## 2 Materials and Methods
### Mathematical Modeling
#### 2.1.4 Simulation Method and Software Tools  
In order to solve the two systems of seven ordinary differential equations (ODEs) created for the two vibrating systems, the ode45 solver from MATLAB(r) was used [16]. As such, two separate functions were created in which the ODEs for each system were written as state-space models. Before the state-space model, however, a simple if/else statement is used to select the correct damping rate for each damper based on Equation (29). For the traditional suspension layout, to determine the damper velocity at each wheel, Equation (28) will be used. When it comes to the hydraulic interconnected suspension,however, we need to use the following equation [2] to correctly calculate the velocities of each suspension unit:  
\[\left[ \begin{array}{c}\mathrm{v}_{1}\\ \mathrm{v}_{2}\\ \mathrm{v}_{3}\\ \mathrm{v}_{4}\end{array} \right]=\mathrm{T}^{-1}.\left[ \begin{array}{c} \mathrm{v}_{\mathrm{f}}\\ \mathrm{v}_{\mathrm{f}}\\ \mathrm{v}_{\mathrm{f}}\\ \mathrm{v}_{\mathrm{f}}\\ \mathrm{v}_{\mathrm{f}}\end{array} \right] \tag{31}\]  
Another software program used to simulate the vibrating response of the vehicle is OptimumLap(r)[17]. The purpose of this program is to calculate the lateral and longitudinal accelerations as well as the velocity of the simulated vehicle along a given track. The software completes this task by using a quasi-steady-state simulation model that reduces the vehicle to a single mass point [17]. In order for this model to be able to approximate the velocity of the vehicle along the given track, the performance envelope of the vehicle is used (Figure 4). The performance envelope of the vehicle is created by stacking on top of each other the friction ellipses of the vehicle at different velocities and then combining them with the traction model of the car. Although more complex lap time simulation methods are available, the complexity of these methods and the large number of variables that must be taken into account can lead to extremely long computation times [18] for a personal computer; therefore, they were deemed unsuitable for the current simulation. Moreover, the correlation of the quasi-steady-state simulation method with more complex simulation methods, as well as with real-life data, reveals that the results have negligible differences (under 1%). The track used in this simulation was created to be both simple to compute and a realistic representation for a part of a Formula Student Track. Using these two principles, the chosen track has the following configuration (Figure 5): the start line is followed by a 60 m straight, at the end of which there is a corner with a radius of 20 m and length of 70 m. At the end of the first corner, a right-hand corner with a radius of 9 m (the minimum corner  
Figure 4: Performance envelope of the BS19 Formula Student car. radius allowed by the FSAE rules) and a length of 31.5 m will follow. After a straight of 20 m, there is a portion of the track for slalom manoeuvres that leads into the final 30 m straight to the finish line. By introducing the parameters of BS 19 in OptimumLap(r), the simulated lap time is 19.62 s. If we graph the car speed (Figure 6), longitudinal acceleration (Figure 7), and lateral acceleration (Figure 8), we get the following results:  
The last input we need in order to have a complete picture of how the models react is a road profile. Naturally, the best signal used in this situation would be one created based on the scanning of the competition track. However, since the configuration of the competition track is changed every year, the collection of such data would be both time-consuming and expensive. In order to solve this problem, a generic but realistic road profile is generated using the method provided by [PERSON] and [PERSON] [19], based on Fourier series. Another important advantage of using this method when generating the road profile is the fact that the method proposed by [PERSON] and [PERSON] can generate roads of different roughness levels.