a new photoelastic investigation

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d 9.0W t b 1

E 1

1

E 2 (3)

f 2 W t e

d 1 (4)

f a f 1 f 2

f 1 f 2(5)

F d f a 2 f 2 f a (6)

The dynamic factor K v used by AGMA relates the total toothload, including the internal dynamic effects, to the transmittedtangential tooth load.

K vF d W t

W t (7)

K v can be determined by the AGMA standard 9 or by theBuckingham dynamic load in this paper.

2.2 Determining the Critical Point. The AGMA technicalpaper 10 refers to four methods of determining the critical pointsin the root llet of a gear tooth. Three of the points are commonlyassociated with the slope angle of the root llet. The Lewis for-mula has already been mentioned above. Niemann proposes mak-

ing a line from the intersection of the action line of the appliedforce and the radial centerline of the teeth, i.e., tangential to thetooth prole at the root llet, and treating that tangential point asa critical point. On the other hand, the Hofer method of calculat-ing tooth strength takes the tangency point of a 30-degree angle asthe critical point in the root llet. The Colbourne point is denedas where the stress function becomes maximal. Equations formu-lated to nd these four critical points are formed into nonlinearone-variable equations and to solve these, the Newton-Raphsonnumerical iteration method appears to be best suited 10 . Oncethe critical points are solved, they can be used to draw lines orparabola on photoelastic images for image processing.

2.3 Formulas for the Bending Stress. The original Lewisformula was devised for the transverse component of the appliedload. In Fig. 1, it is apparent that the load normal to the toothsurface has a tangential and a radial component. The radial com-ponent causes a small compressive stress across the root of thegear tooth. This causes the tensile stress to decrease by a smallamount, and the compressive stress on the opposite side of the

tooth to increase by a slight amount. In most materials, a tensilestress is more damaging than a slightly higher compressive stress.The tensile stress at the llet of this tooth is

6 W t h

t 2b

W r tb

(8)

To provide more scope for comparison, two popular methods of estimating bending stresses, namely, Dolan’s concentration factorand the AGMA equation for bending stress in gear teeth are used.Dolan’s empirical equation 4 is

K D 0.18 t r

0.15

t h

0.45

(9)

and the AGMA equation 9 is

F W t K o K v K s1

bm

K H

Y J (10)

3 Specimens and Experimental Setup

3.1 Specimen. Two full-depth spur gears have 20 and 25teeth, 20 deg pressure angle, the module m 6. AGMA qualityNo. 10. Both gear specimens, 5.8 mm in face width, are made of photoelastic material PSM-1 from Measurements Group Co.,USA. The specimens were cooled in vast oil ow to prevent re-sidual stresses during the cutting process.

3.2 Gear Mechanism. The said two gears operated on acenter distance 135 mm. An AC motor and a V-belt coupled withtwo equal-sized pulleys were used to drive the gear mechanism.The pinion was connected to the drive shaft of the motor. Thedynamometer in the mechanism provided the required load whenthe gear rotated. The entire experiment mechanism is shown inFig. 2.

3.3 The Digital Photoelastic System With Function of RealTime Image Taking. As shown in Fig. 3, the digital photoelas-tic system provided with a Synchronous Trigger and ContinuousImage Taking was used to take real time photoelastic images of the gear teeth in transmission. It is similar to the former system6 except for the following two new parts:

(a) CCD Camera

The replacement CCD camera with 1035 1320 resolution,manufactured by Xillix Co., Canada, provides enhanced imageresolution,

(b) Gated Intensier The gated intensier type VS3-1845, Video Scope Co., USA

discharges the time exposure function for the CCD camera to

Fig. 1 Critical point and applied force on a gear tooth Fig. 2 Photograph of the gear mechanism

366 Õ Vol. 125, JUNE 2003 Transactions of the ASME



which it is attached. It amplies the light intensity and controls thelight gated time, capturing clear images of fast-moving objects.The gated time can reach 62.5 ns.

4 Experimental Procedures

4.1 Installation of Experimental Setup. To ensure steadyperformance and avoid abnormal vibration, a gear drive must berigidly supported and the shafts accurately aligned. The followingdescribes the minimum precautions required to achieve this goal.Shaft mounted drives are mounted as close to the driven equip-ment bearing support as possible to minimize bearing loads due tooverhung load. The motor is aligned to the input shaft, usingshims under the feet to provide support, and then bolted to itsfoundation. To ensure continuous accuracy during operations, thegearing must be sustained in accurate alignment and adequatelylubricated.

4.2 Capturing Photoelastic Images. Static images are fareasier to take than dynamic ones. So, before attempting dynamicimage taking, the experimental setup was installed and the digitalphotoelastic system calibrated as if in preparation for taking static

images. The calibration of the system 11 is to establish the rela-tionship between the gray level Z and fringe order N , which is

N 1

sin 1 Z

Z max

1/2

(11)

where Z max is the maximum response of the system and is theslope of the camera sensitivity curve. A set of N and Z is obtainedby employing the Tardy compensation and least square methodson an unloaded specimen. Then, the material fringe value of thetest specimen is determined. The real time image taking techniqueis similar to that in the author’s former research 6 . The keyfactors to ensuring success with all the required images are theexact timing between images and the positioning of both the trig-ger sensor and the marker for the trigger. First, the photoelectricsensor is properly positioned and then the marker a black sticker

placed on the gear shaft. Image taking begins on receipt of thetrigger signal under software control. If the rst image is triggeredtoo early or too late, the position of the marker is adjusted accord-ingly. Once the marker’s position is decided, the timing betweensubsequent images is set. To give the camera enough time, allimages but the rst are taken at the next turn of the pinion. Tovary the pinion’s rotation speed, the adjustment and regulation isin accordance with the above procedures.

4.3 Experimental Determination of MTBS and CriticalPoint. A series of computer programs based on the routine pack-age residing and previous self-developed programs 12 on theimage analysis system were developed to determine the experi-

mental MTBS and the critical point. With image processing, edgedetection is the rst step in locating the contact point and thecritical point. An upper and/or lower threshold value is introducedinto the image-processing system so that the gray levels aboveand/or below the threshold are removed throughout the image.Because the area around the gear specimen is much darker thanthe gear specimen itself, the thresholding operation can easilydistinguish between them. The Laplacian edge enhancement tech-nique produces sharper edge denition than most other tech-niques. Edge enhancement processes reduce an image into itsedge. The Laplacian lter operation is then used to distinguish the

prole of the specimen from the thresholded image. After theoperation, a prole of one-pixel width is obtained. Coordinates of the prole are accordingly recorded for later use. To obtain anaccurate prole of a spur gear, a numerical iteration program,based on the involute function, using the prole coordinates ex-tracted from the previous processed images, is run to modify theprole of the spur gear until it coincides with the actual prole of the gear specimen. This procedure for detecting the gear prolemust be performed only once on every image. According to two-dimensional photoelastic theory, the difference of in-plane princi-pal stresses is

1 2 N f

b (12)

Along the free boundary of the tooth llet, the principal stress

tangential to the boundary, E , i.e., the tensile bending stress, is

E

N f

b (13)

It is immediately apparent from Eq. 13 that the maximum tensilebending stress can be determined if the maximum fringe order N can be measured. The maximum tensile bending stress at the toothllet is located at the brightest point, or the darkest point on theboundary surrounded by a dark band nearest to the boundary. Forall the experiments performed, the maximum fringe orders on thetooth llet are usually the fractional fringe orders, and to classifythe orders of the fringes, the distinction of dark and bright zonesin the photoelastic fringes near the boundary is the rst thing to beconsidered by a gray level scanning. The end points of the zonewhere the maximum fringe orders occurred, intersecting with the

boundary of the tooth llet, were located on the boundary. Aboutten boundary points with equal arc length between the endpointswere chosen. The corresponding fringe order of the boundarypoint was determined by converting the measured value of thegray level into a fringe order by Eq. 11 . The same procedurewas repeated for the other boundary points to obtain their fringeorders. Then, the magnitude and position of the maximum bound-ary fringe order was then determined by interpolating the fringeorders and coordinates of the boundary points used, respectively.The interpolation scheme used was a piecewise cubic-splineroutine.

5 Results and DiscussionsThe investigation in this paper concentrated on photoelastic im-

aging of the pinion because this is the part more apt to fail. Theimages of the gear teeth were all taken in the dark eld. Dynamicimages of such as transmitted torque, contact position, and rota-tion speed proved very difcult to secure. To ensure covering allthe conditions occurring during the gear meshing process for lateranalysis, over three thousand images on a trial and error basis.Some were stained by lubricant grease. All had the pinion locatedon the right hand side and rotating clockwise. In most cases,analysis by the image processing method was not needed. Formany others, a simple visual examination of the unprocessed im-age yielded interesting information. The images in Fig. 4 clearlyshow that the single-tooth contact has greater MTBS than thedouble teeth contact. So, if because of double teeth contact, the tip

Fig. 3 Digital photoelastic system with function of real timeimage taking

Journal of Mechanical Design JUNE 2003, Vol. 125 Õ 367



Fig. 4 Photograph of photoelastic images for a gear-meshing cycle, n Ä 486 rpm andT Ä 250 N-cm

Fig. 5 Photoelastic images for single-tooth contact and different torques in the static state




load appeared not be the worst load, then the point for experimen-tal investigation should be the single-tooth contact. Figure 5shows images of different torques for single-tooth contact in thestatic state. Apparently, the fringe orders at the tooth llets in-crease as the torque increases. The static state images are clearerthan the dynamic state images. The images in Fig. 6 were taken atdifferent speeds and T 250 N-cm, and show the dynamic effecton the gear teeth. The dynamic load comes into effect as the fringeorders at the tooth llets increase when the rotation speed in-creases. Only representative images selected for further investiga-tion were analyzed by image processing. Figure 7 shows a typicalimage of various dened critical points being determined by theimage processing. The Lewis stresses are much lower than theobserved experimental MTBS because the concentrations of stresses are invariably developed at the llets, and their stressraising effect is not considered in the Lewis equation. The HPSTC

is one base pitch away from the rst point of contact. The contactpoint of single-tooth contact varied between 0.58 and 1.0 basepitch away from the rst point of contact in this experiment. Thelocation of contact point divided by the base pitch for non-dimensionalization is used as the abscissa. Figures 8 and 9 showthat MTBS is increased as the contact point is moved towardHPSTC in both the static and dynamic states. Even in the experi-mental conditions here, when the contact point is at the samelocation, the MTBS is much greater than Dolan’s MTBS. Theauthor’s experimental MTBS is from 18 to 31 percent greater thanDolan’s.

There is little change in the magnitude of MTBS at low rotationspeed. The dynamic load comes into effect obviously when therotation speed increases to around one thousand rpm. Figure 10shows that the MTBS at 950 rpm is 12% higher than the MTBS at177 rpm. The gure also compares the author’s experimentalMTBS with that of both AGMA and Buckingham. The differencebetween AGMA and Buckingham MTBS is the calculation of dy-namic load based on the different approaches. Equation 6 , Buck-

Fig. 6 Photoelastic images for single-tooth contact, different speeds and T Ä 250 N-cm

Fig. 7 Typical image of determining various dened criticalpoints by image processing

Fig. 8 Variation of MTBS with location of contact point for T Ä 200 N-cm in the static state




ingham’s calculation of dynamic load, is substituted for Eq. 7 todetermine K v and AGMA K v is determined by the AGMA stan-dard 9 but Eq. 10 is still used for both to calculate MTBS. Allthe factors in Eq. 10 may be treated as a unity except K v and Y J 13 because the experimental setup and specimens are well de-

signed, manufactured and assembled. The AGMA MTBS is mostclose to the author’s experimental MTBS in all three bending

stresses. The other two are Dolan and Buckingham bendingstresses. Buckingham’s MTBS is far greater than that of AGMAand the MTBS in the author’s experiment due to an overestima-tion of the dynamic load. The experimental data in Fig. 11 indi-cate that the MTBS increases considerably as the transmittedtorque increases. The transmitted torque is found to have a pri-

mary effect on the magnitude of MTBS compared with the othertwo factors of rotation speed and contact point. The general con-clusion to be drawn from test work on dynamic load is that exactcalculation is nearly impossible. Both Figs. 10 and 11 show thatthe difference between the author’s experimental MTBS and thatof AGMA is below 10% and that the author’s experimental MTBSis lower than that of AGMA. For a complicated dynamic problem,the accuracy with which AGMA MTBS ts the author’s experi-mental data is quite satisfactory.

For convenient comparison of the positions of the four criticalpoints, the intersection of the action line of the applied force andthe radial centerline of the tooth serves as the common base point,i.e., the vertex of the Lewis parabola shown in Fig. 1. The angle between the centerline of the tooth and the line made from thecommon base point to each critical point is dened as the positionof the critical point. The smaller the angle, the closer the MTBS isto the gear blank. The numerical method combined image pro-cessing technique was used to locate the theoretically weakestsection for each tooth and for each contact point load position .An examination of Figs. 12– 15 shows that the position of theMTBS obtained from the photoelastic analysis of the pinion speci-men is not located at the weakest section as dened by Lewis, butin general occurs at a point lower closer to the gear blank on the

llet contour. The AGMA technical paper 10 concluded that theLewis point was the highest farthest from the gear blank of thefour critical points, i.e., the Hofer, Lewis, Niemann and Colbournepoints, and that the Colbourne point was presumed best for deter-mining the critical point. Nevertheless, the stress function used todetermine the Colbourne point, which is still based on a simplecantilever stress analysis, is impractical for an actual gear. TheLewis and Colbourne points always lie close to each other, ac-cording both to the results in the AGMA paper and to the author’s

Fig. 9 Variation of MTBS with location of contact point for n Ä 177 rpm and T Ä 250 N-cm

Fig. 10 Variation of MTBS with rotation speed for HPSTC andT Ä 250 N-cm

Fig. 11 Variation of MTBS with transmitted torque for HPSTCand n Ä 950 rpm

Fig. 12 Variation of position of MTBS with location of contactpoint for T Ä 200 N-cm in the static state

Fig. 13 Variation of position of MTBS with location of contactpoint for n Ä 177 rpm and T Ä 250 N-cm




calculation. Therefore, the Colbourne point is not included in thegures featured in this paper. Figures 12–15 also show that theNiemann point is closest to the experimental point and that thedifference in the position between the two points is within 1%.There is no doubt that Niemann is the best of the four methods fordetermining the critical point.

It is notable in Figs. 12–13 that as the load position movedtoward the HPSTC of a tooth, the MTBS increased but movedposition toward the root llet. The trend was the same as in theDolan experiments. Figure 14 shows no evidence for the rotationspeed affecting the position of the MTBS, because any change inthe position of the MTBS produced by changing the rotationspeeds of the pinion is quite small. Figure 15 shows that an in-crease in the magnitude of load may cause the position of theMTBS to move slightly higher away from the root. In this experi-ment, the only factor inuencing the position of the MTBS is thelocation of the contact point during gear meshing.

6 ConclusionsThe experimental MTBS for every contact point during the pro-

cess of gear meshing can be determined owing to the establish-ment of this self-developed system in this paper. No new approachfor determining the magnitude of the MTBS or the critical point inthe root llet is required, because the experimental data can betted by the existing methods. The AGMA standard is the mostsatisfactory for determining the magnitude of MTBS according tothe experimental results in this paper, but not for determining thecritical point. The Niemann’s critical point is the most accurate forthese experimental results. Although the AGMA and Niemann ap-proaches have been around for years, this is the rst time that theiraccuracy for the dynamic bending stress problem of a spur gear

tooth in the root llet by a photoelastic method has been proven.The methods that have been tested in these experiments may becondently employed by gear designers or manufacturers to de-termine the magnitude of MTBS and the critical point for spurgear teeth in transmission.

AcknowledgmentsThis research was supported in part by the National Science

Council Grant No. NSC-87-2212-E-159-003 of the Republic of China. The author wishes to thank Professor Wei-Chung Wang of National Tsing Hua University for the loan of equipment.

Nomenclature

C the common base point used to dene the position of MTBS

E 1 modulus of elasticity of pinion E 2 modulus of elasticity of gearF d dynamic loadK D Dolan’s concentration factorK H load distribution factorK o overload factorK s size factor M effective mass inuence at pitch line of gears

M 1 effective mass acting at pitch line of pinion M 2 effective mass acting at pitch line of gear

N fringe order R1 pitch radius of pinion R2 pitch radius of gear

T transmitted torqueV pitch-line velocity of gears

W t applied tangential forceW r applied radial forceY J geometry factor

b face width of gearsd deformation of teeth at pitch line under applied load

W t e measured error in action of gears

f 1 force required to accelerate the masses as rigid bod-ies

f 2 force required to deform teeth amount of effectiveerror

f a acceleration load on gear teeth f

material fringe valuem modulen rotation speedh height of load position above the weakest sectionr radius of circular llett thickness of tooth at Lewis’ weakest section pressure angle angle dening the position of MTBS

Lewis’ computed combined stress at HPSTC E experimental maximum tensile bending stress F AGMA bending stress at HPSTC 1 in-plane principal stress 2 in-plane principal stress

References1 Lewis, W., 1893, ‘‘Investigation of the Strength of Gear Teeth,’’ Proc. Engr’s

Club Phila., 10 , pp. 16–23.2 Niemann, G., 1978, Machine Element Design and Calculation in Mechanical

Engineering , Vol. II, Gears, Translated by K. Lakshminarayana et al.,Springer-Verlag Berlin Heidelberg, New York.

3 DIN 3990, 1972, Calculation of Load Capacity of Spur, Helical and BevelGears , Deutsche Norm., Berlin.

4 Dolan, T. J., and Broghamer, E. L., 1942, ‘‘A Photoelastic Study of theStresses in Gear Tooth Fillet,’’ University of Illinois Bull. 335.

5 Allison, I. M., and Hearn, E. J., 1980, ‘‘A New Look at the Bending Strengthof Gear Teeth,’’ Exp. Mech., pp. 217–226.

6 Wang, M. J., and Wang, W. C., 1995, ‘‘Investigation of Contact Stresses of aRotating Cam by a Digital Photoelastic Method,’’ Strain J. Brit. Soc. StrainMeasurement, 31 1 , pp. 17–24.

Fig. 14 Variation of position of MTBS with rotation speed forHPSTC and T Ä 250 N-cm

Fig. 15 Variation of position of MTBS with transmitted torquefor HPSTC and n Ä 950 rpm




7 Ozgu ven, H. N., and Houser, D. R., 1988, ‘‘Mathematical Models Used inGear Dynamics-A Review,’’ Sound Vib., 121 3 , pp. 383–411.

8 Buckingham, E., 1949, Analytical Mechanics of Gears , McGraw-Hill Book Company, New York.

9 AGMA 2101-C95, 1995, ‘‘Fundamental Rating Factors and Calculation Meth-ods for Involute Spur and Helical Gear Teeth,’’ American Gear ManufacturersAssociation, Alexandria, Virginia.

10 AGMA 94FTMS1, 1997, ‘‘Computer-Aided Numerical Determination of Hofer, Lewis, Niemann and Colboune Points,’’ American Gear ManufacturersAssociation, Alexandria, Virginia.

11 Wang, W. C., and Chen, T. L., 1989, ‘‘Half-fringe Photoelastic Determinationof Opening Mode Stress Intensity Factor for Edge Cracked Strips,’’Eng. Fract.Mech., 32 , pp. 111–122.

12 Wang, M. J., and Wang, W. C., 1994, ‘‘Transient Thermal Stress Analysis of aNear-edge Elliptical Defect in a Semi-innite Plate Subjected to a MovingHeat Source,’’ Press. Vessels & Piping, 57 , pp. 99–110.

13 AGMA 908-B89, 1997, ‘‘Geometry Factors for Determining the Pitting Resis-tance and Bending Strength of Involute Spur, Helical and Herringbone GearTeeth,’’ American Gear Manufacturers Association, Alexandria, Virginia.


a new photoelastic investigation

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