## 2 Materials and Methods  
### Mathematical Modeling  
Before diving into the mathematical modelling for the two suspension systems, the notations used and the sign convention should be discussed. The traditional suspension model used for comparison is based on the work of professor [PERSON] [11]; therefore, the notations are as follows: body bounce \(x_{i}\) body roll \(\rho_{i}\); body pitch \(\theta_{i}\); wheels hop \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\); and road excitations for each wheel \(y_{1}\), \(y_{2}\), \(y_{3}\), and \(y_{4}\). For the sign convention, we will consider motions towards the ground negative for \(x\), \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\). The values for \(y_{1}\), \(y_{2}\), \(y_{3}\), and \(y_{4}\) follow a similar rule, with the starting point being considered 0 and values above are positive. Body roll \(\rho\) is considered positive when the chassis is rotating towards the right side, while \(\theta\) is considered positive when the chassis is rotating towards the front. One thing to note in the vehicle model proposed by [PERSON] is that the tires are modelled as simple springs. However, this simplification does not have a significant effect due to the small damping values of the tires. Another problem that has to be solved is the use of the lateral and longitudinal accelerations of the vehicle as excitation for the vibrating vehicle model. For this, we need to calculate the moments of inertia for roll and pitch (\(I_{r}\) and \(I_{p}\), respectively). The necessity for this step is given by the kinematic design of the suspension that imposes a given roll and pitch axis around which the sprung mass rotates. However, \(I_{r}\) and \(I_{p}\) can be calculated using the Huygens-Steiner theorem. Applied to the current problem, the equation takes the following form. In order to introduce the effects of the lateral and longitudinal acceleration of the vehicle into the vibrating model, the pitch and roll moments of inertia were calculated with the following formula. \[I=I_{cm}+m\cdot d^{2} \tag{1}\]  
I--resulting moment of inertia;  
\(I_{cm}\)--inertia tensor (\(I_{xx}\) for roll and \(I_{yy}\) for pitch);  
Figure 3: BlueStreamline BS19. d--either the perpendicular distance between the CG and roll axis, or the perpendicular axis between the CG and the pitch axis. Another aspect that needs to be introduced into the vibrating model is the effect of the longitudinal and lateral weight transfer. The method of calculation and the different components of this effect use the methodology presented by [PERSON] [1]. The elastic weight transfer will take the following form:  
\[R_{\text{lat-wt}}=\text{ SM}\cdot\text{a}_{\text{lat}}\cdot\Delta_{\text{z}} \tag{2}\]  
\[P_{\text{long-wt}}=\text{ SM}\cdot\text{a}_{\text{long}}\cdot\Delta_{\text{ PCz}} \tag{3}\]  
where  
SM--suspended mass [kg];  
a\({}_{\text{lat}}\)--lateral acceleration [m/s\({}^{2}\)];  
a\({}_{\text{long}}\)--longitudinal acceleration [m/s\({}^{2}\)];  
\(\Delta_{\text{z}}\)--distance from the CG of the suspended weight to the roll axis [m];  
\(\Delta_{\text{PCz}}\)--distance from the CG of the suspended weight to the pitch axis [m]. The geometric weight is calculated with  
\[WT_{\text{geo-f}}=\frac{\text{ SM}\cdot\frac{\text{ WD}_{\text{SM}}}{100}\cdot\text{a}_{\text{lat}}\cdot\text{RC}_{\text{front}}}{ \text{track}_{\text{front}}} \tag{4}\]  
\[WT_{\text{geo-r}}=\frac{\text{ SM}\cdot\frac{100-\text{ WD}_{\text{SM}}}{100}\cdot\text{a}_{\text{lat}}\cdot\text{RC}_{\text{ rear}}}{\text{track}_{\text{rear}}} \tag{5}\]  
\[WT_{\text{geo-long}}=\frac{\text{ SM}\cdot\text{a}_{\text{long}}\cdot\text{PC}_{\text{h}}}{\text{wheelbase}} \tag{6}\]  
where  
WD\({}_{\text{SM}}\)--weight distribution of the suspended mass [%];  
PC\({}_{\text{h}}\)--pitch center height [m];  
RC\({}_{\text{front/rear}}\)--height of the front/rear roll center [m].