Content. Professur für Steuerung, Regelung und Systemdynamik. Lecture: Vehicle Dynamics Tutor: T. Wey Date: , 11:07:21

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1 1 Content Overview 0. Basics on Signal Analysis 1. System Theory 2. Vehicle Dynamics Modeling 3. Active Chassis Control Systems 4. Signals & Systems 5. Statistical System Analysis 6. Filtering 7. Modeling, Simulation with Matlab/Simulink Date: , 11:07:21

2 2 Vehicle Dynamics Modeling Longitudinal Dynamics Lateral Dynamics Vertical Dynamics 2. Chassis Components Suspension Spring and Shock Anti-roll-bar Steering Date: , 11:07:21

3 z Longitudinal Dynamics Lateral Dynamics Vertical Dynamics vertical 3 yaw pitch y lateral roll x longitudinal Date: , 11:07:21

4 4 Longitudinal Dynamics 1-dimensional (plane) model Supply side Powertrain Drive line Demand side / Driving resistances Rolling resistance Air drag Climbing resistance Vehicle inertia Braking Source: Willumeit Date: , 11:07:22

5 5 Longitudinal Dynamics Equation of motion Body Front axle Rear axle Source: Willumeit Date: , 11:07:22

6 6 Longitudinal Dynamics Driving torque Vehicle inertia Climbing resistance x V H x FG g r dyn s F G sin St Air drag F WL Rolling resistance: contribution by tire, track, skew F WR Date: , 11:07:23

7 7 Longitudinal Dynamics Rolling resistance: Tire Pressure distribution nonsymmetric due to hysteresis MP~e Rolling resistance MP F WR = F U = s Directly proportional to wheel load e F WR = f R F P = F P s Date: , 11:07:23

8 8 Longitudinal Dynamics Rolling resistance: Tire Rolling resistance coefficient fr [-] In general, fr is a function of vehicle speed Measured data Source: Willumeit Date: , 11:07:24

9 9 Longitudinal Dynamics Rolling resistance: Tire Rolling resistance coefficient and tire pressure Measured data Source: Willumeit Date: , 11:07:24

10 10 Longitudinal Dynamics Rolling resistance: Skew Wheel plane and direction of movement differ by slip angle Causation for slip angle Geometrical alignment of front wheels Lateral charge of wheel Curve driving (centripetal) Cambered road Crosswind Rolling resistance at skew F WR = F WR cos F sin Source: Willumeit Date: , 11:07:24

11 11 Longitudinal Dynamics Rolling resistance: Skew For small (<5 ) we have linear relation between and lateral force F =k This yields F WR = F WR k 2 For large slip angle non-linear relation Source: Willumeit Date: , 11:07:25

12 12 2. Chassis Components Longitudinal Dynamics Braking: pitch axis Braking induces longitudinal, vertical and pitch oscillations For identical brake torques front and rear no vertical movement of COG For COG position identical to pitch axis no pitching oscillation Date: , 11:07:25

13 13 Lateral Dynamics Vehicle model Side slip angle Yaw rate Longitudinal and lateral forces at each tire Steering wheel angle (different for left and right front wheel) Date: , 11:07:26

14 14 Lateral Dynamics Single track model Rolling degree of freedom neglected Tire non-linearities neglected Restricted to small angles and distances (for linearity) Date: , 11:07:26

15 15 Lateral Dynamics Single track model Mean wheel loads at front and rear axle Slip angles at front and rear wheels Date: , 11:07:29

16 16 Lateral Dynamics Single track model - Variables Date: , 11:07:31

17 17 Lateral Dynamics Single track model Equations of motion Date: , 11:07:35

18 18 Lateral Dynamics Single track model as AS :Substitution Steering wheel angle t Longitudinal forces F x,i Single track v=r COG x, y, Date: , 11:07:37 Center of gravity motion

19 19 Lateral Dynamics Damping Yaw eigen frequency Implicit stiffness and damping rate by tires 2nd order model Yaw and side slip have same eigen frequency Eigen freq. k: side force delta front to rear Source: Willumeit Date: , 11:07:37

20 20 Lateral Dynamics Yaw eigen frequency: example for typical parameter values 2nd order system behavior slip Source: Willumeit Date: , 11:07:38

21 21 Lateral Dynamics Steady-state Cornering Constant steering angle or constant curve radius Seldom in reality (Autobahn entry). Most of curves are build as clothoid (bending proportional arc length) Very useful as 'artificial' manoeuvre for analyzing lateral dynamics skid-pad is standard for automotive companies Important data are steering angle and the resulting side slip angle clothoid Understeer, Oversteer, Neutral steer Steerable rear axle would allow to compensate side slip angle completely Date: , 11:07:38

22 22 Lateral Dynamics Understeer In technical terms understeer is the condition when the slip angles of the front tires are greater than the slip angles of the rear tires. In other words the front wheel angle is greater than the angle normally required for the turn. The slip angle is the angle of the tire in relation to it's direction of travel. Oversteer Oversteer is the reverse of understeer, that is where the slip angles of the rear tires are greater than the front tires. In cases of oversteer the rear tires tend to describe a larger cornering radius than the front tires. Date: , 11:07:39

23 23 Lateral Dynamics Steady-state cornering =0 ; =0 ; =0 ; F x, F=0 This results in a direct relation between side slip angle and steering angle (cf. slide 17, assume sin =, cos =1-, sin =, cos =1- ) Fa = F a F b F a =F x, R F x, F 1 F y, F F b= F y, R F x, F F y, F 1 At lateral acceleration equal to zero yields a negative side slip angle (assumption: front wheel steering only) Date: , 11:07:39

24 24 Lateral Dynamics Typical behavior of side side slip angle over lateral acceleration for steady-state cornering Source: Willumeit Date: , 11:07:39

25 25 Lateral Dynamics vf lf A Ackermann vehicle Assumption: no slip angles v A lr COG vr r A Instantaneous center of rotation ICR A Date: , 11:07:39

26 26 Lateral Dynamics vf General case with slip angles F lf r v ICR vr R vr R COG lr ki: Side force coefficent F y, F=k F F F y, R =k R R = F R A = R A Date: , 11:07:40

27 27 Lateral Dynamics «1 for rear wheel drive»1 for front wheel drive Steering wheel angle vs. lateral acceleration F x, F F m v2 lr F lf R 1 = A F y,f r l F l R F y, F l F l R F y, R Calculation Measurement: Ford Mondeo δ/ Single-track vehicle Ackermann vehicle Source: ATZ aq / g Date: , 11:07:40

28 28 Lateral Dynamics Understeer, oversteer and neutral in steady-state cornering In real driving conditions non-linear behavior of tire side forces Radius const. Under-/Oversteer definition results from gradient d /daq SingleTrack-SWA Ackermann-SWA Source: Willumeit Date: , 11:07:40

29 29 Lateral Dynamics Test maneuver: vehicle behavior during Power-off in a turn Load change at rear axle Increase in yaw speed Source: ATZ Date: , 11:07:40

30 30 Lateral Dynamics Fy Side force coefficient F y, F=k F F F y, R =k R R k F/ R ~F z, F/ R Wheel load difference Fz Source: Willumeit Date: , 11:07:41

31 31 Lateral Dynamics Kamm'scher Circle Sum of longitudinal forces and lateral forces cannot exceed a certain value max Fp FP: wheel load max: maximum friction coefficient If longitudinal forces are fully utilized, no side forces can be generated and vice versa Fy Fx max F p Date: , 11:07:41

32 32 Lateral Dynamics Steering Wheel Torque: beside steering wheel angle (SWA) the torque is of interest as a design parameter Side force at front wheels Power assist Hydraulic (cf. Chapter 4) Electrical (cf. Chapter 4) Measurement: Ford Mondeo Source: ATZ Date: , 11:07:41

33 33 Vertical Dynamics Quarter Vehicle Model (QVM) No pitch, no roll modeling Mass m1 equal unsprung masses (wheel, tire, arms, brake,...) Mass m2 equal ¼ body mass Spring (linear) Damper (linear) No tire damping z2 m2 c2 d z1 m1 c1 Date: , 11:07:42 z0

34 34 Vertical Dynamics Quarter Vehicle Model (QVM) Equations of motion Mass matrix [ Damping matrix ][ ] [ m1 0 z 1 d d d d 0 m2 z 2 with P 1=c 1 z0 Stiffness matrix ][ ] [ z 1 c c c z 2 c 2 c2 Possible output variables z1(wheel movement) z2 (body movement) Date: , 11:07:42 ][ ] [ ] z1 P = 1 z2 0

35 35 Vertical Dynamics Quarter Vehicle Model (QVM) State space description [ d m1 x = d m2 1 0 d m1 d m2 0 1 c 1 c 2 m1 c2 m2 0 0 ][] c2 m1 c2 x m2 0 0 c1 m1 0 u 0 0 y=[ ] x Eigen values Body eigen frequency (¼ car) Wheel eigen frequency x 4= z 2 ; x 3=z 1 ; x 2= z 2 x1= z 1 Date: , 11:07:43

36 36 Vertical Dynamics Simple 3-dimensional vertical dynamics model Height, pitch, roll modeling Source: Willumeit Date: , 11:07:43

37 37 Vertical Dynamics Simple 3-dimensional vertical dynamics model Body mass distribution No body elasticity Valid only below bending eigen frequency of body Point P: part of body Accelerations at P P' P Source: Willumeit Date: , 11:07:44

38 38 Vertical Dynamics 3D vertical dynamics model: block diagram z 0l t z0r t H1(s) H2z(s) H3(s) H1(s) H2z(s) H4(s) u t z 2P t v t Source: Willumeit z0l,r(t): Road input (partially correlated) H1(s): Quarter vehicle model H2z(s): Pitch model between upper point QVM and P' H3(s): Roll model between left side body and P H4(s): Roll model between right side body and P Date: , 11:07:44

39 39 Vertical Dynamics 3D vertical dynamics model: linear transfer functions H1(s): Quarter vehicle model T 1 H 1 j =c [ I j A ] b H2z(s): QVM from state space description (slide 34) Pitch model between upper point QVM and P' j l ' H 2z j =1 1 e l P: position in x-direction Wheel base 1 Time-delay from front to rear axle = v Date: , 11:07:45

40 40 Vertical Dynamics 3D vertical dynamics model: linear transfer functions H3(s): Roll model between left side body and P (static) s' H 3 j =1 s H4(s): P: position in y-direction Track width Roll model between right side body and P (static) s' H 4 j = s Date: , 11:07:45

41 41 Vertical Dynamics Quarter Vehicle Model PSD (chapter 1, slide 35f) z 0 t z t H(s) Road surface characteristics z0(t) are stationary with normal distribution Vehicle speed is const. Vehicle model behaves linear P zz = H j 2 P z0z0 Date: , 11:07:45

42 42 Vertical Dynamics Road surface characteristics z0(t) Typical PSDs for road surfaces Tarmac Cement Macadam medium Cobblestone medium Dirt road Date: , 11:07:45

43 43 Vertical Dynamics Quarter Vehicle Model PSD H(jω) PSD acceleration road PSD acceleration body Position & acceleration PSD P z0 z0 = 4 P z0z0 H(jω) 2 Date: , 11:07:46

44 44 Vertical Dynamics Seat rail vertical acceleration Source: ATZ Date: , 11:07:46

45 45 Vertical Dynamics Seat rail vertical acceleration: PSD (Power Spectrum Density) Source: ATZ Date: , 11:07:47

46 46 2. Chassis Components Suspension Front axle: McPherson Transverse Link Long. forces: rotation around front bush Lat. forces: radially stiff front lower control arm bush Date: , 11:07:47

47 47 2. Chassis Components Suspension Rear axle: Multilink suspension Rear axle: SLA suspension Maximum load compartment: SLA selected for wagon Lat. forces: radially stiff front lower control arm bush Date: , 11:07:48

48 48 2. Chassis Components Suspension McPherson front axle Source: Ford Date: , 11:07:48

49 49 2. Chassis Components Suspension McPherson front axle: Roll center height Source: Ford Date: , 11:07:49

50 50 2. Chassis Components Suspension Trailing arm SLA rear axle Source: Ford Date: , 11:07:49

51 51 2. Chassis Components Suspension Trailing arm SLA rear axle: Bump steer Source: Ford Date: , 11:07:50

52 52 2. Chassis Components Suspension Multilink rear axle: Roll center height Source: Ford Date: , 11:07:51

53 53 2. Chassis Components m2 Spring and shock m1 c F1(t) Perfect coil spring modeling No spring mass Constant stiffness c 2nd order model s 2 2 Y 1 s s G 1 s = = F 1 s 2 m1 s y1 c m2 c c m1 m 2 c m2 2 Y 2 s s G 2 s = = F 1 s y2 2 m1 s c c m1 m 2 Date: , 11:07:51

54 54 2. Chassis Components Spring and shock Bode Diagram Bode diagram of perfect coil spring Magnitude (db) Above 40Hz is the real spring behavior not covered by perfect model: resonances Possible solution: FEM modeling Phase (deg) Frequency (rad/sec) Date: , 11:07:

55 55 2. Chassis Components Spring and Shock McPherson strut Coil spring Damper inside spring Distributed mass Stiffness c=f( y) Date: , 11:07:51

56 56 2. Chassis Components Spring and shock Nitrogen Hydropneumatic spring Citroën Air spring Force transfer by oil Height control by oil Damper volume change Damping with nitrogen reservoir Spring elements connected to arms Cylinder Date: , 11:07:52

57 57 2. Chassis Components Spring and shock Hydro pneumatic spring Function Date: , 11:07:52

58 58 2. Chassis Components Spring and Shock Damper Hydraulic principle Ideal damper modeling F D= y F1(t) F1(t) m1 m1 y1 Optimized modeling (Maxwell model) d Additional spring d m2 m2 y2 c Date: , 11:07:53 y1 y2

59 59 2. Chassis Components Spring and Shock Damper dynamics: transfer functions Ideal model Y 1 s s2 m2 s d G1 s = = F 1 s m1 m2 s d m1 m2 Y 2 s s 2 d G 2 s = = F 1 s m1 m2 s d m1 m2 Maxwell model 2 m2 2 d s m2 s d c Y 1 s s G1 s = = F 1 s m! m2 2 d s m1 m2 s d m1 m2 c 2 Y 1 s s d G1 s = = F 1 s m! m2 2 d s m1 m2 s d m1 m2 c Date: , 11:07:54

60 60 2. Chassis Components Spring and Shock Measured Damper dynamics Measurements Bode diagram ideal vs. Maxwell model behavior Maxwell Ideal Date: , 11:07:54

61 61 2. Chassis Components Spring and Shock Damper realization Twintube Monotube and twintube Dynamic Damper Properties Hysteresis Different forces in jounce and rebound Friction forces Cavitation Reduced damping at higher frequencies Date: , 11:07:55 Monotube

62 62 2. Chassis Components Spring and Shock Twintube Monotube properties Compression: displaced oil compensated by gas reservoir Damping work by piston valve Prevent cavitation: pre-charge pressure 30 ~ 40 bars in passive state Twintube properties Compression: displaced oil pressed in outer tube through base valve Damping work by base valve Package: diameter Monotube Source: ZF Sachs Date: , 11:07:55

63 63 2. Chassis Components Anti-roll-bar (ARB) Spring connection between wheels of one axle Effect only during inverse z-movement of wheels Increased roll stiffness of vehicle Reduced roll angle and roll speed Date: , 11:07:56

64 64 2. Chassis Components Anti-roll-bar (ARB) Effect on side forces Increasing wheel load difference due to ARB Side force depends non-linear on wheel load ARB as tuning parameter for understeer resp. oversteer Higher ARB stiffness at front axle: understeer tendency Higher ARB stiffness at rear axle: oversteer tendency Date: , 11:07:56

65 65 2. Chassis Components Steering Driver demands Reasonable Steering Wheel Torque (SWT) Sensitivity Auto Return Passive Safety Vehicle demands Steering kinematics Package Cost Source: RWTH Aachen, Kraftfahrzeugtechnik Date: , 11:07:57

66 66 2. Chassis Components Steering Steering rack realization Different gear mechanisms Gear rack shows advantages Source: RWTH Aachen, Kraftfahrzeugtechnik Date: , 11:07:57

67 67 2. Chassis Components Steering Steering rack realization Tie-rods connected via ball joints Attachment points on the outside (a) or in the middle (b) part Source: RWTH Aachen, Kraftfahrzeugtechnik Tie-rods Date: , 11:07:58

68 68 2. Chassis Components Steering Power Assist Hydraulic assist Open-center valve with torsion bar Constant flow pump Source: RWTH Aachen, Kraftfahrzeugtechnik Date: , 11:07:59

69 69 Matlab/Simulink Exercise Single Track Model Simple Simulink model Equation of Motion - assume steering wheel angle is equal to zero Tire forces function of wheel slip and driving situation Date: , 11:07:59

70 70 Matlab/Simulink Exercise Single Track Model Simple Simulink model Simulink modeling of 2nd order differential equation Implement Simulink representation of single track model based on the given equations Date: , 11:08:00

71 71 Matlab/Simulink Exercise Vertical Dynamics A QVM as described in fig. 1 should be modelled in Simulink. Parameters are c1 = 150kN/m c2 = 15000N/m m1 = 30kg m2 = 300kg d = 0kg/s Show for z0(t)=1(t) and z0(t)=sin(t) the body motion In general, the eigen frequency of body and wheel are easily calculated by c2 c 1 c 2 Body = ; Wheel = m2 m1 Mathematically this is not correct. Evaluate graphically the difference between the solution above and the approximation. What happens to the eigen frequency for d = 1400kg/s? Date: , 11:08:00 fig. 1

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