Research on Multi-Objective Optimization of Low Pulsation Unloading Damping Groove of Axial Piston Pump

Abstract

The high- and low-pressure switching of the axial piston pump is realized by the structure of the valve plate, and the buffer-groove structure on the valve plate is very important to reduce the pressure shock and flow fluctuation. In order to optimize the structural parameters of the buffer tank of the piston pump, the influence of the triangular-groove structure on the outlet pressure–flow characteristics was analyzed, and the low-pulsation buffer-groove structure was designed. First, the relationship between the structure of the triangular buffer tank and the output pressure and flow characteristics of the pump was analyzed theoretically. The Computational Fluid Dynamics (CFD) method was used to calculate the pressure and flow characteristics of the whole pump flow field of the triangular-groove structure, and the influence of the buffer-groove structure parameters on the outlet flow pulsation characteristics was studied. The multi-objective optimization algorithm was used to optimize the structure parameters of the buffer tank, and the optimized structure significantly reduced the outlet pressure–flow pulsation of the piston pump. The results show that, after optimization, the width angle of the front and rear triangular groove is 82.3°, the depth angle is 12.7°, the optimized pressure pulsation rate is 0.3%, and the optimized flow pulsation rate is 13.7%.

1. Introduction

The axial piston pump, as a kind of volumetric power hydraulic component, has the advantages of a compact structure, high efficiency, convenient flow adjustment, etc. It is widely used in industry, construction, agriculture, shipping, and aviation, as well as other types of hydraulic systems that require high pressure, a large flow, and precise control. However, the axial piston pump, due to its unique structure and working mode, inevitably faces problems as a result of its work in the process of flow pulsation and pressure impact.

In order to overcome or weaken the effects of flow pulsation and pressure shock, a buffer groove is usually opened in the transition area of the flow distribution disc, which can effectively regulate the pressure change in the piston chamber in the pre-boosting and pre-discharge phases and ensure the stability of the piston pump in the working process. However, the structural design of the cushion groove and parameter matching is often a key indicator of the pressure–flow performance of the piston pump. If the cushion groove is not properly designed, it may lead to increased flow pulsation at the outlet of the piston pump, and the pressure shock inside the pump is enhanced, which affects the stability of the pump. Therefore, although the buffer tank size, location, and structure of the optimized design have a certain engineering value, the pump output flow, pressure, and efficiency performance indicators have been improved accordingly.

In the recent research, scholars have carried out relevant research on the optimal design of buffer tank structures. J. O. Poulmberg [1] from Linkoping University, Sweden, analyzed the pressure shock generated during the high- and low-pressure switching process of piston pump and the flow backflow generated by the distribution link. The pressure shock at the outlet is reduced by optimizing the structural parameters of the buffer tank. Professor Xu Bing’s team [2] of Zhejiang University analyzed the influence law of the key parameters of a buffer tank with a typical structure and output pressure–flow characteristics, analyzed the variation trend of output flow characteristics, and obtained an optimized structure. Hong Haocen and Zhao Chunxiao et al. [3] from Zhejiang University proposed a data-driven multi-parameter optimization method for a buffer tank structure and calculated the optimal solution for that buffer tank structure based on the optimal target of flow pulsation through the data analysis of simulation results. Ericson [4] proposed a multi-parameter optimization algorithm based on the pre-compression cavity method by analyzing the contact cavity of the buffer tank. The optimized buffer tank structure obtained can effectively reduce the flow pulsation and pressure shock of the piston pump. Ma Jien [5] adopted the lumped-parameter method to conduct mathematical modeling of the flow pulsation and pressure shock of the piston pump and optimized the design of the valve plate structure of the piston pump. Liang Dedong [6] adopted the multi-objective genetic algorithm to optimize the depth angle and width angle of the triangular groove of the valve plate and optimized the structure to effectively reduce the fluid noise of the piston pump. Wei Xiuye et al. [7] analyzed the influence of the staggered angle of the valve plate on the flow pulsation of the axial piston pump and studied the influence of the staggered angle of different rotation angles on the pump flow pulsation based on Computational Fluid Dynamics (CFD) simulation. Song Yuning [8] and Gu Fan [9] et al. adopted the CFD simulation method to optimize the flow characteristics of the double V-type unloading groove, and the simulation results effectively reduced the flow pulsation at the pump outlet. Zhai Jiang [10] and Xie Shicong [11] et al. used Pumplinx to conduct a CFD numerical simulation of a certain type of axial piston pump and the axial piston motor of a certain type of static pressure transmission device and conducted an experimental verification of the results. The results show that if the internal flow domain grid division, numerical method, and calculation accuracy are reasonable, CFD technology can effectively predict its internal flow characteristics and can replace testing to a certain extent.

In this study, a multi-objective-based intelligent optimization algorithm is proposed to enhance the adaptive optimization computation of the buffer tank. The purpose of optimizing the buffer tank is to reduce the flow pulsation and pressure shock of the piston pump. Therefore, the content of this study mainly contains the following aspects: The theoretical model of the buffer tank flow is established in Section 2. The structural characteristic parameters characterizing the buffer tank are defined and the simulation model of the internal flow field of the whole pump of the piston pump is established in Section 3 and Section 4, and numerical calculations are carried out for the pressure and flow characteristics of the piston pump. Then, based on the simulation results, the influences of the buffer tank width angle and depth angle and the length on the dynamic characteristics of the pump are analyzed. Finally, the optimal solution of the buffer tank structure is obtained by the multi-objective genetic optimization algorithm and verified by the simulation method. Conclusions are drawn in Section 5. This study focuses on the low-pulsation optimization design of piston pumps, with the buffer-groove structure installed on the distribution plate as the research object. By using computational fluid dynamics methods, the flow characteristics and variation patterns of pumps under multiple different structural parameters are solved. A multi-objective intelligent optimization algorithm is proposed for the structural optimization of buffer slots to enhance the adaptive optimization calculation of buffer slots. Research has shown that optimization algorithms can greatly improve the efficiency of damping-groove optimization, and the optimization effect has been verified through simulation technology. The multi-objective optimization method proposed by this research institute can provide a certain reference for the optimization design of the buffer groove of the distribution plate.

2. Flow Model of Axial Piston Pump

The moving piston and the piston chamber of the cylinder form a sealed cavity. During the operation of the piston pump, the piston reciprocates inside the cylinder. As the flow channel in the piston chamber passes through the distribution plate and the distribution window, there is a pressure difference between the oil in the flow channel of the piston chamber and the oil in the flow channel of the distribution window. During the distribution process, the oil flows in or out, causing the damping hole formed by the piston chamber and the hourglass groove of the distribution plate to produce a throttling effect. The throttling formula, at this time, is as follows [7,10]:

$$q_t=C_v{A}_{hp}\sqrt{\frac{2\left|p_h-p_f\right|}{\rho }}\cdot sign(p_f-p_h)$$

(1)

$$q_n=C_v{A}_{lp}\sqrt{\frac{2\left|p_f-p_l\right|}{\rho }}\cdot sign(p_l-p_f)$$

(2)

where C_v represents the flow coefficient; A_lp and A_hp are the flow areas for the low- and high-pressure regions (unit: mm²), respectively; p_l and p_h are the return oil pressures for the low- and high-pressure regions (unit: MPa), respectively; and ρ is the density of the oil (unit: kg/m³).

As shown in Formulas (1) and (2), the flow areas, A_lp and A_hp, are linearly related to the flow rate through the throttling orifice, and they are important parameters affecting the variation in the flow rate at the distribution stage. Based on the research of predecessors, the core of the structural optimization of the piston pump’s distribution plate lies in optimizing the change process of the flow area. By optimizing the structure of the damping slots, the smoother the change trend of the flow area, the more effective it is at reducing the output flow pulsation of the piston pump, thereby achieving vibration reduction and noise reduction for the piston pump.

Combining the structural model of the triangular groove to analyze its flow area, shown in Figure 1, for the triangle efg, ef is the line connecting the contact point between the piston-chamber outlet groove and the top of the triangular groove; hg is perpendicular to ef; and φ is the included angle of ah on the distribution circle of the outlet groove (cylinder block rotation angle). The triangle efg is the cross-section of the damping groove perpendicular to the plane abc, and its area is as follows:

$$S_{\Delta efg}=ef·\mathrm{gh}/2={(R_f\phi )}^2\cdot t{g}^2{\theta }_{1}tg\frac{{\theta }_2}{2}$$

(3)

The actual flow area is the smallest cross-sectional area through which the fluid flows, which is the projected area triangle efi of the flow line ad over the edge line ef, and the angle between it and the triangle efg is θ. The area is given by the following:

$$S_{\Delta efi}=\frac{a{h}^{2}t{g}^2{\theta }_{1}tg\frac{{\theta }_2}{2}}{tg{\theta }_1\sin \theta +\cos \theta }$$

(4)

$$S_{min}=a{h}^{2}tg{\theta }_1\sin {\theta }_{1}tg\frac{{\theta }_2}{2}$$

(5)

On the other hand, due to the fact that, during the contact process, ef is an arc related to the wide radius, r, of the outlet groove of the piston chamber, the flow area is as follows:

$$S_{min}=R_f^2{\phi }^{2}tg{\theta }_{1}sin{\theta }_{1}tg\frac{{\theta }_2}{2}$$

(6)

$$ef=2ah\cdot tg{\theta }_{1}tg\frac{{\theta }_2}{2}$$

(7)

Therefore, it is necessary to correct the flow area, and the correction factor is as follows:

$$K_s=\frac{\stackrel{⏜}{ef}}{ef}=\frac{r\cdot arctan(2R_f\phi \cdot tg{\theta }_{1}tg\frac{{\theta }_2}{2}/\left(2r\right))}{R_f\phi \cdot tg{\theta }_{1}tg\frac{{\theta }_2}{2}}$$

(8)

Thus, the corrected formula for the flow area of the triangular groove is obtained as follows:

$$S=R_{f}r\phi sin{\theta }_{1}arctan(R_f\phi \cdot tg{\theta }_{1}tg\frac{{\theta }_2}{2}/r)$$

(9)

According to this formula, it can be known that the size of the flow area of the triangular groove is related to its structural parameters, such as the depth angle, θ₁; width angle, θ₂; cylinder block rotation angle, φ; and distribution circle, R_f. The unit of angle is “°”, and the unit of distribution circle is “mm”.

The research object of this paper is a typical axial piston pump with plate distribution, which performs high- and low-pressure oil distribution through the valve plate. The piston cavity is connected with the inlet and outlet through the valve plate to suck and discharge oil. As long as there is a pressure difference between the oil in the piston cavity and the oil distribution window, it will play a throttling role. The throttling formula is as follows:

$$q_i=C_{r}A\sqrt{\frac{2\left|p_f-p_i\right|}{\rho }}sign\left(p_f-p_i\right)$$

(10)

where q_i is the throttling flow rate, C_r is the flow coefficient, A is the flow area(unit: mm²), P_f is the oil pressure in the piston cavity (unit: MPa), p_i is the oil pressure in the flow distribution chamber (unit: MPa), and ρ is the oil density (unit: kg/m³). When the piston is connected with the damping groove, the flow area is small and changes quickly, which has a great influence on the flow and pressure pulsation.

The pressure in the piston chamber will experience two pressure transitions in a piston pump operation cycle: the piston from oil absorption to oil discharge and from oil discharge to oil absorption. The two transitions, namely when the piston is connected with the damping groove, will cause a total of four pressure shocks. It is found that [5], in these two transitions, if the increase/decrease rate of the flow area can keep pace with the increase/decrease rate of the axial movement velocity of the piston pump, the pressure shocks can be greatly reduced.

According to the formula, the flow area of the triangular groove (Figure 1) is mainly determined by the width angle and depth angle of its own structure, the radius of the bottom of the valve plate structure, the radius of the arc of the piston cavity, and the variable of the contact length between the piston cavity and the triangular groove determined by the rotational speed. When calculating the flow area of the valve plate, the vice radius of the spherical flow is very big, so we first simplified the arc ae in a straight line, and then valve tray on damping of the tank into the cylinder was cut to a corner of the plane, following the projection theory. According to the theory, the flow area, S, is always the smallest, so it conforms to the definition of flow area.

As a damping structure, the triangular groove plays a throttling role in the working process of the piston pump; especially when the damping groove and the piston chamber are connected, the throttling effect is more obvious. This will affect the normal flow of oil, thereby affecting the size and pulsation of the outlet flow rate of the piston pump. For triangular grooves, their characteristic parameters are generally divided into width angle, depth angle, and length:

For the width angle, when the width angle is small, the overcurrent area is also relatively small, which leads to a greater damping effect of the groove. Therefore, the resistance of flow backflow increases, and the amount of backflow decreases, thus creating significant pressure shocks. When the width angle is large, the flow area gradually increases, the damping effect decreases, and thus the pressure impact is reduced. When the width angle increases to a specific value, the flow rate and pressure impact reach their minimum, and at this point, the time for the pressure to reach steady state remains basically unchanged. When the width angle continues to increase, the flow area becomes too large, and the damping effect of the groove is greatly weakened, leading to severe flow backflow. Therefore, the time for pressure to reach steady state is significantly shortened, but the pressure impact is significantly increased.

For the depth angle, it mainly affects the jet angle of the oil flowing out of the groove and the outlet pressure of the front section of the waist-shaped groove for oil discharge. When the triangular groove at the front end of the piston chamber and the oil-discharge waist groove are connected, the greater the depth angle, the faster the hydraulic recoil speed of the high-pressure oil, resulting in a change in outlet pressure. When the depth angle is a specific value, the pressure overshoot during oil discharge is the lowest, and the flow pulsation at the oil suction and discharge ports is the smallest. But when the depth angle is too small or too large, the pressure overshoot during oil suction and discharge will significantly increase.

In terms of length, when determining the width angle and depth angle of the triangular groove, the change in length has little effect on the speed and direction of oil flowing through the triangular groove, but it has a significant impact on the outlet pressure of the front section of the oil-discharge waist groove.

3. The Flow Field Model

According to the simplified model, the flow field in the pump cavity of the piston pump was extracted from Space Claim Direct Modeler (SCDM(2024.R1)), which is a preprocessing modeling software. The flow field in the pump cavity of the piston pump was initially extracted, and the diverting field models of each part were assembled according to the actual positions in the pump. The fluid in and out of the oil port, valve plate, and piston are meshed, and the meshed sizes are shown in

Table 1. Grid size of each part of fluid domain.

	Inlet and Outlet	Valve Plate	Piston
Maximum cell size	0.01	0.008	0.005
Minimum cell size	0.001	0.0004	0.0005
Cell size on surfaces	0.005	0.002	0.002
Number of cells	1,391,161
Number of faces	5,035,447
Number of nodes	2,012,674

.

The external load carrying capacity of the oil film is inversely proportional to the thickness of the oil film, proportional to the area of the disk, and proportional to the dynamic viscosity. As deduced from the analysis of the above, the pump size is the same case: it constructs a pair of spherical surface areas of the oil film that is greater than the cone surface area of the oil film; the thickness is the same, so the bearing capacity of the deputy piston pump with a sphere than the cone valve piston pump, and the bearing capacity means that the ability of the traffic impact, presumably with a sphere pair of piston pump flow pulsation, is less than the cone with vice piston pump flow.

4. Analysis of Simulation Results

4.1. Optimization Design and Pulsation Simulation Analysis of Triangular Groove

This paper analyzes the distribution characteristics of a certain type of closed axial piston pump, focusing on the analysis of the distribution disk model with a unidirectional triangular groove. Calculations are performed for models with width angles of 70°, 90°, and 110°, respectively. The structure of the distribution disk and the flow field model are shown in Figure 2.

The computational fluid dynamics method used in this study discretizes the fluid calculation domain of the piston pump and then uses control equations to solve the physical quantities of each discrete element. Numerical calculations are completed by dividing the watershed grid, setting boundary conditions, and controlling the solution parameters.

For watershed grid division, the specific details are as follows:

(1)

Import and export watershed: This watershed is a stationary watershed, with a streamlined surface and not-too-severe bending mutations. Therefore, when dividing the grid, the number of grids can be appropriately reduced. Grid generation is achieved through the universal grid-partitioning technique provided by CFD software (Pumplinx R.4.6.0), and the corresponding grid control parameters are set as follows: maximum grid size, minimum grid size, and face grid size.

(2)

Distribution plate watershed: This watershed is a stationary watershed, with fine structures of pre-boost buffer tanks and pre-release buffer tanks distributed on the distribution surface. Therefore, when dividing the grid, it is necessary to increase the number of grids. This is to better calculate the fluid flow state of the top window of the cylinder block plug hole after passing through the buffer tank, making the calculation results more accurate. Grid generation is carried out through the universal grid-partitioning technique provided by CFD software, and the corresponding grid control parameters are set as follows: maximum grid size, minimum grid size, and face grid size. And the control parameters need to be appropriately reduced.

(3)

Distribution sub oil film watershed: This watershed is a stationary watershed that plays a crucial role in lubrication and leakage control of the distribution sub. Due to its radial thickness reaching the micrometer level, the number of grids needs to be appropriately increased during grid division. Using finite-element preprocessing tools, we divide the mesh here, set the radial thickness to 20 μm, and evenly divide it into four layers.

(4)

Piston rotor basin: This basin is a motion basin, and the motion form of the rotor is composite motion, that is, reciprocating linear motion in the cylinder block plug hole and rotating motion with the cylinder block. For the motion grid, each time step is calculated to reconstruct and deform the watershed boundary according to the laws of motion. Therefore, when dividing the grid, the piston rotor grid template included in CFD software is selected. By setting the piston rotor parameters—number of pistons, rotation center, rotation axis, and grid control parameters (the same as the general grid template)—the grid fully satisfies the motion form of the piston rotor during the working process.

For boundary conditions, the specific details are as follows:

The boundary conditions of computational fluid dynamics are necessary conditions for limiting the computational fluid domain and obtaining the target results through a numerical simulation. For the boundary conditions in this study, pressure boundary conditions are used for both the inlet and outlet, with the pressure inlet (0.35 MPa) and pressure outlet (25 MPa) set, respectively. Considering the leakage of the oil film in the clearance of the distribution pair, the outer edge of the oil film is set as the pressure outlet (0.5 MPa). In addition, except for the pressure boundary, all other boundaries are uniformly set as the solid wall surface, and the rotation speed of the piston pump is set to 1000 rpm.

When simulating, the following assumptions are made: The inlet and outlet basins of the piston pump are laminar flow, the outlet basins are low Reynolds number turbulence, the pre-pressure boosting and pre-pressure relief buffer slots of the distribution plate are high Reynolds number turbulence, and the rotor piston chamber is high Reynolds number turbulence.

Under different buffer-groove structures, the flow rate information at the outlet is extracted. As can be seen from Figure 3, when the width angle is selected to be 70°, the amplitude of the flow rate pulsation is the smallest, with a pulsation rate of 11.82%. However, there is a significant reverse flow of the flow rate during the pressure-rise stage, which results in a larger impact on the output flow [12,13].

Similarly, a simulation analysis is conducted on the buffer-groove structures with different depth angles, and the outlet flow rate information is extracted, as shown in Figure 4. Calculations are performed for models with depth angles of 7°, 8°, and 9°, while keeping the boundary conditions unchanged.

A further simulation analysis is conducted on the buffer-groove structures with different lengths of the triangular grooves, and the outlet flow rate information is extracted, as shown in Figure 5. Calculations are performed for models with lengths of 17 mm, 20 mm, and 23 mm, while maintaining the boundary conditions constant.

In the above figures, the angle represents the rotation angle of the cylinder block. In this study, the piston pump has nine pistons. When the pump rotates one cycle, it experiences nine periodic flow pulsations with the same variation pattern, with a 40° gap between each cycle. In order to better extract the detailed characteristics of flow-pulsation changes, a single pulsation cycle was selected for analysis, that is, the flow variation when the cylinder block rotates over 40°.

Through the above analysis, it can be seen that the depth angle, width angle, and length of the triangular groove all have an impact on the output pressure and flow rate characteristics. On the other hand, since there is no buffer-groove structure machined at the rear end of the hourglass-shaped flow passage of the distributor, the triangular-groove structure will not perform a pre-compression function when switching the rotation direction. A simulation calculation is performed for the triangular-groove buffer structure with a width angle of 70°, a depth angle of 7°, and a length of 20 mm, to solve for the output flow characteristics of the piston pump at a reversal speed of 1000 rpm, as shown in Figure 6. The width angle, depth angle, and length are the important parameters to determine the shape and size of the triangular groove. In general, when the piston window leaves the waist groove of the oil suction chamber, it should be in contact with the triangular groove at the oil outlet, so the length of the groove does not change, thus, remaining constant. According to previous studies, the influence of depth angle and width angle of the triangular groove on the pulsation of the piston pump is complementary. As the depth angle increases, the width angle should decrease, so that the change rate of the flow area can be maintained at a certain level to achieve the optimal pulsation, which also conforms to the theoretical derivation and calculation formula of the flow area of the triangular groove in this paper. Therefore, according to experience, the depth angle is selected as 7° to simplify the procedure and optimize the design of triangular groove only from the angle of width.

In Figure 6, it can be seen that the triangular trough with a width angle of 70° has the most effective influence in suppressing the pump flow pulsation, and the troughs are significantly higher than the other two. The enlarged figure on the right shows that all three have two peaks, but the two peaks of 70° are closer and tend to become stable faster; in other words, the fluctuations are smaller (

Table 2. Different width angles correspond to pump flow pulsation data.

Width angle of triangular groove (°)	110	90	70
Mean outlet flow (L/min)	550	551	550
Flow pulsation amplitude (L/min)	75	70	65
Flow pulsation rate (%)	13.64	12.70	11.82

).

In the range of 70°–110°, the smaller the width angle of the triangular groove, the better the suppression effect on the flow pulsation of the piston pump.

As shown in Figure 7, from the crest, when the width angle is greater than 70°, crest 1 decreases with the increase in the width angle. When the width angle is less than 70°, the crest 1 decreases with the decrease in the width angle. However, no matter how the width angle of the triangular trough changes, wave peak 2 remains unchanged. From the point of view of the trough, it is still a 70° width angle that is needed to achieve the best effect. The outlet flow pulsation of the 70° pump is optimized by 7% compared with that of the 90° pump. Meanwhile, it is found that the width angle of the triangular groove has no effect on the average outlet flow of the pump, and it has little effect on the outlet flow pulsation crest 2 of the pump.

As shown in Figure 8, compared with the outlet flow rate, the pump outlet pressure pulsation rate can reach 0.5% or even lower, far less than the flow pulsation rate. It can also be seen from the figure above that the width angle of triangular groove has less than 0.01 Mpa influence on the amplitude of pressure pulsation at the pump outlet, while the suppression of pressure pulsation by triangular groove with a width angle of 70° is still optimal.

4.2. Multi-Objective Optimization of Damping Grooves Based on the N, SGA-II Method

The Nondominated Sorting Genetic Algorithm II (NSGA-II) is a classic algorithm in the field of multi-objective optimization that is known for its high diversity and strong convergence properties. It has been widely applied in engineering fields. This paper employs a selection strategy based on binary tournament selection, and the genetic strategy adopts a simulated binary crossover strategy, the principle of which is

$$\begin{array}{l}{x}_{t+1,1}=0.5[(1+{\beta }_i)x_{t,1}+(1-{\beta }_i)x_{t,2}]\\ x_{t+1,2}=0.5[(1-{\beta }_i)x_{t,1}+(1+{\beta }_i)x_{t,2}]\end{array}$$

(11)

In the text, x_t,1 and x_t,2 represent the offspring individuals in the t-th generation, while x_t+1,1 and x_t+1,2 are the individuals produced by crossover in the (t + 1)-th generation. β_i is the crossover operator of the algorithm. This paper employs a polynomial mutation strategy, the principle of which is as follows:

$$x_{t+1}=x_t+\delta (x_{\max }-x_{\min })$$

(12)

where X_max and X_min are the upper and lower bounds of the current individual’s corresponding objective, and δ is the mutation operator. The parameter settings related to this algorithm are shown in

Table 3. Parameter settings related to NSGA-II.

Parameter Name	Value Range
Population size	100
Number of generations/iterations	100
Crossover probability	0.9
Mutation probability	0.1
Simulated binary crossover parameters	20
Polynomial mutation parameters	20

[9].

In this paper, the NSGA-II algorithm is used for the multi-objective optimization of damping-groove structure parameters. For the NSGA-II algorithm, a larger population size provides a wider search space, so for this example, the population size is set to 100. Through multiple calculations, it is found that when the number of iterations is greater than 100, the example converges, so the number of iterations is set to 100. The efficiency of cross-pairing and the efficiency of mutation operations are used for optimization fine-tuning, which is set to 20 in this example. In addition, a higher crossover probability helps to explore new regions of the solution space, while a higher mutation probability helps to maintain population diversity, so the crossover probability is set to 0.9, and the mutation probability is set to 0.1 in this example.

The key structural parameters of the damping groove are known, including the depth angle, width angle, and length of damping groove. Therefore, the constraint range of optimization parameters is shown in

Table 4. Multi-objective optimization variable table for damping groove.

Optimization Objective	Optimized Value Range	Unit
Trigonal-groove width angle	[50, 150]	°
Depth angle of triangular groove	[5, 20]	°
Triangular-groove length	20	Mm

. Taking the outlet pressure shock and flow pulsation as optimization objectives, the bidirectional buffer tank configuration was solved by multi-parameter optimization. The center symmetry design was adopted for the allocation of the buffer tank, and the NSGA-II algorithm was adopted for multi-objective optimization [6,14]. The optimization variables are shown in Table 4.

The triangular groove is independently optimized to reduce the outlet flow pulsation of the piston pump. Based on the previous analysis of the pressure–flow characteristics of the piston pump [14], when the bidirectional configuration is added, it mainly affects the output flow backflow characteristics and leakage characteristics. Therefore, the objectives of the optimization strategy are set as the amplitude of the flow pulsation, the maximum flow, the local minimum of the flow curve, the rising gradient of the flow curve, and the descending gradient of the flow-inversion curve.

The flow pulsation amplitude calculation formula is as follows:

$$D_{fr}=\max \left(q_{out}\right)-\min \left(q_{out}\right)$$

(13)

The formula for calculating the maximum value of stream scene is as follows:

$$D_{qmax}=\max \left(q_{out}\right)$$

(14)

The local minimum calculation formula of the flow curve is as follows:

$$D_{qmin\prime }=\min \prime \left(q_{out}\right)$$

(15)

The ascending gradient calculation formula of the flow curve is as follows:

$$D_{dq1}=\frac{q_{out1}-q_{out2}}{t_{q_{out1}}-t_{q_{out2}}}$$

(16)

The descending gradient calculation formula of the flow-inversion curve is as follows:

$$D_{dq2}=\frac{q_{out3}-q_{out4}}{t_{q_{out3}}-t_{q_{out4}}}$$

(17)

where $D_{fr}$ is the amplitude of flow pulsation, $D_{qmax}$ is the maximum flow rate, $D_{qmin\prime }$ is the local minimum of the flow curve, $D_{dq1}$ is the rising gradient of the flow curve, and $D_{dq2}$ is the descending gradient of the flow-reversal curve.

The objective function of optimization is as follows:

(18)

The calculation results converge to the global optimum of the optimal pulsation rate (within the value range). The comparison between the optimized results and the pressure and flow characteristics of the original structure is shown in Figure 9 and Figure 10.

The optimal solution is selected based on the minimum-distance selection method, and the distance between each solution on the Pareto front plane and the ideal point is calculated. The optimization results show that, for the two-quadrant piston pump structure that needs to be adjusted forward and backward, the configuration of the plate buffer groove should adopt the center symmetry configuration, and the triangle groove has better pressure and flow characteristics with the same optimal structural parameters. The optimized structural parameters are as follows: front/rear triangular-groove width angle, 82.3°; and depth angle of the front/back triangle groove, 12.7°. The same structural parameter configuration of the front triangle groove and the back triangle groove can ensure the same pressure and flow characteristics during positive and negative rotation. The comparison of the pressure–flow characteristics with the original structure shows that the flow pulsation rate of the optimized structure is 13.7%, which is 0.5% lower than before optimization (flow pulsation rate is 14.2%). The pressure pulsation is 0.3%, which is 0.1% lower than before optimization (0.4%), and the pulsation amplitude is 20.09 MPa.

5. Conclusions

This study focuses on the optimization design of buffer tanks. By using computational fluid dynamics simulation methods, numerical simulations were conducted on multiple different combinations of buffer tank structures, and a detailed analysis was conducted on the strong coupling relationship between the structural parameters of buffer tanks and the pressure–flow characteristics of plunger pumps. Then, a multi-objective intelligent optimization algorithm was proposed to adaptively optimize the buffer tank and obtain the optimal buffer tank configuration. The conclusion is as follows:

• Simulate and compare the individual characteristic parameters of the buffer tank: For the width angle of the buffer tank, when selecting a width angle of 70°, the amplitude of flow pulsation is the smallest, with a pulsation rate of 11.82%. For the depth angle of the buffer tank, when the depth angle is 8°, the flow pulsation amplitude of the pump is the smallest. For the length of the buffer tank, when the length is 23 mm, the flow pulsation amplitude of the pump is the smallest.
• By using multi-objective optimization algorithms to optimize the configuration of the distribution plate buffer groove, the optimized width angle of the buffer groove is 82.3°, and the depth angle is 12.7°. The comparison of pressure–flow characteristics with the original structure shows that the optimized structure has a flow pulsation rate of 13.7%, which is 0.5% lower than before optimization (flow pulsation rate of 14.2%). The pressure fluctuation is 0.3%, which is 0.1% lower than before optimization (0.4%), and the fluctuation amplitude is 20.09 MPa.

In the future, research on the experimental verification of optimization models will be conducted by the author, focusing on exploring the unloading characteristics and multi-objective optimization design of buffer tanks with various structural types, such as triangular, moment, and combination. Additionally, the dynamic characteristics of the oil film within the friction pair of the pump’s flow distribution will be solved, and adaptive optimization research will be carried out on the key design dimensions of the flow distribution pair.