# Optimization of Damping in a Semi-Active Car Suspension System with Various Locations of Masses
## 2 Methods  
The random, artificial road profile can be generated through various methods, based on: (1) linear filtering, (2) superposition of harmonics, or (3) inverse fast Fourier transformation of discretized power spectra density (PSD) [55]. One of the most commonly implemented [56; 57; 58; 59; 60; 61], the superposition of harmonics (or sinusoidal approximation), we also used. The longitudinal road profiles were generated using the original MATLAB code, implementing the method of superposition of harmonics in the spatial domain according to Equations (1) and (2):  
\[z(x)=\sum_{i=1}^{N=(\Omega_{\mathrm{U}}-\Omega_{\mathrm{L}})/\Delta\Omega}Z_{i }\cos(\Omega_{i}x+\varphi_{i})\text{ with } \tag{1}\]  
\[Z_{i}=\sqrt{2G_{\mathrm{d}}(\Omega_{i})\Delta\Omega} \tag{2}\]  
where \(N=(\Omega_{\mathrm{U}}-\Omega_{\mathrm{L}})/\Delta\Omega\)--number of harmonic samples (in our case, \(N\) = 1000); \(\Omega_{\mathrm{U}}\), \(\Omega_{\mathrm{L}}\)--upper and lower angular spatial frequencies [rad/m] in the PSD spectrum; \(\Delta\Omega\)--the width of each frequency band; \(Z_{i}\)--amplitude of the \(i\)th harmonic; \(\Omega_{i}=\Omega_{\mathrm{L}}+(i-1)\Delta\Omega\)--angular spatial frequency of the \(i\)th harmonic; \(\varphi_{i}\)--phase angle of the \(i\)th harmonic uniformly distributed in the range from 0 to \(2\pi\); and \(G_{\mathrm{d}}(\Omega_{i})\)--displacement PSD at angular spatial frequency \(\Omega_{i}\). The ISO 8608 standard recommends the lower and upper limits of angular spatial frequencies (=2\(\pi\)m) equal to \(2\pi\)\(\times\) 0.01 rad/m and \(2\pi\)\(\times\) 10 rad/m for general on-road measurements, respectively [62]. To obtain various road profiles that more closely resemble real roads, we changed the linear fitting of \(G_{\mathrm{d}}(\Omega)\) proposed by ISO [62] to two split fittings offered by [PERSON] [63] in Equation (2), and we modified it by reducing the amplitude of the frequencies higher than \(\Omega_{2}\) and lower than 0.04 \(\times\) 2\(\pi\) rad/m:  
\[G_{\mathrm{d}}(\Omega)=\left\{ \begin{array}{l l}G_{\mathrm{d}}(\Omega_{0}) \Omega^{-1}&\text{for}&\Omega\leq 0.04\times 2\pi\text{ rad/m}\\ G_{\mathrm{d}}(\Omega_{0})\Omega^{-w_{1}}&\text{for}&0.04\times 2\pi\text{ rad/m} \leq\Omega\leq\Omega_{1},\\ G_{\mathrm{d}}(\Omega_{0})\Omega^{-w_{2}}&\text{for}&\Omega_{1}\leq\Omega \leq\Omega_{2}\\ G_{\mathrm{d}}(\Omega_{0})\Omega^{-w_{3}=5}&\text{for}&\Omega_{2}\leq\Omega \end{array} \right. \tag{3}\]  
where \(w_{i}\) is waviness. The reference frequency \(\Omega_{0}\) = 1 rad/m, the lower break frequency \(\Omega_{1}\) = 0.21 \(\times\) 2\(\pi\) rad/m, and the higher break frequency \(\Omega_{2}\) = 1.22 \(\times\) 2\(\pi\) rad/m produced a minimal error for the Swedish road network [63]. Additionally, a Welch-type windowing function with the value of the exponent equal to 10 was used to minimize the appearance of abrupt shifts at the connections between profile segments [64]. The profile generated in this way applied to the left side, and the profile for the right side was formed by randomly \begin{table}
\begin{tabular}{c c c c} \hline
**Road Type** & **Car Model** & **Remarks** & **Refs.** \\ \hline Main road profile & 3D model with driver & Optimized damping and stiffness for passive, semi-active, and active suspensions & [52] \\ \hline Bump 50 mm & Half-car & C-K figures; genetic algorithm; various control methods & [53] \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._increasing or decreasing it by 20%. For comparison, identical sequences of random numbers were used to generate each profile. The Range Rover Evoque dynamic response on road irregularities was analyzed on the full-car model. Our developed dynamic model included ten masses (the car's mass; four wheels' masses; and the masses of a driver, three passengers, and a baggage box) and moments of inertia about the \(X\) and \(Y\) axes [65].