MCombined Effects of Lead Crowning and Assembly … because of FEM high permeation in the usual design process. Khoshnaw and Ahmed [9] ... root stresses of helical gear pairs obtained

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 24 (2016) pp. 11681-11694

© Research India Publications. http://www.ripublication.com

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Combined Effects of Lead Crowning and Assembly Deviations on Meshing

Characteristics of Helical Gears

Carlo Gorla1, Francesco Rosa1,a, Edoardo Conrado1, Yonatan A Tesfahunegn2

1 Politecnico di Milano, Department of Mechanical Engineering, via La Masa 1, I-20156 Milan, Italy. 2 Engineering Optimization & Modeling Center (EOMC), School of Science and Engineering, Reykjavik University,

Menntavegur 1, IS-101 Reykjavik, Iceland. Corresponding author

Abstract

This paper describes a FEM based approach aimed to

investigate the effects of lead and profile modifications in

combination with machining and assembly deviations on

meshing characteristics of helical gears. The proposed

approach allows introducing any three dimensional tooth

flank modifications (i.e. profile, longitudinal or more

complex modifications) along with manufacturing and

assembly deviations. In the paper, a sample gear set will be

used to investigate the combined effects of lead crowning and

assembly deviations on tooth root stresses, contact pattern,

load-sharing ratio and transmission error. The approach has

been validated with experimental data and its application is

illustrated through sample case studies. On the basis of the

obtained results, some general conclusions on the effects of

lead crowning and assembly deviations in helical gears useful

for gear designers are finally drawn.

Keywords: Gears, Tooth Flank modification, Transmission

error

INTRODUCTION

Growing requirements about load carrying capacity, noise

emissions and vibrations as well as the continuous technical

improvement in the field of gear manufacturing are

demanding the development of suitable designing and

optimization methods for tooth flank modifications. The

basic requirement of these methodologies is an approach able

to accurately determine the effects produced by tooth flank

modifications on the so-called gear meshing characteristics,

such as contact patterns and pressures, bending stresses [1],

transmission errors [2] and load-sharing ratios [3]. A gear

designer has to consider the effects of tooth flank

modifications on all these characteristics in order to be able

to predict how each choice influences the overall gear

behaviour. In fact, it is very difficult to fulfil simultaneously

all the design requirements since a potential solution for one

requirement might not be beneficial for another one. A

durability issue may rise as a consequence of the

implementation of a potential noise reduction solution and

vice versa. For this reason, an approach capable to analyse all

these aspects simultaneously can greatly help gear designers.

A further consideration is that tooth flank modifications,

especially longitudinal tooth modifications, are generally

introduced in order to compensate, along with elastic

deflections, the detrimental effects of manufacturing and

assembly deviations. Therefore, the effect of these deviations,

that affects significantly the gear meshing characteristics,

should be taken into account in the analysis.

Many methods have been presented and discussed in the past

decades to predict gear meshing characteristics. Among these

methods, the more commonly adopted are influence factors,

according to the gear design guides described in handbooks

and standards, such as the well-known ISO 6336 and AGMA

2001-D04 Standards. Influence factors defined in standards

have been the subject of extensive investigations regarding

their calculation and their influence on gear load capacity,

but their applicability is limited and, generally, their results

are not extremely accurate.

On the other hand, alternative numerical approaches are

continuously investigated and proposed in order to evaluate

in more detail gear meshing characteristics, taking into

account actual operating conditions (that mainly results in

misalignments), as well as profile and face modifications,

that are spreading thanks to the improvement of the

capabilities of gear cutting and finishing machines.

Practically, one of the main steps in studying a geared

transmission consists in the development of an accurate

model of its gear pairs, capable to represent also the main

effects of the other components of the transmission on the

gears themselves.

Although several other approaches can be found in literature

(see e.g. [4-8]), analysis based on the Finite Element Method

is one of the most commonly adopted approaches to face this

problem because of FEM high permeation in the usual design

process.

Khoshnaw and Ahmed [9] deeply investigated spur gear root

stress by means of a bi-dimensional FE model applying

concentrated loads in several positions on tooth active flank.

Tesfahunegn et al. [10] investigated by means of nonlinear 2D

finite element analyses the influence of the shape of tooth

profile modifications on several gear meshing characteristics

of spur gears. Conrado and Davoli [1] studied by means of 3D

FEM models the effects of tooth geometry and gear body on

the distribution of tooth root stresses in a spur gear pair.

Atanasovska et al. [11-13] presented a three dimensional spur

gear FEM model reproducing only a part of the gear wheels.

Li [14-15] presented an extensive and detailed analysis of his

numerical and experimental investigations of spur gear

meshing. In the numerical investigations, gear teeth contact is

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faced with a general purpose algorithm in order to determine

actual load distribution and misalignments as well as profile

and face modifications are considered. More recently, Roda

et al [16-17] presented a 3D FEM model reproducing spur

gear meshing, including also the shafts (by means of beam

elements), in order to reproduce more accurately

misalignments due to shaft bending.

On the other hand, very few papers analyse in such a detail the

meshing of helical gears including misalignments and active

flank modifications. Wagaj and Kahraman [18] performed a

parametric study of perfectly aligned helical gears to

“quantify the changes in the contact and bending stresses as a

function of tooth profile modification”, by considering “only

six-tooth segments of both gears”. S. J. Park and W. S. Yoo

[19] studied the meshing of modified helical gears by using

the FE approach. They faced the problem of introducing

profile modifications, but did not consider face modifications

and misalignments.

In this paper, a FE procedure able to determine the main

meshing characteristics of helical gears with tooth flank

modifications as well as manufacturing and assembly

deviations is presented. Although the procedure is of general

nature, here the focus will be on the combined effects of

longitudinal modifications, in particular circular lead

crowning, and of assembly deviations. This procedure has

been validated with experimental data on the combined

influence of lead modifications and assembly errors on tooth

root stresses of helical gear pairs obtained by Hoatait and

Kahraman [20]. Moreover, the application of this procedure

has been illustrated by the analysis of sample helical gear

pairs performed in order to investigate the combined effects of

lead modifications and assembly deviations on GMCs.

ASSEMBLY DEVIATIONS

Assembly deviations (ADs) can be defined generically as

deviations of gear sets from their perfectly aligned conditions.

From a practical point of view, ADs may result from a

combination of different sources such as shaft, housing and

bearing deflections. No matter from where the ADs come

from, they introduce changes in the gear system behaviour.

These changes can result in an increment of contact and/or

bending stresses, as well as of transmission error and, then, in

vibration and noise.

More in detail, several approaches have been conceived to

define gear assembly deviations; nevertheless, two are the

most widespread. The most common is based on ISO/TR

10064-3:1996 and on AGMA 915-3-A99. In this approach the

shaft parallelism deviations are divided into two types: in-

plane deviations and out-of-plane deviations. These deviations

are defined, respectively, in the common plane of axes and in

the skew plane, i.e. a plane perpendicular to the common

plane of axes. On the other hand, in several papers (e.g. [14]

and [20]), the misalignments are defined in a plane parallel to

the operating plane of action and in a plane perpendicular to

this one. Also in this case, the most effective way of studying

misalignments of a gear set consists in “decomposing” them

in two rotations about two axes (Figure 1): an axis n that is

normal to the plane of action, and an axis t parallel to cross

section plane. This approach is very suitable for the

CAD/CAE systems, since it practically requires defining an

axis and rotating a part (pinion and/or gear) about it, two tasks

that all the CAD/CAE systems can easily accomplish.

Figure 1: Reference system adopted to define misalignments.

This is the reason why, following this last approach, we

hereafter introduce misalignments by rotating a member of the

gear set about these two axes. In order to ease the calculations

a third axis (z) has been introduced, so that it is possible to

define a reference system tnz. Introducing this reference

system, the ADs can be also managed by means of the matrix

techniques usually adopted to change coordinate system. The

rotation about n axis is called εn, while the rotation about t

axis is called εt. In Appendix 1, a general relation (i.e. that

hold even if rotations are not small) between the adopted

approach and the definitions of misalignment of the ISO and

AGMA standards is presented. If rotations are small, the

above mentioned set of relations simplifies in the following

equations:

𝑓Σ𝛿 = |− sin 𝛼 ∙ 𝜀𝑛 + cos 𝛼 ∙ 𝜀𝑡|

𝑓Σ𝛽 = |cos 𝛼 ∙ 𝜀𝑛 + sin 𝛼 ∙ 𝜀𝑡|

where εn and εt (expressed in radiant) are the rotations about n

and t axis respectively, fΣδ is the in-plane deviation and fΣβ is

the out-of-plane deviation defined by the standards (see

Appendix 1 for more details).

Even if this procedure has been conceived to model gear axis

misalignments, it is worth noting that it can be used to

simulate a deviation of gear axis with respect to tooth

geometrical axis, as explained in more detail in the next

sections.

FEA PROCEDURE

In order to fulfil the research objectives, and to reduce

modelling time, a semi-automatic procedure has been

developed using Python language. This procedure is capable

of importing and pre-processing gear geometries as well as of

post-processing the results of the analysis. The flow chart in

Figure 2 shows the three main steps of this procedure. Since



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this procedure has been conceived to help gear designers in

evaluating several geometrical solutions in several operating

conditions, the great majority of the steps hereafter described

have been completely automated.

Figure 2: Modelling procedure overview.

Pre-Processing:

In the first phase, the three dimensional geometrical models of

the gears are generated starting from the design data.

In order to reach an adequate precision in tooth geometry

definition, specific software has been developed in MATLAB

(see [3], [21] and [22]). This software computes a cloud of

points on the tooth flank and root fillet and writes them into a

text file in a format that CAD software (e.g. Pro-Engineer)

can read and import. This text file may also contain coordinate

resulting from the measurement of an actual gear. In the CAD

system, these points are first interpolated using splines in

sketches. Flank and root surfaces are then generated by

“interpolating” these curves. By means of common modelling

techniques, the models of the wheels and the pinion are

generated and finally exported in a neutral file format.

In this stage it is possible to introduce several geometrical

deviations of the single wheel. In particular, it is possible to

introduce pitch error. It is also possible to realize a different

micro-geometry for each flank; in the followings, anyhow, the

same micro-geometry is adopted for all the teeth.

Even tough, eccentricity and parallelism deviation between

hub and teeth axis may be introduced. Nevertheless, these

deviations will be more efficiently introduced in the assembly

stage, so that a change of the values of these deviations does

not require the generation of a new model, but simply the

change of the rotation axes, i.e. a change of the boundary

conditions in the FE model.

Finally gear and pinion geometrical models are imported in a

FE software, ABAQUS/CAE [23].

Once the geometrical models are created, each wheel (gear

and pinion) is divided and exported in two parts for the sake

of controlling the mesh generation in ABAQUS/CAE; the first

part contains the active portion of the teeth and the second

part contains the remaining portion of the gear body. Figure 3

shows these two parts.

Figure 3: Gear geometry partitioning.

In order to use hexahedral elements, the resulting geometry is

subdivided in quadrilateral regions, paying particular care to

their form, as to say avoiding shapes that could lead automatic

mesh generation algorithms to generate unacceptably distorted

element (Figure 3). The two parts are then meshed, refining

the contact surfaces and the zones where a high stress gradient

is expected, as shown in Figure 4. Finally, the parts are tied by

surface-based tie constraint technique [3].

After having defined material properties, the gear and the

pinion are assembled as shown in Figure 5.

At this stage the ADs are introduced. ADs are categorized in

parallel and angular ADs. The parallel ADs are related to a

translation of the gear axes due to such as centre distance

deviations and eccentricities. The angular ADs are associated

with angular rotational errors of the gear axes, and they are

applied using angles 𝜺𝒕 and 𝜺𝒏. In order to apply 𝜺𝒕 and 𝜺𝒏

rotations, datum axes 𝒕 and 𝒏 are also created.

Once the gear and pinion are assembled and misaligned, the

next step is to define the contact surfaces of pairs of teeth. The

contact between tooth flanks will be then handled by

ABAQUS using a general purpose contact algorithm, so that

the actual contact areas and stresses are determined. The teeth

pairs contact analysis will be therefore executed without

introducing any simplification.

Figure 4: Gear mesh.



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Figure 5: Assembled gear pair.

For a systematic and automatic post-processing of results, the

regions from which GMCs will be extracted are located on the

pinion finite element model in the pre-processing phase (see

Table 1). How the GMCs are extracted is discussed in post-

processing sub-section.

The last step in pre-processing consists in applying load and

boundary conditions. These conditions are defined so that it is

possible to automatically run a sequence of static analysis.

The application of the prescribed conditions is summarized in

the following key-points.

Creation of two reference points at the gear and pinion

centres.

Connect the two reference points to the hubs by means of

Multi-Point Constraints (MPC). The application of these

constraints to the hubs is equivalent to neglecting shaft

deformations.

Apply the torque on the reference point of the pinion

about its axis.

Apply the constraints:

o Displacements (U1, U2 and U3) on the two reference

points;

o Rotations (UR3) about the gear reference point.

The selection of the reference system with respect to which

these constraints are defined is of great importance.

If these constraints are defined in the reference system of the

gear, the above described misalignments are actually applied,

since gear rotation and geometrical axis are coincident. On the

other hand, if they are defined with respect to the global

reference system, the gear rotates about the theoretical axis,

but its geometrical axis is rotated, hence a misalignment

between gear bore and teeth is simulated. This approach can

be obviously generalized by defining an additional reference

system to define a different gear rotation axis, in order to

simulate a misaligned gear-set, the gear of which rotates about

an axis different form the gear geometrical axis. Similar

considerations can be done for pinion constraints: it is

possible to simulate its rotation about a different axis by

creating another adequate reference system.

Table 1: Finite Element model zones from which the data

used to evaluate GMCs were extracted.

GMC Location

TE Hub of pinion

MPRS Root fillet of tooth that completes a mesh cycle

CPRESS Flank surface of tooth that completes a mesh cycle

Analysis:

The complete analysis is composed of a sequence of static

analyses, performed loading the system in a predetermined set

of positions. After each static analysis the constraints that

prevent the gear rotation and the torque applied on pinion are

removed. In this way, it is possible to automatically move the

system to the position for the next analysis. Constraints are

then re-established, and the system is ready for a new static

analysis. This procedure is automatically completed by

ABAQUS/STANDARD. Following this procedure, it is

possible to analyse the whole meshing of a teeth pair,

analysing it in any position. It is worth noting that these

analyses are static, and therefore they do not include any

dynamic effect. On the other hand, even if a dynamic analysis

would be performed considering the gear set only, it will not

supply reliable information about the dynamic behaviour of

the actual gear mounted in a transmission, since these results

are greatly influenced by inertia, stiffness and damping of the

other components of transmission (shafts, bearings, …). The

presented approach is therefore suited to determine the static

TE (in the following, only static TE will be considered, and

then the adjective ‘static’ will be omitted).

The analysis is performed by the commercial solver

Abaqus/Standard. It creates an output database file (ODB) that

contains model information and analysis results in terms of

assembly of part instances. These results have to be converted

in terms of design variables, in order to obtain the GMCs.

Post-Processing: The GMCs data are extracted from regions listed in Table 1.

Because of the huge amount of data, a software tool to extract

these data from the ODB file has been developed using

Abaqus Scripting Interface (ASI), which is an extension of the

Python object-oriented programming language. These data are

extracted and directly saved in text data files; in this section

the relationships used to determine each GMC are presented.

Before continuing, it is worth explaining how these results

were extracted and analysed.

Starting from contact pressures, at each meshing position, the

contact pressures at nodes on the contact surfaces are given as

Abaqus/Standard output. Several nodes are generally in

contact depending on mesh size: in this sample case, about

five nodes are in contact in each cross section (see Figure 6).

The contact pressure values of all the contacting nodes were

stored dividing them depending on the transverse cross

section where they were located. Then, in each transverse

cross section, the nodes where the contact pressure reaches its

maximum value are extracted from the stored data. This



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procedure is repeated for all the analysed meshing positions,

so that, finally, a contour plot can be drawn, in order to show

the maximum contact pressure values as a function of the

pinion roll angle and the position along the face, i.e. on the

plane of action of the helical gear pair. These distributions of

contact pressures can be used, on the one hand, to calculate

the stress distribution below the contact surface for each

meshing position allowing the determination of the stress

histories which the gear material undergoes and the

subsequent application of multiaxial fatigue criteria for the

determination of the pitting load carrying capacity (see e.g.

[24] and [25]). On the other hand, the envelope of the area in

the contact plane where contact pressures are different from

zero represents a contact pattern on the plane of action of the

helical gear pair.

Figure 6: Contact pressures

Load sharing ratio is defined as the fraction of the sum of the

normal contact force born by each teeth pair. The resultant of

the contact pressure (CPRESS) acting on each tooth pair is

automatically computed by Abaqus at each position.

Transmission error (TE) is determined according to its

definition as follows:

𝑻𝑬 = 𝜽𝟏 −𝒁𝟐

𝒁𝟏

𝜽𝟐

where 𝒁𝟏,𝟐 are the number of teeth of the pinion and the wheel

respectively and 𝜽𝟏,𝟐 denotes their angular rotations with

respect to the ideal contact configuration. The pinion rotation

is derived from the history output of the rotational degree of

freedom (UR3) of the reference point of the pinion hub, while

the wheel position is predetermined since the wheel hub

rotation is constrained.

The maximum value of the maximum principal stress at the

tooth root is adopted as an indicator of bending stress level.

First, the whole meshing cycle is examined in order to locate

the meshing position where the maximum principal root stress

(MPRS) reaches its maximum. Then, in this meshing position,

the values of MPRS are extracted at the diameter and in the

cross section of the node where it reaches the maximum value.

The MPRS and CPRESS data are stored as a function of their

position with respect to face and radius (Figure 6) so that it

will be easier to analyse them. In such data arrangement the

results can be treated in different ways, for example, variation

of MPRS along the face width at constant fillet radius or vice-

versa.

TEST CASES

The test cases here presented were taken from the

experimental work of Hoatait and Kahraman [20] in order to

have the possibility to validate the model with their

experimental results. These tests have been realized pairing

the same pinion with three gears with different micro-

geometries and introducing controlled ADs. Table 2 lists the

main geometric data of these gear pairs. The pinion had a

nominal tooth profile crown modification of 12 µm in the

involute direction, while it had no lead modification. The

mating wheels were not modified in the involute direction, but

they had different amounts of circular lead crowning FC (0,

12 µm and 25 µm). Since the same pinion was mated with all

these three wheels, the three pairs are named FC0, FC12, and

FC25, referring to the lead crown of 0 µm, 12 µm and 25 µm

respectively.

Table 2: Main data of the sample gear pair geometry

Parameter Pinion Gear

Normal module [mm] 2.04

Normal pressure angle [deg] 16.0

Helix angle [deg] 32.5

Pitch diameter [mm] 150.0

Base diameter [mm] 142.02

Minor diameter [mm] 142.43

Major diameter [mm] 153.74 153.24

Number of teeth 62 62

Face width [mm] 20 20

Circular tooth thickness [mm] 2.46

Root fillet radius [mm] 1.34

Centre distance [mm] 150

Involute crown [µm] 12 0

Several of the experiments (listed in Table 3) were simulated

in order to analyse the combined effect of lead modifications

and assembly deviations as well as to validate the

implementation of the above described FEA procedure.

The validation of FE models was based on the variation of

root stresses due to the combined effects of lead modifications

and assembly deviations of helical gear pairs. In particular,

experimental results concerning the time history of the tooth

root strains along the face width, measured by means of strain

gauges, were compared with the ones obtained from the

numerical simulations. The experimental and the numerical

results were in good agreement.

RESULTS

The main defects of misaligned helical gear pairs are edge

contact, vibrations, noise, and not favourable bending stress



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distribution along the face width. The presence and the extent

of these defects can be analysed by means of GMCs, such as

contact pattern, load sharing ratio, transmission error and

bending stresses. In the following sections, the GMCs of the

previously described gear pairs, calculated by means of the

procedure described in the previous sections, are presented

and compared in order to analyse the combined effects of ADs

and lead modifications from the gear design view point.

Contact Pattern:

One of the main concern of design and manufacturing of

helical gear pairs with parallel axes is the edge contact of

tooth surfaces that can be caused by misalignment. The

presence of edge contacts is checked by the determination of

the tooth contact pattern.

The main objective of this section is to understand the

combined effect of lead modifications and assembly

deviations on the contact pattern of the helical gear pairs here

considered. The results about the contact pattern were derived

from the calculated contact pressures, but quantitative results

about contact pressures are not here discussed since a more

refined mesh than the one used in the present study, shown in

Figure 4, should be used in order to make an accurate analysis.

Figure 7 shows the contour plots of maximum contact

pressures as a function of face width and roll angle with three

levels of lead modification and four levels of ADs at 200 Nm.

In this figure, across the column the lead modification varies

and along the row the AD changes. Looking at this diagram, it

is possible to see an envelope of the contact pressures that

represents the tooth contact pattern on the plane of action

resulting from the whole engagement cycle.

In the cases with perfectly aligned gear pairs, this envelope

has a parallelogram shape for the uncrowned gear pair, while

it has an elliptical shape for both the crowned gear pairs. The

parallelogram shape of the contact pattern of the FC0 model is

due to the presence of edge contacts when the tooth pair enters

and exits meshing, whereas the elliptical shape for FC12 and

FC25 means that lead modifications concentrate the contact

pattern in the central part of the face width avoiding the

problem of edge contacts. Moreover, it can be observed that

in the upper and the lower part of the contour plots there are

white areas which are related to the reduction of contact ratio

due to lead modifications.

Regarding the models with ADs, shown in the second row of

Figure 7, it can be seen that for the case without lead

crowning the contact pattern changes to a triangular shape

with a significant edge contact. In the FC12 model, the

contact pattern still has an elliptical shape, but it is sliced at

the beginning of the face width due to the presence of a light

edge contact. The elliptical envelope of FC25 is slightly

changed to one side of the face width, but, with this amount of

lead crowning, there is no edge contact. In general, we can see

that in FC0 and FC12 there is edge contact when the mating

pair exiting the engagement action. This means that the 12 µm

lead modification is not enough to completely avoid edge

contact, while the 25 µm resolves the situation also for AD of

εn=0.002. This founding is in accordance with the

considerations below developed analysing TE. The other

remark that can be made is that the contact length is decreased

by the introduction of ADs as will be discussed in the next

section.

Figure 7: Contact patterns on the plane of action at 200 Nm for various amounts of circular lead crowning without assembly

deviations, in the first row, and with assembly deviations, in the second row.

Load-Sharing Ratio: Figure 8 shows the calculated Load Sharing Ratios (LSRs) at

four different load levels, i.e. input torques, for three different

amounts of the circular lead crowning. Opposite to what

happens in spur gears, lead crowning in helical gears causes

also a reduction of the contact ratio and a variation of the LSR.

In the left column of Figure 8 the LSRs for gears without ADs

are shown. It can be seen that, although for low load levels (i.e.

at 100 Nm and 200 Nm) the longitudinal modifications causes

an increase of the maximum load carried by a tooth and a

reduction of the contact ratio, for high values of the input

torque (i.e. 300 Nm and 600 Nm) the LSRs for the three

solutions are quite similar, i.e. the detrimental effect on the

LSR introduced by lead crowning is negligible.

If the ADs are introduced (right column of Figure 8) in the

solutions FC0, there is a significant increment of the

maximum load carried by a tooth and a reduction of the

contact ratio. In the FC12 the ADs cause the same effects, but



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the extent is reduced. In the FC25 solution, the effect of ADs

is practically negligible. Nevertheless, comparing the different

solutions and taking into account the effects of ADs, the FC25

solution at low loads is the one with the maximum load

carried by a single tooth and the minimum contact ratio, but

for high loads where the situation is pretty much the same.

Transmission Error:

The results presented in this section consider only the variable

part of the TE (practically, the average value has been

subtracted from the TE resulting from the FEM simulations),

since the constant part does not excite system vibrations.

Figure 9 shows the TE for all the considered gear sets. As

expected, the lower TE occurs without any face crowning for

the perfectly aligned gear set, since a 12 µm profile

modification is always present. According to MAAG [26],

Niemann [27] and Dudley [28], this profile modification

should imply a minimum TE when the input torque is about

150 Nm. Anyhow, since the contact pattern in helical gears

evolves “diagonally” on the tooth flank, the face crowning

acts somehow as a profile modification at the beginning and at

the end of meshing of each teeth pair. Therefore, the

minimum TE can occur at different loads.

The last row of Figure 9 shows a summary of all the graphs in

Figure 9: it shows the peak-to-peak value of the TE (PPTE) as

a function of input torque for all the considered gear sets,

without and with ADs.

(a) FC0 —εn=0 rad (e) FC0 — εn=0.002 rad

(b) FC12 — εn=0 rad (f) FC12 — εn=0.002 rad

(c) FC25 — εn=0 rad (g) FC25 — εn=0.002 rad



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(d) Comparison at 200 Nm with εn=0 rad (h) Comparison at 200 Nm with εn=0.002 rad

Figure 8: Load Sharing Ratio (LSR) variation against pinion roll angle without assembly deviations, in the right column (a-d),

and with assembly deviations, in the left column (e-h), for various amounts of cicular lead crowning.






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(d) Comparison with εn=0 rad for different torques (h) Comparison with εn=0.002 rad for different torques

Figure 9: Transmission Error (TE) variation against pinion roll angle without assembly deviations in the right column (a-c) and

with assembly deviations, in the left column (e-g), for various amounts of circular lead crowning. In the last row (d and h),

comparison of the Peak-to-Peak Transmission Error (PPTE) variation against pinion torque.






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(d) Comparison at 200 Nm with εn=0 rad (h) Comparison at 200 Nm with εn=0.002 rad

Figure 10: Maximum Principal Root Stress (MPRS) variation along tooth face width without assembly deviations, in the right

column (a-d), and with assembly deviations, in the left column (e-h), for various amounts of cicular lead crowning.

If the gear set are not misaligned, the PPTE of FC0 increases

as the load increase, since the profile modification is not

capable to absorb gear teeth deformations that increase with

load. The gear set with a 12 µm face crowning exhibits a

clear minimum of the PPTE at about 200 Nm, while the gear

set with a 25 µm face crowning shows a minimum of PPTE at

a higher load close to 300 Nm. These numerical results are in

agreement with the well-known fact that face crowning in

helical gears acts also as a profile modification because of

gradual engagement of helical gear teeth.

Concerning the effect of the assembly deviations, it can be

clearly seen that their effect on PPTE decreases as the amount

of the face crowning increase. The gear pair without any

crowning shows a significant increase of the PPTE above and

below 200 Nm where a minimum of the PPTE comes out.

This effect of the assembly deviations seem is likely to be due

to a different contact pattern and its interaction with a

different portion of tooth profile modifications. The PPTE of

the gear set with a 12 µm face crowning exhibits a slight

increase of the PPTE with respect to the perfectly aligned

condition, but without any significant change in its trend

against the applied torque. In the case of FC25 the changes

induced by misalignments are practically negligible.

Bending Stresses: Figure 10 shows the maximum principal root stress (MPRS)

plots obtained at four load levels for two levels of ADs and

for three different amounts of lead crowning.

In the left column of Figure 10 the variation of MPRS along

the face width of perfectly aligned gear sets are shown. It can

be clearly observed that, for both gears with and without lead

modifications, the curves have similar trends at each different

load. In perfectly aligned gear pairs, as expected, lead

modifications causes an increase in the central part of the face

width and a decrease at the ends. As can be seen in the last

row of Figure 10, where a comparison at 200 Nm is shown,

the increase of the maximum value of the MPRS along the

face width for both the crowned gear pairs is relevant respect

to the maximum value of MPRS for uncrowned gears.

Coming to misaligned gear pairs (see the right column of

Figure 10), in the gear pair FC0 the maximum MPRS

increased significantly due to AD. Looking at the other

models (FC12 and FC25), it can be noticed that their MPRS

peak is practically identical when εn=0.002, and that the

increase of the MPRS is lower. This implies that introducing

lead modification reduces stresses induced by ADs. The other

difference between FC12 and FC25 is the locations of

maximum MPRS along the face width: 3.38 mm and 5.76 mm,

respectively. This means that if the gear designer wants to

keep the maximum MPRS in the middle of the face, a bigger

lead modification should be used. In other words, the

maximum bending stress zone can be kept in the middle of the

face only if an adequate face modification is applied; if the

gear pair will work above the design AD amount, the

maximum MPRS will move from the middle of the face.

Practically, from bending strength point of view, a 12 µm face

crowning seems more favourable, since it results in an MPRS

increase close to increase of the 25 µm face crowning solution

if the actual AD corresponds to εn=0.002, while the MPRS

increase of the FC12 solution is lower if εn is lower. However,

these considerations could be a conflicting choice if the

interest is in minimizing the gear noise as discussed in the

previous sections.

Discussion: In the previous sub-sections, the behaviour of a sample set of

helical gear pairs has been analysed from fourth main points

of view: contact pattern, bending stresses, load-sharing ratio

and transmission error. All these aspects have to be taken into

account and integrated in a unique design. In particular, with

respect to the considered parametric set of helical gear pairs, it

has been shown that

• from bending strength point of view, a 12 µm face

crowning seems more favourable, since it results in

an MPRS increase close to increase of the 25 µm

face crowning solution if the actual AD corresponds

to εn=0.002, while the MPRS increase of the FC12

solution is lower if εn is lower.

• from the contact pattern point of view, it has been

shown that for load levels over 200 Nm the only

solution able to avoid edge contacts is the one with a

lead crowing of 25 µm, while with a lead crown of

12 µm, although in the perfectly aligned condition



11691

there is no edge contact, when ADs are introduced

the edge contact cannot be completely avoided.

• The gear set with a 12 µm face crowning exhibits a

clear minimum of the PPTE at about 200 Nm, while

the gear set with a 25 µm face crowning seems to

have a minimum PPTE at about 300 Nm. It is also

worth noting that if the gear sets are misaligned, the

gear set with a 12 µm face crowning seems to exhibit

a slight reduction of the PPTE with respect to the

gear without any crowning, and both have a

minimum at about 200 Nm. On the other hand, the

gear set with 25 µm exhibits a minimum PPTE at

higher loads.

As usual, a design solution has different impacts on different

characteristics of the gear. The final design has hence to be

decided on the basis of the actual operating conditions (loads

and ADs) and on the resulting characteristics that are more

important for the considered application. For example, if the

gear set will mainly operate at 200 Nm and the misalignment

will be lower than 0.002 rad, the FC12 solution seems to be

more suitable, since the PPTE of this gear set is minimum at

200 Nm, but when εn=0.002 the risk of edge contact is still

quite high. On the other hand, if the gear set will operate at

higher loads, the FC25 solution appears to be more suitable.

The drawback is that if the gear set will operate with lower

loads and/or lower ADs, the TE and the MPRS can be higher

than expected.

CONCLUSIONS

In this paper, a FEM based approach developed in order to

investigate the combined effects of assembly deviations and

lead modifications on meshing characteristics of helical gears

has been presented.

In the described approach, the misalignments are defined in

the plane of action, because their subdivision in this plane is

more significant from gear meshing point of view. Since

standards define shaft parallelism deviations in the so-called

“common plane of axes”, a set of equations that relate the ISO

with the proposed definitions has also been derived. The

proposed approach is very flexible since it allows taking into

account easily tooth flank modifications, assembly deviations

(misalignments of axes of rotations and centre distance

deviation) as well as manufacturing deviations (e.g.

misalignments between teeth and gear hub). Therefore, the

presented approach is suitable for being integrated in an

optimization procedure, in which the goal function is defined

by weighing synthetic measures of gear meshing

characteristics (such as maximum root bending stress,

maximum contact pressure, peak-to-peak transmission

error, …) so that a suitable compromise between the

maximization of load carrying capacity and the minimization

of noise and vibrations can be achieved.

As expected, bending stresses, transmission error and contact

pressures are greatly influenced by the amount of lead

modification and by assembly deviations. Although the edge

loading situation is eliminated, excessive lead modification

amounts could result in an increase of bending stresses and

contact pressures. Under a more practical and general

perspective, these considerations confirm that design choices

can be made only if the goal is clear and explicitly stated. In

the examined sample cases, for example, the FC12 solution

seems to be more suitable if the aim is to reduce root stress,

while the FC25 solution will likely result in a gear set with a

smoother meshing cycle. These considerations confirm the

needing of a software tool capable to contemporaneously

determine the meshing characteristics of a gear set in its actual

operating conditions.

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APPENDIX :

Rotations εn, εt and assembly deviations (fΣβ, fΣδ) according

to standards.

The main aim of this Appendix is to determine a general

relation between rotation εn, εt and standard deviations fΣβ, fΣδ

(Figure 11). Even though these rotations are usually small,

these relationships will be first derived in a general case, and

then linearized.

Furthermore, even if this approach has been conceived to

introduce in-plane and out-of-plane deviations, it will be

shown that it can be used to introduce another sort of gear

geometrical deviation, i.e. gear rotation axis deviation with

respect to gear geometrical axis.

Only the gear will be rotated, since we are interested in

determining the relative position between gear and pinion, and

not in defining the position of the members of a gear-set with

respect to a theoretical reference frame. In fact, the ISO and

AGMA standards define the misalignments on the basis of a

reference system built on the basis of pinion and gear relative

position.

First the gear set is assembled in its nominal position.

Secondly the gear axis is rotated about t and n axes. Finally,

the common plane of axes (“defined by using the longer of the

two bearing spans (L) and one of the bearings on the other

shaft. If the bearing spans are the same, use the pinion shaft

and a wheel bearing.” [AGMA]) is determined on the basis of

the position of three bearing centres and then the out-of-plane

deviation fΣβ (“measured in the “skew plane” which is

perpendicular to the common plane of axes” [AGMA]) and

the in-plane deviation fΣδ (“measured in the common plane of

axes” [AGMA]) are determined on the basis of the fourth

bearing centre position. In other words, the out-of-plane

deviation is the distance between the common plane of axes

and the fourth bearing centre, while the in-plane deviation is

the distance between the projection of the fourth bearing

centre on the common plane of axes and a line parallel to the

first axis (defined by the first two bearings) passing through

the third bearing.



11693

Figure 11: Shaft parallelism deviations according to AGMA

915-3 A99.

In order to develop general relationships, the following

geometrical entities have been considered as indicated in

Figure 12:

A: operating centre distance;

α: pressure angle;

Z1: Z coordinate of the centre of the first

bearing of pinion;

L1: pinion bearing span;

Z2: Z coordinate of the centre of the first

bearing of gear;

L2: gear bearing span;

ZK: Z coordinate of the origin of reference

system tnz;

YK: Y coordinate of the origin of reference

system tnz;

It is worth noting that centre distance is defined before

applying rotations, when the two axes are still parallel. It is

therefore convenient to introduce centre distance error in this

stage; in other words, A is the operating centre distance.

After having the assembled gear set, the gear axis is rotated.

At this stage, it is worth noting that the order in which

rotations are applied is important, since εn and εt can assume

any value.

The equation of the theoretical axis of the gear in global

reference (XYZ) is

𝒓2(1)

= [

0𝐴

𝑍2 + 𝑡1

]

where t is the parameter.

Since the coordinate transformation matrix between

references XYZ and tnz is

𝑴12 = [

sin 𝛼 𝑐𝑜𝑠𝛼 0 0cos 𝛼 −sin 𝛼 0 𝑌𝐾

0 0 1 𝑍𝐾

0 0 0 1

]

the equation of the theoretical axis of the gear in reference tnz

is

𝒓2(2)

= 𝑴12−1 ∙ 𝒓2

(1)

Hence the gear axis rotated about n axis (in tnz reference) is

expressed by the following equation:

�̃�2,𝑛(2)

= [

1 0 0 00 cos 𝜀𝑛 − sin 𝜀𝑛 00 sin 𝜀𝑛 cos 𝜀𝑛 00 0 0 1

] ∙ 𝒓2(2)

This axis has then to be rotated about t axis:

�̃�2,𝑛𝑡(2)

= [

cos 𝜀𝑡 0 sin 𝜀𝑡 00 1 0 0

− sin 𝜀𝑡 0 cos 𝜀𝑡 00 0 0 1

] ∙ �̃�2,𝑛(2)

and, finally, it can be represented in the global reference:

�̃�2(1)

= 𝑴12 ∙ �̃�2,𝑛𝑡(2)

It is now possible to determine bearing centre positions.

Figure 12: Adopted reference systems (XK = 0).

Pinion

𝑻𝟎 = [

00𝑍1

1

] 𝑻𝟏 = [

00

𝑍1 + 𝐿1

1

]

Gear in its theoretical position

𝑸𝟎 = 𝒓2(1)

|𝑡=0

= [

0𝐴𝑍2

1

] 𝑸𝟏 = 𝒓2(1)

|𝑡=𝐿2

= [

0𝐴

𝑍2 + 𝐿2

1

]



11694

Misaligned gear

𝑷𝟎 = �̃�2(1)

|𝑡=0

𝑷𝟏 = �̃�2(1)

|𝑡=𝐿2

The next step is to determine the common plane of axes, i.e.

the plane passing through the centres of the two bearings of

gear-set element with the shorter bearing span and the centre

of one of the bearing of the other element. In order to unify

the formulas, in the following calculations, the three bearing

centres used to determine the common plane of axes will be

named H1, H2 (belonging to the same gear member) and H3,

while the last bearing centre (H4) will be used to determine

deviations. Practically, if the bearing span of the pinion is

shorter:

𝑯1 = 𝑻0 𝑯2 = 𝑻1 𝑯3 = 𝑷0 𝑯4 = 𝑷1

otherwise:

𝑯1 = 𝑷0 𝑯2 = 𝑷1 𝑯3 = 𝑻0 𝑯4 = 𝑻1

Hence, the coefficients of the equation of the common plane

of axes (𝑎 ∙ 𝑋 + 𝑏 ∙ 𝑌 + 𝑐 ∙ 𝑍 + 𝑑 = 0) are:

𝑎 = |𝑌𝐻2

− 𝑌𝐻1𝑌𝐻3

− 𝑌𝐻1

𝑍𝐻2− 𝑍𝐻1


|

𝑏 = |𝑋𝐻2

− 𝑋𝐻1𝑋𝐻3

− 𝑋𝐻1



|

𝑐 = |𝑋𝐻2

− 𝑋𝐻1𝑋𝐻3

− 𝑋𝐻1

𝑌𝐻2− 𝑌𝐻1

𝑌𝐻3− 𝑌𝐻1

|

𝑑 = −[𝑎 𝑏 𝑐] ∙ [

𝑋𝐻1

𝑌𝐻1

𝑍𝐻1

]

The distance of the fourth bearing centre from this plane is

hence:

𝑑Σ𝛽 = |𝑎 ∙ 𝑋𝐻4

+ 𝑏 ∙ 𝑌𝐻4+ 𝑐 ∙ 𝑍𝐻4

+ 𝑑

√𝑎2 + 𝑏2 + 𝑐22 |

Finally, the deviation 𝑓Σ𝛽can be computed as 𝑓Σ𝛽 = |𝑑Σ𝛽 𝐿⁄ |

where L is the relevant span according to standards.

The last step is to determine the in-plane deviation (𝑓Σδ).

First we need to determine point H’4 that is the projection of

point (H4) on common plane of axes:

𝑯4′ = 𝑯𝟒 + [𝑎 𝑏 𝑐]𝑻 ∙ 𝑡0

where

𝑯𝑛 = [𝑋𝐻𝑛𝑌𝐻𝑛

𝑍𝐻𝑛]𝑻

and

𝑡0 = −𝑎 ∙ 𝑋𝐻4

+ 𝑏 ∙ 𝑌𝐻4+ 𝑐 ∙ 𝑍𝐻4

+ 𝑑

𝑎2 + 𝑏2 + 𝑐2

Second, we need the unit vector (ur’) of line r’ that is parallel

to line (H2-H1) and passes through point H3. Since these two

lines are parallel, their unit vector is identical. The unit vector

ur’ can be hence determined as follows:

𝒖𝑟′ =𝑯2 − 𝑯1

|𝑯2 − 𝑯1|

Finally, the distance between point and line r’ can be

determined by means of the following equation:

𝑑Σδ = |𝒖𝑟′ × (𝑯′4 − 𝑯3)|

and, hence, the corresponding standard misalignment

is: 𝑓Σ𝛿 = 𝑑Σ𝛿 𝐿⁄ .

These equations are very general. The drawback is that they

cannot be easily handled. Nevertheless, these equations can be

greatly simplified by assuming that rotations are small (i.e.

assuming that sin 𝜀 ≅ 𝜀 , cos 𝜀 ≅ 1 and that second order

terms are negligible).

The final result of this simplification is:

𝑑Σ𝛿 = 𝐿 ∙ |− sin 𝛼 ∙ 𝜀𝑛 + cos 𝛼 ∙ 𝜀𝑡|

𝑑Σ𝛽 = 𝐿 ∙ |cos 𝛼 ∙ 𝜀𝑛 + sin 𝛼 ∙ 𝜀𝑡|

and hence

𝑓Σ𝛿 = |− sin 𝛼 ∙ 𝜀𝑛 + cos 𝛼 ∙ 𝜀𝑡|

𝑓Σ𝛽 = |cos 𝛼 ∙ 𝜀𝑛 + sin 𝛼 ∙ 𝜀𝑡|

where 𝜀𝑛 and 𝜀𝑡 are expressed in radiant.

It is worth noting that, if rotations are small enough

(|𝜀| < ~ 1 2⁄ °), the only relevant geometrical parameter is the

bearing span L. The influence on standard misalignments of

all the other dimensions is negligible, and, hence, they are not

likely to influence gear meshing characteristics. Furthermore,

if the rotations are small, their application order is no longer

influent, i.e. the two rotations can be applied in any sequence.

Documents

MCombined Effects of Lead Crowning and Assembly … because of FEM high permeation in the usual design process. Khoshnaw and Ahmed [9] ... root stresses of helical gear pairs obtained