vehicle dynamics

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Technical Seminar Series - Vehicle Dynamics August 2006

ArvinMeritor Quarterly Technical ArvinMeritor Quarterly Technical Seminar Series – Part IISeminar Series – Part II

Vehicle Dynamics Vehicle Dynamics

Troy Tech CenterJuly 31, 2006

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ObjectivesObjectives

• To introduce the basic concepts in vehicle dynamics, focusing on vehicle handling and stability

• To demonstrate the use of TruckSim software in simulating in the dynamics of trucks and tractor-semi-trailer combinations

• To identify the vehicle parameters that are pertinent to vehicle dynamics• Identify parameters required as inputs to TruckSim

• Propose improvements to standard laboratory tests

• To identify the vehicle tests and associated vehicle performance metrics used in assessing the handling performance of vehicles• Propose improvements to skid pad tests

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Presentation OutlinePresentation Outline

• Basic Tire Behavior

• Basics of Vehicle Dynamics: Steady-State Cornering

• Vehicle Tests for Handling Performance

• Using TruckSim to Simulate Vehicle Dynamics

• Inputs to TruckSim: Vehicle Parameters

• Examples of Using Simulation Results in Generating Vehicle Handling Performance Metrics

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Basic Tire BehaviorBasic Tire Behavior

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SAE Tire Axis SystemSAE Tire Axis System

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Definition of Slip Ratio during BrakingDefinition of Slip Ratio during Braking

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Generation of Longitudinal ForceGeneration of Longitudinal Force

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Longitudinal Force vs. Slip RatioLongitudinal Force vs. Slip Ratio

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Effect of Road Surface on Longitudinal Effect of Road Surface on Longitudinal Force Adhesion/Friction CoefficientForce Adhesion/Friction Coefficient

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Generation of Lateral Force and Aligning Generation of Lateral Force and Aligning TorqueTorque

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Lateral Force and Aligning Torque vs. Lateral Force and Aligning Torque vs. Slip AngleSlip Angle

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Comparing Cornering Force and Camber Comparing Cornering Force and Camber ThrustThrust

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Combined Slip: Friction Circle DiagramCombined Slip: Friction Circle Diagram

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Combined Slip: Cornering Force and Combined Slip: Cornering Force and Aligning Torque vs. Longitudinal ForceAligning Torque vs. Longitudinal Force

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Factors Affecting Tire Forces and Factors Affecting Tire Forces and MomentsMoments

• Slip Ratio, Slip Angle, Inclination Angle

• Normal (Vertical) Force

• Road Surface

• Tire Inflation Pressure

• Speed of Travel

• Tire Wear

• Tread Pattern

• Tire Construction (Bias Ply vs. Radial Ply)

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Key Idea from Basic Tire Behavior: Key Idea from Basic Tire Behavior:

• Control of vehicle dynamics implies controlling the following 4 variables at each wheel:• Tire Slip Ratio

• Tire Slip Angle

• Tire Inclination Angle

• Tire Normal Force

• The above variables determine the friction forces between the tire and the ground

• How do we distribute the friction forces among all the wheels to get the desired vehicle behavior?

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Basics of Vehicle Dynamics: Basics of Vehicle Dynamics: Steady-State CorneringSteady-State Cornering

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SAE Vehicle Axis SystemSAE Vehicle Axis System

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Vehicle-Fixed SAE CoordinateVehicle-Fixed SAE Coordinate System: System: Symbols and DefinitionsSymbols and Definitions

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Rigid Sprung Mass Equations of MotionRigid Sprung Mass Equations of Motion

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Simplified Equations of MotionSimplified Equations of Motion of the of the Rigid Sprung Mass Rigid Sprung Mass

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Bicycle Model for Lateral DynamicsBicycle Model for Lateral Dynamics

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Bicycle Model for Lateral DynamicsBicycle Model for Lateral Dynamics

• Assumptions• Constant forward velocity, u

• No suspension, no vehicle roll or pitch

• Front wheel steer angle is the average of LH and RH steer

• Motions are small perturbations from an initial trim condition

• All angles are small

• Roadway is flat and level

• Tire lateral forces are linear functions of tire slip angles

• Neglect tire aligning moment

• Neglect lateral load transfer

• Vehicle is symmetric with respect to x-z plane

• Consider only “fixed control” response, i.e., steer angle input

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Bicycle Model: KinematicsBicycle Model: Kinematics

• Front slip angle:

• Rear slip angle:

• C.G. lateral acceleration

ff u

rav δα −+=

u

rbvr

−=α

urva y +=•

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Bicycle Model Equations of MotionBicycle Model Equations of Motion

• Tire force-vs-slip angle relation:

• Derivation of equation of motion along the lateral direction:

rryrffyf CFCF αα −=−= ;

yryfyy FFFam +==∑rrff CCurvm αα −−=+

•)(

ffrfrf Cv

u

CCr

u

bCaCmuvm δ=

++

−++

• )()(

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• Derivation of equation of motion along the yaw direction

yryfzzz FbFaMrI −==∑•

rrffzz CbCarI αα +−=•

ffrfrf

zz Cavu

CbCar

u

CbCarI δ=

−+

++

• )()( 22

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• Let the state variables be

• Coupled equations of motion are:

),( rv

ffrfrf Cv

u

CCr

u

bCaCmuvm δ=

++

−++

• )()(

ffrfrf

zz Cavu

CbCar

u

CbCarI δ=

−+

++

• )()( 22

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• Standard, first-order matrix form of the state equations:

f

zz

f

f

zz

rf

zz

fr

frrf

I

Cam

C

r

v

uI

CbCa

uI

aCbC

uum

aCbC

um

CC

r

v

dt

d δ

+

+−−

−−+−

=

)()(

)()(

22

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Steady-State ResponseSteady-State Response

• Steady-state response determined by setting the first derivatives of the state variables to zero

• Yaw rate gain

• Lateral acceleration gain

rf

rfssf

CCba

ubCaCmba

ur

)(

)()(

2

+−

−+=

δ

rf

rfssf

y

CCba

ubCaCmba

ua

)(

)()(

2

2

+−

−+=

δ

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Steady-State ResponseSteady-State Response

• If we define the understeer coefficient as

• Yaw rate gain


g

uKba

ur

usssf2

)( ++=

δ

g

uKba

ua

usssf

y

2

2

)( ++=

δ

rf

fr

r

zr

f

zfus CCba

aCbCgm

C

F

C

FK

)(

)(

+−

=−=

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Steady-State CorneringSteady-State Cornering

rff u

rba ααδ +−+= )(rff R

L ααδ +−=

ff u

rav δα −+=u

rbvr

−=α

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Equilibrium Equations during Equilibrium Equations during Steady-State CorneringSteady-State Cornering

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Steady-State Handling EquationSteady-State Handling Equation

rff R

L ααδ +−=

rryrffyf CFCF αα −=−=

−+=

r

yr

f

yff C

F

C

F

R

Lδ

rumL

aFrum

L

bF yryf ==

rumC

La

C

Lb

R

L

rff

−+= //δ

gR

u

C

F

C

F

R

L

r

zr

f

zff

2

−+=δ

gR

uK

R

Lusf

2

+=δ

g

ru

C

F

C

F

R

L

r

zr

f

zff

−+=δ

r

zr

f

zfus C

F

C

FK −=

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Characteristic Speed and Critical SpeedCharacteristic Speed and Critical Speed

• Characteristic speed (understeer vehicle) – the speed at which the steer angle required to maintain the turn radius is equal to twice the Ackermann steer angle

• Critical speed (oversteer vehicle) – the speed at which the steer angle to maintain the turn radius is equal to zero

uschar K

Lgu =

uscrit K

Lgu

−=

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Steady-State Response to Steer InputsSteady-State Response to Steer Inputs

• Steer angle versus speedgR

uK

R

Lusf

2

+=δ

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• Curvature response

guK

L

R

usssf2

1/1

+=

δ

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• Yaw rate gain

guK

L

ur

usssf2

+=

δ

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guK

L

ua

usssf

y

2

2

+=

δ

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Vehicle Tests for Handling PerformanceVehicle Tests for Handling Performance

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Constant Radius TestConstant Radius Test

)/( gaK

y

fus ∂

∂=

δ

g

aK

R

L yusf +=δ

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Constant Speed TestConstant Speed Test

g

aK

g

a

u

Lg yus

yf +

=

2δ

usy

f Ku

Lg

ga+=

∂∂

2)/(

δ

uscrit K

Lgu

−=2

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Constant Speed Test: Handling Constant Speed Test: Handling DiagramsDiagrams

)/( fy

us uLrg

aK δ−⋅−=

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Constant Steer Angle TestConstant Steer Angle Test

g

a

L

K

LRyusf −=

δ1

L

K

ga

R us

y

−=∂∂

)/(

)/1(

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Constant Steer Angle Test : Constant Steer Angle Test : Understeer Gradient PredictionUndersteer Gradient Prediction

• Understeer gradient can be predicted by using an ADAMS or TruckSim model of the vehicle

• Based on constant steer angle test

• Perform 2 simulations of vehicle response to a specified step steer input, each simulation having a different initial velocity

• No need to model driver steering control, no need to model vehicle speed (drive torque) control

( ) ( )g

aK

R

L

g

aK

R

L y

us

y

usf2

2

1

1

+=+=δ

( ) ( )

−

−=

−−=

21

1

22

2

2121

12 )/1/1(

u

a

u

a

aa

gL

aa

RRgLK yy

yyyyus

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Constant Steer Angle Test : Constant Steer Angle Test : Understeer Gradient PredictionUndersteer Gradient Prediction

Steady-State Values: 70 kph 80 kph 90 kph 100 kph Vehicle Speed (km/hr) 69.95 79.94 89.94 99.20Lateral Acceleration (g's) 0.26 0.29 0.31 0.33Yaw Rate (degrees/sec) 7.63 7.28 6.95 6.66Vehicle Roll Angle (degrees) 4.18 4.85 5.32 5.68Vehicle Slip Angle (degrees) -0.73 -1.06 -1.38 -1.66

Percent Overshoot: 70 kph 80 kph 90 kph 100 kph Lateral Acceleration (pct.) 3.86 6.29 9.61 12.81Yaw Rate (pct.) 15.26 21.01 25.82 30.08Vehicle Roll Angle (pct.) 5.45 9.11 10.72 13.70Vehicle Slip Angle (pct.) 25.31 33.34 42.99 52.48

Response Time: 70 kph 80 kph 90 kph 100 kph Lateral Acceleration (sec) 0.33 0.33 0.34 0.34Yaw Rate (sec) 0.22 0.20 0.19 0.17Vehicle Roll (sec) 1.15 1.12 1.09 1.07Vehicle Slip (sec) 0.83 0.79 0.79 0.79

Handling Performance Metrics 75 kph 85 kph 95 kph Average Understeer Gradient (deg/g) 10.80 9.18 8.18 9.39Vehicle Roll Gradient (deg/g) 27.22 21.62 20.21 23.02

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Take-Away: Steady-State CorneringTake-Away: Steady-State Cornering

• Understeer gradient:

• Steady-state handling equation:

• A critical speed exists when a vehicle is oversteer• Characteristic speed is a measure of understeer• Steady-state characteristics are important performance

metrics of vehicle handling capabilities• Understeer gradient determines steady-state response• Standard tests are available for measuring understeer

gradient

r

zr

f

zfus C

F

C

FK −=

gR

uK

R

Lusf

2

+=δ

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Other Factors Affecting UndersteerOther Factors Affecting Understeer

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• Lateral load transfer / roll moment distribution

• Tire camber (may be induced by vehicle roll)

• Roll steer

• Lateral force compliance (steer and camber)

• Aligning torque compliance (steer and camber)

• Aligning torque

• Tractive force (FWD vs. RWD)

• Steering system compliance

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50



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Understeer Budget: Example 1Understeer Budget: Example 1

• Prof. Barak’s example (SAE Seminar)

• Tire cornering stiffness and weight distribution contributes 50% to vehicle understeer

• Aligning torque compliance steer contributes 23% to vehicle understeer

Factor Front (deg/g) Rear (deg/g) Front – Rear (deg/g)

Tire Cornering Stiffness 3.24 1.86 1.38 Aligning Torque 0.065 -0.088 0.153 Aligning Torque Compliance 0.665 0.038 0.627 Lateral Force Compliance 0.20 0.11 0.09 Roll Camber (with aligning torque) 0.86 0.46 0.40 Roll Steer 0.35 0.23 0.12 Understeer Gradient 5.38 2.61 2.77

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Understeer Budget: Example 2Understeer Budget: Example 2

• J. C. Dixon, Tires, Suspension and Handling (SAE Publication, 1996):

• Tire cornering stiffness and weight distribution contributes 14% to vehicle understeer

• Aligning torque compliance steer contributes 29% to vehicle understeer

Factor Front (deg/g) Rear (deg/g) Front – Rear (deg/g)

Tire Cornering Stiffness 7.2 6.6 0.6 Aligning Torque 0.1 -0.1 0.2 Aligning Torque Compliance 1.3 0.1 1.2 Lateral Force Compliance 0.2 0.2 0.0 Roll Camber 1.2 0.0 1.2 Roll Steer 0.5 -0.5 1.0 Understeer Gradient 10.5 6.3 4.2

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Take-Away: Factors Affecting UndersteerTake-Away: Factors Affecting Understeer

• Understeer is not determined by the ratio of axle load to tire cornering stiffness alone

• Suspension design can have a big impact on vehicle understeer/oversteer characteristics

• The steady-state handling equation derived from the linear bicycle model can still be used as long as we account for other effects that contribute to the understeer gradient

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Limit Handling Performance due to Limit Handling Performance due to Nonlinear Tire CharacteristicsNonlinear Tire Characteristics

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Limit Handling due to Tire NonlinearitiesLimit Handling due to Tire Nonlinearities

• Motivation: load sensitivity of rear axle tire lateral force• Normal force increases due to lateral load transfer

• Increase in normal force results in increase in required slip angle to maintain the same level of required lateral force

• Increase in slip angle results in decrease in cornering force

• Change in cornering compliance may result in oversteer

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• Basic kinematic relation between steer angle and tire slip angles still applies

• Four possible cases, as lateral acceleration increases:• Initial oversteer, becomes more oversteer (vehicle spins)

• Initial understeer, becomes more understeer (vehicle plows)

• Initial understeer, becomes oversteer (vehicle spins)

• Initial oversteer, becomes understeer (vehicle plows)

rff R

L ααδ +−= 21 ααδ −+=R

Lf

58



Case 1: OS

Case 2: US

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Case 3: US OS Case 4: OS US

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Ideal US

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Take-Away: Limit HandlingTake-Away: Limit Handling

• Vehicle US/OS characteristics are not constant due to nonlinearities in the tire force-deflection relations

• Other factors such as body roll will change the US/OS characteristics of the vehicle during operations

• Commercial vehicles, due to their high C.G. locations, are more susceptible to changes in US/OS

• Consider understeer gradient as the slope of (α1 – α2) vs. (Ay/g) curve

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Steady-State Handling of Tractor-Semi-Trailer Steady-State Handling of Tractor-Semi-Trailer CombinationsCombinations

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Steady-State Handling Model of Tractor-Semi-Steady-State Handling Model of Tractor-Semi-Trailer CombinationTrailer Combination

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Handling Equation for the TractorHandling Equation for the Tractor

gR

uK

R

Ltus

tf

2

,+=δ

gR

u

C

W

C

W

R

L

r

r

f

ftf

2

−+=

αα

δ

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Handling Equation for the Semi-TrailerHandling Equation for the Semi-Trailer

gR

uK

R

Lsus

s2

,+=Γ

gR

u

C

W

C

W

R

L

s

s

r

rs2

−+=Γ

αα

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Trailer Articulation Angle GainTrailer Articulation Angle Gain

• 2 Modes of Instability Possible• Tractor Jackknife• Trailer Swing

• Note: stability analysis results are first-order approximations from linear model

( ) ( )( ) ( )gRuKRL

gRuKRL

tust

suss

f2

,

2,

++

=Γδ

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Tractor JackknifeTractor Jackknife

• 2 Cases Possible

• Case 1: and

• Case 2: and

and

• Critical Speedtus

tcrit K

Lgu

,−=

0, <tusK 0, >susK

0, <tusK 0, <susK

( ) ( )tstussus LLKK <,,

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Tractor SwingTractor Swing

• and

and

• Critical Speed

tus

tcrit K

Lgu

,−=

0, <tusK 0, <susK

( ) ( )tstussus LLKK >,,

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Using TruckSim to Simulate Vehicle DynamicsUsing TruckSim to Simulate Vehicle Dynamics

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Vehicle Test or Computer Simulation?Vehicle Test or Computer Simulation?

• Vehicle tests are more appropriate when:• Test vehicle for a specific product is available

• Objective is to identify potential problems in normal operation of a specific vehicle

• A subjective evaluation is required from a driver or passenger

• Problem requires high fidelity modeling such that modeling and simulation requires too much time

• Operator safety (e.g., crash or rollover events) is not an issue

• Etc.

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Vehicle Test or Computer Simulation?Vehicle Test or Computer Simulation?

• Modeling and simulation are more appropriate when:• Test vehicle for a specific product is not available

• Objective is to identify the cause of performance problems

• A sensitivity study on one or more design variables is desired

• Design synthesis: evaluate candidate designs and answer many “what if” questions

• Problem of concern requires low or moderate fidelity models such that modeling and simulation can be accomplished in a reasonable amount of time

• Repeatable tests of rollover or crash events are desired

• Accident reconstruction

• Etc.

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TruckSimTruckSim

• Simulate dynamic behavior of trucks, buses, and tractor-semi-trailer combinations

• Simulate response of vehicle to driver inputs such as steering, braking, and acceleration

• Simulate response of vehicle to environment such as rough roads, wind

• Includes provisions for interfacing with Matlab/Simulink to simulate the response of the vehicles with active controls (e.g., active suspension or steering)

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Why Use TruckSim?Why Use TruckSim?

• Pre-defined vehicle models – no need to create a model from scratch; requires user to input vehicle parameters

• Fast runtime – vehicle models are represented by ordinary differential equations (ODE’s) using a minimum number of independent variables

• Easy to use interface –interfaces are intuitive, and can be navigated like a web browser

• What If Analysis – vehicle design parameters can be changed quickly, hence, design decisions regarding vehicle dynamics can be made quicker

• Mature product – developed by UMTRI in 1980’s

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TruckSim ModelsTruckSim Models

• Single-unit truck or bus• 1 or 2 steer axles• Single or tandem drive axles• Front: solid axle or independent suspension• Rear: solid axle suspension• Frame twist feature available with custom license

• Tractor-semi-trailer combination• Tractor: 2 or 3 axles, all solid axle suspensions• Semi-trailer: 1, 2, or 3 axles, all solid axle suspensions

• A-train doubles• Double trailers with single trailer axles

• Special configurations possible with custom license

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TruckSim: Typical ScenariosTruckSim: Typical Scenarios

• Constant Radius Test

• Step Steer Test

• Double Lane Change

• Straight-line braking (constant-µ or split-µ)

• Braking while turning

• Acceleration

• Rollover: fish-hook maneuver

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TruckSim: Run Control ScreenTruckSim: Run Control Screen

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TruckSim: Vehicle Configuration ScreenTruckSim: Vehicle Configuration Screen

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TruckSim: Tractor Screen (3 axles)TruckSim: Tractor Screen (3 axles)

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TruckSim: Trailer Screen (3 axles)TruckSim: Trailer Screen (3 axles)

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TruckSim: Trailer Payload ScreenTruckSim: Trailer Payload Screen

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TruckSim: Tractor Screen (2 axles)TruckSim: Tractor Screen (2 axles)

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TruckSim: Tractor Sprung Mass ScreenTruckSim: Tractor Sprung Mass Screen

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TruckSim: Solid Axle Kinematics ScreenTruckSim: Solid Axle Kinematics Screen

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TruckSim: Axle Lateral .vs. Roll MotionTruckSim: Axle Lateral .vs. Roll Motion

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TruckSim: Suspension ScreenTruckSim: Suspension Screen

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TruckSim: Suspension ComplianceTruckSim: Suspension Compliance

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TruckSim: Suspension DampingTruckSim: Suspension Damping

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TruckSim: Auxiliary Roll StiffnessTruckSim: Auxiliary Roll Stiffness

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TruckSim: TiresTruckSim: Tires

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TruckSim: Tire Data - Longitudinal ForcesTruckSim: Tire Data - Longitudinal Forces

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TruckSim: Tire Data - Lateral ForcesTruckSim: Tire Data - Lateral Forces

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TruckSim: Tire Data - Aligning MomentTruckSim: Tire Data - Aligning Moment

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TruckSim: Tire Model – Pacejka CoefficientsTruckSim: Tire Model – Pacejka Coefficients

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TruckSim: Steering SystemTruckSim: Steering System

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TruckSim: Steered Wheel KinematicsTruckSim: Steered Wheel Kinematics

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TruckSim: Brake SystemTruckSim: Brake System

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TruckSim: Brake Torque Data from DynoTruckSim: Brake Torque Data from Dyno

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TruckSim: Truck with PowertrainTruckSim: Truck with Powertrain

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TruckSim: 4WD PowertrainTruckSim: 4WD Powertrain

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TruckSim: Engine Torque MapTruckSim: Engine Torque Map

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TruckSim: Transmission Gear RatiosTruckSim: Transmission Gear Ratios

102


TruckSim: Transmission Shift ScheduleTruckSim: Transmission Shift Schedule

103


TruckSim: Viscous DifferentialTruckSim: Viscous Differential

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TruckSim: X-Y Plots (Post-Processing) TruckSim: X-Y Plots (Post-Processing)

105


Embedding a TruckSim Model in Embedding a TruckSim Model in Matlab/SimulinkMatlab/Simulink

106


TruckSim: Animation (Post-Processing)TruckSim: Animation (Post-Processing)

107


TruckSim Results: Constant Radius TestTruckSim Results: Constant Radius Test

108


TruckSim Results: Constant Speed TestTruckSim Results: Constant Speed Test

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TruckSim Results: Constant Steer Angle TestTruckSim Results: Constant Steer Angle Test

Lateral Acceleration Yaw Rate

110


Summary of Suspension Characterization Summary of Suspension Characterization Inputs Required in TruckSimInputs Required in TruckSim

• Suspension kinematics• Axle steer vs. axle roll• Axle dive vs. wheel travel• Wheel recession vs. jounce• Lateral motion vs. jounce• Lateral motion vs. axle roll• Toe and camber settings• Axle steer vs. axle wrap• Axle steer vs. wheel travel• Left wheel vs. right wheel

steer angle (Ackerman)• Spring and shock motion

ratios

• Suspension compliance• Spring force vs. displacement• Shock force vs. velocity• Auxiliary roll stiffness• Axle lateral stiffness• Axle fore-aft stiffness• Toe angle vs. Fx• Steer angle vs. Fy• Steer angle vs. Mz• Camber angle vs. Fx• Camber angle vs. Fy• Camber angle vs. Mz

111


Questions?Questions?

vehicle dynamics

Documents

vehicle dynamics inputs

dynamics of trucks

vehicle parameters examples

iiseminar series

cornering force

slip ratiolongitudinal

tire forces

generation of lateral