# Semi-Active System of Vehicle Vibration Damping
## 3 Vehicle Model  
The considered mathematical vehicle model serves the purpose of analyzing the vibration in the vertical plane. Figure 7 shows the scheme of the mechanical system, which was adopted for the equations describing the vehicle vibration to be formulated. In order to simplify this description, the following were adopted:  
* Elastic suspension elements have linear characteristics;
* Vehicle wheels are in constant contact with the road surface;
* Elastic-dissipative features of the suspension are mapped with the help of elements located directly above the wheels;
* The vehicle mass is distributed symmetrically relative to the longitudinal axis. To describe the vehicle vibration, coordinates were selected that determine the vehicle displacements relative to the equilibrium position. Figure 7 shows the selected marked coordinates and the basic model parameters. The equations of the vehicle vibration are defined with the following formulae:  
\[\mathrm{m}\bar{\mathrm{z}}+\sum_{\mathrm{i=1}}^{4}(\mathrm{S}_{\mathrm{i}}+ \mathrm{T}_{\mathrm{i}})=0, \tag{11}\]  
\[\mathrm{J}_{\mathrm{x}}\bar{\Phi_{\mathrm{x}}}+\mathrm{b}_{1}(\mathrm{S}_{ \mathrm{1}}+\mathrm{T}_{\mathrm{1}})+\mathrm{b}_{2}(\mathrm{S}_{\mathrm{2}}+ \mathrm{T}_{\mathrm{2}})-\mathrm{b}_{1}(\mathrm{S}_{\mathrm{3}}+\mathrm{T}_{ \mathrm{3}})-\mathrm{b}_{2}(\mathrm{S}_{\mathrm{4}}+\mathrm{T}_{\mathrm{4}})+ \mathrm{M_{1}}^{\mathrm{S}}+\mathrm{M_{2}}^{\mathrm{S}}=\mathrm{M}_{\mathrm{x}}, \tag{12}\]  
\[\mathrm{J}_{\mathrm{y}}\bar{\Phi_{\mathrm{y}}}-\mathrm{a}_{1}(\mathrm{S}_{ \mathrm{1}}+\mathrm{T}_{\mathrm{1}})+\mathrm{a}_{2}(\mathrm{S}_{\mathrm{2}}+ \mathrm{T}_{\mathrm{2}})-\mathrm{a}_{1}(\mathrm{S}_{\mathrm{3}}+\mathrm{T}_{ \mathrm{3}})+\mathrm{a}_{2}(\mathrm{S}_{\mathrm{4}}+\mathrm{T}_{\mathrm{4}})= \mathrm{M}_{\mathrm{y}}, \tag{13}\]  
\[\sum_{\mathrm{i=1}}^{4}\mathrm{m}_{\mathrm{0}}\bar{\mathrm{z}}_{\mathrm{0i}}- \sum_{\mathrm{i=1}}^{4}(\mathrm{S}_{\mathrm{i}}+\mathrm{T}_{\mathrm{i}})+ \sum_{\mathrm{i=1}}^{4}(\mathrm{S}_{\mathrm{0i}}+\mathrm{T}_{\mathrm{0i}})+ \sum_{\mathrm{i=1}}^{4}\mathrm{F_{i}}^{\mathrm{S}}=0, \tag{14}\]  
where \(\mathrm{S}_{\mathrm{i}}\), \(\mathrm{T}_{\mathrm{i}}\), \(\mathrm{i=1,\dots,4}\)--elastic and dissipative forces acting between the \(\mathrm{i}\)-th wheel and the suspension, so that  
\[\mathrm{S}_{\mathrm{i}}:=\mathrm{k}_{\mathrm{i}}\mathrm{U}_{\mathrm{i}},\ \mathrm{T}_{\mathrm{i}}=\mathrm{f}(\mathrm{V}_{\mathrm{i}};\mathrm{I}_{\mathrm{ i}}),\ \mathrm{i=1,\dots,4}, \tag{15}\]  
forces resulting from the tire deformation  
\[\mathrm{S}_{\mathrm{0i}}=\mathrm{k}_{\mathrm{0i}}\mathrm{U}_{\mathrm{0i}},\ \mathrm{T}_{\mathrm{0i}}=\mathrm{c}_{\mathrm{0i}}\mathrm{V}_{\mathrm{0i}},\ \mathrm{i=1,\dots,4}, \tag{16}\]  
Figure 6: Schematic plot of dissipative characteristic of the semi-active damper \(\mathrm{T}=\mathrm{f}(\mathrm{v};\mathrm{\tau})\); \(\mathrm{T}\) —friction force, \(\mathrm{v}\) —velocity of damper deformation, \(\mathrm{\tau}\) —damping parameter. \[ \begin{array}{l}\text{U}_{1}=\text{z}+\text{b}_{1}\Phi_{\text{x}}-\text{a}_{1} \Phi_{\text{y}}-\text{z}_{01}\\ \text{U}_{2}=\text{z}+\text{b}_{2}\Phi_{\text{x}}+\text{a}_{2}\Phi_{\text{y}}- \text{z}_{02}\\ \text{U}_{3}=\text{z}-\text{b}_{1}\Phi_{\text{x}}-\text{a}_{1}\Phi_{\text{y}}- \text{z}_{03}\\ \text{U}_{4}=\text{z}-\text{b}_{2}\Phi_{\text{x}}+\text{a}_{2}\Phi_{\text{y}}- \text{z}_{04}\end{array} , \tag{17}\]  
\[ \begin{array}{l}\text{V}_{1}=\text{\ }\dot{\text{z}}+\text{b}_{1}\dot{\Phi}_{ \text{x}}-\text{a}_{1}\dot{\Phi}_{\text{y}}-\dot{\text{z}}_{01}\\ \text{V}_{2}=\dot{\text{z}}+\text{b}_{2}\dot{\Phi}_{\text{x}}+\text{a}_{2}\dot {\Phi}_{\text{y}}-\dot{\text{z}}_{02}\\ \text{V}_{3}=\dot{\text{z}}-\text{b}_{1}\dot{\Phi}_{\text{x}}-\text{a}_{1} \dot{\Phi}_{\text{y}}-\dot{\text{z}}_{03}\\ \text{V}_{4}=\dot{\text{z}}-\text{b}_{2}\dot{\Phi}_{\text{x}}+\text{a}_{2}\dot {\Phi}_{\text{y}}-\dot{\text{z}}_{04}\end{array}