Automotive Differential Dynamics

Automotive Differential

Prof. M.Sc. Ing. Marco A. Z. BarretoEmail: [email protected]

Summary

• Introduction

• Components.

• Differential working.

• Basic Differential Mathematical Model.

• Differential Mathematical Model.

• Failure Mode Analyses

• Axial Forces on Gear.

• Wheel Speed and Forces

• New Differential Designs.

• Q&A.

Introduction

Internal Wheel Path

External Wheel Path

The objective conditions of vehicle travel require that its wheels move at different velocities. This iscaused by the fact that paths traveled by each of the wheels are not the same.[1] The drivingwheels accommodated the inequality in the velocities by use a differential.

The modern differential was invented in 1827 by Pecquer and was first used on steam tractorengines for the same reason that it is still used today.[5]

Definition:

The Differential is the power-dividing unit in general form it is a mechanism with a single input and two outputlink. [1]

Differential are subdivided:

• Interwheel differential, installed in driving axle and distributing the power among its right and left wheels.[1]

• Interaxle differential, installed between two axles or between two cluster of axles or between one axle anda cluster of axles.[1]

• Interside differential, installed between left and side of the vehicle.[1]

Differential types:

• Bevel-gear differential or Spur-gear.

• Symmetrical differential or Asymmetrical differential.

• Open Differential, Self-locking Differential, Limited Slip Differential, Lockable Differential, etc.

Components

Components of Differential [2]:

1- Bevel Drive Gear.

2- Differential Housing

3- Bevel Pinions

4- Side Gear/ Drive Bevel Gear

5- Differential Axles (Pin/Spider)

6- Axle Shafts

Components of Differential [3]:

Material supplied by Prof. E. Kirchner

1- Ring Gear2- Screw Fitting3- Left Housing half4- Right Housing half5- Four Differential Axles(pin/spider)6- Bevel Pinions7- Side Gear/ Drive Bevel Gear8- Axle Shafts9- Central Bodies10- Elastic Circlip11- Thrust Washer12- Taper Roller Bearing13- Screw Fitting

Differential Working

The Power Path on Differential:

Drive Gear -> Driven Gear -> Differential House -> Differential Axle Spider -> Bevel Pinions ->Side Gears

Differential Working:

Basic Differential Dynamic Model[4]

)(21

SDGRSDGLDREa

)(21

SDGRSDGLHSE

)(21 SDGRSDGLBEV a

)(22 SDGLSDGRBEV a

MDRE

MSDGRMSDGL

DRE

SDGRSDGL

HSE

BEV1

BEV2

SDGLSDGR ;

Differential has 2 degree of freedom and theremaining rotation angles can be correlated withgeneralized coordinates by:

Definition:DRE – Rotation angle of drive gear.

HSE – Rotation angle of house.

BEV1/2 – Rotation angle of gear bevel 1 and 2.

SDGR – Rotation angle of side gear right.

SDGL – Rotation angle of side gear left.MDRE – Torque on drive gear.MSDGR – Resistive torque on side gear rightMSDGL – Resistive torque on side gear left

2

3

4

1

IDRE

IHSE

ISDGL ISDGR

IBEV1

• Kinetic Energy - Lagrange’s Equations

Definition:IDRE – Inertia of drive gear.IHSE – Inertia of house + components.IBEV1/2 – Inertia of bevel gear 1 and 2.ISDGR – Inertia of side gear right.ISDGL – Inertia of side gear left.

26

121

ii

iIT

]

[21

222

211

2222

BEVBEVBEVBEV

SDGRSDGRSDGLSDGLHSEHSEDREDRE

II

IIIIT

])()(

)(41)(

4[

21

2222

2221

2

22221

SDGLSDGRBEVSDGRSDGLBEVSDGRSDGR

SDGLSDGLSDGRSDGLHSESDGRSDGLDRE

aIaII

IIaIT

5

6

• Considering ISDGL = ISDGR and IBEV1 = IBEV2 rewrite the equations:

SDGRSDGLSDGRSDGL BAT )(21 22

22

21 2

44aIIIaIA BEVSDG

HSEDRE

22

21 2

44aIIaIB BEV

HSEDRE

Where the abbreviations:

• Principles of d’Alembert and Jourdain, virtual work:

SDGRSDGRSDGLSDGLDREDREe MMMW

SDGRSDGRDRESDGLSDGLDRESDGRSDGRSDGLSDGLSDGRSDGLDREe MaMMaMMMaMW )

2()

2()(

2111

• Generalized torques:

)2

( 1SDGLDRESDGL MaMq )

2( 1

SDGRDRESDGR MaMq

• Lagrangian equation of motion:

kkk

qTTdtd

7

8

9

10

11

12 13

SDGRDRESDGRSDGL

SDGLDRESDGRSDGL

MaMAB

MaMBA

1

1

2121

• Appling Lagrangian Equation Of Motion:

• Vehicle Straight ahead with constant speed:

0SDGL 0SDGR.constSDGLSDGR SDGRSDGLDRE MMaM 121

• Vehicle Straight ahead in accelerate:

SDGRSDGLDRESDGRSDGL MMaMBABA 1)()( SDGHSEDRE IIaIBA22

)(21

SDGLSDGRSDGRSDGL MMABBA )()( 224)( aIIBA BEVSDG

224)( aIIAB BEVSDG

SDGRSDGLDRE MMaM 1 and SDGRSDGL MM

• Vehicle Start Curve with constant torque:

constaMDRE 1 SDGRSDGL MMand

14

15

16

17

18

22

21 2

44aIIIaIA BEVSDG

HSEDRE

22

21 2

44aIIaIB BEV

HSEDRE

8

9

Differential Dynamic Model

DREDREDREDRETDRE IMArFM 1

• Drive Gear:

• Driving Gear + Housing:

• Bevel Pinion Gear:

HSEHSEHSEPINTDRNT IMArFrF 21

BEVBEVPINHSEBEVTBEVT IMArFrF /43

432 TTT FFF

rDRE

rDRN

rPIN

rBEVrBEV

19

20

21

22

FT4

FT2

FT3

BEV

• Side Gear Left:

• Side Gear Right:

SDGLSDGLSDGLSDGLSDGLT IMAMrF 4

rSDGL rSDGR

SDGRSDGRSDGRSDGRSDGRT IMAMrF 3

Definition:rDRE – radius of drive gear.rDRN – radius of driven gear.rPIN – radius axle spider bevel contact area.rBEV – radius of bevel gear.rSDGR – radius of side gear right.rSDGL – radius of side gear left.

Definition:MADRE – Losses on drive gear axle.MAHSE – Losses on driven gear axle.MAHSE/PIN – Losses on bevel pinion gear axle.MASDGR – Losses on side gear right axle.MASDGL – Losses on side gear left axle.

The dynamic model include the friction loss from differential components and it can help tounderstand the effect separated.

23

24

• Considering equations 19 - 20:

HSE

DRE

HSE

DREa1HSEHSEDREHSEDREPINTDRE IaIMAaMArFaM 21121

DRE

DRN

rra1

• Merger of equations 21, 22 and 23:

• Merger of equations 21, 22 and 24:

BEV

SDGRBEVBEVSDGRSDGRSDGR

BEV

SDGRPINHSESDGR

SDGRT r

rIIMAr

rMAMrF222 /2

BEV

SDGLBEVBEVSDGLSDGLSDGL

BEV

SDGLPINHSESDGL

SDGLT r

rIIMAr

rMAMrF222 /2

• Combination of equations 22 - 23 and 22 - 24:

22

23

24

21

21

222122/

11HSEHSEDREBEVBEVSDGLSDGLSDGLPINHSEHSEDRESDGLDRE IaIaIIMAaMAMAaMAMaM

21

21

222122/

11HSEHSEDREBEVBEVSDGRSDGRSDGRPINHSEHSEDRESDGRDRE IaIaIIMAaMAMAaMAMaM

BEV

SDGR

BEV

SDGL

rr

rra

222

1PIN

SDGR

PIN

SDGL

rr

rr

25

26

• Solving the right hand side of equations 25 - 26, Kinematic Approach:

34C

BEVBEVC rvv4

BEVBEVC rvv3

• Considering the other components:

SDGLSDGLrv4

SDGRSDGRrv3

PINHSEC rv

• Replace on first equations:

BEVBEVPINHSESDGLSDGL rrr

BEVBEVPINHSESDGRSDGR rrr

• Considering both equation to achieve:

BEVBEVSDGRSDGRSDGLSDGL rrr 2BEV

SDGRSDGR

BEV

SDGLSDGLBEV r

rr

r22

PINHSESDGRSDGRSDGLSDGL rrr 2PIN

SDGRSDGR

PIN

SDGLSDGLHSE r

rr

r22

27

28

2aSDGRSDGLBEV

2SDGRSDGL

HSE

• Considering the right wheel locked and input the movement by house:

22 aHSEBEVHSESDGL 2

22 22 aa SDGRSDGLBEVSDGLSDGR

• Considering the house locked and input the movement by one wheel:

0BEVSDGRSDGLHSE

• For vehicle in straight ahead:2aSDGRSDGLBEV

2SDGRSDGL

HSE

• Replace 27 -28 on right hand side from equations 25 – 26, make ISDGR = ISDGL and include 1 BevelPinion Gear:

SDGRSDGLSDGRHSE

SDGRDRE

SDGRBEVSDGLHSE

SDGLDRE

SDGLBEVSDGLSDGL BAIaIaIIaIaIIM44

244

2212

2

212

2

SDGRSDGLSDGLHSE

SDGLDRE

SDGLBEVSDGRHSE

SDGRDRE

SDGRBEVSDGRSDGR ABIaIaIIaIaIIM44

244

2212

2

212

2

• For the Left hand side from equations 25 – 26:

SDGRSDGLSDGLPINHSEHSEDRESDGLDRE BAMAaMAMAaMAMaM 2/11 2

21

22

SDGRSDGLSDGRPINHSEHSEDRESDGRDRE ABMAaMAMAaMAMaM 2/11 2

21

22

SDGRSDGLSDGLDRE BAMaM21

SDGRSDGLSDGRDRE ABMaM21

Withouta Loss

• The torque losses can be distinct in two groups, first group:

RESHSEDREDRE MMAaMAaM21

2211 MRES – Resultant Torque on Driven Axle

Main Items:

• Taped Roller Bearing (Drive Axle and Driven Axle)• Gear Mesh Losses• Drag Fluid Resistance

Friction loss of pinion bearings accountsfor about 50% of whole power loss at thedifferential and the main factors areagitating resistance 30%, slidingresistance 5% and rolling resistance 65%.

2/ aMA PINHSE

SDGMA

• The torque losses can be distinct in two groups, second group:

Positive for - BEV or FT3 >FT4 on bevel pinion reference.

Negative for + BEV or FT3 <FT4 on bevel pinion reference.

FT4

FT2

FT3

BEV

Positive for HSE> SDG or FT3 >FT4 on bevel pinion reference.

Negative for HSE< SDG or FT3 <FT4 on bevel pinion reference.

Main Items:

• Friction between Pinion and Bevel Gear• Friction between Bevel Gear and House• Gear Mesh Losses• Liquid drag torque on lubricant

Main Items:

• Friction between Side Gear and House• Gear Mesh Losses• Liquid drag torque on lubricant

• The two main equation can be write:

SDGRSDGLSDGLPINHSESDGLRES BAMAaMAMM 2/2

SDGRSDGLSDGRPINHSESDGRRES ABMAaMAMM 2/2

29

30

• Friction Components:

Failure Mode Analyses

• Failure Mode Analyses

Axial Forces on Gears

cwpa FQ costan´'

cwcwa rgT

rgTQ costan

8costan

)2)(4(00'

cwa rgTQ costan4

0'

• Axial Force on Side Gear Mesh:

• For a Four Pinions

• For a Two Pinions

• Axial Force on Side Gear

31

32

cwpa FQ sintan´'

cwcwa rgT

rgTQ sintan

4sintan

)1)(4(00'

cwa rgTQ costan2

0'

• Axial Force on Bevel Pinion Gear Mesh:



• Axial Force on Bevel Pinion Gear

33

34


)]sintancos(sin1[cos)tan(

80'

wkww

ck

ga r

TQ 35

)]sintancos(sin1[cos)tan(

40'

wkww

ck

ga r

TQ 36



Equations 31 to 34 do not considererimportant items like gear tooth frictionand motion of contact point along thepressure line. The influences of twoitems was include on equations 35and 36 and it can be verified ongraphic above.

Wheel Speed and Forces

CoG

vx

vy

v

VX

VY

VX

VY

VX

VY

VX

VY

CoG

lF

lR

bR /2

vX

vY

v

bF /2 bF /2

bR /2

• Wheel Speed on Curve:

cosVVX sinVVY

Vehicle Speed 2F

FLXFRXbVV

2R

RLXRRXbVV

RRLYRRY lVV

FFLYFRY lVV

Rotation Speed

22 )()( YXRES VVV

Resultant Speed

VRLX

VRLY

VFLX

VFRX

VRRX

VFLY

VRRY

VFRY

“Rolling Resistance is the mechanical energy converted into heat by tire moving for unit distance on roadway”.

• Wheel Resistance Force (Rolling Resistance) [6]:

SAE J2452 – Coastdown Test (from 115kph to 15kph)

)( 2cvbvaFPF ZRoll

P – Tire Pressure.Fz – Tire Vertical Load.v – Linear Speed.

• Wheel Resistance Force (Uneven Road Force) [7]:

X

WF

X

Runeven0

• The uneven in the road are cushioned andabsorbed by tire.

• The spring work does not have a effect on thewheel resistance because the energyintroduce on compression are regained onrebounding.

• The damper work relative to the covereddistance constitutes the additional componentof wheel resistance.

X

M

Distance x

Distance x

Spr

ing

Forc

eD

ampe

rFor

ce

Whe

elFo

rce

introducedenergy

regainedenergy

damperwork

springwork

• Wheel Resistance Force (Plastic Road Force) [7]:

RplasticWZPlastic fFF ,

A – The deformation of the road surfacecaused by the wheel rolling under load onplastic road.

B – When the wheel rolls through a plasticroad surface, masses of soil are pushed infront of the tire until they are thickened ordisplaced to side.

C – A further but smaller partial resistance isproduced due to friction between the tiresidewall and road material.

Road Surface fr - value

new firm asphalt; concrete, small pavement;cobblestone pavement 0.005 – 0.015

rolled; firm gravel; wear down; washboard gravel 0.02 – 0.03

tarred, wear down, washboard gravel 0.03 – 0.04

very well dirt roads 0.04 – 0.05

dirt roads 0.05 – 0.15

sand 0.15 – 0.35

• Wheel Resistance Force (Component Resulting from Slip):

Lateral Force:

wheelplane

motiondirection

lateralforce

roll resistanceforce

centrifugalforce

LF

RollF

v

CoG

vx

vy

v

sinLF cosRollF

FL

FR

RL

RR

cossin RollLR FFF

• Wheel Resistance Force (Toe-Angle and Camber Angle):

)sin(LTOE FF

CAMBERF

toe-in toe-out

positive camber negative camber

• Drive Shafts with Difference Torsional Stiffness

)( SDGLWHLTLSDGL kMSDGRSDGLSDGLPINHSESDGLRES BAMAaMAMM 2/2

SDGRSDGLSDGRPINHSESDGRRES ABMAaMAMM 2/2 )( SDGRWHRTRSDGR kM

)2

(2mrITrFM WHLWHLBLRLSDGL

)2

(2mrITrFM WHRWHRBRRRSDGR

Considering a vehicle in straight line and vehicle resistance force divided between the tractive wheels.

Definition:

IWHL – Inertia of left wheel.IWHR – Inertia of right wheel.m – Vehicle mass.kTL – Stiffness of left drive shaft.kTR – Stiffness of right drive shaft.TBR – Drag brake torque right.TBL– Drag brake torque Left.FRL – Drag force on left wheel.FRR – Drag force on right wheel.

37

38

39

40

• Results of Simulation – Different Drive Shaft Stiffness

Increasethe friction

5% of Stiffnessdifference

SameStiffness

Reduce theinput torque Reduce the

input torque

• Results of Simulation – Different Drive Shaft Stiffness



• Results of Simulation – Drag Brake Torque

Increase the dragbrake torque

Considering the same drive shaft stiffness onleft and right sides, the drag brake torque wasintroduced on one wheel side and increasedto see the influence on bevel gear rotationangle. The simulations were performedwithout a friction of differential components.

The drag brake torque on one wheel sidemake the bevel gear rotate but the rotationdifference is smaller between the side gearsand it can be verified on right side graphic.

New Differential Design

• New Neumayer Tekfor Differential:

• New Schaeffler Differential:

• New Powertrain:

• Hybrid Powertrain Transmission

• Electric Powertrain Transmission

• Bibliography:

[1] Driveline Systems of Ground Vehicles, Theory and Design. A.F. Andreev; V.I.Kabanau; V.V.Vantsevich.

[2] Automotive Transmissions Fundamentals, Selection, Design and Application. H. Naunheimer; B. Bertsche; J.Ryborz; W. Novak.

[3] Leistungsübertragung inFahrzeuggetrieben Grundlagen der Auslegung, Entwicklung und Validierung vonFahrzeuggetrieben und deren Komponenten. E. Kirchner.

[4] Ground Vehicle Dynamics. K. Popp; W. Schiehlen

[5] Race Car Vehicle Dynamics. W.F.Milliken; D.L.Milliken

[6] The Pneumatic Tire. A.N.Gent; J.D.Walter.

[7] Lecture Longitudinal Dynamics of Vehicles (IKA – institut kraftfahrwesen Aachen). H. Wallentowitz.

[8] Lecture Notes - Vehicle Dynamics. G. Rill.

[9] Video Youtube - http://www.youtube.com/watch?v=p3rGjyvNTF0

Backup

• Results of Simulation (without a friction)

• Results of Simulation (With Friction)

• Results of Simulation (Brake Drag Torque)

• Results of Simulation (With Friction)

Failure Mode Analyses

X0

Y0

x

y

x

y

090

0180

1

2

Automotive Differential Dynamics

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