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Modeling of Torque Vectoring Drives for Electric
Vehicles: a Case Study
Franciscus L.J. van der Linden Jakub Tobolář
German Aerospace Center (DLR), Institute of System Dynamics and
Control, 82234 Wessling,
Germany{Franciscus.vanderLinden,Jakub.Tobolar}@dlr.de
Abstract
This paper shows some aspects of the implementa-tion of a gear
model with losses, nonlinear elasticityand forcing errors in the
Modelica language utilizingconcept of replaceable functions. Using
such gearmodel for a torque vectoring drive modeling, a casestudy
about a powertrain dynamic behavior in a sim-plified vehicle model
is carried out. The total vehiclemodel is analyzed in several
detail stages of the pow-ertrain reaching from a fixed efficiency
with constantspring stiffness to a model using nonlinear losses
andnonlinear tooth stiffness. Subsequently, the simula-tion results
of such levels of modeling detail provingtendency to drive line
oscillation are presented anddiscussed.
Torque Vectoring Drive, Gearing, Vehicle Dynam-ics
1 Introduction
To design the gearing solution for an electrical ve-hicle,
different gear topographies are typically ana-lyzed utilizing
computer simulations in an early de-sign stage. To perform such
studies efficiently, it isimportant that the gearing topographies
can easilybe designed and integrated into the vehicle modelswhich
are used for maneuverability tests assessingthe driving
quality.
A solution enabling such gear topography designand vehicle
integration was introduced recently by(van der Linden, 2015). To
prove the usability ofthat concept also for more complex gearing
topogra-phies, an electric vehicle powertrain configurationwith
controlled torque vectoring device was chosenin this paper – a
future-oriented solution particu-larly suitable to actively
influence the dynamic be-havior of the vehicle, such as using
active yaw ratecontrol. Such a torque vectoring drive (TVD)
con-figuration is used e. g. in experimental cars like theVISIO.M
(Gwinner et al., 2014) and allows for veryhigh vectoring torques
with a small electric motor.
Figure 1. Torque vectoring drive consisting of a differ-ential,
superimposing unit and spur gear train. Note thatonly single
planets are shown for simplification of the cal-culations.
A graphical overview of the gearing solution is shownin Figure
1.
After giving an overview of some implementationaspects of the
method in Section 2, the present studywill continue with a TVD
model decription in Sec-tion 3. Here, a gear with constant
elasticity andconstant efficiency will be first introduced as a
ref-erence model. For further investigations, the modelcomplexity
will be successively increased with dif-ferent loss models as well
as nonlinear elasticity andbacklash in the gearing. Utilizing a
simple vehiclemodel, briefly referred in Section 4, the
simulationresults will be discussed in detail in Section 5.
2 Gear model description
The gear models used in this analysis base on previ-ous work
which consisted of the simulation of elas-tic ideal gears (van der
Linden, 2012). These mod-els have been extended with various loss
modelsand elasticity models according to (van der Linden,
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2015). An overview of the forces and torques calcu-lated in this
publication are shown in Figure 2. Theforces and torques (FxA, FyA,
τA, FxB , FyB and τB)in this Figure are calculated using the
integral ofthe forces of a complete cycle of the meshing tooth.By
supplying also an internal gear force element,epicyclic gears can
be modeled as well. Since thederivation of all the theory goes
beyond the scopeof this paper, only brief outline is given in the
fol-lowing sections. For a detailed description, pleaserefer to the
abovementioned paper. Since the de-rived model of gear teeth
contact is purely planar, itis implemented using the
PlanarMechanics toolbox(Zimmer, 2012). This allows the use of
standard pla-nar parts and enables a good transferability of
forcesand torques into the 3D world
2.1 Friction force implementation
Since in gear dynamics different friction models areoften used,
a decision was made to implement thefriction using a replaceable
function structure.
The friction is implemented using a state machineto be able to
handle friction during the Forward,Backward and Stuck mode. To
switch between thesemodes, two transition modes are used:
StartForwand StartBackw. A tearing variable sa is used inthe Stuck
mode to calculate the forces to keep themodel stuck. This approach
is similar to the frictionimplementation of (Otter et al., 1999).
The gearfriction force Ft is calculated using specialized
func-tions based on e. g. gear meshing speed, operationalmode of
the gear contact, contact angle and the radiiof the gear
wheels.
The concept of replaceable functions allows for aquick selection
between the different friction mod-els: no friction, viscous
friction, specified efficiency,Coulomb friction, friction according
to the DIN 3990specifications (DIN 3990 Teil 4, 1987) and a
frictionimplementation from Niemann and Winter (1989).The DIN 3990
friction and the friction to Niemannand Winter both define the
friction as a function ofthe speed and loading of the gear.
Furthermore, a continuous friction model which isnot based on a
state machine is implemented. Thisimplementation uses a regularized
friction model tosmooth the discontinuity of the gear friction. It
isimplemented using
Ft = µ|Fn|tanh|vmesh|
vmesh,0. (1)
In this equation, vmesh is the relative speed of thegears at the
meshing point and vmesh,0 the character-istic meshing speed which
is chosen small comparedto the nominal meshing speed. Using this
regular-ization, event chattering of gear systems with manygear
contacts can be avoided. However, it must be
noted that in this case no stiction can take place, asthe
friction is zero at zero speed.
In this paper, a fixed friction coefficient will beused as this
method is heavily used in the design ofgear transmissions for
powertrain analysis, togetherwith the friction implementation to
the DIN 3990norm due to a good match with measured frictionresults
in a previous publication (van der Linden,2015).
2.2 Elasticity implementation
Similar to the variability of friction methods usedin the
modeling of gears, also the gear elasticity isdescribed in many
ways. In most cases, a nonlinearrelation between normal forces Fn
and deformationof the gear at contact is present. To incorporatethe
different stiffness models known from literature,also the
elasticity is implemented using another setof replaceable
functions. These functions calculatethe normal contact force Fn
from multiple modelinputs like the mesh deformation and speed,
gearradii, thickness of the wheels and wheel positions.The position
of the gear wheels makes it possibleto include a position dependent
gear stiffness whichcan be used to simulate the effect of meshing
teethor the effect of a damaged tooth.
2.3 Forcing error implementation
To simulate forcing errors like misalignment ofthe gear wheels,
manufacturing errors or damagedtooth, a position dependent forcing
error is addedto the overall gear deformation. In Figure 3,
thedeformation between the gears is given by ∆AB =∆AB,0 + ∆AB,e. In
this equation, ∆AB,e is the elas-tic deformation of the gear
contact as discussed inSection 2.2. Adding the forcing distance
∆AB,0 givesthe total gear deformation.
Also in this case, replaceable functions are usedto implement
several cases: a forcing error definedby the misalignments of the
gears, a forcing errordefined by a Fourier-series and a table-based
inter-polation. All these methods define the forcing erroras a
function of the position of each gear wheel.
2.4 Graphical representation of gears
The graphical representation of the gears isan important way to
check proper geome-try of the gear. Therefore, visualizers
fromModelica.Mechanics.MultiBody.Visualizers areused to visualize
the gear wheels. The results ofsuch exemplary 3D representations
can be seen inFigure 1 and Figure 5. The parameters needed forthe
visualization, such as gear radius or thickness,are directly taken
from the gear model parameters.
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dAlA
dB lBFn
Ft
F
φgear
φcontact
φB
αA
x
y
αB
φA
Gear BGear A
τA
τB
FyA
FxA
FyB
FxB
Figure 2. Free body diagram of two involute gears. In the figure
ωA < 0 and Gear A drives Gear B.
∆AB,e
∆AB,0
Figure 3. Forcing error excitation using a position de-pendent
forcing distance ∆AB,0
2.5 Implementation of constraint equa-
tions
The gear model as depicted in Figure 2 needs a de-fined distance
between point A and point B for acorrect calculation of the forces
and torques. How-ever, when a constraint equation is defined in
eachgear contact model, epicyclic gear sets result in mod-eling
problems since the planet–sun distances andsun–ring distance is
defined double, thus leading toan overconstrained system.
Therefore, these con-straints cannot be defined in the model
itself, butmust be defined using the PlanarMechanics library.This
mimics the behavior of real gears: a gear con-nection itself has no
constraining equations. Thegear wheel positions are defined by the
bearing ar-rangement of the transmission.
An example of a simple epicyclic gear set is shownin Figure 4
and Figure 5. The structure of the modelis similar to the gear
construction. Hence, each bear-ing, mass and carrier is modeled in
Modelica just likein a real system.
Figure 4. Modelica model of simple epicyclic gear
con-figuration
Figure 5. Graphical representation of the epicyclic
gearconfiguration shown in Figure 4.
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3 Construction of a torque vec-
toring drive in Modelica
The models as described in Section 2 are used tobuild a complete
torque vectoring drive consistingof a Ravigneaux differential,
superimposing gear andspur gear train.
3.1 Ravigneaux differential
The Ravigneaux differential is used to allow for dif-ferent
speeds of the car tires. Compared to commonopen differentials with
bevel gears, using a Ravi-gneaux differential allows for a smaller
constructionenvelope combined with lower losses. The Model-ica
representation of the differential is shown in Fig-ure 6. The
Ravigneaux differential uses four gearinstances, together with a
carrier which houses twoconnecting planets.
For simplicity reasons, only one set of planets isused in this
analysis. When using all planet setsof a full Ravigneaux gear, the
system would – inthe case of rigid gear connections – lead to an
over-determined system. In the case of elastic gears, thedifferent
gear stages which can switch between fric-tion modes can lead to
heavy event chattering of theplanets.
To compensate the stiffness reduction caused bythe lower number
of modeled planets, thicker gearswith higher masses are
incorporated.
The reduction of number of planets would leadto an unbalanced
gear model as long as the massesof the planets are considered as
well. But since themodel in Figure 6 is purely rotationally coupled
(seerotational flanges on both left and right side), thiseffect
does not apply. On the contrary, if the bear-ing forces are
studied, this planet number reductionwill lead to wrong results. In
this case, all planetmasses should be added to balance the system
andthey must be rotationally coupled with the planetwhich is driven
by the gear connections.
3.2 Superimposing gear
The superimposing gear uses the input torque to cre-ate a torque
difference between the output flanges.Also in this case, only a
single planet is modeledinstead of all planets. The stiffness and
mass arecompensated to mimic all planets as depicted in Sec-tion
3.1. In Figure 7, the setup of the gear is shown.For the
superimposing gear, four gear instances areneeded.
3.3 Overall TVD model
Connecting the Ravigneaux differential, superim-posing gear and
spur gear train together, a complete
Figure 6. Ravigneaux differential
Figure 7. Superimposing gear
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ConferenceSeptember 21-23, 2015, Versailles, France
DOI10.3384/ecp15118151
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Table 1. TVD model configurations under investigation
Configuration DescriptionFixed eta Constant spring constant,
con-
stant gear efficiencyFixed eta withBacklash
Nonlinear spring constant andbacklash, constant gear
effi-ciency
DIN 3990 etawith Backlash
Nonlinear spring constant andbacklash, efficiency using toDIN
3990
TVD is generated as shown in Figure 8. To repre-sent the
connection elasticity between the drives, ro-tational
stiffness-damping elements are added fromthe Modelica standard
library.
In the overall TVD model, ten gear connectionsare included.
In the simulations which are presented in Sec-tion 5, three
different configurations are analyzedaccording to Table 1.
The spring constant of the gear instances isset to 20×106 N/m/mm
(Newton per meter permillimeter gear width), and the gear damping
to20×103 Ns/m/mm for all gear connections. Thecoupling between the
superimposing gear, Ravi-gneaux gear and spur gear train have a
stiffness of107 Nm/rad and a damping of 105 Nms/rad,
respec-tively.
4 Vehicle model
To analyze the TVD model in typical vehicle driv-ing maneuvers,
a vehicle model has to be utilized.For the sake of simplicity, a
planar vehicle modelwas introduced which moves in the horizontal
plane,thus enabling longitudinal, lateral and yaw motiononly.
Additionally, a six degrees of freedom mass(i. e. three positions
and three rotations) was joinedto the planar vehicle body. By
taking into account
Figure 8. Complete TVD consisting of Ravigneaux dif-ferential,
superimposing gear and spur gear train
the forces on this mass, wheel load variation dueto vehicle mass
transformation between wheels dur-ing braking, accelerating and
cornering are enabled.The tire models allow slip and are based on
the workof Zimmer and Otter (2010).
A small size electric vehicle with rear-wheel driveis considered
for simulation. Its mass is about1000 kg with wheelbase of 2.6 m
and track of 1.45 m.
The powertrain of the vehicle consists of TVD asdescribed above,
the main motor which applies themain driving torque, and a
differential motor whichdivides torques to the wheels of one axle.
Utilizingthe differential motor control, the torque
vectoringfunctionality can be realized. Finally, driveshaft
el-ements are considered as well to additionally incor-porate their
elasticity. An overview of the model isshown in Figure 9. To mimic
the electrical time con-stant of the motors, a first order system
with a timeconstant of 10−3 s for both motors is used.
5 Simulation results
Using the vehicle model with a free steering setup(free steering
wheel), an acceleration maneuver issimulated. The main motor torque
is given as aramped signal, and the differential motor torque asa
changing signal as shown in Figure 10.
5.1 Elastic drive shafts
The wheel torque of the right driveshaft during themaneuver is
shown in Figure 11. It can be observedthat the results of the
constant efficiency and theDIN 3990 efficiency are differs
significantly. Thefixed efficiency cases (97% per gear stage) show
abehavior which is intuitively expected of the TVD:the differential
torque of the differential motor is am-plified and split between
the two axles.
Introducing the DIN 3990 friction model, the re-sults yield – in
contrast to the constant efficiency– an oscillating output torque.
This is caused bythe fact that due to the pre-load of the
differential,
Figure 9. Vehicle model with motor configuration anddriveshaft
elasticity. The right bottom section of the di-agram (with the
planarToMultibody element) enables ananimation where the drive is
fixed to the car.
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0 2 4 6 8 10
−50
0
50
Time/ s
Torq
ue/
Nm
Figure 10. Motor torques during maneuver. The mainmotor toque (
) has a ramp of 1 s to 15 Nm , the differ-ential motor ( ) is
controlled with a changing referencetorque.
0 2 4 6 8 10−400
−200
0
200
Time / s
Torq
ue
/N
m
Figure 11. Wheel torques of the right driveshaft (refer-ence
stiffness) with different friction and elasticity models:A fixed
efficiency without backlash ( ), fixed efficiencywith backlash ( )
and a friction law to the DIN 3990standard ( ).
combined with low rotational velocities, lead to highfriction.
Due to this high friction combined with thepre-load, the gear can
get in the stuck mode (thisphenomenon is also described and
measured for airpath actuators by Ahmed et al. (2012)).
The combination of the drivetrain elasticity withhigh friction
leads to a stick-slip problem resultingin highly varying torques.
Note that without fur-ther measurements and / or experience on TVD,
itcannot be concluded which of the friction modelscorrectly
represents the real system.
Analyzing the speed of the differential motor de-picted in
Figure 12, the stick-slip problem is alsoevident. High rotational
accelerations are caused bythe fast variation of the motor speed,
which can leadto high loads on the rotor of the motor. This canlead
to fatigue damage of the motor.
0 2 4 6 8 10−20
0
20
Time / s
Spee
d/
rad
s−1
Figure 12. Speed of the differential motor using drive-shafts
with the reference stiffness. Different friction andelasticity
models are shown: A fixed efficiency withoutbacklash ( ), fixed
efficiency with backlash ( ) anda friction law to the DIN 3990
standard ( ).
0 2 4 6 8 10−400
−200
0
200
Time / s
Torq
ue
/N
m
Figure 13. Wheel torques of the right driveshaft withdifferent
friction and elasticity models. A ten times in-creased stiffness of
the driveshafts is used. A fixed effi-ciency without backlash ( ),
fixed efficiency with back-lash ( ) and a friction law to the DIN
3990 standard( ).
5.2 Stiff driveshafts
Using driveshafts with a significant higher stiffnessand
damping, the stick-slip problems described inSection 5.1 can be
avoided. In presented examplewith increased stiffness, a ten times
higher frictionand damping has been used w.r.t. the nominal
sit-uation. The wheel torques of the right rear wheel(see Figure
13) behave as expected, also for TVDusing the DIN 3990 friction
model. Moreover, thehigh peaks in motor velocity of the
differential gearare eliminated, cf. Figure 12 and Figure 14.
5.3 Simulation of eccentricities
Eccentricities are common in most gear wheels andare often
caused by manufacturing tolerances. Tosimulate a non-perfect drive,
an eccentricity of 10 µmis added to both gear wheels of the first
stage ofthe spur gear train. This eccentricity excites the
Modelling of Torque-Vectoring Drives for Electric Vehicles: a
Case Study
156 Proceedings of the 11th International Modelica
ConferenceSeptember 21-23, 2015, Versailles, France
DOI10.3384/ecp15118151
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0 2 4 6 8 10
−5
0
5
Time / s
Spee
d/
rad
s−1
Figure 14. Speed of the differential motor using drive-shafts
with ten times increased stiffness. Different frictionand
elasticity models are shown: a fixed efficiency with-out backlash (
), fixed efficiency with backlash ( )and a friction law to the DIN
3990 standard ( ).
0 2 4 6 8 10−130
−128
−126
−124
Time / s
Torq
ue
/N
m
Figure 15. Wheel torques with and without gear eccen-tricities:
All simulation results have a fixed efficiency anda nonlinear
stiffness. The different lines show a simula-tion without
eccentricity( ), simulation with the nom-inal stiffness ( ) and a
simulation with an increasedstiffness( ).
gear train leading to vibrations. In this simulation,a constant
torque of 5 Nm is applied to the differ-ential motor to keep the
Ravigneaux differential outof the stuck mode. The main drive motor
is drivenwith a constant load for a constant acceleration.
In Figure 15, the wheel torques of a simulationwith a stiff
driveshaft with eccentricity, an elasticdriveshaft with
eccentricity and an elastic driveshaftwithout eccentricity are
given. Analyzing the sim-ulation results, it is clear that this
eccentricity hasa high-frequent impact on the wheel torques. Witha
nominal driveline, the torque variations are lowerat high
velocities as for a stiff driveline. A detailedview of this
vibration is shown in Figure 16. Thehigh frequent vibrations of the
gear seem to excitethe drivetrain for the nominal stiffness
drivetrain.
3 3.05 3.1 3.15 3.2 3.25
−128
−126
−124
Time / s
Spee
d/
rad
s−1
Figure 16. Speed of the differential motor with andwithout gear
eccentricities: All simulation results have afixed efficiency and a
nonlinear stiffness. The differentlines show a simulation without
eccentricity( ), simu-lation with the nominal stiffness ( ) and a
simulationwith an increased stiffness( ).
6 Discussion
During performed simulations, we realized that –due to the large
number of switching componentsand high gear stiffness – the
proposed model chal-lenges common numerical solvers like DASSL
orRadau IIA. Finding consistent restart conditions af-ter an event
can be hard, since this often directlytriggers a next event in an
adjacent gear connection.
In some cases, it is therefore advisable to use aregularized
friction model as presented in Section2.1. Such friction models can
help to avoid eventsand make a simulation progress even if very
com-plex gearing configurations are analyzed. However,most of these
problems can be avoided by modelingthe real-life world more
accurately. As an exam-ple, in this paper, spring-damper models
betweenthe differential, superimposing gear and spur geartrain have
been added to mimic the stiffness of theconnections. This avoided
many problems with thesimulation results.
7 Conclusion
In this paper, different techniques for gear modelingwere
presented and adopted for a torque vectoringdrive which was
analyzed in complete car model sim-ulations. Applying such gear
modeling techniques,which include losses, nonlinear elasticity and
forcingerrors, a various level of gear detail can be selectedwhich
proved to significant influence the simulationresults.
The higher level of modeling detail is particularlyimportant
when investigating torque and speed os-cillation issues which can
be useful for e. g. drivelinedesign. Then, simple fixed efficiency
based frictionmodels are insufficient. In contrast, DIN 3990 or
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similar friction models are required to capture sucheffects.
Furthermore, it was shown that both insuf-ficient torsional
stiffness of the drive shafts and stickslip in the gear lead to
such large torque oscillationswithin a complete driveline.
The influence of gear eccentricities on the drive-line was shown
for driveshafts of different elasticity.It is shown that the a
higher stiffness of this shaftincreases the load on this axle. This
shows that atrade-off must be made between the load caused
byeccentricities and the load caused by the stick-slipeffect, as a
high driveshaft elasticity lowers the loadcaused by stick-slip
effects, but increases the loadcaused by a stiffer shaft.
The drawback of some presented models can bean increased
simulation effort due to large numberof events. For such cases, the
regularized frictionmodel proved to be possible alternative.
It is worth mentioned that the model and the sim-ulation results
were not validated so far. Especially,the damping coefficient is
largely unknown and notwell researched at the moment. It is
advisable topush the experimental research to get a usable
esti-mate of the gear damping properties in the future.
References
F. S. Ahmed, S. Laghrouche, and M. El Bagdouri.Overview of the
modelling techniques of actuator non-linearities in the engine air
path. Proceedings of theInstitution of Mechanical Engineers, Part
D: Journalof Automobile Engineering, 227(3):443–454, September2012.
ISSN 0954-4070. doi:10.1177/0954407012453905.
DIN 3990 Teil 4. Tragfähigkeitsberechnung von Stirn-rädern;
Berechnung der Freßtragfähigkeit, 1987.
Philipp Gwinner, Michael Otto, and Karsten Stahl.Lightweight
Torque-Vectoring Transmission for theElectric Vehicle VISIO.M. In
COFAT 2014,March 2014. URL
http://mediatum.ub.tum.de/doc/1226683/1226683.pdf.
Gustav Niemann and Hans Winter. Maschinenelemente:Band 2:
Getriebe allgemein, Zahnradgetriebe - Grund-lagen, Stirnradgetriebe
(German Edition). Springer,1989. ISBN 3-540-11149-2.
M Otter, H Elmqvist, and S E Mattsson. Hybridmodeling in
Modelica based on the synchronous dataflow principle. In Computer
Aided Control Sys-tem Design, 1999. Proceedings of the 1999
IEEEInternational Symposium on, pages 151–157,
1999.doi:10.1109/CACSD.1999.808640.
Franciscus L. J. van der Linden. Modelling of Elas-tic Gearboxes
Using a Generalized Gear ContactModel. In Proceedings of the 9th
InternationalMODELICA Conference, pages 303–310, Munich,November
2012. Linkoping University Electronic
Press.doi:10.3384/ecp12076303.
Franciscus L. J. van der Linden. Modeling of gearedpositioning
systems: An object-oriented gear contactmodel with validation.
Proceedings of the Institutionof Mechanical Engineers, Part C:
Journal of Mechan-ical Engineering Science, June 2015. ISSN
0954-4062.doi:10.1177/0954406215592056.
Dirk Zimmer. A Planar Mechanical Library for Teach-ing Modelica.
In Proceedings of the 9th Interna-tional Modelica Conference, pages
681–690, Munich,November 2012. Linköping University Electronic
Press.doi:10.3384/ecp12076681.
Dirk Zimmer and Martin Otter. Real-time mod-els for wheels and
tyres in an object-orientedmodelling framework. Vehicle System
Dynam-ics, 48(2):189–216, February 2010. ISSN 0042-3114.
doi:10.1080/00423110802687596.
URLhttp://www.tandfonline.com/doi/abs/10.1080/
00423110802687596.
Modelling of Torque-Vectoring Drives for Electric Vehicles: a
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158 Proceedings of the 11th International Modelica
ConferenceSeptember 21-23, 2015, Versailles, France
DOI10.3384/ecp15118151