## 2 Materials and Methods
### Mathematical Modeling
#### 2.1.3 Damper Modelling  
Although previously the damping rate was described as a linear function of the compression/extension velocity, most automotive dampers are modified to have different damping rates based on both the direction and value of the damper's compression/extension velocity.  
The asymmetry of the damper when it comes to the compression vs. the extension motion is a deliberate design choice. The necessity of this design is described by [PERSON] and [PERSON] [12], by pointing out that at compression the energy of the system is stored in the spring. This means that less damping is required at this point as opposed to the rebound motion of the suspension when the energy released by the spring has to be dissipated in order to avoid excessive oscillations of the wheel. Usually, this asymmetry means that the rebound damping rate is between 2 and 4 times higher [12; 13; 14; 15] than the damping rate during compression.  
On top of the asymmetry discussed previously, there is another departure from the linear model given by the internal architecture of the damper. Although there are pressure valves on the piston of the damper, they usually do not open until a certain velocity is reached (also known as the knee velocity). Before this point, the hydraulic liquid inside the damper will flow around the piston resulting in a higher damping rate. Once the knee velocity is reached, the pressure valves open, allowing more flow inside the damper leading to a reduction of the damping rate. Usually, however, this is a useful characteristic because the lower velocity damping rate can be used to control the motion of the car, while the lower damping rate, which happens at higher damper velocity, is used to control the motion of the wheel over road irregularities.  
The value of the damping rate as a function damper velocity can be calculated using the following formula:  
\[\mathrm{F_{c}}=\left\{\begin{array}{ll}\mathrm{C_{D}}\cdot(1-\mathrm{e_{D}}) \cdot\
u^{\lambda}&0<\mathrm{v}<1\\ \mathrm{C_{D}}\cdot(1-\mathrm{e_{D}})\cdot\left[\mathrm{l^{\lambda}}\cdot \left(1-\frac{1}{\mathrm{r}}\right)+\frac{\
u^{\lambda}}{\mathrm{r}}\right]& \mathrm{v}>1\\ \mathrm{C_{D}}\cdot(1-\mathrm{e_{D}})^{-1}\cdot(-\
u^{\lambda})&-1<\mathrm{v}< 0\\ \mathrm{C_{D}}\cdot(1-\mathrm{e_{D}})^{-1}\cdot\left[\mathrm{l^{\lambda}}\cdot \left(1-\frac{1}{\mathrm{r}}\right)+\frac{-\
u^{\lambda}}{\mathrm{r}}\right]& \mathrm{v}<-1\end{array}\right. \tag{29}\]  
and  
\[\mathrm{C_{D}}=\ \xi\text{,}2\cdot\sqrt{\mathrm{K}\cdot\mathrm{m}} \tag{30}\]  
where  
\(\mathrm{F_{c}}\)--damping force;  
\(\mathrm{C_{D}}\)--damping rate;  
\(\mathrm{e_{D}}\)--asymmetry factor (desired ratio between the compression and rebound forces);  
\(\lambda\)--progressive factor (0 for Coulomb friction, 1 for viscous friction 2 for quadratic damping);  
\(\mathrm{l}\)--knee velocity;  
\(\mathrm{r}\)--desired ratio between low and high velocity damping;  
\(\mathrm{v}\)--damper piston velocity (positive for compression and negative for rebound);  
\(\mathrm{\xi}\)--damping ratio;  
\(\mathrm{K}\)--the spring rate from which the energy is dissipated;  
\(\mathrm{m}\)--sprung mass acting on the damper.  
The dampers where there is a compressive/rebound asymmetry will be used for all dampers present on the traditional suspension layout. However, on the hydraulic interconnected suspension layout, only the bounce suspension unit will have a damper with asymmetry to avoid roll and pitch inconsistencies. The force-velocity graphs for all the dampers in the models are shown in Appendix B.