Dynamic lubrication characteristic analysis method for plunger pair of aviation fuel plunger pump

Abstract

The invention discloses a method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump. The method comprises the steps of 1, constructing a plunger pair kinematics and dynamics model which is used for establishing a plunger pair hydrodynamic lubrication model and a plunger dynamic force balance model; step 2, constructing and solving a plunger pair lubrication model used for establishing a plunger dynamic force balance model; and step 3, solving a plunger full-cycle dynamic force balance model and numerical values for analyzing the change conditions of the minimum oil film thickness, the pressure peak value and the leakage flow of a plunger pair in the working process of the plunger pump, and providing guidance for structure and performance optimization of the plunger pump. The method closely fits the reality, has the characteristics of high prediction accuracy and high stability, can truly reflect the real condition of lubrication of the plunger by an oil film in the working process of the plunger pump, and provides certain guidance for design and performance optimization of the aviation fuel plunger pump.

Description

Technical Field

[0001] The invention relates to the technical field of aviation fuel plunger pumps, and in particular to a method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump. Background Art

[0002] The aviation fuel plunger pump is the "heart" of the fuel supply system in the aircraft engine. It has harsh working environment conditions and has high pressure, high speed, wide temperature range (-20℃~120℃) and other working conditions. With the continuous improvement of the main engine's requirements for aircraft engine performance, higher requirements are put forward for the service performance of the fuel plunger pump under extreme working conditions. Among them, the three key friction pairs of the plunger pump play the role of sealing and supporting, and their lubrication and friction and wear performance directly determine the performance, life and reliability of the plunger pump. However, the analysis of the dynamic lubrication characteristics of the plunger pair of the domestic aviation fuel plunger pump still has problems such as unreasonable boundary conditions in the process of model establishment, large differences from the actual situation, and the over-simplification of the theoretical model, which cannot accurately reflect the complex dynamic lubrication under real working conditions. Therefore, the study of the lubrication support characteristics of the three key friction pairs of the plunger pump (plunger pair, slipper pair and distribution pair) has important engineering practice value. Summary of the invention

[0003] In order to overcome the above technical problems, the purpose of the present invention is to provide a method for analyzing the dynamic lubrication characteristics of the plunger pair of an aviation fuel plunger pump. The method is closely in line with reality, has the characteristics of high prediction accuracy and strong stability, and can truly reflect the actual situation of the oil film lubrication of the plunger during the operation of the plunger pump, providing certain guidance for the design and performance optimization of the aviation fuel plunger pump.

[0004] The technical solution adopted by the present invention is:

[0005] A method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump comprises the following steps:

[0006] Step 1: Construct the kinematic and dynamic models of the plunger pair to establish the plunger pair fluid dynamic lubrication model and the plunger dynamic force balance model;

[0007] Step 2: Construct and solve the plunger pair lubrication model to establish the plunger dynamic force balance model;

[0008] Step 3: The full-cycle dynamic force balance model and numerical solution of the plunger are used to analyze the changes in the minimum oil film thickness, pressure peak, and leakage flow of the plunger pair during the operation of the plunger pump, providing guidance for the structure and performance optimization of the plunger pump.

[0009] The step 1 is specifically as follows:

[0010] 1) Establish the plunger kinematic model

[0011] With the spherical joint as point o(o′), the piston motion coordinate system xyzo and the sliding shoe motion coordinate system x′y′o′ parallel to the swash plate surface are established respectively;

[0012] Since the plunger pump has a periodic working law, the starting position of the moving parts is artificially specified, and the upper dead point A0 is the starting position of the plunger movement;

[0013] When the plunger rotates with the cylinder body at any angle from the top dead point A0 When reaching point A, the displacement s of the plunger ball head along the axis of the plunger cavity relative to the initial position A0 is kz for:

[0014]

[0015] Where: R—radius of plunger distribution circle; β—inclination angle of inclined plate;

[0016] The plunger rotates with the cylinder at any angle When , the contact length between the plunger and the plunger cavity is:

[0017] l F = l Fmin -s k (2)

[0018] Where: l Fmin —The minimum contact length corresponding to the plunger being at the top dead point A0;

[0019] When the plunger moves from the top dead center to the bottom dead center B0, the relative displacement change value of the plunger is H k for:

[0020] H k =2Rtanβ(3)

[0021] The displacement of the plunger relative to the initial position A0 is s k Taking the derivative of time t, we can get the velocity v of the plunger when it reciprocates in the plunger cavity. kz :

[0022]

[0023] Where: ω—cylinder rotation angular velocity;

[0024] The speed v of the plunger when it reciprocates in the plunger cavity k Taking the derivative of time t, we can get the acceleration a of the plunger when it reciprocates in the plunger cavity. k :

[0025]

[0026] The linear velocity v of the plunger when it moves in a circle around its own axis in the plunger cavity kx :

[0027] v kx =ω k r k (6)

[0028] Where: k — angular velocity of plunger rotation (considered to be consistent with the angular velocity of cylinder body rotation); r k — plunger radius;

[0029] 2) Establish plunger dynamics model

[0030] The forces acting on the plunger include the pressure of the plunger chamber oil on the plunger, the inertia force of the plunger along the axis of the plunger chamber, the friction between the plunger and the bushing, the interaction force of the sliding shoe on the plunger, the centrifugal force of the plunger around the axis of rotation of the cylinder body, the hydraulic support force of the plunger auxiliary oil film on the plunger, and the solid contact force between the plunger and the bushing. All the forces acting on the plunger are divided into three types: external force, fluid force and contact force.

[0031] 21) External forces

[0032] The external forces include the pressure of the plunger chamber oil on the plunger, the inertia force of the plunger along the axis of the plunger chamber, the friction between the plunger and the bushing, the interaction force of the sliding shoe on the plunger, the centrifugal force of the plunger around the axis of rotation of the cylinder body, and the torque generated by each force. k y k z k o k , the coordinate system rotates with the cylinder body, so that z k The axis is always located on the axis of the plunger cavity, y k The axis always points radially outward, and the origin of the coordinate system is o k It is always placed at the beginning of the fluid film interface. Assuming that the starting position of the plunger pair is on one side of the displacement chamber, the fluid film moves along z k The piston reciprocates in the cylinder hole as the cylinder body rotation angle changes. The starting position of the fluid film interface will also change accordingly. Based on this, the analysis of the external forces / torques acting on the piston is as follows:

[0033] The force of the oil in the plunger chamber on the plunger:

[0034]

[0035] Where: p r —Instantaneous pressure of oil in plunger chamber; d k — plunger diameter;

[0036] exist When the plunger cavity is connected to the outlet of the plunger pump, p r At the outlet pressure p out Pulse nearby; When the plunger cavity is connected to the inlet of the plunger pump, p r At the inlet pressure p in Pulsating nearby;

[0037] Inertia force of the plunger along the axis of the plunger cavity:

[0038] F ak =-m k ·a k (8)

[0039] Where: m k — plunger mass; a k —Acceleration of the plunger during reciprocating linear motion;

[0040] The friction between the plunger and the bushing (since this friction is much smaller than the combined external force, only the shear flow effect is considered when the fluid flows in a plane):

[0041]

[0042] Where: μ is the dynamic viscosity coefficient of the fluid at 20°C; v k —The speed of the plunger when it reciprocates in the plunger cavity; h k —Difference between plunger hole radius and plunger radius; l F —Contact length between plunger and bushing;

[0043] The friction force of the sliding shoe pair on the plunger (since this friction force is much smaller than the combined external force, only the effect of fluid shear flow when the oil film is constant is considered):

[0044]

[0045] Where: v s —Linear velocity of the sliding shoe when it makes circular motion on the surface of the swash plate; h s —The distance between the sliding shoe and the swash plate when working; r1—The outer radius of the sliding shoe bottom surface; r2—The radius of the oil chamber at the center of the sliding shoe bottom surface;

[0046] The centrifugal force of the plunger around the cylinder axis:

[0047] F ωk =m k ·ω 2 ·R(11)

[0048] In the space rectangular coordinate system x k y k z k o k In the middle, the plunger moves along zk The net force in the direction is:

[0049] F Ak =F Dk +F ak +F Tk (12)

[0050] The interaction force between the sliding shoe and the plunger is perpendicular to the swash plate plane and its magnitude satisfies:

[0051] F sk =F ks (13)

[0052] F sk Along the space rectangular coordinate system x k y k z k o k The y in k With z k Directional decomposition, we can get:

[0053]

[0054] in

[0055] F skz =F Ak (15)

[0056] F sky =F Ak ·tanβ(16)

[0057] In summary, the external forces acting on the plunger are in the spatial rectangular coordinate system x k y k z k o k Medium k With y k The component of force in the direction is:

[0058]

[0059] The external forces acting on the plunger are related to the spatial rectangular coordinate system x k y k z k o k x in k With y k The component moment in the direction of the coordinate axis is:

[0060]

[0061] Where: z Rk —The distance between the plunger ball head and the spatial rectangular coordinate system o k distance;

[0062] l se —The distance between the plunger ball head and the plunger mass center;

[0063] 22) Fluid Force

[0064] The oil film hydraulic support force / torque of the plunger pair can be divided into: k ,y k The force in the direction of x k ,y k The moment of the shaft, take an infinitely small piece of oil film in the plunger oil film, assume that the pressure of the small piece of oil film is a constant value p, and the area of ​​the oil film is dx k dz k , then the force exerted by the small piece of oil film on the plunger is pdx k dz k .

[0065] The total hydraulic support force of the piston auxiliary oil film on the piston is obtained by numerically integrating the micro-element hydraulic support force in the circumferential direction and the axial direction:

[0066]

[0067] Plunger auxiliary oil film at x k and k The hydraulic support torque of the plunger in the direction is:

[0068]

[0069] The fluid force on the plunger is mainly the hydraulic support force and torque generated by the plunger auxiliary oil film on the plunger;

[0070] 23) Contact force

[0071] When the oil film thickness at each point of the plunger pair is determined, the elastic deformation can be expressed by the following formula:

[0072]

[0073] In order to ensure the stability of the calculation process of the plunger pair pressure field distribution, a reasonable assumption is made: h kmin Determined by the surface roughness of the plunger pair, making simplified assumptions about the pressure and the deformation of the plunger, the contact pressure is expressed as follows:

[0074]

[0075] Where: d k —Outer diameter of the plunger; d c —Inner diameter of bushing; h el —Elastic deformation of the plunger; E k — elastic modulus of the plunger;

[0076] The contact pressure of the bushing on the plunger along the coordinate system x is obtained by two-dimensional numerical integration. k ,y k Directional contact forces and moments;

[0077]

[0078] The contact force is introduced to consider the extreme situation that the plunger is in direct contact with the bushing due to excessive tilt of the plunger during movement, so as to make the calculation result more realistic.

[0079] The step 2 is specifically as follows:

[0080] 1) Reynolds equation for plunger pair

[0081] Expand the plunger auxiliary oil film into a plane shape, where x k The coordinate is the circumferential direction of the plunger, z k The coordinate is the axial direction of the plunger, H k The coordinate is the thickness direction of the plunger auxiliary oil film, and the axial length of the expanded oil film is the contact length l between the plunger and the cylinder body. F , the length in the circumferential direction is the circumference of the plunger 2πr k , the oil film characteristics of the plunger pair that is expanded into a plane rectangle will be analyzed and calculated in the rectangular coordinate system. Since it is assumed that the fuel medium is incompressible, its density remains unchanged, ρ = constant, and the Reynolds equation in the rectangular coordinate system is:

[0082]

[0083] Where: v kx —The speed of the oil film along the circumference of the plunger is related to the rotation of the plunger;

[0084] v kz —The speed of the oil film along the axial direction of the plunger is related to the reciprocating speed of the plunger in the plunger cavity;

[0085] 2) Modeling of plunger oil film thickness

[0086] In the tilt plunger geometry model, section 1 is the rectangular coordinate system x k y k z k o k Medium k o k y k The section 2 is a section along the axis of the plunger cavity and coincides with the x k o k y k Surface distance l F a cross section of

[0087] Among them, k Point is the origin of the coordinate system, x k is the direction of the tangential velocity of the plunger when it moves in a circular motion with the cylinder body, y k is the direction of the normal velocity of the plunger when it moves in a circular motion with the cylinder body, z k is the direction of the plunger cavity axis. On section 1, the position of the section circle center o1 is represented by a set of eccentric distances (e1, e2), where e1 represents the distance between point o1 and o k Click on x k The eccentricity of the direction, e2 represents the eccentricity of point o1 relative to o k Point at y k Similarly, on section 2, the position of the center o2 of the section circle is represented by a set of eccentric distances (e3, e4);

[0088] Since the plunger will tilt to a certain extent during its reciprocating motion, the sections 1 and 2 of the plunger are strictly speaking an ellipse. However, considering that the clearance between the plunger pair is much smaller than the plunger radius, the ellipse is approximated as a circle with a radius of the plunger radius r. k .

[0089] In particular, the spatial rectangular coordinate system xyzo presents the following coordinates:

[0090] x=φ k r k ,y=h,z=z k (30)

[0091] Select a point between section 1 and section 2 that is the origin of the coordinate system. k Distance z k For any cross section, the position of the center point o3 of the cross section circle can be expressed by a set of eccentric distances (e x ,e y ) indicates that, from the geometric proportions of similar triangles, we can conclude that:

[0092]

[0093] In the formula,

[0094]

[0095] We can get:

[0096]

[0097] Where: z k —With the coordinate origin o k The distance; l F —Contact length between plunger and bushing;

[0098] This formula can also be replaced by:

[0099]

[0100] Similarly, we can get:

[0101]

[0102] or

[0103]

[0104] x k Rotate the starting axis counterclockwise φ k The current oil film thickness value is calculated from the angle, where the oil film thickness can be obtained from the geometric relationship between Bo3 and the plunger radius:

[0105] h=Bo3-r k (37)

[0106] Where:

[0107]

[0108] Substituting formula (37) into formula (38), we get:

[0109]

[0110] In the same way, a normal extrusion velocity field can be defined, and the time derivative vector of the plunger eccentricity is The function of, that is, the rate of change of the oil film thickness at any point of the plunger pair can be expressed as:

[0111]

[0112] 3) Discretized Reynolds equation of plunger pair

[0113] The finite difference method is used to solve the two-dimensional Reynolds equation, which is generally divided into three steps: a. Generate a discrete grid; b. Construct a discrete equation; c. Solve the system of equations;

[0114] First, the expanded two-dimensional rectangular plunger pair solution domain is meshed and the solution domain is divided along the circumferential x k Direction and axis z k The direction is divided into m and n equidistant intervals, and the solution domain range is defined as 0≤x k ≤2πr k ,0≤z k ≤l F ,The control domain of the computational grid is 61 × 21 equidistant nodes, and the oil film pressure can be obtained using the half-step central difference format;

[0115] Then, according to the basic idea of ​​finite difference and half-step central difference, the Reynolds equation of the plunger pair is solved in the control volume, and the discrete form of the Reynolds equation of the plunger pair is obtained as follows:

[0116]

[0117] In the pair Performing half-step central finite difference gives:

[0118]

[0119] By replacing each term with A, B, C, D, F, G, and H, the equation can be simplified to:

[0120] Ap i+1,j +Bp i-1,j +Cp i,j+1 +Dp i,j-1 -Ep i,j =H (43)

[0121] in:

[0122]

[0123] Finally, solve the Reynolds equation after the piston pair is discretized;

[0124] The discretized Reynolds equation is transformed into the form of ax=b, where the coefficient matrix a is a strictly diagonally dominant matrix. Therefore, the super-relaxation iteration method is used to obtain:

[0125]

[0126] In the formula, the range of the relaxation factor ω is ω∈(1,2);

[0127] For the iteration termination condition, it is given according to the relative convergence criterion:

[0128]

[0129] When the relative change of two adjacent pressure fields is less than the given allowable error, it is considered that the pressure field has basically stopped changing, and the iteration is terminated. ε is the convergence accuracy. If ε is set too large, it will lead to a large calculation error. If it is too small, it will increase the number of iterations and even cause iterative divergence. Therefore, a reasonable error range must be selected to set the relative error ε to 10 -3 ; When the pressure at all points of the plunger auxiliary oil film has the relative error, the pressure value at each point of the oil film obtained is the distribution of the plunger auxiliary pressure field when the cylinder rotation angle is at this position.

[0130] 4) Initial conditions and boundary conditions of plunger pair

[0131] 41) Initial conditions of plunger pair

[0132] The initial conditions of the plunger pair are the values ​​of the eccentricity of the two end faces (e1, e2, e3, e4) corresponding to the start of the plunger movement and the value of its oil film thickness h. The oil film thickness of the plunger pair is generally related to the assembly clearance of the plunger-bushing and the surface roughness. The initial value of the plunger pair oil film thickness h is set to 30μm, and the initial value of the eccentricity of the two end faces is set to:

[0133]

[0134] 42) Plunger side boundary conditions

[0135] The boundary conditions of the plunger pair can be divided into Reynolds boundary conditions in the axial and circumferential directions;

[0136] The axial boundary condition is the pressure value corresponding to the beginning and end of the fluid film interface during the reciprocating motion of the plunger. The pressure at the beginning of the fluid film corresponds to the instantaneous pressure of the plunger cavity, and the pressure at the end of the fluid film corresponds to the pressure of the rotor cavity, which can be expressed by the following formula:

[0137]

[0138] Where: p r —Instantaneous pressure in the plunger cavity; p e —Pressure in the rotor cavity; x—Circumferential direction of the plunger in the two-dimensional rectangular plane, ranging from 0 to 2πr k ;

[0139] The circumferential boundary conditions are periodic boundary conditions. The plunger is expanded into a two-dimensional rectangular plane, so the pressure at the starting position of the plunger surface is equal to the pressure at the ending position of the plunger surface, which can be expressed by the following formula:

[0140] p(z,0)=p(z,2πr k )(49)

[0141] 5) Simulation of instantaneous pulsating pressure in the plunger cavity

[0142] Cylinder rotation angle When the plunger cavity is connected to the outlet of the plunger pump, the instantaneous pressure of the oil in the plunger cavity is p r At the outlet pressure p out Pulsation near the cylinder rotation angle When the instantaneous pressure of the oil in the plunger chamber is connected to the inlet of the plunger pump, the instantaneous pressure of the oil in the plunger chamber is p r At the inlet pressure p in Pulsating nearby;

[0143] Taking the pre-pressurization and pre-depressurization of the vibration-damping triangular groove into consideration, based on the structural parameters of the designed plunger pump and the empirical formula obtained from relevant literature, the variation law of the instantaneous pressure of the plunger chamber oil with the rotation angle of the cylinder body is simulated;

[0144] The conclusion of the instantaneous pressure change law of the oil in the plunger chamber can be drawn as follows:

[0145] (1) Due to the pre-pressurization effect of the vibration reduction triangular groove, the cylinder body rotates at an angle During the process, the instantaneous pressure of the oil in the plunger chamber gradually increased from 0.44MPa to 16.2MPa, and During the process, the pressure gradually decreases to around 15MPa at the outlet of the plunger cavity;

[0146] (2) Due to the pre-pressure reduction effect of the vibration reduction triangular groove, the cylinder body rotates at an angle During the process, the instantaneous pressure of the oil in the plunger chamber gradually increased from 15.2MPa to 19.5MPa, and During the process, the plunger cavity inlet pressure gradually decreases to around 0.5MPa.

[0147] The step three is specifically as follows:

[0148] The force balance model is solved by using finite difference and Newton-Raphson methods;

[0149] 1) Modeling of plunger dynamic force balance model

[0150] During the operation of the plunger pump, the plunger components are mainly subjected to: the pressure of the plunger chamber oil on the plunger, the inertia force of the plunger during the reciprocating motion, the friction of the plunger pair, the centrifugal force of the plunger in the circular motion with the cylinder body, the friction of the slipper pair on the plunger, the interaction force of the slipper on the plunger, the hydraulic support force of the plunger pair oil film on the plunger, and the solid contact force of the bushing on the plunger when the plunger tilts. These forces / torques are combined to establish the dynamic force balance equation of the cylinder body during the 0°~360° rotation process, as shown in the following formula:

[0151]

[0152] Where: e—eccentricity of the center points of the two cross sections of the plunger (e1, e2, e3, e4);

[0153] —Eccentricity derivative of the center points of the two sections of the plunger

[0154] F ek —Total external force on the plunger; F fk —Fluid force on the plunger; F ck —Contact force on the plunger;

[0155] M ek —Total external torque on the plunger; M fk —Fluid torque on the plunger; M ck —Contact torque on the plunger;

[0156] For a plunger pump with a constant speed, the normal inertia force of the plunger at each point on the oil film is much smaller than the hydraulic support force. Therefore, the normal inertia force of the oil film is ignored. Equations (50) and (51) can be transformed into:

[0157]

[0158] The resultant force f k and resultant moment M k Along x k Direction and y k The direction is decomposed into:

[0159]

[0160] The dynamic force balance model (54) of the plunger during the whole cycle is a nonlinear equation system, and there is no root formula. Therefore, the Newton-Raphson method is used to solve the approximate roots of the model and the derivative of the change of the eccentricity of the two cross sections is used. As the iteration variable, the iteration format of the Newton-Raphson method for the plunger pair is:

[0161]

[0162] In the formula, It can be expressed as:

[0163] or

[0164] in, and Combining into a 4×4 Jacobian matrix, replaced by J, the specific form is as follows:

[0165]

[0166] For equation group (54), given the piston rotation angle in the cylinder The initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ), and set the Newton-Raphson iteration convergence condition f k <10 -3 ,available Moment And through the time step Δt, the eccentricity at the next moment can be obtained The specific formula is as follows:

[0167]

[0168] By repeating the calculation in this way, the eccentricity value corresponding to each angle during the cylinder body rotation of 360° can be obtained.

[0169] Since the fluid force in equation (54) / Torque There is no analytical solution, which makes it difficult to solve each element in the Jacobi matrix. Therefore, the finite difference method of the central difference format is used to differentiate the values ​​of two or more adjacent numerical points to replace the partial derivatives in the partial differential equation. The specific difference format is as follows:

[0170]

[0171] Where, i = 1, 2, 3, 4, j = 1, 2, 3, 4;

[0172] 2) Solution of plunger dynamic force balance model

[0173] Cylinder rotation angle The initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ) can be any given value, but it must be less than the plunger-sleeve clearance and the force balance equation can be iteratively converged. The initial eccentricity of the plunger is set to (e 10 ,e 20 ,e 30 ,e 40 )=(0,0,0,0)μm, and it should be noted that when the cylinder body rotates 360° to reach the initial position for the second time, its eccentricity (e1, e2, e3, e4) is not necessarily exactly equal to the initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ). Therefore, it is necessary to continue the iteration until the plunger pump reaches the rated speed or stable working state, and the resultant force / torque corresponding to the same cylinder rotation angle remains unchanged, and the plunger eccentricity meets the following convergence conditions to terminate the iteration and complete the solution of the dynamic force balance model;

[0174]

[0175] Where: N is the number of revolutions of the cylinder.

[0176] Beneficial effects of the present invention:

[0177] The present invention combines the kinematic analysis and force analysis of the plunger, sliding shoe, cylinder body and other components, as well as the plunger auxiliary fluid dynamic lubrication model, and the changing law of the instantaneous pulsation pre-pressure increase and pre-pressure reduction of the plunger cavity oil when the vibration-damping triangular groove acts, to numerically solve the dynamic force balance equation of the plunger, establish the plunger auxiliary fluid dynamic lubrication model, simulate the lubrication characteristics of the plunger auxiliary oil film of the plunger pump under different working conditions, construct the corresponding relationship between the oil film state and the plunger wear, and give a reasonable inference on the cause of abnormal wear of the plunger and the cylinder bore from the perspective of the plunger auxiliary oil film lubrication, which can provide certain theoretical guidance for in-depth understanding of the plunger auxiliary oil film lubrication characteristics, improving the plunger-cylinder body structure, and improving related performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0178] Figure 1 It is a schematic diagram of the movement of the plunger and the sliding shoe.

[0179] Figure 2 It is a schematic diagram of plunger external force analysis.

[0180] Figure 3 It is a schematic diagram of the oil film in the gap between the plunger and the plunger hole.

[0181] Figure 4 It is the expansion diagram of the oil film along the generatrix of the plunger hole.

[0182] Figure 5 It is a schematic diagram for deriving the plunger pair oil film thickness.

[0183] Figure 6 It is a schematic diagram of grid division and difference format.

[0184] Figure 7 It is a schematic diagram of the oil film pressure distribution of the plunger pair of the plunger pump.

[0185] Figure 8 This is a graph showing how the instantaneous pressure of the oil in the plunger chamber changes with the cylinder rotation angle.

[0186] Fig. 9 It is the flow chart for solving the dynamic force balance model of the plunger throughout its entire cycle.

[0187] Fig.10 The center of the plunger section 1 is x k Graph showing the variation pattern of the eccentricity e1 in the direction.

[0188] Fig.11 The center of the plunger section 1 is at y k Graph showing the variation pattern of eccentricity e2 in direction.

[0189] Fig.12 The center of the plunger section 2 is x k Graph showing the variation pattern of eccentricity e3 in direction.

[0190] Fig.13 The center of the plunger section 2 is at y k Graph showing the variation pattern of the eccentricity e4 in the direction.

[0191] Fig.14 is the force exerted on the plunger k Graph showing the law of change of directional force.

[0192] Fig.15 is the pressure exerted on the plunger k Graph showing the law of change of directional force.

[0193] Fig.16 is the force exerted on the plunger k Graph showing the law of change of directional torque.

[0194] Fig.17 is the pressure exerted on the plunger k Graph showing the law of change of directional torque.

[0195] Fig.18 It is the distribution change diagram of the plunger oil film thickness.

[0196] Fig.19 It is the change diagram of the piston auxiliary oil film pressure distribution. DETAILED DESCRIPTION

[0197] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0198] A method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump comprises the following steps:

[0199] Part 1: Constructing the kinematic and dynamic model of the plunger pair

[0200] The first part is specifically:

[0201] 1) Establish the plunger kinematic model

[0202] During the operation of the aviation fuel plunger pump, the plunger reciprocates along the axis of the cylinder in the plunger cavity, and at the same time, it makes a circular motion around the center line of the cylinder as the cylinder rotates. In addition, the plunger also rotates around its own axis while revolving. Figure 1 The schematic diagram of the movement of the plunger and the slipper during the operation of the plunger pump is shown in Figure 1. The plunger motion coordinate system xyzo and the slipper motion coordinate system x′y′o′ parallel to the swash plate surface are established with the spherical joint as point o(o′).

[0203] Since the plunger pump has a periodic working law, the starting position of the moving parts can be artificially specified. Therefore, this paper stipulates that the upper dead point A0 is the starting position of the plunger movement.

[0204] When the plunger rotates with the cylinder body at any angle from the top dead point A0 When reaching point A, the displacement s of the plunger ball head along the axis of the plunger cavity relative to the initial position A0 is kz for:

[0205]

[0206] Where: R is the radius of the plunger distribution circle; β is the inclination angle of the inclined plate.

[0207] The plunger rotates with the cylinder at any angle When , the contact length between the plunger and the plunger cavity is:

[0208] l F = l Fmin -s k (2)

[0209] Where: l Fmin —The minimum contact length corresponding to when the plunger is at the top dead center A0.

[0210] When the plunger moves from the top dead center to the bottom dead center B0, the relative displacement change value of the plunger is H k for:

[0211] H k =2Rtanβ(3)

[0212] The displacement of the plunger relative to the initial position A0 is s k Taking the derivative of time t, we can get the velocity v of the plunger when it reciprocates in the plunger cavity. kz :

[0213]

[0214] Where: ω—cylinder rotation angular velocity.

[0215] The speed v of the plunger when it reciprocates in the plunger cavity k Taking the derivative of time t, we can get the acceleration a of the plunger when it reciprocates in the plunger cavity. k :

[0216]

[0217] The linear velocity v of the plunger when it moves in a circle around its own axis in the plunger cavity kx :

[0218] v kx =ωk r k (6)

[0219] Where: k — angular velocity of plunger rotation (considered to be consistent with the angular velocity of cylinder body rotation); r k —Plunger radius.

[0220] 2) Establish plunger dynamics model

[0221] During the operation of the aviation fuel plunger pump, the plunger reciprocates along the axis of the plunger cavity, and rotates in a circular motion around the cylinder axis with the cylinder body. After being acted upon by the circumferential friction of the oil in the plunger cavity, the plunger will also rotate around its own axis in the plunger hole. The forces acting on the plunger include the pressure of the plunger cavity oil on the plunger, the inertia force of the plunger along the axis of the plunger cavity, the friction between the plunger and the bushing, the interaction force of the sliding shoe on the plunger, the centrifugal force of the plunger around the axis of rotation of the cylinder body, the hydraulic support force of the plunger auxiliary oil film on the plunger, and the solid contact force between the plunger and the bushing. All the forces acting on the plunger are divided into three types: external force, fluid force and contact force.

[0222] 21) External forces

[0223] The external forces mainly include the pressure of the plunger cavity oil on the plunger, the inertia force of the plunger along the axis of the plunger cavity, the friction between the plunger and the bushing, the interaction force of the sliding shoe on the plunger, the centrifugal force of the plunger around the axis of rotation of the cylinder body, and the torque generated by each force. Figure 2 As shown, establish the space rectangular coordinate system x k y k z k o k , the coordinate system rotates with the cylinder body, so that z k The axis is always located on the axis of the plunger cavity, y k The axis always points radially outward. The origin of the coordinate system is o k It is always placed at the beginning of the fluid film interface. Assuming that the starting position of the plunger pair is on one side of the displacement chamber, the fluid film moves along z k The piston reciprocates in the cylinder bore as the cylinder body rotation angle changes, and the starting position of the fluid film interface changes accordingly. Based on this, the analysis of the external forces / torques acting on the piston is as follows.

[0224] The force of the oil in the plunger chamber on the plunger:

[0225]

[0226] Where: p r —Instantaneous pressure of oil in plunger chamber; d k —Plunger diameter.

[0227] exist When the plunger cavity is connected to the outlet of the plunger pump, p r At the outlet pressure p out Pulse nearby; When the plunger cavity is connected to the inlet of the plunger pump, p r At the inlet pressure p in Pulsating nearby.

[0228] Inertia force of the plunger along the axis of the plunger cavity:

[0229] F ak =-m k ·a k (8)

[0230] Where: m k — plunger mass; a k —The acceleration of the plunger during reciprocating linear motion.

[0231] The friction between the plunger and the bushing (since this friction is much smaller than the combined external force, only the shear flow effect is considered when the fluid flows in a plane):

[0232]

[0233] Where: μ is the dynamic viscosity coefficient of the fluid at 20°C; v k —The speed of the plunger when it reciprocates in the plunger cavity; h k —Difference between plunger hole radius and plunger radius; l F —Contact length between plunger and bushing.

[0234] The friction force of the sliding shoe pair on the plunger (since this friction force is much smaller than the combined external force, only the effect of fluid shear flow when the oil film is constant is considered):

[0235]

[0236] Where: v s —Linear velocity of the sliding shoe when it makes circular motion on the surface of the swash plate; h s —The distance between the sliding shoe and the swash plate when working; r1—The outer radius of the sliding shoe bottom surface; r2—The radius of the oil chamber in the center of the sliding shoe bottom surface.

[0237] The centrifugal force of the plunger around the cylinder axis of rotation:

[0238] F ωk =m k ·ω 2 ·R(11)

[0239] In the space rectangular coordinate system x k y k zk o k In the middle, the plunger moves along z k The net force in the direction is:

[0240] F Ak =F Dk +F ak +F Tk (12)

[0241] The interaction force between the sliding shoe and the plunger is perpendicular to the swash plate plane and its magnitude satisfies:

[0242] F sk =F ks (13)

[0243] F sk Along the space rectangular coordinate system x k y k z k o k The y in k With z k Directional decomposition, we can get:

[0244]

[0245] in

[0246] F skz =F Ak (15)

[0247] F sky =F Ak ·tanβ(16)

[0248] In summary, the external forces acting on the plunger are in the spatial rectangular coordinate system x k y k z k o k Medium k With y k The component of force in the direction is:

[0249]

[0250] The external forces acting on the plunger are related to the spatial rectangular coordinate system x k y k z k o k x in k With y k The component moment in the direction of the coordinate axis is:

[0251]

[0252] Where: z Rk—The distance between the plunger ball head and the spatial rectangular coordinate system o k distance;

[0253] l se —The distance between the plunger ball head and the plunger center of mass.

[0254] 22) Fluid Force

[0255] Fluid force analysis mainly includes the hydraulic support force, viscous friction force and other forces of the plunger oil film on the plunger and the torque generated. When solving the pressure distribution of the plunger oil film, the complex Simpson formula is used to calculate the double integral through the numerical integration method, and further calculate the parameters such as the plunger friction force, oil film bearing capacity and leakage. Since the viscous friction force in the circumferential direction is not considered, and the viscous friction force in the axial direction of the plunger has been discussed in the external force part, only the effect of the hydraulic support force is discussed in terms of fluid force.

[0256] The oil film hydraulic support force / torque of the plunger pair can be divided into: k ,y k The force in the direction of x k ,y k Take an infinitely small piece of oil film from the piston pair oil film, assume that the pressure of the small piece of oil film is a constant value p, and the area of ​​the oil film is dx k dz k , then the force exerted by the small piece of oil film on the plunger is pdx k dz k .

[0257] The total hydraulic support force of the piston auxiliary oil film on the piston is obtained by numerically integrating the micro-element hydraulic support force in the circumferential direction and the axial direction:

[0258]

[0259] Plunger auxiliary oil film at x k and k The hydraulic support torque of the plunger in the direction is:

[0260]

[0261] The fluid force acting on the plunger is mainly the hydraulic support force and torque generated by the plunger auxiliary oil film on the plunger.

[0262] 23) Contact force

[0263] Due to centrifugal force, the interaction force between the sliding shoe and the plunger, etc., the plunger will tilt during movement. If the tilt angle is too large, the oil film of the plunger pair will rupture, and the plunger will be in direct contact with the bushing. At this time, the bushing will produce a solid contact force on the plunger. For the calculation of contact pressure, it is necessary to solve the elastic contact equation. LarsOlems proved through experiments that considering the contact pressure can ensure the stability of the numerical calculation process. Therefore, the contact pressure is estimated using an empirical formula. When the oil film thickness at each point of the plunger pair is determined, the elastic deformation can be expressed by the following formula:

[0264]

[0265] In order to ensure the stability of the calculation process of the plunger pair pressure field distribution, a reasonable assumption is made: h kmin Determined by the surface roughness of the plunger pair. Making simplified assumptions about the pressure and the deformation of the plunger, the contact pressure can be expressed as follows:

[0266]

[0267] Where: d k —Outer diameter of the plunger; d c —Inner diameter of bushing; h el —Elastic deformation of the plunger; E k —Elastic modulus of the plunger.

[0268] By performing two-dimensional numerical integration of the contact pressure, the contact pressure of the bushing on the plunger along the coordinate system x can be obtained. k ,y k Directional contact forces and moments.

[0269]

[0270]

[0271] The contact force is introduced to consider the extreme situation that the plunger is in direct contact with the bushing due to excessive tilt of the plunger during movement, so as to make the calculation result more realistic.

[0272] Part 2: Constructing a plunger pair lubrication model

[0273] 1) Reynolds equation for plunger pair

[0274] The gap between the plunger surface and the bushing surface determines the oil film thickness of the plunger pair. The gap of the plunger pair is generally from a few microns to tens of microns, and the plunger radius is generally from a few millimeters to more than ten millimeters. Therefore, the oil film thickness of the plunger pair is much smaller than the plunger radius. Figure 3 The oil film between the plunger and the bushing is shown in the figure. Now the plunger auxiliary oil film is expanded into a plane shape. The expanded oil film is as follows Figure 4 As shown ink The coordinate is the circumferential direction of the plunger, z k The coordinate is the axial direction of the plunger, H k The coordinate is the thickness direction of the plunger auxiliary oil film. The axial length of the expanded oil film is the contact length l between the plunger and the cylinder body. F , the length in the circumferential direction is the circumference of the plunger 2πr k The oil film characteristics of the plunger pair that is a rectangular plane will be analyzed and calculated in the rectangular coordinate system. Since the fuel medium is assumed to be incompressible, its density remains unchanged, ρ = constant. The Reynolds equation in the rectangular coordinate system is:

[0275]

[0276] Where: v kx —The speed of the oil film along the circumference of the plunger is related to the rotation of the plunger;

[0277] v kz —The speed of the oil film along the axial direction of the plunger is related to the reciprocating speed of the plunger in the plunger cavity.

[0278] 2) Modeling of plunger oil film thickness

[0279] In the tilt plunger geometry model, section 1 is the rectangular coordinate system x k y k z k o k Medium k o k y k The section 2 is a section along the axis of the plunger cavity and coincides with the x k o k y k Surface distance l F A cross section of. Among them, o k Point is the origin of the coordinate system, x k is the direction of the tangential velocity of the plunger when it moves in a circular motion with the cylinder body, y k is the direction of the normal velocity of the plunger when it moves in a circular motion with the cylinder body, z k In section 1, the position of the center o1 of the section circle is represented by a set of eccentric distances (e1, e2), where e1 represents the distance between o1 and o. k Click on x k The eccentricity of the direction, e2 represents the eccentricity of point o1 relative to o k Point at y k The eccentricity in the direction; similarly, on section 2, the position of the center o2 of the section circle is represented by a set of eccentricity distances (e3, e4).

[0280] Since the plunger will tilt to a certain extent during its reciprocating motion, the sections 1 and 2 of the plunger are strictly speaking an ellipse. However, considering that the clearance between the plunger pair is much smaller than the plunger radius, the ellipse is approximated as a circle with a radius of the plunger radius r. k . In particular, the spatial rectangular coordinate system xyzo presents the following coordinates:

[0281] x=φ k r k ,y=h,z=z k (30)

[0282] Select a point between section 1 and section 2 that is the origin of the coordinate system. k Distance z k Any cross section, such as Figure 5 As shown, the position of the center point o3 of the cross-section circle can be expressed by a set of eccentric distances (e x ,e y ) indicates that, from the geometric proportions of similar triangles, we can conclude that:

[0283]

[0284] In the formula,

[0285]

[0286] We can get:

[0287]

[0288] Where: z k —With the coordinate origin o k The distance; l F —Contact length between plunger and bushing.

[0289] This formula can also be replaced by:

[0290]

[0291] Similarly, we can get:

[0292]

[0293] or

[0294]

[0295] x k Rotate the starting axis counterclockwise φ k The current oil film thickness value is calculated from the angle, where the oil film thickness can be obtained from the geometric relationship between Bo3 and the plunger radius:

[0296] h=Bo3-r k (37)

[0297] Where:

[0298]

[0299] Substituting formula (37) into formula (38), we get:

[0300]

[0301] In the same way, a normal extrusion velocity field can be defined, and the time derivative vector of the plunger eccentricity is The function of, that is, the rate of change of the oil film thickness at any point of the plunger pair can be expressed as:

[0302]

[0303] Part III: Solving the plunger pair lubrication model

[0304] 1) Discretized Reynolds equation of plunger pair

[0305] The finite difference method is used to solve the two-dimensional Reynolds equation, which is generally divided into three steps: a. generate discrete grids; b. construct discrete equations; c. solve the system of equations.

[0306] First, the expanded two-dimensional rectangular plunger pair solution domain is meshed and the solution domain is divided along the circumferential x k Direction and axis z k The direction is divided into m and n equidistant intervals, and the solution domain range is defined as 0≤x k ≤2πr k ,0≤z k ≤l F ,The control domain of the computational grid is 61×21 equidistant nodes. The oil film pressure can be obtained using the half-step central difference format. Figure 6 The figure shows the meshing and difference format of the solution domain.

[0307] Then, according to the basic idea of ​​finite difference and half-step central difference, the Reynolds equation of the plunger pair is solved in the control volume, and the discrete form of the Reynolds equation of the plunger pair is obtained as follows:

[0308]

[0309] In the pair Performing half-step central finite difference gives:

[0310]

[0311] By replacing each term with A, B, C, D, F, G, and H, the equation can be simplified to:

[0312] Ap i+1,j +Bp i-1,j +Cp i,j+1 +Dp i,j-1 -Ep i,j =H (43)

[0313] in:

[0314]

[0315] Finally, the Reynolds equation after discretization of the plunger pair is solved.

[0316] The discretized Reynolds equation is transformed into the form of ax=b, where the coefficient matrix a is a strictly diagonally dominant matrix. Therefore, the super-relaxation iteration method is used to obtain:

[0317]

[0318] Wherein, the value range of relaxation factor ω is ω∈(1,2).

[0319] For the iteration termination condition, it is given according to the relative convergence criterion:

[0320]

[0321] When the relative change of two adjacent pressure fields is less than the given allowable error, it can be considered that the pressure field has basically stopped changing, and the iteration is terminated. ε is the convergence accuracy. If ε is set too large, it will lead to a large calculation error. If it is too small, it will increase the number of iterations and even cause iterative divergence. Therefore, a reasonable error range must be selected. The relative error ε is set to 10 -3 When the pressure at all points of the plunger auxiliary oil film is equal to the relative error, the pressure value at each point of the oil film is the pressure field distribution of the plunger auxiliary when the cylinder body rotation angle is at this position.

[0322] 2) Initial conditions and boundary conditions of the plunger pair

[0323] 21) Initial conditions of plunger pair

[0324] The initial conditions of the plunger pair are the values ​​of the eccentricity (e1, e2, e3, e4) of the two end faces corresponding to the beginning of the plunger movement and the value of its oil film thickness h. Figure 7 As shown in the figure, the oil film thickness of the plunger pair is generally related to the assembly clearance between the plunger and the bushing and the surface roughness. Based on the plunger-bushing clearance of 30 μm in the plunger pump model designed in Chapter 2, the initial value of the plunger pair oil film thickness h is set to 30 μm. The initial value of the eccentricity of the two end faces is set to:

[0325]

[0326] 22) Plunger side boundary conditions

[0327] The boundary conditions of the plunger pair can be divided into Reynolds boundary conditions in the axial and circumferential directions, such as Figure 7 shown.

[0328] The axial boundary condition is the pressure value corresponding to the beginning and end of the fluid film interface during the reciprocating motion of the plunger. The pressure at the beginning of the fluid film corresponds to the instantaneous pressure of the plunger cavity, and the pressure at the end of the fluid film corresponds to the pressure of the rotor cavity, which can be expressed by the following formula:

[0329]

[0330] Where: p r —Instantaneous pressure in the plunger cavity; p e —Pressure in the rotor cavity; x—Circumferential direction of the plunger in the two-dimensional rectangular plane, ranging from 0 to 2πr k .

[0331] The circumferential boundary condition is a periodic boundary condition. Since the plunger is unfolded into a two-dimensional rectangular plane, the pressure at the starting position of the plunger surface is equal to the pressure at the ending position of the plunger surface, which can be expressed by the following formula:

[0332] p(z,0)=p(z,2πr k )(49)

[0333] 3) Simulation of instantaneous pulsating pressure in the plunger cavity

[0334] Since the pressure boundary condition at the beginning of the fluid film corresponds to the instantaneous pressure of the plunger cavity, the change of the instantaneous pressure of the plunger cavity has a great influence on the solution of the Reynolds equation of the plunger pair.

[0335] For the simulation calculation of the instantaneous pulsating pressure of the plunger cavity, you can refer to the changing law of the instantaneous pressure of the plunger cavity during the operation of the plunger pump:

[0336] Cylinder rotation angle When the plunger cavity is connected to the outlet of the plunger pump, the instantaneous pressure of the oil in the plunger cavity is p r At the outlet pressure p out Pulsation near the cylinder rotation angle When the instantaneous pressure of the oil in the plunger chamber is connected to the inlet of the plunger pump, the instantaneous pressure of the oil in the plunger chamber is p r At the inlet pressure p in Pulsating nearby.

[0337] Now, taking into account the pre-pressurization and pre-depressurization of the vibration-damping triangular groove, based on the structural parameters of the designed plunger pump and the empirical formula obtained from relevant literature, the variation law of the instantaneous pressure of the plunger chamber oil with the rotation angle of the cylinder body is simulated. Figure 8 .

[0338] Depend on Figure 8 The conclusion of the instantaneous pressure change law of the oil in the plunger chamber can be drawn as follows:

[0339] (1) Due to the pre-pressurization effect of the vibration reduction triangular groove, the cylinder body rotates at an angle During the process, the instantaneous pressure of the oil in the plunger chamber gradually increased from 0.44MPa to 16.2MPa, and During the process, the pressure gradually decreases to around 15MPa at the plunger cavity outlet.

[0340] (2) Due to the pre-pressure reduction effect of the vibration reduction triangular groove, the cylinder body rotates at an angle During the process, the instantaneous pressure of the oil in the plunger chamber gradually increased from 15.2MPa to 19.5MPa, and During the process, the plunger cavity inlet pressure gradually decreases to around 0.5MPa.

[0341] Part 4: Plunger full cycle dynamic force balance model and numerical solution

[0342] During the operation of the plunger pump, the plunger component is mainly subjected to three types of forces: external force, fluid force, and contact force. These three forces / torques are combined to establish a dynamic force balance model. Since the established equations have no analytical solution, finite difference and Newton-Raphson methods are used to solve the force balance model.

[0343] 1) Modeling of plunger dynamic force balance model

[0344] During the operation of the plunger pump, the plunger components are mainly subjected to: the pressure of the plunger chamber oil on the plunger, the inertia force of the plunger during the reciprocating motion, the friction of the plunger pair, the centrifugal force of the plunger during the circular motion of the cylinder body, the friction of the slipper pair on the plunger, the interaction force of the slipper on the plunger, the hydraulic support force of the plunger pair oil film on the plunger, and the solid contact force of the bushing on the plunger when the plunger tilts. These forces / torques are combined to establish the dynamic force balance equation of the cylinder body during the 0°~360° rotation process, as shown in the following formula:

[0345]

[0346] Where: e—eccentricity of the center points of the two cross sections of the plunger (e1, e2, e3, e4);

[0347] —Eccentricity derivative of the center points of the two sections of the plunger

[0348] F ek —Total external force on the plunger; F fk —Fluid force on the plunger; F ck —Contact force on the plunger;

[0349] M ek —Total external torque on the plunger; M fk —Fluid torque on the plunger; M ck —Contact torque on the plunger.

[0350] For a plunger pump with a constant speed, the normal inertia force of the plunger at each point on the oil film is much smaller than the hydraulic support force. Therefore, the normal inertia force of the oil film is ignored. Equations (50) and (51) can be transformed into:

[0351]

[0352]

[0353] The resultant force f k and resultant moment M k Along x k Direction and y k The direction is decomposed into:

[0354]

[0355] The dynamic force balance model (54) of the plunger during the whole cycle is a nonlinear equation system, and there is no root formula. Therefore, the Newton-Raphson method is used to solve the approximate roots of the model and the derivative of the change of the eccentricity of the two cross sections is As the iteration variable. Therefore, the iteration format of the Newton-Raphson method for the plunger pair is:

[0356]

[0357] In the formula, It can be expressed as:

[0358] or

[0359] in, and Combining into a 4×4 Jacobian matrix, replaced by J, the specific form is as follows:

[0360]

[0361] For equation group (54), given the piston rotation angle in the cylinder The initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ), and set the Newton-Raphson iteration convergence condition f k <10 -3 ,available Moment And through the time step Δt, the eccentricity at the next moment can be obtained The specific formula is as follows:

[0362]

[0363] By repeating the calculation in this way, the eccentricity value corresponding to each angle during the cylinder body rotation of 360° can be obtained.

[0364] Since the fluid force in equation (54) / Torque There is no analytical solution, which makes it difficult to solve each element in the Jacobi matrix. Therefore, the finite difference method with the central difference format is used to replace the partial derivatives in the partial differential equation by differentiating the values ​​of two or more adjacent numerical points. The specific difference format is as follows:

[0365]

[0366] Among them, i=1,2,3,4, j=1,2,3,4.

[0367] 2) Solution of plunger dynamic force balance model

[0368] Cylinder rotation angle The initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ) can be any given value, but it must be less than the plunger-sleeve clearance and the force balance equation can be iteratively converged. The initial eccentricity of the plunger is set to (e 10 ,e 20 ,e 30 ,e 40 )=(0,0,0,0)μm. It should be noted that when the cylinder body rotates 360° and reaches the initial position for the second time, its eccentricity (e1, e2, e3, e4) is not necessarily exactly equal to the initial eccentricity (e 10 ,e 20 ,e 30 ,e 40). Therefore, it is necessary to continue the iteration until the plunger pump reaches the rated speed or stable working state, and the resultant force / torque corresponding to the same cylinder rotation angle remains unchanged, and the plunger eccentricity meets the following convergence conditions to terminate the iteration and complete the solution of the dynamic force balance model. The specific process is as follows Fig. 9 shown.

[0369]

[0370] Where: N is the number of revolutions of the cylinder.

[0371] Part V: Examples

[0372] 1) Simulation parameters

[0373] During the operation of the plunger pump, the rotation angle of the cylinder changes, which will cause the force state of the plunger at each rotation angle to be different, and accordingly different degrees of tilt will occur, which is ultimately characterized by the continuous change of the eccentricity parameters (e1, e2, e3, e4) of the center of the two cross-sections of the plunger. The variation law of the plunger oil film thickness, pressure, leakage flow, etc. can be calculated based on the eccentricity parameters (e1, e2, e3, e4). To this end, the present invention will give the relevant parameters of the designed pump and the simulation results of the plunger lubrication under the corresponding parameters and conduct relevant analysis. The specific parameters are shown in Table 1:

[0374] Table 1 Partial parameters of the simulation of lubrication characteristics of the plunger pair

[0375]

[0376]

[0377] Set the swash plate angle to 15°, the speed to 1500r / min, the inlet pressure to 0.5MPa, the outlet pressure to 15MPa, and the initial eccentricity of the plunger to (e 10 ,e 20 ,e 30 ,e 40 )=(0,0,0,0)μm, combined with the instantaneous pressure change law of the plunger chamber oil, solve the plunger dynamic force balance equation. The relative error between the plunger position when the cylinder body rotates to 721° and the position when it rotates to 1081° is less than 10 -3 , the calculation process converges, and the motion state of the plunger is basically stable.

[0378] 2) Simulation results

[0379] 21) Analysis of the simulation results of plunger eccentricity

[0380] After the calculation converges, the center of the plunger section 1 is at x k The eccentricity e1 in the direction changes with the cylinder rotation angle as shown in the following Fig.10 As shown, the center of section 1 is at y k The eccentricity e2 in the direction changes with the cylinder rotation angle as shown in the following Fig.11 As shown; the center of the plunger section 2 is at x k The eccentricity e3 in the direction changes with the cylinder rotation angle as shown in the following Fig.12 As shown, the center of section 2 is at y k The eccentricity e4 in the direction changes with the cylinder rotation angle as shown in the following Fig.13 shown.

[0381] It can be seen that the plunger section 1 and section 2 are at x k The eccentricity e1 and e3 in the y direction change with the cylinder body rotation angle in a sinusoidal periodic curve. k The eccentricity e2 and e4 in the direction change with the cylinder body rotation angle in a cosine periodic curve. k The direction is still y k There is solid contact between the plunger cavity bushing in all directions. This is because the oil viscosity is too low, the oil film is not tough, and the hydraulic support force is too small, which causes the oil film to rupture.

[0382] 22) Analysis of plunger force simulation results

[0383] The piston is subjected to x k Directional force Fig.14 As shown, the k Directional force Fig.15 As shown; the plunger is subjected to x k Direction torque Fig.16 As shown, the k Direction torque Fig.17 shown.

[0384] It can be seen that the oil film support force and torque on the plunger in the oil discharge area are much greater than the oil film force and torque on the oil suction area. This is because in the oil discharge area, the plunger is subjected to the interaction force from the sliding shoe perpendicular to the swash plate. The force along the axial direction of the plunger balances the axial forces of the oil pressure, inertia, friction, etc. on the plunger. When the plunger reciprocates in the plunger cavity, the oil film support force generated by the tilt at a certain angle can balance the radial force. The hydraulic support force of the oil film is in the relative coordinate system x of the plunger. k The force component in the direction is basically a sine curve, which mainly balances the friction force of the oil film of the sliding shoe pair and a part of the radial force component of the sliding shoe reaction force. The hydraulic support force of the oil film is in the relative coordinate system y of the plunger. kThe force component in the direction is basically a cosine curve. This part of the force mainly balances the centrifugal force of the plunger and the other part of the radial force component of the reaction force of the sliding shoe. The amplitudes of these two force components in the oil discharge area are much larger than those in the oil suction area. Similar to the hydraulic support force curve of the plunger auxiliary oil film, the plunger auxiliary oil film has a great influence on the x k ,y k The hydraulic support torque balances the torque generated by the radial component of the reaction force of the slipper, the centrifugal force of the plunger and the friction force of the slipper.

[0385] As the cylinder rotates, the hydraulic support force of the plunger auxiliary oil film increases with the increase of the external force on the plunger and decreases with the decrease of the external force, and some contact forces appear. This is because when the external force increases, the plunger deviates toward the wall of the plunger cavity. At this time, the oil film thickness decreases and the pressure peak increases, thereby offsetting the external force. When the hydraulic support force of the plunger auxiliary oil film is not enough to offset the external force, the plunger will come into solid contact with the wall of the plunger cavity, thereby generating contact force to offset it.

[0386] 23) Analysis of the simulation results of plunger oil film thickness field

[0387] The plunger oil film thickness can intuitively display the plunger's tilt posture. Fig.18 This is the distribution diagram of the plunger oil film thickness at intervals of 45° from 0° to 360°.

[0388] It can be seen that the minimum value of the plunger auxiliary oil film thickness distribution in the oil discharge area is more serious than that in the oil suction area. This is because the plunger's oil film hydraulic support force / torque in the oil discharge area needs to balance a larger external force / torque. When the plunger is at the top dead center, the instantaneous pressure of the plunger chamber oil is small, and the thickness distribution of the plunger auxiliary oil film is relatively uniform. As the cylinder body rotates to about 10°~190°, the plunger is in the oil discharge state, and the instantaneous pressure of the plunger chamber oil is about 15MPa. Due to the large force on the plunger and the high degree of inclination, the oil film is uneven, and even the oil film thickness is close to 0μm, which is in a state of rupture. As the cylinder body rotates to about 190°~370°, the plunger is in the oil suction state, and the instantaneous pressure of the plunger chamber oil is about 0.5MPa. At this time, the plunger is under small force, the degree of inclination is low, the oil film is uniform, and the oil film thickness distribution is also relatively stable.

[0389] 24) Analysis of the simulation results of the piston oil film pressure field

[0390] The plunger auxiliary oil film pressure can intuitively display the force condition of the plunger. Fig.19 This is the distribution diagram of the plunger auxiliary oil film pressure at every 45° from 0° to 360°.

[0391] By comparison, it is found that the maximum value of the plunger auxiliary oil film pressure field distribution in the oil discharge area is larger than the maximum value in the oil suction area. This is because the pressure at the bottom of the plunger cavity is much greater in the oil discharge area than in the oil suction area. At the same time, the oil pressure at the bottom of the plunger is also much greater than the pressure at the position of the plunger ball head, which makes the boundary pressure at the bottom of the plunger greater, the force is worse, and it is more likely to have a pressure spike. And it can be seen from the figure that the oil film pressure is zero in some areas. This is because the plunger is tilted to a large extent in this area, causing the oil film to rupture, and the plunger is in solid contact with the wall of the plunger cavity, so that there is no oil film pressure and hydraulic support force. This is why the plunger pump is prone to friction and wear in actual working conditions.

[0392] The above gives the simulation results of the thickness field and pressure field of the plunger pair oil film. It can be seen that the plunger is subjected to greater force in the oil discharge area, and the oil film needs to bear greater pressure. The bearing capacity of the plunger pair oil film directly affects the working pressure of the plunger pump. Therefore, when designing the plunger pump, it is necessary to pay attention to the design of the plunger pair. Since the simulation assumes an isothermal model, the effect of temperature on the oil film characteristics is not included. Therefore, when the plunger is in the oil discharge area, the simulation results are in an unreal state.

[0393] Aiming at the lubrication characteristics of the plunger pair of an aviation fuel plunger pump, the present invention combines the kinematic analysis and force analysis of the plunger, the sliding shoe, the cylinder body and other components, the plunger pair fluid dynamic lubrication model, and the changing law of the instantaneous pulsation pre-pressure increase and pre-pressure reduction of the plunger cavity oil when the vibration reduction triangular groove acts, numerically solves the dynamic force balance equation of the plunger, establishes the plunger pair fluid dynamic lubrication model, simulates the lubrication characteristics of the plunger pair oil film during the working process of the plunger pump by using MATLAB, constructs the corresponding relationship between the oil film state and the plunger wear, and finally provides a reasonable diagnosis method for the cause of abnormal wear of the plunger and the cylinder bore from the perspective of the plunger pair oil film lubrication.

Claims

1. A method for analyzing the dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump, characterized in that: The steps include: Step 1: Construct the kinematic and dynamic models of the plunger pair to establish the plunger pair fluid dynamic lubrication model and the plunger dynamic force balance model; Step 2: Construct and solve the plunger pair lubrication model to establish the plunger dynamic force balance model; Step 3: The full-cycle dynamic force balance model and numerical solution of the plunger are used to analyze the changes in the minimum oil film thickness, pressure peak, and leakage flow of the plunger pair during the operation of the plunger pump, providing guidance for the structure and performance optimization of the plunger pump.

2. The method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump according to claim 1, characterized in that: The step 1 is specifically as follows: 1) Establish the plunger kinematic model With the spherical joint as point o(o′), the piston motion coordinate system xyzo and the sliding shoe motion coordinate system x′y′o′ parallel to the swash plate surface are established respectively; The starting position of the moving parts is artificially specified, with the top dead point A0 as the starting position of the plunger movement; When the plunger rotates with the cylinder body at any angle from the top dead point A0 When reaching point A, the displacement s of the plunger ball head along the axis of the plunger cavity relative to the initial position A0 is kz for: Where: R—radius of plunger distribution circle; β—inclination angle of inclined plate; The plunger rotates with the cylinder at any angle When , the contact length between the plunger and the plunger cavity is: l F ＝l Fmin -s k (2) Where: l Fmin —The minimum contact length corresponding to the plunger being at the top dead point A0; When the plunger moves from the top dead center to the bottom dead center, the relative displacement change value H of the plunger k for: H k =2Rtanβ (3) The displacement of the plunger relative to the initial position A0 is s k Taking the derivative of time t, we can get the velocity v of the plunger when it reciprocates in the plunger cavity. kz : Where: ω—cylinder rotation angular velocity; The speed v of the plunger when it reciprocates in the plunger cavity k Taking the derivative of time t, we can get the acceleration a of the plunger when it reciprocates in the plunger cavity. k : The linear velocity v of the plunger when it moves in a circle around its own axis in the plunger cavity kx : v kx =ω k r k (6) Where: k — angular velocity of plunger rotation (considered to be consistent with the angular velocity of cylinder body rotation); r k — plunger radius; 2) Establish plunger dynamics model The forces acting on the plunger include the pressure of the oil in the plunger chamber on the plunger, the inertia force of the plunger along the axis of the plunger chamber, the friction force between the plunger and the bushing, the interaction force of the slipper on the plunger, the centrifugal force of the plunger around the axis of rotation of the cylinder body, the hydraulic support force of the plunger auxiliary oil film on the plunger, and the solid contact force between the plunger and the bushing; all the forces acting on the plunger are divided into three types: external force, fluid force and contact force.

3. The method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump according to claim 2, characterized in that ； 21) External forces The external forces include the pressure of the plunger chamber oil on the plunger, the inertia force of the plunger along the axis of the plunger chamber, the friction between the plunger and the bushing, the interaction force of the sliding shoe on the plunger, the centrifugal force of the plunger around the axis of rotation of the cylinder body, and the torque generated by each force. k y k z k o k , the coordinate system rotates with the cylinder body, so that z k The axis is always located on the axis of the plunger cavity, y k The axis always points radially outward, and the origin of the coordinate system is o k It is always placed at the beginning of the fluid film interface. Assuming that the starting position of the plunger pair is on one side of the displacement chamber, the fluid film moves along z k The piston reciprocates in the cylinder hole as the cylinder body rotation angle changes. The starting position of the fluid film interface will also change accordingly. Based on this, the specific external forces / torques on the piston are as follows; The force of the oil in the plunger chamber on the plunger: Where: p r —Instantaneous pressure of oil in plunger chamber; d k — plunger diameter; exist When the plunger cavity is connected to the outlet of the plunger pump, p r At the outlet pressure p out Pulse nearby; When the plunger cavity is connected to the inlet of the plunger pump, p r At the inlet pressure p in Pulsation nearby; Inertia force of the plunger along the axis of the plunger cavity: F ak =-m k ·a k (8) Where: m k — plunger mass; a k —Acceleration of the plunger during reciprocating linear motion; Friction between plunger and bushing: Where: μ is the dynamic viscosity coefficient of the fluid at 20°C; v k —The speed of the plunger when it reciprocates in the plunger cavity; h k —Difference between plunger hole radius and plunger radius; l F —Contact length between plunger and bushing; The friction force of the sliding shoe pair on the plunger: Where: v s —Linear velocity of the sliding shoe when it makes circular motion on the surface of the swash plate; h s —The distance between the sliding shoe and the swash plate when working; r1—The outer radius of the sliding shoe bottom surface; r2—The radius of the oil chamber at the center of the sliding shoe bottom surface; The centrifugal force of the plunger around the cylinder axis: F ωk =m k ·ω 2 ·R (11) In the space rectangular coordinate system x k y k z k o k In the middle, the plunger moves along z k The net force in the direction is: F Ak =F Dk +F ak +F Tk (12) The interaction force between the sliding shoe and the plunger is perpendicular to the swash plate plane and its magnitude satisfies: F sk =F ks (13) F sk Along the space rectangular coordinate system x k y k z k o k The y in k With z k Directional decomposition, we can get: in F skz =F Ak (15) F sky =F Ak ·tanβ (16) In summary, the external forces acting on the plunger are in the spatial rectangular coordinate system x k y k z k o k Medium k With y k The component of force in the direction is: The external forces acting on the plunger are related to the spatial rectangular coordinate system x k y k z k o k x in k With y k The component moment in the direction of the coordinate axis is: Where: z Rk —The distance between the plunger ball head and the spatial rectangular coordinate system o k distance; l se —The distance between the plunger ball head and the plunger mass center; 22) Fluid Force The oil film hydraulic support force / torque of the plunger pair can be divided into: k ,y k The force in the direction of x k ,y k The moment of the shaft, take an infinitely small piece of oil film in the plunger oil film, assume that the pressure of the small piece of oil film is a constant value p, and the area of ​​the oil film is dx k dz k , then the force exerted by the small piece of oil film on the plunger is pdx k dz k . The total hydraulic support force of the piston auxiliary oil film on the piston is obtained by numerically integrating the micro-element hydraulic support force in the circumferential direction and the axial direction: Plunger auxiliary oil film at x k and k The hydraulic support torque of the plunger in the direction is: The fluid force on the plunger is mainly the hydraulic support force and torque generated by the plunger auxiliary oil film on the plunger; 23) Contact force When the oil film thickness at each point of the plunger pair is determined, the elastic deformation can be expressed by the following formula: h kmin Determined by the surface roughness of the plunger pair, making simplified assumptions about the pressure and plunger deformation, the contact pressure is expressed as follows: Where: d k —Outer diameter of the plunger; d c —Inner diameter of bushing; h el —Elastic deformation of the plunger; E k — elastic modulus of the plunger; The contact pressure of the bushing on the plunger along the coordinate system x is obtained by two-dimensional numerical integration. k ,y k Directional contact forces and moments; 4. The method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump according to claim 1, characterized in that: The step 2 is specifically as follows: 1) Reynolds equation for plunger pair Expand the plunger auxiliary oil film into a plane shape, where x k The coordinate is the circumferential direction of the plunger, z k The coordinate is the axial direction of the plunger, H k The coordinate is the thickness direction of the plunger auxiliary oil film, and the axial length of the expanded oil film is the contact length l between the plunger and the cylinder body. F , the length in the circumferential direction is the circumference of the plunger 2πr k , for the oil film characteristics of the plunger pair that is expanded into a plane rectangle, it will be analyzed and calculated in the rectangular coordinate system, ρ = constant, and the Reynolds equation in the rectangular coordinate system is: Where: v kx —The speed of the oil film along the circumference of the plunger is related to the rotation of the plunger; v kz —The speed of the oil film along the axial direction of the plunger is related to the reciprocating speed of the plunger in the plunger cavity; 2) Modeling of plunger oil film thickness In the tilt plunger geometry model, section 1 is the rectangular coordinate system x k y k z k o k Medium k o k y k The section 2 is a section along the axis of the plunger cavity and coincides with the x k o k y k Surface distance l F a cross section of Among them, k Point is the origin of the coordinate system, x k is the direction of the tangential velocity of the plunger when it moves in a circular motion with the cylinder body, y k is the direction of the normal velocity of the plunger when it moves in a circular motion with the cylinder body, z k is the direction of the plunger cavity axis. On section 1, the position of the section circle center o1 is represented by a set of eccentric distances (e1, e2), where e1 represents the distance between point o1 and o k Click on x k The eccentricity of the direction, e2 represents the eccentricity of point o1 relative to o k Point at y k Similarly, on section 2, the position of the center o2 of the section circle is represented by a set of eccentric distances (e3, e4); The radius is the plunger radius r k . The spatial rectangular coordinate system xyzo presents the following coordinates: x=φ k r k ,y=h,z=z k (30) Select a point between section 1 and section 2 that is the origin of the coordinate system. k Distance z k For any cross section, the position of the center point o3 of the cross section circle can be expressed by a set of eccentric distances (e x ,e y ) indicates that, from the geometric proportions of similar triangles, we can conclude that: In the formula, We can get: Where: z k —With the coordinate origin o k The distance; l F —Contact length between plunger and bushing; This formula can also be replaced by: Similarly, we can get: or x k Rotate the starting axis counterclockwise φ k The current oil film thickness value is calculated from the angle, where the oil film thickness can be obtained from the geometric relationship between Bo3 and the plunger radius: h=Bo3-r k (37) Where: Substituting formula (37) into formula (38), we get: In the same way, a normal extrusion velocity field is defined, and the time derivative vector of the plunger eccentricity is The function of, that is, the rate of change of the oil film thickness at any point of the plunger pair can be expressed as: 3) Discretized Reynolds equation of plunger pair The finite difference method is used to solve the two-dimensional Reynolds equation, which is generally divided into three steps: a. Generate a discrete grid; b. Construct a discrete equation; c. Solve the system of equations; First, the expanded two-dimensional rectangular plunger pair solution domain is meshed and the solution domain is divided along the circumferential x k Direction and axis z k The direction is divided into m and n equidistant intervals, and the solution domain range is defined as 0≤x k ≤2πr k ,0≤z k ≤l F ,The control domain of the computational grid is 61 × 21 equidistant nodes, and the oil film pressure can be obtained using the half-step central difference format; Then, according to the finite difference and half-step central difference, the Reynolds equation of the plunger pair is solved in the control volume, and the discrete form of the Reynolds equation of the plunger pair is obtained as follows: In the pair Performing half-step central finite difference gives: By replacing each term with A, B, C, D, F, G, and H, the equation can be simplified to: Ap i+1,j +Bp i-1,j +Cp i,j+1 +Dp i,j-1 -Ep i,j =H (43) in: Finally, solve the Reynolds equation after the piston pair is discretized; The discretized Reynolds equation is transformed into the form of ax=b, where the coefficient matrix a is a strictly diagonally dominant matrix. Therefore, the super-relaxation iteration method is used to obtain: In the formula, the range of the relaxation factor ω is ω∈(1,2); For the iteration termination condition, it is given according to the relative convergence criterion: When the relative change of two adjacent pressure fields is less than the given allowable error, it is considered that the pressure field has basically stopped changing, and the iteration is terminated. ε is the convergence accuracy. If ε is set too large, it will lead to a large calculation error. If it is too small, it will increase the number of iterations and even cause iterative divergence. Choose a reasonable error range and set the relative error ε to 10 -3 ; When the pressure at all points of the plunger auxiliary oil film is the relative error, the pressure value of each point of the oil film is the pressure field distribution of the plunger auxiliary when the cylinder body rotation angle is at this position; 4) Initial conditions and boundary conditions of plunger pair 41) Initial conditions of plunger pair The initial condition of the plunger pair is the value of the eccentricity (e1, e2, e3, e4) of the two end faces corresponding to the start of the plunger movement and the value of its oil film thickness h. The oil film thickness of the plunger pair is generally related to the assembly clearance and surface roughness of the plunger-bushing. The initial value of the eccentricity of the two end faces is set as: 42) Plunger side boundary conditions The boundary conditions of the plunger pair can be divided into Reynolds boundary conditions in the axial and circumferential directions; The axial boundary condition is the pressure value corresponding to the beginning and end of the fluid film interface during the reciprocating motion of the plunger. The pressure at the beginning of the fluid film corresponds to the instantaneous pressure of the plunger cavity, and the pressure at the end of the fluid film corresponds to the pressure of the rotor cavity, which can be expressed by the following formula: Where: p r —Instantaneous pressure in the plunger cavity; p e —Pressure in the rotor cavity; x—Circumferential direction of the plunger in the two-dimensional rectangular plane, ranging from 0 to 2πr k ; The circumferential boundary conditions are periodic boundary conditions. The plunger is expanded into a two-dimensional rectangular plane, so the pressure at the starting position of the plunger surface is equal to the pressure at the ending position of the plunger surface, which can be expressed by the following formula: p(z,0)=p(z,2πr k ) (49) 5) Simulation of instantaneous pulsating pressure in the plunger cavity Cylinder rotation angle When the plunger cavity is connected to the outlet of the plunger pump, the instantaneous pressure of the oil in the plunger cavity is p r At the outlet pressure p out Pulsation near the cylinder rotation angle When the instantaneous pressure of the oil in the plunger chamber is connected to the inlet of the plunger pump, the instantaneous pressure of the oil in the plunger chamber is p r At the inlet pressure p in Pulsating nearby.

5. The method for analyzing dynamic lubrication characteristics of a plunger pair of an aviation fuel plunger pump according to claim 1, characterized in that: The step three is specifically as follows: The force balance model is solved by using finite difference and Newton-Raphson methods; 1) Modeling of plunger dynamic force balance model During the operation of the plunger pump, the plunger components are mainly subjected to: the pressure of the plunger chamber oil on the plunger, the inertia force of the plunger during the reciprocating motion, the friction of the plunger pair, the centrifugal force of the plunger in the circular motion with the cylinder body, the friction of the slipper pair on the plunger, the interaction force of the slipper on the plunger, the hydraulic support force of the plunger pair oil film on the plunger, and the solid contact force of the bushing on the plunger when the plunger tilts. These forces / torques are combined to establish the dynamic force balance equation of the cylinder body during the 0°~360° rotation process, as shown in the following formula: Where: e—eccentricity of the center points of the two cross sections of the plunger (e1, e2, e3, e4); —Eccentricity derivative of the center points of the two sections of the plunger F ek —Total external force on the plunger; F fk —Fluid force on the plunger; F ck —Contact force on the plunger; M ek —Total external torque on the plunger; M fk —Fluid torque on the plunger; M ck —Contact torque on the plunger; Ignore the inertial force normal to the oil film Equations (50) and (51) can be transformed into: The resultant force f k and resultant moment M k Along x k Direction and y k The direction is decomposed into: The dynamic force balance model (54) of the plunger during the whole cycle is a nonlinear equation system. The Newton-Raphson method is used to solve the approximate roots of the model and the derivatives of the changes in the eccentricity of the two cross sections are used as As the iteration variable, the iteration format of the Newton-Raphson method for the plunger pair is: In the formula, It can be expressed as: in, and Combining into a 4×4 Jacobian matrix, replaced by J, the specific form is as follows: For equation group (54), given the piston rotation angle in the cylinder The initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ), and set the Newton-Raphson iteration convergence condition f k <10 -3 ,available Moment And through the time step Δt, the eccentricity at the next moment can be obtained The specific formula is as follows: By repeating the calculation in this way, the eccentricity value corresponding to each angle during the cylinder body rotation of 360° can be obtained. The finite difference method using the central difference format is used to differentiate the values ​​of two or more adjacent numerical points to replace the partial derivatives in the partial differential equation. The specific difference format is as follows: Where, i = 1, 2, 3, 4, j = 1, 2, 3, 4; 2) Solution of plunger dynamic force balance model Cylinder rotation angle The initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ) is an arbitrary value, but it must satisfy the conditions that it is less than the plunger-sleeve clearance and the force balance equation can be iteratively converged. The initial eccentricity of the plunger is set to (e 10 ,e 20 ,e 30 ,e 40 )=(0,0,0,0)μm, and it should be noted that when the cylinder body rotates 360° to reach the initial position for the second time, its eccentricity (e1, e2, e3, e4) is not necessarily exactly equal to the initial eccentricity (e 10 ,e 20 ,e 30 ,e 40 ), therefore, it is necessary to continue the iteration until the plunger pump reaches the rated speed or stable working state, and the resultant force / torque corresponding to the same cylinder rotation angle remains unchanged, and the plunger eccentricity meets the following convergence conditions to terminate the iteration and complete the solution of the dynamic force balance model; Where: N is the number of revolutions of the cylinder.

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