Analysis of the lateral dynamics of the vehicle (stability and step steer manoeuvre) of a Linearized vehicle model with roll motion is done in this project. considering a high double wishbone suspension for front and a semi-trailing arm suspension for rear, firstly we derive the roll camber coefficient and camber stiffness benefiting from a multibody dynamics simulation (MSC ADAMS) validated with experimental values. On the other hand, cornering stiffness and aligning moment are computed through the tire diagrams using Carsim software. Root loci of the system with the objective of investigating the vehicle stability as a function of the vehicle speed is plotted with varying the speed V. Furthermore, a step steer manoeuvre will be performed in which the time history of the variables and the trajectory with respect to the rigid vehicle model will be compared. In the last part, the effect of suspension stiffness and damping variation on the vehicle stability will be examined. *Only the first ten pages of the project is presented here; if you are interested to study the rest of this document, please do not hesitate to contact e via saeid.ghaffari@studenti.polito.it.

1. Department of Mechanical
and Aerospace Engineering
Analysis of the Stability and Step steer Maneuver of
a Linearized Vehicle Model with Roll Motion
Saeid Ghaffari
Professor: Nicola Amati

2. 2
Methodology
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion
Data set
Front suspension characteristics – high DWB
Rear suspension characteristics – semi-trailing arm suspension
Roll centre height
Approaches for defining the front and rear suspension stiffness and damping
Dynamic response of the stability analysis (root loci)
Study the motion with locked controls of the vehicle following a step steering input

3. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
3
Methodology
Characteristic Value (unit)
Density of air 1.206 (kg/m3)
Front area 2.3 (m2)
Lateral aerodynamic force
coefficient (𝐶 𝑦) 𝛽
-0.5
Aerodynamic coefficient about z
axis (𝐶 𝑀𝑧) 𝛽
-0.05
Starting from a D-class sedan with the following characteristics, we will able to find the requirements of the analysis.
Data set
Tire Characteristic Value (unit)
Cornering stiffness 𝐶1 (front) 144446 (N/rad)
Cornering stiffness 𝐶2 (rear) 102764 (N/rad)
Camber stiffness 𝐶 𝛾1 (front) -4892 (N/rad)
Camber stiffness 𝐶 𝛾2 (rear) -3317 (N/rad)
Aligning stiffness 𝑀𝑧1 𝛼 (front) 8978 (Nm/rad)
Aligning stiffness 𝑀𝑧2 𝛼 (rear) 8318 (Nm/rad)
Real examples:
Alfa Romeo 159
Audi A4
BMW 3 series
Mercedes-Benz C-Class
-120
-100
-80
-60
-40
-20
0
Camberthrustcoefficient(N/deg)
Vertical load (N)
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion

4. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
4
Methodology
Characteristic Value (unit)
Acceleration of gravity 9.81 (m/s2)
Sprung mass 1370 (kg)
Unsprung mass
Front axle
80 (kg)
Unsprung mass
rear axle
80 (kg)
Yaw moment of inertia, 𝑱 𝒛 2315.3 (kgm2)
Roll moment of inertia, 𝑱 𝒙 671.3 (kgm2)
Mixed moment of inertia,
𝑱 𝒙𝒛
+90 (kgm2)
Wheel base 2.780 (m)
Front axle to CG 1.110 (m)
Height of CG 0.520 (m)
Wheel track 1.550 (m)
Moments and products of inertia of the unladen sprung mass are very important in our calculations. From CarSim, the moments and
product are all taken about the mass center of the sprung mass and are based on a vehicle axis system in which the X axis (longitudinal)
and Y axis (lateral) are parallel with the ground when the vehicle is at rest on a flat level surface, and the Z axis (vertical) points up in
parallel with the gravity vector; inasmuch as we have no information about 𝑱 𝒙𝒛 , we can use the data related to real cars.
Data set
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion

5. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
5
Methodology
Front suspension characteristics – high DWB
𝜸 𝟏 𝝋 : front axle camber angle variation with respect to roll angle
According to the suspension characteristic obtained from
ADAMS/Car analysis, in small roll angle the roll camber
coefficient is around + 0.8, which has a good agreement
with the literature for this type of suspension.
The rate of change in wheel inclination angle with respect
to change in vehicle roll angle.
By convention, a positive roll angle is a rotation about a
longitudinal axis such that the right side moves down and
the left, up.
Roll Camber coefficient
𝜸 𝟏 𝝋 = 𝟎. 𝟖
The camber change and roll steer depend on the geometry of the
suspension system.
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion

6. Ref. The Automotive Chassis, Jörnsen Reimpell,Helmut Stoll,Jürgen W. Betzler, chapter 3, page 185
The average roll camber factors for the following axles are:
The mean value of the wheel outside of a bend and the
inside one together with the kinematic body roll angle gives
the roll camber coefficient.

7. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
7
Methodology
𝜹 𝟏 𝝋 : front axle toe angle variation with respect to roll angle
Front suspension characteristics – high DWB
The rate of change in steer angle with respect to change in vehicle roll angle.
A positive roll steer acts in the same direction as the front steering angle, steered by steering wheel, whereas a negative roll steer acts in
the opposite direction. From the results obtained from ADAMS/Car in small roll angles,
Steady-state cornering with the front and rear roll steer 𝛼 𝑓 and 𝛼 𝑟
Roll Steer
𝜹 𝟏 𝝋 = 𝟎. 𝟒𝟐
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion

8. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
8
Methodology
Rear suspension characteristics – semi-trailing arm suspension
𝜸 𝟐 𝝋 : rear axle camber angle variation with respect to roll angle
According to the suspension characteristic obtained from
ADAMS/Car analysis, for small roll angle the roll camber
coefficient is around + 0.63.
Roll Camber coefficient
𝜸 𝟐 𝝋 = 𝟎. 𝟔𝟑
The wheel on the outside of the bend goes into positive
camber and the one on the inside into negative camber.
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion

9. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
9
Methodology
𝜹 𝟐 𝝋 : rear axle toe (steer) angle variation with respect to roll angle
Rear suspension characteristics – semi-trailing arm suspension
From the results obtained from ADAMS/Car
in small roll angles,
Steady-state cornering with the front and rear roll steer 𝛼 𝑓 and 𝛼 𝑟
Roll Steer
𝜹 𝟐 𝝋 = 𝟎. 𝟎3
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion

10. Chassis Design-A
Automotive Engineering - A.Y. 2018-2019
Department of Mechanical
and Aerospace Engineering
10
Methodology
Roll centre height
The position of Roll center changes during roll motion. The roll center is then a fixed point only for small angles about
the symmetric position (vanishing roll angle). In the case of large roll angles the roll center, and the roll axis as well, lies
outside the symmetry plane of the body.
By creating a suspension assembly for front and rear of the same
type we introduced before, we can find the roll centre height in static
vehicle characteristic (small roll angle).
So we can say that the roll centre height of the body in the normal
plane passing from the CG is 153 mm.
Position Suspension
Roll centre height
(mm)
Rear Semi-trailing arm 77
Front High DWB 204
ℎ = ℎ 𝐶𝐺 − 153 ℎ = 367 𝑚𝑚
Vertical distance from the CG to the roll axis (h) is:
Roll centre height is a very important parameter in this
study, and increasing the height will increase the risk of
vehicle instability.
Analysis of the Stability and Step steer Maneuver of a Linearized Vehicle Model with Roll Motion