Non-Standard

Cylindrical Gears

Bryan N.V. ParsonsDouglas WaltonLaurentia Andrei Gabriel Andrei

AbstractThis paper examines three gear geometries,

each of which has special features not found onstandard cylindrical gears. The first are high pres-sure angle spur gears where the pressure anglehas been maximized at the expense of contactratio. High pressure angles result in higher forces,but the stresses are less and are mostly compres-sive, leading to good fatigue resistance. The sec-ond design is a curved face width gear, giving ahigher contact ratio compared to spur gears andlower contact and bending stresses. These gearsare similar to double helical gears in that there areno axial forces. The third gear is a tapered facewidth gear, in which the addenda, dedenda andtooth width are all tapered. The benefits are thattooth engagement is gradual and compensationfor center distance variations can be achieved byrelative axial adjustment. All these gears weremade of plastic materials.

IntroductionThe annual production of plastic gears now

outnumbers the production of metal gears (morethan 500 million per year for cars alone).However, the design of plastic gears relies heavi-ly on the experience gained from many years ofdesigning and manufacturing steel gears. Work inthe United Kingdom and Romania has focused onincreasing the transmissible power density ofpolymer gears. The basic geometry used for poly-mer gears is mainly determined from standardtooth proportions as recommended in steel gearstandards. The work described sought to providegears of superior performance designed to non-standard forms where pressure angles, addendaand dedenda are chosen for maximum efficiencyand load carrying capacity. The forms suggestedhere are novel in concept and differ considerablyfrom conventional practice.

The authors are primarily concerned with thedesign and practical aspects of plastic and poly-mer composite gears. The aim is to improve theperformance of these gears in order that the trans-missible power levels are raised so that plasticgears can be employed in a wider range of prod-ucts. Historically, plastic gears have followedsteel gear practice in terms of tooth proportions.However, since the primary manufacturingmethod is a molding process, almost any profilecan be considered (i.e. tooth forms are not a func-tion of the cutting process). This provides anopportunity for designing novel gear forms.

Wear is the predominant mode of failure fordry running gears, but root bending fatigue or

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pitch line fracture are also commonplace (Ref.1). In the case of lubricated plastic gears, pittingcan also arise. Plastic gears are particularly sus-ceptible to temperature. In the case of dry run-ning plastic gears, large friction forces will arise,and these can lead to high temperatures. It is notuncommon for the combined bulk and flash tem-peratures to exceed the melting temperature ofthe plastic, which will result in high wear rates(Ref. 2). Even if the temperatures are below themelting point, the mechanical properties of plas-tics are affected by heat and, for example, theelastic modulus decreases markedly with tem-perature, altering the contact ratio and load shar-ing. Thus, any design procedures and innova-tions which might reduce running temperaturesand/or reduce tooth stresses in plastic gears areworth examining.

The gears described in this paper challengethe accepted wisdom of cylindrical gear designfirst by examining the effects of pressure angle,the benefits of a curved face width gear and theadvantages of tapered face width gears, whichpermit gradual tooth engagement without theneed for a helical tooth form.

Management SummaryCurved face width (CFW) spur gears are not popular in the gear

industry. But these non-metallic gears have advantages over stan-dard spur gears: higher contact ratio, higher tooth stiffness, andlower contact and bending stresses.

CFW gears also provide better operating features. Tooth heightdecreases in the sections away from the gear center, so a lower slid-ing friction is expected with consequences on the gear’s thermalbehavior. They axially locate each other, improving meshing whenmisaligned. There also are no axial forces inherent in helical gears.

For spur gears, load sharing follows the classic “top hat” shapewith a sudden change when moving from partial load to full load.With a CFW gear, tooth contact and load changing are gradual like ahelical gear. This should lead to smoother running, quieter gearscompared to spurs.

But CFW gears are difficult to design and mount in gear trains. Forcertain tooth geometries, the gear train is sensitive to center distancevariations.

Gears with high pressure angles (> 20°) are generally viewed asunsuitable due to lower contact ratios and higher noise levels. Thesegears, however, benefit from having stronger tooth forms and lowercontact stresses and are more efficient due to shorter sliding dis-tances. With dry-running plastic gears, the increased efficiencymeans lower running temperatures and, because of plastic’sresilience, no noticeable noise difference. If the gears are fully lubri-cated, the high pressure angles result in higher entrainment veloci-ties, leading to advanced lubrication. The authors tested 30° pressureangle gears and “extreme” 40° pressure angle gears with measure-able improvements in strength and running temperatures.

Tapered face width gears can adjust backlash by axial movementof one of them. Such adjustment is much easier than backlash adjust-ment via radial center distance adjustment. The tapered face widthprinciple can be applied to spur and helical internal and externalgears and rack and pinion sets, so it is a versatile means of employ-ing backlash control in, for example, automotive steering systems.

Gabriel Andrei is a professor of mechanicalengineering and vice dean inthe Department of MachineElements & Tribology at the University of Galati, locatedin Romania.

Laurentia Andrei is an associate professor inthe Department ofDescriptrive Geometry,Mechanisms & Tolerances atthe University of Galati.

Douglas Walton is a professor in the Depart-ment of Mechanical Engi-neering at the University ofBirmingham in Birmingham,U.K. Walton is head of theVehicle Technology ResearchCenter and teaches engineer-ing design. His area of exper-tise is in plastic and uncon-ventional gearing.

Bryan N.V. Parsons is a research associate in theDepartment of MechanicalEngineering at the Universityof Birmingham.

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High Pressure Angle GearsThe aim: To investigate the potential of high

pressure angle gear tooth profiles in polymer andpolymer composite materials. A preliminarystudy showed that such tooth profiles could lowercontact and bending stresses and increase effi-ciencies compared with gears of standard propor-tions. Studies on tooth forms showed that wearfor gears molded with high 25° pressure angleswas less than at standard (20°) or low (14.5°)pressure angles (Ref. 3). This raised questions asto what extent the pressure angle might beincreased still further to reduce wear. A briefstudy was conducted for this application to inves-tigate effects of pressure angle on sliding veloci-ty, contact stress, tooth bending stress and effi-ciency. High pressure angle (30° and 40°) gearswere injection molded from acetal (the bench-mark material), a glass-filled nylon (high per-formance polymer composite) and PEEK (to per-mit high temperature operation). Tests were per-formed under dry and oil-lubricated conditions.

High pressure angle gear geometry results ina reduction of the addendum and the eliminationof the top land.

It also results in an increase in the radius ofthe involute at the pitch point, (see Fig. 1), whichwill manifest itself as a reduction in both the con-tact and bending stresses (the section modulusincreasing with pressure angle) as well as areduction in sliding velocities. Figure 2 shows thetransmissible torque per mm face width againsttooth number at constant Hertzian contact stressfor a 40°, 20° and 14.5° pressure angle gears (18teeth is about the minimum for this 40° pressureangle). The 20° and 14.5° pressure angle gearsboth have a contact ratio of 2. Also, Table 1shows the effects of changing pressure angle oncontact ratio and center distance extendibility.

The Lewis form factor for these gears isshown in Figure 3. This factor becomes question-able as an indication of tooth bending strength forthe 40° pressure angle gear since radial load isnot considered. The line of action of the forceacting on the teeth for the majority of the contactperiod falls within the extent of the tooth root.This implies there is no bending of the tooth formand that both flanks of the tooth are in compres-sion. This factor might be important for a numberof brittle materials that have good strength incompression but break under moderate tension.Preliminary Finite Element Analysis (FEA) indi-

Non-Standard Cylindrical Gears

Figure 2—Comparison of torque capacity against toothnumber for different pressure angles.

Figure 3—Showing Lewis form factor against tooth numbers for gears of varying pressureangles.

Figure 4—Showing variation of efficiency with tooth numbers for different pressure angles.

Table 1 Pressure angle (deg) 20 30 40Contact ratio 1.65 1.35 1.12Center distance extendibility (mm) 1.43 0.98 0.37

(30-tooth spur gear, module 2 mm, standard proportions)

40° PA 20° PA CR = 214.5° PA CR = 2

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Torq

ue N

m/m

m

Tooth Number

0 5 10 15 20 25 30 35 40 45 50

0 10 20 30 40 50 60

20° PA standard40° PA20° PA CR =2

20° PA CR = 240° PA14.5° PA CR = 2

Tooth Number

Tooth Number0 5 10 15 20 25 30 35 40 45 50

99

98.5

98

97.5

97

96.5

96

Foam

Fac

tor

Effic

ienc

y %

0.3

0.25

0.2

0.15

0.1

0.05

0

Figure 1—Showing (a) a standard 20° profile compared to(b) a 40° high pressure angle profile.

(a) (b)

99.5

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cates that the profile of the 40° pressure anglegear teeth shows an improvement in tooth bend-ing stress compared to the standard profile, evenallowing for the difference in load sharing.

An unexpected result was the discovery thatthe apparent efficiency increased with pressureangle (Fig. 4). In nearly every case, contact andbending stresses and efficiency for the high pres-sure angle gear indicate improved performance.This higher efficiency mostly results from thereduced contact ratio of the high pressure angledesign. If gears with low pressure angles weremodified, such as by reducing their outside diam-eters, the claimed efficiency benefits wouldlargely, possibly completely, disappear.

Still, high efficiency has two rewards forplastic gears. One is economy, and the secondreward is a lower running temperature for a giventransmitted power. Material properties deterio-rate with increasing temperature so that, at lowtemperatures, the material will be stiffer andstronger. High pressure angle gears are not nor-mally favored because of higher normal loads fora given transmitted torque, low contact ratios andhigher noise. Our preliminary studies show thatother advantages (such as low stresses, higherefficiency) might more than outweigh these con-siderations, resulting in a higher transmittedpower for a given size of gear.

Finally, the higher resilience of plasticsmeans that the operating noise levels are less thanthose for steel. Thus, noise considerations maynot be a problem. During the subsequent tests,mesh temperatures, efficiency and noise wererecorded. Average dynamic coefficients of fric-tion will be back-calculated from measured effi-ciencies, so the results can be compared to gearsmade from the same materials, but of standard 20°pressure angle form. It will be possible to take thedesign a stage further and include high pressureangles with a helical tooth form. This would resultin even higher transverse pressure angles as well asimproved load sharing. Should the results provepromising, other materials might be investigated,such as ceramics. Thus, if high pressure angle gearsare successful, there will be considerable futureresearch and development potential.

Curved Face Width GearsCurved face width gears (Fig. 8), have been

developed in Russia (Ref. 4) and by Gleason inthe U.S., but, due to their complex geometry, fewengineers are aware of their existence. The par-ticular advantages these gears have over standardspur gears are higher contact ratios and lower

Figure 5—Gear tooth flank generation.

A A

β β

Mi

CM

Mi

B

Mf

RbCB

Mf

Rb

Figure 6—Showing the solid primitives used to simulate the curved face width gear teethgeneration.

Figure 7—Simulation of the tooth cutting process.

Figure 8—Pictorial representation of the curved facewidth spur gear, with modified geometry.

tool 2tool 1

blank

β β

α α

blank tool 2tool 1

β βcutting edges

24

Ø64

a) b)

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lower sliding velocities and, when run in oil,enhanced lubrication conditions were expected.The shape of these gears makes the design of amolding die commercially impractical. Hencethis proposed design is suitable only formachined gears, plastic or metal. Such gearsmust be mounted with close control of relativeaxial position, generally not required for spur orhelical parallel-axis gears (Ref. 5).

The kinematics of the gear generationprocess. Two different cutters were used for theconcave and convex tooth flank generation.Figure 5 illustrates the kinematics of the genera-tion process, where the tools are working in thehalf-width plane of the blank. The normal sectionsof the cutters show straight lines for the imaginaryrack-cutter flanks, with a zero pressure angle. Thedesigned geometry for the tooth flanks is based onthe following kinematic principles:

• the cutting tool performs rotational motion (A)about its inclined axis, leading to a curved toothalong the gear face width and a variation intooth height;

• the gear being generated is rotated about its axis (B) and is provided with translational motion(C), tangential to the base circle of the gear(Rb), in order to obtain the rolling motionrequired for involute tooth form generation.

Two phases are necessary in order to generatethe entire gear. The cutting process starts with thetooth-by-tooth concave flank generation (Fig. 5a)followed by the convex flank generation (Fig.5b), where a new tool is used, properly posi-tioned relative to the already cut concave flank.

Simulation of the gear generation process.The modified geometry of the curved face widthspur gears described here requires a theoreticalinvestigation before the gears can be made. Anumerical simulation of the gears’ generation andmesh, based on the traditional conjugate surfacegeneration theory, was developed by one of theauthors (Ref. 6). The complexity of the gear toothgeometry implied complex mathematical calcula-tions and large computer programs. The authorsused the advantages of solid modeling techniquesin order to obtain a complete representation of thegear. The gear design model can be further used forthe gear generation error analysis, for finite ele-ment analysis and for rapid prototyping.Figure 11—Path of contact after 3° rotation.

Figure 10—Path of contact after 1° rotation.

Non-Standard Cylindrical Gearscontact and bending stresses, due to the doublecurvature of the tooth flanks. They axially locateeach other, offering better meshing in plane mis-alignment conditions, and there are no axialforces, which are inherent in helical gears. Thedifficulty in gear train mounting and, for certaintooth geometries, a sensitivity to center distancevariations are the only two serious disadvantages.

The curved face width spur gear described inthis article differs from previous curved facewidth gears in that they have a modified geome-try with a variable tooth height and width alongthe gear face width. Reduced bending stresses,

Figure 9—Path of contact for the initial gear toothcontact.

To simulate the kinematic generation of thecurved face width gear flanks, the blank and thetools are modeled by cylindrical and conicalsolid primitives using solid modeling methodswith conveniently chosen dimensions. Work pre-viously done by the authors on measuring andcomparing the performance of plastic gears ledto a “standard” gear being used that has a 20°pressure angle, 30 teeth of module 2 mm andaddendum equal to the module. In order that thecurved face width gear could be compared tostandard geometries, the first curved face widthgears were made the same size. Thus, a gearblank of 64 mm diameter and 24 mm face width(Fig. 6) was made. The rotational motion of thetwo cutters about their own axes was replaced bythe solids “Tool 1” and “Tool 2,” obtained byrotating the tool’s normal sections around thesame axis, using the REVOLVE command. Therotational motion of the blank was performedincrementally by using a ROTATE3-D com-mand, with an angular increment; its translation-al motion was also done incrementally.

The tool successively intersects the blankuntil the generation of a tooth flank is completed(Fig. 7). The removal of the “material” is doneusing a Boolean subtraction operation, per-formed by the SUBTRACT command. The auto-matic development of the procedure enables adynamic view of the process and a representa-tion of the gear to be made (Fig. 8).

Gear meshing simulation. A theoreticalanalysis of the gears in mesh shows that the con-tact line depends on the magnitude of the trans-mitted torque and on the gear alignment. Whenthe concave side of the pinion tooth enters mesh,contact starts at the tooth dedendum, at the endof the gear face width and extends to the mid-point of the face width, leading to a concavecurve of contact. Similarly, contact on a tooth’sconvex flank determines a convex curve of con-tact, starting at the tooth dedendum, at the gear’smid-section. The facilities offered by the solidmodeling technique enabled the gear mesh to besimulated, in order to investigate the path of con-tact. Using the virtual models and the dynamicsimulation of the gear meshing, the theoreticalpath of contact was obtained from a virtualmodel of the ideal gear train.

Figures 9–11 show several representations ofthe path of contact on the pinion teeth’s convexflanks (a 0.1 mm interference was used for thepath representation). The initial contact shown inFigure 9, obtained by hand positioning of the

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Table 2Torque Von Mises stresses Maximum deflection[Nm] [N/mm2] [mm]4 1.18 0.0146 1.77 0.0218 2.36 0.02710 2.96 0.03512 3.55 0.041

virtual gear, corresponds to an arbitrary pointalong the theoretical line of contact. The geartrain was next rotated by 1° (Fig. 10) and then 3°(Fig. 11), to obtain the contact path during gearmeshing. Analyzing the tooth contact, it is obvi-ous the “contact line” is a spatial curve, whichchanges in curvature during the mesh cycle.

Bending stresses and deflections in curvedface width gears. The authors designed a modi-fied geometry for the plastic curved face widthspur gears in order to enhance the load capacityand to avoid excessive tooth deflection. The data-base obtained from the gear generation simula-tion was used to analyze the gear tooth resistance,using finite element analysis procedures. Stressesand stiffness of a curved face width gear werecompared to those of a standard spur gear.

Curved and straight spur gears were modeledusing COSMOS/M set to the following geometri-cal parameters: modulus 2 mm, 30 teeth and 24mm face width. Poisson’s ratio for the plasticmaterial used in the gear’s manufacture, Ertalon66SA, was 0.3; and the tensile modulus of elas-ticity was 3,450 MPa. The analysis was devel-oped for two teeth in mesh under the appliedloads given in Table 2 and shown in Figure 12.

Figure 12—Showing the applied loads.

y

x

Table 3Torque Von Mises stresses Maximum tooth[Nm] [N/mm2] deflection [mm]4 10.78 0.0176 16.47 0.0228 21.55 0.03310 27.59 0.04312 32.64 0.051

Table 2 lists the results of the tooth bendingstresses and deflections for a standard region ofthe curved face width gear—a continuum divid-ed into 18,348 linear tetrahedral elements inter-connected at 3,721 nodes. For the maximumload applied (12 Nm), Figures 13 and 14 illus-trate the distribution of the Von Mises stressesand the tooth deflections, respectively.

Figure 13 shows that maximum stresses areconcentrated on the dedendum of the tooth withload applied close to its addendum. Figure 14shows that maximum tooth deflection appears inthe gear mid-face width, at the tooth addendum.Due to variable tooth height and width, the toothstiffness can differ, as shown by the variation oftooth deflection along the gear face width. It canbe seen that, close to the gear face width ends,the tooth deflection decreases by approximately20% compared to the gear mid-section toothdeflection. A similar analysis, done for thecurved face width gears with the load applied onthe concave flanks, showed higher bendingstresses concentrated on the tooth dedendum andsimilar maximum displacements of the toothaddendum, at the gear’s mid-face width.

The analysis of the spur gear tooth bendingresistance was also developed, for the sameapplied loads. The results are given in Table 3and show the high bending stresses developedcompared to the curved face width gears; thebending stresses developed in standard spurgears being approximately 10 times higher thanthe curved gear tooth stresses. The result of thefinite element analysis on the bending resistanceis summarized in Figure 15.

Tapered GearsThe tapered gear design utilizes the principle

of access and recess action within a single gear.At the mid-plane, the geometry of the tooth formis substantially standard. The gear has a lineartaper so that the profile is modified to accessaction at one end of the gear and recess action atthe other. The matching meshing gear must havea similar taper and the access region must meshwith the recess portion of the mating gear. Thiscondition can be satisfied when the direction ofthe taper is inverted (Fig. 16). For spur gears,this profile results in the flanks of the teeth beingslightly helical and a small end thrust will begenerated when torque is transmitted.Interestingly, the direction of the axial force is

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Non-Standard Cylindrical Gears

Figure 15—Comparison of maximum stresses in curved face width and standard spur gears.

Torque

[N/mm2]

Von

Mis

es s

tres

ses

[Nm]2 4 6 8 10 12

curved tooth-load applied on the convex flankcurved tooth-load applied on the concave flankspur gear tooth

30

20

10

0

Figure 13—Von Mises stress distribution.

Figure 14—Distribution of curved face width tooth deflections.

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the same for both drive and overrun conditions. Another feature of the gear profile is the abil-

ity of the gears to compensate for center distancevariation or backlash by adjusting the relativeaxial position of the gears. This results from thetaper in the tooth width. The taper of the gears isin all proportions linear with axial position, sothat the addendum, dedendum and tooth width allchange proportionally (Fig.17). This produces ashape that has a natural draft angle on all exter-nal surfaces, facilitating the manufacture of thesegears by some molding process. This processmight be cavity molding for plastic gears or sin-tering or forging for metal gears.

The contact ratio varies across the plane ofthe meshing gears, and the slight helical profileof the flanks provides a small phase differencebetween the ends of the gear. The important fea-ture is that the contact ratio changes smoothly asthe mesh progresses so that the teeth are gradual-ly loaded and unloaded. This feature should helpto make these gears quiet in operation. The beamstrength of the tooth profile should be compara-ble with conventional gears’ tooth profiles. Thesecond moment of area is that the tooth root islarger for the tapered gear profile than for a stan-dard tooth.

The tapered gear can also be made helical(Fig. 18) with a different helix angle on the oppo-site flanks of the teeth. In this case, left- andright-handed helical gears are required. A specialcase is where the helix angle is made so that oneflank of the tooth is straight while the other ishelical. For example, an axial reaction forcemight be generated while the gear pair is drivingand no such force is generated during overrun.

ConclusionsThree novel gear types are examined in this

paper, each offering some benefits over conven-tional cylindrical gears. High pressure anglegears of the same materials have withstood high-er loads than gears of standard geometries. Noisehas not been a problem. Curved face width gearsare shown to provide lower stresses compared tospur gears of the same dimensions. These gearsare manufactured by cutting the plastic, andthough this form of production results in lessaccurate gears, these gears run satisfactorily anddemonstrate their capacity to operate at highertorque levels than conventional spur gears.Tapered face width gears are modeled and man-ufactured using rapid prototyping techniques.The main advantages of the tapered gears are the

ease of manufacture by molding techniques,including sintering and forging. Kinematically,they offer a degree of action that effectively pro-vides mesh overlap. The gears provide toleranceto mold shrinkage due to the adjustability of themesh by axial displacement. r

References1. Walton, D, A.B. Cropper and D. Weale. “Failure Mode Analysisof Plastic Gears,” Proceedings of International TribologyConference, Nagasaki, Japan, 2001.2. Hooke, C.J., K. Mao, D. Walton, A. Breeds, and S.N. Kukureka,“Measurement and Prediction of the Surface Temperature inPolymer Gears and its Relationship to Gear Wear,” TranslationASME Japan of Tribology, 92-Trib–12, Oct. 1992.3. White, J. Ph.D. Thesis, “Design and Evaluation of PolymerComposite Spur Gears,” The University of Birmingham, U.K.,1999.4. Sidorenko, A.K. “70 NKMZ” Gear Train (in Russian),Mashinostroienie, Moscow, 1984.5. Andrei, L., G. Andrei, A. Epureanu, N. Oancea, and D. Walton.“Numerical Simulation and Generation of Curved Face WidthGears,” International Journal of Machine Tools and Manufacture,42 (2002), pp. 1–6.6. Andrei, L. “Study of the Plastic Curved Face Width Spur GearsGeneration and Behaviour,” Ph.D. Thesis, The University ofGalati, Romania, 2001.

Figure 16—Showing two tapered gears in mesh.

Figure 17—Comparison of the extremes of profile for thetapered gear.

Figure 18—Showing a pair of helical tapered gears.

t = tooth width thin endT = tooth width fat endh = change in addendumd= change dedendum

d

h

T

t

This paper was printedwith the friendly permis-sion of the previous pub-lisher, VDI-Verlag ofDusseldorf, Germany. Itwas printed in 2002 inthe proceedings for theInternational Conferenceon Gears, VDI report1665, Vol. 1, pp. 311–326.