## 4 Estimation of the Parameters of Mathematical Models  
By performing the experiments and using the Simulink Parameter Estimation tool in the Matlab environment, it was possible to parametrise the models which describe the oscillating movement of the spool, taking mixed friction into account. The major parameters of the spool pair were determined from the authors' own measurements and from data taken from the literature [14; 31] (Table 2). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Name** & **Value** & **Unit** \\ \hline \(\mathrm{m_{c}}\) & mass of spool + mass of lubricant + 1/3 of the mass of springs & 0.18 & (kg) \\ \hline \(\mathrm{c_{sz}}\) & equivalent stiffness of the springs & 1632 or 5846 & (N/m) \\ \hline \(\mathrm{m_{su}}\) & mass of spool & 0.172 & (kg) \\ \hline \(\mathrm{d_{t}}\) & piston diameter & \(20\times 10^{-3}\) & (m) \\ \hline h & radial play in the spool–sleeve system & \(15\times 10^{-6}\) & (m) \\ \hline l & length of the piston & \(32\times 10^{-3}\) & (m) \\ \hline \(\mathrm{\mu}\) & dynamic viscosity of the lubricant (HL68 or Azolla ZS22 at & \(612\times 10^{-4}\) or \(198\times 10^{-4}\) & (N*s/m\({}^{2}\)) \\  & a temperature of 20 \({}^{\circ}\)C) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selected parameters of the spool pair. Figure 11: Comparison of acceleration amplitude of oscillation of the directional control valve spool for each measurement series as a function of the frequency of exciting oscillation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**(Equivalent Stiffness of Springs in** & \(\mathbf{a_{k0}}\) **(m)** & \(\mathbf{d_{k0}}\) **(m)** & \(\mathbf{k_{t}}\) **(N/m)** & **Objective Function** \\ \hline
**N/m, Hydraulic Oil)** & & & & \\ \hline \hline \(\mathrm{(1632,\,HL68)}\) & 0.00356 & 0.00565 & 46.86 & \(4.1084\times 10^{7}\) \\ \hline \(\mathrm{(5846,\,HL68)}\) & \(1.9854\times 10^{-6}\) & 0.00732 & 100 & \(7.618\times 10^{7}\) \\ \hline \(\mathrm{(5846,\,Azolla\,ZS22)}\) & \(2\times 10^{-6}\) & \(8\times 10^{-6}\) & 100 & \(1.0015\times 10^{8}\) \\ \hline \(\mathrm{(1632,\,Azolla\,ZS22)}\) & \(7.1717\times 10^{-5}\) & 0.0032 & 99.985 & \(3.1135\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of estimated parameters in model 1. Using the same software tool, the parameters in model 2 were estimated and described using Equation (23) (Table 4). However, as noted above, if fluid friction is assumed to be the only type of friction in the spool pair, generating the force described by Equation (12), significant differences between the results of a simulation based on this assumption and the results of experimentation will arise. The values of estimated parameters and the corresponding values of the objective function are presented in Table 6.  
The minimization of the value of the objective function, defined as the weighted sum of squared errors, was used as a criterion to verify the correctness of the test results. The default features of the Simulink Parameter Estimation tool were used to make a mathematical model based on a deterministic approach that included the estimated parameter values [32]:  
\[\chi^{2}(a)=\sum_{i=1}^{K}\left[\frac{y_{i}-y(x_{i};a)}{\sigma_{i}^{2}}\right], \tag{33}\]  
where \(x_{i}\) and \(y_{i}\) are the coordinates of the experimental points for \(x\)--time and \(y\)--acceleration amplitude, \(\sigma_{i}\) is the standard deviation of \(y_{i}\), \(K\) is the number of points and \(y(x_{i};\,a)\) is a fitted curve. An additional criterion was the ratio between the estimated amplitudes of oscillation acceleration and the experimentally measured values, \(a_{est}/a_{pom}\), which should be as close