Determination of the dynamic characteristics of a gear pump by the load variation method using special bench systems

Introduction. The main disadvantages of pumps are the acoustic noise that accompanies their operation, the vibration of the body, connected pipelines, and the resulting structural damage.

The traditional way of improving pump designs, used in mechanical engineering and consisting in smoothing out the uneven flow of the liquid, is hampered by the lack of means of direct measurement of fluid flow at the pump outlet. This limits any experimental evaluation of the design efficiency and finishing work on pumps. In addition, this limits the analytical description of the pump as a source of vibrations required by hydraulic system developers to provide the target dynamic quality of the systems by themselves.

For this reason, the research on the modeling of dynamic processes in hydromechanical systems [1][2] and the application of new approaches to solving problems of hydraulic system dynamics [3][4] based on physical principles are pressing issues.

Materials and Methods. The implementation of V. P. Shorin's method of load variation [4] is presented. It consists in determining pressure pulsations behind a pump interacting with several bench systems, whose dynamic characteristics are calculated in advance. Further, the processed frequency-response functions of the pressure using the impedance of the bench systems are recalculated into the pump's own dynamic characteristics — pulsation performance and impedance. The development of the approach provides that special bench systems are proposed. They take into account the connecting fittings, adapters and internal channels of the pump, which makes it possible to more accurately determine the dynamic characteristics of pumps, as well as to verify the applicability of the pump model used in the form of an equivalent source of vibrations.

Research Results

Method for determining the dynamic characteristics of a gear pump

Models and calculation of dynamic characteristics of special bench systems

Basic options of bench systems at the liquid outflow during pump tests were considered. In concentrated parameters, the bench system can implement active flow-dependent, inertial, and elastic loads. In distributed parameters, the bench system can be provided in the form of an extended cylindrical pipeline implementing an active frequency-independent non-reflective load.

Pump models in the form of equivalent source of flow fluctuations (ESFF) or pressure (ESPF) are determined using constant flow source Q₀, an ideal source of pressure fluctuations p (or flow q) and internal impedance Z_и [4]. Figure 1 shows pump models in the form of an equivalent source of pressure fluctuations (and flow) with an attached system.

In this connection, the task of selecting a bench system is to provide such system parameters at which the best conditions for high measurement accuracy are realized. The task of selecting the parameters of the attached system should be carried out when the impedance changes in the range from -∞ to +∞.

Computational models of impedances of special bench systems:

• with active throttle load (throttle):
  
  (1)
  
  
• with active distributed load (extended pipeline):
  
  (2)
  
  
• with capacitive load (cavity):
  
  (3)

The formulas indicate: Z_др(Q₀) — impedance of the throttle; ReZ_тр(Q₀) — real part of the impedance of the pipeline, including the connecting line, fittings, internal channels in the pump and throttle; ReZ_лам(Q0), ReZ_турб(Q0), ReZ_м.с.(Q0) — real part of the impedance of the fittings and the internal channel in the pump and cavity with laminar, turbulent fluid flow behind the pump and local resistances, respectively; f_k — frequency k-th of the harmonic component of the vibration spectrum behind the pump; ρ — density of the working environment; l_Σ — pipeline length, including the length of main line 6, connecting fittings 5 and 7 and internal channels of the pump and throttle (not shown in Fig. 7); S_тр — cross-sectional area of the pipeline; L — “inertia” of the connecting fittings and the internal channel in the pump and cavity; С — elasticity of the cavity; k — number of the harmonic component of the vibration spectrum; j — imaginary unit

An active frequency-independent load was implemented using a bench system with an extended pipeline length of 106 m.

The dependences of the impedance (module and phase) on the frequency were calculated according to the models of bench systems using formulas (1)–(3). Table 1 for the 1st harmonic shows the regression functions of the impedance module of the bench system Z_i(f) and phases β_i(f) for the analyzed bench systems.

The background information for determining the dynamic characteristics of the pump was the measurement of pressure at the pump outlet. An example of an oscillogram and a pressure pulsation spectrum at the pump outlet in the bench system with an active distributed load is shown in Figure 3. Operating mode of the pump in this case is as follows: average pressure P₀ = 23,8 MPa , drive shaft speed n = 1000 rpm.

The amplitude-frequency response of pressure pulsations at the pump outlet in the speed range of the pump drive shaft n=500–1250 rpm is shown in Figure 4. The regression functions built in the Microsoft Excel environment using the Trendline tool are plotted with a solid line.

The analysis of the oscillogram, spectrogram and amplitude-frequency response of pressure pulsations showed the following: the type of process was polyharmonic steady; the number of recorded harmonics was 8, analyzed¹ — 4; the amplitude range of pressure fluctuations was 0.28–0.52 MPa; the drive shaft frequency range was 500−1250 rpm, the analyzed harmonic components of the spectrum 83–833 Hz.

Fourier series expansion was used for harmonic analysis of periodic signals [7]. To decompose into a Fourier series, sections of the pressure oscillogram measured relative to the reference signal at the rotor frequency were used, i.e., one section of the oscillogram was extracted for a full rotation of the rotor. The reference signal was recorded at a fixed gear position, set at the beginning of recording the pressure waveform for five bench systems. In this way, the amplitude-frequency (FRF) A_k(ω) and phase-frequency (PFC) φ_k(ω) characteristics were determined.

Calculation of the spectrum of excited vibrations in various bench systems and determination of approximating for individual harmonic components of pressure fluctuations

Based on the experimental determination of pressure in five bench systems: with a throttle (no.1–3), with a cavity (no.4) and an extended pipeline (no.5) at the pump outlet, the spectra of excited vibrations were calculated. Table 2 shows the regression functions of the amplitudes A_p(f) and the initial phases of pressure pulsations for the 1st harmonic.

Calculation of the dynamic characteristics of the pump

To calculate the dynamic characteristics of the pump (impedance and pulsation performance of the source), it is required to conduct an experiment with at least two dynamic loads². Therein, according to the model of an equivalent source of flow fluctuations refined by the authors using analytical dependences p_i(f) and Z_i(f) from Tables 1 and 2, the pump impedance is calculated from the formula:

(4)

where Z_u — pump impedance;

— pump impedance module;

— real and imaginary part of pump impedance;

φ_u — pump impedance argument;

Z₁, Z₂ — input impedance of dynamic loads no. 1–2;

— impedance arguments of dynamic loads no. 1–2;

p₁, p₂ — pressure pulsations behind the pump in bench system no. 1–2;

— amplitudes of flow pulsations behind the pump in bench system no. 1–2;

φ₁, φ₂ — initial phases of pressure pulsations of dynamic loads no. 1–2.

Then, using the analytical dependences from Tables 1 and 2, the pulsation capacity of the pump was determined according to the model of an equivalent source of vibrations refined by the authors (variable component of volumetric flow q_и and pressure p_и):

(5)

where q_u, p_u — variable component of the volume flow and pressure of the pump;

A_qu, A_pu — pump flow and pressure pulsation amplitude;

φ_q, φ_p — argument of the variable component of the volume flow and pressure of the pump.

Of the calculated characteristics according to ESFF (q_u(f)) and ESPF (p_u(f)) models, using formula (5), only one will be stable, since the equivalent oscillation source has an independent dynamic characteristic only for one of the parameters (pulsating flow or pressure).

The calculation of impedance Z_u and the variable component of the volume flow q_и of the pump was carried out according to formulas (4)–(5) for five dynamic loads, i.e., using ten different combinations of these loads: “1_2”, “1_3”, “2_3”, “1_4”, “2_4”, “3_4”, “1_5”, “2_5”, “3_5”, “4_5” (Table 1).

For further calculation and analysis, a characteristic of the pump flow pulsation amplitudes A_qи was obtained from ten flow pulsation amplitude curves. The values of the amplitudes A_qи (at each frequency) were calculated using the formula from source [8]:

where — average amplitude of pulsating flow;

ΔA_qи — confidence interval of the amplitude of pump flow pulsations

σ — standard deviation of the amplitude of the pulsating flow rate from the average value;

m — number of dynamic load combinations (m = 10);

t — Student's coefficient (t = 2,262, was calculated for the confidence level 0.95).

The confidence interval of amplitudes ΔA_qи was formed from errors (instrument and method) and the degree of dependence of the dynamic characteristics of the pump A_q(f) on the bench system. The instrument error consisted of the error of determining the pressure and phases, the method error included the error of the unaccounted influence of the dynamic properties of the drive mechanical system.

The phases of flow pulsations φ_q and their deviations were calculated in a similar way. When calculating the phases, combinations “1_2” and “1_5” were excluded due to the degeneration of the variable component of the volume flow of the pump q_и when using close values of characteristics at the same frequency: φ₁ ≈ φ₂ and β₁ ≈ β₂.

The results of calculating the intrinsic dynamic characteristic of the pump in the form of pulsation performance and impedance are shown in Figure 5. Phases, for convenience, are presented in degrees.

The amplitude of the flow pulsations at the 1st harmonic has a monotonically increasing character. With an increase in the harmonic number, the slope of the flow pulsation amplitude curve decreases to an almost constant value to the 4th harmonic. The values of amplitudes A_q at the higher harmonics are by an order of magnitude less than at the 1st harmonic.

The pump impedance module at the 1st harmonic has a monotonically increasing character.

To evaluate the realized dynamic loads, the criterion of matching the impedance module of the oscillation source and the input impedance module of the dynamic load was used [99, 10]. The impedance modules should be comparable so that the quantitative values of the dynamic characteristics of the pump were determined with minimal error. For comparison, ratio was used. At , the dynamic load was considered consistent with the source.

Due to the difference in the curve shapes of the pump impedance modules and dynamic loads in the frequency range of 83–417 Hz, it was rational to use average ratio , that provided considering the contribution of most of the frequency range, i.e., ratio according to the 1st harmonic would be:

• for active throttle load no.1 — 1.16;
• for active throttle load no.2 — 0.8;
• for active throttle load no.3 — 2.65;
• for capacitive load no.4 — 4.64;
• for active distributed load no.5 — 0.35.

That is, the dynamic loads of bench systems no. 1, 2 were the most consistent with the source of vibrations. Thus, the dynamic characteristic of the pump, calculated using dynamic loads of bench systems no.1, 2 (“1_3”, “2_3”, “1_4”, “2_4”, “1_5”, “2_5”) was determined with a minimum error.

Evaluation of the stability of the dynamic characteristics of the pump from the characteristics of bench systems

Stability of the dynamic characteristics of the pump was evaluated using mathematical statistics tools [8]. Thus, the stability of the characteristic A_qи was estimated through calculating the ratio of the confidence interval at frequency f to the average value of the characteristic at this frequency³. The resulting relative deviation Δ_q should not exceed the specified deviation Δ_допq:

where — average median value of the amplitude of pump flow pulsations;

Δ_допq — specified relative deviation of the pump flow rate pulsation amplitude from the average value.

With relative deviation the stability of characteristic A_qи was maximum. Based on the experience of other researchers, it is preferable to select value Δ_допq within 5–30 % [11–17].

For a single-digit evaluation of the dynamic characteristics of the pump, it is rational to use the value of the relative deviation. The calculation of such a value is possible through estimating the average median value of relative deviations in the frequency range of each harmonic

Value Δ_qMe is efficiently used, if

i.e., when the confidence interval ΔA_qи does not exceed 30% of the average amplitude of pulsating

The results of assessing the stability of the dynamic characteristics of the pump from the characteristics of the bench systems4 were: according to the 1st harmonic — 2 %, according to the 2nd — 8 %, according to the 3rd — 15 %, according to the 4th — 48 %. The results of the calculation of the dynamic characteristics of the pump showed that for four harmonics in the frequency range of 500–2500 rpm, the gear pump under study should be considered according to the model of an equivalent source of flow fluctuations.

Checking the applicability of the pump model used as an equivalent source of flow fluctuations

To test the developed approach for evaluating the dynamic characteristics of the pump, a calculated determination of pressure pulsations in bench systems with a throttle, a cavity and an extended pipeline at the pump outlet was performed. Using the found dynamic characteristics of the pump, the amplitudes of pressure pulsations in the analyzed bench systems were calculated, which were compared to the experimental ones. The results of the calculation of amplitudes A_p(f) in a bench system with an extended pipeline are presented below. The amplitudes of pressure pulsations p_i were calculated in accordance with the ESFF model from the formula:

(6)

The results of calculating the amplitudes of pressure pulsations behind the pump are shown in Figure 6. The dotted line indicates the amplitudes obtained experimentally, and the solid line — calculated from formula (6).

Convergence of the results of the verification calculation was evaluated according to the formula of deviation of the difference between the experimental and calculated values from the experimental data. The results of the test calculation for the bench systems with a throttle, a cavity and an extended pipeline at the pump outlet showed that according to the 1st harmonic, the relative deviation from the experimental data for the considered bench systems did not exceed 10 %, the minimum relative deviation was characteristic of a bench system with an extended pipeline at the pump outlet — 2%. The amplitudes of the pressure pulsations of the 2nd–4th harmonics were within the corridor calculated from the formula (where — the average amplitude of the pulsating pressure).

• assessment of the dynamic characteristics of the gear pump during the entire service life;
• development of a four-pole model of a gear pump as an equivalent source of vibrations, taking into account the mechanical drive and hydraulic output systems;
• development of a gear pump model as an equivalent source of vibrations, taking into account its design features, the displacement process, reverse water hammer in the chamber between teeth, the work of the locked volume, and leakage from the injection area to the suction area.

Here, it is required to expand the nomenclature of the tested volumetric pumps for the entire class of volumetric displacement machines to confirm the operability of the proposed approach.