# Improving Performance of a Resonant String-Based Pulsation Attenuator in Hydraulic Systems
## 2 Modelling of the Resonant String-Based Hydraulic Pulsation Attenuator (RSHPA)
### Performance Evaluation Method  
The hydraulic system shown in Figure 2 can be expressed by:  
\[\left[\begin{array}{c}P_{1}(s)\\ Q_{1}(s)\end{array}\right]=\mathbf{A_{1}}\mathbf{A_{2}}\mathbf{A_{3}}\left[ \begin{array}{c}P_{4}(s)\\ Q_{4}(s)\end{array}\right]=\left[\begin{array}{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right]\left[\begin{array}{c}P_{4}(s)\\ Q_{4}(s)\end{array}\right] \tag{10}\]  
where \(a_{11}\)-\(a_{22}\) denote four elements of the total transfer matrix of the system. If the RSHPA is replaced by a pipeline, the system can be expressed by:  
\[\left[\begin{array}{c}P_{1}^{\prime}(s)\\ Q_{1}^{\prime}(s)\end{array}\right]=\mathbf{A_{1}}\mathbf{A_{22}^{\prime}} \mathbf{A_{3}}\left[\begin{array}{c}P_{4}^{\prime}(s)\\ Q_{4}^{\prime}(s)\end{array}\right]=\left[\begin{array}{cc}a_{11}^{\prime}& a_{12}^{\prime}\\ a_{21}^{\prime}&a_{22}^{\prime}\end{array}\right]\left[\begin{array}{c}P_{4}^ {\prime}(s)\\ Q_{4}^{\prime}(s)\end{array}\right] \tag{11}\]  
where the symbols with \(<\)\(>\) denote the parameters expressed after replacing RSHPA with a pipeline.  
Insertion loss (IL) is a key evaluation index to judge the variation conditions at the input of the system load, and it is defined as [16]:  
\[H_{IL}(s)=20\log\left|\frac{P_{4}\prime(s)}{P_{4}(s)}\right|. \tag{12}\]  
It can be observed that the larger the value of IL is, the better the suppression performance of the RSHPA is. In general, the value of IL is expected to be positive, which means the pulsation to the load is suppressed compared to that without RSHPA. It is notable that IL is not only determined by the performance of the RSHPA, but also affected by the other hydraulic components and pipelines in the hydraulic system. Thus, IL can reflect the performance of RSHPA in a practical fluid power system.  
Figure 3: Equivalent circuit of the string array in the RSHPA.  
The frequency characteristics of the pressure ratio (PR) of load to source can be used to evaluate whether the pulsation from the source is amplified at the load position. PR is formulized as:  
\[H_{PR}(s)=20\log\Biggl{|}\frac{P_{4}(s)}{P_{1}(s)}\Biggr{|}. \tag{13}\]  
In contrast to IL, a negative value of PR indicates that the source pulsation is suppressed at the load. The smaller the PR is, the better the source pulsation is suppressed at the load. The frequency characteristics of input flow is determined by the pump. Therefore, the input flow before and after installing the RSHPA stays the same as:  
\[Q_{1}(s)=Q_{1}^{\prime}(s). \tag{14}\]  
The frequency characteristics of load (throttle valve) is assumed to be unchanged, thus [17]:  
\[\frac{P_{4}(s)}{Q_{4}(s)}=\frac{P_{4}^{\prime}(s)}{Q_{4}^{\prime}(s)}=\frac{2 P_{0}}{Q_{0}} \tag{15}\]  
where \(P_{0}\) and \(Q_{0}\) denote the average pressure and flow at the inlet of the load. Combining Equations (10)-(15), IL and PR can be converted to the following form:  
\[H_{IL}(s)=20\lg\Biggl{|}\frac{2P_{0}a_{21}+a_{22}Q_{0}}{2P_{0}a_{21}^{\prime} +a_{22}^{\prime}Q_{0}}\Biggr{|} \tag{16}\]  
\[H_{PR}(s)=20\log\Biggl{|}\frac{2P_{0}}{2P_{0}a_{11}+a_{12}Q_{0}}\[PERSON]{|}. \tag{17}\]  
In the later section, Equations (17) and (18) are used to optimize the performance of RSHPA.