Bergamo University Italy 12 th -14 th June 2012 Professor Mike Blundell Phd, MSc, BSc (Hons), FIMechE, CEng Lecture 7- Full Vehicle Modelling

Contents Underlying Theory (Bicycle Model Approach) Understeer and Oversteer Modelling Strategies (Lumped Mass, Swing Arm, Roll Stiffness, Linkages) Vehicle Body Measurements and Influences

Underlying Theory - Bicycle Model The simplest possible representation of a vehicle manoeuvering in the ground plane (bicycle model) Weight transfer Tyre lateral force characteristics as a function of tyre load GRF O1O1 X1X1 Y1Y1 O 2 G 2 Y2Y2 X2X2 V x2 FyFy FyFy V y2  z2 3

Vehicle Handling “Handling” is –different to maximum steady state lateral acceleration (“grip”) –much less amenable to a succinct definition –“a quality of a vehicle that allows or even encourages the operator to make use of all the available grip” –(Prodrive working definition) –Emotional definitions like: “Confidence” (Consistency/Linearity to Inputs) “Fun” (High Yaw Gain, High Yaw Bandwidth) “Fluidity” (Yaw Damping Between Manoeuvres) “Precision” (Disturbance Rejection) Courtesy of

 “Inertia Match” is the relationship between the CG position, wheelbase and yaw inertia. At the instant of turn in:  = C af a f a t / I zz v = C af a f t / m Combining these velocities gives an “instant centre” at a distance c behind the CG: c = I zz / ma Noting that I zz = m k 2 Thus if c is equal to b then 1 = k 2 / ab a b c Vehicle Handling u v

k 2 / ab therefore describes the distance of the centre of rotation with respect to the rear axle. a b c It is referred to as the “Dynamic Index” DI fraction is length ratio c / b DI > 1 implies c > b DI < 1 implies c < b Magic Number = 0.92 Vehicle Handling

Vehicle Handling – Understeer and Oversteer 7 Forward Speed (V x ) Lateral Acceleration (A y ) Roll Angle Lateral Acceleration (g) Yaw Rate (deg/s) Roll Rate (deg) Trajectory ( Y (mm) vs. X (mm)) For pure cornering (Lateral Response) the following outputs are typically studied: Typical lateral responses measured in vehicle coordinate frame

Cornering at Low-Speed 8  Centre of Turn Centre of Mass oo ii R t L Assuming small steer angles at the road wheels to avoid scrubbing the wheels The average of the inner and outer road wheel angles is Known as the Ackerman Angle

Steady State Cornering 9 Start with a simple ‘Bicycle’ model explanation The model can be considered to have two degrees of freedom (yaw rotation and lateral displacement) No roll! In order to progress from travelling in a straight line to travelling in a curved path, the following sequence of events is suggested: 1.The driver turns the hand wheel, applying a slip angle at the front wheels 2.After a delay associated with the front tyre relaxation lengths, side force is applied at the front of the vehicle 3.The body yaws (rotates in plan), applying a slip angle at the rear wheels

Steady State Cornering (continued) 10 rr  ff  Centre of Turn c b R After a delay associated with the rear tyre relaxation lengths, side force is applied at the rear of the vehicle Lateral acceleration is increased, yaw acceleration is reduced to zero

Bicycle Model Simplified 11 The bicycle model can be described by the following two equations of motion: c b αrαr δ αfαf X Y m a y Fr y Ff y Ff y + Fr y = m a y Ff y b - Fr y c = 0

Understeer and Oversteer 12 Neutral Steer Path Disturbing force (e.g. side gust) Acting through the centre of mass Understeer Path Oversteer Path Olley’s Definition (1945)

Understeer and Oversteer 13 Understeer promotes stability Oversteer promotes instability (spin) Neutral Steer Understeer Oversteer

The Constant Radius Test 14 The procedure may be summarised as: Start at slow speed, find Ackerman angle Increment speed in steps to produce increments in lateral acceleration of typically 0.1g Corner in steady state at each speed and measure steering inputs Establish limit cornering and vehicle Understeer / Oversteer behaviour The constant radius turn test procedure can be use to define the handling characteristic of a vehicle (Reference the British Standard) ayay V 33 m a y = V 2 / R

Understeer Gradient 15 At low lateral acceleration the road wheel angle d can be found using: Where: δ = road wheel angle (deg) K = understeer gradient (deg/g) A y = lateral acceleration (g) L = track (m) R = radius (m) δ (deg) Lateral Acceleration (g) Ackerman Angle Understeer Oversteer K = Understeer Gradient It is possible to use results from the test to determine Understeer gradient Use steering ratio to establish road wheel angle d from measured hand wheel angles

Limit Understeer and Oversteer Behaviour 16 δ (deg) Lateral Acceleration (g) Limit Understeer Limit Oversteer Vehicle 1 Vehicle 2 Neutral Steer Vehicle Speed (kph) Understeer Oversteer Critical Speed 2 Characteristic Speed δ (deg)

Consideration of Cornering Forces using a Roll Stiffness Approach 17 m a y V F Ry F Fy -m a y V F Ry F Fy  F y = m a y Where a y is the centripetal acceleration acting towards the centre of the corner  F y - m a y = 0 Where –m a y is the d’Alembert Force

Free Body Diagram Roll Stiffness Model During Cornering 18 Representing the inertial force as a d’Alembert force consider the forces acting on the roll stiffness model during cornering as shown cm m a y RC rear h Y Z X Roll Axis RC front F FIy F FIz F FOy F FIz F ROy F ROz F RIy F RIz K Tr K Tf

Forces and Moments Acting at the Roll Axis 19 F FOy RC rear Y Z X RC front F FIy F FIz F ROy F ROz F RIy F RIz K Tr K Tf F FRCy F RRCy M RRC M FRC cm m a y h Roll Axis F RRCy F FRCy M RRC M FRC

Forces and Moments (continued) 20 Consider the forces and moments acting on the vehicle body (rigid roll axis) A roll moment (m a y.h) acts about the axis and is resisted in the model by the moments M FRC and M RRC resulting from the front and rear roll stiffnesses K Tf and K Tr F FRCy + F RRCy - m a y = 0 M FRC + M RRC - m a y. h = 0 The roll moment causes weight transfer to the inner and outer wheels

Forces and Moments (continued) 21 RC rear Y Z X RC front DF FzM DF RzM M RRC M FRC Inner Wheels Outer Wheels t f t r ΔF FzM = component of weight transfer on front tyres due to roll moment ΔF RzM = component of weight transfer on rear tyres due to roll moment

Forces and Moments (continued) 22 Taking moments for each of the front and rear axles gives: It can be that if the front roll stiffness K Tf is greater than the rear roll stiffness K Tr there will be more weight transfer at the front (and vice versa) It can also be seen that an increase in track will obviously reduce weight transfer

Forces and Moments (continued) 23 Consider again a free body diagram of the body roll axis and the components of force acting at the front and rear roll centres cm m a y h Roll Axis F RRCy F FRCy b c This gives:

Forces and Moments (continued) 24 From this we can see that moving the body centre of mass forward would increase the force, and hence weight transfer, reacted through the front roll centre (and vice versa) We can now proceed to find the additional components, DF FzL and DF RzL, of weight transfer due to the lateral forces transmitted through the roll centres Taking moments again for each of the front and rear axles gives:

Forces and Moments (continued) 25 It can be that if the front roll centre height h f is increased there will be more weight transfer at the front (and vice versa) We can now find the resulting load shown acting on each tyre by adding or subtracting the components of weight transfer to the front and rear static tyre loads ( F FSz and F RSz ) F FOy RC rear Y Z X RC front F FIy F FIz F ROy F ROz F RIy F RIz cm m a y Roll Axis This gives: F FIz = F FSz - DF FzM – DF FzL F FOz = F FSz + DF FzM + DF FzL F RIz = F RSz - DF RzM - DF RzL F ROz = F RSz + DF RzM + DF RzL

Loss of Cornering Force due to Nonlinear Tyre Behaviour 26 At this stage we must consider the tyre characteristics The tyre cornering force F y varies with the tyre load F z but the relationship is not linear Lateral Force F y Vertical Load F z ΔFyΔFy     Inner Tyre Load Static Tyre Load Outer Tyre Load

Loss of Cornering Force (continued) 27 The figure above shows a typical plot of tyre lateral force with tyre load at a given slip angle The total lateral force produced at either end of the vehicle is the average of the inner and outer lateral tyre forces From the figure it can be seen that DF y represents a theoretical loss in tyre force resulting from the averaging and the nonlinearity of the tyre Tyres with a high load will not produce as much lateral force (in proportion to tyre load) compared with tyres on the vehicle

More weight transfer at either end will tend to reduce the total lateral force produced by the tyres and cause that end to drift out of the turn At the front this will produce Understeer and at the rear this will produce Oversteer The Effect of Weight Transfer on Understeer and Oversteer 28 Drift Increase front weight transfer - Understeer Drift Increase rear weight transfer - Oversteer

In summary the following changes could promote Understeer: Increase front roll stiffness relative to rear. Reduce front track relative to rear. Increase front roll centre height relative to rear. Move centre of mass forward The Effect of Weight Transfer (continued) 29

Case Study - Vehicle Modelling Study LINKAGE MODEL LUMPED MASS MODEL SWING ARM MODEL ROLL STIFFNESS MODEL

The following are typical of the tests which have been performed on the proving ground: (i) Steady State Cornering - where the vehicle was driven around a 33 metre radius circle at constant velocity. The speed was increased slowly maintaining steady state conditions until the vehicle became unstable. The test was carried out for both right and left steering lock. (ii) Steady State with Braking - as above but with the brakes applied at a specified deceleration rate ( in steps from 0.3g to 0.7g) when the vehicle has stabilised at 50 kph. (iii) Steady State with Power On/Off - as steady state but with the power on (wide open throttle) when the vehicle has stabilised at 50 kph. As steady state but with the power off when the vehicle has stabilised at 50 kph. (iv) On Centre - application of a sine wave steering wheel input (+ / - 25 deg.) during straight line running at 100 kph. (v) Control Response - with the vehicle travelling at 100 kph, a steering wheel step input was applied ( in steps from 20 to 90 deg. ) for 4.5 seconds and then returned to the straight ahead position. This test was repeated for left and right steering locks. (vi) I.S.O. Lane Change (ISO 3888) - The ISO lane change manoeuvre was carried out at a range of speeds. The test carried out at 100 kph has been used for the study described here. (vii) Straight line braking - a vehicle braking test from 100 kph using maximum pedal pressure (ABS) and moderate pressure (no ABS). Vehicle Handling Tests

Following the guidelines shown performing all the simulations with a given ADAMS vehicle model, a set of results based on recommended and optional outputs would produce 67 time history plots. Given that several of the manoeuvres such as the control response are repeated for a range of steering inputs and that the lane change manoeuvre is repeated for a range of speeds the set of output plots would escalate into the hundreds. This is an established problem in many areas of engineering analysis where the choice of a large number of tests and measured outputs combined with possible design variation studies can factor the amount of output up to unmanageable levels. MANOEUVRES - Steady State Cornering, Braking in a Turn, Lane Change, Straight Line Braking, Sinusoidal Steering Input, Step Steering Input, DESIGN VARIATIONS - Wheelbase, Track, Suspension,... ROAD SURFACE - Texture, Dry, Wet, Ice, m-Split VEHICLE PAYLOAD - Driver Only, Fully Loaded,... AERODYNAMIC EFFECTS - Side Gusts,... RANGE OF VEHICLE SPEEDS - Steady State Cornering,... TYRE FORCES - Range of Designs, New, Worn, Pressure Variations,... ADVANCED OPTIONS - Active Suspension, ABS, Traction Control, Active Roll, Four Wheel Steer,... Computer Simulations

Double Lane Change Manoeuvre 30 m25 m 30 m15 m A B C A times vehicle width m B times vehicle width m C times vehicle width m

Lane Change Simulation

Determination of Roll Stiffness

Roll Moment (Nmm) FRONT SUSPENSION Roll Angle (deg)

Modelling the Steering System Steering column part Revolute joint to vehicle body Steering motion applied at joint REV COUPLER MOTION Steering rack part TRANS Translational joint to vehicle body Front suspension

Modelling the Steering System Motion on the steering system is ‘locked’ during the initial static analysis Downward motion of vehicle body and steering rack relative to suspension during static equilibrium Connection of tie rod causes the front wheels to toe out

Modelling the Steering System COUPLER

Modelling a Speed Controller Dummy transmission part located at mass centre of the body FRONT WHEELS REV TORQUE COUPLER

Comparison with Track Test (Lane Change)

Case Study – Dynamic Index Investigation Tests Performed at the Prodrive Fen End Test Facility: Coordinated by Damian Harty Coventry University Subaru Vehicle

Calibration of Dynamic Index High DI = 1.02Mid DI = 0.92Low DI = 0.82 Front Ballast48 kg27 kg5.5 kg Rear Ballast57 kg29 kg0 kg Central Ballast40.5 kg90 kg140 kg Vehicle Ballast Conditions:

Calibration of Dynamic Index Excel Spreadsheet ADAMS Simulation Prodrive Inertia Rig (Quadrifiler)

Calibration of Dynamic Index ADAMS Quadrifiler Simulation:

Tests Performed at the Prodrive Fen End Test Facility: Basalt Strip X2 Lane change (50MPH) 0.3g and 0.8g Step steer Sine wave steering input increased frequency (50MPH) Lift off and turn in Lane change (60MPH) 3 Expert Drivers (Prodrive) 1 Experienced Automotive Engineer (Coventry University) 5 Non-Expert Student Drivers (Coventry University) 3 Settings of Dynamic Index (0.82, 0.92 and 1.02) Proving Ground Tests

Driving on Wet Basalt Non-Expert Expert Driver

ADAMS Simulations

Example Results Proving Ground Results ADAMS Results

Subjective Assessment Example Questionnaire

Subjective Assessment Example Questionnaire

Subjective Assessment

Driver 1 Driver 2 Driver 3 Subjective Assessment

Driver 1 Driver 2 Driver 3 Subjective Assessment

Conclusions Dynamic index (DI) is an important modifier of vehicle handling performance. Subjective assessment indicates a DI of 0.92 is desirable. Experienced drivers may prefer a more “agile” vehicle with a low DI. Non-expert drivers may prefer a more “forgiving” car with a high DI. A detailed validated multi-body systems model of a vehicle allows in depth analysis of responses that may be difficult to measure on the proving ground. Subjective/objective correlation remains a challenge in vehicle dynamics

Tutorial 8 – Planning Full Vehicle Models Demonstration of Roll Stiffness Model in Solver File Fiala and Road Data Files AView Demonstrations of Lane Change Parameter changes such as CM height