imece2008-67411

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1 Copyright © 2008 by ASME

Proceedings of IMECE2008

ASME International Mechanical Engineering Congress & ExpositionOctober 31 - November 6, 2008, Boston, MA

IMECE2008-67411

MODELING AND EXPERIMENTAL VALIDATION OF ACTIVE LIMITED SLIPDIFFERENTIAL CLUTCH DYNAMICS

Vladimir Ivanović, Joško Deur, Zvonko Herold

University of Zagreb, Faculty of Mechanical Engineering andNaval Architecture, I. Lučića 5, HR-10002 Zagreb, Croatia

Matthew Hancock, Francis Assadian

Jaguar Cars Ltd., Banbury Rd. Gaydon,Warwickshire CV35 0XJ, UK

ABSTRACTThis paper deals with modeling of an Active Limited Slip

Differential (ALSD) which comprises a wet clutch actuated by

an electromechanical mechatronic system consisting of a DC

motor and ball-ramp mechanism. The structure of the proposed

ALSD model is divided in two subsystems: (i) clutch axial

force development model and (ii) clutch torque development

model. The former includes DC motor dynamics; gear box

kinematics and backlash; ball-ramp mechanism kinematics,

friction, and compliance; fluid squeeze speed dynamics; and

clutch pack axial compliance and damping. The latter includes

structural compliance and inertias, as well as dynamic friction

effects. Each submodel has a pure physical structure. Thisfacilitates model parameterization through a series of relatively

simple experimental procedures, which are also described in

the paper. The final model has been validated under various

operating conditions by using an ALSD test rig. The validation

results point to a good modeling accuracy.

INTRODUCTIONModern vehicles are being progressively equipped with

various vehicle dynamics mechatronic systems in order to

improve the driving safety and the joy of drive. One of these

systems is Active Limited Slip Differential (ALSD), which

application is growing in all segments of modern passenger

vehicles both with and without All-Wheel-Drive (AWD)systems [1]. Compared to the traditional open differential, the

ALSD additionally comprises an actively controlled multi-plate

wet clutch that connects the differential casing and one of the

output shafts. In this way the clutch provides the ability of

active control of locking torque between the two output shafts,

which can be effectively utilized to improve the vehicle traction

and yaw stability performances [2]-[4]. As such, the ALSD

forms an integral part of the yaw stability control system, and a

good knowledge of the ALSD dynamic behavior is crucial for

the control system design and tuning. A physical structure of

the model is preferable in order to facilitate mode

parameterization and provide a good basis for model-based

analyses of the system. The latter can be an effective tool

during various optimization procedures when considering

either control or system design aspects.

This paper presents results of experimentally supported

work on modeling and validation of an electro-mechanically

actuated ALSD system. The emphasis is on modeling of the

clutch dynamics.

NOMENCLATURE

a, b = Outer and inner radii of clutch disc (m) c1 = Ball and ramp stiffness (Nm/rad)

c2 = Driveline torsional stiffness (Nm/rad)

cc2 = Clutch torsional stiffness (N/m)

ccl = Clutch axial stiffness (N/m)

crs = Clutch actuator reset spring stiffness (N/m)

F app = Clutch pack applied force (N)

F c = Clutch pack axial contact force (N)

h = Fluid film thickness (m)

ia = Clutch DC motor armature current (A)

iaR = Clutch DC motor armature current reference (A)

J 1 = Clutch DC motor inertia (kgm2)

J 2 = Driveline structural inertia (kgm2)

K br = Ball and ramp mechanism reduction ratiod cl = Clutch axial damping (Ns/m)

K g 1 = DC motor gear box gear ratio

K g 2 = Final drive gear ratio

K t = DC motor torque constant (Nm/A)

La = Clutch DC motor armature inductance (H)

N f = Number of active friction surfaces

Ra = Clutch DC motor armature resistance (Ω)r e = Equivalent radius of cutch disc (m)

t = Time (s)

Proceedings of IMECE20082008 ASME International Mechanical Engineering Congress and Exposition

October 31-November 6, 2008, Boston, Massachusetts, USA

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T c = Clutch torque (Nm)

T C = Coulomb friction torque (Nm)

T f.br = Ball and ramp friction torque (Nm)

T f 1 = Clutch DC motor friction torque (Nm)

T f 2 = Driveline friction torque (Nm)

T m1 = Clutch actuator DC motor torque (Nm)

T m2 = Driving motor torque (Nm)T S = Static friction torque (Nm)

ua = Clutch DC motor armature voltage (V)

α b1 = Clutch actuator gear box backlash (rad)

α b2 = Differential backlash (rad)

α m1 = Clutch actuator DC motor position (rad)

α m2 = Driving motor position (rad)

δ = Stribeck exponent

µ = Clutch friction coefficient

σ 0 = Bristle horizontal stiffness coefficient

σ 1 = Bristle horizontal damping coefficient

σ 2 = Viscous friction coefficient

τ delay = Clutch torque response delay (ms)

υ sp = Clutch separator plate temperature (°C)ω l = Speed of left differential output shaft (rpm)

ω m2 = Driving motor sped (rad/s)

ω r = Speed of right differential output shaft (rpm)

ω s = Clutch slip speed (rpm)

ω s = Stribeck speed

ALSD = Active Limited Slip Differential

AWD = All Wheel Drive

DC = Direct Current

SM = Servo Motor

SYSTEM DESCRIPTIONA principal scheme of the considered ALSD system is

shown in Fig. 1. The particular ALSD has a common structure,

where one of the output shafts (in this case the left shaft) is

connected to the differential case by means of a controllable

multi-plate wet clutch. The clutch operates at relatively low slip

speeds (typically up to 120 rpm), while the maximum clutch

torque is approximately 2500 Nm.

An electric DC motor is used to engage the clutch through

a gear reduction and ball-ramp mechanism that converts the

motor torque into a high clutch pack axial force. Asschematically illustrated in Fig. 1, the ball-ramp mechanism

consists of two discs (input and output disc) with oppositely

arranged grooves with defined slope (ramp) and balls placed in

the grooves. The input disc is fixed to the motor shaft, while

the output disc is rotationally fixed to the housing. Relative

rotation between the discs forces balls to drive up the ramp thus

increasing the distance between the two discs, i.e. transforming

the rotation into the axial movement. The ratio between the DC

motor torque and the axial force depends on the gear box and

ball-ramp reduction ratio, which is defined by the ramp angle

The output disc is axially connected to the press plate, which

directly compresses the clutch pack and thus locks the

differential. A reset spring is placed between the ball-rampoutput disc and the differential housing. It returns the

mechanism into its initial position in the case of power supply

SM

DC

TORQUE

SENSOR

BALL

RAMP

T m2

T c , ω

l =0

ω r

ω m2

F app

K g2

GEAR

BOX α m1

Fig. 1. Principal scheme of ALSD test rig

1 - Active Limited Slip Differential

2 - Direct-drive electric servo motor

3 - Connecting shaft

4 - Torque measuring system lever

5 - Force sensor

6 - Clutch actuator DC motor

7 - Incremental encoder

8 - Chopper box (clutch motor chopper, signal amplifiers

power supply)

9 - Industrial Pentium III PC, electric motor power supply

and control subsystem

Fig. 2. Photographs of ALSD test rig

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failure, thus opening the differential.

For the purpose of experimental

characterization, model validation, and control of

the particular ALSD a test rig has been designed

[5]. Photographs of the implemented rig are shown

in Fig. 2. In order to provide measurements of the

clutch torque, the clutch-side output shaft isgrounded by means of a force sensor. The input

shaft is driven by a torque servo motor providing

accurate control of the clutch slip speed up to 30

rpm. The clutch activation is based on feed-back

control of clutch actuator DC motor armature

current control.

The measured variables are the clutch torque

T c, the clutch driving motor speed ω m2 (i.e. the clutch slip speed

ω s=K g2-1ω m2), the clutch motor position α m1, and the clutch DC

motor armature current ia. In addition, the clutch DC motor

armature temperature υ a and the differential fluid temperature

υ fluid are also measured.

SYSTEM MODELINGFig. 3 shows a principle scheme of the ALSD model. It can

be divided into two subsystems: (i) clutch axial force

development model and (ii) clutch torque development model.

The input to the axial force model is the DC motor armature

voltage ua, while the outputs are the clutch pack axial contact

force F c and the fluid film thickness h. The outputs from the

axial force development model are fed to the clutch torque

development model together with the driving motor torque T m2.

The clutch torque development model outputs the actual value

of the clutch torque T c. Implementation of each model is

considered in detail below.

Axial force development modelFig. 4 shows a schematic and a corresponding block

diagram representation of the axial force development model.

The model can be divided into two parts. The first part relates

to the DC motor drive, which comprises a DC motor model

with included motor inertia J 1 and motor friction T f 1, gearbox

reduction ratio K g 1, gearbox backlash 2α b1, and ball-ramp

mechanism friction T f.br . The second part includes ball-ramp

reduction ratio K br , reset spring stiffness crs, the fluid squeeze

speed process )( app F h& , and axial compliance and damping of

clutch friction material ccl and d cl , respectively. Due to very

high reduction ratio between the motor shaft and the press

plate, i.e. very low press plate speeds, the effect of press plate

mass m can be neglected. The two model parts are connected

through ball-ramp mechanism compliance with the stiffness

coefficient c1.

DC motor model. The permanent-magnet DC motor can be modeled by using the common set of equations [7]. For the

known armature voltage ua, the armature current ia can be

obtained based on the following first-order differential equation

(see Nomenclature):

amva

aaaaa u K dt

di Li R =++ 1)()( ω ϑ ϑ . (1)

The motor torque T m1 is proportional to the armature current ia

aat m i K T ⋅= )(1 ϑ . (2)

The motor speed ω m1 is given by

111

11.11

11 )(

−− −−−= g g br f f mm K T f K T T T

dt

d J α

ω , (3)

where T f 1 is the motor bearing friction torque, T f.br K g1-1 is the

ball-ramp friction torque referred to the DC motor shaft, f (α ) is

the backlash function (see Fig. 4b), and T 1 K g1-1 is the ball-ramp

torque referred to the DC motor shaft. The backlash function

f (α ) provides zero reactive ball-ramp friction torque on the

motor shaft while the mechanism drives through the backlash.

The motor friction torque T f 1 can be modeled in different

ways based on static or dynamic friction models such as Dah

or LuGre models [6]. Following [8] it has been decided to

apply a dynamic friction model, which when compared to staticmodels includes compliance of asperity contacts with stiffness

coefficient σ 0 at the zero friction force. The LuGre model has

the following form:

z g dt

dz

m

mm

)( 1

101

ω

ω σ ω −= , (4)

12101 m f dt

dz z T ω σ σ σ ++= , (5)

where z is asperity deflection state variable, and g( ω m1 ) is the

sliding friction function given byδ

ω ω ω smeT T T g C S C m

/1

1)()( −

−+= , (6)

The model has a compact structure without switching logic andis able to provide an accurate friction dynamics description

The parameters can be relatively easily experimentally

identified, as it is demonstrated below.

Ball-ramp friction torque and compliance. The ball-ramp mechanism assembly consists of the ball-ramp

mechanism itself and two additional thrust needle bearings in

order to facilitate the mechanism functionality. The design of

the ball-ramp mechanism itself is similar to a ball thrus

AXIAL FORCE

DEVELOPMENT

DC motor

gear box

ball-ramp mechanism

reset spring

fluid film dynamics

clutch plate axial compliance

CLUTCH TORQUE

DEVELOPMENT

inertia

structural complianceclutch friction dynamics

ua

F c

T c

T m2

h

Fig. 3. Schematic representation of overall clutch model, (b) axial forcdevelopment subsystem and (c) clutch torque development subsystem.

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bearing. For the practical reasons, the ball-ramp mechanism has

reduced number of balls when compared to a ball thrust bearing

of similar size, which accordingly may result in higher friction

and compliance. The total friction is expected to be a sum of

the friction contributions from each bearing and ball-rampmechanism. Since the maximum ball-ramp axial force can be

very high (>30kN at maximum clutch torque), the friction

torque will be high. Thus, a precise modeling of ball-ramp

friction is important for the overall model accuracy.

According to [9] the bearing Coulomb friction torque can

be calculated as:

ab fbC F d

T ⋅⋅=2

. µ , (7)

where µ b is the coefficient of friction, d is bore diameter of

bearing, and F a is axial force. The coefficient of friction for a

typical thrust ball bearing is 0.0013 and for a typical needle

thrust bearing it equals 0.005. Due to different number of balls

between a standard bearing and the ball-ramp mechanism, the

real value of the particular ball-ramp mechanism friction

coefficient needs to be obtained by experimental identification.

The friction model can be implemented in a form of a

static or a dynamic model. The Dahl dynamic model1 has been

1 The LuGre model (4)-(5) is an extension of the Dahl model (8)-(9) with

the Stribeck friction effect (T S ≠T C ). Since the Stribeck effect is not emphasized

for ball bearings (T S ≈T C ), the Dahl model is a good choice for ball-ramp

friction.

primarily developed for the purpose of modeling the bal

bearing friction [10], and it, thus, appears as a natural candidate

for ball-ramp fiction modeling. The structure of the Dahl mode

reads [6]:i

appbr C

br br z

F T dt

dz

−=

)(.0

ω σ ω , (8)

z T br f 0. σ = , (9)

where ω br is the ball-ramp relative speed, i the is shaping factor

(usually equal to 1), and T C.br is the Coulomb friction, which is

function of the axial force F app.

The ball-ramp compliance can be modeled according to the

theory of elastic deformation of bearings [9]. This approach has

not been considered in this work due to lack of information on

the actuator components. The compliance can be, however

simply obtained by experimental identification.

Reset spring. The reset spring is well known

Belleville spring (see Fig. 5). The Belleville spring force-displacement characteristics can be calculated as:

+

−⋅

−⋅⋅

⋅⋅−

⋅⋅= 1

2)1(

4)(

22 t

s

t

h

t

s

t

h

t

s

D

st E s F

α µ , (10)

where s is spring displacement, E is Young’s modulus o

elasticity, h = H -d ,

M

rsc

1c

ppv

0≅m

2v

1f T

1J

1cc

1cd

h

1T 1ω 1α

1mT 1mω 1mα

1g K

1F

2v hv pp += &

2 x

br K

ppd

appF 12 bα

au

R L

ai

br f T .

1 x 1v

ball-ramp

appF

au1mω

s

1+

-

• appF

pv

-

+

Reset

spring

ball-

ramp

limiter

+

+gear box

backlash

ball-ramp

compliance

br K 1c

br K

11

−g K

11

−g K

DC

motor

Wet clutch

axial

dynamics

ppd

1T

c F

a

b

limiter

ball-ramp

friction

11

−g K

br ω

appF br f T .

×

α bl

α bl

f (α )

1

Fi . 4. Axial force develo ment model: a schematic and b block dia ram re resentation.

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−

−

+

−⋅=

δ δ

δ

δ

δ

π α

ln

2

1

1/

112

,

−

−⋅= 1

ln

1

ln

61

δ

δ

δ π β , δ = D/d .

For the parameters of the particular spring, used in the ball-

ramp mechanism, the nonlinear characteristic illustrated in Fig.

5 is obtained.

Ball-ramp limiter.

The ball-ramp limiter is modeled as

illustrated in Fig. 4b. It represents an elastic element with high

stiffness that generates reactive force, which opposes the reset

spring and applied force. The reactive force is obtained from

press plate position d pp.

Wet clutch axial dynamics. The wet clutch axialdynamics include a fluid film model and clutch axial

compliance. Fig. 6a shows the block diagram representation of

the axial dynamics model [11]. The fluid film model is usually

based on the Reynolds equations, extended with Partir and

Cheng flow factors related to asperity roughness, and force

balance equations where the applied force F app

is balanced by

the reaction forces of the fluid and asperities ( F c). The model

solves the fluid film squeeze speed h& of one active friction

surface, which is than multiplied by number of active friction

surfaces N f in order to obtain resultant speed of the press plate

v pp. Another output of the model is the asperity reaction force

F c which is used to calculate the contact friction torque

contribution (see next subsection). The clutch pack axial

compliance model is defined by clutch pack stiffness ccl and

damping d cl and does not include the deformation of asperity

roughness. The model is rather complex and requires large

parameterization efforts.

Therefore, in order to provide a simple, but still physical

model, which will be easy to parameterize experimentally, asimplification of the full model is proposed as shown in Fig.

5b. The fluid film model is reduced to the clutch pack free-play

block and axial damping represented by the

damping coefficient d e. The clutch pack

free-play is modeled by using standard

dead zone element. The axial damping

emulates the fluid squeeze dynamics acting

similarly as viscous friction. Note that

when the fluid is squeezed out and the

process of clutch pack deformation takes

place, the damping coefficient provides

damping to the deformation process. Since

the damping coefficient for the two zones(free-play and clutch pack deformation)

may not be equal or even similar, a

switching logic can be easily introduced if

needed. In the particular case, the

switching logic is not considered, since the

satisfactory simulation results have been

obtained with the constant coefficient. The

damping coefficient can be obtained based

on experimental results in order to provide

accurate motor speed response during driving through the free-

play zone.

The clutch axial compliance is represented by an

equivalent clutch stiffness coefficient ccl.e, which when

compared to the clutch stiffness ccl of the full model includes

asperity roughness stiffness as well. The axial clutch pack

force, which is used to calculate the contact friction torque T cis denoted by F c at the output of the block related to clutch

equivalent stiffness. Note that when the free-play is absent, the

applied force F app reaches the axial force F c.

Clutch torque development modelThe schematic representation of the clutch torque

development sub-model is shown in Fig. 7. This model is

developed for the configuration of the ALSD test rig. The

structure can, though, be readily modified for real ALSD-based

driveline configurations (cf. [14]). The model comprises

structural inertias J 2, final drive ratio K g 2, gear backlash 2α b2structural compliance c2 (e.g. half shafts), and a clutch friction

model defined by the clutch torsional stiffness cc2. The clutch

•

•appF

ppv +-

s

1

cl c

-

+

f N

h&

Fluid

film

model

•appF

ppv

s

1

ecl c .

-+

ed 1

c F

fp x

Clutch axial

compliance

Clutch

packfree play

Axial

dynamics

dampingcoefficien

a b

Clutch pack axial

compliance

ppd

c F 1/d

cl

Fig. 6. Wet clutch axial dynamics: full model (a) and simplified model (b).

d

D

H t

a

b s [mm]

F [ N ]

0 1 2 30

500

1000

1500

2000

2500

Fig. 5. Illustration of Belleville spring (a) and chara-cteristic obtained for particular spring parameters(b).

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friction model can be implemented in different ways, as

outlined below.

• Static friction model, where the clutch friction is proportional to the applied force and the clutch coefficient

of friction (c.f.[11])

• Karnopp model can be used (see e.g. [6], [12], [14]) inorder to avoid numerical inefficiency of the static model

in the zero slip speeds region.

• Dynamic friction models, such as reset-integrator friction

model ([6],[12]). The reset integrator model can beregarded as an extension of the Coulomb friction model

with a (linear) spring which accounts for compliance of

asperity contact. The existence of the linear stress-strain

curve of friction contact (instead of a nonlinear stress-

strain curve in the Dahl/LuGre model) may be utilized in

order to decrease the order of the overall model. Namely,

structural compliance is characterized by a linear stress-

strain curve and thus any structural compliance that exists

in the system and does not directly relates to the clutch

torsional compliance (e.g. structural compliance denoted

by c2 in Fig.7), can be simply incorporated into the

friction model. In this way the model can be simplified

because some small inertias such as J 2’ can be omitted.• Dynamic clutch model extended with fluid film thickness

and asperity roughness dynamics (see e.g. [11]), which is

based on the fluid film model briefly

outlined in the previous subsection.

The reset integrator model has been chosen to

model the clutch friction motivated by the above

outlined advantage related to the possibility of

reduction of the model order. The block diagramof the reset integrator friction model is given in

Fig. 8. The model is typically used in its standard

form (see dashed box in the figure) related to the

case of constant normal force, i.e. constant

maximum presliding bristle deflection z 0. The maximum

presliding deflection is calculated as z 0= T C /σ 0.c, where T C isthe friction potential and σ 0.c is the equivalent clutch torsiona

stiffness that includes clutch and structural torsional stiffness

The clutch torque T c is calculated as actual bristle deflection z

multiplied by the stiffness σ 0.c. Note that damping coefficien

σ 1.c is given for the sake of generality, but in the particular case

it equals zero. Since the normal force is varying in the

particular case, the model has been extended in order to provide full functionality. The extension includes calculation o

the variable friction potential T C based on normal force F cnumber of friction surfaces N f , equivalent clutch radius r e, and

coefficient of friction µ . The extension additionally includes

bristle deflection reset logic (denoted by BDR in Fig. 8) in

order to provide correct operation at decrease of the normal

force F c, which is not provided by the model standard form

The bristle deflection reset logic is as follows. When the

normal force F c is decreased and the new friction potential is

lower than the actual level of friction, i.e. z 0

BDR

BDR:

Fig. 8. Block diagram of reset integrator friction model

with variable friction potential.

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inductance ( La). The armature resistance Ra is determined based

on Eq. (1) as Ra=ua/ia, from steady-state experiments (dia/dt =0),

where the armature voltage is kept constant at various values

and the motor is at standstill (ω m1 = 0 which implies uemf = 0),

while the armature current is measured. The experiments were

conducted for a wide range of armature temperatures υ a.

The armature inductance La is determined from thearmature current step response, where the motor is kept at

standstill (ω m1=0). The armature current response corresponds

to the 1st order lag response with time constant T a = La/ Ra (see

Eq. (1)). The inductance can be, thus, calculated as La = T a Ra

from the identified armature time constant and the armature

resistance.

Motor voltage constant. Estimation of the motorvoltage constant K v is based on steady-state experiments where

the armature voltage is kept constant at various preset values,

while the actual steady-state values of armature current and

motor speed are measured. According to the steady-state form

of Eq. (1), the motor voltage constant can be expressed as:( ) 1/ maaav Riu K ω −= .Some alternative methods are described in [8].

Torque constant. The torque constant K t is relation between the armature current and the motor torque, and it is

temperature dependant. Therefore, the torque constant is

determined by measuring the motor torque at various armature

currents and temperatures, which requires a torque measuring

apparatus.

Motor inertia and friction torque. The motor inertiaand the friction torque have been estimated based on the motor

starting and stopping experimental procedure [13]. The motorstarting (I) and stopping (II) equations are given by

I mm f t a J T K i .1111 )( ω ω &=− , (12)

II mm f ω J ωT .1111 )( &−= . (13)

The system (11), (12) represents a homogenous linear system

of two equations and two unknowns. The solution of the

system reads:

)(

)(1

)(

1.1

1.111

m II m

m I m

t am f

K iT

ω ω

ω ω ω

&

&−

= , (14)

)(

)()(

1.1

1111

m II m

m f m

T J

ω ω

ω ω

&

−= . (15)

In order to obtain "noise-free" time derivatives of the speed

curves, )(,.1 t II I mω & , the experimentally recorded starting and

stopping motor speed traces ω m1(t ) are interpolated by fourth-

order polynomials. The polynomial coefficient are then used to

calculate the time derivatives of the speed curves )(,.1 t II I mω & .

Since the friction torque in Eqs (11) and (12) is a function of

the speed, the time-domain curves )(,.1 t II I mω & are transformed

to the speed-domain curves )( 1,.1 m II I m ω ω & , and then used to

calculate the parameters based on Eqs. (13) and (14).

Friction torque estimated based on Eq. (13) is used to

determine the Coulomb friction force F C and viscous term

coefficient σ 1 in the dynamic LuGre model given by Eqs. (4)-

(6). The remaining parameters (σ 0, T S , δ , and ω s) are

determined by non-linear least-squared optimization method inorder to fit the experimentally recorded presliding deflection

curve obtained from a breakaway experiment.

Ball-ramp friction and reset springIdentification of ball-ramp friction and reset spring

characteristics can be conducted based on hysteretic process

curve that gives relation between the motor torque T m1 and the

motor position α m1. The curve can be obtained either by

ramping the motor position or motor current upwards and than

back downwards. It is preferable to remove the clutch pack

during the experiment in order to obtain the curve for a wide

range of actuator position α m1. Fig. 9 shows the experimentally

recorded curve obtained by ramping the motor current. Thehysteretic curve is caused by contributions of the motor and

ball-ramp friction. The curve also shows that the reset spring is

pre-tensioned, which caused shift of the curve in the vertica

direction, i.e. the direction of the motor torque. The initial rese

spring force is denoted by F 0.

The ball-ramp friction T f.br is determined from the obtained

hysteretic curve by compensating the contribution of the

previously identified motor bearing friction T f 1, as illustrated in

Fig. 9. Since the experiment is conducted with removed clutch

pack, the ball-ramp axial force is equal to the reset spring force

F 0 (see Fig. 4). This fact can be used to estimate the ball-ramp

friction coefficient µ b from Eq. (7).

Ball-ramp and clutch pack compliance, backlash andfree-play

The ball-ramp stiffness c1, the clutch pack stiffness ccl , the

clutch pack free-play x fp, and the gear box backlash α bl (cf. Fig

4) have been determined experimentally by ramping the motor

current upwards and then back downwards. In order to estimate

the parameters, the recorded results are presented as motor

torque vs. motor position curve shown in Fig. 10. First part of

the curve (denoted by 1) relates to the gear box backlash and

clutch pack free-play referred to the motor shaft angle

(α bl +K g 1 K g 2 x fp). In this part the motor torque is used to drive

the mechanism through the free-play, which includes

overcoming motor friction and compression of the reset spring.The second part of the curve (denoted by 2) relates to ball-

ramp mechanism deformation and the clutch pack compression

process. This process is characterized by a hysteretic curve

resulted by the ball-ramp friction. The hysteresis becomes

wider as the motor torque increases, caused by increase of ball-

ramp friction due to increase of the clutch applied force, which

directly acts as axial force on the ball-ramp mechanism. As

illustrated in Fig. 10, the total stiffness ccl +c1, which includes

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contributions of the ball-ramp mechanism and the clutch pack,

corresponds to the gradient of the hysteretic curve. Note that

the identified stiffness as illustrated in Fig. 10 is referred to the

motor shaft side.

In order to separate the gear box backlash from the clutch

pack free-play and the ball-ramp from the clutch pack

compliance, an additional identical identification experiment

needs to be conducted, where the ball-ramp output is fixed (cf.Fig. 4a).

Structural and clutch torsional complianceStructural and clutch torsional compliance have been

estimated experimentally in a straightforward way. The driving

motor torque T m2 is sinusoidally varied while the clutch is fully

locked (cf. Fig. 7). The equivalent compliance that includes

structural and clutch torsional compliance is determined as the

gradient of the motor torque vs. motor position curve, T m2(α m2)

MODEL VALIDATIONThe overall ALSD model has been validated with respect

to experimentally recorded clutch activation and deactivation

responses. The driving motor speed was controlled at a presetvalue in the range from 0.5 to 25 rpm. The clutch activation

was simulated by applying armature current step requests. For

the purpose of validation, the ALSD model has been extended

with the actual driving motor speed controller and the clutch

motor armature current controller, as used on the test rig. The

clutch pack free-play was not compensated for in al

experiments (maximum free-play). Each experiment was

repeated at least three times. Since an excellent repeatability

was observed, only one of the experiments was taken for the

model validation. Note that the recorded experimenta

responses are given primarily to asses the model accuracy and

not the clutch control performance. The overall performance

can be significantly improved by using more refined clutchcontrol, which is beyond the scope of this paper .

Fig. 11 shows model validation results during clutch

activation and deactivation at the slip speed of 25 rpm and

three different armature current requests. At a glance, a good

qualitative and quantitative modeling accuracy can be

observed.

After stepwise change of armature current reference, there

exists a pure delay in the clutch torque signal. The pure delay is

caused by the clutch pack free-play, i.e. some time is needed to

drive the actuator mechanism (motor + ball-ramp + press plate

through the free-play. Note that the pure delay changes with the

current request. The larger the request the shorter the pure

delay. However, the pure delay is not directly proportional tothe current request due to the limited battery voltage (12V in

the particular case). Namely, the back-electromotive force

reduces the armature resistance voltage proportionally to the

motor speed, thus causing armature current and motor torque

drop. The model predicts the pure delay very well. Only at the

very low armature current request (Fig. 11c), there is some pure

delay prediction error which can be attributed to the small

actuator motor torque when compared to the magnitude of

motor/gear friction.

The pure delay phase is followed by the process of the

clutch pack compression. The clutch axial force is developed

and accordingly the clutch torque increases. Note that the

simulated clutch torque response at the beginning of the

engagement is sharper than in the experiment. This is because

the viscous friction is not included in the particular version of

the model and a constant rough value of clutch stiffness

coefficient is used.

The clutch torque transient exhibits a lag behavior, which

is predicted very well by the model. The lag behavior is

characterized by a relatively large overshoot followed by a wel

damped settling. This behavior is caused by the actuator

dynamics during clutch pack compression. The damping is

T m 1

[ N m ]

ω m 1

[ r a d / s ]

α m1

[°]

T f 1

at ω m1.1

T f 1 at ω m1.2

ω m1.2

ω m1.1

T f .br

Reset spring force F

referred to motor shaft

F 0 x itotal

0 400 800 1200 1600 20000

0.02

0.04

0.06

0.08

-30

-20

-10

0

10

20

α m1

[°]

0 400 800 1200 1600 2000

Fig. 9. Illustration of hysteretic process curve used for ball-ramp friction estimation.

0 400 800 1200 1600

0

0.2

0.4

0.6

0.8

T m 1

[ N m ]

α m1

[°]

gear backlash +

clutch pack free-playccl + c

1

DC motor +

ball-ramp friction

1 2

Fig. 10. Illustration of hysteretic process curve used for

clutch compliance and free-play estimation.

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provided by ball-ramp friction. It can be observed that the

damping is somewhat larger in the simulation response, which

might be explained by existence of vibrations that may act as

dither effect, thus reducing the friction to some extent. The

clutch torque steady-state value is predicted rather well in the

case of low and middle current request. In the case of high

request a larger error is present, which may be attributed to theoverestimated friction in the ball-ramp mechanism.

The clutch is deactivated by requesting zero armature

current, where the reset spring forces the mechanism to go into

its initial position, and thus opens the clutch. The simulated

clutch torque response is accurate. Only the motor position

transient is somewhat slower in the case of low and middle

torque requests.

Fig. 12 shows model validation results during clutch

activation and deactivation at the very low slip speed of 0.5

rpm and high armature current request. A very good correlation

between experiment and simulation can be observed. The

clutch actuator (DC motor) transient is similar as in the

previously explained case of slip speed of 25 rpm. The maindifference can be observed in the clutch torque response, which

is now significantly slower. This is result of the structural

compliance and low clutch speed (see Fig. 7). During the clutch

engagement, the clutch becomes locked (uncontrollable) as

long as the input torque is smaller than the clutch friction

potential. Similar scenario can occur under the real operating

conditions on car, where the slip speed is equal to a half of the

left/right wheel speed diference and the structural compliance

corresponds to the half shafts compliance [14].

The accuracy of the axial force development model has

been additionally validated by applying various armature

current ramps upwards and downwards. The experiment is

similar to the ones used for the identification of the axial

compliance and the free-play (Fig. 10). The validation results

are shown in Fig. 13. A very good correlation can be observed.

CONCLUSIONA control oriented model of an electromechanically

actuated Active Limited Slip Differential has been developed

and experimentally validated. The model includes two separate

subsystems: (i) axial force development model which includes

the clutch actuator and the clutch axial dynamics, and (b) clutch

torque development model. The clutch torque development

model has been developed for the specific configuration of the

ALSD test rig, but it can be easily rearranged for the

configuration as on car (cf. [14]).

The model has a physical structure. All the parameters can

be easily experimentally identified following the outlined

methods. Therefore, the model can be suitable for control

system optimization purposes, as well as for various model-

based analyses. The model transient behavior has been

experimentally validated with respect to various clutch

activation and deactivation responses. The validation results

point out that the model can accurately predict all dominant

effects of the clutch torque dynamics: (i) pure delay effect and

0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1-500

0

500

1000

1500

2000

2500

3000

t [s]

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1-500

0

500

1000

1500

2000

2500

t [s]

Experiment Simulation

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6-200

0

200

400

600

800

1000

1200

t [s]



(a) High armature current request (iaR

= 12 A)

100iaR

[A] 100ia [A] 50ua [V] 50α m1 [rad] T c [Nm]

(b) Medium armature current request (iaR = 7 A)

(c) Low armature current request (iaR

= 2.5 A)

Fig. 11. Model validation results during clutch activati

and deactivation scenario at slip speed s = 25 rpm a

different armature current requests.

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(ii) lag transient behavior with overshoot, during both clutch

activation and deactivation. The latter includes the observed

dependence of the clutch torque transient time response on the

slip speed during the clutch engagement (cf. [14]). Namely, the

transient time response becomes longer with decrease of the

slip speed as a result of the ALSD structural torsional

compliance.

In order to further enhance the overall modeling accuracy,

the model can be extended with a clutch thermal model, and

related temperature and slip speed dependence of the clutchfriction coefficient of friction. A fully physical fluid film model

may also be considered.

ACKNOWLEDGMENTS It is gratefully acknowledged that this work has been supported

by Jaguar Cars Ltd.

REFERENCES[1] Gassmann, T., Barlage, J.A.,. 2004, "Electronic Torque Manager

(ETM): An Adaptive Driveline Torque Management System",

SAE paper No. 2004-01-0866.

[2] Hancock, M., 2006. "Vehicle Handling Control Using ActiveDifferentials", Ph.D. Thesis, University of Loughborough, UK.

[3] Cheli, F., M. Giaramita, M. Pedrinelli and G. Sandoni, G. C.Travaglio, 2005, "A new control strategy for a semi-active

differential (Part II)", 16th IFAC World Congress.

[4] Piyabongkarn, D., Lew, J., Grogg, J. and Kyle, R., 2006,"Stability-Enhanced Traction and Yaw Control using Electronic

Limited Slip Differential", SAE 2006-01-1016.

[5] Ivanović, V., Herold, Z., Deur, J., Hancock , M., and Assadian, F.,2008, "Experimental Setups for Active Limited Slip Differential

Dynamics Research", SAE paper No. 2008-01-0302.

[6] Armstrong-Hélouvry, B., Dupont, P., Canudas de Wit, C., 1994"A survey of models, analysis tools and compensation methods

for the control of machines with friction", Automatica, 40, 419

425.

[7] Leonhard, W., 2001, "Control of Electrical Drives", 3rd EditionSpringer Verlag, Berlin.

[8] Pavković, D., Deur, J., Jansz, M., Perić, N., 2003, "ExperimentaIdentification of Electronic Throttle Body", Proceedings of 10th

European Conference on Power Electronics and Applications

(EPE 2003), Toulouse, France.

[9] Eschmann, P., Hasbargen, I., Weigand, K., 1985, "Ball andRoller Bearings: Theory, Design, and Application", John Wiley

and Sons Ltd.

[10] Dahl, P.R., 1977, "Measurement of Solid Friction Parameters oBall Bearings", Proceedings of Sixth Annual Symposium on

Incremental Motion, Control Systems and Devices, University

of Illinois, ILO. pp. 49-60.

[11] Deur, J., Petrić, J., Asgari, J., Hrovat, D., 2005, "Modeling oWet Clutch Engagement Including a Thorough Experimenta

Validation", SAE paper No. 2005-01-0877.

[12] Deur, J., Asgari, J., Hrovat, D, 2006, "Modeling and Analysis oAutomatic Transmission Engagement Dynamics-Nonlinear Case

Including Validation", ASME Journal of Dynamic and

Measurement, and Control , Vol. 128, pp 251-262.

[13] Deur, J., Božić, A., Perić, N., 1999, "Control of electric drivewith elastic transmission, friction and backlash – experimenta

system", Automatika, Vol. 40, Nos.3-4, pp. 129-137.

[14] Deur, J., Hancock, M., Assadian, F., 2008, "Modeling of ActiveDifferential Dynamics", Proceedings of 2008 ASME

International Mechanical Engineering Congress and Exposition

(IMECE 2008), Boston, MA, 2008.

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6-500

0

500

1000

1500

2000

2500

3000

t [s]


100iaR

[A] 100ia [A] 50u

a [V] 50α

m1 [rad] T

c [Nm]

Fig. 12. Model validation results during clutch activation and

deactivation scenario at slip speed s = 0.5 rpm and high

armature current request.

0 200 400 600 800 1000 1200 1400 16000

2

4

6

8

10

12

14

16

18

20

i a R

[ A ]

α m1

[°]

Experiment SimulationdiaR

/dt = 1A/s

5 A/s

10 A/s

15 A/s

20 A/s

Fig. 13. Model validation results of axial force developmesubsystem of clutch activation and deactivation at differe

armature current ramps.

imece2008-67411

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