National%20Technical%20University%20of%20Athens%20Diploma%20Thesis%20

1
National Technical University of AthensDiploma
ThesisComputational 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

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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

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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

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Chapter 2 Equations
12
Forces and Torques
13
6X6 System of differential Equations
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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

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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

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Front wheel steering

• Function of front wheel steering angle with
  respect to time
• INPUT Duration of steering T Final angle
  dffinal Degree of the polynomial r
• OUTPUT Front wheel steering angle function with
  respect to time df(t)
• IF tltT THEN
• k(dffinal/Tr)tr
• ELSE
• kdffinal
• END
• df(t)k

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Introduction of 4-wheel steering

• drg(df(t))
• dr?df(t)

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Criterion for the loss of tire adhesion

• Maximum alowable lateral force for each tire,
  according to the friction circle consept

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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

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User interface
21
User Interface
22
User interface
23
User interface
24
User interface
25
Chapter 4 Simulations

• Test 1 General case
• Weight distribution 60 front - 40 rear

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Simulations

• Test 1 General case
• Weight distribution 60 front - 40 rear

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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

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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 Nsec/m
  (2500 to 3500)

• 40

47
Simulations

• Test 4 like Test 2.1
• BUT
• Increase of damping coefficient per 1000 Nsec/m
  (2500 to 3500)

• 41

• 54

48
Simulations

• Test 4 like Test 2.1
• BUT
• Increase of damping coefficient per 1000 Nsec/m
  (2500 to 3500)

• 42

49
Simulations

• Test 4 like Test 2.1
• BUT
• Increase of damping coefficient per 1000 Nsec/m
  (2500 to 3500)

• 43

50
Simulations

• Test 4 like Test 2.1
• BUT
• Increase of damping coefficient per 1000 Nsec/m
  (2500 to 3500)

• 44

• 55

51
Simulations

• Test 4 like Test 2.1
• BUT
• Increase of damping coefficient per 1000 Nsec/m
  (2500 to 3500)

• 45

52
Simulations

• Test 5 like Test 2.1
• BUT
• Increase of stiffness coefficient of anti-roll
  bars per 2000 Nm/rad (3500 to 5500)

• 41

53
Simulations

• Test 5 like Test 2.1
• BUT
• Increase of stiffness coefficient of anti-roll
  bars per 2000 Nm/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

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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 (F1F2-1500
  N, F3F4-1000 N)

• 58

• 60

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Simulations

• Test 7 like Test 1
• BUT
• Braking forces applied on all wheels (F1F2-1500
  N, F3F4-1000 N)

• 34

58
Simulations

• Test 8 like Test 1
• BUT
• Braking forces applied on rear wheels
  (F3F4-1000 N)

• 56

• 60

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Simulations

• Test 8 like Test 1
• BUT
• Braking forces applied on rear wheels
  (F3F4-1000 N)

• 34

60
Simulations

• Test 9 like Test 1
• BUT
• Driving torque applied on rear wheels (F3F4824
  N)

• 56

• 58

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Simulations

• Test 9 like Test 1
• BUT
• Driving torque applied on rear wheels (F3F4824
  N)

• 34

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Expanding possibilities