Latest advancements in the lubricant simulations of geared systems: a technology ready for industrial applications

Abstract

Energy efficiency represents an important aspect of mechanical design. Despite their long history, gears still play a determinant role in several applications ranging from the automotive, to the aeronautical sectors. The more and more stringent regulations in terms of efficiency have encouraged the gearbox manufacturers to increase the investments to achieve more efficient designs leading to energy saving, reduction of pollutant emission and increased reliability related to the reduction of the operating temperatures. A decrease of the power losses allows also a downsize and a reduction of the weight of the system, with an increase in the power density and performances. Engineering tools allowing a comparison of different design solutions already during the design stage can pave the way to a real transition to a sustainable future. Most available models are based on empirical relations and dimensional analyses resulting to be accurate only as far as the geometry and operating conditions reflect the ones used to calibrate the models. With the developments in computational performances the research started to focus on numerical approaches. However, while most of the numerical approaches have been proved to be sufficiently accurate to capture the power losses of geared systems, the high computational effort required for their application to real gearboxes is still hurting with the industrial practice. Moreover, new phenomena related to new lubricant (e.g aeration, channeling, circulation) could be not captured/simulated with the standard available models. In this paper the latest advancements to overcome both the computational effort issue and the lack of specific models are shown with practical industrial case studies.

Full text

ÜBERSICHTSARBEITEN/REVIEW ARTICLES
https://doi.org/10.1007/s10010-023-00698-z
Forschung im Ingenieurwesen (2023) 87:1181–1191
Latest advancements in the lubricant simulations of geared systems:
a technology ready for industrial applications
Franco Concli1·MarcoN.Mastrone
1
Received: 10 March 2023 / Accepted: 20 July2023 / Published online: 22 August 2023
© The Author(s) 2023
Abstract
Energy efficiency represents an important aspect of mechanical design. Despite their long history, gears still play a deter-
minant role in several applications ranging from the automotive, to the aeronautical sectors. The more and more stringent
regulations in terms of efficiency have encouraged the gearbox manufacturers to increase the investments to achieve more
efficient designs leading to energy saving, reduction of pollutant emission and increased reliability related to the reduction
of the operating temperatures. A decrease of the power losses allows also a downsize and a reduction of the weight of
the system, with an increase in the power density and performances. Engineering tools allowing a comparison of different
design solutions already during the design stage can pave the way to a real transition to a sustainable future. Most available
models are based on empirical relations and dimensional analyses resulting to be accurate only as far as the geometry and
operating conditions reflect the ones used to calibrate the models. With the developments in computational performances
the research started to focus on numerical approaches. However, while most of the numerical approaches have been proved
to be sufficiently accurate to capture the power losses of geared systems, the high computational effort required for their
application to real gearboxes is still hurting with the industrial practice. Moreover, new phenomena related to new lubricant
(e.g aeration, channeling, circulation) could be not captured/simulated with the standard available models. In this paper
the latest advancements to overcome both the computational effort issue and the lack of specific models are shown with
practical industrial case studies.
Franco Concli
franco.concli@unibz.it
1Faculty of Engineering, Free University of Bolzano/Bozen,
Piazza Università, 39100 Bolzano, Italy
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1182 Forschung im Ingenieurwesen (2023) 87:1181–1191
Neuste Fortschritte bei der Schmierstosimulation von Getriebesystemen: eine industrietaugliche
Technologie
Zusammenfassung
Die Energieeffizienz ist ein wichtiger Aspekt der mechanischen Entwurf. Getriebe spielen immer noch eine entscheidende
Rolle in verschiedenen Anwendungen, von der Automobilindustrie bis hin zur Luftfahrt. Die immer strengeren Vorschriften
in Bezug auf den Wirkungsgrad haben die Getriebehersteller dazu veranlasst, ihre Produkte zu verbessern, um effizientere
Konstruktionen zu erreichen, die zu Energieeinsparungen, einer Verringerung der Schadstoffemissionen und einer höheren
Zuverlässigkeit aufgrund der Senkung der Betriebstemperaturen führen. Eine Verringerung der Energieverluste ermöglicht
auch eine Verkleinerung und Gewichtsreduzierung des Systems bei gleichzeitiger Erhöhung der Leistungen. Technische
Hilfsmittel, die einen Vergleich verschiedener Konstruktionslösungen bereits in der Entwurfsphase ermöglichen, können
den Weg für einen echten Übergang zu einer nachhaltigen Zukunft ebnen. Die meisten verfügbaren Modelle beruhen
auf empirischen Beziehungen und Dimensionsanalysen, die nur insoweit genau sind, als die Geometrie und die Betriebs-
bedingungen denen entsprechen, die zur Kalibrierung der Modelle verwendet wurden. Mit den Entwicklungen bei den
Rechenleistungen begann sich die Forschung auf numerische Ansätze zu konzentrieren. Die meisten numerischen Ansät-
ze haben sich zwar als hinreichend genau erwiesen, um die Leistungsverluste von Getrieben zu erfassen, aber der hohe
Rechenaufwand, der für ihre Anwendung auf reale Getriebe erforderlich ist, stellt immer noch eine Einschränkung dar.
Darüber hinaus können neue Phänomene im Zusammenhang mit neuen Schmierstoffen (z.B. Belüftung, Kanalisierung,
Zirkulation) mit den verfügbaren Standardmodellen nicht erfasst/simuliert werden. In diesem Beitrag werden die neues-
ten Fortschritte zur Überwindung des Problems des Rechenaufwands und des Mangels an spezifischen Modellen anhand
praktischer industrieller Fallstudien vorgestellt.
1Introduction
The power losses of gearboxes can be classified into two
main categories. The first one comprises the load-dependent
power losses, i.e. the one related to friction and proportional
to the transmitted load. Those losses arise mainly in the
contact surfaces of bearings (sub-index B) and gears (sub-
index G). The second category of losses are the so-called
load-independent power losses (sub-index 0), that arise from
the interaction of the lubricant (mixture) and the moving/
rotating components [1]. PLS refers to the losses of seals,
while PXto the losses of other generic components such
clutches or synchronizes.
PL=PLG +PLG0+PLB +PLB 0+PLS +PLX (1)
While for the load-dependent losses accurate analyti-
cal models can be used, the load-independent losses are
more complex to be quantified due to the large amounts
of factors involved in the dissipation mechanism, ranging
from the lubricant type (mineral oil, synthetic oil, grease)
to the lubrication condition (dip lubrication, oil injection,
spray etc.). Moreover, with the “green-tribology”, targeted
to environmentally friendly consumption of resources and
energy, and the consequent development of new eco-lubri-
cants, new phenomena (e.g. aeration, channeling, circula-
tion etc.) have been observed. With these premises, the real
challenge is not any more to have only time-efficient mod-
els for predicting the power dissipation, but also reliable
model to capture the real phenomena causing that power
dissipation. The goal of the present paper is to shed light
on the latest possibility offered by the numerical approaches
for the simulation of the lubrication of gearboxes. In par-
ticular, different solvers and mesh handling techniques for
the simulation of various operating conditions (churning
lubrication, cavitation, aeration and grease-lubrication) as
well as geometrical configurations (spur gears, bevel gears,
planetary gears, multistage gearboxes) will be detailed de-
scribed.
2 State of the art
2.1 Analytical models
The load-independent power losses of gears PLG 0can
be subdivided three sub-categories: churning (PLG0;C ),
windage (PLG0;W ) and squeezing losses (PLG 0;S )
PLG0=PLG 0;C +PLG0;W +PLG0;S (2)
Churning and windage are two physical phenomena that
differ by the fact that the former is related to a fluid mixture
(multiple phases), the latter involves only one single phase
fluid. Churning losses are predominant in dip lubrication,
while windage has an impact only when gears strongly in-
teract with air (e.g. injection lubrication). These two phe-
nomena are complementary. Squeezing is instead related to
sudden volume variations in the mating region and the re-
lated pressure gradients. The first studies on this topic date
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Forschung im Ingenieurwesen (2023) 87:1181–1191 1183
back to Soo and Princeton [2], Daily and Nece [3], and
Mann and Marston [4] whose works involved the churning
of smooth disks. The first scholar dealing with a (single)
rotating gear was Ohlendorf [5]. Niemann [1] improved
the formulation by Ohlendorf to consider high immersion
depths and low tangential speeds. Terekhov [6] expanded
the application field to a wider range of operating conditions
and included additional influencing parameters such the ra-
dial wall distance and the volumes of the housing. Even
if the model according to Terekhov can be probably con-
sidered the first quasi-complete model, it does not consider
some important parameters such as, for example, the nor-
mal backlash and the helix angle. Similar approaches were
proposed by Boness [7], Lauster and Boos [8], Mauz [9],
and Walter [10]. Mauz’s model seems to be, even nowa-
days, 40 years later, the most complete available model.
Some years later, Changenet et al. [11] proposed an ex-
perimentally derived model based on dimensional analysis.
Another analytical model is proposed in the ISO/TR 14179-
1[12] standard. Kahraman et al. [13–16] proposed a physic-
based fluid mechanics model to predict the churning power
losses of a gear pair. For what concerns the windage losses,
the first studies were performed in the ’80 by Anderson
and Loewenthal [17] and successively by Dawson [18].
Starting from the results of Butsch [19], Maurer [20]and
Terekhov [6] proposed equations for the squeezing losses.
Diab et al. [21] derived an empirical model for the es-
timation of such losses. More recently, Quiban et al. [22]
focused their investigation on windage on spiral bevel gears.
As previously mentioned, the analytical/empirical model
have the big advantage to be very lean (from a computa-
tional perspective). However, the range of applicability is
very limited, they did not provide any information about
the lubricant distribution, and are not suitable to take into
account the new phenomena related to the adoption of new
lubricants and additives.
2.2 CFD studies on gears
The relatively recent significant increase in the computa-
tional resources had promoted a fast grown of the numeri-
cal approaches. Among them, Computational Fluid Dynam-
ics (CFD) is actually the most widespread method for the
analysis of systems involving fluid flows, including geared
transmissions. The main reason why CFD has lagged be-
hind with respect to other Computer-Aided Engineering
(CAE) tools is related to the high complexity of theunderly-
ing behaviour, which precludes a description of fluid flows
that is at the same time economical and sufficiently accu-
rate. However, despite these general considerations, CFD
has gained a lot of of interest also in industrial practice,
leading to the reduction of time and costs of new designs,
and to the possibility to study systems where controlled ex-
periments are particularly complex to perform. The main
CFD approach relies on Finite Volumes (FV). The do-
main of interest (computational domain, i.e. the internal
volume of the gearbox in case of lubrication simulations),
is discretized with a very high number of elementary vol-
umes (cells of the mesh). For each cell it is possible to
solve three conservation (or governing) Partial Differential
Equations (PDE)s for mass (Eq. 3), momentum (Eq. 4),
and energy (Eq. 5). These PDEs (Navier-Stokes equations)
are converted into algebraic equations that are solved itera-
tively [23].
@
@t +r.U/=0 (3)
@.U/
@t +r.UU/=
g−rp+2
3rU+rrU+rUT
(4)
@e
@t +r.eU/=gU −r.pU/−r2
3.rU/U
+rrU+rUTU+r.rT/+Q
(5)
An application of the FV method for gear lubrication sim-
ulations requires specific mesh-handling techniques to deal
with the topological changes of the domain in consequence
of the gear rotation. The simplest approach to study a single
rotating gear is the so called Moving Reference Frame
(MRF). It is a steady state approximation in which dif-
ferent cell regions move at different rotational speeds. This
method is also called “frozen rotor approach”: it does not
consider the relative motion of a rotating zone with respect
an adjacent stationary zone, i.e. the mesh remains fixed.Ad-
ditional terms related to the Coriolis and centripetal accel-
erations are added to the momentum Eq. 4. This approach is
particularly used in turbo-machinery, but it has been applied
also to single rotating gears. Very simple models consider-
ing a single sector of spur gears with periodic boundary
conditions were considered by Al-Shibl et al. [24], Pallas
et al. [25], Chaari et al. [26], and Marchesse et al. [27].
The scholars exploited this approach to study the windage
power losses of single spur gears. Sometimes even a 2D
simplification of the sector was adopted, leading to an over-
simplification of the physics and to an underestimation of
the power losses. Webb et al. [28] and Turner et al. [29]
applied the same approach on spiral bevel gears. Cavotta
et al. [30] studied the churning power losses of a rotating
disk analysing the influence of different turbulence mod-
els. Bianchini et al. [31,32] applied the MRF technique
to a planetary gearbox for aero-engine applications. How-
ever, in order to apply this modeling technique, the backlash
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1184 Forschung im Ingenieurwesen (2023) 87:1181–1191
was hugely increased so that the gears did not engage any
more. Lu et al. [33] implemented a numerical model of
a spiral bevel gear pair to study the lubrication and temper-
ature characteristics of an intermediate gearbox with splash
lubrication. Also in this case the gears meshing was ne-
glected. Apart from the impossibility to model the gear
meshing, the MRF gives just the regime solution of an
unsteady problem. In order to improve the results and de-
scribe the transient start-up, the sliding mesh approach (also
called rigid mesh motion RMM) can be an alternative. The
domain is split, like in the MRF approach, in several cylin-
drical regions. Differently from the MRF approximation,
the motion is reproduced by rotating one portion of the
mesh instead of the reference system only. Despite the ca-
pability to model transient regimes, also the RMM cannot
deal with the gear meshing. Santra et al. [34] exploited
this method to study the lubrication in the transmission of
a tractor. Concli et al. [35,36]andGorlaetal.[37]ap-
plied the RMM to investigate the load-independent power
losses of spur and helical single gears. The results showed
good agreement with experimental measurements. Fondelli
et al. [38] used such technique to analyse the oil jet im-
pinging on a single spur gear to evaluate the power losses
in oil jet lubrication. Hildebrand et al. [39] studied the in-
fluence of gear geometry and circumferential speed on the
heat dissipation under atmospheric conditions. In this case,
the center distance was augmented in order to apply the
sliding mesh, namely to eliminate the gear engagement. An
alternative that overcome the limitation of not being able
to model the gear engagement is represented by the over-
set mesh (also called overlapping grids method, or chimera
framework), in which, the different meshes are not comple-
mentary but overlapping. Each body has its integral mesh.
A common background mesh serves as reference grid for
the motions of the other grids. The field variables are inter-
polated between the grids. In this way the meshing process
is significantly simplified and completed independently for
each sub-region. However, the interpolation causes numer-
ical error and loss of accuracy. Klier et al. [40] applied the
over-set mesh approach to study the influence of the oil
filling level on the lubrication of two spur gears in a cylin-
drical domain. Cho et al. [41] implemented a CFD model
based on the over-set mesh approach of a planetary gearbox.
Renjith et al. [42] analysed the power losses and the lubri-
cation in a differential system, demonstrating the capability
Fig. 1 LRA: 2 subsequent meshes – the grids differs significantly in terms of element size
of the CFD model to provide useful information for the
optimization of the internal layout of the gearbox. Arisawa
et al. [43] implemented a numerical model to investigate
the influence of the shrouds on the churning and windage
power losses in bevel gears. Saegusa and Kawai [44] stud-
ied the oil flow in an automotive manual transmission. The
results of predictions of lubrication performance and churn-
ing losses displayed a good correlation with experimental
data. Deshpande et al. [45] analysed oil jet lubrication in
helical and spur gears. The authors have used this approach
to study the squeezing losses [46,47].
An alternative approach that allows to consider the gear
meshing is the re-meshing. This approach foresees a defor-
mation of the mesh to accomplish the boundary motions
up to a certain threshold after which some elements are re-
placed. On the one hand, this approach does not need the
interpolation of the field variables, hence ensuring high ac-
curacy. On the other hand, the regeneration of the mesh can
result in a higher computational cost. The re-meshing could
be either “local” or “global” depending regions enabled for
re-meshing.
In the Local Remeshing Approach (LRA), only the ele-
ments having low quality are deleted and re-created. This
method is effective but the mesh update can lead to ex-
tremely small elements that limits the maximum allowable
time-step to ensure the numerical stability leading to signif-
icant decreases of the computational performances (Fig. 1).
Gorla et al. [48] and Concli et al. [49–51] applied the
LRA to study the power losses of spur gears. The nu-
merical results showed good agreement with experimental
data. Following a similar approach, Liu et al. [52–56]and
Hildebrand et al. [57] focused on the oil flow prediction
and its validation with high-speed camera recordings . The
implemented CFD models could capture the experimen-
tally observed oil flow considerably well. Similar works on
spur gear pairs were done by Burberi et al. [58], Fondelli
et al. [59], Li et al. [60], Korsukova et al. [61]. More re-
cently, also bevel gears (Hu et al. [62], Lu et al. [63]),
hypoid gears (Peng et al. [64,65]), and orthogonal face
geardrive(Daietal.[66]) were simulated with CFD. It
emerges that while different types of gears could be mod-
elled with LRA, this approach requires massive paralleliza-
tion to contain the computational costs. In this regard, Con-
cli et al. [67–71] implemented efficient models in an open
source environment based on the the Global Remeshing
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Forschung im Ingenieurwesen (2023) 87:1181–1191 1185
Fig. 2 GRA: 2 subsequent meshes – the grids are comparable
Approach (GRA) (Fig. 2). The main idea is to perform
a global substitution of the entire grid rather than substitut-
ing only the low quality elements. While the domain to be
re-meshed results higher, the better control over the mesh
size and quality and the consequent allowable time-steps
lead to much better computational performances. The main
drawback of the GRA is related to the fact that it is ef-
ficient only as long as the computational domain can be
discretized with extruded grids (e.g. Type-C back-to-back
test rig, cycloidal gears, spur planetary gears). If this is not
the case, as in helical or bevel gears, the great advantages
of this mesh-handling strategy cannot be exploited. Indeed,
the complete re-meshing with tetrahedrons does not give
any benefit with respect to the LRA.
While the introduction of numerical tools allowed to
overtake the limitations of the analytical/empirical models
for predicting the load-independent power losses of gears
and the lubricant distribution, the analysis of geared systems
is particularly challenging due to the topological modifica-
tion of the computational domain caused by the gear ro-
tations. Methods as the MRF and the sliding mesh can be
used only for a single rotating gear. The over-set mesh is
not sufficiently accurate to predict the oil flows and power
losses. The re-meshing approaches (both LRA and GRA)
seems to be the only options for reliable efficiency predic-
tions. While the LRA is computationally not efficient in
any condition, the GRA ensures computational gains only
for simple extrudable geometries [71,72]. Moreover, the
more and more frequent adoption of additives and spe-
cial lubricants is promoting the occurrence of new physical
phenomena that, both with traditional approaches and with
experimental measurements, are difficult to be explained,
predicted and quantified. With these premises, the authors
have, on the one side, further developed the GRA method
to expand the efficient domain of applicability also to non-
extrudable geometries, as helical and bevel gears, and com-
plex systems as multistage and planetary gearboxes, on the
other side, have implemented new solvers capable of mod-
elling phenomena such aeration (bubble entrapment in the
lubricant), cavitation (transition from liquid to vapour of
the lubricant due to local pressure decreases), channeling
and circulation (phenomena typical of non-Newtonian flu-
ids). The next paragraphs are devoted to briefly explain the
theory and showing possible applications of such new com-
putational tools.
3 New approaches for industrial
applications
In the next sub-section, 4 practical applications are shown.
Specifically, the study of an industrial multistage helical
gearbox exploiting the new GRAMC mesh handling tech-
nique, the effect of cavitation of the lubricant on the load-
independent power losses in an FZG back-to-back test rig,
the different loss mechanisms in grease lubrication and the
impact of aeration both on losses and lubricant flows in
bearings.
3.1 Enabling complex simulation via GRAMC &AMI:
a multistage helical gearbox
In order to be able to effectively apply any FV numeri-
cal method to a real (complex) industrial gearbox, an ef-
ficient mesh-handling strategy is fundamental. In this re-
gard, the authors have developed a new recursive approach
called Global Remeshing Approach with Mesh Clustering
(GRAMC). It relies on the generation of a set of meshes that
covers one entire engagement. This set of grids is stored in
database and is recursively reused. Once the set of meshes
(typically 10 to 20 – but specific procedures based on mesh
quality and design parameters can be implemented to deter-
mine the optimum number of meshes) has been computed,
it is sufficient to coherently set the time-handling libraries
and rotational speeds to investigate the desired operating
conditions. This algorithm was introduced to overcome the
high computational effort required for the CFD simulation
of domains that are not-extrudable, as helical and bevel
gears.
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1186 Forschung im Ingenieurwesen (2023) 87:1181–1191
Fig. 3 Multistage helical gear-
box [74]: aexperimental setup;
bmeshing domains; cpressure
fields and identification of the
loss mechanisms
The analysed system is shown in Fig. 3a. It is a two
stage gearbox having a total transmission ratio of 4.76. An
Arbitrary Mesh Interface (AMI) was used to split the do-
main (Fig. 3b) and simplify the meshing procedure (use of
non conformal grids). Exploiting the GRAMC approach, 12
grids were created for the first sub-domain (input stage) and
9 for the second one (output stage). Fig. 4shows the grid-
update procedure. The 1st grid for the input domain (Mesh
1) is substituted after few time-steps and covers 1=12th of
engagement. On the contrary, for the output stage, which
rotational speed is slower, the grid (Mesh A) can is sub-
stituted every three input stage grid updates. The results in
terms of power losses differs from the experimental mea-
sures by less than 10% for each tested condition in the range
3000–9000rpm and 40–60 ıC. Fig. 3cshowsthepressure
fields: it is interesting to appreciate the capability of the
present model to highlight both the squeezing (pressure gra-
dients in the mating region) as well as the windage mech-
anisms (pressure peaks at the tooth tips) as main sources
of loss. Moreover, it is interesting to notice that with the
Fig. 4 Wor kflow o f t h e GRAMC
algorithm [74]
present approach it was possible to perform each simula-
tion, parallelized among 32 cores (8INTEL Xeon®Gold
6154 CPU, 4 Cores, 3GHz-384GFLOPs) in just about 3
hours (to reach the regime condition where the efficiency
was evaluated). Other studies from the authors have high-
lighted that the present approach based on GRAMC can lead
to a reduction of the computational effort of about −93%
between LRA and GRA [67] and a further −96% between
GRA and GRAMC [73]. As an example, the simulation of
the FZG back-to-back test rig (reference case) with dip lu-
brication, took 2466min with the LRA, 172.5 min with the
GRA and only 7.5min with the GRAMC with a net speed
up of x328 times!
3.2 Enabling the study of caviting systems
Even if cavitation is a lower-order phenomenon in gear-
boxes, for some specific operating conditions their effects
are significantly affecting the efficiency of the system. In
this regard an FZG back-to-back test rig was used as refer-
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Forschung im Ingenieurwesen (2023) 87:1181–1191 1187
ence. The testing gearbox was completely filled with lubri-
cant. Being the oil not compressible, its pressurization, if
cavitation do not occur, should not have any impact on the
load independent power losses. Experimental tests by Otto
et al. [75] shown a significant power loss difference for the
not-pressurized system vs. pressurized one. The scholars do
not explain this strange evidence. With CFD it is possible
to keep the cavitation phenomenon into account by simply
adding a source term mimic the phase change rate to the
mass conservation equation.
@˛
@t +r.˛U/+r.˛Uc.1−˛// =Pm
v
(6)
where ˛is the volume fraction, the sub-indexes land v
stand for liquid and vapour phases, and Pmis the specific
mass transfer. This could be estimated according to differ-
ent models. The most common one is the model by Kunz
et al [76]. Ucis an artificial supplementary velocity field
that is defined in the vicinity of the interface in such a way
that the local flow steepens the gradient of the volume frac-
tion function and the interface resolution is improved. Cav-
itation is the transition from liquid to vapour phase when
the vaporization pressure is achieved. This change of state
has the effect of limiting the minimum value of pressure.
By doing this, the power loss decrease could captured very
well. The comparison between the measured data in terms
of power losses and the numerical predictions shows a dis-
crepancy that results averagely below 2%.
Fig. 5 Resistant torque with and without over-pressure: experimental vs. CFD
Fig. 5clearly shows the effect of cavitation: by limiting
the minimum pressure on the rear flank to the vaporization
value, the suction contribution to the losses disappears re-
ducing the total dissipation. Also the lubricant fluxes result
completely different: instead of being sucked-in again in
the successive teeth pocket, the oil is thrown away axially.
3.3 Enabling the study of grease lubrication
mechanisms
While in case of cavitation the implementation of the new
physical behaviour was possible by simply adding a source
term in the mass conservation equation, in presence of
grease lubrication new rheological models should be con-
sidered. The Newtonian assumption of describing the fluid
behaviour with 2 parameters (density and viscosity) is not
any more sufficient. Grease, in fact, exhibits a non-New-
tonian behaviour; therefore additional quantities have to
be specified to fully describe its properties. In past stud-
ies the authors have demonstrated that the Herschel-Bulk-
ley model (a combination of both the Bingham Plastic and
Power Law models) is capable to ensure accurate result for
what concern grease lubrication of gears [77].Thegreaseis
assumed to behave as a very viscous fluid at low-strain rates
(PPc) while when the critical value Pcis overtaken,
the viscosity is described using a power law.
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1188 Forschung im Ingenieurwesen (2023) 87:1181–1191
Fig. 6 Different power loss mechanisms at different filling levels: channelling vs. circulation
For PPc
=0
P2− P
PC+K.2−n/ +.n −1/P
PC(7)
For P>Pc
=0
P+KP
Pcn−1
(8)
Kis the grease consistency factor and nis the power law
index.
As an example, let consider the FZG back-to-back test
rig. Experimental measures by Stemplinger et al [78]have
shown that after a certain threshold, a further increase of
the amount of lubricant (grease) do not produce a further
increase in the power losses (as for oil dip lubrication).
The specifically developed numerical approach was able to
accurately reproduce the power loss measurements giving
at the same time an insight on the physical behaviour. Two
mechanisms have been observed: channelling at low filling
levels, and circulation at high filling levels. The former
refers to a condition in which almost no lubricant is present
between the teeth, thus providing an insufficient amount of
grease in the mating region. A gap originates between the
gears and the sump. The grease is squeezed out axially and
does not come back in the engagement region. On the other
side, circulation occurs at higher filling levels and refers to
the condition in which the grease completely wets the teeth.
In this situation a certain amount of fresh grease moves from
the sump to the rotating wheels region and is circulated
around. The flattening of the power loss curve at high filling
levels (Fig. 6) can be related to the transition between the
two phenomena. The comparison between the measured
data in terms of power losses and the numerical predictions
shows a discrepancy that results averagely below 5%.
3.4 Enabling the study of the aeration phenomenon
In mechanical gearboxes, the motion of the gears, but also
the one of the rollers causes of the bearings, especially in
case of lubricants with additives, promotes the formation
of a mixture of the lubricant and the air present in the
system. Aeration is the physical phenomenon by which air
is entrapped with in a liquid (in this case the lubricant).
Three main aeration types can be found in the literature: (1)
entrained air, (2) foam, and (3) dissolved air [79]. Entrained
air refers to suspended bubbles. The level of aeration is
the balance between the rate of incorporation of air and its
release. The latter leads to the foaming effect; the air, which
has a lower density with respect to the lubricant, rises to the
free-surface forming thin liquid lamellae which thickness
depends on the surface tension. These phenomenon heavily
influences the lubrication properties changing the average
behaviour of the lubricant mixture and, consequently, its
effectiveness and the amount of power losses. In order to
take this effect into account, the aeration analytical model
according to [80] was implemented in OpenFOAM®.As
for cavitation, the mass conservation equation is modified
with an additional source term defined as
Sg=CairASs2Pt−Pd
(9)
where Ggis the volume of air per unit of time, Cair is a cal-
ibration parameter, ASthe free surface area at each cell,
is the lubricant density, and Ptrepresents the turbulent
forces and Pdconsiders the stabilizing forces and depends
from the turbulence characteristic length scale. Such model
was successfully applied to study the behaviour of a ref-
erence taper roller bearing at increasing rotational speed.
Experimental acquisitions via Particle Image Velocimetry
K
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Forschung im Ingenieurwesen (2023) 87:1181–1191 1189
Fig. 7 Comparison between the standard-, the aerated-solver and the
PIV measurements in terms of velocity fields [82]
(PIV) [81,82] allow to reconstruct the tangential velocity
field between the cage and the outer ring. While for low
speeds, where the effect of aeration is negligible, the stan-
dard solver can effectively predict the velocity and pressure
fields and, therefore, the power losses, with an increasing
speed more and more air bubbles are entrapped in the lubri-
cant. As shown in Fig. 7, at 2100 rpm the standard solver
is not any more capable to reproduce the real conditions
while the new solver, that includes the aeration mechanism,
is aligned with the PIV data.
4Conclusion
The increasing demand for reliable power losses models for
the manufacturing of more and more efficient gearboxes has
led many research groups to focus on the development of
predictive tools for the load-independent power losses. The
adoption of flexible numerical tools that can be customized
according to the specific needs allows on the one hand to
capture mostly every physical behaviour and, thus, properly
modelling lubrication and estimate power losses and effi-
ciency. In this regard, new solvers for cavitation, aeration
and grease-lubrication have been developed by the author
and proved to be effective in reproducing the experimen-
tal evidence. On the other hand, the recent development of
effective mesh handling strategies has paved the way for
a massive application of such techniques to industrial cases
considering that now the computational effort is compatible
with the industrial practice. With the presented approach,
the simulation effort is reduced by a factor of x328 with
respect to the standard available approaches. The simula-
tion of a reference single stage cylindrical spur gear stage
could be now solved in few minutes while about 10 years
ago, with the approaches available at that time, the solution
required some days.
Funding Open access funding provided by Libera Università di
Bolzano within the CRUI-CARE Agreement.
Open Access This article is licensed under a Creative Commons At-
tribution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long as
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vide a link to the Creative Commons licence, and indicate if changes
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permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view
a copy of this licence, visit http://creativecommons.org/licenses/by/4.
0/.
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