1. National Technical University of Athens Diploma Thesis «Computational Simulation of the road behaviour of a vehicle by use of a non- linear six-degree of freedom model» George Mavros Supervisor: Assoc. Prof. Dr.-Ing K.N.Spentzas Athens, September 2000

2. Purpose of the report The aim is to create a versatile computational tool in order to predict, analyse and understand all aspects of the road behaviour of a vehicle. The whole analysis is based on an advanced non-linear six-degree of freedom model

3. Procedure Selection of vehicle model - consideration of all important phenomena Equations Programming Simulations - Testing

4. Chapter 1: Definition of Model Six degree of freedom, non-linear model with four wheels and provision for 4- wheel steering q y V U x p W r z

5. Aerial View x -c c a y O llift b Clift -d d

6. Side View lair x hairside h z

7. Frontal View y hairfr z

8. Forces and torques Gravitational forces - provision for inclined road Lateral tire forces due to slip angle Longitudinal forces due to acceleration or braking - rolling resistance Aerodynamic forces (drag, lateral and lift) Spring and damping forces from the suspension All consequent torques plus torque from anti- roll bars

9. Simplifications (1) Vehicle is symmetrical towards XZ plane Total mass is constant No Camber angles or other kinds of inclination are introduced Tire contact patch does not change Unsprung mass is added to sprung mass

10. Simplifications (2) Lateral and longitudinal tire adhesion coefficients are equal to eachother Steering angle on the right wheel(s) is equal to the steering angle on the left wheel(s) The function of Lateral tire force with respect to slip angle is linear

11. Chapter 2: Equations

12. Forces and Torques

13. 6X6 System of differential Equations

14. Chapter 3: Algorithms & Routines Main objectives: Solving the system of equations arithmetically Definition of the function of front wheel steering angle with respect to time Introduction of 4-wheel steering Introduction of criteria for the loss of tire adhesion

15. Solving the system of equations All routines are created in the MATLAB environment The system of differential equations is solved by the means of a 4th order - single step RUNGE KUTTA method The results obtained are extremely close to the ones obtained when using the built-in MATLAB function: «ode45»

16. Front wheel steering Function of front wheel steering angle with respect to time INPUT: Duration of steering: T; Final angle: δ ffinal ; Degree of the polynomial: r OUTPUT: Front wheel steering angle function with respect to time: δ f (t) IF t<T THEN k=(δ ffinal /T r )*t r ELSE k=δ ffinal END δ f (t)=k

17. Introduction of 4-wheel steering δ r =g(δ f (t)) δ r =λ*δ f (t)

18. Criterion for the loss of tire adhesion Maximum alowable lateral force for each tire, according to the friction circle consept:

19. Criteria for the total loss of roadholding at each end of the vehicle Criterion for the total loss of roadholding at the front of the vehicle: Criterion for the total loss of roadholding at the rear of the vehicle:

20. User interface

21. User Interface

22. User interface

25. Chapter 4: Simulations Test 1 : General case Weight distribution 60% front - 40% rear

26. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

27. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

28. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

29. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

30. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

31. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

32. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

33. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

34. Simulations Test 1 : General case Weight distribution 60% front - 40% rear 57 59 61

35. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

36. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

37. Simulations Test 1 : General case Weight distribution 60% front - 40% rear

38. Simulations Test 1.1 : General case Weight distribution 60% front - 40% rear Longer period of running, which shows a bigger part of the orbit 39

39. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 38

40. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 46

41. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 47 52 54

42. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 48

43. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 49

44. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 50

45. Simulations Test 2.1: like Test 1.1 BUT: Weight distribution 40% front - 60% rear 51

46. Simulations Test 4: like Test 2.1 BUT: Increase of damping coefficient per 1000 N*sec/m (2500 to 3500) 40

47. Simulations Test 4: like Test 2.1 BUT: Increase of damping coefficient per 1000 N*sec/m (2500 to 3500) 41 54

48. Simulations Test 4: like Test 2.1 BUT: Increase of damping coefficient per 1000 N*sec/m (2500 to 3500) 42

49. Simulations Test 4: like Test 2.1 BUT: Increase of damping coefficient per 1000 N*sec/m (2500 to 3500) 43

50. Simulations Test 4: like Test 2.1 BUT: Increase of damping coefficient per 1000 N*sec/m (2500 to 3500) 44 55

51. Simulations Test 4: like Test 2.1 BUT: Increase of damping coefficient per 1000 N*sec/m (2500 to 3500) 45

52. Simulations Test 5: like Test 2.1 BUT: Increase of stiffness coefficient of anti-roll bars per 2000 N*m/rad (3500 to 5500) 41

53. Simulations Test 5: like Test 2.1 BUT: Increase of stiffness coefficient of anti-roll bars per 2000 N*m/rad (3500 to 5500) 44

54. Simulations Test 6: like Test 2.1 BUT: Increase of stiffness coefficient of springs per 4000 N/m (front) & 5000 N/m (rear) (19000 & 18000 to 23000) 41 47

55. Simulations Test 6: like Test 2.1 BUT: Increase of stiffness coefficient of springs per 4000 N/m (front) & 5000 N/m (rear) (19000 & 18000 to 23000) 50 44

56. Simulations Test 7: like Test 1 BUT: Braking forces applied on all wheels (F1=F2=- 1500 N, F3=F4=- 1000 N) 58 60

57. Simulations Test 7: like Test 1 BUT: Braking forces applied on all wheels (F1=F2=- 1500 N, F3=F4=- 1000 N) 34

58. Simulations Test 8: like Test 1 BUT: Braking forces applied on rear wheels (F3=F4=- 1000 N) 56 60

59. Simulations Test 8: like Test 1 BUT: Braking forces applied on rear wheels (F3=F4=- 1000 N) 34

60. Simulations Test 9: like Test 1 BUT: Driving torque applied on rear wheels (F3=F4=824 N) 56 58

61. Simulations Test 9: like Test 1 BUT: Driving torque applied on rear wheels (F3=F4=824 N) 34

62. Expanding possibilities