Research on Extended Transfer Matrix of Centrifugal Pumps
Abstract: This study explores the dynamic transmission characteristics of centrifugal pumps by introducing torque parameters of rotating shafts into the analysis of hydraulic systems. Focusing on the torque excitation response of pump shafts, the research examines the numerical transfer matrix and evaluates internal disturbance sources to better understand the system's dynamic behavior. The findings demonstrate that the basic experimental methods, data processing techniques, and mode identification of transmission coefficients are still valid and effective. The observed torque fluctuations show a symmetrical pattern similar to water pressure, although influenced by additional factors. The extended transfer matrix reveals distinct characteristics between torque transmission and hydraulic parameters, with transmission coefficients showing a linear relationship over certain frequency ranges. Disturbance source analysis further confirms the reliability of the method from an application perspective.
Keywords: Centrifugal Pump System, Extended Transfer Matrix, Dynamic Transmission Characteristics, Torque Fluctuation, Hydraulic System
The research on the transmission matrix of hydraulic machinery has been significantly advanced by studies on the POGO phenomenon in aerospace engineering. However, the investigation of hydraulic machinery’s transfer matrix is primarily focused on dynamic stability, especially in the frequency domain. Common assumptions in calculating dynamic fluctuations include uniform flow, positive pressure, negligible flow velocity (as it is much smaller than the wave speed), plane wave assumption (due to duct cross-section being smaller than the wavelength), and linear transmission. Acoustic and excitation techniques also ensure accurate and efficient experimentation. Most previous research has used transfer matrices containing pressure and flow fluctuation vectors. While this approach captures essential aspects of the problem, it is considered a "basic matrix" suitable for early-stage research. As the study progresses, extending the transfer matrix to include additional dynamic parameters like pump shaft torque, rotational speed, and guide vane opening becomes both necessary and feasible. However, such extensions increase experimental complexity. Therefore, a step-by-step approach based on research objectives, test facilities, and data processing capabilities is recommended.
This paper presents an experimental study conducted at the EPFL_IMHEF Institute of Hydraulic Engineering and Mechanics in Lausanne, Switzerland, focusing on the dynamic transmission characteristics of pump shaft torque. The study discusses the fundamental features of torque-actuated responses, the derivation of the transfer matrix, and internal disturbance sources. It explores the correlation between pump shaft torque fluctuations and hydraulic parameter fluctuations, aiming to establish their characteristic transmission relationships.
1. Experimental Setup and Data
1.1 Experimental Setup
Figure 1 shows the PF4 test bench used in the EPFL_IMHEF study. The test machine is a Francis pump turbine with a specific speed of nq = 39 (power ratio speed ns = 3.13nq), an impeller diameter of 152 mm, and seven blades. A 30 kW DC motor drives the pump, maintaining a nearly constant angular velocity through a governor. At 2000 rpm, the optimal conditions were Q = 9.5 L/s and E = 85 J/kg. The hydraulic pipeline includes two pressure steel pipes with inner diameters of 100 mm (high pressure) and 150 mm (low pressure). Three pressure sensors are installed on both sides of the pipes, and two water acoustic filters are connected to the test pressure pipe to ensure zero impedance boundary conditions. Valve actuators are placed at various points along the pressure pipe, and the experiment uses one of them. Pressure sensors are quartz piezoelectric type, capable of adjusting signal gain to produce voltage outputs corresponding to water pressure fluctuations. Rotor torque is measured using strain gauges calibrated on the pump shaft.
1.2 Key Concepts
To ensure accurate data processing and application, it is crucial to understand key concepts. This includes identifying excitation shocks and wave transmission methods.
1.2.1 Excitation Oscillation
A rotary valve actuator, driven by an induction motor connected to a frequency converter, generates periodic perturbations in the hydraulic system. The actuator creates excitation signals by blocking the jet at a given frequency, with an air slot acting as a buffer. The exciter is small and has minimal impact on the system.
1.2.2 Identification of Transmission Waves
Flow fluctuations in a hydraulic pipe are indirectly determined using the characteristic acoustic impedance of the pipe. The wave velocity is calculated using the oscillatory patterns of pressure signals from three pressure sensors. This process assumes a plane wave and handles fluctuations below 250 Hz.
1.3 Data Processing
Time-domain data collected via computer acquisition equipment serves as the original data. With a sampling frequency of 1024 Hz, 240 analysis windows are defined over a 120-second excitation period. Each window lasts 0.5 seconds and contains 512 samples. Using a Hanning window-weighted Fourier transform, the transfer function of water pressure data can be organized to calculate wave velocity and indirect traffic complex coefficient qx(n). Data processing reduces calculation errors and typically selects a reference channel opposite the excitation side. The Fourier transform of time-domain moment data is similar to the water pressure signal.
2. Torque Fluctuation Response
Pump impellers link the torque fluctuation of the drive shaft and the pressure fluctuation of the hydraulic system. Although these signals originate from different physical media, they respond differently to the same external disturbance, resulting in distinct characteristics. Figure 4 shows the response under the same excitation conditions as Figure 3. The y-axis displays changes in frequency f and amplitude ΔT (in Nm/40). The torque variation shows a nearly symmetric distribution at corresponding frequencies but differs from hydraulic pressure in its stimulated frequency distribution, which is also influenced by other factors, such as the pump shaft rotation frequency. The figure shows a concentrated frequency band around 33 Hz, matching the experimental apparatus’s rotational frequency. The power spectrum of the torque signal resembles a triangular shape like water pressure but with a much lower amplitude. Using the low-pressure side of the actuator results in a higher response amplitude compared to the high-pressure side due to greater influence on suction head. This phenomenon highlights the close relationship between the moment response and energy conversion within the pump impeller.
3. Transfer Matrix
3.1 Matrix Expression
Under the assumption of linear propagation and no external force, pump shaft torque fluctuation parameters are embedded in the hydrodynamic wave vector (p, q). The transfer characteristic from one vector to another is expressed as:
(3)
Where p, q, T represent water pressure, flow rate, and pump shaft torque fluctuation, respectively; the superscripts “4†and “3†indicate signals from channels 4 and 3 (see Figure 1), i.e., the outlet or inlet of the pump; [M] is the extended transfer matrix. In equation (3), p, q, T, mi, j are complex coefficients of frequency. Matrix coefficients are named based on their physical meanings: m11, m22 are water pressure and flow transfer coefficients; m12 is hydraulic impedance coefficient; m21 is hydraulic admittance coefficient; m31 and m32 are moment admittance coefficients. Coefficients m11, m12, m21, m22 describe the hydraulic system’s characteristics, while m31, m32 reveal energy conversion information between flow and mechanical impeller. These coefficients can be normalized using the characteristic impedance z4 on the high-pressure side and the water flow v in the impeller area. The complex coefficient matrix corresponds to 3 linear equations with 6 unknowns. If two sets of independent fluctuation data are available, all mi and j coefficients can be calculated using linear equations.
3.2 Experimental Results
Figure 5 shows experimental calculation results under steady-state conditions: specific energy E = 80.2 J/kg, flow Q = 11.4 L/s, speed 2000 rpm. The x-axis is frequency f, and the y-axis shows dimensionless complex function matrix coefficients m11, m12, m21, m22, m31, and m32. Solid lines represent real parts, and dotted lines represent imaginary parts. The coefficients m11, m12, m21, m22 correspond to the former matrix, while m31 and m32 are discussed here. These coefficients are much smaller in magnitude than the hydraulic system coefficients and show small amplitude fluctuations across a wide frequency range, except at integral multiples of the rotational frequency. This aligns with earlier discussions on torque-induced responses. At lower frequencies, excluding the rotational frequency perturbation, m31 and m32 are approximately linear with respect to frequency. The real part of m31 reflects the relationship between hydraulic energy and torque, while the imaginary part of m32 expresses the inertial effect of water flow on torque.
4. System Disturbance Source Analysis
Equation (3) can also be considered a homogeneous transmission equation, implying that internal disturbance sources in the experimental setup are negligible compared to external forced excitation. The equation can also include perturbation source vectors. If the acoustic transmission of a water pump is expressed by reflection sources (pS4, qS4) and (pS3, qS3) on both sides of the tested pump, the disturbance of torque fluctuation can be regarded as a disturbance acting on the pump shaft TS. The above transfer equation then becomes:
The location and properties of the disturbance sources are random, and internal disturbances can be analyzed using the above homogeneous transmission equations. Figure 6 shows different frequency responses of torque ripple. The solid line corresponds to external excitation, and the dotted line corresponds to theoretical calculations under the same experimental conditions. The x-axis is frequency f (Hz), and the y-axis is torque fluctuation amplitude ΔT (Nm). Figure 7 shows the internal disturbance of torque as a function of frequency. Large pulses are related to integer multiples of the shaft rotation frequency, such as 33 Hz or 7 × 33 Hz (note that the number of pump impeller blades is 7). Graphically, the disturbance source TS is a replica of the torque frequency response without external excitation.
5. Conclusion
This paper introduces pump torque fluctuation parameters to expand the basic transfer matrix, demonstrating that the use of basic experimental methods, data processing, and pattern recognition of coefficients remains feasible. The observed torque excitation response shows that although the fluctuation frequency is influenced by other factors, it still exhibits a nearly symmetrical distribution like water pressure. The calculated extended transfer matrix reveals different characteristics between torque and hydraulic parameters, with torque-related coefficients showing a linear relationship over certain frequency ranges. Disturbance source analysis not only applies the extended transfer matrix but also verifies the correctness of the transmission characteristics determination method.
Acknowledgment: This study was completed at the Institute of Hydrodynamics and Hydraulics at the Federal Institute of Technology in Lausanne, Switzerland (EPFL_IMHEF), and the authors would like to thank the colleagues there for their support.
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