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Figure 4.21 - Wheel free-body diagram
wheel center wheel wheel wheel x wheel wheel wheel
M I
T F R I
(4.48)wheel x wheel wheel
wheel
T F R
I (4.49)
By calculating the angular acceleration of the wheel when subjected to the moments described above, one can once again use the Matlab Simulink environment advantages of looped systems, differentiation and integration. By differentiating the angular acceleration one obtains the wheel angular velocity, which serves as an input to the longitudinal slip ratio calculation as described in chapter 3.
39 behavior accounting to develop the slip angle that provides the maximum tire lateral force in specified conditions of load and inclination.
To achieve this condition the tie-rod pickup point of the upright is positioned with an offset in the y axis relative to the intersection of the steering axis with the wheel center plane in top view (Figure 4.22):
Figure 4.22 - Steering Geometry
The solid lines represent the nominal position of the steering system with no wheels steered.
The dashed lines represent a linear displacement of the rack.
In order to model this, advantage was taken of the Matlab Simulink environment once again.
Simulink presents in its package a variant that allows the modelling of multi-body systems for dynamic and kinematic analysis. This is the Simmechanics library.
Since the driver input in the model is the steering wheel angle, δSW, one can first of all transform it in rack displacement by using the c-factor:
factor
SWrack displacement c
(4.51) A multibody system contemplating the front suspension, the steering rack and tie-rods, is modelled in Simmechanics. The rack is assembled in the model using a prismatic joint with one degree of freedom for the rack displacement to be introduced – done using a joint actuator (Figure 4.23). A joint sensor is then placed at the wheel centers to measure the rotation relative to the z axis.Figure 4.23 - Steering system in Simmechanics
40
4.7.2 Sideslip and slip angles calculation
As defined in chapter two, the sideslip angle of the vehicle can be calculated as follows:
tan
1 y xv
v (4.52)
This variable is useful for vehicle performance analysis but for the calculation of slip angles the side-slip angles of each wheel will be used (Figure 4.24), as defined in (Jazar 2008).
Figure 4.24 - Slip angle
i i i
SA (4.53)
tan
1 yi ixi
v
v (4.54)
The value of the steering angle is obtained from the Simmechanics model but the sideslip angle of each wheel is calculated as in Eq. (4.54).
Figure 4.25 - Wheel velocities schematic
Knowing the velocities of the vehicle’s center of gravity (Figure 4.25) and according to (Beer, Johnston et al. 2006) one can calculate the velocities at the wheels (xi and yi are the position of the wheels measured in the vehicle coordinate system):
41
yi y i
xi x i
v v x
v v y
(4.55)
4.7.3 Inclination angle calculation
The inclination angle as defined in chapter three will be calculated using the displacement of the wheel and also the wheel orientation. As the steering axis is usually not vertical, the steering of the wheel causes the inclination angle to suffer changes according to mentioned geometry. The inclination angle also changes with the wheel travel due to the suspension design. These two contributions happen at the same time but as described in (Neves 2012) they can be treated independently and added to give the final contribution.
The commonly known inclination angle gain with vertical motion is considered in static conditions equal to the camber gain, and also to the variation of the KPI angle. In order to calculate the KPI angle change, the method described in (Blundel and Harty 2004) as the 3-point method will be used. This method states that it is possible to find a valid position of a point if the position of three other points is known and also the distance from these points to the unknown point. This is useful when treating a suspension wishbone where we know the chassis pickup-points and the length of the wishbone arms.
Figure 4.26 - 3-point method schematic
Looking at Figure 4.26, the goal of this method is to calculate points 1 and 2 knowing points 4, 5, 6, 7 and 3 and also their distance to the unknown points.
2 2 2
2
1 3 1 3 1 3 1 3
2 2 2
2
1 4 1 4 1 4 1 4
2 2 2
2
1 5 1 5 1 5 1 5
2 2 2
2
2 3 2 3 2 3 2 3
2 2 2
2
2 6 2 6 2 6 2 6
2 2 2
2
2 7 2 7 2 7 2 7
2 2 2
1 2 1 2 1 2 1 2
. .
d x x y y z z
L x x y y z z
L x x y y z z
d x x y y z z
L x x y y z z
L x x y y z z
S t x x y y z z d
(4.56)
42
Solving of the first six equations always finds two solutions, so, to filter this, the last equation is introduced as a constraint. The distance, d1-2, is known as it is the distance between the pickup-points of the upright.
By solving these equations for several positions of the point 3, which represents the wheel center, one finds the successive positions of the upright pickup-points through the wheel travel. Using trigonometry one can find the gain in KPI angle and caster angle (Figure 4.27).
Figure 4.27 - KPI angle calculation
1 2 1
1 2
tan
y yKPI z z
(4.57)
As the suspension of the FST 05e is designed with nearly linear and symmetric KPI angle gain one can define a coefficient to be multiplied by the wheel travel in order to know the KPI angle gain. For this the equations are only solved for the static position and then with a wheel travel of 10 mm.
1010
z static z static mm
wheel wheel
KPI KPI
IA KPI
z z
(4.58)
The same can be done to calculate the gain in caster angle.
The contribution of the steering axis rotation for the inclination angle gain is presented next and also the calculation of the inclination angle by adding all the contributions:
(1 cos ) sin
(1 cos ) sin
FR FR
FR
FL FL
FL
IA KPI
IA KPI
(4.59)
43
, ,
, ,
, ,
, ,
( )
( )
( )
(
FR static FR wheel FR static
wheel FR
FR static FL wheel FL static
wheel FL
RR static RR wheel RR static
wheel
RL static RL wheel
wheel
IA IA
IA IA z z
z
IA IA
IA IA z z
z
IA IA IA z z
z
IA IA IA z
z
RL zstatic
)
(4.60)
4.7.4 Force and moment calculation
The tire lateral forces and self-aligning moments are calculated using the Pacejka model developed, all the parameters present in the calculation were described above:
, , ,
Fy f SA SL FZ IA (4.61)
, ,
Mz f SA FZ IA (4.62)