New calculation method for the load capacity of bevel and hypoid gears based on loaded tooth contact analysis

Prof. Dr.-Ing. Bernd-Robert Höhna, Prof. Dr.-Ing. Karsten Stahlb, Dr.-Ing. Christian Wirthc

FZG-Forschungsstelle für Zahnräder und Getriebebau, Technical University of Munich, Boltzmannstraße 15, 85748 Garching, Germany

a hoehn@fzg.mw.tum.de, b stahl@fzg.mw.tum.de, c wirth@fzg.mw.tum.de

Keywords: bevel gears, hypoid gears, load capacity, calculation method, tooth contact analysis

Abstract. At the FZG (Gear Research Centre, Munich, Germany) a research project was carried out to analyze the influence of the hypoid offset on the load capacity of bevel gears by systematic theoretical and experimental investigations. For the experimental investigations two types of bevel gears were designed, one for the pitting tests and one for the tooth root tests. The results of the tooth root tests showed as expected an increasing load capacity with higher offsets. In contrast the pitting tests showed an increasing, but after reaching a maximum, a decreasing load capacity with higher offsets.

Regarding the test results a new calculation method was developed that is based on a loaded tooth contact analysis (LTCA). The method is able to consider the local stresses on the flank and in the tooth root. The local strength values are derived out of the standard ISO 6336 for the calculation of helical gears.

For bending the local geometry of the tooth root is considered to adopt the strength values of helical gears to bevel and hypoid gears. As a result the local safety factors might be calculated along the face width of pinion and wheel. For pitting the local sliding conditions are taken into account in order to appraise the local lubrication conditions as well as the risk of crack initiations due to shear stresses and higher contact temperatures. The recalculation of the test showed for both types of failure a good correlation between the test results and the calculated values.

Introduction

Hypoid gears are bevel gears with nonintersecting axes where the hypoid offset is defined as the shortest distance between these two axes. Hypoid gears are preferred over bevel gears without offset if aspects of gear noise or of installation space are in focus. Pitting and tooth root breakage are still the two most frequent failure types occurring in practical applications of bevel and hypoid gears. There are several national and international standards for the calculation of the load capacity of these gears such as DIN 3991, AGMA 2003 and ISO 10300. But up to now these standards do not cover bevel gears with offset. Although the occurring stresses on hypoid gears can be calculated with higher sophisticated methods that are based on a tooth contact analysis (TCA), there is no commonly used method to evaluate different designs of flank forms regarding local safety factors. This is because the local strength for bending and pitting is not known. For this reason a research project was carried out at FZG (Gear Research Centre, Munich, Germany) to analyze the influence of the hypoid offset on the load capacity of bevel gears by systematic theoretical and experimental investigations. Furthermore a new calculation method should be developed to predict the local risk of pitting and the risk of tooth root breakage on the base of a loaded TCA.

Experimental tests

For the experimental investigations two types of bevel gears were designed, one for the pitting tests and one for the tooth root tests. The aim of the two different gear designs was to examine

pitting isolated from tooth root breakage. Thus, the gears for the tooth root tests were designed with offsets a = 0 / 15 mm / 31,75 mm with a relatively small module (mmn = 2,2…2,5 mm), the gears for the pitting tests with offsets a = 0 / 15 mm / 31,75 mm / 44 mm with a bigger module (mmn = 3,5…4,2mm). All wheels had the same outer diameter de2 = 170 mm. The gear sets were made of 18CrNiMo7-6 case hardened and finish ground. For each geometry the S/N-curve was determined by approximately 20 tests.

Results of the pitting tests. On each examined variant pitting occurred isolated from tooth root breakage. Beside the pitting failures micro pitting appeared. With increasing offset, the micro pitting area on the flank grew faster and bigger. In Fig. 1 (left side) a pinion flank of the geometry variant with a = 15 mm is shown after 11 and 27 million pinion revolutions at T1 = 300 Nm.

It was proven that the flank form deviation which occurs due to micro pitting affects the load distribution on the flank during the lifetime. This leads to locally changing load conditions and thus influences pitting. In test with torques close to the endurance limit of the gear set micro pitting had a larger influence because of the large runtime.

11 mio. pinion load cycles

27 mio. pinion load cycles

66 mio. pinion load cycles

70 mio. pinion load cycles

Fig. 1: left: micro pitting and pitting on pinion flank (T1 = 300 Nm, a = 15 mm) right: initial pitting at the addendum of the pinion toe (T1 = 300 Nm, a = 15 mm)

Whereas at the geometry variant with a = 0 mm (non hypoid) micro pitting occurred mainly at the dedendum of the flank, for the hypoid variants (a = 15 mm / a = 31,75 mm / a = 44 mm) micro pitting could be documented over the whole active flank. The same applies to pitting. On the right side of Fig. 1 the condition of the pinion flank (a = 15 mm) is shown for two load cycles. Micro pitting could affect pitting in such way that the initial pitting occurs also at the addendum of the toe which is in contrast to helical gears.

Although on the wheel flanks of the hypoid variant with an offset a = 44 mm pitting was also detected, it was not possible to evaluate the load capacity of the wheels because of the very few numbers of failures.

Fig. 2 shows the endurance limit for pitting for all tested variants. For each torque the maximum Hertzian pressure was calculated by means of a loaded tooth contact analysis (LTCA) with BECAL [3]. BECAL [3] (Bevel Gear Calculation) was developed by the FVA (Forschungsvereinigung

0

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0 15 31.75 44

offset [mm]

pini

on to

rque

[Nm

]

0

350

700

1050

1400

1750

2100

cont

act s

tres

s [N

/mm

²]

endurance torque - pinionmax. contact stress (LTCA)

-22%

-6%

-0%

Ref

.

Fig. 2: endurance limits of the test gears

Antriebstechnik). It considers the deflections between pinion and wheel, which are caused by the elasticity of the teeth, bearings, shafts and housings. In contrast to the increasing pinion endurance torques the occurring maximum stresses on the flanks are decreasing. For example the maximum stress on the flank of variant a = 44 mm for the endurance limit is 22% smaller than at the non hypoid variant.

Results of the tooth root tests. All examined variants failed by the decisive type of failure tooth root breakage. Fig. 3 shows a tooth root breakage on a pinion that can be regarded as representative for all tests. The crack initiation was detected in all cases close to the 30°-tangent to the tooth fillet. In Fig. 4 the endurance limits of the tested variants are compared. As it can be seen the tests showed an increasing load capacity with higher offsets as expected. The corresponding maximum tooth root stresses are for all variants within a range of ± 3.5%.

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offset [mm]

pini

on to

rque

[Nm

]

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300

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900

1200

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1800

toot

h ro

ot s

tres

s [N

/mm

²]

endurance torquemax. tooth root stress (LTCA)

+7%

+7%

Ref

.

Fig. 3: typical tooth root breakage Fig. 4: endurance limits of the test gears

New local calculation method for pitting

Method and results. The new local calculation method is based on the results of the loaded tooth contact analysis with BECAL [3]. The local strength values are derived from ISO 6336 [5] for the calculation of helical gears, which has the advantage that the strength values on helical gears are statistically well ensured.

For pitting a criteria was developed that allows to compare the occurring Hertzian pressure at a certain point in the contact area with the strength values of helical gears from ISO 6336 [5]. Investigations showed that it is necessary to regard the local sliding conditions in the point of contact in order to evaluate the whole flank.

The known fact [6, 7] that on helical gears there is a higher pitting resistance in the area of positive slip is now regarded in the new local calculation method by the so called slip factor ZS. The effect of the hypoid specific sliding lengthwise the contact line on the pitting strength is quantified by the newly introduced hypoid factor ZHyp. The local pitting safety factor is calculated according to Eq. 3 with the local contact stress and the local permissible contact stress.

vilokHiH K,,, σσ = (1)

iHypiSNTWv

RLXH

HiHP

ZZZZZ

ZZZS

,2,1,2,1

2,1min

2,1lim2,1,

⋅

⋅=σ

σ (2)

2,1,

min2,1,2,1,

)1(

iH

HiHPiH

SS

σσ =

= (3)

σH,i [N/mm²] local contact stress σH,lok,i [N/mm²] local nominal contact stress Kv [-] dynamic factor acc. to [2]

σH,lim [N/mm²] allowable stress number for contact stress

acc. to [4] σHP,i1,2 [N/mm²] local permissible contact stress ZHyp,i [-] hypoid factor ZL [-] lubricant factor acc. to [4] ZNT1,2 [-] life factor acc. to [4] ZR [-] roughness factor acc. to [4] ZS,i1,2 [-] slip factor Zv [-] speed factor acc. to [4] ZW [-] work-hardening factor acc. to [4] ZX [-] size factor acc. to [4] SH,i1,2 [-] local safety factor for contact stress SHmin1,2 [-] minimum safety factor acc. to [4]

The recalculation of the pitting tests showed a very good agreement of the predicted with the real pitting areas on the flanks. Fig. 5 shows for four different load cycle numbers the pinion flank conditions in comparison to the predicted pitting areas that are characterised by having a safety factor SH,i < 1 (T1 = 300 Nm). As it can be seen not only the place of the initial pitting is calculated correctly but also the pitting dynamics are predicted reasonably.

5,5 mio. pinion load cycles 11 mio. pinion load cycles

27 mio. pinion load cycles 70 mio. pinion load cycles

Fig. 5: pitting during the test and calculated state of the flank (T1 = 300 Nm, a = 15 mm)

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offset [mm]

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rque

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endurance torque - testendurance torque - calculation

Fig. 6: endurance limits of the test and

calculated endurance limits

Fig. 6 shows the comparison between the endurance limits of the tests and the calculated endurance limits. The calculated endurance limit is per definition reached when the lowest local safety factor becomes SH,i = 1. Peaks of pressure at the edge of the flank are not allowed in this consideration. As it can be seen there is a quite good correlation for all regarded offsets.

Slip factor. The slip factor was introduced in order to evaluate the dependency between the

relative sliding and the permissible contact stresses on the flank. The strength values that are used in ISO 6336 [5] for the load capacity of helical gears are only valid for negative slip. To use those values for a calculation at any point of the flank the influence of slip has to be considered.

Fig. 7: velocities of hypoid gear flank

1,,

2,,

1, 1isenkt

isenkt

isenkw

w−=ζ ;

2,,

1,,

2, 1isenkt

isenkt

isenkw

w−=ζ (4)

4047

247

2,1,2,1, += isenkiSZ ζ and 175.11 , ≤≤ iSZ (5)

0

0,4

0,8

1,2

1,6

-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4

slip perpendicular to contact line [-]

slip

fact

or, Z

S [-

]

Fig. 8: slip factor

wt,senk1,2 [mm/s] tangential velocity perp. to the contact line ζsenk,i1,2 [-] slip perpendicular to the contact line ZS,i1,2 [-] slip factor

toe toe toe toe

The relative motion of pinion and wheel leads to tangential speed components that are illustrated in Fig. 7. The circumferential velocity at any point on the flank can be projected in a plane parallel to the flank surface. This component is called the tangential surface velocity. Defining the tangential velocity perpendicular to the contact line as the projection of tangential surface velocity perpendicular to the contact line the slip perpendicular to the contact line can be expressed by Eq. 4.

Experimental investigations on roller tests by Rösch [6] showed a decreasing permissible stress with increasing negative slip. [6, 7] list possible reasons for that phenomenon. Based on this knowledge the slip factor ZS was implemented as shown in Fig. 8 and Eq. 5.

Hypoid factor. The relative motion between the flanks of hypoid gears with nonintersecting axes leads to a sliding component lengthwise the contact line. However helical gears and bevel gears without offset have only a sliding component that is perpendicular to the contact line. Therefore the influence of the lengthwise sliding on the pitting resistance has to be taken into account when the strength values of helical gears should be used for the calculation of hypoid gears. Whereas the amount of lengthwise sliding is approximately constant the perpendicular sliding component is scaled significantly along the tooth profile similar to helical gears. For this reason only the lengthwise component has to be regarded to evaluate the different sliding conditions on hypoid gears in comparison to helical gears. This component leads to higher sliding velocities and causes therefore higher temperatures in the contact point.

As Fig. 2 illustrates, the permissible stresses are decreasing with an increasing offset and a so increasing sliding component lengthwise to the contact line. Based on the test results the hypoid factor ZHyp was introduced to evaluate the lengthwise sliding on hypoid gears. ZHyp was empirically determined and should cover the following phenomena:

1) Influence of the contact temperature on the lubricating film thickness. 2) Influence of the contact temperature on the function of the oil. 3) Influence of the temperature on the material strength. 4) Influence of the temperature distribution on the thermal stresses. 5) Influence on the fracture mechanics on the surface. The hypoid factor can be calculated according to Eq. 6. Hereby the sliding velocity lengthwise to

the contact line is assumed to have a negative influence on the pitting resistance as described above. The reference value is the sum of the tangential velocities perpendicular to the contact line (see Fig. 7). These components are considered to have a positive effect on the pitting resistance because of their influence on the oil film thickness. As Fig. 9 shows the hypoid factor is ZHyp = 1 for bevel gears with intersecting axes and becomes smaller for higher offsets. For practical applications the lower limit of ZHyp = 0.6 is usually not reached.

−−=

Σ

15.03.01,,

,,,

isenk

ipargiHyp v

vZ (6)

and 16.0 , ≤≤ iHypZ

ZHyp,i [-] hypoid factor vΣ,senk,i1,2 [mm/s] sum of velocities perp. to contact line vg,par,i1,2 [mm/s] sliding speed lengthwise the contact line 0

0,2

0,4

0,6

0,8

1

1,2

0 0,5 1 1,5 2vgpar / vΣsenk [-]

hypo

id fa

ctor

, ZH

yp [-

]

a=0

a=15

a=15

a=31,75

a=44

a=31,75

a=0tooth rootvariants

pitti

ng v

aria

nts

Fig. 9: hypoid factor

New calculation method for tooth root breakage

Like the new method for pitting the local calculation of the load capacity for bending is based on the results of BECAL [3]. The strength values are derived from the standard ISO 6336 [5] for helical gears. For each mesh position the tooth root stresses (Eq. 7) are compared with the local

permissible strength values that are determined by Eq. 8. Because of the specific methods used by BECAL [3] to calculate the local tooth root stresses it is possible to determine the factors in Eq. 8 analogous to ISO 10300 [4]. Thus the local safety factors can be considered along the whole mesh of one pair of teeth. It is important that always the smallest of all calculated safety factors is the decisive value for the bending load capacity.

As it can be seen in Fig. 10 the new calculation method also has a very good correlation regarding the calculated endurance torques and the endurance torques derived from the tests.

vlokiFiF K2,1,,2,1, σσ = (7)

2,1min

2,12,12,12,1,2,12,1,

F

NTXRrelTirelTFEiFP S

YYYYδσσ = (8)

2,1,

2,1min2,1,2,1,

)1(

iF

FiFPiF

SS

σσ =

= (9)

SF,i1,2 [-] local safety factor for bending stress SFmin1,2 [-] minimum safety factor acc. to [4] σFE1,2 [N/mm²] allowable stress number acc. to [4] σF,i1,2 [N/mm²] local tooth root stress σF,i,lok1,2 [N/mm²] local nominal tooth root stress σFP,i1,2 [N/mm²] local permissible tooth root stress YδrelT,i1,2 [-] local relative sensitivity factor acc. to [4] YRrelT1,2 [-] relative surface factor acc. to [4] YX1,2 [-] size factor acc. to [4] YNT1,2 [-] life factor acc. to [4]

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offset [mm]pi

nion

torq

ue [N

m]

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h ro

ot s

tres

s [N

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+7%

+7%

Ref

.

Fig. 10: calculated endurance limits and endurance limits of the tests

Summary

At FZG (Gear Research Centre, Munich, Germany) systematic investigations have been carried out in order to develop a new calculation method for the pitting and bending load capacity of bevel and hypoid gears [1, 2]. By means of a huge number of test gears the new calculation method was statistically ensured as far as possible. Having the machine settings of a gear set it is now possible to use the results of a loaded tooth contact analysis by BECAL [3] to evaluate on the one hand the local risk of pitting on the flank. On the other hand the fatigue limit and the endurance limit for pitting and bending can be determined. This method is not restricted to bevel gears without offset but also for hypoid gears with nonintersecting axes.

References

[1] Wirth, Ch.: Entwicklung eines Berechnungsverfahrens zur Grübchen- und Zahnfußtragfähigkeit von Hypoidrädern. Forschungsvereinigung Antriebstechnik e.V., Heft 887, Frankfurt 2009.

[2] Wirth, Ch.: Zur Tragfähigkeit von Kegelrad- und Hypoidgetrieben. Diss. TU München, 2008.

[3] Hutschenreiter, B.: Ergänzung der Berechnungsmöglichkeit von BECAL für Flanken mit negativem Höhenballigkeitsradius und Kopfrücknahme. Abschlussbericht und Handbuchergänzung D zur Version 3.4.0. FVA-Forschungsheft Nr. 548, Forschungsvereinigung Antriebstechnik e.V., 2007.

[4] ISO 10300: Calculation of Load Capacity of Bevel Gears. Part 1 2001, Part 2-3 2003.

[5] ISO 6336: Calculation of Load Capacity of Spur and Helical Gears. Parts 1-3, 5, 1996.

[6] Rösch, H.: Untersuchungen zur Wälzfestigkeit von Rollen – Einfluss von Werkstoff, Wärmebehandlung und Schlupf. Diss. TU München, 1976.

[7] Hertter, T.: Rechnerischer Festigkeitsnachweis der Ermüdungstragfähigkeit vergüteter und einsatzgehärteter Stirnräder. Diss. TU München, 2003.