Straight tooth planetary gear train dynamic modeling method for mixed lubrication

Abstract

The invention provides a straight tooth planetary gear train dynamic modeling method based on mixed lubrication. In the gear operation process, a meshing interface rough peak and an oil film coexist and are generally in a mixed lubrication state. Firstly, tooth surface lubrication characteristics are solved based on a line contact mixed lubrication model; then, on the basis of lubrication characteristics, rough peak rigidity and oil film rigidity are solved respectively, tooth surface contact rigidity is obtained, then, Hertz contact rigidity is replaced to solve gear meshing rigidity, and meanwhile, tooth surface meshing damping (contact damping) is solved. And finally, the influence of the oil film on the gap is taken into account, a coupling model of planetary gear train dynamics and mixed lubrication is established, and the tooth surface dynamic response and the tooth surface lubrication characteristic are obtained through numerical iteration. According to the method, the influence of meshing interface roughness, fluid and elastic deformation is comprehensively considered, the dynamic model closer to the actual state is established, and a dynamic theoretical basis is provided for vibration control and life prediction of the planetary gear system.

Description

Technical Field

[0001] The invention relates to the field of gear dynamics, and in particular to a dynamics modeling method for a spur gear planetary gear train with mixed lubrication. Background Art

[0002] As a core component of transmission systems in many industries such as aerospace, automobiles, and robotics, the dynamic characteristics of traditional planetary gearboxes significantly affect the overall performance of the equipment. As the industry's requirements for high-end equipment performance continue to increase, lubrication has received more and more attention due to its outstanding ability to reduce friction and improve efficiency during the transmission process. In actual working processes, the gear meshing interface is often in a mixed lubrication state. Therefore, in-depth research on the mechanism of mixed lubrication in gear transmission systems and analysis of the interaction between mixed lubrication and gear dynamics are effective means to solve technical problems such as vibration reduction and noise reduction, optimization of transmission efficiency, and extension of service life.

[0003] In view of the above situation, the present invention provides a dynamic modeling method for a spur planetary gear train with mixed lubrication. Summary of the invention

[0004] The purpose of the present invention is to propose a dynamic modeling method for a spur gear planetary gear system with mixed lubrication, comprehensively consider the influence of the meshing interface roughness, fluid and elastic deformation, establish a dynamic model that is closer to the actual state, and provide a dynamic theoretical basis for the vibration control and life prediction of the planetary gear system.

[0005] To achieve the above object, the technical solution of the present invention is: a method for dynamic modeling of a spur gear planetary gear train with mixed lubrication, comprising the following steps:

[0006] S1. Construct a line contact mixed lubrication model for spur planetary gear train;

[0007] S2. Determine the contact stiffness and contact damping of the tooth surfaces based on the lubrication characteristics between the tooth surfaces, and obtain the meshing stiffness and meshing damping;

[0008] S3. Considering the influence of oil film on the internal excitation of the gear system, a coupling model of mixed lubrication and planetary gear system dynamics is established;

[0009] S4. Perform coupled iterative solution to obtain the tooth surface lubrication characteristics and gear dynamic response.

[0010] Preferably, the S1 is specifically:

[0011] The line contact mixed lubrication model is a one-dimensional unified Reynolds equation, expressed as

[0012]

[0013] Where x' is the direction of fluid flow; ρ is the density of the lubricating oil; η is the viscosity of the lubricating oil; p and h are the oil film pressure and film thickness, respectively; u r =(u 1 +u 2 ) / 2 is the tooth surface entrainment speed, u 1 and u 2 It represents the speed of the two tooth surfaces relative to the meshing point; t represents time.

[0014] In the meshing contact area, when the rough peaks are in contact, the film thickness is 0, and the Reynolds equation degenerates into

[0015]

[0016] The film thickness equation between tooth surfaces is:

[0017]

[0018] In the formula, h 0 represents the rigid body displacement; v(x',t) represents the elastic deformation of the gear tooth surface; δ 1 and δ 2 Represents the roughness of the two tooth surfaces; R is the comprehensive curvature radius, expressed as

[0019] R=(1 / R 1 +1 / R 2 ) -1

[0020] In the formula, R 1 and R 2 They represent the curvature radii of the two gears at the meshing point respectively;

[0021] The load equation is

[0022]

[0023] In the formula, p c (x') and p l (x') are the roughness peak pressure and fluid pressure respectively; (x') are the roughness peak pressure and fluid pressure respectively; i ' n and x' represent the inlet and outlet positions of the contact region;

[0024] out

[0025] The boundary conditions are

[0026]

[0027] Among them, p(x i ' n ,t) represents the contact pressure at the entrance; p(x' out ,t) represents the contact pressure at the outlet;

[0028] Preferably, the viscosity and density of the lubricating oil are expressed as

[0029] η=η 0 exp((Inη 0 +9.67)(-1+(1+5.1×10 -9 p) z ))

[0030]

[0031] Where η 0 is the environmental viscosity; ρ 0 represents the environmental density; z is the viscosity pressure index.

[0032] Preferably, in S2, the tooth surface contact stiffness is determined as follows:

[0033] Under mixed lubrication, fluid and roughness peaks coexist, and the tooth surface contact stiffness k m Expressed as

[0034] k m =k o +k c

[0035] In the formula, k o Indicates the oil film stiffness; k c represents the asperity peak stiffness;

[0036] The contact of a single asperity peak can be divided into three stages: elastic stage, elastic-plastic stage and plastic stage;

[0037] In the elastic stage, the contact stiffness k e It is expressed as:

[0038] k e =2E'β 0.5 δ c 1.5

[0039] Where β is the roughness peak radius; δ c is the deformation of the roughness peak; E' is the equivalent elastic modulus, expressed as:

[0040]

[0041] In the formula, v 1 and v 2 Respectively represent the Poisson's ratio of the two gears; E 1 and E 2 Respectively represent the elastic modulus of the two gears;

[0042] In the elastic-plastic stage, the contact stiffness k ep It is expressed as:

[0043] k ep =2.2352E'β 0.5 δ c1 0.5 (δ c / δ c1 -1) 0.27

[0044] In the formula, δ c1 is the critical value of elastic deformation turning into elastic-plastic deformation, expressed as

[0045] δ c1 =(πKH / 2E') 2 β

[0046] In the formula, K = 0.454 + 0.41v is a constant coefficient; v is Poisson's ratio; H is the surface hardness;

[0047] When the deformation exceeds the critical value δ c2 =110δ c1 When it turns to the plastic stage, the contact stiffness k p It is expressed as:

[0048] k p =2πβH

[0049] The total roughness peak contact stiffness of the contact interface is:

[0050]

[0051] Where n s is the rough peak distribution density; A n is the legal contact area; l is the distance from the surface average height plane to the roughness peak average plane; φ(z) is the roughness peak probability density function that obeys Gaussian distribution;

[0052] The oil film stiffness is expressed as

[0053]

[0054] Where, L a is the roughness peak load ratio; B is the bulk modulus of the lubricating oil, expressed as

[0055]

[0056] In the formula, p h represents the average stress of the oil film; B 0 represents the bulk modulus of the oil film at ambient pressure; B 0 ' is the rate of change of bulk modulus.

[0057] Preferably, in S2, the gear meshing stiffness is determined by the potential energy method, and under mixed lubrication conditions, the contact stiffness replaces the Hertzian contact stiffness and is expressed as:

[0058]

[0059] In the formula, k bg (g = 1, 2) represents the bending stiffness of the driving and driven wheels respectively; k sg (g=1,2) represents the shear stiffness of the driving and driven wheels respectively; k ag (g=1,2) represents the compression stiffness of the driving and driven wheels respectively; k fg (g = 1, 2) represents the root fillet foundation stiffness of the driving and driven wheels respectively; k h represents the Hertzian contact stiffness; k fg (g=1, 2) represent the tooth root fillet correction coefficients of the driving and driven wheels respectively.

[0060] Preferably, in S2, the contact damping between gear pairs is expressed as

[0061]

[0062] Where, L is the tooth width; Δx' is the distance between adjacent nodes; α is the pressure angle; I is the total number of discrete points in the contact area;

[0063] The contact damping between the gear pairs is the meshing damping.

[0064] Preferably, S3 is specifically:

[0065] The lumped parameter method is used to establish the translation-torsion dynamics model of the planetary gear train. The differential equation of motion is derived based on Newton's second law. The differential equations of each component are expressed as follows:

[0066] Sun gear differential equation:

[0067]

[0068] The differential equation of motion of the nth planetary gear is:

[0069]

[0070] Differential equation of planet carrier motion:

[0071]

[0072] Differential equation of motion of internal gear ring:

[0073]

[0074] In the formula, m j and I j(j=s,p n ,c,r) represents the mass and moment of inertia of the sun gear, planet gear, planet carrier and inner gear ring, x j ,y j , and u j are the vibration displacements of each component in the x direction, y direction and around its own axis; k jl and c jl (l = x, y, u) represent the support stiffness and support damping of each component in the corresponding direction; k jn ,c jn The meshing stiffness and meshing damping in the corresponding directions when the nth planetary gear meshes with the sun gear and the inner gear ring; r bj (j=s,p,r) is the base circle radius of each component; r c Indicates the radius of the center distribution circle of the planet carrier; T s and T c are the torques of the sun gear and the planet carrier respectively; ω c represents the angular velocity of the planetary gear; ψ jn (j=s, r) are the position angles between the nth planetary gear and the sun gear and the inner ring gear, respectively, expressed as:

[0075]

[0076] In the formula, α s and α r are the meshing angles of the external and internal meshing pairs, respectively; Indicates the position angle of the nth planetary gear;

[0077] Under lubrication conditions, the presence of the oil film will cause the clearance to decrease, and the nonlinear displacement function can be expressed as:

[0078] f(δ jn )=δ jn -b jn +h mjn (j=s,r)

[0079] Where b jn are the tooth side clearances between the external and internal gear pairs, h mjn are the average film thickness between each gear pair; δ jn is the relative displacement between the gear pairs, expressed as:

[0080] δ sn =-(x s -x spn )sinψ sn +(y s -y spn )cosψ sn +u s +uspn

[0081] δ rn =-(x r -x prn )sinψ rn +(y r -y prn )cosψ rn +u r -u prn

[0082] In the formula, x spn ,y spn and u spn They represent the displacement of the planetary gear in the x, y and u directions when the planetary gear is meshed externally; prn ,y prn and u prn They represent the displacement of the planetary gear in the x, y and u directions when it meshes with the inner gear ring.

[0083] Considering the elastic deformation between the planet gear and the planet carrier, δ cnx ,δ cny and δ cnu Expressed as

[0084]

[0085] In the formula, δ cnx ,δ cny and δ cnu They represent the relative displacement between the planet carrier and the planet gear along the lower x, y and u directions respectively.

[0086] The above equations are arranged and expressed in matrix form:

[0087]

[0088] In the formula, q=(x s ,y s ,u s ,x c ,y c ,u c ,x r ,y r ,u r ,x pi ,y pi ,u pi ,) T (i=1,2,…n) is the generalized coordinate vector; M is the mass matrix; G is the helical matrix; K b ,K m and K ω are the support stiffness matrix, meshing stiffness matrix and centripetal stiffness matrix respectively; F is the force vector; the comprehensive damping matrix Ct It is expressed as:

[0089] C t =C+C m

[0090] In the formula, C m is the contact damping matrix; C is the Rayleigh damping matrix, expressed as:

[0091] C=ε·M+ζ·(K b +K m )

[0092] In the formula, ε and ζ are the mass proportional coefficient and stiffness proportional coefficient respectively;

[0093] Dynamic meshing force F between tooth surfaces d It is expressed as:

[0094]

[0095] Preferably, S4 is specifically: in order to obtain the lubrication characteristics and dynamic response of the tooth surface, a numerical method is used to solve, and the solution process is as follows:

[0096] S4.1. Determine the basic parameters of lubricating oil and gear, discretize the contact tooth profile into 100 points, and calculate the curvature radius and entrainment speed of each meshing point;

[0097] S4.2, solve the inter-tooth load distribution and mixed lubrication model;

[0098] S4.3, calculate the contact stiffness and contact damping, so as to obtain the meshing stiffness and meshing damping;

[0099] S4.4. Use the Longo Kutta method to solve the differential equations of the planetary gear train dynamics. When the convergence accuracy ε is satisfied F When , the loop ends and the gear dynamic response and tooth surface lubrication characteristics are output; otherwise, repeat step S4.2 until the convergence accuracy is met.

[0100] Preferably, the S4.2 is as follows:

[0101] The load distribution factor is expressed as:

[0102]

[0103] In the formula, ε α K is the overlap degree; 1 Indicates the engagement point; K 2 It represents the meshing point; C represents the meshing point when entering single-tooth meshing; D represents the meshing point when exiting meshing; P represents the meshing point.

[0104] In the solution of the mixed lubrication model, the contact area is set to x'∈[-4,1.5] and discretized into 130 points. When the film thickness is less than 10nm, it is judged as a rough peak contact, and the simplified Reynolds equation is used to calculate the rough peak pressure. Among them, the initial Hertz stress is used as the initial iteration condition, DC-FFT is used to calculate the elastic deformation, and the semi-system method is combined with the progressive grid encryption method for iterative solution. When the pressure accuracy ε p When it is satisfied, the iteration ends and the parameters such as oil film pressure, oil film thickness and roughness peak load ratio are output. p It is expressed as:

[0105]

[0106] In the formula, represents the contact pressure of node i at the kth iteration;

[0107] Preferably, the convergence accuracy ε in S4.4 is F It is expressed as:

[0108]

[0109] In the formula, is the dynamic meshing force at the mth iteration.

[0110] Compared with the prior art, the present invention has the following beneficial effects:

[0111] Considering the coexistence of rough peaks and oil film on the meshing interface during gear operation, they are generally in a mixed lubrication state. First, the lubrication characteristics of the tooth surface are solved based on the line contact mixed lubrication model. Then, based on the lubrication characteristics, the rough peak stiffness and the oil film stiffness are solved respectively to obtain the tooth surface contact stiffness, and then the gear meshing stiffness is solved instead of the Hertz contact stiffness, and at the same time, the tooth surface meshing damping is solved. Finally, taking into account the influence of the oil film on the clearance, a coupling model of the planetary gear system dynamics and mixed lubrication is established, and through numerical iteration, the tooth surface dynamic response and tooth surface lubrication characteristics are obtained. The present invention comprehensively considers the influence of the meshing interface roughness, fluid and elastic deformation, and establishes a dynamic model that is closer to the actual state, providing a dynamic theoretical basis for vibration control and life prediction of the planetary gear system. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] Figure 1 The gear contact interface of an embodiment of the present invention;

[0113] Figure 2 A rough peak contact interface provided by an embodiment of the present invention;

[0114] Figure 3 A coupled planetary gear train dynamics model provided by an embodiment of the present invention;

[0115] Figure 4 A numerical solution process provided by an embodiment of the present invention;

[0116] Figure 5 It is a schematic diagram of inter-tooth load distribution of the present invention;

[0117] Figure 6 The time-varying meshing stiffness of the outer meshing of the planetary gear system provided by the embodiment of the present invention;

[0118] Figure 7 The external meshing damping of the planetary gear system provided by the embodiment of the present invention;

[0119] Figure 8 The external meshing dynamic meshing force distribution of the planetary gear system provided by the embodiment of the present invention;

[0120] Fig. 9 The outer meshing transmission error distribution of the planetary gear system provided by the embodiment of the present invention;

[0121] Fig.10 The external meshing vibration velocity distribution of a planetary gear system provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0122] It should be noted that the following detailed descriptions are illustrative and are intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present application belongs.

[0123] A dynamic modeling method for spur gear planetary gear train with mixed lubrication, based on Figures 1 to 4 , specifically including the following steps:

[0124] Step 1: Construct a line contact mixed lubrication model for a spur planetary gear train and obtain parameters such as oil film pressure, oil film thickness, and roughness peak load ratio;

[0125] like Figure 1 As shown, the line contact mixed lubrication model is a one-dimensional unified Reynolds equation, expressed as

[0126]

[0127] Where x' is the direction of fluid flow; ρ is the density of the lubricating oil; η is the viscosity of the lubricating oil; p and h are the oil film pressure and film thickness, respectively; u r =(u 1 +u 2 ) / 2 is the tooth surface entrainment speed, u 1 and u 2 It represents the speed of the two tooth surfaces relative to the meshing point; t represents time.

[0128] In the meshing contact area, when the rough peaks are in contact, the film thickness is 0, and the Reynolds equation degenerates into

[0129]

[0130] The film thickness equation between tooth surfaces is:

[0131]

[0132] In the formula, h 0 represents the rigid body displacement; v(x',t) represents the elastic deformation of the gear tooth surface; δ 1 and δ 2 Represents the roughness of the two tooth surfaces; R is the comprehensive curvature radius, expressed as

[0133] R=(1 / R 1 +1 / R 2 ) -1

[0134] In the formula, R 1 and R 2 They represent the curvature radii of the two gears at the meshing point respectively;

[0135] The load equation is

[0136]

[0137] In the formula, p c (x') and p l (x') are the roughness peak pressure and fluid pressure respectively; x i ' n and x' out Indicates the entry and exit positions of the contact area;

[0138] The boundary conditions are

[0139]

[0140] Among them, p(x i ' n ,t) represents the contact pressure at the entrance; p(x' out ,t) represents the contact pressure at the outlet;

[0141] Step 2: Determine the tooth surface contact stiffness and contact damping based on the lubrication characteristics between the tooth surfaces, and then obtain the meshing stiffness and meshing damping;

[0142] like Figure 2 As shown, under mixed lubrication, fluid and roughness peaks coexist, and the tooth surface contact stiffness is expressed as

[0143] k m =k o +k c

[0144] In the formula, ko Indicates the oil film stiffness; k c represents the asperity peak stiffness.

[0145] The contact of a single rough peak can be divided into three stages: elastic stage, elastic-plastic stage and plastic stage. In the elastic stage, the contact stiffness k e It is expressed as:

[0146] k e =2E'β 0.5 δ c 1.5

[0147] Where β is the roughness peak radius; δ c is the deformation of the roughness peak; E' is the equivalent elastic modulus, expressed as:

[0148]

[0149] In the formula, v 1 and v 2 Respectively represent the Poisson's ratio of the two gears; E 1 and E 2 Respectively represent the elastic modulus of the two gears;

[0150] In the elastic-plastic stage, the contact stiffness k ep It is expressed as:

[0151] k ep =2.2352E'β 0.5 δ c1 0.5 (δ c / δ c1 -1) 0.27

[0152] In the formula, δ c1 is the critical value of elastic deformation turning into elastic-plastic deformation, expressed as

[0153] δ c1 =(πKH / 2E') 2 β

[0154] In the formula, K=0.454+0.41v is the constant coefficient; v is the Poisson's ratio; H is the surface hardness.

[0155] When the deformation exceeds the critical value δ c2 =110δ c1 When it turns to the plastic stage, the contact stiffness k p It is expressed as:

[0156] k p =2πβH

[0157] Through the above analysis, it can be seen that the total roughness peak contact stiffness of the contact interface is:

[0158]

[0159] Where n s is the rough peak distribution density; A n is the statutory contact area; l is the distance from the surface average height plane to the average plane of the roughness peak; φ(z) is the probability density function of the roughness peak that obeys the Gaussian distribution.

[0160] The oil film stiffness is expressed as

[0161]

[0162] Where, L a is the roughness peak load ratio; B is the bulk modulus of the lubricating oil, expressed as

[0163]

[0164] In the formula, p h represents the average stress of the oil film; B 0 represents the bulk modulus of the oil film at ambient pressure; B 0 ' is the rate of change of bulk modulus.

[0165] The gear meshing stiffness is determined by the potential energy method. It should be noted that under mixed lubrication conditions, the contact stiffness replaces the Hertzian contact stiffness and is expressed as:

[0166]

[0167] In the formula, k bg (g = 1, 2) represents the bending stiffness of the driving and driven wheels respectively; k sg represents shear stiffness; k ag represents the compression stiffness; k fg Indicates the basic stiffness of the tooth root fillet; k h represents the Hertzian contact stiffness; k fg Indicates the tooth root fillet correction factor.

[0168] The contact damping between gear pairs is expressed as

[0169]

[0170] Where, L is the tooth width; Δx' is the distance between adjacent nodes; α is the pressure angle; I is the total number of discrete points in the contact area;

[0171] The contact damping between the gear pairs is the meshing damping.

[0172] Step 3: Considering the influence of oil film on the internal excitation of the gear system, a coupling model of mixed lubrication and planetary gear system dynamics is established;

[0173] like Figure 3 As shown in the figure, the translation-torsion dynamic model of the planetary gear train is established by the lumped parameter method, and the differential equation of motion is derived based on Newton's second law. The differential equations of each component are expressed as follows:

[0174] Sun gear differential equation:

[0175]

[0176] The differential equation of motion of the nth planetary gear is:

[0177]

[0178] Differential equation of planet carrier motion:

[0179]

[0180] Differential equation of motion of internal gear ring:

[0181]

[0182] In the formula, m j and I j (j=s,p n ,c,r) represents the mass and moment of inertia of the sun gear, planet gear, planet carrier and inner gear ring, x j ,y j , and u j are the vibration displacements of each component in the x direction, y direction and around its own axis; k jl and c jl (l = x, y, u) represent the support stiffness and support damping of each component in the corresponding direction; k jn ,c jn The meshing stiffness and meshing damping in the corresponding directions when the nth planetary gear meshes with the sun gear and the inner gear ring; r bj (j=s,p,r) is the base circle radius of each component; r c Indicates the radius of the center distribution circle of the planet carrier; T s and T c are the torques of the sun gear and the planet carrier respectively; ω c represents the angular velocity of the planetary gear; ψ jn (j=s, r) are the position angles between the nth planetary gear and the sun gear and the inner ring gear, respectively, expressed as:

[0183]

[0184] In the formula, αs and α r are the engagement angles of the external engagement pair and the internal engagement pair respectively; represents the position angle of the nth planet gear.

[0185] In the case of lubrication, the existence of the oil film will cause the clearance to decrease, so the non-linear displacement function is expressed as:

[0186] f(δ jn ) = δ jn -b jn +h mjn (j = s, r)

[0187] In the formula, b jn are the backlash between the external engagement pair and the internal engagement gear pair respectively, and h mjn are the average film thicknesses between the respective gear pairs; δ jn is the relative displacement between the respective gear pairs, and is expressed as:

[0188] δ sn = -(x s - x spn )sinψ sn +(y s - y spn )cosψ sn + u s + u spn

[0189] δ rn = -(x r - x prn )sinψ rn +(y r - y prn )cosψ rn + u r - u prn

[0190] In the formula, x spn , y spn and u spn represent the displacements of the planet gear in the x, y, and u directions during external engagement respectively; x prn , y prn and u prn represent the displacements of the planet gear in the x, y, and u directions during engagement with the internal gear ring respectively

[0191] Considering the elastic deformation between the planet gear and the planet carrier, then δ cnx , δ cny and δ cnu are expressed as

[0192]

[0193] In the formula, δ cnx , δ cny and δ cnu They represent the relative displacement between the planet carrier and the planet gear along the lower x, y and u directions respectively.

[0194] The above equations can be arranged into matrix form:

[0195]

[0196] In the formula, q=(x s ,y s ,u s ,x c ,y c ,u c ,x r ,y r ,u r ,x pi ,y pi ,u pi ,) T (i=1,2,…n) is the generalized coordinate vector; M is the mass matrix; G is the helical matrix; K b ,K m and K ω They are the support stiffness matrix, meshing stiffness matrix and centripetal stiffness matrix respectively; F is the force vector; the comprehensive damping matrix is ​​expressed as:

[0197] C t =C+C m

[0198] In the formula, C m is the contact damping matrix; C is the Rayleigh damping matrix, expressed as:

[0199] C=ε·M+ζ·(K b +K m )

[0200] Where ε and ζ are the mass proportional coefficient and stiffness proportional coefficient respectively.

[0201] Dynamic meshing force F between tooth surfaces d It is expressed as:

[0202]

[0203] Step 4: Perform coupled iterative solution to obtain the tooth surface lubrication characteristics and gear dynamic response.

[0204] like Figure 4 As shown, in order to obtain the lubrication characteristics and dynamic response of the tooth surface, a numerical method is used to solve the problem. The solution process is as follows:

[0205] (1) Determine the basic parameters of lubricating oil and gear, discretize the contact tooth profile into 100 points, and calculate the curvature radius and entrainment speed of each meshing point;

[0206] (2) Figure 5 As shown, the inter-tooth load distribution and mixed lubrication model are solved. The load distribution coefficient can be expressed as:

[0207]

[0208] In the formula, ε α K is the overlap degree; 1 Indicates the engagement point; K 2 It represents the meshing point; C represents the meshing point when entering single-tooth meshing; D represents the meshing point when exiting meshing; P represents the meshing point.

[0209] In the solution of the mixed lubrication model, the contact area is set to x'∈[-4,1.5], discretized into 130 points. When the film thickness is less than 10nm, it is judged as a rough peak contact, and the simplified Reynolds equation is used to calculate the rough peak pressure. Among them, the initial Hertz stress is used as the initial iteration condition, DC-FFT is used to calculate the elastic deformation, and the semi-system method is combined with the progressive grid encryption method for iterative solution. When the pressure accuracy ε p When it is satisfied, the iteration ends and the parameters such as oil film pressure, oil film thickness and roughness peak load ratio are output. p It is expressed as:

[0210]

[0211] In the formula, represents the contact pressure of node i at the kth iteration;

[0212] (3) Calculate the contact stiffness and oil film damping, and then obtain the meshing stiffness and meshing damping;

[0213] (4) The Longo-Kutta method is used to solve the differential equations of the planetary gear train dynamics. When the convergence accuracy ε is satisfied F When , the loop ends and the gear dynamic response and tooth surface lubrication characteristics are output. Otherwise, repeat step (2) until the convergence accuracy is met. The convergence accuracy ε F Expressed as

[0214]

[0215] In the formula, is the dynamic meshing force at the mth iteration.

[0216] The following is a specific example to illustrate the effect of the technical solution of this embodiment:

[0217] Taking NGW 11-6 planetary gearbox as an example, the gear dynamic response is analyzed. Its specific parameters are as follows:

[0218] Table 1 Basic parameters of spur gear planetary gear train

[0219]

[0220] The present invention takes the meshing of the sun gear and the planetary gear as an example, and the following results can be obtained through the above numerical calculation method. Figure 6 , 7 are the time-varying mesh stiffness and mesh damping of gears respectively; Figure 8 , 9 are the dynamic meshing force and dynamic transmission error of gears respectively; Fig.10 is the vibration velocity. It can be seen that under lubrication conditions, the time-varying stiffness of the gear increases, the dynamic transmission error decreases, and the dynamic meshing force and the vibration velocity fluctuation amplitude decrease.

[0221] Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present invention.

Claims

1. A dynamic modeling method for a spur gear planetary gear train with mixed lubrication, characterized in that: The steps include: S1. Construct a line contact mixed lubrication model for spur planetary gear train; S2. Determine the contact stiffness and contact damping of the tooth surfaces based on the lubrication characteristics between the tooth surfaces, and obtain the meshing stiffness and meshing damping; S3. Considering the influence of oil film on the internal excitation of the gear system, a coupling model of mixed lubrication and planetary gear system dynamics is established; S4. Perform coupled iterative solution to obtain the tooth surface lubrication characteristics and gear dynamic response.

2. A method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 1, characterized in that: The S1 is specifically: The line contact mixed lubrication model is a one-dimensional unified Reynolds equation, expressed as Where x' is the direction of fluid flow; ρ is the density of the lubricating oil; η is the viscosity of the lubricating oil; p and h are the oil film pressure and film thickness respectively; u r =(u1+u2) / 2 is the tooth surface entrainment speed, u1 and u2 represent the speeds of the two tooth surfaces relative to the meshing point; t represents time; In the meshing contact area, when the rough peaks are in contact, the film thickness is 0, and the Reynolds equation degenerates into The film thickness equation between tooth surfaces is: In the formula, h0 represents the rigid body displacement; v(x',t) represents the elastic deformation of the gear tooth surface; δ1 and δ2 represent the roughness of the two tooth surfaces; R is the comprehensive curvature radius, which is expressed as R=(1 / R1+1 / R2) -1 In the formula, R1 and R2 represent the curvature radius of the two gears at the meshing point; The load equation is In the formula, p c (x') and p l (x') are the roughness peak pressure and fluid pressure respectively; (x') are the roughness peak pressure and fluid pressure respectively; i ' n and x' represent the contact region entry and exit positions; out The boundary conditions are Among them, p(x i ' n ,t) represents the contact pressure at the entrance; p(x' out ,t) represents the contact pressure at the outlet.

3. A method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 2, characterized in that: The viscosity and density of lubricating oil are expressed as η=η0exp((Inη0+9.67)(-1+(1+5.1×10 -9 p) z )) Where η0 is the ambient viscosity; ρ0 represents the ambient density; and z is the viscosity-pressure index.

4. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 1, characterized in that: In S2, the tooth surface contact stiffness is determined specifically as follows: Under mixed lubrication, fluid and roughness peaks coexist, and the tooth surface contact stiffness k m Expressed as k m =k o +k c In the formula, k o Indicates the oil film stiffness; k c represents the asperity peak stiffness; The contact of a single asperity peak can be divided into three stages: elastic stage, elastic-plastic stage and plastic stage; In the elastic stage, the contact stiffness k e It is expressed as: k e =2E'β 0.5 d c 1.5 Where β is the roughness peak radius; δ c is the deformation of the roughness peak; E' is the equivalent elastic modulus, expressed as: In the formula, v1 and v2 represent the Poisson's ratio of the two gears respectively; E1 and E2 represent the elastic modulus of the two gears respectively; In the elastic-plastic stage, the contact stiffness k ep It is expressed as: k ep =2.2352E'β 0.5 d c1 0.5 (d c / d c1 -1) 0.27 In the formula, δ c1 is the critical value of elastic deformation turning into elastic-plastic deformation, expressed as d c1 =(πKH / 2E') 2 b In the formula, K = 0.454 + 0.41v is a constant coefficient; v is Poisson's ratio; H is the surface hardness; When the deformation exceeds the critical value δ c2 =110δ c1 When it turns to the plastic stage, the contact stiffness k p It is expressed as: k p =2πβH The total roughness peak contact stiffness of the contact interface is: Where n s is the rough peak distribution density; A n is the legal contact area; l is the distance from the surface average height plane to the roughness peak average plane; φ(z) is the roughness peak probability density function that obeys Gaussian distribution; The oil film stiffness is expressed as Where, L a is the roughness peak load ratio; B is the bulk modulus of the lubricating oil, expressed as In the formula, p h represents the average oil film stress; B0 represents the bulk modulus of the oil film at ambient pressure; B0' is the rate of change of bulk modulus.

5. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 1, characterized in that: In S2, the gear meshing stiffness is determined by the potential energy method. Under mixed lubrication conditions, the contact stiffness replaces the Hertz contact stiffness and is expressed as: In the formula, k bg (g = 1, 2) represents the bending stiffness of the driving and driven wheels respectively; k sg (g=1,2) represents the shear stiffness of the driving and driven wheels respectively; k ag (g=1,2) represents the compression stiffness of the driving and driven wheels respectively; k fg (g = 1, 2) represents the root fillet foundation stiffness of the driving and driven wheels respectively; k h represents the Hertzian contact stiffness; k fg (g=1, 2) represent the tooth root fillet correction coefficients of the driving and driven wheels respectively.

6. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 1, characterized in that: In S2, the contact damping between gear pairs is expressed as Where, L is the tooth width; Δx' is the distance between adjacent nodes; α is the pressure angle; I is the total number of discrete points in the contact area; The contact damping between the gear pairs is the meshing damping.

7. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 1, characterized in that: The S3 is specifically: The lumped parameter method is used to establish the translation-torsion dynamics model of the planetary gear train. The differential equation of motion is derived based on Newton's second law. The differential equations of each component are expressed as follows: Sun gear differential equation: The differential equation of motion of the nth planetary gear is: Differential equation of planet carrier motion: Differential equation of motion of internal gear ring: In the formula, m j and I j (j=s,p n ,c,r) represents the mass and moment of inertia of the sun gear, planet gear, planet carrier and inner gear ring, x j ,y j , and u j are the vibration displacements of each component in the x direction, y direction and around its own axis; k jl and c jl (l = x, y, u) represent the support stiffness and support damping of each component in the corresponding direction; k jn ,c jn The meshing stiffness and meshing damping in the corresponding directions when the nth planetary gear meshes with the sun gear and the inner gear ring; r bj (j=s,p,r) is the base circle radius of each component; r c Indicates the radius of the center distribution circle of the planet carrier; T s and T c are the torques of the sun gear and the planet carrier respectively; ω c represents the angular velocity of the planetary gear; ψ jn (j=s, r) are the position angles between the nth planetary gear and the sun gear and the inner ring gear, respectively, expressed as: In the formula, α s and α r are the meshing angles of the external and internal meshing pairs, respectively; Indicates the position angle of the nth planetary gear; Under lubrication conditions, the presence of the oil film will cause the clearance to decrease, and the nonlinear displacement function can be expressed as: f(d jn )=d jn -b jn +h mjn (j=s,r) Where b jn are the tooth side clearances between the external and internal gear pairs, h mjn are the average film thickness between each gear pair; δ jn is the relative displacement between the gear pairs, expressed as: d sn =-(x s -x spn )sinψ sn +(y s -y spn )cosψ sn +u s +u spn d rn =-(x r -x prn )sinψ rn +(y r -y prn )cosψ rn +u r -u prn In the formula, x spn ,y spn and u spn They represent the displacement of the planetary gear in the x, y and u directions when the planetary gear is meshed externally; prn ,y prn and u prn They represent the displacement of the planetary gear in the x, y and u directions when it meshes with the inner gear ring. Considering the elastic deformation between the planet gear and the planet carrier, δ cnx , δ cny and δ cnu Expressed as In the formula, δ cnx ,δ cny and δ cnu They represent the relative displacements between the planet carrier and the planet gear along the lower x, y and u directions respectively; The above equations are arranged and expressed in matrix form: In the formula, q=(x s ,y s ,u s ,x c ,y c ,u c ,x r ,y r ,u r ,x pi ,y pi ,u pi ,) T (i=1,2,…n) is the generalized coordinate vector; M is the mass matrix; G is the helical matrix; K b ,K m and K ω are the support stiffness matrix, meshing stiffness matrix and centripetal stiffness matrix respectively; F is the force vector; the comprehensive damping matrix C t It is expressed as: C t =C+C m In the formula, C m is the contact damping matrix; C is the Rayleigh damping matrix, expressed as: C=ε·M+ζ·(K b +K m ) In the formula, ε and ζ are the mass proportional coefficient and stiffness proportional coefficient respectively; Dynamic meshing force F between tooth surfaces d It is expressed as:

8. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 1, characterized in that: Specifically, S4 is as follows: In order to obtain the lubrication characteristics and dynamic response of the tooth surface, a numerical method is used to solve the problem. The solution process is as follows: S4.

1. Determine the basic parameters of lubricating oil and gear, discretize the contact tooth profile into 100 points, and calculate the curvature radius and entrainment speed of each meshing point; S4.2, solve the inter-tooth load distribution and mixed lubrication model; S4.3, calculate the contact stiffness and contact damping, so as to obtain the meshing stiffness and meshing damping; S4.

4. Use the Longo Kutta method to solve the differential equations of the planetary gear train dynamics. When the convergence accuracy ε is satisfied F When , the loop ends and the gear dynamic response and tooth surface lubrication characteristics are output; otherwise, repeat step S4.2 until the convergence accuracy is met.

9. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 5, characterized in that: The S4.2 is as follows: The load distribution factor is expressed as: In the formula, ε α is the overlap degree; K1 is the meshing point; K2 is the meshing point; C is the meshing point when entering single-tooth meshing; D is the meshing point when exiting meshing; P is the meshing point; In the solution of the mixed lubrication model, the contact area is set to x'∈[-4,1.5] and discretized into 130 points. When the film thickness is less than 10nm, it is judged as a rough peak contact, and the simplified Reynolds equation is used to calculate the rough peak pressure. Among them, the initial Hertz stress is used as the initial iteration condition, DC-FFT is used to calculate the elastic deformation, and the semi-system method is combined with the progressive grid encryption method for iterative solution. When the pressure accuracy ε p When is satisfied, the iteration ends and the output parameters include oil film pressure, oil film thickness and roughness peak load ratio; where ε p It is expressed as: In the formula, represents the contact pressure at node i at the kth iteration.

10. The method for dynamic modeling of a spur gear planetary gear train for mixed lubrication according to claim 6, characterized in that: The convergence accuracy ε in S4.4 F It is expressed as: In the formula, is the dynamic meshing force at the mth iteration.

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