A Kinematic Analysis of Meshing Polymer Gear Teeth (1)

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A kinematic analysis of meshing polymer gear teethM Karimpour1,K D Dearn2∗, and D Walton2

1Mechanics of Materials Division, Department of Mechanical Engineering, Imperial College London,

South Kensington Campus, London, UK 2Power Transmission Laboratory, School of Mechanical Engineering, The University of Birmingham, Edgbaston,

Birmingham, UK

The manuscript was received on 16 December 2009 and was accepted after revision for publication on 30 April 2010.

DOI: 10.1243/14644207JMDA315

Abstract: This article describes an investigation into the contact behaviour of polymeric geartransmissions using numerical finite element (FE) and analytical techniques. A polymer gearpair was modelled and analysed using the ABAQUS software suite and the analytical results werecalculated using the BS ISO 6336 rating standard. Before describing the results, the principlesof the strategies and methods employed in the building of the FE model have been discussed.The FE model dynamically simulated a range of operating conditions. The simulations showedthat the kinematic behaviour of polymeric gears is substantially different from those predictedby the classical metal gear theory. Extensions to the path of contact occur at the beginning andend of the meshing cycle. These are caused by large tooth deflections experienced by polymergear teeth, as a result of much lower values of stiffness compared to metallic gears. The prema-ture contact (occurring at the beginning of the meshing cycle) is hypothesized to be a factor inpitch line tooth fractures, whereas the extended contact is thought to be a factor in the extreme

wear as seen in experiments. Furthermore, the increase in the path of contact also affects theinduced bending and contact stresses. Simulated values are compared against those predictedby the international gear standard BS ISO 6336 and are shown to be substantially different. Thisis particularly for the case for bending stresses, where analytically derived values are indepen-dent of contact stiffness. The extreme tooth bending and the differences between analytical andnumerical stresses observed in all the simulations suggest that any future polymeric gear-rating standard must account for the effects of load sharing (as a result of tooth deflection) and friction(particularly in dry-running applications).

Keywords: spur gears, polymers, steels, friction, temperature, kinematics

1 INTRODUCTION

Polymeric materials were first employed in gearing applications in the 1950s and have developed intoa large range of applications. The majority of thesetend to be in motion control (low load, temperature,and speeds). However, the development of the superengineering polymers and polymer gearing technol-ogy has pushed the application limits further intomoderate power transmission functions. The benefits

∗Corresponding author: Power Transmission Laboratory, School of Mechanical Engineering, The University of Birmingham, Edgbas-

ton, Birmingham B15 2TT, UK.

email: [email protected]

of employing such polymers in gearing applications

emanate from their low cost, low weight, resilience,and ability to run dry. A reduced load capacity and,crucially, a poor temperature resistance tend to limitthe use of such gears.

High temperatures reduce the mechanical prop-erties of polymers much more than metals. One of the consequences of this is a reduction in the trans-mission accuracy of polymer gears, giving rise totransmission errors caused by the large tooth deflec-tions. Loss of contact ratio due to the following twofactors should not be ignored in performance specifi-cation, namely a lower manufacturing accuracy grade

and an allowance for thermal and hydroscopic expan-sion. Klein Meuleman et al. used a quasi-static FEmodel to simulate transmission errors in a polymericgear set over a range of operational conditions [1].

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102 M Karimpour, K D Dearn, and D Walton

With this, empirical measurements, and an analyticalmodel, they were able to minimize composite tooth-to-tooth transmission errors through novel geometricmodifications.

This was not the first time FEA had been usedto analyse the kinematic behaviour of non-metallicgears. Walton et al. used the FE method to study load-sharing effects and were among the first tosuggest that, as a result of thermal softening, thecontact ratio of a polymeric gear set was greaterthan that predicted by theory [2]. Using a non-dimensional analysis based on a variety of oper-ational parameters, the actual contact conditionsin terms of a contact ratio were determined by a gear elasticity parameter. This work then led tofurther studies on the beneficial effects of perfor-mance and profile modification, such as backlash

allowance [3] and tip relief [4], in non-metallicgears.

The FE method was also employed by Senthilvelanand Gnanamoorthy to assess the effect of tooth fil-let radius on gear performance [5]. They employed abasic single-tooth model, loaded only at the tip, anddisregarded contact ratio effects, calculating an equiv-alent line load. This indicated, as would be expected,higher bending stress levels in teeth with smaller filletradii. There were similar increases in tooth deflec-tions, having the combined effect of shortening thegear life.

Van Melick used both FEM and analytical methodsto investigate the influence of stiffness on dissimilarmaterials, gear kinematics, and stresses (a steel gearis typically 70 times stiffer than an equivalent poly-oxymethylene polymer gear) [6]. He suggested thatpolymeric gear kinematics are different to those pro-duced by the classical gear theory, with the effect of increasing stresses and, through an extended path of contact, influencing the wear resistance of the plasticgear. An interesting aspectof this study was the discov-ery of a reciprocating motion at the root of the drivengear as the teeth disengage at the end of the meshing cycle. The FEA model used employed a quasi-static

solution.FEA has been used for some time to model the com-

plexities of the steel gear theory. One of the mostdetailed and accurate FEA simulation on metallicgears was arguably conducted by Mao [7]. He utilizedFE to investigate the effects of micro-geometry modi-fications on the reduction of transmission errors andfatigue damage in a metallic helical gear set. To achievethe maximum possible geometrical accuracy, insteadof importing a model into the FE software (ABAQUS),the gear geometry was mathematically generatedusing Python Script. Advanced surface-based tie tech-

niques between nodes were utilized to obtain a high-quality mesh for contact. The novel aspect of this wasthat instead of a multi-simulation technique (quasi-static), the whole gear meshing process was achieved

by one simulation, making it possible to study thereal rolling and sliding contact. Having developed anaccurate FE model, shaft misalignment and assem-bly deflection effects on gear surface durability and

transmission error were also studied. This revealedan effective way of reducing transmission error andgear surface wear damage through micro-geometry modification (i.e. crowning, tip relief, and leadcorrection).

2 THEORY

The complexities of polymer gearing offer an idealapplication for the finite-element method. Many researchers have used it to study the fundamentalkinematic and kinetic behaviours of polymer gears. Inrecent years, as computational power has increased,along with the sophistication of commercial finite-element software, researchers have employed numer-ical techniques not only to study the behaviour of polymeric gear transmissions, but also to modify andoptimize gear trains for specific applications.However,these polymer gear simulations are all based on quasi-static solutions (as detailed above). The following section discusses current rating methods for non-metallic gears and discusses those that are deemedmost appropriate to the unique behaviour of plasticgears. The development of a dynamic non-linear FE

model to study the kinematic behaviour of a poly-mer gear transmission is then described. Employing a dynamic solution allowed the whole meshing cycleto be continuously simulated, resulting in a betterunderstanding of tooth bending effects, and the rami-fications of these effects for other aspects of polymergear performance. In addition to this, it provided ananswer as to whether gear-rating standards,developedspecifically for metallic gears, could be used to designand rate polymer gears.

2.1 Design and rating standardsGear-rating procedures adopt different assumptionsto give the best approximation of stresses within andaround gear teeth; hence, it is inevitable that differ-ent procedures will predict different values according to the assumptions made. In a comparison of avail-able procedures for the rating of non-metallic gears(BS6168, ESDU68001, and Polypenco),Walton and Shiobserved large discrepancies between different meth-ods and suggested that an experimental investigation

was required to assess which rating standard was themost accurate [8]. Following on from this, Cropper

compared the stress levels predicted by a similar rangeof rating standards against an FE analysis considering load sharing and showed BS ISO 6336 (method B) tobe the most accurate [9].

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Kinematic analysis of meshing polymer gear teeth 103

2.2 Bending stress estimation

All bending stress analyses are based on the Lewisbending equation, which has gone through many minor modifications since its first introduction. For

instance, ISO 6336:1996 [10] has adopted the exactlocation of maximum stress experimentally found by Hoffer [11]. ISO 6336 (methods B and C) as well as

ANSI/AGMA 2001 take into account the stress concen-tration at the tooth root fillet, which is ignored by theother methods. A unique characteristic of method B inISO 6336 is that it enables the tooth form factor to becalculated from first principles (an extensive process,butit accounts for anytoothgeometry); other methodsuse tabulated data instead.

2.3 Contact stressesThe basis of all contact stress calculations is the classi-cal Hertzian contact analysis. Stress levels are usually calculated at the pitch point where the sliding fric-tion may be assumed to be zero as well as negligiblebending deflection due to contact. The only differ-ence between the standards is the load-sharing effect.

ANSI/AGMA 2001 assumes no load sharing, whereasISO 6336 does consider a load-sharing factor in thecalculation of contact stress. When considering poly-meric materials, load sharing cannot be disregardedand hence ISO 6336:1996 would seem to be the most

accurate available method. When comparisons aremade against a standard in this article, they are madeagainst ISO 6336:1996 method B (from this pointforward referred to as the Standard).

3 FINITE-ELEMENT MODEL FOR PLASTIC GEARTOOTH KINEMATIC ANALYSIS

All gear theories as well as the gear-rating standardsare based on rigid-body kinematics. This assumes thatno significant deformation occurs in the tooth, pre-serving the involute profile. This is probably a valid

assumption in the metallic gear theory; however, it isunlikely to be the case for plastic gears in all but thelightest of applications.

An accurate two-dimensional model of the Birm-ingham benchmark gear was developed and importedinto a state-of-the-art FE package (ABAQUS). A sum-mary of the gear geometry is given in Fig. 1. In orderto reduce the computational time of the simulation,only ten meshing teeth were modelled, with two pairsof mating teeth meshed to a higher mesh density to enable the extraction of accurate data (shown inFig. 2). The mesh in this region has relatively small

elements dimensions (i.e. approximately 0.04 mm).In addition to this, the use of a structured mesh of quadrilateral elements following the involute profileof the teeth and enhanced hour-glass control adds

Fig. 1 Geometric specification of the ‘Birmingham

Benchmark’ gear geometry

to the accuracy of the solution. The size of the ele-ments was determined based on the simulation of metallic gear teeth from reference [12]. As the polymermaterial used in the simulations has a lower stiff-

ness than the material used by Wang and Howard, thefield variable gradients are much smaller than those of metallic gears. Hence, the same mesh density with astructured mesh throughout the tooth has been used.This will be conservative in terms of mesh refinementbut should guarantee accurate results. The rest of theteeth were modelled with a coarse mesh to increasecomputational efficiency.

As the stress elements do not have rotational degreesof freedom, an advanced surface-based tie technique

was utilized to attach the nodes at the gear hubto a rigid-body shaft, later used to apply rotationalloads and boundary conditions (also shown in Fig. 2).

In order to minimize fluctuations between contact-ing surfaces, an inertia load was assigned to theshaft to dampen unwanted vibrations. The elementsused (CPE4R) were four-node bilinear plain strainquadrilaterals with reduced integration [13].

Solutions were obtained in three steps, namely approach, loading, and rotation, selected to reducethe occurrence of contact inaccuracies and vibration.

A smooth step profile was defined as custom ampli-tude to apply all the boundary conditions and loadsas smoothly as possible. Boundary conditions andloads were applied at the drive shaft reference points

(centres), according to the solution step sequence.Four separate surfaces were defined on each mesh-ing gear, to isolate each zone of the meshing cycle.Theboundary conditions were selected as follows.



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Fig. 2 (a) Loads and boundary conditions and (b) generated mesh representing the gear

1. Initial step: Boundary conditions were applied toconstrain completely the driving and driven gears.

2. Approach: A rotational constraint was removedfrom the driving gear, then a rotational boundary condition was activated, closing the ‘gap’ betweeninteracting tooth flanks. The magnitude of rotation

was equivalent to the theoretical backlash for thebenchmark geometry.

3. Loading: A rotational constraint was removed fromthe driven gear (thus both gears were now free to

rotate). Opposing rotational moments were thenapplied on the driving and driven gears, essentially resisting each other.

4. Rotation: Rotational boundary conditions wereapplied about the drive shaft centres. The magni-tude of the rotations allowed at least one pair of gear teeth to complete one meshing cycle.

Simulations were conducted for combinationsshown in Table 1. Load-sharing data were smoothedusing the Stavitzky–Golay [14] method implemented

within MATLAB. This was to compensate for the fluc-

tuations caused by the inevitable numerical errorsand contact inaccuracies (this was a result of the way the contact algorithm is defined in ABAQUS). Mate-rial properties for the simulations were taken from the

Table 1 Simulation schedule

E (GPa) T (Nm) µ Notes

Test 1 3.1 5 0Test 2 3.1 8 0Test 3 3.1 11 0Test 4 3.1 15 0Test 5 2.5 8 0 Analogous to POM (Delrin® 500)

at 35 ◦CTest 6 2 8 0 Analogous to POM (Delrin® 500)

at 50 ◦CTest 7 1 8 0 Analogous to POM (Delrin® 500)

at 90 ◦CTest 8 3.1 8 0.3Test 9 3.1 8 0.5

polymer manufacturer Du Pont [15]. For each simula-tion, Poisson’s ratio was taken as 0.35 and the materialmodel used was isotropic linear elastic.

4 RESULTS

With the FE model completely defined, simulation

and analysis were initiated to test a variety of com-mon operational conditions for polymer gears. These

were specifically selected to examine the explicit kine-matic responses and to assess the suitability of using



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Fig. 3 Extensions to the path of contact extracted from

the FE model simulating an applied load of 7 N m(E = 3.1 GPa)

the Standard for the rating of polymeric gears. Stresses were then examined and compared against thosepredicted by the Standard, highlighting the shortcom-ings of this rating method. Finally, the effects thatoperational conditions (for example, temperature andfriction) hadon theperformance of thegears are given.

4.1 Kinematics

4.1.1 Path of contact

The classical gear theory predicts that the path of contact of a pair of meshing gears lies on a straightline (effectively being the locus of all contact points)between the first (FPC) and last points of tooth con-tact (LPC) (governed by the addendum radii). Figure 3shows the locus of contact points extracted fromsimulation 2 (specified in Table 1), plotted againstthe theoretical benchmark geometry. It exhibits, asexpected, a line of contact between the addendumradii as predicted by the theory. However, there are

also periods of premature contact (Fig. 4) occurring before the theoretical FPC and extended contact afterthepredicted LPC. Theextensions do notcoincide withthe linear section of the line but are almost perpen-dicular to it. The extended contact (at the LPC) liesalmost exactly on the addendum radius of the wheel,suggesting that this extraordinary contact occurs onthe involute area of the tooth flank. This was not thecase for the premature contact region, suggesting thatcontact occurs outside the involute flank on the toothtip.

The main reason for this behaviour lies with the

verylargetoothdeflections observed in polymeric gearteeth, deformations that are much greater than thosefound in metallic gearing. These deflections signifi-cantly alter the (theoretical) tooth geometry and are

Fig. 4 A schematic of the premature contact caused by

large tooth deflections

governed by themechanical propertiesof the polymer.

This can be verified by the results presented by Wang and Howard [12]. It was shown that in soft metallicgears (i.e. aluminium gears), the load share ratio pro-file differs from the theoretical curve and that withgreater load, the extent of premature and extendedcontact would be increased.

During the premature contact stage, the tip of thetooth and top land of the wheel make heavy contact

with the pinion tooth in the proximity of its pitch point.Contact then continues up the tooth flanks until they are tangent to one another, followed by ‘normal’ con-tact until the theoretical LPC. This was likely to be the

result of a deflection ‘lag’ (i.e. it was the result of suc-cessive teeth already being deflected). After this point,the deflected teeth attempt to return to their originalform. In doing so, the tip of the pinion slides along the flank of the wheel in the direction of the pitchpoint. This was the reciprocating motion that was firstsuggested by van Melick and is discussed below.

Van Melick [6] suggested that the reciprocation of the tooth tip at the mating root accounts for the dis-tinctive wear patterns described by Breeds et al . [16],such that they govern the mechanisms of wear in poly-meric gearing. This is a strong statement given that

Walton et al. [3] reported that wear was the predomi-

nant form of failure in polymer gears.Giventhe locality of the premature contact region (i.e. at the pitch lineof the pinion), this may also be a contributory factorin pitch line fractures (PLFs). This is particularly inter-esting considering the work of Cropper [9], who notedthat for the benchmark geometry (i.e. when the pinionand wheel are geometrically identical) PLF occurs only on the pinion. This,however, requires further research.

4.1.2 Load sharing

Thefractionof theapplied load transmitted byindivid-

ual gear teeth was governed by the theoretical contactratio, which, in case of the benchmark geometry, was1.65. Physically, this means that for approximately two-thirds of the meshing cycle, two pairs of teeth



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Fig. 5 The load share ratio against the roll angle predicted by simulation 2

carry the load, whereas for the remainder a single paircarries it. The division of the applied load between dif-ferent meshing tooth pairs is defined as the load shareratio. The Standard defines the load share fraction (formetal gears) as 1/3, 2/3, and 3/3 (i.e. at the first pointof contact one-third of the load is carried by a mesh-ing pair and this increases to two-thirds at the point of single tooth contact where finally the entire load is car-ried). The reverse of this then occurs as the tooth pairruns out of the mesh.

Figure 5 shows the load share ratio plotted againstthe roll angle in simulation 2. It reaches a maximum of approximately 0.8, suggesting that single tooth con-tact does not occur in polymeric gears under suchconditions, pushing the real contact ratio above 2.This clearly invalidates the load sharing predicted by the Standard. The parabolic form of the load shareratio also suggests that deflections are shared evenly between the meshing pair, which is in agreement withthe work of Klein Meuleman et al. [1]. With the max-imum load occurring at the pitch line of the teeth, itis in this area where maximum contact and bending stresses are expected. Another interesting aspect of Fig. 5 is the increase in the roll angle (the rotationalangle of gear body from FPC to LPC). Theoretically,

the roll angle of the benchmark geometry should be19.84◦; however, this value increases in the simulationsby 30per cent to25◦. The reason for this is the cumula-tive effect of the extensions to the path of contact andultimately the large tooth deflections.

4.2 Stress analysis

In the following sections, Von Mises stress was chosenas the scalar representative of the stress state at a givenpoint. At the tooth root, this would translate into themagnitude of tensile stresses caused by bending. In

the case of the contact patch, this would represent thestresses caused by contact between the mating toothflanks. As in the Standard, this could then be com-pared with the limits of the material in order to provide

Fig. 6 Position of nodes used to locate the position of

maximum: (a) bending and (b) contact stresses

a rating for it as a gear material (i.e. Von Mises yieldcriteria). The nodes from which the predicted stresses

were extracted from the tooth flank and root are shownin Fig. 6.

4.2.1 Maximum bending stress

The extracted bending stress predicted by simulation 2for each node is plotted against the roll angle in Fig. 7.



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Fig. 7 Bending stresses at various nodes against the roll angle predicted by simulation 2

It takes a similar form to the load-sharing graph above;it also shows the location of the nodes from which data

were taken. The maximum stress occurs at the ‘centre’of the fillet radius (node 2539) and when the point of contact is at the pitch point. This is not, however, thepoint at which the Standard predicts that maximumstress will occur. The Standard predicts that the great-est stress is induced at the tooth tip (i.e. the first or lastpoint of contact).

The difference between the assumption made inthe Standard and the location of maximum stress pre-

dicted in the simulation has a significant effect on thepredicted bending stress values. The simulation pre-dicts a maximum bending stress of 25.9 MPa, whereasthe Standardcalculates it tobe 38 MPa.This is a signifi-cant difference of around 46 per cent. Tooth deflectionis the most likely explanation for this discrepancy.The Lewis equation, on which the Standard is based,

assumes a bending stress maximum at tip loading (being the point at which the lever arm is greatest);however,itdoesnotconsiderthatatthispoint,theloadis being carried by (at least) two teeth pairs reducing the tangential force. The effect of load sharing reducesthe magnitude of the applied load, with the peak at thepitch point, where maximum stress occurs.

4.2.2 Maximum contact stress

A similar approach was adopted to establish the posi-

tion of maximum contact stress across the tooth flank.Figure 8 shows the corresponding stress profile plot-ted against the roll angle along with the nodes along the tooth flank. Once again, the maximum stress valueoccurred at the pitch point. A maximum contact stressvalue of 38.69 MPa was predicted. This is in closeagreement with the Standard that predicts a value of

Fig. 8 Contact stresses at various nodes against the roll angle predicted by simulation 2



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Fig. 9 Induced stress contours during meshing simulations 2 (left) and 4 (right)

contact stress of 37.57 MPa (a difference of 2.9 percent).

Caution should, however, be exercised when using the Standard to calculate contact stresses. Thesingle-pair tooth contact factor, used in the Standard,converts thecontact stresscalculatedat thepitchpointto that at the inner point of contact [17]. This may gosome way to explaining the discrepancy but certainly requires further examination. The stress contours withthegearteethduringloading,duringsimulations2and4, are shown in Fig. 9.

It is also interesting to note the manifestation of theextended contact points in Fig. 8. Evidence of prema-ture contact appears in the double peak seen in theplot of node 2139 and the single peak in node 2143

(representing initial contact when the tip/top land of the wheel tooth collides with the pitch point of thepinion tooth). Extended contact is apparent in theincrease in the stresses seen in node 2093.

4.2.3 Effect of applied loads

The analysis conducted to assess the effect of load onthe contact and bending stresses revealed some inter-esting results. Four separate simulations were carriedout to simulate a range of applied loads. A compar-

ison of the stresses generated from the simulationsand those calculated using the Standard is given inTable 2. The values of contact stress found using thesimulations correlate well with the Standard’s values.



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Table 2 A comparison of FE and BS ISO 6336-derived

stresses for a range of loads

Contact stress (MPa) Bending stress (MPa)

Torque

(Nm)

FE

simulation

BS ISO

6336

FE

simulation

BS ISO

6336

Test 1 5 31.57 29.71 17.13 23.80Test 2 7 38.70 37.58 25.94 38.08Test 3 10 44.20 44.06 34.36 52.36Test 4 15 48.75 49.71 42.31 66.64

This is not the case for the bending stresses however, where discrepancies between simulated and the Stan-dard values increase with load. This makes the Stan-dard (part 3) even less suitable for rating polymer gearsin demanding applications.

These stresses should be carefully interpreted, asthey are those that occur at the pitch point. It has,however, already been shown thatbecause of extendedcontact (specifically the interference of tooth tip andmating flank), high contact stress peaks occur during the early and later stages of the meshing cycle. Thestresses shown in Table 2 do not account for these.

Figure 10 shows the kinematic effect of increasing the loads on meshing gear teeth. It indicates that themaximum load carried by a single gear tooth pairdecreases as the applied load increases. This impliesthat the higher loads induce greater tooth deflections,increasing the real contact ratio and load sharing.There are other kinematic effects induced by thesedeflections, namely larger extensions to the path of thecontact. Figure 11 compares the paths of contact foreach of the load conditions specified in Table 1. Thissuggests that greater tooth deflections cause prema-ture contact to occur earlier (it therefore follows that

Fig. 11 Effect of applied loading on the path of contact

initial contact occurs closer to thepitch point)and thatextended contact lasts longer. Thus, the overall lengthof the contact path is increased.

4.3 Effects of operating conditions

4.3.1 Temperature

The effect of temperature in this case is taken to rep-resent the reduction in the material stiffness becauseof elevated operating temperatures. This is a simpli-fication of the physical manifestation of high geartemperatures; however, in terms of the kinematicbehaviour of the gear teeth, the reduction in modulushas a strong influence. Table 3 summarizes the reduc-tion in stiffnesswith temperature rise. These values areassigned to the materials specified in simulations 2, 5,

Fig. 10 Effect of applied loading (torque) on load sharing



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Table 3 Material properties as a function of gear body

temperature

Modulus of elasticity (GPa)

Torque(Nm)

Equivalentgear body temperature(◦C)

Test 2 3.1 7 23Test 5 2.5 7 35Test 6 2 7 50Test 7 1 7 90

Material data taken from reference [13].

6, and 7 (all transmit a load of 7 N m), which are basedon the homo-polymer polyoxymethylene as specifiedby Du Pont [15].

A decrease in stiffness increases the level of loadsharing experienced by individual gear teeth (Fig. 12)and has a similar effect on the path of contact. Theinfluence of temperature on kinematic behaviour is

similar to the effects of increasing load, as shown inFig. 10. A comparison of simulated and Standard derived

bending and contact stresses is shown in Fig. 13.Once again, contact stresses correlate well with oneanother and decrease with a reduction in stiffness(an effect that was expected as the Standard derivedcontact stresses account for contact stiffness). Thishas implications in terms of gear operating perfor-mance, particularly at elevated temperatures. Thiscould explain why Kono [18] observed an increase in

Fig. 12 Effect of increasing temperature (i.e. the reduction in modulus) on load sharing

Fig. 13 Comparison of contact and bending stresses versus Young’s modulus, at the pitch point



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Fig. 14 Effect of friction on load sharing

Fig. 15 Comparison of contact and bending stresses versus coefficient of friction

gear durability during his experimental programme at

150 ◦C. Interestingly, bending stresses do not correlate well. FE-determined bending stresses increase withstiffness, whereas the Standard derived stresses areoverestimated and remain constant. This is becausethe Standard bending stress is, in theory, dependentonly on geometry and load. For BS ISO 3663 to becomea better approximation of polymeric gear bending stresses, it should incorporate a material factor that

would account for load sharing due to tooth bending.

4.3.2 Friction

The Standard does not consider the effects of friction,because it is written for metallic gears that are alwayslubricated and where frictional forces are low. No con-sideration is given to the effect friction will have on

contact and bending stresses of polymer gears. Yet,

Walton et al. [19] have shown that coefficients of fric-tion in dry-running polymeric gears can be as high as0.8. Frictional forces induced by coefficients of fric-tion of this magnitude must affect the position andmagnitude of contact stresses and to a lesser extentthe stresses in bending. Simulations 2, 8, and 9 wereexecuted, transmitting a load of 7 N m with increas-ing coefficients of friction. These were conducted toinvestigate theeffect of increasing friction on thestressand kinematic behaviour of the gears.

The gear theory dictates that at the pitch point, thegears experience only a rolling action (with no slid-

ing velocity) and so assumes a negligible frictionalinfluence. Figure 14 suggests that for (dry-running)polymeric gears this is not the case. It shows thatfrictional forces decrease the maximum load share



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ratio and hence increases transmitted loads and sub-sequent contact stresses. Friction does not seem toaffect the kinematic behaviour of the gears and hasa negligible effect on the length of the path of contact.

Similar trends, shown in Fig. 15, are observed whenFE and the Standard derived stresses are comparedagainst increasing coefficients of friction. As expected,Standard derived stresses show no variation with fric-tion. The FE simulations increasingly overestimatecontact stresses compared with those calculated fromthe Standard. FE contact stresses show a strongdepen-dence on friction. Bending stresses are also shown toincrease with friction but remain below stress levelspredicted by the Standard.

Therefore, for the Standard to become a more accu-rate means of specifying polymeric gears, a frictionalfactor should be incorporated into the rating equa-

tions to account for tangential forces, particularly indry-running applications. The dominance of contactstresses (being greater than the bending stresses) may also contribute to wear being the predominant failuremechanism in dry-running polymeric gears.

5 CONCLUSIONS

This article has simulated the kinematic and kineticbehaviour of a pair of dry-running, similar-material,

non-metallic gears running under a variety of operating conditions. The stresses extracted from thefinite-element models have been compared to thosecalculated from the BS ISO 6336 rating Standard(method B). The main conclusions are as follows.

1. The assumptions made by the classical gear the-ory and inherited by most common gear-rating standards, specifically those of negligible toothdeflections and frictional effects, are not valid fordry-running non-metallic gears that have highfriction coefficients.

2. Employing a non-linear FE technique to develop a

dynamic simulation of the meshing cycle enabledthe entire kinematic history of the cycle to berecorded (including the load-sharing effects andincreases to the path of contact) and also allowedthe frictional effects to be investigated during contact. These were shown to have a significantinfluence on FE-derived stresses.

3. Contact is shown to occur outside the theoreti-cal line of contact, increasing the roll angle of thegears.

4. Premature contact extends the beginning of thecontact line; the physical result of this is an impact

between the tip of the driven gear and the regionof the pitch point of the driver gear. This inducesabnormally high levels of contact stress and couldbe a contributory factor in PLFs.

5. Extended contact and van Melicks’ suggestedreciprocating motion at the tooth root forpolymer–steel contacts wereconfirmed in the caseof all the polymer gear meshes simulated here.

6. The extensionsto the pathof contact increased theroll angles predicted by the theory.7. For the conditions simulated, load-sharing ratios

were always below 1. This implied that the realcontact ratios were always above 2, despite thetheoretical prediction of 1.65.

8. As the Standard does not consider tooth deflec-tions, reducing the Young’s modulus of thepolymer (as would be the case at elevated oper-ational temperatures) has no effect on Standard-predicted bending stresses. It does, however, withthe FE-derived bending stresses. Increasing thestiffness of the modelled gears is likely to cause

the simulated bending stresses to converge withthose of the Standard.

9. Frictional effects cannot be disregarded in dry-running polymer gears where they were shownto have a significant effect on the induced toothstresses.

10. Loss of contact ratio due to allowance for expan-sion and lower tooth accuracy need to be consid-ered.

The above conclusions point to the need for a spe-cific polymer gear Standard that accounts for theidiosyncrasies that are not based on metallic gear-

rating methods. The authors have evidence to show that the lack of applicable design data and a rating Standard, in the public domain, is preventing thisnovel form of gearing from being fully exploited.

© Authors 2010

REFERENCES

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APPENDIX 1

Notation

E Young’s modulus (GPa)r a addendum radius

T torque (N m)

µ coefficient of frictionϕ theoretical pressure angle (degree)

Subscripts

1 pinion2 wheel

APPENDIX 2. BS ISO 6336 (METHOD B)

A2.1 Part 2: calculation of surface durability

Surface durability is assessed according to nominalcontact stress levels calculated at the pitch point orthose at the inner point of single tooth contact. Thenominal contact stress, σ H , and the permissible con-tact stress, σ HP , can be calculated using equations (1)and (2), respectively.

σ H = Z Bσ HO K AK V K H αK H β (1)σ HO = Z H Z E Z β Z ε

F t

D 1b

u + 1u

(2)

A2.1.1 Z H : zone factor for contact stress

The zone factor accounts for the influence of toothflank curvature at the pitch point and converts tan-gential force to a normal force at the pitch point(equation (3))

Z H =

2cosβb cosα wt

cos2 αt sin α wt(3)

A2.1.2 Z E : elasticity factor for contact stress

The elasticity factor accounts for the influence of material properties (equation (4)).

Z E =

1

π{[(1 − v 21)/E 1] + [(1 − v 22)/E 2]}(4)

A2.1.3 Z ε: contact ratio factor

The contact ratio factor, Z ε, considers the influence of the transverse contact and overlap ratios on the tooth

surface load capacity (equation (5)).For spur gears

Z ε =

4 − εα3

(5)

The transverse contact ratio, εα, is given inequation (6) and the overlap ratio in equation (7).

Transverse base pitch

P bt = mt π cosαt Length of path of contact

g α = 12

d 2a1 − d 2b1 ±

d 2a1 − d 2b1

− a sin αwt

(positive sign for external gears)



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Transverse contact ratio

εα = g α

P bt (6)

Transverse base pitch

εβ =b sin β

πmn(7)

A2.1.4 Z B and Z D : single-pair tooth contact factors

The single pair tooth contact factors transform thecontact stress at the pitch point to the inner pointof single pair tooth contact ((if Z B > 1 or Z D > 1),equation (8)).

M 1 = tan αwt

(d 2a1/d 2b1) − 1 − (2π/z 1)

×

(d 2a2/d 2b2) − 1 − (εα − 1)(2π/z 2)

M 2 =tan αwt

(d 2a2/d

2b2) − 1 − (2π/z 2)

× (d 2a1/d

2b1) − 1 − (εα − 1)(2π/z 1)

Z B = 1 if M 1 1 Z D = 1 if M 2 1

Z B = M 1 if M 1 > 1 Z D = M 2 if M 2 > 1(8)

A2.2 Part 3: calculation of tooth bending strength

Nominal tooth bending strength is calculated at thetooth root using a modified form of the Lewis bending equation (given in equation (9)). In a similar manner tothat outlined above, multiplying the nominal bending

stress by a series of correctional factors gives a permis-sible stress level (equation (10)). Again, using ISO 6336for polymeric gears, most of the correctional factors

were set to unity

σ F = σ FOK AK V K H αK H β (9)

σ FO = Y F Y s Y βF t

bmn(10)

A2.2.1 Y F : tooth form factor

The tooth form factor accounts for tooth shape

(equation (11)).

Y F =(6hFe /mn) cosαFen

(S Fn/mn)2 cosαn(11)

Calculation of hFe

d n = mnz n d bn = d n cos αnd an

= d n

+ d a

− d εαn

=εα

cos2

βb

d en = 2 z

|z |

d an

2

2−

d bn

2

2

−πd cos β cos αn|z | (εαn − 1)

2

+

d bn

2

2

αen = cos−1

d bn

d en

γ e =0.5π+ 2 tan αn x

z n

+ inv αn − inv αen

αFen = αen − γ e h fe

mn= 0.5z n

cosαn

cosαFen− cos

π

3 − θ

+ 0.5ρ fp

mn− G

cos θ

Calculation of tooth root normal chord

S Fn

mn= z n sin

π

3 − θ

+

√ 3

G

cos θ − ρ fp

mn (12)

An accurate value of θ is required to evaluate hFe ,S Fn, and αFen; it can be calculated through iterationusing equation (13) and an iterative seed value of π/6(suggested by BS 6168: 1987 [20]).

Calculation of θ

E = π4

mn − h fp tan αn − (1 − sin αn)ρ fp

cosαn

where ρ fp is the basic rack root fillet radius, S pr theundercut factor (set to zero for polymer gears), and h fpthe dedendum of the basic rack

G = ρ fpmn

− h fpmn

+ x

where x is the profile shift coefficient (zero in the caseof polymer gears)

Base helix angle

βb = sin−1(sin β cos αn)

z n =z

cos2 βb cos β

Virtual number of teeth of a helical gear

H = 2G z n

π

2 − E

mn

− π

3



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Hence

θ = 2G z n

tan θ − H (13)

A2.2.2 Y S : stress correction factor

The stress correction factor (equation (14)) convertsthe nominal bending stress to local tooth root stressand accounts for the following:

(a) the stress amplifying effect of the section changeat the tooth root fillet radius;

(b) the conversion of the calculated bending stressinto a true bending stress, accounting for theexact location of the applied load, at the point of maximum loading

Y S = (1.2 + 0.13L)q [1/1.21+(2.3/L)]s (14) where

L = S FnhFe

, q s =S Fn

2ρF

ρ f

mn= ρFp

mn+ 2G

2

cos θ(z n cos2 θ − 2G )


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