# Evaluation of Dynamics of a Freight Wagon Model with Viscous Damping
## 3 Vibration Damping Models
### Viscous Friction--The Maxwell Model  
The Maxwell model is, along with the Kelvin-Voigt model, the basic mathematical description of the viscoelastic behavior of a solid material. It consists of a spring and a damper connected in series [22]. In this model, the spring represents the Young's modulus and the damper the viscosity of the material. It allows for modeling stress relaxation. One can divide the total strain into one for the spring (\(\epsilon_{1}\)) and one for the dashpot (\(\epsilon_{2}\)) [23]. The standard form of the Maxwell model's constitutive equation is (2):  
\[\sigma+\frac{\eta}{E}\dot{\sigma}=\eta\dot{\epsilon}, \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Body mass—empty (laden) & 7500 kg (40,770 kg) \\ Body moment of inertia \(l_{xx}\)—empty (laden) & 9600 kg\(\cdot\)m\({}^{2}\) (35,000 kg\(\cdot\)m\({}^{2}\)) \\ Body moment of inertia \(l_{yy}\)—empty (laden) & 85,500 kg\(\cdot\)m\({}^{2}\) (220,000 kg\(\cdot\)m\({}^{2}\)) \\ Body moment of inertia \(l_{zz}\)—empty (laden) & 85,500 kg\(\cdot\)m\({}^{2}\) (220,000 kg\(\cdot\)m\({}^{2}\)) \\ Wheelset mass & 1925 kg \\ Wheelset moment of inertia \(l_{xx}\) & 950 kg\(\cdot\)m\({}^{2}\) \\ Wheelset moment of inertia \(l_{yy}\) & 150 kg\(\cdot\)m\({}^{2}\) \\ Wheelset moment of inertia \(l_{zz}\) & 950 kg\(\cdot\)m\({}^{2}\) \\ Wheelset base & 5.2 m \\ Wheel radius & 0.5 m \\ Spring longitudinal stiffness \(k_{x}\) & 12 \(\times\) 10\({}^{6}\) N/m \\ Spring lateral stiffness \(k_{y}\) & 0.64 \(\times\) 10\({}^{6}\) N/m \\ Spring vertical stiffness \(k_{z}\) & 1.8 \(\times\) 10\({}^{6}\) N/m \\ Vertical damping (one viscous damper) \(c\) & 100 \(\times\) 10\({}^{3}\) Ns/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the two-axle freight wagon model. where \(\sigma\)--stress, \(\dot{\sigma}\)--stress rate, \(\epsilon\)--strain, \(\dot{\epsilon}\)--strain rate, \(\eta\)--viscosity, \(E\)--Young modulus. The use of a spring and a damper series configuration in the Maxwell model extends beyond the analysis of the behavior of viscoelastic materials. As with the Kelvin-Voigt model used in vehicle dynamics generally to model two separate suspension elements, namely a spring and a damper working in parallel and simultaneously, the Maxwell's series configuration is employed to represent the operation of a hydraulic damper only. In this model, a series spring relates to the mounting elements and also takes into account the effect of oil compressibility [24], so this model allows for a more accurate representation of the operation of an actual hydraulic damper. Hence, the EN 13802 standard applicable to railway suspension hydraulic dampers [25] requires providing parameters of the Maxwell model: dynamic stiffness (3) and dynamic damping rate (4), both depending on the phase shift, which is typically greater than in the Kelvin-Voigt model. \[k=\frac{F_{0}}{d_{0}}\sqrt{1+\tan^{2}\Phi}, \tag{3}\]  
\[c=\frac{k}{2\pi f\tan\Phi}, \tag{4}\]  
where \(F_{0}\)--damper force amplitude at sinusoidal motion, \(d_{0}\)--damper displacement amplitude at sinusoidal motion, \(\Phi\)--force-displacement phase shift at sinusoidal motion, \(f\)--excitation frequency. A scheme of the Maxwell model, relating to the suspension damper, is shown in Figure 7.  
The equation of motion of a dashpot rod of mass \(m\), stiffness \(k\), and damping constant \(c\) is expressed in the following form [26]:  
\[m\ddot{x}_{2}=-c\dot{x}_{2}-k(x_{2}-x_{1}), \tag{5}\]  
Neglecting mass of the piston and rearranging:  
\[\frac{c}{k}\ddot{x}_{2}+x_{2}=x_{1}, \tag{6}\]  
Assuming harmonic input \(x_{1}\) of amplitude A, the solution of the equation for a zero initial condition is:  
\[x_{2}=\frac{k^{2}}{c^{2}\omega^{2}+k^{2}}A\sin(\omega t)-\frac{\omega ck}{c^{2} \omega^{2}+k^{2}}A\cos(\omega t), \tag{7}\]  
The damping force in [PERSON]'s model can be calculated by differentiating expression (7) and multiplying it by the damping constant \(c\).