_ I

¢-

4.m

o

e, 0.12

0

eC

o

.__

a. 0.08

0.04 _

0.00

1000

I

3000 50O0

I I

7000 9000

Rotating speed, RPM

/

I I I

11 000 13 000 15 000

Figure 5.--Contour diagram showing the effect of pinion hob offset and rotating speed on gear

dynamic stress. (The hob offset for the gear is determined from eq. 1. The optimal static case

hob offset is shown by the dashed line.)

0.25 --

/- Maximum allowable

• 0.20

_¢

0)

=: 0.15

O

.Q

O

_- 0.10

0

Q.

0.05 -- ,,-- Minimum allowable

/

," value (0.015 in.)

o.oo I----_---_ l I

0 4000 8 000 12 000 16 000

Rotating speed, RPM

Figure 6.--Determining pinion hob offset to balance dynamic

tooth strength of pinion and gear at different speeds.
values in Figs. 4 and 5 occur at an offset greater than the optimal static

value for all speeds.

We can optimize the pinion hob offset to balance the dynamic

strength of the gear set by minimizing the difference between the

pinion and gear dynamic stresses. Figure 6 shows how this optimum

pinion hob offset varies with the rotation speed. At most speeds (except

near the resonance of 13500 rpm), the best value lies between the optimal static value, 0.159 inch (4.04 ram) and the maximum allowable value,

0.213 inch (5.41 nun). At the resonance speed, the best value was about

60 percent of the optimum static value.

The analysis presented in this paper can be used to determine the

hob offset required to balance the dynamic tooth strength of gears operated at extended center distance. The amount of offset depends on the

intended operating speed range of the gear pair. For a gear set which will

operate over a range of speeds, an averaged hob offset value can be used

to balance dynamic strength.

CONCLUSIONS

DANST-PC, a new version of the NASA gear dynamics code, was

used to study the dynamic stress of non-standard spur gears cut with

offset hobs. The study was appli_l to gear pairs operated at a center

distance greater than the standard value. The operating speed of the transmission was varied over a broad range to evaluate speed effects on the

dynamic response. The following conclusions were obtained from the

investigation:

1. Over most of the speed range studied, the best value for hob

offset is greater than the optimum value obtained from the static procedure. The hob offset tends to make the pinion teeth longer and the gear

teeth shorter.

2. For gears operating over a range of speeds, a weighted average

hob offset value can be used to balance the dynamic strength of pinion

and gear.

3. The best hob offset to minimize dynamic stress in the pinion is

different from that required to minimize dynamic stress in the gear.

4. The analysis developed in this study can be used to determine the

hob offset value to balance the dynamic tooth strength of the pinion and

gear. The balanced design will provide a higher load capacity gear

system.

Lin, H.H., Townsend, D.R, and Oswald, EB., 1989, "Dynamic Loading of Spur Gears with Linear or Parabolic Tooth Profile Modifications"

Proc. of ASME 5th Int'l Power Trans. and Gearing Conf., Chicago,

Illinois, Vol. 1, pp. 409-419.

Lin, H.H., Lee, C., Oswald, EB., and Townsend, D.P., 1993,

"Computer-Aided Design of High-Contact-Ratio Gears for Minimum

Dynamic Load and Stress," ASME Journal of Mechanical Design,

Vol. 115, pp. 171-178.

Mabie, H.H., Walsh, E.J., and Bateman, V.I., 1983, Determination

of Hob Offset Required to Generate Non-Standard Spur Gears with

Teeth of Equal Strength, Mechanism and Machine Theory, Vol. 18,

No. 1, pp. 181-192.

Mabie, H.H. and Reinholtz, C.E, 1987, Mechanisms and Dynamics of Machinery, Chapter 5, Fourth edition, John Wiley & Sons, Inc.,

New York.

Mabie, H.H., Rogers, C.A., and Reinholtz, C.E, 1990, Design of

Nonstandard Spur Gears Cut by a Hob, Mechanism and Machine Theory,

Vol. 25, No. 6, pp. 635-644.

Oswald, EB., Rebbechi, B., Zakrajsek, J.J., Townsend, D.P., and

Lin, H.H., 1991, Comparison of Analysis and Experiment for Dynamics

of Low-Contact-Ratio Spur Gears, Proc. of ASME 13th Biennial

Conference on Mechanical Vibration and Noise, Miami, Florida,

pp. 263-269.

Oswald, EB., Lin, H.H., Liou, C.H., and Valco, M.J., 1993, Dynamic Analysis of Spur Gears Using Computer Program DANST, AIAA

paper 93-2295, AIAMSAE/ASME/ASEE 29th Joint Propulsion Conference, Monterey, California.

Oswald, EB., Townsend, D.P., Rebbechi, B., and Lin, H.H., 1996,

Dynamic Forces in Spur Gears---Measurement, Prediction, and Code

Validation, Proposed for presentation at ASME 7th International Power

Transmission and Gearing Conference, San Diego, CA, 1996.

Shigley, J.E., and Mitchell, L.D., 1983, "Mechanical Engineering

Design:' 4th edition, Chap. 13, McGraw-Hill Book Company.

Townsend, D.P., editor, 1991, "Dudley's Gear Handbook" McGrawHill Book Company, Chapter 4, pp. 4.81.

Walsh, E.J., and Mahie, H.H., 1971, A Simplified Method for

Determining Hob Offset Values in the Design of Nonstandard Spur Gears,

Proc. of 2nd Applied Mechanism Conference, Oklahoma State University, Stillwater, Oklahoma, pp. 23-1 to 23-18.

REFERENCES

Comell, R., 1981, Compliance and Stress Sensitivity of Spur

Gear Teeth, ASME Journal of Mechanical Design, Vol. 103, No. 2,

pp. 447-459.

6

REPORT DOCUMENTATION PAGE I Form OMBNo. Approved 0704-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

September 1996 Technical Memorandum

4. TITLE AND SUBTITLE

Balancing Dynamic Strength of Spur Gears Operated at

Extended Center Distance

6. AUTHOR(S)

Hsiang Hsi Lin, Chuen-Huei Liou, Fred B. Oswald, and Dennis P. Townsend

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Lewis Research Crater

Cleveland, Okio 44135-3191

and

Vdaide Propulsioo Dirv.ctoratc

U.S. Army Research l.atboratory

Clcvdand, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

Nadoaal Aeronautics md Space Administration

Washiagum, D.C. 20546.4XX)1

aad

U.S. Anny Research Laboratory

Adelphi, Marflaad 20783-1145

5. FUNDING NUMBERS

WU-505--62-36

1L162211A47A

8. PERFORMING ORGANIZATION

REPORT NUMBER

E--10255

10. SPONSORI NG/IK)NITORING

AGENCY REPORT NUMBER

NASA TM-107222

ARL--TR-1087

11. SUPPLEMENTARY NOTES

Prepared for the Seventh International Power Transmission andGearing Conference sponsored by the American Society of Mechanical

Engineers, Sm Diego, California, October 6-9, 1996. Hsiang Hsi Lin and Chuen-Huei Liou The University of Memphis, Memphis,

Tennessee 38152; Fred B. Oswald and Dennis P. Townsend, NASA Lewis Research Center. Responsible person, Fred B. Oswald,

organization code 2730, (216) 433-3957.

12,1. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 37

This publicadon is available from the NASA Center for AeroSpace Information., (301) 621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This paper presents an analytical study on using hob offset to balance the dynamic tooth strength of spur gears operated

at a center distance greater than the standard value. This study is an extension of a static study by Mabie and others. The

study was limited to the offset values that assure the pinion and gear teeth will neither be undercut nor become pointed.

The analysis presented in this paper was performed using DANST-PC, a new version of the NASA gear dynamics code.

The operating speed of the transmission influences the amount of hob offset required to equalize the dynamic stresses in

the pinion and gear. The optimum hob offset for the pinion was found to vary within a small range as the speed changes.

The optimum value is generally greater than the optimum value found by static procedures. For gears that must operate

over a wide range of speeds, an average offset value may be used.

14. SUBJECTTERMS

Gears; Spur gears; Hob offset

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