# Semi-Active System of Vehicle Vibration Damping
## 5 Simulation Research of Vehicle Vibration  
Vibration simulations for a medium-sized passenger car are considered here, the model of which was mentioned in Section 3. The following parameter values regarding this vehicle have been adopted: \(\mathrm{m}=1250\) kg--mass of the vehicle body; \(\mathrm{m}_{0}=25\) kg--reduced mass of the wheel and suspension elements; \(\mathrm{J}_{\mathrm{x}}=\mathrm{m}\mathrm{\rho}_{\mathrm{x}}\mathrm{{}^{2}}\), \(\mathrm{\rho}_{\mathrm{x}}=0.6\) mm--moment of the body inertia relative to the longitudinal axis; \(\mathrm{J}_{\mathrm{y}}=\mathrm{m}\mathrm{\rho}_{\mathrm{y}}\mathrm{{}^{2}}\), \(\mathrm{\rho}_{\mathrm{y}}=1.15\) m--of the body inertia relative to the transverse axis; \(\mathrm{a}_{1}=1.4\) m, \(\mathrm{a}_{2}=1.45\) m, \(\mathrm{b}_{1}=\mathrm{b}_{2}=0.725\) m--measurements defining location of the body mass center relative to the wheels; \(\mathrm{h}_{\mathrm{s}}=0.5\) m--height of the vehicle mass center above the road surface (Figure 8); \(\mathrm{k}_{1}=\mathrm{k}_{2}=14.5\) kN/m--rigidity of front and rear wheel springs; \(\mathrm{k}_{0}=200\) kN/m--stiffness of the wheel tire; \(\mathrm{c}_{0}=2.5\cdot 10^{5}\) Ns/m--coefficient of tire damping; \(\kappa_{1}=\kappa_{2}=1.5\cdot 10^{3}\) Ns/rad--rigidity of front and rear stabilizers. The vehicle considered is equipped with the magneto-rheological dampers (MR dampers) whose characteristic is shown in Figure 6. The following parameters were adopted, shown in the Figure 10: \(\mathrm{T}_{\mathrm{A}}=400\) N, \(\mathrm{V}_{\mathrm{A}}=0.0075\) m/s, \(\mathrm{c}_{\mathrm{min}}=210\) Ns/m, \(\mathrm{c}_{\mathrm{g}}=3\cdot\mathrm{c}_{\mathrm{min}}\), \(\mathrm{I}_{\mathrm{max}}=2\) A. In the MR damper, the parametric variable represents the current flowing through the winding of the coil located at the damper's side [11; 31]. On the basis of the presented data, the damper characteristics have been formulated (1) as follows:  
\[\mathrm{f}(\mathrm{V};\tau):=\mathrm{c}_{\mathrm{min}}\mathrm{V}+[\mathrm{f}_ {\mathrm{max}}(\mathrm{V})-\mathrm{c}_{\mathrm{min}}]\tau, \tag{66}\]  
if  
\[\mathrm{f}_{\mathrm{max}}(\mathrm{V}):=\left\{ \begin{array}{cc}\mathrm{T}_{ \mathrm{A}}\mathrm{V}&\mathrm{gdy}&|\mathrm{V}|\leq\mathrm{V}_{\mathrm{A}}\\ \mathrm{T}_{\mathrm{A}}+\mathrm{c}_{\mathrm{g}}(|\mathrm{V}|-\mathrm{V}_{ \mathrm{A}})\mathrm{sign}\mathrm{V}&\mathrm{gdy}&|\mathrm{V}|>\mathrm{V}_{ \mathrm{A}}\end{array} \right., \tag{67}\]  
\[\tau:=\frac{\mathrm{I}}{\mathrm{I}_{\mathrm{max}}},\;\mathrm{I}\in[0,\mathrm{I }_{\mathrm{max}}]\;\tau\in\mathrm{R}^{1}, \tag{68}\]  
where I--intensity of the current flowing through the coil winding. Moreover, the assumption was made that the vehicle moves along the road, the roughness of which is described by the following function: \(\mathrm{\xi}(\mathrm{s})=\mathrm{\xi}_{0}\sin 2\pi\mathrm{\xi}\), where \(\mathrm{\xi}_{0}=5\) mm, \(\mathrm{L}=12\) m, s--the stretch of road covered. The vehicle is moving at variable velocity, the plot of which is shown in Figure 11. Another assumption was that the road's uneven surface under the left-side and right-side wheels is not exactly the same but shifted in phase by \(\mathrm{\cdot s}=\mathrm{a}_{1}\). In the presented paper, harmonic excitation was used. In this type of research, excitation based on the road profile in accordance with ISO 8608 [32; 33] is often used. Figure 10: Adopted characteristic of the MR damper, T—friction force, v—velocity of damper deformation, \(\mathrm{c}_{\mathrm{min}}\)—minimum value of the damping coefficient, \(\mathrm{c}_{\mathrm{g}}\)—maximum value of the damping coefficient, \(\mathrm{c}_{\mathrm{i}}\)—i-th value of the damping coefficient, \(\mathrm{T}_{\mathrm{A}}\)—value of the friction force at point A, \(\mathrm{V}_{\mathrm{A}}\)—value of the velocity of damper deformation at point A.