# Improving Performance of a Resonant String-Based Pulsation Attenuator in Hydraulic Systems
## 4 Optimization of RSHPA
### Installation Position Optimization  
This section aims to find the most suitable installation location of the RSHPA in a straight pipe line system, so as to achieve the optimal attenuation effect of the RSHPA. The optimization goal is still 220 Hz and the geometrical and physical parameters in the optimization process remain the same as before. The only variable is the position of the RSHPA in the straight pipeline system.  
As mentioned, the effective frequency band can be extended by deploying multiple strings with uniformly-spaced resonant frequency. In addition, the resonant frequency of the RSHPA can also be identified by experiments. The identification steps can be summarized as: (1) Tune the rotational speed of the driving motor of the pump to the designed points. The pulsation frequency is calculated by Equation (26). (2) Transform the pressure data at the outlet of the RSHPA into the frequency domain by fast Fourier transform (FFT) and observe the power spectrum in real-time. (3) Fine tune the rotational  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & \(\mathbf{l_{serial}}\) (\(\mathbf{\mathrm{m}}\)) & \(\mathbf{l_{parallel}}\) (\(\mathbf{\mathrm{m}}\)) & \(\mathbf{D}\) (\(\mathbf{\mathrm{m}}\)) & \(\mathbf{g}\) & \(N_{RS}\) & \(\mathbf{\Theta}\) (\(\mathbf{\mathrm{mm}}\)) & \(\mathbf{T_{i}}\)(\(\mathbf{\mathrm{N}}\)) \\ \hline Search range & [0.2, 0.35] & [0.2, 0.5] & [0.08, 0.15] & [1, 30] & [2, 5] & [0.1, 0.4] & [1, 50] \\ \hline Constraint condition & & \(T_{i}/[\pi(\Theta_{i}/2)^{2}]<[\sigma_{SAFE}]\), \(N_{RS}D\leq 0.15\) (for the parallel case) & & & & & \\ \hline Original value & 0.2 & 0.2 & 0.08 & 5 & 2 & 0.2 & 50 \\ \hline Serial optimization & 0.32 & & 0.15 & 1 & 5 & 0.1 & 0.52 \\ \hline Parallel optimization & & 0.35 & 0.15 & 1 & 5 & 0.4 & 7.7 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameter information for multiple-chamber in specific frequency optimization.  
Figure 14: Specific frequency optimization for serial and parallel combinations of RSHPA (220 Hz).  
speed of the motor until the power spectrum captures an obvious decrease at the peaks. (4) Inversely estimate the tension on the string and finely tune the tension until the pulsation can be attenuated at the designed operating frequency.  
Figure 15 shows the comparison results of the theoretical curve and the measured data with the uncertainty represented by the error bar. The average deviation in attenuation strength is of 22.6 dB. The horizontal coordinate denotes the distance between the pump and the RSHPA. Through calculation, it is found that when the RSHPA is installed at the inlet, the middle of the pipeline or near the load, the insertion loss of the RSHPA at 220 Hz can both reach the maximum value. In the experiment, some positions were selected for insertion loss measuring and the measured data are given with the error bar. The test results show that the measured values are basically consistent with the theoretical values.  
In order to further explore the universality of RSHPA layout, 150 Hz and 390 Hz are selected as the optimization objectives. Figure 16 shows the influence of RSHPA layout on insertion loss at 150 Hz and 390 Hz. The uncertainty in attenuation strength for them is 17.2 dB and 21.1 dB, respectively. It can be seen that although RSHPA has extreme value points in the middle of the pipeline, not all of them are maxima or even minima. Therefore, optimizing IL at different frequency would have different optimal position and each case should be analyzed by the numerical model using the optimization algorithm.  
Figure 16: Optimization of installation position of RSHPA (150 Hz and 390 Hz).  
Figure 15: Optimization of installation position of RSHPA (220 Hz).