## 3 Model Validation
### Model Validation  
The experimental results at 4000 rpm are presented in Figure 5a,b. The peak-to-peak value of the swash plate oscillation are 0.9\({}^{\circ}\), 0.7\({}^{\circ}\), and 0.6\({}^{\circ}\) at the three load flow rates, 1.5 L/min, 4.5 L/min, and 11.0 L/min, respectively. The average swash plate angles of the three sets of results are 2.7\({}^{\circ}\), 4.6\({}^{\circ}\), and 12.3\({}^{\circ}\), respectively. It can be observed, through the structure of the variable displacement mechanism, that a higher load flow rate leads to a larger average angle of the swash plate, which also leads to a smaller average volume of the actuator piston chamber. To evaluate the variable displacement mechanism dynamic response, the oscillation ratio is defined as the ratio of angle oscillation amplitude per pressure pulsation amplitude, as in Equation (34). The fast Fourier transform (FFT) results of the oscillation ratio at 4000 rpm are presented in Figure 5c for comparison with the simulation results. The ratios of the three sets of experiments are approximately 7.5, 5.5, and 4.0, respectively. The result can be interpreted as confirmation that the larger the volume of the actuator piston chamber is, the larger the resulting oscillation ratio of the variable displacement mechanism. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Pump: & \\ Discharge pressure, \(P_{o}\) (MPa) & 21 \\ Rated speed range (r/min) & 3500–4500 \\ Pumping piston diameter, \(d_{p}\) (mm) & 7.7 \\ Piston rotation radius, \(r_{pc}\) (mm) & 27 \\ Outlet chamber, \(V_{0}\) (m\({}^{3}\)) & \(3\times 10^{-5}\) \\ Chamber before valve spool, \(V_{1}\) (m\({}^{3}\)) & \(1.9\times 10^{-7}\) \\ Chamber behind valve spool, \(V_{2}\) (m\({}^{3}\)) & \(3.9\times 10^{-8}\) \\ Actuator piston chamber, \(V_{3}\) (m\({}^{3}\)) & \(2\times 10^{-7}\) \\ Actuator piston diameter, \(d_{a}\) (mm) & 8 \\ Valve diameter, \(d_{v}\) (mm) & 2.4 \\ Valve spring stiffness, \(k_{v}\) (N mm\({}^{-1}\)) & 50 \\ Valve spool mass, \(m_{v}\) (g) & 0.9 \\ Bias spring stiffness, \(k_{v}\) (N-mm\({}^{-1}\)) & 22 \\ Swash plate moment of inertia, \(J_{s}\) (Kg-m\({}^{2}\)) & \(4.2\times 10^{-5}\) \\ Maximum swash plate angle, \(\varphi_{\max}\) (\({}^{\circ}\)) & 15.3 \\ \hline Test rig: & \\ Inlet pressure (MPa) & 0.5 \\ Oil density, \(\rho\) (Kg-m\({}^{-3}\)) & 833 \\ Oil bulk modulus, \(K\) (MPa) & 1000 \\ Oil temperature (\({}^{<}\)C) & 20–30 \\ Oil Dynamic viscosity, \(\mu\) (Pa-s) & \(8.33\times 10^{-3}\) \\ Sampling frequency (Hz) & 10240 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of test rig. Figure 4: Measurement schematic of the swash plate angle. \[\varepsilon=\frac{\Phi_{\varphi}-\min\bigl{(}\Phi_{\varphi}\bigr{)}}{\max\bigl{(} \Phi_{\varphi}\bigr{)}-\min\bigl{(}\Phi_{\varphi}\bigr{)}}\Bigg{/}\frac{\Phi_{ pt}-\min\bigl{(}\Phi_{pt}\bigr{)}}{\max\bigl{(}\Phi_{pt}\bigr{)}-\min\bigl{(}\Phi_{ pt}\bigr{)}} \tag{34}\]  
where \(\Phi_{\varphi}\) is the amplitude of the swash plate oscillation angle, \(\Phi_{pt}\) is the amplitude of the outlet chamber pressure pulsation, and \(\varepsilon\) is the oscillation ratio. As the rated working speed range of this high-speed pump is from 3500 rpm to 4500 rpm, the simulation and experiments are carried out in the frequency interval from 500 Hz to 700 Hz, which is equivalent to 3340 rpm to 4670 rpm. Figure 6a shows the comparison of the improved model results and the common model results at the flow rate 1.5 L/min. The common model is based on previous research on the piston pump system [20; 21; 22; 23; 24; 25; 26; 27]. There are significant differences between the two results. To further validate the accuracy of the improved model, the results of three sets of experiments and simulations are shown in Figure 6b. The load flow rates of the three sets are 1.5 L/min, 4.5 L/min, and 11.0 L/min, and the average volume of the actuator piston chamber of the three series of experiments is less than 5 \(\times\) 10\({}^{-7}\) m\({}^{3}\).