# Improving Performance of a Resonant String-Based Pulsation Attenuator in Hydraulic Systems
## 4 Optimization of RSHPA
### Geometry Optimization
#### 4.1.2 Specific Frequency Optimization  
To enhance the performance of RSHPA at specific frequency, the natural frequency of the strings should be tuned to the corresponding frequency. Here, 220 Hz is selected as the optimized goal. The search ranges, original values and optimized values calculated using the APSO algorithm are given in Table 5 where \(T_{i}\) and \(\Theta_{i}\) denotes the applied tension and diameter of the \(i\)-th string. The symbol \([\sigma_{SAFE}]\) denotes the allowable stress of the strings. It can be observed that APSO is still the most computationally efficient between three algorithms, and reduces the running time by 22.2% and 37.1% compared to that of CPSO and IPSO algorithms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & _I (m)_ & _D (m)_ & \(g\) & \(\Theta_{i}(\mathbf{mm})\) & \(\rho_{0}(\mathbf{kg/m^{3}})\) & \(\mu_{0}(\mathbf{Pa\text{-}s})\) \\ \hline Search range & [0.2, 0.35] & [0.05, 0.15] & [1, 30] & [0.1, 0.4] & 890 & 0.034 \\ \hline Original value & 0.25 & 0.08 & 5 & 0.4 & 890 & 0.0344 \\ \hline Optimized value & 0.35 & 0.15 & & & & \\ \hline Running time & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter information used in Figure 7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & _I (m)_ & _D (m)_ & \(g\) & \(\Theta_{i}(\mathbf{mm})\) & \(T_{i}(\mathbf{N})\) \\ \hline Search range & [0.2, 0.35] & [0.05, 0.15] & [1, 30] & [0.1, 0.4] & [1, 50] \\ \hline Constraint condition & & \multicolumn{4}{c}{\(T_{i}/[\pi(\Theta_{i}/2)^{2}]<[\sigma_{SAFE}]\)} \\ \hline Original value & 0.25 & 0.088 & 1 & 0.2 & 1.8 \\ \hline Optimized value & 0.35 & 0.15 & 10 & 0.4 & 7.7 \\ \hline Running time & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter information used in Figure 8.  
The natural resonant frequency of strings is formulized as [22]:  
\[f_{in}=\frac{n}{2l}\sqrt{\frac{T_{i}}{\delta}} \tag{26}\]  
where \(\delta\) denotes the linear density of the strings. In this paper, \(\delta\) is calculated by \(\delta=\rho\pi\Theta^{2}/4\) where \(\rho\) denotes the bulk density and it is determined by the material of the commercial string. In this paper, \(\rho\) is 8110 kg/m\({}^{3}\). Parameter \(f_{in}\) denotes the \(n\)-th order frequency response of the \(i\)-th string. In general, \(f_{in}\) is equal to the pulsation frequency of the noise source, which is the axial piston pump in this paper. The pulsation frequency is calculated by [10]:  
\[f_{pul}=\frac{n_{p}Z}{60} \tag{27}\]  
where \(n_{p}\) denotes the rotational speed of the motor and \(Z\) denotes the piston number of the pump. It is assumed that all strings are consistent. The objective function is given as follows:  
\[OF_{3}(l,D,g,\Theta_{i},T_{i})=-IL_{220}. \tag{28}\]  
Figure 8 demonstrates the comparison results using the original and optimized values in the specific frequency optimization case. It is observed that the attenuation performance at the desired 220 Hz is enhanced obviously, which corresponds to the second-order response of the strings. Additionally, the other order responses of the strings are also improved, indicating that the enhancement is useful for all responses even if we only optimize one of them. Note that the effective frequency band for enhancement of the desired frequency is narrow. Thus, the fabrication error and parameter drift in a practical device would easily make the characteristics of the RSHPA deviate from the design objective. Two methods can be used to solve the above problem. One takes advantage of the peaks generated by the expansion chamber, which has a wide effective band, such as the effect at 490 Hz in Figure 8. However, the peaks caused by the strings and the expansion chamber are difficult to both maximize at any specified frequency. It can be observed in Figure 9 that the curve in magenta, which is obtained by tuning the peaks generated by the expansion chamber (without tuning the parameters of strings), has the same peak frequency as the other cases.