# On the Influence of Suspension Geometry on Steering Feedback
## Materials and Methods
### Case 3  
The third case introduces the half-shaft inclination \(\xi\) in the transverse plane (Figure 5):  
Figure 4: Case 2: kingpin angle \(>0\).  
In a FWD car with an independent suspension system, the differential output is not necessarily aligned with the wheel axis. In any case, it is usually connected to the spindle by means of a half shaft (or driveshaft) and two constant velocity (CV) joints, with an inclination angle \(\xi\). Given the nature of the CV joints, the driving torque remains constant along the half shaft. A secondary torque (Mp, Figure 6) is established as a function of the angle of inclination of the axle half shaft \(\xi\).  
With reference to the S point, the power balance is:  
\[\mathrm{M}_{2}\cdot\omega_{\mathrm{hub}}=\mathrm{C}_{\mathrm{m}}\cdot\omega_{ \mathrm{m}} \tag{8}\]  
where \(\omega_{\mathrm{hub}}\) is the angular velocity at the wheel hub and \(\omega_{\mathrm{m}}\) and \(\mathrm{C}_{\mathrm{m}}\) are the angular velocity and output torque at the differential.  
And again, due to the nature of the CV joint:  
\[\omega_{\mathrm{m}}=\omega_{\mathrm{hub}} \tag{9}\]  
So:  
\[\mathrm{M}_{2}=\mathrm{C}_{\mathrm{m}} \tag{10}\]  
Figure 5: Case 3: inclined half shaft.  
Figure 6: Power balance and secondary torque.  
The equations are the same as case 3, but the M1 component in the xz plane now changes to:  
\[\text{M}_{\text{1c}}=\text{M}_{\text{1}}\text{-}\text{cos}(\text{v})=\text{Fx-KO- cos}(\text{v}) \tag{16}\]  
On the xy plane:  
\[\text{M}_{\text{1\parallel}}=\text{M}_{\text{1c}}\text{-}\text{cos}(\text{ \lambda})=\text{Fx-KO-cos}(\text{v})\text{cos}(\text{\lambda}) \tag{17}\]  
Then the torque module on the steering axis is:  
\[=\text{Fx}\ [\text{KO-cos}(\text{v})\ \text{cos}(\text{\lambda})\ +\ \text{r}\ \sqrt{\text{sin2 }\ (\text{\lambda})[\text{cos2 }\ (\text{\xi})+1-2\ \text{cos}(\text{\xi})]\ +\ \text{cos2 }\ (\text{\lambda})\text{sin}(\text{\xi})}] \tag{18}\]  
From the formulas above, therefore, the moment around the steering axis is a function of the longitudinal force at ground, the inclination angle of the half shafts \(\xi\), and the typical parameters of the steering geometry, such as the kingpin (\(\lambda\)) and caster angles (\(\
u\)), as well as the kingpin offset (KO).  
Now, considering the complete front axle of the car, the two sides are connected by the steering rack. In a straight-line driving condition at constant speed, the Fx forces, as well as the M1 moments on both sides, are equal in module but opposite in direction, so the contribution to the torque steer is zero. The moment M2 does not cause any torque steer at all if the torque transmitted by the differential is the same on both sides, as it is the case for an open differential, and if the geometry is symmetric in terms of steering axis position (caster and kingpin angles) and half shaft inclination angle \(\xi\). If an LSD is installed and there is a torque difference transmitted across the differential, this contribution can become relevant.  
A torque vectoring system (TV) is based on a purposely designed delivery of the drive torques to each individual wheel. It is well known as a powerful tool to improve the balance and stability of the vehicle as the difference in terms of longitudinal force at ground level can be used to generate a moment controlling yaw, i.e., rotation around the vertical axis. However, when TV is applied on the front axle, the effects in terms of torque steer can be seen as similar to those caused by the action of a self-locking differential.