## 3 Suspension Design and Modelling
### Modelling of the Hydroneumatic Suspension
#### 3.2.4 Lugre Friction Model  
For the hydropneumatic suspension system, the friction force generated by the seal and guiding elements significantly impacts the vibration calculation. Many researchers have studied Coulomb friction characteristic in cylinders or actuators [24], but such as Karnopp and LuGre model [25; 26] have been proved to be more effective on many occasions. [PERSON] and [PERSON] [26] studied the dynamic behaviors of friction using a hydraulic cylinder. They proposed a modification to the LuGre model to improve the simulation accuracy. The LuGre friction model takes the surface contact as the bristles' elastic deformation under the micro-structure as shown in Figure 5. It can provide a comprehensive description of the static and dynamic characteristics of the friction between the piston and the cylinder. Whether the friction model parameters can be effectively and quickly identified from a large amount of experimental data is of great significance for improving the accuracy of the suspension model. This paper chooses the genetic algorithm (GA) to identify the friction model parameters because it has a comprehensive search range and is easy to obtain the optimal global solution.  
The LuGre model is described by:  
\[\left\{\begin{array}{ll}F_{f}=\sigma_{0}z+\sigma_{1}\frac{dz}{dt}+\sigma_{2 }v\\ \frac{dz}{dt}=v-\frac{\sigma_{0}z}{\dot{g}(v)}|v|\\ g(v)=F_{c}+(F_{s}-F_{c})e^{-\left(\frac{\pi}{\hbar}\right)^{2}}\end{array}\right. \tag{10}\]  
where \(\sigma_{0}\) is the stiffness for a spring-like behavior for small displacements, \(\sigma_{1}\) and \(\sigma_{2}\) are the damping coefficient that relates to pre-sliding and viscous friction states, \(z\) is the average bristle deflection, \(g(v)\) is [PERSON]'s curve in the function of sliding velocity, \(v_{s}\) is the Stribeck velocity, \(F_{s}\) and \(F_{c}\) are the static friction force and Coulomb force.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Hydraulic Variables** & **Symbol** & **Unit** & **Value** \\ \hline Pre-charge gas pressure & \(P_{0}\) & Pa & \(7\times 10^{5}\) \\ Pre-charge gas volume & \(V_{0}\) & m\({}^{3}\) & \(2.64\times 10^{-4}\) \\ Length of the damping pipe & \(l\) & m & 0.5 \\ Diameter of the damping pipe & \(d\) & m & \(7.5\times 10^{-3}\) \\ Sectional area of the damping pipe & \(A_{C}\) & m\({}^{2}\) & \(4.4\times 10^{-5}\) \\ Density of the oil & \(\rho\) & Kg/m\({}^{3}\) & \(0.87\times 10^{3}\) \\ Gas polytrophic exponent & \(n\) & & 1.3 \\ Kinematic viscosity of the oil & \(\mu\) & Pa \(\cdot\) s & 10 \\ \hline \end{tabular}
\end{table}
Table 1: Initial value of hydropneumatic variables.  
Figure 5: LuGre friction model.  
LuGre friction model includes four static parameters and two dynamic parameters. The GA program shown in Figure 6 indicates the whole process of parameter identification. The first step is to identify the static parameters \(F_{s}\), \(F_{c}\), \(v_{s}\) and \(\sigma_{2}\). In the static case, it is assumed that the bristles' deformation has reached the maximum and that is to say the \(\frac{dz}{dt}=0\). In this step, the friction force can be expressed as:  
\[F=\left(F_{c}+(F_{s}-F_{c})e^{-\left(\frac{v}{\sigma_{2}}\right)^{2}}\right) \text{sign}(v)+\sigma_{2}v \tag{11}\]  
The dynamic parameters identification is based on the static parameters result. Above four identified static parameters are brought into the Lugre friction model, and then the dynamic parameters \(\sigma_{0}\) and \(\sigma_{1}\) are also identified by the GA program.  
\[F=\sigma_{0}z+\sigma_{1}v-\frac{\sigma_{0}\sigma_{1}z|v|}{\left(F_{c}+(F_{s}- F_{c})e^{-\left(\frac{v}{\sigma_{2}}\right)^{2}}\right)}+\sigma_{2}v \tag{12}\]  
In the Matlab GA program, the maximum generation is 200, the population size is 200, the reproduction ratio is 0.7, and the mutation probability is 0.03. The identified values of these parameters are shown in Table 2.