## 5 Discussion and Vibration of the Pump Housing
pressure levels at specific harmonics, particularly those associated with cavitation or other operational inefficiencies, are evident as distinct peaks in the spectrum. Figure 12c presents the frequency spectrum of the suction chamber pressure, with magnitude expressed in bars, as a function of harmonic order. The displayed pressure range is 0 to 3.6 bar. Peaks at specific harmonics suggest potential resonant frequencies or characteristic pressure oscillations within the pump. These measurements, acquired across a range of operating conditions, allow for the analysis of the harmonic content to identify potential resonance issues. Figure 12d presents the frequency spectrum of the suction chamber pressure, expressed in decibels (dB), as a function of harmonic order. The displayed range is 0 to 120 dB. These dB levels represent the magnitude of the pressure fluctuations at each harmonic. Elevated dB levels at specific harmonics may indicate potential issues such as cavitation or mechanical imbalances within the pump. The analysis of these harmonics helps identify characteristic frequencies in the suction chamber pressure signal, which can be related to specific pump behaviors. The set of Equations (1)-(11) that describe the time-varying behavior of the pump under different operating conditions can be written in the following form:  
\[\frac{dy_{i}}{d\phi}=f_{i}(\phi,y_{1},y_{2},\dots,y_{N}) \tag{13}\]  
where i is in the interval from 1 to N, and N is the number of differential equations that make up the system. Given the complexity of the differential equations of the working process, numerical methods have been applied that do not require derivations from the equations. Such methods are, for example, Runge-Kutta and Euler-Cauchy. The basic criterion for choosing a method is the accuracy, duration, and stability of the calculation. The specific boundary conditions adopted to include the effects of fluid properties are the relationship between the pressure, density, and isothermal compressibility coefficient of the fluid:  
\[\left(\frac{dp}{d\rho}\right)_{T}=\left(\frac{\partial p}{\partial\rho}\right) _{T}=\frac{1}{\varepsilon_{T}\rho} \tag{14}\]  
All the pressures and vibrations were measured in parallel every approx. 0.09\({}^{\circ}\) of the pump drive shaft (exactly 4096 times per one revolution of the shaft). An optical encoder with 1024 pulses per revolution was used as an incremental angle encoder. The angle encoder pulses were quadrupled using the angle encoder interface on the ADS 2000 system/1, 4, 5/so that 4096 pulses were obtained per shaft revolution. The total number of measured data was (4 + 1) \(\times\) 4096 = 20,480 per revolution (cycle), i.e., 204,800 for 10 consecutive cycles. The number of samples of 4096 was not chosen randomly, but was deliberately set to apply the fast Fourier transform (FFT) of the measured signals. Measurements were made for seven operating modes. The pressure pulsations in the pressure chamber depend on the number of cylinders, which is obvious in this case, since it is a pump with eight cylinders (Figure 11). Pronounced pressure fluctuations in axial piston pumps around the maximum and minimum pressure points arise from a combination of internal and instrumental factors. Internal factors include inertial effects, where pistons accelerate and decelerate within their cylinders and their inertia creates significant forces. At the points of maximum and minimum pressure, the rate of change in piston velocity is greatest, leading to strong inertial forces that contribute to pressure surges. Rapid changes in piston speed compress and expand the fluid inside the cylinders and connecting lines. This rapid change in volume translates directly into pressure fluctuations. The sudden opening and closing of valves within the pump control system results in sudden changes in the direction and magnitude of flow. This can cause significant pressure transients, especially around extreme pressure values where the flow changes most rapidly. Leakage across the valve face can vary with pressure.