# Improving Performance of a Resonant String-Based Pulsation Attenuator in Hydraulic Systems
## 2 Modelling of the Resonant String-Based Hydraulic Pulsation Attenuator (RSHPA)  
### Modelling  
Figure 1 demonstrates a 3-dimensional model of the RSHPA [10]. The structure is composed of an expansion chamber and multiple parallel strings. The strings can be resonant with the fluid pulsation whose frequency is equal to its natural resonant frequency. Then the strings absorb the pulsation energy and transform it to the vibration, which finally becomes the friction heat energy. The expansion chamber suppresses the pulsation by neutralizing the transmitting and reflecting waves via the abrupt cross-sectional area. Thus, RSHPA has potential to work for both wide band and specified frequency conditions. In this paper, the RSHPA is installed in a typical hydraulic system, as shown in Figure 2 where the power source and load are a pump and a throttle valve, respectively. The uppercase and lowercase letters of the pressure and flow represent the parameters used at the frequency and time domains, respectively. The transfer matrix of the upstream pipeline, RSHPA and the downstream pipeline are defined as \(\mathbf{A}_{1}\), \(\mathbf{A}_{2}\) and \(\mathbf{A}_{3}\), respectively. The transfer matrix of pipelines can be obtained by taking Laplace transform of their dynamic functions. The detailed derivation can refer to the literature [10]. Here, we provide the transfer matrix of pipelines directly as follows. Figure 1: 3D model of the RSHPA. Figure 2: Schematic diagram of a hydraulic system with the resonant string-based hydraulic pulsation attenuator (RSHPA). \[\mathbf{A_{1}}=\mathbf{A_{3}}=\left[ \begin{array}{cc}\mathrm{ch}\Gamma(s)&Z_{ \mathrm{c}}(s)\mathrm{sh}\Gamma(s)\\ \frac{\mathrm{sh}\Gamma(s)}{Z_{\mathrm{c}}(s)}&\mathrm{ch}\Gamma(s)\end{array} \right] \tag{1}\]  
where \(\mathrm{sh}(\cdot)\) and \(\mathrm{ch}(\cdot)\) denote the hyperbolic sine and cosine functions, \(s\) denotes the complex variable. To analyze the characteristics of RSHPA at the frequency domain, the complex number \(s\) can be substituted by \(s=j\omega\) where \(\omega\) denotes the angular frequency of the system. The angular frequency can be calculated by \(\omega=2\pi f\) where \(f\) denotes the system frequency. \(\Gamma(s)\) denotes the propagation operator, and \(Z_{\mathrm{c}}(s)\) denotes the characteristic impedance of the pipeline. \(\Gamma(s)\) and \(Z_{\mathrm{c}}(s)\) are given as [11]:  
\[\Gamma(s)=\frac{sL_{i}}{c_{0}}\sqrt{1+\frac{8\pi v_{0}}{A_{p}s}} \tag{2}\]  
\[Z_{\mathrm{c}}(s)=\frac{\rho_{0}c_{0}}{A_{p}}\sqrt{1+\frac{8\pi v_{0}}{A_{p}s }}. \tag{3}\]  
where \(L_{i}\)\((i=1,2)\) denotes the length of the pipeline, \(v_{0}\) denotes the kinematic viscosity of the fluid, \(A_{p}=\pi d^{2}/4\) denotes the sectional area of the pipeline, \(d\) denotes the inner diameter of the pipeline, \(\rho_{0}\) denotes the density of the fluid, and \(c_{0}\) denotes the speed of wave propagation in fluid. Note that the propagation of the fluid satisfies the plane wave condition and the viscous friction loss of the fluid is consequently assumed to be proportional to the fluid speed [11]. For the RSHPA, the transfer matrix is calculated as [10]  
\[\mathbf{A_{2}}=\left[ \begin{array}{cc}1+\frac{(Z_{\mathrm{C}_{ ECA}}+Z_{string})(R_{\mathrm{ECA}}+Z_{\mathrm{L}_{ECA}})}{Z_{\mathrm{C}_{ ECA}}Z_{string}}&R_{ECA}+Z_{\mathrm{L}_{ECA}}\\ \frac{Z_{\mathrm{C}_{ECA}}+Z_{string}}{Z_{\mathrm{C}_{ECA}}Z_{string}}&1\end{array} \right] \tag{4}\]  
where \(R_{ECA}\), \(Z_{\mathrm{C}_{ECA}}\) and \(Z_{\mathrm{L}_{ECA}}\) denote the equivalent resistance, capacitive impedance and inductive impedance of the expansion chamber, respectively, and \(Z_{string}\) denotes the equivalent impedance of the string array. It is noted that the derivation method of the attenuator model is essentially a lumped-parameter method, which believes that each hydraulic element, such as a pipeline, can be electrically modelled by a combination of a resistance, inductance and a capacitance.