# Numerical and Experimental Analysis of Pressure Pulsation Attenuator Based on Helmholtz Resonator
## 3 Theoretical Model
### Model of the Hydraulic Circuit with the HTPA  
The HTPA is connected to the hydraulic circuit, as shown in Figure 6.  
Where \(p_{i}\), \(p_{d}\), and \(p_{o}\) are pump outlet pressure, load front-end pressure, and load back-end pressure; \(q_{i}\), \(q_{d}\), and \(q_{o}\) are pump outlet flow rate, load front-end flow rate and load back-end flow rate. In this paper, the method of insertion loss \(IL\) is used to evaluate the effect of pressure pulsation suppression. The higher the value of insertion loss at the corresponding frequency, the better the effect of pressure pulsation suppression at that frequency. Insertion loss is defined as the ratio of the pressure pulsation \(p_{d}\) at the front end of the load in a hydraulic piping system without an HTPA to the pressure pulsation \(p^{\prime}_{d}\) at the front end of the load with an HTPA:  
\[IL=20 log\left|\frac{p^{\prime}_{d}}{p_{d}}\right| \tag{12}\]  
The insertion loss visually reflects the effect of pressure pulsation suppression at the load end of the pipeline before and after the installation of the HTPA, while considering the effect of the HTPA and the whole system on the pressure pulsation suppression effect of the pipeline. Considering that pipeline A and pipeline B are long, the distribution parameter method is adopted to establish the transfer matrix \(A\) and \(B\), as shown in Equation (13) [27]:  
\[W= \begin{pmatrix}W_{11}&W_{12}\\ W_{21}&W_{22}\end{pmatrix} = \begin{bmatrix}ch[L\Gamma(s)]&Z_{A}(s)sh[L\Gamma(s)] \\ sh[L\Gamma(s)]/Z(s)&ch[L\Gamma(s)]\end{bmatrix}  \tag{13}\]  
where \(\Gamma(s)\) is the propagation constant, \(\Gamma(s)=\sqrt{s(s+R_{p})}/c\); \(c\) indicates the speed of sound, \(c=\sqrt{K_{e}/\rho}.Z(s)\) is the characteristic impedance of the pipeline, \(Z(s)=\rho c^{2}\)\(\Gamma(s)/A_{p}s\). \(A_{p}\) is the cross-sectional area of the pipeline. \(L\) is the length of the pipeline. Figure 5: Thin-walled small-bore throttle valve load. Figure 6: The hydraulic circuit with the HTPA. Replace the HTPA with an equal length of pipeline C into the circuit. Similarly, the transfer matrix \(C\) of pipeline C is built according to the Equation (13). The relative friction coefficient in the above takes the value \(R_{\texttt{P}}\) using a frequency-dependent model, introducing the coefficient of determination \(\chi\):  
\[\chi=r\sqrt{\frac{\omega}{\mu}} \tag{14}\]  
The corresponding relative friction coefficient values \(R_{\texttt{P}}\) are taken according to different \(\chi\) values [33]:  
\[\left\{ \begin{array}{ll}R_{\texttt{P}}=8\mu/r^{2}&\chi<1.3\\ R_{\texttt{P}}=\omega\psi\sqrt{\Psi^{2}+2}&1.3<\chi<38\\ R_{\texttt{P}}=\omega\psi\sqrt{2\mu\omega}/r&38<\chi<68\\ R_{\texttt{P}}=0&\chi>68\end{array} \right. \tag{15}\]  
where \(\psi=\chi/\left(\chi^{2}-\sqrt{2}\chi+1\right)\), \(r\) is the radius of the pipeline, \(\omega\) is the angular frequency of the pressure pulsation of the fluid in the pipeline and \(\mu\) is the kinematic viscosity of the fluid. The load pressure is set by a throttling valve, which is simplified to a straight pipeline of length \(L_{D}\) and throttling load model. The transfer matrix \(D\) of the pipeline D is built according to Equation (13). The relationship between \(Q_{d}(s)\) and \(P_{d}(s)\) can be obtained:  
\[ \begin{pmatrix}P_{d}(s)\\ Q_{d}(s)\end{pmatrix} = \begin{pmatrix}D_{11}&D_{12}\\ D_{21}&D_{22}\end{pmatrix}\begin{pmatrix}P_{o}(s)\\ Q_{o}(s)\end{pmatrix}  \tag{16}\]  
\[\frac{Q_{d}(s)}{P_{d}(s)}=\frac{D_{21}P_{o}(s)+D_{22}Q_{o}(s)}{D_{11}P_{o}(s)+ D_{12}Q_{o}(s)}=\frac{D_{21}Z_{R}(s)+D_{22}}{D_{11}Z_{R}(s)+D_{12}} \tag{17}\]  
Substitute Equation (11) into Equation (18):  
\[\frac{Q_{d}(s)}{P_{d}(s)}=\frac{2\Delta P_{d}D_{21}+Q_{d}D_{22}}{2\Delta P_{d} D_{11}+Q_{d}D_{12}} \tag{18}\]  
The dynamic characteristic matrix of the input and load-side output of the system after installation of the HTPA is \(U=A\times C\times B\):  
\[ \begin{pmatrix}P_{i}(s)\\ Q_{i}(s)\end{pmatrix}  = \begin{pmatrix}U_{11}&U_{12}\\ U_{21}&U_{22}\end{pmatrix}\begin{pmatrix}P_{d}(s)\\ Q_{d}(s)\end{pmatrix}