# Improving Performance of a Resonant String-Based Pulsation Attenuator in Hydraulic Systems
## 4 Optimization of RSHPA
### Geometry Optimization
#### 4.1.1 Wide-Band Optimization  
As mentioned in the literature [2], the frequency of most hydraulic pulsation is less than 1 kHz; thus, this frequency band is selected as the optimization objective in this paper. To optimize the wide-band performance, we construct two objective functions using the sum of uniformly-spaced frequency attenuation effects, and they are formulized as:  
\[OF_{1}(l,D)=-\sum_{i=1}^{1000}IL_{i} \tag{24}\]  
\[OF_{2}(l,D)=-\sum_{i=1}^{1000}\left(k_{1}\frac{IL_{i}-IL_{\min}}{IL_{\max}-IL_{ \min}}+k_{2}\frac{PR_{\max}-PR_{i}}{PR_{\max}-PR_{\min}}\right)\,,\,k_{1}+k_{2}=1 \tag{25}\]  
where \(IL_{i}\) and \(PR_{i}\) denote the \(IL\) and \(PR\) values when the frequency is \(i\). The subscripts max and min denote the corresponding maximum and minimum values during iteration. It is notable that the attenuation effects judged by \(IL\) and \(PR\) are normalized in Equation (25), in order to adjust their contributions in the objective function via two constants \(k_{1}\) and \(k_{2}\). In addition, two objective functions are both minimization problems.  
Figure 7 demonstrates the comparison result using \(OF_{1}\) and the related parameter information is given in Table 4 where \(g\) denotes the total number of strings. The original values given in the table are the same as the parameter values of the existing RSHPA in our lab. It is observed that the optimized \(IL\) is nearly larger than the original results during the entire given frequency band. This result agrees with the optimization objective. Additionally, although the objective function only contains the items of \(IL\), the performance of \(PR\) is also improved. Thus, improving the performance of \(IL\) for wide band conditions can be a simple way to improve the performance of \(PR\) in the same condition. One should note that the optimized values of the length and diameter of the RSHPA reach the upper bound of their search ranges, indicating that the larger volume of the RSHPA is, the better the performance for wide band conditions. However, the geometry of RSHPA will be definitely restricted in a practical fluid power system due to the factors of limited installation space, cost, etc. The other phenomenon to note is that the attenuation performance from the string-fluid resonators (see narrow-band peaks) is somewhat reduced and even becomes worse after optimization. This result reveals that whether the string-fluid resonator can well work not only depends on its performance but is also affected by the expansion chamber. Therefore, we can reasonably infer that if a hydraulic system needs to suppress the noise at specific frequency using RSHPA, the wide band optimization is not suitable.  
Figure 7: Attenuation performance of single-chamber RSHPA in \(OF_{1}\).  
The last row of Table 4 provides the running time of solving the optimization problem using three PSO-based algorithms. Three algorithms all finally converge to a close result. The proposed APSO consumes the least time and reduces time by 23.5% and 33.4% compared to CPSO and IPSO, respectively. This numerical result validates the superiority of APSO in computational efficiency. To validate whether the result is globally optimal, we use the traversal search method to calculate again and obtain the same parameters. This result indicates that the global search ability of the APSO algorithm is not weakened in the optimization problem given in this paper. In the rest of this study, the optimized results are all examined by the traversal search method.  
As introduced in Section 2.2, a larger value of _IL_ and a smaller value of _PR_ are beneficial to a hydraulic silencer, especially when _IL_ is larger than 0 and _PR_ is smaller than 0. According to our repeated tests, the optimized values using different combinations of \(k_{1}\)(0.1, 0.3, 0.5, 0.7 and 0.9) and \(k_{2}\) are all close to the results computed using \(OF_{1}\). Thus, it may be concluded that the objective function only containing _IL_ items or both _IL_ and _PR_ items will lead to a same result.