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An algorithm for robust gear modifications design Dipl. Ing. ETH Hanspeter Dinner, EES Gear GmbH, Switzerland, h.dinner@ees-gear.ch

Dr. Ing. ETH Ulrich Kissling, KISSsoft AG, Switzerland, ulrich.kissling@kisssoft.ch

Introduction

The design of gear modifications is one of the key tasks for high performance gears, where

high performance may be defined e.g. as a maximized torque capacity, minimized vibration

level, highest scuffing resistance or lowest wear risk. Gear modifications, or the gear micro

geometry, are assessed in the tooth contact analysis TCA where the true gear geometry

(considering both gear macro and micro geometry) is combined with the true gear

misalignment (which again is a function of many parameters as shown below) to find

- Line load distribution to calculate KHβ e.g. along ISO6336-1, Annex E

- Contact stress distribution to check e.g. against stress peaks

- Transmission error, to assess the gear vibration excitation

- Local sliding speeds and pressure, to assess the risk of wear in low speed gears

- Local lubricant film thickness to assess the risk of micropitting

Figure 1: Input parameters and results of the tooth contact analysis

The TCA allows for the assessment of the effectiveness of a gear modification. Yet, the

obvious question remains “what is the best gear modification”. A range of papers has been

published on the subject of gear modifications, where the different types available and their

effect e.g. on the transmission error of the loaded mesh are described. These papers have in

common that they usually look at one particular correction at operating condition only.

However, if the profile and lead modifications in a gear pair is to be optimized, different

amounts of each modification should be combined and checked for suitability for a range of

operating torque levels. This means that we have to find a combination of modifications that

- Lead to a good level in a particular parameter (e.g. a low variation in the transmission

error or a low KHβ value) at the nominal torque level

Algorithm for Robust Gear Modifications Design

- Result in little variation in the level of this particular parameter if the torque level

changes, meaning, it is a robust design

Furthermore, the different modifications in lead and profile direction may be combined in

any manner and it may not always be intuitive which combination is the best. A suitable

solution to the above problem is to calculate the TCA for different combinations of

modifications, for different sizes of the respective modification and for different load levels

automatically. Then, for each combination, parameters of interest (e.g. PPTE or KHβ) are

found and may be assessed by the gear designer. This also means that the final selection of

the “best” solution is up to the gear designer and is not left for the calculation algorithm to

decide, as the importance of each parameter is, remains and has to be a subjective choice

based on experience and design philosophy. While this approach is not highly refined, it

shows to be highly effective as the below examples will illustrated.

Profile modifications

Profile modifications are either limited to the root or tip area or cover the whole tooth

height. The former are called tip and root relief, and different types are possible. The later

may be a pressure angle modification or profile crowning. The definition of profile

corrections may be found in ISO21771:2007.

Profile modifications have an effect on the gear strength rating along ISO6336:2006 in the

sense that they affect the theoretical contact ratio and hence the contact stress under load.

However, it is recommended not to consider the profile modifications in the gear rating for

pitting and bending. Furthermore, they have a considerable effect on the scuffing rating e.g.

along ISO/TR13989:2000 where they much affect the flash temperature at the start and end

of mesh. They also have an effect on micropitting rating, e.g. along ISO15144:2010, method

B where the profile modifications are considered. If method A (where the local contact

pressure from a tooth contact analysis is used to calculate the EHD film thickness) is used,

then, obviously, the profile modifications have a direct influence as they affect the contact

stresses calculated.

In case of poorly lubricated gears such as dry running plastic gears or slow running, highly

loaded girth gears they strongly affect the local pressure, in particular at the start and end of

mesh and hence the wear rate. Furthermore, profile corrections are applied to avoid point-

surface-origin macropitting in the root of a driving pinion [8].

Another focus of the profile modifications typically is to design them such that the vibration

excitation during the gear meshing is minimized even if load levels centre distance and gear

quality vary. This excitation is typically assessed by means of the variation of the

transmission error, the peak to peak transmission error PPTE and it’s Fourier analysis or the

variation of the resulting meshing / bearing reaction forces (as the bearing forces ultimately

are responsible for housing excitation), see e.g. [1], [3], [4], [5].

Figure 2: Left: Profile modification in K-chart (blue) and permissible error (red). A barreling is

superimposed with a progressive tip relief. Right: Linear tip and root relief definition along

ISO21771:2007.

Lead modifications

Typically, three types of lead modifications are combined in a mesh as listed below. Each of

them serves a distinctive, different purpose

1) Helix angle modification: account for the shaft deflection at design load level

2) Crowning: account for variations in shaft deflection e.g. if the load varies or if

machining errors in the housing are present

3) End relief: ensure that at extreme load levels (when gears are severely misaligned for

a short time), no stress concentrations occur at the end of the face width

4) Variable corrections to compensate uneven thermal expansion of the gears in case of

e.g. turbo gears

Typically, the end relief is applied on the gear (assuming the pinion has the lower face width)

only. The helix angle modification and the crowning may be distributed between the two

gears in the mesh. Below, the resulting flank modification without and with superimposed

profile directions are shown. The lead modifications are then assessed e.g. through the

calculation of KHβ along ISO6336-1:2006, Annex E.

Figure 3: Gear modifications. Left: profile and lead modifications. Right: only lead

modifications, where a helix angle modification is superimposed to a crowning and a

progressive end relief on both sides of the flank.

Application example, automotive transmission

Let us consider a gear mesh in an automotive transmission. The gear has a high theoretical

contact ratio of εα=1.72. The gear modifications should be optimized such that the PPTE is

minimized for different torque levels, starting at 50% nominal torque up to 110% nominal

torque. In parallel, there is the interest to achieve a low contact stress, allowing for a higher

power density in the mesh resulting in a lower gear mass. To maximize the performance, the

best combination of tip/root relief, profile crowning and pressure angle modification has to

be found. Just by guessing it is unlikely to find a good solution. A systematic search by

varying all this parameters stepwise has to be performed. Figure 4 shows the user window,

where 3 groups of modifications can be defined including the stepwise variation. All possible

combinations are then analyzed, giving results in a table as shown in the Figure 4.

Figure 4: Left: Set up of the definition of profile modification groups using software [7].

Right: Resulting PPTE for the different combinations of modifications for different load

levels.

The display of the different results such as PPTE, KHβ, εa, Micropitting safety,... for the

different modification variants shows clearly the tendencies. The gear designer can choose

carefully his optimum solution, having a low PPTE combined with modest Hertzian pressure.

Figure 5: PPTE for each modification and for different load levels (differently colored curves).

Areas with low PPTE for all torque levels are indicated.

Figure 6: Tooth contact stress for the different modifications combinations. The variations

are considerable. For the highest load level, the stress range is 1350MPa to 1915MPa, a

variation of +42% with respect to the lowest value.

From the graphics above, it can be seen which modification shows both low absolute values

and a low variation in value. This design is then ideal, meaning favorable (low absolute PPTE)

and robust (not susceptible to variations in load ).

Application example, plastic gears

One of the key design problems with plastic gears, especially those running in dry condition,

is wear [9]. Wear is the most common failure mode in plastic gear and the wear of plastic

gears may greatly be improved through optimized modifications [10]. Furthermore, plastic

gears are often used in medical devices, vehicles (as actuators), kitchen appliances or

consumer electronics where a low noise and vibration level is desirable. On the other hand,

applying profile modifications on plastic gears is simple and has no impact on the

manufacturing costs. Applying lead modifications however is a major challenge and may be

limited to helix angle modifications.

In this example, we seek an optimal modification to reduce the wear of the gear while

achieving also a low PPTE. The gears are made of thermoplastic POM and they are running

without lubrication, having a specific wear rate of kW=1.03 mm^3/Nm/10e6. The gear data

used (reference profile) is for a high contact ratio gear, which has superior wear

performance to start with, see e.g. [11]. The wear calculation follows e.g. and may be

described as follows:

Wear depth in [mm] on a point on the gear flank, basic formula (product of specific wear

rate, times pressure, times sliding distance):

δw (mm) = KLNPδmetric (mm2/N) * P(N/mm2) * V(mm/s) * T(s)

δw (mm) = 0.001*KLNPδ (mm3/Nm) * P(N/mm2) * V(mm/s) * T(s)

From the above basic formula, we find the wear depth per gear:

δw_i = 0. 001*KLNPδ _i * F * vg / b / vp_i; i = 1,2

Or, using the specific sliding ζ = vg/vp_i :

δw_i = 0. 001*KLNPδ _i * F * ζ_i / b; i = 1,2

And finally, the wear after n load cycles

δw_i = n * 0. 001*Kfactorδ_i * F / b * ζ_i; i = 1,2

Where

F (N) Load

b (mm) Face width

vp1, vp2 (mm/s) Velocity tangential to the flank of gear1, gear2

vp1,2*∆t (mm) Moving distance of a point on the flank (Gear1, 2)

vg (mm/s) = vp1 - vp2 Sliding velocity

∆t (s) Time

vg*∆t (mm) Sliding distance

ζ Specific sliding

A = b*(vpi*∆t) Surface

P = F / b/(vpi*∆t) Pressure

δw (mm) Wear depth (mm)

i Index for gear No, i=1, 2

Figure 7: Left: local wear rate (blue lines) and resulting wear after operating life (not to scale)

of original gear geometry. Right: original gear geometry

In the optimization, we will combine different amount of tip and root relief with different

correction diameters to find the wear rate for each case as shown below.

Figure 8: Flank wear on gear 1 and gear 2 combined for an operating time of 200μm for

different torque levels and different modifications.

The resulting wear on the flank of one of the gears in the gear mesh before and after

optimization is shown in the next figure. It can be seen that by applying a combined

correction, the total wear can be reduced to about half of the original value, effectively

doubling the gear lifetime at no additional manufacturing cost.

Figure 9: Flank wear on gear 1 in the mesh for nominal torque and 2000h operation. Left:

wear of original gear of approximately 0.6μm. Right: wear of gear with optimized

modifications of approximately 0.3μm.

Application example, high speed stage of a wind gearbox

Consider a high speed stage (HSS) in a wind turbine gearbox with a rated power of 3MW.

The smaller gear, the pinion, is driven by the larger gear, the wheel. The shafts are both

supported by a non-locating bearing on one side (the side facing the rotor of the turbine)

and a locating, paired taper roller bearing in X arrangement on the other side (the side facing

the generator). The wind turbine operates in a wide range of wind speed, resulting in a wide

range of torque levels acting in the HSS. Typical load show that most of the time is spent at a

torque level between 80% to 120% of the nominal torque Tnom. However, even in

operation, peak loads may reach 140% or 150% of the rated torque. Furthermore, extreme

load cases may reach 200% or more percent of the rated torque, however, this torque levels

are rare.

Here, the key objective is to find lead modifications

1) That result in a low KHβ value in the operating range of the torque

2) Allows for the change in shaft and bearing deflection due to the load changes

3) Which ensures that at the peak loads, no stress concentrations at the edges of the

face width occur.

We will split the above three requirements as follows

- Requirement 1) and 2) will be resolved through a combination of the helix angle

modification and crowning. To optimize the modification, we will use the

optimization tool

- Requirement 3 )will be resolved through an end relief, which will be applied based on

experience

Furthermore, we will check the effectiveness of the modifications in two distinctive torque

domains

- Torque domain A: operating load range, from 80% to 150% nominal torque. Here, we

want to achieve a low KHβ value and a good stress distribution.

- Torque domain B: extreme load range, at 200% nominal torque. Here, we want to

ensure that there are no stress concentrations at the end of the face width.

For the purpose of this study, we will put all modifications on the pinion and none on the

wheel and we will assume that the face width of both gears is equal. The deformation of the

bearings due to the gear forces is based on a non-linear bearing stiffness which is calculated

from the bearing inner geometry. The bearing stiffness is non-linear due to two effects

1) There is a finite bearing clearance which results in a horizontal line in the

displacement-force curve. This finite bearing clearance is calculated from the

assembly clearance (here, for the CRB, C3), the fit between the races and the shaft

and housing, the temperature of the bearing races and the centrifugal forces on the

races.

2) With increasing load, more rolling elements get into contact, effectively increasing

the stiffness with increasing load

Figure 10: Left: non-linear bearing stiffness curve of the CRB on the output shaft. Right: Non-

linear radial (orange) and axial (blue) bearing stiffness curve for the larger of the two TRB on

the output shaft

Figure 11: High speed stage in a wind turbine gearbox

The shaft deflection considers these bearing stiffness effects. The results are calculated using

a non linear FEM solver (meaning that the equilibrium of all external forces is solved in the

Driving wheel

Cylindrical roller bearing

Taper roller bearing

Taper roller bearing

Cylindrical roller bearing

Driven gear

Taper roller bearing

Taper roller bearing

deformed state of the shaft), considering the shaft by means of Timoshenko type beam

elements and considering the bearings by means of a non linear spring / gap element.

Figure 12: Deformations in xy (orange) and zy (blue) plane. Left: Shaft deformation with C0

assembly clearance in the CRB. Right: Shaft deformation without clearance in the CRB.

Considering nominal torque and without any modifications, we find a contact stress

distribution as shown below, left side. Using the shaft calculation, we find an approximate

value for the helix angle modification Cβ=80μm and from experience, we estimate a suitable

crowning of Cb=20 μm and find a contact stress distribution as shown below, right side. We

can see that the contact pattern has improved but is not optimized. Furthermore, only

nominal load condition (100% of nominal torque) has been considered till now whereas in

wind gearboxes, a load level slightly higher than nominal torque should be considered in the

design of the gears.

Figure 13: Contact stress for nominal torque in plane of contact, without gear modifications.

Top left: No gear misalignment. Top right: With gear misalignment.

Figure 14: Contact stress for nominal load in plane of contact, with gear modifications. Left:

Experience based lead modifications only. Right: Experience based lead and profile

modifications.

Figure 15: Set up of optimization tool. Torque range considered is 80% to 150% of nominal

torque.

Finally, let us consider the effect of an end relief, which is of particular importance in case of

overloads. From the TCA for a peak load of 250% nominal torque, we find stress

concentrations at the end of the face width (see left side of above figure). After we apply a

curved end relief (here, over 10% of the face width on each side) at both sides of the face

width, we can reduce this stress concentration. Obviously, the KHβ increases (!) with this

modification as now, the effective face width (the face width experiencing load under

nominal load condition) is reduced.

Crowning should be

varied between 20μm

and 50μm

Helix angle modification

should be varied

between 70μm and 120

μm

Torque should be varied

between 80% to 140% of

nominal torque

Four steps each should

be considered

Figure 16: Contact stress in plane of contact for 250% nominal torque. Left: Stress

concentration at the end of the face width. Right: No stress concentration at the end of the

face width after an end relief has been applied.

Application example, sun-planet mesh in a mill gearbox

Horizontal (e.g. for sugar mills) or vertical (e.g. for cement mills) output planetary stages

transmit very high power levels at very low speed, resulting in large planetary stages having

with ratios d/b>1.00 – in particular on the sun gear. Due to the torsional wind up of the

planet carrier and the sun shaft, corrections in the sun-planet mesh are applied, the basic

considerations are explained e.g. in [14] or [15]. Care has to be taken not to select to high

levels of crowning, as the resulting loss in load carrying face width is severe. The design

objective is hence to find a combination of helix angle correction and crowning resulting in a

low KHβ value for different planet pin deflections and sun gear torsion, an objective that has

been reportedly achieved in the industry [13]. In this example, a gear face width of

b=430mm was applicable and the helix angle and the crowning on the sun gear were

selected as variables (while an end relief on the planet, having somewhat lower face width,

was applied).

The load level was varied in five steps between 90% and 130% of nominal load.

The crowning was varied between 10µm and 50µm and the helix angle modification

between -150µm and -190µm, both in five steps, giving a total of 125 calculations (25

different combinations of modifications, each for five different load levels). The result of

these are shown in the below graphics:

Figure 17: Resulting KHβ values for different load levels and different corrections

It can be seen that for modification 3:2:-, the lowest KHβ values result and that there, the

variation of the KHβ value is also small (KHβ is not much affected by the load level). For this

correction, the crowning is 30μm and the helix angle correction is -170μm.

Figure 18: Left: Lead corrections on planet gear. Right: Lead corrections on sun gear (lead

corrections in ring gear are not shown).

If we now look at the tooth contact pattern, the contact stress and KHβ for different load

levels (nominal load, lowest load and highest load) with the above corrections applied, we

find that

- KHβ (not considering manufacturing errors fma and fHβ) is low for all load cases

- KHβ varies only very little with the variation of the external loads

- For no load conditions, stress concentrations at the edge of the face width exists (also

due to the end relief)

We may therefore conclude that the design is robust, see the below images for the results at

lowest, nominal and highest load!

For 100% load and 240μm tilt

For the face load distribution factor KHβ we find:

KHβ =wmax/wm=2097N/mm/1980N/mm=1.06

For the contact stress in the plane of contact and the contact pattern on the sun we find:

Figure 19: Contact stress in plane of contact and contact pattern on sun gear for nominal

load.

For 90% load and scaled tilt, as lowest load

For the face load distribution factor KHβ we find:

KHβ =wmax/wm=1920N/mm/1782N/mm=1.08

For the contact stress in the plane of contact and the contact pattern on the sun we find:

Figure 20: Contact stress in plane of contact and contact pattern on sun gear for lowest load.

For 130% load and scaled tilt, as highest load

For the face load distribution factor KHβ we find:

KHβ =wmax/wm=2861N/mm/2574N/mm=1.11

For the contact stress in the plane of contact and the contact pattern on the sun we find:

Figure 21: Contact stress in plane of contact and contact pattern on sun gear for highest

load.

Summary

In the above four examples, general guidelines on the use and purpose of profile and lead

modifications have been presented. It is pointed out that the key to a successful design is

not only to optimize the modifications for a certain load level, but to take into account a

range of operating loads and selected overloads. While for overload cases, experience based

application of gear modifications makes sense, a search algorithm to find a robust

modification design for the whole range of operating loads is proposed.

We consider it critical that such an algorithm does not propose a single solution claiming it is

the best, but that the algorithm presents all solutions found for the designer to assess, e.g.

using graphics as shown above. While such an assessment could be automatic based on the

numerical results and user defined weights for each parameter, we strongly believe that

gear design is, remains and should be also intuitive. Above, a search algorithm, suitable

graphics for this intuitive assessment of the results and the application of the method in

various industries for different optimization targets are presented.

References

[1] Kissling, Effects of Profile Corrections on Peak-to-Peak Transmission Error, Gear

Technology, July 2010

[2] Houser, Hariant, Ueda, Determining the source of gear noise, Gear Solutions,

February 2004

[3] Houser, Harianto, Profile Relief and Noise Excitation in Helical Gears

[4] Palmer, Fish, Evaluation of Methods for Calculating Effects of Tip Relief on

Transmission Error, Noise and Stress in Loaded Spur Gears, Gear Technology,

January/February 2012

[5] Oswald, Townsend, Influence of Tooth Profile Modification on Spur Gear Dynamic

Tooth Strain, NASA Technical Memorandum 106952

[6] Kissling, Raabe, Calculating Tooth Form Transmission Error, Gear Solutions,

September 2006

[7] KISSsoft 03-2012

[8] Errichello, Hewette, Eckert, Point-Surface-Origin Macropitting Caused by Geometric

Stress Concentration

[9] Beermann, Estimation of lifetime for plastic gears, to be published

[10] KISSsoft AG, Wear on plastic gears, not published

[11] Venkatesan, Rameshkumar, Sivakumar, Simulation of Wear for High-Contact-

Ratio-Gear, Gear Technology India, April-June 2012

[12] Feulner, Verschleiss trocken laufender Kunststoffgetriebe, Lehrstuhl

Kunststofftechnik, Erlangen, 2008.

[13] Hess, Entwicklungstendenzen bei grossen Planetengetrieben für den

industriellen Einsatz, DMK2005

[14] MAAG Taschenbuch, 2nd Edition

[15] Amendola, Amendola, Yatzook, Longitudinal Tooth Contact Pattern Shift, Gear

Technology, May 2012