## 4 Estimation of the Parameters of Mathematical Models
\begin{table}
\begin{tabular}{c c c} \hline \hline
\begin{tabular}{c} **(Equivalent Stiffness of Springs in N/m,** \\ **Hydraulic Oil)** \\ \end{tabular} & **h (m)** & **Objective Function** \\ \hline \hline (1632, HL68) & \(3.1171\times 10^{-5}\) & \(2.5752\times 10^{7}\) \\ \hline (5846, HL68) & \(8.4188\times 10^{-6}\) & \(2.9586\times 10^{7}\) \\ \hline (5846, Azolla ZS22) & \(1.9375\times 10^{-5}\) & \(2.2014\times 10^{7}\) \\ \hline (1632, Azolla ZS22) & \(8.7237\times 10^{-5}\) & \(2.9153\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values of estimated parameters in model 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline \begin{tabular}{c} **(Equivalent Stiffness of Springs in** \\ **N/m, Hydraulic Oil)** \\ \end{tabular} & \
\begin{tabular}{c} **C\({}_{1}\) (\(\sqrt{s}\)/m)** \\ \end{tabular} &
\begin{tabular}{c} **C\({}_{2}\) (s/m)** \\ \end{tabular} & **Objective Function** \\ \hline (1632, HL68) & \(10.0536\) & \(9.6365\) & \(1.67\times 10^{7}\) \\ \hline (5846, HL68) & \(9.9998\) & \(9.9906\) & \(1.81\times 10^{7}\) \\ \hline (5846, Azolla ZS22) & \(9.9998\) & \(10.0046\) & \(1.72\times 10^{7}\) \\ \hline (1632, Azolla ZS22) & \(9.9684\) & \(15.9903\) & \(8.98\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of estimated parameters in model 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \begin{tabular}{c} **(Equivalent Stiffness of Springs in** \\ **N/m, Hydraulic Oil)** \\ \end{tabular} & \
\begin{tabular}{c} **H\({}_{0}\)** \\ \end{tabular} &
\begin{tabular}{c} **K\({}_{\mathbf{V}}\)** \\ \end{tabular} & **Objective Function** \\ \hline (1632, HL68) & \(0.16025\) & \(0.77652\) & \(2.4495\times 10^{7}\) \\ \hline (5846, HL68) & \(0.10885\) & \(-0.047217\) & \(7.4157\times 10^{7}\) \\ \hline (5846, Azolla ZS22) & \(0.10039\) & \(-0.0008\) & \(9.8865\times 10^{7}\) \\ \hline (1632, Azolla ZS22) & \(0.03823\) & \(0.025445\) & \(3.1519\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Values of estimated parameters in model 3.  
to 1 as possible. The values of the objective function and the ratio \(a_{est}/a_{pom}\), averaged for the frequency range of 10-100 Hz, are shown in Table 7.  
A comparison of the values of the objective function for different equivalent values of stiffness for the springs and hydraulic oils with different viscosities is shown in Figure 12.  
A comparison of the time waveforms of the acceleration of spool oscillations among the results of the experiments and the simulations after estimation is presented in Figure 13. The figures show a comparison of the estimated parameters of model 1 (13), model 2 (23), model 3 (28a) and model 4 (29) for the measurement series where equivalent spring stiffness equalled 1632 N/m and the oil used was HL68 oil at a temperature of 20 \({}^{\circ}\)C.  
The adequacy of the models for describing the oscillating motion of the spool can be assessed by comparing the amplitudes of acceleration of spool oscillations recorded during testing with the amplitudes calculated from the parametrised model. The ratio of these amplitudes should be as close to 1 as possible. Figures 14-17 show this ratio for the different values of centring spring stiffness and the different hydraulic oils used in the testing. According to the results, after parametrisation and estimation of the constant coefficients C1 and C2, model 2 (23) produced a considerably more accurate description of the oscillation of the spool in the directional control valve sleeve than model 1 (13), model 3 (28a) or model 4 (29) across the entire range of frequencies tested. As seen in Table 7 and Figure 9, for model 2 the averaged value of the objective function was the smallest and the averaged value of the \(a_{est}/a_{pom}\) ratio was closest to 1. Likewise, the graphs in Figures 14-17 indicate that the ratio between the amplitudes of spool oscillation acceleration from the models after estimating the selected parameters and those measured during the tests was closest to 1 when using model 2.