modeling of torque vectoring drives for electric vehicles: a case … of torque-vectoring... ·...

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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151

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: StartForw

and 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.

Modelling of Torque-Vectoring Drives for Electric Vehicles: a Case Study

152 Proceedings of the 11th International Modelica ConferenceSeptember 21-23, 2015, Versailles, France

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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.

Session 2D: Automotive Applications 1

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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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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



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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.

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modeling of torque vectoring drives for electric vehicles: a case … of torque-vectoring... ·...

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