The performance of an aerospace fuel gear pump is directly related to the stability and reliability of the engine. As fuel gear pumps are developing towards higher rotational speeds and greater power, problems such as reduced volumetric efficiency, flow fluctuations, vibration, and noise have seriously hindered the improvement of their performance. In the production trial of arc gear pumps, it has been found that the number of gear teeth has a significant impact on their performance. Therefore, this article takes the circular-arc fuel gear pump as the research object to study the influence of the number of gear teeth on the flow characteristics in the cavity of the gear pump.
1. Introduction
The aerospace fuel system is closely related to providing a stable and powerful boost to the engine. As a key component of the fuel system, the performance of the aerospace fuel gear pump directly affects the stability and reliability of the engine’s operation. With the continuous development of fuel gear pumps towards higher rotational speeds and greater power, a series of problems such as decreased volumetric efficiency, flow fluctuations, vibration, and noise have seriously hindered the enhancement of the performance of fuel gear pumps. Scholars have conducted a large amount of research on these issues. For example, Zhu et al. analyzed the lubrication characteristics and contact behavior of the gear pump sliding bearings to explore the reliability of dynamic pressure lubrication in gear pumps operating at high rotational speeds and low medium viscosities. Li Geqiang et al. conducted numerical analysis on double circular-arc helical gears through Fluent and compared the flow characteristics with those of involute helical gear pumps with the same geometric parameters. Zhao et al. studied the CCHGP type gear pump and explained based on the fluid mechanics of the lumped parameter method that this gear pump can reduce or even eliminate flow pulsation.
In terms of the influence of gear tooth shape and rotational speed on the flow characteristics of the pump, Szwemin et al. analyzed the impact of gear eccentricity on leakage and pressure accumulation on the circumference for different gap shapes and sizes. Tankasala et al. determined a multi-objective optimization method for gear size and flow change mechanisms to analyze the instantaneous flow and flow non-uniformity of the pump. Huang et al. concluded that pumps with larger moduli, larger face widths, or smaller numbers of teeth have a larger displacement, but it may lead to more severe flow rate fluctuations. Zhou Lanmei et al. studied the flow pulsation of external meshing gear pumps with different numbers of teeth and obtained the conclusion that an increase in the number of teeth can improve the flow quality of the gear pump. Xu Wengang et al. established a dynamic model of pressure pulsation in the single and double tooth meshing process of the gear pump, discussing the modulation effect of rotational frequency on the pressure pulsation signal and the influence of different working conditions on pressure pulsation. With the continuous improvement of various computer-aided software and various test methods, Liu et al. proposed a suppression method of non-circular gear variable speed drive to analyze the flow pulsation components of the elliptical gear pump. Zhao simulated the multiple cracks of the gear pump gear through the wavelet finite element method to identify the location and size of the multiple cracks of the gear pump gear. Mucchi et al. determined the variable excitation load from the internal pressure evolution of the external gear pump tooth space through the experimentally evaluated model.
Currently, to meet the requirements of high performance and high stability of aerospace fuel gear pumps in extreme environments, the structure of the main components of the gear pump, namely the gears, is developing in the direction of multi-toothing and helicalization. In the production trial of circular-arc gear pumps, it is found that the number of gear teeth has a great influence on their performance. Therefore, this article takes the circular-arc fuel gear pump as the research object, in which the tooth top and tooth root of the gear pump adopt a smooth circular arc, and the middle is connected in an involute manner. The influence of the number of gear teeth on the flow characteristics in the cavity of the gear pump is studied. The structure of this article is as follows: Based on the previous work, a three-dimensional model of the gear pump with 6 to 9 teeth is established, and the mesh division and mesh independence verification are carried out. Through the RNG k-ε turbulence model, a three-dimensional unsteady numerical analysis is performed. The accuracy of the numerical simulation is verified through experiments. The mathematical model of the instantaneous flow rate of the gear pump is established to analyze the influence of the number of gear teeth on the instantaneous flow characteristics of the gear pump. The pressure distribution in the pump cavity and the force characteristics of the gears are discussed to enhance the understanding of its internal flow characteristics.
2. Calculation Model
Taking the gear pump with 7 teeth as an example, the circular-arc-involute-circular-arc is adopted as the end face profile of the gear, and the establishment of the end face profile equation can be found in detail in reference [16]. Referring to reference [17], it is more favorable to select a rack pressure angle of 14.5°. When the pressure angle is 14.5°, the minimum number of teeth to avoid undercutting is 6, and thus the calculation model of the circular-arc gear pump with 6 to 9 teeth is established. The influence of the number of teeth on the flow performance in the cavity of the pump under different operating conditions (import and export pressure difference ΔP = 2 MPa – 10 MPa, rotational speed n = 1500 r/min – 3000 r/min) is analyzed.
The gear geometry parameters of 6 to 9 teeth are shown in Table 1. In Table 1, Z represents the number of gear teeth, β represents the helical angle, and A represents the center distance. Keeping the outer diameters of the gears with 4 different numbers of teeth consistent, Re = 34.08 mm, and the other geometric parameters of the gears with 6 to 9 teeth are as follows: gear width B = 20 mm, inlet diameter Dm = 18 mm, outlet diameter Dcor = 15 mm, radial clearance δ = 0.02 mm, and tooth pitch angle τ = 2π / Z.
It can be seen from Figure 2 that as the number of gear teeth gradually increases, in order to make the axial coincidence degree 1, the value of the helical angle β decreases with the increase of the number of teeth. Under the same outer diameter parameters, the number of transition chambers increases from the inlet end to the outlet end, but the effective volume in the pump cavity decreases.
3. Numerical Simulation
3.1 Mesh Division and Dynamic Mesh Model
As shown in Figure 3, the entire mesh area is divided into the inlet, gear cavity, and outlet sections. Considering that the tooth gap between the two gears and the radial gap between the gear and the pump cavity are very small, the quadrilateral structured mesh is adopted in the gear rotation domain. Considering the high requirement of the turbulence model for the number of meshes and the influence of the number of meshes on the simulation results, taking the 7-tooth circular-arc gear pump as an example, referring to the mesh independence verification method in reference [20], by adjusting the mesh division scale, six calculation models with different numbers of meshes are generated respectively.
Under the working conditions of 8 MPa and 2000 r/min, as shown in Figure 4, when the number of meshes is less than 1.24×10^6 (point C), the calculation results will have large fluctuations. Compared with point F, the average flow error of point A and point B is 5.29% and 2.52% respectively; in the process of the number of meshes gradually increasing from point C to point F, the change of the calculation results is relatively gentle. Compared with point F, the average outlet flow calculation result errors of point C, point D, and point E are 0.52%, 0.25%, and 0.03% respectively.
On the basis of ensuring accuracy and saving computing resources as much as possible, the final number of meshes of the 7-tooth gear pump calculation model is determined to be 1.24×10^6. As shown in Figure 3, 360 layers of meshes are arranged in the circumferential direction, 60 layers of meshes are arranged in the axial direction, 18 layers of meshes are arranged at the radial clearance, and 36 layers of meshes are arranged at the tooth gap, totaling 40 sets of overlapping meshes. According to a similar mesh division scale, the final number of meshes of the 6-tooth, 8-tooth, and 9-tooth gear pump calculation models is 1.41×10^6, 1.36×10^6, and 1.32×10^6 respectively.
The transmission medium is aviation kerosene, and the relevant physical parameters are: ρ = 800 kg/m^3, μ = 0.007 Pa·s. It is set that the gear pump rotates a total of 10 turns, among which each turn rotates 360 steps, and each step rotates 1°. If the relative pressure difference of the monitoring point in the corresponding time step between two adjacent periods is less than 5%, it is considered that the periodic requirement is met, and the result of the last turn is taken for analysis.
3.2 Control Equations and Turbulence Model
The fluid model and turbulence model can be directly called in PumpLinx. The turbulence parameters are obtained from the Navier-Stokes equations, and the RNG k-ε model is used to solve the gear pump cavity. The RNG k-ε model uses the statistical technique of the renormalization group to correct the turbulent viscosity, which can better handle the flows with large curvatures, strong rotations, and high strain rates. The solid wall adopts a non-slip wall, and the near-wall area adopts a standard wall function. The contribution of the pulsating expansion in the compressible turbulence is not considered, and the influence of gravity on the flow field is ignored. Pressure boundary conditions are set at the inlet and outlet. The transport equations of the turbulent kinetic energy k and dissipation rate ε of this model are as follows:
In the equations, is the generation term of the turbulent kinetic energy k caused by the average velocity gradient; is the generation term of the turbulent kinetic energy k caused by buoyancy; , , and are empirical constants respectively; and are the Prandtl numbers corresponding to the turbulent kinetic energy k and dissipation rate ε respectively; and are the user-defined source terms respectively.
4. Experimental Verification
To verify the accuracy of the numerical simulation method, a 7-tooth gear pump is taken as a prototype for experimental verification, and a closed test bench of the circular-arc gear pump is built to test the outlet volume flow of the pump under 5 different working pressures. The pump is driven by an AC motor, and the test system consists of two parts: the connection from the oil tank to the pump inlet and the connection from the pump outlet to the oil tank. Figures 5 and 6 are the schematic diagrams of the gear pump test bench, and the test instruments include a Siemens Beide frequency conversion motor with a power of 7.5 kW; according to the motor configuration, pressure, and flow requirements, the frequency converter model is S200-G7.5/P11T4B, the pressure sensor model is MIK-P30030MPa, with a measurement accuracy of 0.5%; the flowmeter model is LC-A2-0.2/AIGFI, with a range of 0.3 – 3 m^3/h, and a measurement accuracy level of 0.5; the data collector is of USB3200 type, with a sampling rate of up to 500 ks/s; the throttle valve diameter is DN20, and the safety valve diameter is DN10; the test medium is hydraulic oil.
During the test, the inlet pressure of the prototype (5) is fixed at 0, and as the motor (6) drives the prototype to operate, the hydraulic oil is delivered into the hydraulic pipeline from the oil tank (1). The pressure control valve (9) is placed on the outlet side of the pump to facilitate adjusting the outlet pressure of the pump. To ensure the safety of the test, the outlet pressure of the prototype is adjusted to the minimum value through the pressure control valve. After the system operates stably, the outlet pressure is gradually adjusted through the pressure control valve, the pressure in the pipeline is detected by the pressure gauge (4/7), the flow in the pipeline is detected by the flowmeter (8), and finally, the pressure and flow values are transmitted to the data collector in the form of analog signals, and the data collector transmits them to the upper computer, and the data is processed and displayed on the visual interface of the upper computer.
Figure 7 is the comparison diagram of the numerical prediction and experimental results of the circular-arc gear pump at 2000 r/min, and each point in the figure corresponds to the outlet flow under the corresponding import and export pressure difference. It can be seen from Figure 7 that under different import and export pressure difference operating conditions, the numerical calculation value is higher than the experimental value, and the error is within 4.21%. The volume flow obtained by the numerical calculation is in good agreement with the experimental value. With the increase of the import and export pressure difference, the deviation between the experimental and numerical prediction volume flow values increases. The main reason is that the pressure difference has a great influence on its volumetric efficiency, and the volumetric loss increases sharply with the increase of the pressure difference. In addition, the numerical calculation ignores the mechanical losses caused by bearings, gears, and mechanical seals, as well as the error of simplifying the three-dimensional model of the circular-arc gear pump rotation domain. In general, using the RNG k-ε turbulence model to predict the performance of the pump has good accuracy and reference value.
5. Results and Analysis
5.1 Influence of the Number of Teeth on the Outlet Flow Characteristics of the Circular-Arc Gear Pump
Figure 8 shows the relationship between the outlet flow and volumetric efficiency of the 6-tooth to 9-tooth gear pumps under different working conditions. The histogram (corresponding to 2, 4, 6, 8, and 10 MPa of this node respectively, the same for the broken line graph) corresponds to the outlet flow on the left coordinate axis, and the broken line graph corresponds to the volumetric efficiency on the right coordinate axis. The volumetric efficiency of the gear pump is defined as:
The theoretical displacement per revolution of the circular-arc gear pump [17] is:
In the formula, is the simulated value of the average outlet flow of the pump per unit time, Q is the theoretical value of the outlet flow of the pump per unit time, is the radius of the addendum circle, and R is the radius of the pitch circle.
It can be seen from Figure 8 that under the condition of keeping the same outer diameter of the gear, the number of teeth has a great influence on the outlet flow of the pump, and the outlet flow and volumetric efficiency of the pump with different numbers of teeth show a linear decreasing trend with the increase of the import and export pressure difference.
Taking the working condition of 2000 r/min as an example, when the import and export pressure difference increases from 2 MPa to 10 MPa, the average outlet flow of the 6-tooth pump has the largest decrease, followed by the 8-tooth pump, which decreases by 2.29 L/min and 1.34 L/min respectively, and the volumetric efficiency decreases by 10.19% and 7.25% respectively; the average outlet flow of the 9-tooth pump has the smallest decrease, which is 0.61 L/min, and the volumetric efficiency decreases by 3.61%; the 7-tooth pump has a relatively small decrease.
The relationship between the flow pulsation coefficient and the number of teeth under different working conditions is shown in Figure 9. The flow pulsation coefficient is defined as:
In the formula, and are the maximum and minimum outlet flows respectively, and is the average outlet flow. Figure 9 shows that the number of teeth has a significant influence on the flow pulsation of the pump. In the process of the import and export pressure difference increasing from 2 MPa to 10 MPa, the flow pulsation coefficients of pumps with different numbers of teeth all show an increasing trend. Taking the working condition of 2000 r/min as an example, the increase of the 6-tooth pump is the largest, which is 0.43; the increase of the 7-tooth pump is the smallest, which is 0.03; the increases of the 8-tooth and 9-tooth pumps are 0.35 and 0.07 respectively. At the same import and export pressure difference, the 7-tooth and 9-tooth pumps can maintain a relatively low flow pulsation.
5.2 Influence of the Number of Teeth on the Flow Characteristics in the Cavity of the Circular – Arc Gear Pump
Taking the simulation data of the 6 – 9 tooth pumps under the working conditions of 8MPa and 2000r/min as an example, the influence of the number of teeth on the flow characteristics and performance in the cavity of the pump is compared and analyzed. Figure 10 shows the outlet instantaneous flow characteristic curves of the 6 – 9 tooth gear pumps within one cycle.
It can be seen from Figure 10 that the outlet instantaneous flow of pumps with different numbers of teeth changes periodically, and the number of pulsations of the pump outlet flow is equal to the number of gear teeth. The number of teeth has a significant influence on the outlet flow pulsation of the pump. Among them, the flow drop values from the peak to the trough of the 7 – tooth and 9 – tooth pumps are relatively small, decreasing by 2.84L/min and 1.76L/min respectively. The flow drops from the peak to the trough of the 6 – tooth and 8 – tooth pumps are relatively large, decreasing by 5.53L/min and 4.90L/min respectively. The 6 – 9 tooth gear pumps all have different degrees of secondary flow pulsation.
When the number of teeth increases to 9, its pulsation characteristic curve gradually changes from an “M” – shaped peak to a “V” – shaped peak, and the pulsation amplitude decreases. In the process of the number of teeth gradually increasing, the change of the flow pulsation amplitude within one cycle does not show a strictly linear decreasing trend with the increase of the number of teeth, and the decreasing amplitudes are different. As shown in Figures 8 and 9, the average outlet flow of pumps with different numbers of teeth does not show a strictly linear decreasing trend with the increase of the number of teeth. Taking the working conditions of 8MPa and 2000r/min as an example, the average outlet flows of the 6 – 9 tooth pumps are 15.29L/min, 17.54L/min, 13.81L/min, and 14.93L/min respectively, and the volumetric efficiencies are 64.93%, 86.30%, 72.08%, and 85.63% respectively.
The pulsation of the fluid and the fluid output are mainly related to the geometric shape of the flow channel, the velocity and flow direction of the fluid. The outlet pulsation of the gear pump is determined by the superposition of the shape of the outlet control volume and the periodically changing fluid velocity direction. The outlet instantaneous flow of the circular – arc gear pump is not only related to the number of teeth and the volume of the tooth cavity control area, but also related to factors such as the tooth pitch angle τ, the helical angle β, and the meshing angle φ of the two gears (these parameters change immediately with the change of the number of teeth). The following is the mathematical expression of the outlet instantaneous flow of the circular – arc gear pump from a mathematical perspective.
The circular – arc helical gear is regarded as being superimposed by countless circular – arc spur gears with a thickness of dx, and the instantaneous flow model of the helical gear can be obtained by integrating the instantaneous flow of the circular – arc spur gear with a thickness of dx. As shown in Figure 11, let one end face of the circular – arc helical gear pump be the reference plane (setting the lagging end face of the wheel tooth on the other end entering meshing as the positive reference system), then the angle corresponding to the meshing point of the same tooth at a position x away from the end face is:
In the formula, θ is the angle corresponding to the meshing point of the same tooth at the reference plane. It is assumed that the angle corresponding to the meshing point of the wheel tooth at the reference plane (exactly at the node) is 0.
The theoretical instantaneous flow of the circular – arc gear pump obtained by the swept area method [17] is:
Substituting equation (5) into equation (6), the instantaneous flow of the circular – arc spur gear pump with a thickness of dx at a position x away from the end face can be obtained as:
Equation (7) is a continuous function with a period of 2Nπ/Z (N is a positive integer), and the function is symmetrical about the axis of symmetry within one period. The change of the instantaneous flow function within one period can be expressed as the function characteristics of the instantaneous flow in the complete interval. By superimposing countless instantaneous flow waveforms staggered by an angle β in turn, the instantaneous output flow of the circular – arc gear pump is formed, and a periodic instantaneous flow waveform is obtained.
Figure 13 shows the comparison between the theoretical instantaneous flow curve and the numerical simulation instantaneous flow within one meshing cycle. Under the working conditions of 8MPa and 2000r/min, within one tooth pitch angle cycle, the coincidence degree of the two is about 83.28%.
According to formula (14), the maximum and minimum values of the theoretical instantaneous flow within one tooth pitch angle cycle are obtained, and the theoretical instantaneous flow pulsation coefficients of the 6 – 9 tooth gear pumps are 4.50×10⁻⁵, 3.03×10⁻⁵, 4.04×10⁻⁵, and 3.15×10⁻⁶ respectively, which reflects the advantage of the circular – arc gear pump in low flow pulsation from the side. With the increase of the number of teeth, the theoretical calculation instantaneous flow pulsation coefficient generally shows a decreasing trend.
However, the instantaneous flow pulsation coefficient of the numerical simulation is increased to varying degrees compared with the theoretical instantaneous flow pulsation coefficient, which is related to factors such as the extremely high pressure difference between the inlet and outlet in the modeling and simulation, the leakage caused by the tooth/radial clearance; due to the existence of the helical angle, the fluid pressures distributed in the cavities at the same height in the front and rear parts of the gear are different; due to the slight difference in the tooth profile angle, the phase delay appears on the front and rear surfaces of the gear; turbulence, secondary flow, and eddy current appear near the gear and the inlet and outlet along with the gear meshing. With the increase of the number of teeth, these adverse effects can be weakened to a certain extent.
Considering that the flow pulsation frequency (tooth frequency ) is positively correlated with the rotational speed and the number of teeth of the gear. Under high rotational speeds, the number of teeth should be reduced as much as possible to reduce the frequency of flow pulsation and increase the stability of the operation of the gear pump and the system. The simultaneous minimization of the flow pulsation coefficient and the flow pulsation frequency is the best situation for the flow quality of the pump. When the operating conditions are determined, theoretically, only under a specific number of teeth can the pressure pulsation of the circular – arc gear pump be well eliminated. Too many or too few teeth will have obvious impacts.
5.3 Influence of the Number of Teeth on the Pressure Characteristics in the Cavity of the Circular – Arc Gear Pump
The central section in the tooth width direction of the gear is taken as the pressure monitoring surface. Figure 14 shows the schematic diagram of the pressure distribution in the cavity of the 6 – 9 tooth gear pumps when they rotate by τ/4 respectively. It can be seen from the figure that with the increase of the number of teeth, the number of cascaded transition chambers in the gear cavity gradually increases. Taking the first rotation position in Figure 14(a – d) as an example, the pressure differences between the independent chambers of the 6 – 9 tooth pumps are 4.38MPa, 3.50MPa (for the 6 – tooth pump between 2D – 3D and 3D – 4D); 2.65MPa, 2.51MPa, 2.82MPa (for the 7 – tooth pump between 2D – 3D, 3D – 4D, and 4D – 5D); 1.99MPa, 2.01MPa, 2.01MPa, 1.93MPa (for the 8 – tooth pump between 1D – 2D, 2D – 3D, 3D – 4D, and 4D – 5D); 2.00MPa, 2.01MPa, 1.98MPa, 2.01MPa (for the 9 – tooth pump between 1D – 2D, 2D – 3D, 3D – 4D, and 4D – 5D). The pressure difference between adjacent independent chambers decreases step by step, and the pressure drop gradient gradually decreases. Moreover, with the increase of the number of teeth, the pressure distribution in the independent chambers is more and more the same when the gear rotates by τ/4.
During the gear meshing process, the areas of the high – pressure and low – pressure regions in the pump cavity change periodically. Due to the high import and export pressure difference, the shear force on the fluid element and the relative movement of the fluid particles increase, and the continuity of the flow decreases accordingly, resulting in a large pressure difference change between the fluids. At the inlet end, due to the increase in the variable volume of the bottom cavity hole, a local vacuum is formed. Under the tooth gap opening, eddy currents are likely to occur. In addition, the pressure difference between the inlet end and the first entering 1D/1S control volume is large, and the fluid velocity in the extrusion area is prone to distortion. Vortex flows appear below the cavity walls of the left and right gears. As the gear rotates, part of the vortex flow moves along with the gear rotation, forming a vortex within the effective volume between the gear and the cavity wall, and is distributed axially. The fluid velocity at this location is greater than that of the surrounding fluid, which enhances the flow instability. With the increase of the number of teeth, the position where the vortex concentrates gradually changes from near the center of the effective volume to the edge of the cavity wall, which has a positive effect on the stability of the fluid movement in the cavity. Part of the vortex flow at the inlet end remains there, disturbing the fluid that subsequently enters the gear cavity, causing wear to the gear cavity wall and impact losses, and some also leaks along the flow channel.
As the gear continues to rotate, the vortex continues to evolve towards the outlet end. Due to the interaction of the left and right gears, a pair of counter – rotating vortices are formed at the front and rear of the outlet end. Due to the effects of the helical angle and the gear width, the flow patterns and sizes of this pair of counter – rotating vortices at the front and rear of the outlet are different and are not symmetrical along the central axis, which affects the unbalance of the fluid pressure on the two gears. This unbalance is caused by the structure of the circular – arc gear itself. Although theoretically, the circular – arc gear pump can “completely eliminate” the flow pulsation, due to the evolution of the vortex pairs between the gear cavity walls, the unbalanced forces on the two gears, and the leakage at the gaps, the fluid in the cavity of the circular – arc gear pump shows instability.
In addition, the influences of tooth gap leakage and radial leakage on the pressure characteristics in the cavity are mainly considered. The leakage between the two gears depends on the opening area and the pressure difference between the upper and lower parts of the opening. Different numbers of teeth result in different action lines (the contact points of the two meshing gears) of a pair of meshing helical gears, and the changes in the opening area are also different. For radial leakage (the radial clearance of this model is set to 0.02mm), since the geometric shape of the clearance is not a single thin – layer lubrication clearance, the flow here can be laminar or turbulent (depending on the Reynolds number, and the hyperbolic tangent function is used to realize the transition between laminar and turbulent flows, which is specifically represented in ). Therefore, the radial leakage model [4] uses the modified leakage equation:
In reference [4], is defined as the flow coefficient, is the orifice opening area between the tooth top and the cavity wall, and ρ is the fluid density. With the increase of the number of teeth, the between adjacent transition chambers decreases, which has a certain inhibitory effect on radial leakage. Therefore, appropriately increasing the number of teeth can effectively suppress the flow instability caused by the pressure difference during the operation of the pump.
5.4 Influence of the Number of Teeth on the Excitation Force of the Circular – Arc Gear
The two gears are symmetrically distributed and rotate synchronously in opposite directions. Taking the driven gear as the research object, its radial force and axial force are analyzed.
Figure 15 shows the pulsation amplitude diagrams of the forces on the 6 – 9 tooth driven gears within one cycle. The radial force is the resultant force of the components in the x and y directions, and the axial force takes the absolute value of the force along the z – axis. It can be seen from Figure 15 that the radial and axial forces on the gears with different numbers of teeth change periodically, and the number of force pulsations of the gears is in a multiple relationship with the number of teeth. Assuming that the hydraulic oil is incompressible, this is mainly because the periodic movement of the gear pump leads to the periodic change of the volume of each control body (1D – 9D, 1S – 9S), so the internal flow field of the pump changes periodically.
The change of the gear force is related to the gear meshing and the interaction between the gear and the fluid in the cavity. With the increase of the number of teeth, decreases, first decreases and then increases. For the resultant force F, with the increase of the number of teeth, the maximum value of the 9 – tooth gear is 71.38% smaller than that of the 6 – tooth gear, and the maximum value of is 79.37% smaller. Within one cycle, with the increase of the number of teeth, the radial and axial forces on the gear gradually decrease, and the pulsation amplitude gradually stabilizes. Since the 6 – tooth gear has fewer transition chambers, the pressure difference between the transition chamber and the high – pressure chamber at the outlet is large, and a backflow phenomenon is likely to occur, which exerts an opposite force on the gear. Moreover, the effective volume in the cavity of the 6 – tooth gear increases, and flow separation and secondary flow phenomena are more likely to occur.
By performing wavelet transform on the radial and axial forces through Matlab, under the condition that the bandwidth and center frequency are set to cmor1 – 1.5, the scale is converted into frequency, the corresponding continuous wavelet coefficients are solved, and the wavelet responses (pulsation intensities) of the original functions (the radial and axial forces on the gear) in the time and frequency domains are obtained. Figure 16a shows the pulsation frequency spectrum diagram of the radial force on the 6 – 9 tooth gears within one cycle, showing the time – scale change, frequency – scale change, intensity, and phase distribution of the wavelet response of the radial force on the gear within one cycle.
Each pair of gear meshing forms a tooth frequency cycle. When the gear rotates one week, a total of Z pairs of gears alternately mesh to form a rotational frequency cycle. Within one rotational frequency cycle, processes such as meshing, single/double tooth meshing, and disengagement occur, resulting in multiple sudden changes in the effective volume. This is specifically presented in the low – and high – frequency parts of the figure. As the gear rotates, the positive and negative phases of the wavelet response alternate, and the alternating frequency is related to the multiple of the number of teeth. Therefore, the periodic change of the gear radial force is related to the gear meshing. It can be seen from the above figure that the main frequency of the radial force pulsation of the gears with different numbers of teeth all appears at 1 times the tooth frequency, where the vibration amplitude is the largest, indicating that this is the main influencing factor of the gear force pulsation. The wavelet response intensity decreases step by step from low frequency to high frequency, and the frequencies at which the secondary frequencies appear are all multiples of the tooth frequency. Therefore, the change of the wavelet response intensity is related to the meshing rotation of the two gears and the dynamic – static interference between the gear and the pump cavity.
