# Semi-Active System of Vehicle Vibration Damping
## 5 Simulation Research of Vehicle Vibration
The assumption was made that parameters of the virtual damper, included in Formula (40), are determined on the basis of non-dimensional damping coefficients amounting to \(\gamma_{x}{}^{\text{sky}}=0.413;\gamma_{\varphi_{\text{x}}}{}^{\text{sky}}=0.814; \gamma_{\varphi_{\text{y}}}{}^{\text{sky}}=0.470\). Due to the comparative character of the performed tests, it was assumed that the reference vehicle is equipped with classic shock absorbers (dampers) whose bi-directional characteristic is determined by two constants: \(c_{1}=500\)Ns/m when the shock absorber is squeezed, and \(c_{2}=2.5\cdot c_{1}\) when the shock absorber is stretched. While using the presented model, the vehicle motion on the described road was simulated with the velocity given in Figure 11, and the drive time amounted to 30 s. It was assumed that at the initial moment (t = 0), all coordinates and corresponding velocities (see: Equations (11-14)) were equal to zero. Figure 12 shows a scheme of the supply system for the MR damper coil, i.e., of the system featuring the parametric variable \(\tau\), which is the current flowing through the coil. Figure 11: Present waveform of vehicle velocity. The waveforms for the currents \(\mathrm{I}_{\mathrm{L}},\;\mathrm{i}=1,\ldots,4\) flowing through coils in the dampers, determined according to the system of equations. The following parameter values of the coil supply systems were adopted \(\mathrm{R}=2\;\Omega,\;\mathrm{L}=6\;\mathrm{mH},\;\mathrm{I}_{\mathrm{max}}=2\; \mathrm{A}\). \[\mathrm{Li}_{\mathrm{i}}+\mathrm{RI}_{\mathrm{i}}=\mathrm{RI}_{\mathrm{i}}^{ \mathrm{opt}}, \tag{69}\]  
where \(\mathrm{I}_{\mathrm{i}}^{\mathrm{opt}}\) is computed on the basis of the damper characteristic described by Formulas (66-68):  
\[\mathrm{I}_{\mathrm{i}}^{\mathrm{opt}}=\mathrm{f}^{-1}\big{(}\mathrm{V}_{ \mathrm{i}},\;\mathrm{T}_{\mathrm{i}}^{\mathrm{opt}}\big{)},\;\mathrm{i}=1, \ldots,4, \tag{70}\]  
To evaluate the body vibrations, an indicator was adopted whose values determine the sum of the acceleration squares in the four points of the body above the wheels (Figure 6):  
\[\mathrm{W}_{1}:=\left\{\int_{0}^{\mathrm{t}_{\mathrm{end}}}\sum_{\mathrm{i}= 1}^{4}\mathrm{a}_{\mathrm{i}}^{2}(\mathrm{t})\mathrm{dt}\right\}^{\frac{1}{2}}, \tag{71}\]  
where \(\mathrm{a}_{\mathrm{i}}\)--accelerations of the enumerated points, \(\mathrm{i}=1,\ldots,4\). Additionally, the value of an indicator characterizing change in the wheels' pressure on the road surface was also determined:  
\[\mathrm{W}_{2}=\left\{\int_{0}^{\mathrm{t}_{\mathrm{end}}}\sum_{\mathrm{i}=1} ^{4}\mathrm{N}_{\mathrm{i}}^{2}(\mathrm{t})\mathrm{dt}\right\}^{2}, \tag{72}\]  
if  
\[\mathrm{N}_{\mathrm{i}}(\mathrm{t})=\big{(}\mathrm{S}_{0\mathrm{i}}(\mathrm{t })+\mathrm{T}_{0\mathrm{i}}(\mathrm{t})\big{)}\frac{1}{\mathrm{Q}_{0\mathrm{i} }},\;\mathrm{i}=1,\ldots,\;4, \tag{73}\]  
where forces are described by Formula (16), and \(\mathrm{Q}_{0\mathrm{i}}\)--static wheel pressure. Three vibration simulations were conducted for the vehicle moving under the conditions described above. During the first simulation, the vehicle was equipped with classic dampers whose characteristic is given earlier in point 4. The Figure 13 shows the change in the wheels' pressure on the road surface on the front axle, and Figure 14 shows the change in vertical accelerations over the front axle. Figure 13: Change in the wheels’ pressure on the road surface on the front axle without control. The second simulation was conducted with regard to the vehicle equipped with the MR dampers whose characteristic is given in Figure 11; optimal values of forces in the dampers were determined in this simulation in accordance with the first criterial function \(\mathcal{K}^{(1)}\).