The Multibody Systems Approach to Vehicle Dynamics

Transcription

1 The Multibody Systems Approach to Vehicle Dynamics A Short Course Lecture 6 Modelling and Analysis of the Full Vehicle Professor Mike Blundell Phd, MSc, BSc (Hons), FIMechE, CEng

2 Course Agenda Day 1 Lecture 1 Introduction to Vehicle Dynamics Lecture 2 Multibody Systems Simulation Software Day 2 Lecture 3 Modelling and Analysis of Suspension Systems Lecture 4 Tyre Characteristics Lecture 5 Tyre Modelling Day 3 Lecture 6 Modelling and Analysis of the Full Vehicle Discussion and Wrap Up Close

3 Contents Lecture 6 Modelling and Analysis of the Full Vehicle Underlying Theory (Bicycle Model Approach) Understeer and Oversteer Modelling Strategies (Lumped Mass, Swing Arm, Roll Stiffness, Linkages) Vehicle Body Measurements and Influences

4 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 Fluidity Precision (High Yaw Gain, High Yaw Bandwidth) (Yaw Damping Between Manoeuvres) (Disturbance Rejection) Courtesy of

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

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

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

8 Cornering at Low-Speed t L Centre of Mass d o Assuming small steer angles at the road wheels to avoid scrubbing the wheels R Centre of Turn d i δ o L (R 0.5 t) δ i L (R 0.5 t) The average of the inner and outer road wheel angles is Known as the Ackerman Angle L δ R

9 Steady State Cornering 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

10 Steady State Cornering (continued) 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 c b a r a f d R Centre of Turn

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

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

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

14 The Constant Radius Test The constant radius turn test procedure can be use to define the handling characteristic of a vehicle (Reference the British Standard) a y a y = V 2 / R V 33 m 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

15 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 δ (deg) Ackerman Angle 180 PI L R Understeer K = Understeer Gradient Oversteer Lateral Acceleration (g) At low lateral acceleration the road wheel angle d can be found using: Where: 180 δ PI L R K a y δ = road wheel angle (deg) K = understeer gradient (deg/g) A y = lateral acceleration (g) L = track (m) R = radius (m)

16 Limit Understeer and Oversteer Behaviour δ (deg) Limit Understeer δ (deg) 180 L 2 PI R Vehicle 1 Vehicle 2 Neutral Steer Lateral Acceleration (g) Limit Oversteer 180 PI L R Understeer Oversteer Characteristic Speed Vehicle Speed (kph) Critical Speed

17 Consideration of Cornering Forces using a Roll Stiffness Approach -m a y F Ry F Fy V V F Ry F Fy m a y 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

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

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

20 Forces and Moments (continued) 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

21 Forces and Moments (continued) M RRC RC rear 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 DF RzM Outer Wheels DF RzM t r M FRC Inner Wheels RC front Y X Z DF FzM DF FzM t f

22 Forces and Moments (continued) Taking moments for each of the front and rear axles gives: F FzM M t FRC f ma y.h K KTf K Tf Tr 1 t f F RzM M t RRC r ma y.h K KTr K Tf Tr 1 t r 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

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

24 Forces and Moments (continued) 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 F RRCy RC rear ΔF FzL = component of weight transfer on front tyres due to lateral force Δ F RzL = component of weight transfer on rear tyres due to lateral force h r DF RzL t r Inner Wheels DF RzL F FRCy DF FzL Outer Wheels RC front h r DF FzL t f ΔF ΔF Taking moments again for each of the front and rear axles gives: FzL RzL F F FRCy RRCy hf tf hr tr c h f ma y.h bc tf b h r ma y.h bc tr

25 Forces and Moments (continued) 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 ) RC rear Roll Axis cm F ROy m a y This gives: F ROz F RIz F RIy RC front Z F FOy 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 X F FIy F FIz Y F FIz

26 Loss of Cornering Force due to Nonlinear Tyre Behaviour 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 ΔF y Inner Tyre Load Static Tyre Load Outer Tyre Load Vertical Load F z

27 Loss of Cornering Force (continued) 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

28 The Effect of Weight Transfer on Understeer and Oversteer 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 Drift Increase front weight transfer - Understeer Drift Increase rear weight transfer - Oversteer

29 The Effect of Weight Transfer (continued) 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

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

31 Vehicle Handling Tests 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).

32 Computer Simulations 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,...

33 Double Lane Change Manoeuvre 30 m 25 m 25 m 30 m 15 m A C B A times vehicle width m B times vehicle width m C times vehicle width m

34 Lane Change Simulation

35 Determination of Roll Stiffness Rear Roll Centre SPH Applied Roll Angle Motion CYL Rear Roll Centre Front Roll Centre Front Roll Centre CYL INPLANE Applied Roll Angle Motion INPLANE SPH INPLANE INPLANE

36 Determination of Roll Stiffness Roll Moment (Nmm) FRONT SUSPENSION Roll Angle (deg)

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

38 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

39 Modelling the Steering System COUPLER COUPLER

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

41 Comparison with Track Test (Lane Change)

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

43 Calibration of Dynamic Index Vehicle Ballast Conditions: High DI = 1.02 Mid DI = 0.92 Low DI = 0.82 Front Ballast 48 kg 27 kg 5.5 kg Rear Ballast 57 kg 29 kg 0 kg Central Ballast 40.5 kg 90 kg 140 kg

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

45 Calibration of Dynamic Index ADAMS Quadrifiler Simulation:

46 Proving Ground Tests 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)

47 Proving Ground Tests Driving on Wet Basalt Expert Driver Non-Expert

48 ADAMS Simulations

49 ADAMS Simulations

50 Yaw rate (g) Lateral Acceleration (m/s2) Example Results Yaw rate 0.7G Step Steer Yaw rate vs Lateral Acceleration vs Time Time (s) Proving Ground Results ADAMS Results

51 Subjective Assessment Example Questionnaire

52 Subjective Assessment Example Questionnaire

53 Score from 10 Subjective Assessment Subjective Analysis Average Results for each DI

54 Score from 10 Subjective Assessment 10 DI 0.82 Subjective Analysis Damian Harty Driver 1 Driver 2 Driver 3 Lee Adcock David Lapworth 1 0

55 Score from 10 Subjective Assessment 10 DI 0.92 Subjective Analysis Damian Harty Driver 1 Driver 2 Driver 3 Lee Adcock David Lapworth TURN IN Confidence Accuracy Body Slip Control Rate of Change Angle Lateral Gain/Grip Feel

56 Score from 10 Subjective Assessment 10 DI 1.02 Subjective Analysis Damian Harty Driver 1 Driver 2 Driver 3 Lee Adcock David Lapworth TURN IN Confidence Accuracy Body Slip Control Rate of Change Angle Lateral Gain/Grip Feel

57 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