An analysis method for dynamic characteristics of multi-stage planetary gear structure

Abstract

The invention discloses a method for analyzing the dynamic characteristics of a multi-stage planetary gear structure, comprising the following steps: establishing nonlinear dynamic models of a single-stage spur gear transmission system according to a concentrated mass method; analyzing the relative displacement relationship and the motion transmission relationship between two adjacent components, and establishing a dynamic equation of a three-stage planetary gear transmission system based on a Lagrange equation; establishing a nonlinear dynamic equation of a three-stage planetary gear transmission based on time-varying meshing stiffness, a dynamic transmission error, and a meshing phase; and analyzing the dynamic characteristics of a three-stage planetary gear transmission system based on a Runger-Kutta method; the analysis method comprehensively considers nonlinear influencing factors such as time-varying meshing stiffness and dynamic transmission error, combines a concentrated mass method, a Lagrange equation, and a Runger-Kutta method, and applies the method to solving the dynamic characteristics of a three-stage planetary gear transmission system, thereby improving the solution accuracy and efficiency of the multi-stage planetary gear transmission system, and having important significance for improving the meshing stability, load-bearing capacity, and reducing friction loss of the gear transmission system.

Description

Technical Field

[0001] The invention relates to the field of planetary gears, and in particular to an analysis method for dynamic characteristics of a multi-stage planetary gear structure. Background Art

[0002] Planetary gear transmission is widely used in aviation, shipbuilding, automobile, military, machinery and other fields due to its compact structure, strong load-bearing capacity and low bearing load. In the field of heavy-duty transmission, multi-stage planetary gear transmission is usually required to achieve large transmission ratio and high output torque.

[0003] Due to the complex structure of multi-stage planetary gear systems, many coupling factors, and very complex dynamic modeling, the depth and breadth of theoretical analysis of multi-stage planetary gear dynamics are still lacking. Compared with single-stage planetary gear transmissions, the dynamic behavior of multi-stage planetary gear systems is more complex. In the existing technology, there is no suitable analysis method that can grasp the main contradictions of the system and characterize its dynamic coupling relationship, so as to establish an applicable multi-stage planetary gear dynamic model and study its dynamic performance. Summary of the invention

[0004] In order to solve the above problems, the present invention provides a method capable of performing dynamic characteristic analysis on a multi-stage planetary gear.

[0005] In order to achieve the above object, the present invention provides a method for analyzing the dynamic characteristics of a multi-stage planetary gear structure, comprising the following steps:

[0006] Step S1: establishing a nonlinear dynamic model of a single-stage spur gear transmission system according to a concentrated mass method;

[0007] Step S2: Analyze the relative displacement relationship and motion transmission relationship between two adjacent components, and establish the dynamic equation of the three-stage planetary gear transmission system based on the Lagrange equation;

[0008] Step S3: establishing a nonlinear dynamic equation of a three-stage planetary gear transmission based on time-varying meshing stiffness, dynamic transmission error, and meshing phase;

[0009] Step S4: Analyze the dynamic characteristics of the three-stage planetary gear transmission system based on the Runger-Kutta method.

[0010] As a preferred technical solution, the specific steps of establishing the nonlinear dynamic model of the single-stage spur gear transmission system according to the concentrated mass method in step S1 are as follows:

[0011] Step S11: Establish a moving coordinate system fixed to the planet carrier, convert the displacement, velocity, acceleration and other motion quantities of each component in the generalized coordinates into the velocity, displacement, acceleration and other motion quantities in the moving coordinates. The component of vector r in the moving coordinates of the planet carrier is x c,y c , whose components in the fixed coordinate system are x s ,y s ,ω c represents the angular velocity of the moving coordinate system fixed to the planet carrier, then θ=ω c t, according to the geometric relationship in the figure, the relationship is expressed as:

[0012]

[0013] Taking the first-order derivative of the above formula, we get the expression of velocity in generalized coordinates:

[0014]

[0015] By taking the first-order derivative of the above formula and performing parameter substitution, we can obtain the expression of acceleration in generalized coordinates:

[0016]

[0017] Therefore, in the planet carrier follower coordinate system, the acceleration expression of each component of the planetary gear train in the generalized coordinate system can be obtained by coordinate transformation. In the planetary gear train dynamics analysis model, the sun gear-planet gear-ring gear meshing exists in the fixed coordinate system oij and the rotating coordinate system oξη. The position vector of each component of the planetary gear train in the fixed coordinate system is expressed as:

[0018] r i =x i i+y i j,i=s,c,r,1,...,n

[0019] According to the coordinate transformation, the acceleration of each component of the planetary gear train in generalized coordinates can be obtained:

[0020]

[0021] Step S12: According to the geometric positions and motion relationships between the components of the planetary gear train, the relative displacement relationship of the components is analyzed as follows:

[0022] Relative displacement of sun gear and planetary gear in the meshing line direction:

[0023] δ sn =(x n -x s )sinψ sn +(y s -y n )cosψ sn +u s +u n

[0024] ψ sn =ψn -α s

[0025] In the formula, x s ,y s 、u s are the linear displacements of the sun gear in the x-direction, y-direction, and meshing line direction respectively; n ,y n 、u n are the linear displacements of the planetary gear in the x-direction, y-direction and meshing line direction respectively; ψ n is the planet gear position angle, α s is the sun gear-planet gear meshing angle.

[0026] Relative displacement of the inner ring gear and the planetary gear in the meshing line direction:

[0027] δ rn =(x n -x r )sinψ rn +(y r -y n )cosψ rn +u r -u n

[0028] ψ rn =ψ n -α r

[0029] In the formula, x r ,y r 、u r are the linear displacements of the inner gear ring in the x-direction, y-direction and meshing line direction respectively; α r is the meshing angle of the inner ring gear and the planetary gear.

[0030] Planet carrier-planet gear in x direction, y direction, planet carrier tangent direction (u c )Relative displacement:

[0031] X direction: δ cnx =x c -x n -u c cosψ n

[0032] Y direction: δ cny =y c -y n +u c cosψ n

[0033] u c To: δ cnu =(x n -x c)sinψ n +(y c -y n )cosψ n +u c

[0034] In the formula, x c ,y c 、u c They are the x-, y- and tangential displacements of the planet carrier, respectively.

[0035] Step S13: Establish the planetary gear train motion differential equation, the specific steps are as follows:

[0036] The equation of motion of the sun gear is:

[0037]

[0038] The equation of motion of the planet carrier is:

[0039]

[0040] Ring gear motion equation:

[0041]

[0042] The equation of motion for the nth planetary gear:

[0043]

[0044] In the formula, m i ,I i 、r i (i=s,c,r,1,...,n) is the mass, moment of inertia and base circle radius of each component (the planet carrier is the center radius); k i is the translational support stiffness of each component; k sn , k rn are the meshing stiffness of sun gear-planet gear and planet gear-ring gear respectively; k st , k ct , k rt The tangential support stiffness of the sun gear, planet carrier, and ring gear are

[0045] As a preferred technical solution, the specific steps of establishing the three-stage transmission system dynamics model in step S2 are as follows: Step S21: Analyze the coupling relationship between the three-stage planetary gear system, that is, there are mutual forces and torques between the first-stage planet carrier and the second-stage sun gear, the first-stage inner gear ring and the second-stage inner gear ring, the second-stage planet carrier and the third-stage sun gear, the second-stage inner gear ring and the third-stage inner gear ring, and obtain their relative displacement relationship:

[0046] (17) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0047]

[0048] (18) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0049]

[0050] (19) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0051]

[0052] (20) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0053]

[0054] (21) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is:

[0055]

[0056] (22) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is:

[0057]

[0058] (23) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is:

[0059]

[0060] (24) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0061]

[0062] (25) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0063]

[0064] (26) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0065]

[0066] (27) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0067]

[0068] (28) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0069]

[0070] (29) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0071]

[0072] (30) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0073]

[0074] (31) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0075]

[0076] (32) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0077]

[0078] Step S22: According to the Lagrange equation, the kinetic energy and potential energy equations of the system and the dynamic model of the three-stage planetary gear transmission system are established respectively, and the expressions thereof are as follows:

[0079] Kinetic energy equation:

[0080]

[0081] Potential energy equation:

[0082]

[0083] The dynamic equations of the first-stage planet carrier are:

[0084]

[0085] The dynamic equation of the first-stage internal gear ring is:

[0086]

[0087] The dynamic equation of the first-stage sun gear is:

[0088]

[0089] The dynamic equations of the first-stage n planetary gears are:

[0090]

[0091] The dynamic equations of the second-stage planet carrier are:

[0092]

[0093] The dynamic equation of the second-stage internal gear ring:

[0094]

[0095] The dynamic equation of the second-stage sun gear is:

[0096]

[0097] The dynamic equations of the second-stage n planetary gears are:

[0098]

[0099] The dynamic equations of the third-stage planet carrier are:

[0100]

[0101] The dynamic equation of the third-stage internal gear ring:

[0102]

[0103] The dynamic equation of the third-stage sun gear is:

[0104]

[0105] The dynamic equation of the nth planetary gear of the third stage is:

[0106]

[0107] After sorting, the undamped system dynamics equation is obtained in matrix form:

[0108]

[0109] The superscripts Ⅰ, Ⅱ, and Ⅲ of the parameters respectively represent the first, second, and third stages of the three-stage planetary gear train; c ,I r ,Is ,I p are the moments of inertia of the planet carrier, ring gear, sun gear and planet gear, m p is the mass of the planetary gear, r r ,r s ,r p are the base circle radii of the ring gear, sun gear and planet gear respectively, r c is the equivalent radius of the planet carrier, u c ,u r ,u s are the displacements of the planet carrier, ring gear, and sun gear, respectively, and u n is the displacement of the nth planetary gear, k sn ,k rn are the meshing stiffness of the nth planetary gear, sun gear and inner gear ring, respectively, s ,α r are the meshing angles between the sun gear and the planet gear, and between the inner ring gear and the planet gear, respectively, and

[0110] δ sn =u s -u c cosα s +u n

[0111] δ rn =-u c cosα r -u n

[0112] δ sn , δ rn They are the displacement components of the nth planetary gear, sun gear and inner ring gear along the meshing line. are the coupling relative displacements between levels 1-2 and 2-3 respectively; k 12 ,k 23 is the coupling stiffness between levels 1-2 and 2-3, The coupling radius between 1-2 and 2-3, respectively, k r is the tangential support stiffness of the gear ring, T r ,T c ,T s is the torque of the ring gear, planet carrier and sun gear.

[0113] As a preferred technical solution, the specific steps of building a three-stage gear transmission system dynamic model based on time-varying meshing stiffness, dynamic transmission error, and meshing phase in step S3 are as follows:

[0114] Step S31: A dynamic model of a three-stage gear transmission system is established based on time-varying meshing stiffness, dynamic transmission error, and meshing phase:

[0115] The dynamic equations of the first-stage planet carrier are:

[0116]

[0117] The dynamic equation of the first-stage internal gear ring is:

[0118]

[0119] The dynamic equation of the first-stage sun gear is:

[0120]

[0121] The dynamic equations of the first-stage i planetary gears are:

[0122]

[0123] The dynamic equations of the second-stage planet carrier are:

[0124]

[0125] The dynamic equation of the second-stage internal gear ring:

[0126]

[0127] The dynamic equation of the second-stage sun gear is:

[0128]

[0129] The dynamic equations of the second-stage n planetary gears are:

[0130]

[0131] The dynamic equations of the third-stage planet carrier are:

[0132]

[0133] The dynamic equation of the third-stage internal gear ring:

[0134]

[0135] The dynamic equation of the third-stage sun gear is:

[0136]

[0137] The dynamic equation of the nth planetary gear of the third stage is:

[0138]

[0139] The system vibration differential equations are organized into matrix form:

[0140]

[0141] In the above formula, M is the equivalent mass matrix, q is the generalized coordinate matrix, C is the damping matrix, K(t) is the time-varying stiffness matrix, e(t) is the static transfer error matrix, and F is the load matrix. The superscripts Ⅰ, Ⅱ, and Ⅲ of each parameter are the first, second, and third stages of the three-stage planetary gear train, respectively; I c ,I r ,I s ,I p are the moments of inertia of the planet carrier, ring gear, sun gear and planet gear, m p is the mass of the planetary gear, r r ,r s ,r p are the base circle radii of the ring gear, sun gear and planet gear respectively, r c is the equivalent radius of the planet carrier, u c ,u r ,u s are the displacements of the planet carrier, ring gear, and sun gear, respectively, and u n is the displacement of the nth planetary gear, k sn ,k rn are the meshing stiffness of the nth planetary gear, sun gear and inner gear ring, respectively, s ,α r They are the meshing angles between the sun gear and the planetary gears, and between the internal gear ring and the planetary gears.

[0142] Step S32: Calculate the time-varying meshing stiffness, and the calculation formula is:

[0143]

[0144] Where, T m is the meshing period, γ sn is the phase between the nth planetary gear and the first planetary gear (γ s1 =0), γ sr is the phase difference between the internal and external meshing, when the number of planetary gear teeth is odd or even respectively, γ sr Take 1 / 2 and 0 respectively, ε r ,ε s They are the internal and external tooth overlap respectively.

[0145] Step S33: Calculate the meshing phase, and the calculation formula is:

[0146] γ sn =γ rn =ψ n Z r / 2π

[0147] Among them, γ sn is the external meshing phase of the nth planetary gear, γ rnis the internal meshing phase of the nth planetary gear, ψ n is the circumferential angle between the nth planetary gear and planetary gear 1, the planet carrier rotates counterclockwise as + and clockwise as -.

[0148] Step S34: Calculate the dynamic transmission error, the calculation formula is:

[0149]

[0150] Where, e(t) is the static transmission error of the gear, ω f and ψ f are the gear rotation frequency and initial phase, ω m and are meshing frequency and meshing initial phase respectively. For planetary gears, the internal and external meshing errors can be expressed as:

[0151]

[0152] In the formula, e rn (t), e sn (t) are the internal and external meshing transmission errors of the gears, E sn 、E rn are the amplitudes of the total cumulative deviations of the internal and external meshing tooth pitches, respectively; are the amplitudes of the tangential deviation of the inner and outer meshing single tooth pairs respectively; ω s ,ω m are the sun gear rotation frequency and meshing frequency respectively; is the initial phase of internal and external meshing; ψ s It is the initial phase of the sun's rotation; is the internal and external phase difference.

[0153] As a preferred technical solution, step S4 solves the dynamic characteristics of the three-stage planetary gear transmission system based on the Runger-Kutta method, and the specific method is as follows:

[0154] The fourth-order Runger-Kutta algorithm is:

[0155]

[0156] For the kinetic equation:

[0157]

[0158] Convert it to the following form:

[0159]

[0160] make I is the unit matrix of the same order as M, and the above formula can be written in matrix form:

[0161]

[0162] make but Rewrite the above formula as:

[0163]

[0164] The dynamic response of the planetary gear system is analyzed using numerical integration methods.

[0165] The beneficial effects of the present invention compared to the prior art are as follows: the analysis method provided by the present invention comprehensively considers nonlinear influencing factors such as time-varying meshing stiffness and dynamic transmission error, combines the concentrated mass method, Lagrange equation and Runger-Kutta method, and applies them to solving the dynamic characteristics of a three-stage planetary gear transmission system, thereby improving the accuracy and efficiency of solving the multi-stage planetary gear transmission system, which is of great significance to improving the meshing smoothness, load-bearing capacity and reducing friction loss of the gear transmission system. BRIEF DESCRIPTION OF THE DRAWINGS

[0166] Figure 1 is a flow chart of a method for analyzing dynamic characteristics of a multi-stage planetary gear structure provided by an embodiment of the present invention;

[0167] Figure 2 is a schematic diagram of coordinate transformation provided by an embodiment of the present invention;

[0168] Figure 3 is a diagram of a planetary gear train analysis model provided by an embodiment of the present invention;

[0169] Figure 4 is a three-stage planetary gear transmission coupling relationship diagram provided by an embodiment of the present invention;

[0170] Figure 5 is a simplified structural diagram of a three-stage transmission system provided by an embodiment of the present invention;

[0171] Figure 6 This is a diagram of a torsional dynamics model of a planetary gear train provided in one embodiment of the present invention. DETAILED DESCRIPTION

[0172] 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.

[0173] Reference Figure 1The embodiment provides a method for analyzing the dynamic characteristics of a multi-stage planetary gear structure, comprising the following steps:

[0174] S10: According to the concentrated mass method, the nonlinear dynamic model of the single-stage spur gear transmission system is established respectively;

[0175] It should be understood that the structure of the planetary gear system is relatively complex. The single-stage planetary gear transmission system is composed of components such as an inner ring gear, a planet carrier, a sun gear, and a planetary gear. The relative motion and position relationship analysis between the components is somewhat difficult. According to the complexity of the planetary gear system and the specific analysis requirements, the dynamic model of the spur gear planetary transmission in this embodiment adopts a pure torsion model.

[0176] Specifically, step S101: establishing a planetary gear follower coordinate system;

[0177] The generalized coordinates used in the dynamic analysis of planetary gears are the moving coordinate system fixed to the planet carrier. In order to express the displacement, velocity, acceleration and other motion quantities of each component under the generalized coordinates, coordinate transformation is required. Figure 2 , Figure 2 This is a schematic diagram of coordinate transformation. The component of vector r in the planetary frame moving coordinate is x c ,y c , whose components in the fixed coordinate system are x s ,y s ,ω c represents the angular velocity of the moving coordinate system fixed to the planet carrier, then θ=ω c t, according to the geometric relationship in the figure, the following relationship can be obtained:

[0178]

[0179] Taking the first-order derivative of the above formula, we get the expression of velocity in generalized coordinates:

[0180]

[0181] By taking the first-order derivative of the above formula and performing parameter substitution, we can obtain the expression of acceleration in generalized coordinates:

[0182]

[0183] Therefore, in the planet carrier moving coordinate system, the acceleration expression of each component of the planetary gear system in the generalized coordinate system can be obtained through coordinate transformation.

[0184] Reference Figure 3 , Figure 3 It is a planetary gear train dynamics analysis model. The sun gear-planet gear-ring gear meshing exists in the fixed coordinate system oij and the rotating coordinate system oξη. The position vectors of the components of the planetary gear train in the fixed coordinate system are expressed as:

[0185] r i =x i i+y i j,i=s,c,r,1,...,n

[0186] According to the coordinate transformation, the acceleration of each component of the planetary gear train in generalized coordinates can be obtained:

[0187]

[0188] Step S102: Relative displacement analysis of components of the planetary gear train

[0189] Analyzing the relative displacement relationship of each component is to determine the elastic force between the components, which is also the premise for establishing the motion equation of each component. Specifically, the relative displacement between the components is as follows:

[0190] Relative displacement of sun gear and planetary gear in the meshing line direction:

[0191] δ sn =(x n -x s )sinψ sn +(y s -y n )cosψ sn +u s +u n

[0192] ψ sn =ψ n -α s

[0193] In the formula, x s ,y s 、u s are the linear displacements of the sun gear in the x-direction, y-direction, and meshing line direction respectively; n ,y n 、u n are the linear displacements of the planetary gear in the x-direction, y-direction and meshing line direction respectively; ψ n is the planet gear position angle, α s is the sun gear-planet gear meshing angle.

[0194] Relative displacement of the inner ring gear and the planetary gear in the meshing line direction:

[0195] δ rn =(x n -x r )sinψ rn +(y r -y n )cosψ rn +u r -un

[0196] ψ rn =ψ n -α r

[0197] In the formula, x r ,y r 、u r are the linear displacements of the inner gear ring in the x-direction, y-direction and meshing line direction respectively; α r is the meshing angle of the inner ring gear and the planetary gear.

[0198] Planet carrier-planet gear in x direction, y direction, planet carrier tangent direction (u c )Relative displacement:

[0199] X direction: δ cnx =x c -x n -u c cosψ n

[0200] Y direction: δ cny =y c -y n +u c cosψ n

[0201] u c To: δ cnu =(x n -x c )sinψ n +(y c -y n )cosψ n +u c

[0202] In the formula, x c ,y c 、u c They are the x-, y- and tangential displacements of the planet carrier, respectively.

[0203] Step S103: Establishing the differential equation of motion of the planetary gear train

[0204] The equation of motion of the sun gear is:

[0205]

[0206] The equation of motion of the planet carrier is:

[0207]

[0208] Ring gear motion equation:

[0209]

[0210] The equation of motion for the nth planetary gear:

[0211]

[0212] In the formula, m i ,I i 、r i (i=s,c,r,1,...,n) is the mass, moment of inertia and base circle radius of each component (the planet carrier is the center radius); k i is the translational support stiffness of each component; k sn , k rn are the meshing stiffness of sun gear-planet gear and planet gear-ring gear respectively; k st , k ct , k rt They are the tangential support stiffness of the sun gear, planet carrier and ring gear respectively.

[0213] Expressed in matrix form:

[0214]

[0215] Where M is the system generalized mass matrix, K b is the support stiffness matrix, K m is the gear meshing stiffness matrix, q is the system generalized coordinate matrix, which is expressed as follows:

[0216] q=(x s ,y s ,u s ,x c ,y c ,u c ,x r ,y r ,u r ,x1,y1,u1,...x N ,y N ,u N )

[0217] G ω , K ω is the gyro matrix and diagonal matrix caused by acceleration; when the planet carrier speed is not very high, G ω , K ω Compared with other parameter matrices, it is small and can be ignored.

[0218] S20: Establishing a torsional dynamics model for a three-stage planetary gear transmission system

[0219] Construct a concentrated parameter model for the dynamic model of the gear transmission system: discretize the model, concentrate the mass of each component of the gear transmission system, and convert the interaction between the components into elastic force. Use the mechanical idea to convert the original model into a relatively simple and easy-to-calculate discrete mass-spring vibration system. For the three-stage planetary gear system, the connection between the front-stage planet carrier and the rear-stage sun gear can be represented by a torsion spring. Therefore, the establishment of the torsional dynamics model of the multi-stage planetary gear system can be obtained by using the inter-stage coupling method based on the torsional dynamics model of the single-stage planetary gear system, such as Figure 4 shown.

[0220] Step S201: Relative displacement relationship analysis

[0221] In order to accurately analyze the dynamic characteristics of the three-stage planetary gear transmission system, the coupling relationship between the three-stage planetary gear system must be considered, that is, there are mutual forces and torques between the first-stage planet carrier and the second-stage sun gear, the first-stage internal gear ring and the second-stage internal gear ring, the second-stage planet carrier and the third-stage sun gear, and the second-stage internal gear ring and the third-stage internal gear ring.

[0222] According to the inter-stage coupling relationship, the following relative displacement relationship can be obtained:

[0223] (33) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0224]

[0225] (34) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0226]

[0227] (35) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0228]

[0229] (36) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is:

[0230]

[0231] (37) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is:

[0232]

[0233] (38) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is:

[0234]

[0235] (39) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is:

[0236]

[0237] (40) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0238]

[0239] (41) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0240]

[0241] (42) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0242]

[0243] (43) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0244]

[0245] (44) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is:

[0246]

[0247] (45) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0248]

[0249] (46) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0250]

[0251] (47) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0252]

[0253] (48) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is:

[0254]

[0255] Step S202: Establishing a torsional dynamics model of a three-stage planetary gear transmission system

[0256] To simplify the problem, a three-stage planetary gear transmission system is established and the following assumptions are made:

[0257] 1. The gear body of the planetary gear train is a rigid body;

[0258] 2. The elasticity of meshing gears is represented by springs, and damping is ignored;

[0259] 3. Ignore the influence of friction between meshing tooth surfaces;

[0260] 4. Gear rings at all levels are rigidly supported.

[0261] For the three-stage planetary gear transmission system, considering that the gear ring bodies of each stage are rigid bodies and rigid supports, the response can be set to 0 or the corresponding motion equation can be removed during calculation and solution, and the vibration differential equation group of the gear transmission system can be established as follows.

[0262] The dynamic equations of the first-stage planet carrier are:

[0263]

[0264] The dynamic equation of the first-stage internal gear ring is:

[0265]

[0266] The dynamic equation of the first-stage sun gear is:

[0267]

[0268] The dynamic equations of the first-stage n planetary gears are:

[0269]

[0270] The dynamic equations of the second-stage planet carrier are:

[0271]

[0272] The dynamic equation of the second-stage internal gear ring:

[0273]

[0274] The dynamic equation of the second-stage sun gear is:

[0275]

[0276] The dynamic equations of the second-stage n planetary gears are:

[0277]

[0278] The dynamic equations of the third-stage planet carrier are:

[0279]

[0280] The dynamic equation of the third-stage internal gear ring:

[0281]

[0282] The dynamic equation of the third-stage sun gear is:

[0283]

[0284] The dynamic equation of the nth planetary gear of the third stage is:

[0285]

[0286] After sorting, the undamped system dynamics equation is obtained in matrix form:

[0287]

[0288] Where M is the equivalent mass matrix, M = diag(M1,M2,M3); x is the generalized coordinate matrix, x = [x Ι ,x Ⅱ ,x Ⅲ ]; F is the load array, F = [F Ι ,F Ⅱ ,F Ⅲ ].

[0289]

[0290]

[0291]

[0292]

[0293]

[0294]

[0295]

[0296]

[0297]

[0298]

[0299]

[0300]

[0301]

[0302] k 12 is a 6-row 6-column matrix, whose 1st row and 3rd column element is The remaining elements are 0; k 13 is a zero matrix with 6 rows and 7 columns;

[0303] k 21 is a 6-row 6-column matrix, and its 3rd row and 1st column element is The remaining elements are 0;

[0304] k 23 is a 6-row 7-column matrix, whose first row and third column element is The remaining elements are 0;

[0305] k 31 It is a zero matrix with 7 rows and 6 columns;

[0306] k 32 is a 7-row 6-column matrix, and its 3rd row and 1st column element is The remaining elements are 0.

[0307] The superscripts Ⅰ, Ⅱ, and Ⅲ of each parameter represent the first, second, and third stages of the three-stage planetary gear train respectively; c ,I r ,I s ,I p are the moments of inertia of the planet carrier, ring gear, sun gear and planet gear, m p is the mass of the planetary gear, r r ,r s ,r p are the base circle radii of the ring gear, sun gear and planet gear respectively, r c is the equivalent radius of the planet carrier, u c ,u r ,u s are the displacements of the planet carrier, ring gear, and sun gear, respectively, and u n is the displacement of the nth planetary gear, k sn ,k rn are the meshing stiffness of the nth planetary gear, sun gear and inner gear ring, respectively, s,α r are the meshing angles between the sun gear and the planet gear, and between the inner ring gear and the planet gear, respectively, and

[0308] δ sn =u s -u c cosα s +u n

[0309] δ rn =-u c cosα r -u n

[0310] δ sn , δ rn They are the displacement components of the nth planetary gear, sun gear and inner ring gear along the meshing line. are the coupling relative displacements between levels 1-2 and 2-3 respectively; k 12 ,k 23 is the coupling stiffness between levels 1-2 and 2-3, The coupling radius between 1-2 and 2-3, respectively, k r is the tangential support stiffness of the gear ring, T r ,T c ,T s is the torque of the ring gear, planet carrier and sun gear.

[0311] S30: Establish the nonlinear dynamic equations of the three-stage planetary gear transmission system based on time-varying meshing stiffness, dynamic transmission error, and meshing phase;

[0312] Step S301: Dynamic equation modeling

[0313] According to the gear error analysis, the gear transmission error is approximated as a harmonic function of the shaft frequency and the tooth frequency, and the trapezoidal wave is used to represent the time-varying characteristics of the gear meshing stiffness. On this basis, according to the meshing phase relationship of the three-stage planetary gear system, the time-varying stiffness and transmission error of each gear meshing are determined, and the lumped parameter method is used to establish the dynamic model of the three-stage planetary gear system based on the Lagrange equation.

[0314] The schematic diagram of the three-stage transmission system is as follows Figure 5 As shown in the figure, for the spur planetary transmission system, without considering the translational freedom of each component of the system, its torsional dynamics analysis model is as follows: Figure 6 As shown in Figure 2, the model takes into account errors and time-varying stiffness. In order to simplify the problem reasonably, the following assumptions are made when establishing the torsional dynamics model of the planetary gear transmission system:

[0315] 1. The gear body and planet carrier are rigid bodies;

[0316] 2. Gear meshing elasticity is represented by springs;

[0317] 3. The parameters of each planetary gear are the same and they are evenly spaced around the circumference;

[0318] 4. Ignore the influence of tooth surface meshing friction;

[0319] 5. All gear rings at each level are rigidly supported

[0320] 6.The system damping is linear viscous damping.

[0321] For multi-degree-of-freedom systems, the Lagrange equation is used to establish a dynamic model:

[0322]

[0323]

[0324] L=TU

[0325] According to the Lagrange equation:

[0326]

[0327] in, (i=r,c,s,1,2,…,N) is the generalized coordinate. Substituting it into the system vibration differential equation, we can get it.

[0328] The dynamic equations of the first-stage planet carrier are:

[0329]

[0330] The dynamic equation of the first-stage internal gear ring is:

[0331]

[0332] The dynamic equation of the first-stage sun gear is:

[0333]

[0334] The dynamic equations of the first-stage n planetary gears are:

[0335]

[0336] The dynamic equations of the second-stage planet carrier are:

[0337]

[0338] The dynamic equation of the second-stage internal gear ring:

[0339]

[0340] The dynamic equation of the second-stage sun gear is:

[0341]

[0342] The dynamic equations of the second-stage n planetary gears are:

[0343]

[0344] The dynamic equations of the third-stage planet carrier are:

[0345]

[0346] The dynamic equation of the third-stage internal gear ring:

[0347]

[0348] The dynamic equation of the third-stage sun gear is:

[0349]

[0350] The dynamic equation of the nth planetary gear of the third stage is:

[0351]

[0352] The system vibration differential equations are organized into matrix form:

[0353]

[0354] In the above formula, M is the equivalent mass matrix, q is the generalized coordinate matrix, C is the damping matrix, K(t) is the time-varying stiffness matrix, e(t) is the static transfer error matrix, and F is the load matrix. The superscripts Ⅰ, Ⅱ, and Ⅲ of each parameter are the first, second, and third stages of the three-stage planetary gear train, respectively; I c ,I r ,I s ,I p are the moments of inertia of the planet carrier, ring gear, sun gear and planet gear, m p is the mass of the planetary gear, r r ,r s ,r p are the base circle radii of the ring gear, sun gear and planet gear respectively, r c is the equivalent radius of the planet carrier, u c ,u r ,u s are the displacements of the planet carrier, ring gear, and sun gear, respectively, and u n is the displacement of the nth planetary gear, k sn ,k rn are the meshing stiffness of the nth planetary gear, sun gear and inner gear ring, respectively, s ,αr are the meshing angles between the sun gear and the planet gear, and between the inner ring gear and the planet gear, respectively, and

[0355] δ sn =u s -u c cosα s +u n +e sn (t)

[0356] δ rn =-u c cosα r -u n +e rn (t)

[0357] e sn (t), e rn (t) and δ sn , δ rn They are the transmission error and displacement components of the nth planetary gear, sun gear and inner ring gear along the meshing line direction. are the coupling relative displacements between levels 1-2 and 2-3 respectively; k 12 ,k 23 is the coupling stiffness between levels 1-2 and 2-3, The coupling radius between 1-2 and 2-3, respectively, k r is the tangential support stiffness of the gear ring, T r ,T c ,T s is the torque of the ring gear, planet carrier and sun gear.

[0358] Step S302: Determine the meshing phase relationship

[0359] The meshing phase relationship of the planetary gears can be determined by the number of gear teeth and the position of the planetary gears. Figure 4 As shown in the figure, when the planet carrier rotates counterclockwise by an angle ψ, planet gear 1 moves to the position of planet gear 2, and the number of meshing times completed between the gears is ψZ r / 2π, so the planetary gear meshing phase relationship is:

[0360] γ sn =γ rn =ψ n Z r / 2π

[0361] Among them, γ sn is the external meshing phase of the nth planetary gear, γ rn is the internal meshing phase of the nth planetary gear, ψ n is the circumferential angle between the nth planetary gear and planetary gear 1, the planet carrier rotates counterclockwise as + and clockwise as -.

[0362] Step S303: Calculate meshing stiffness

[0363] The time-varying characteristics of meshing stiffness are represented by trapezoidal waves. The dynamic response of the trapezoidal wave time-varying stiffness system can be obtained by numerical solution. When the number of planetary gear teeth is even, the time-varying phases of the internal and external meshing stiffness are considered to be the same; when the number of planetary gear teeth is odd, the time-varying phases of the external meshing stiffness differ by half a meshing cycle. The time-varying stiffness of the internal and external meshing of each planetary gear is:

[0364] k sn (t) = k s1 (t-γ sn T m )

[0365] k rn (t) = k r1 (t-(γ sn +γ sr )T m )

[0366] Where, T m is the meshing period, γ sn is the phase between the nth planetary gear and the first planetary gear (γ s1 =0), γ sr is the phase difference between the internal and external meshing, when the number of planetary gear teeth is odd or even respectively, γ sr Take 1 / 2 and 0 respectively, ε r ,ε s They are the internal and external tooth overlap respectively.

[0367] Step S304: Error analysis

[0368] Error excitation is one of the main internal excitation factors in the gear system. If there are errors in gear processing and assembly, the gear meshing will deviate from the theoretical ideal position. Due to the time-varying error, this deviation forms a displacement-type excitation in the meshing process. The gear error can be divided into tooth profile error and pitch error. The total tangential deviation of the gear is the sum of the cumulative total pitch deviation and the single tooth tangential deviation. The gear tolerance is determined by the gear manual.

[0369] When studying the gear torsional dynamics, the radial error of the gear can be ignored. Then the static transmission error of the gear is the total tangential deviation of the gear. The time-varying nature of the gear transmission error can be approximated as the superposition of the harmonic functions of the shaft frequency and the tooth frequency:

[0370]

[0371] Where, e(t) is the static transmission error of the gear, ω f and ψ f are the gear rotation frequency and initial phase, ω m and are meshing frequency and meshing initial phase respectively. For planetary gears, the internal and external meshing errors can be expressed as:

[0372]

[0373] In the formula, e rn (t), e sn (t) are the internal and external meshing transmission errors of the gears, E sn 、E rn are the amplitudes of the total cumulative deviations of the internal and external meshing tooth pitches, respectively; are the amplitudes of the tangential deviation of the inner and outer meshing single tooth pairs respectively; ω s ,ω m are the sun gear rotation frequency and meshing frequency respectively; is the initial phase of internal and external meshing; ψ s It is the initial phase of the sun's rotation; is the internal and external phase difference.

[0374] S40: Solving the dynamic characteristics of a three-stage planetary gear transmission system based on the Runger-Kutta method;

[0375] The basic idea of ​​the Runger-Kutta method is to construct a formula using a linear combination of values ​​at a certain point so that after expanding it according to the Taylor formula, it has as many terms as possible that are exactly the same as the Taylor expansion of the solution to the initial value problem, so as to determine the parameters and ensure the high accuracy of the formula. The fourth-order Runger-Kutta algorithm is as follows:

[0376]

[0377] The three-stage series planetary gear system dynamic differential equation established in the present invention is a second-order nonlinear differential equation containing time-varying parameters. Before solving the equation, the time-varying parameters must be solved and the equation must be reduced to a standard ordinary differential equation initial value problem of the first-order derivative.

[0378] For the kinetic equation:

[0379]

[0380] Convert it to the following form:

[0381]

[0382] make I is the unit matrix of the same order as M, and the above formula can be written in matrix form:

[0383]

[0384] make but Rewrite the above formula as:

[0385]

[0386] The dynamic response of the planetary gear system is solved by numerical integration method. In this embodiment, the 4-5th order Runge-Kutta variable step numerical integration method in Matlab is used to directly call the ode45 function to solve the differential equations, which has high calculation accuracy.

[0387] In addition, an embodiment of the present invention further provides a computer-readable storage medium, wherein the computer-readable storage medium may store a program, which, when executed, includes part or all of the steps of any method for analyzing the dynamic characteristics of a multi-stage planetary gear structure recorded in the above method embodiment.

[0388] In addition, each functional unit in each embodiment of the present invention may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0389] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable memory. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution can be embodied in the form of a software product, which is stored in a memory and includes several instructions for a computer device (which can be a personal computer, a server or a network device, etc.) to perform all or part of the steps of the method described in each embodiment of the present invention. The aforementioned memory includes: U disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), mobile hard disk, magnetic disk or optical disk and other media that can store program codes.

[0390] A person skilled in the art can understand that all or part of the steps in the various methods of the above embodiments can be completed by instructing related hardware through a program, and the program can be stored in a computer-readable memory, and the memory can include: a flash drive, a read-only memory (English: Read-Only Memory, abbreviated as: ROM), a random access memory (English: Random Access Memory, abbreviated as: RAM), a magnetic disk or an optical disk, etc.

[0391] An exemplary flow chart for analyzing the dynamic characteristics of a multi-stage planetary gear structure according to an embodiment of the present invention is described above with reference to the accompanying drawings. It should be noted that the large amount of details included in the above description are only exemplary illustrations of the present invention, rather than limitations of the present invention. In other embodiments of the present invention, the method may have more, fewer or different steps, and the order, inclusion, function, etc. of the steps may be different from those described and illustrated.

Claims

1. A method for analyzing the dynamic characteristics of a multi-stage planetary gear structure, characterized in that: The following steps are involved: Step S1: establishing a nonlinear dynamic model of a single-stage spur gear transmission system according to the concentrated mass method; the specific steps of establishing a nonlinear dynamic model of a single-stage spur gear transmission system according to the concentrated mass method in step S1 are as follows: Step S11: Establish a moving coordinate system fixed to the planet carrier, convert the displacement, velocity, and acceleration motion of each component in the generalized coordinates into the velocity, displacement, and acceleration motion in the moving coordinates. The component of vector r in the moving coordinates of the planet carrier is x c ,y c , whose components in the fixed coordinate system are x s ,y s ,ω c represents the angular velocity of the moving coordinate system fixed to the planet carrier, t represents time, then θ=ω c t, the relationship is expressed as: Taking the first-order derivative of the above formula, we get the expression of velocity in generalized coordinates: By taking the first-order derivative of the above formula and performing parameter substitution, we can obtain the expression of acceleration in generalized coordinates: Therefore, in the planet carrier follower coordinate system, the acceleration expression of each component of the planetary gear train in the generalized coordinate system can be obtained by coordinate transformation; in the planetary gear train dynamics analysis model, the sun gear-planet gear-ring gear meshing exists in the fixed coordinate system oij and the rotating coordinate system oξη; the position vector of each component of the planetary gear train in the fixed coordinate system is expressed as: r i =x i i+y i j,i=s,c,r,1,...,n According to the coordinate transformation, the acceleration of each component of the planetary gear train in generalized coordinates can be obtained: Step S12: According to the geometric positions and motion relationships between the components of the planetary gear train, the relative displacement relationship of the components is analyzed as follows: Relative displacement of sun gear and planetary gear in the meshing line direction: d sn =(x n -x s )sinψ sn +(y s -y n )cosψ sn +u s +u n ψ sn =ψ n -a s In the formula, x s ,y s 、u s are the linear displacements of the sun gear in the x-direction, y-direction, and meshing line direction respectively; n ,y n 、u n are the linear displacements of the planetary gear in the x-direction, y-direction and meshing line direction respectively; ψ n is the planet gear position angle, α s is the sun gear-planet gear meshing angle; Relative displacement of the inner ring gear and the planetary gear in the meshing line direction: d rn =(x n -x r )sinψ rn +(y r -y n )cosψ rn +u r -u n ψ rn =ψ n -a r In the formula, x r ,y r 、u r are the linear displacements of the inner gear ring in the x-direction, y-direction and meshing line direction respectively; α r is the meshing angle between the inner ring gear and the planetary gear; Planet carrier-planet gear in x direction, y direction, planet carrier tangent u direction c Relative displacement: x-direction: δ cnx = x c - x n - u c cos ψ n y - direction: δ cny = y c - y n + u c cos ψ n uc direction: δ cnu =(x n -x c )sinψ n +(y c -y n )cosψ n +u c In the formula, x c ,y c 、u c are the displacements of the planet carrier in the x-direction, y-direction and tangential line respectively; Step S13: Establish the planetary gear train motion differential equation, the specific steps are as follows: The equation of motion of the sun gear is: The equation of motion of the planet carrier is: Ring gear motion equation: The equation of motion for the nth planetary gear: In the formula, m i ,I i 、r i is the mass, moment of inertia and base circle radius of each component, where i = s, c, r, 1, ..., n; k i is the translational support stiffness of each component, where i = s, c, r, 1, ..., n; k sn , k rn are the meshing stiffness of sun gear-planet gear and planet gear-ring gear respectively; k st , k ct , k rt are the tangential support stiffness of the sun gear, planet carrier, and ring gear respectively; Step S2: Analyze the relative displacement relationship and motion transmission relationship between two adjacent components, and establish the dynamic equation of the three-stage planetary gear transmission system based on the Lagrange equation; the specific steps of establishing the three-stage transmission system dynamic model in step S2 are as follows: Step S21: Analyze the coupling relationship between the three-stage planetary gear system, that is, there are mutual forces and torques between the first-stage planet carrier and the second-stage sun gear, the first-stage inner gear ring and the second-stage inner gear ring, and the second-stage planet carrier and the third-stage sun gear, and the second-stage inner gear ring and the third-stage inner gear ring, and obtain their relative displacement relationship: (1) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is: (2) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is: (3) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is: (4) The relative displacement between the first stage gear ring and the second stage gear ring The projection of the direction is: (5) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is: (6) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is: (7) The relative displacement between the first-stage planet carrier and the second-stage sun gear The projection of the direction is: (8) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is: (9) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is: (10) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is: (11) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is: (12) The relative displacement between the second stage gear ring and the third stage gear ring The projection of the direction is: (13) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is: (14) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is: (15) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is: (16) The relative displacement between the second-stage planet carrier and the third-stage sun gear The projection of the direction is: Step S22: According to the Lagrange equation, the kinetic energy and potential energy equations of the system and the dynamic model of the three-stage planetary gear transmission system are established respectively, and the expressions thereof are as follows: Kinetic energy equation: Potential energy equation: The dynamic equations of the first-stage planet carrier are: The dynamic equation of the first-stage internal gear ring is: The dynamic equation of the first-stage sun gear is: The dynamic equations of the first-stage n planetary gears are: The dynamic equations of the second-stage planet carrier are: The dynamic equation of the second-stage internal gear ring: The dynamic equation of the second-stage sun gear is: The dynamic equations of the second-stage n planetary gears are: The dynamic equations of the third-stage planet carrier are: The dynamic equation of the third-stage internal gear ring: The dynamic equation of the third-stage sun gear is: The dynamic equation of the nth planetary gear of the third stage is: After sorting, the undamped system dynamics equation is obtained in matrix form: The superscripts Ⅰ, Ⅱ, and Ⅲ of the parameters respectively represent the first, second, and third stages of the three-stage planetary gear train; c ,I r ,I s ,I p are the moments of inertia of the planet carrier, ring gear, sun gear and planet gear, m p is the mass of the planetary gear, r r ,r s ,r p are the base circle radii of the ring gear, sun gear and planet gear respectively, r c is the equivalent radius of the planet carrier, u c ,u r ,u s are the displacements of the planet carrier, ring gear, and sun gear, respectively, and u n is the displacement of the nth planetary gear, k sn ,k rn are the meshing stiffness of the nth planetary gear, sun gear and inner gear ring, respectively, s ,α r are the meshing angles between the sun gear and the planet gear, and between the inner ring gear and the planet gear, respectively, and δ sn =in s -in c cosα s +in n δ rn =-in c cosα r -in n δ sn , δ rn are the displacement components of the nth planetary gear, sun gear and inner gear ring along the meshing line direction; are the coupling relative displacements between levels 1-2 and 2-3 respectively; k 12 ,k 23 is the coupling stiffness between levels 1-2 and 2-3, The coupling radius between 1-2 and 2-3, respectively, k r is the tangential support stiffness of the gear ring, T r ,T c ,T s is the torque of the ring gear, planet carrier and sun gear; Step S3: Establishing a three-stage planetary gear transmission nonlinear dynamic equation based on time-varying meshing stiffness, dynamic transmission error, and meshing phase; the specific modeling steps of step S3 for establishing a three-stage gear transmission system dynamic model based on time-varying meshing stiffness, dynamic transmission error, and meshing phase are as follows: Step S31: A dynamic model of a three-stage gear transmission system is established based on time-varying meshing stiffness, dynamic transmission error, and meshing phase: The dynamic equations of the first-stage planet carrier are: The dynamic equation of the first-stage internal gear ring is: The dynamic equation of the first-stage sun gear is: The dynamic equations of the first-stage n planetary gears are: The dynamic equations of the second-stage planet carrier are: The dynamic equation of the second-stage internal gear ring: The dynamic equation of the second-stage sun gear is: The dynamic equations of the second-stage n planetary gears are: The dynamic equations of the third-stage planet carrier are: The dynamic equation of the third-stage internal gear ring: The dynamic equation of the third-stage sun gear is: The dynamic equation of the nth planetary gear of the third stage is: The system vibration differential equations are organized into matrix form: In the above formula, M is the equivalent mass matrix, q is the generalized coordinate matrix, C is the damping matrix, K(t) is the time-varying stiffness matrix, e(t) is the static transfer error matrix, and F is the load matrix; the superscripts Ⅰ, Ⅱ, and Ⅲ of each parameter are the first, second, and third stages of the three-stage planetary gear train, respectively; I c ,I r ,I s ,I p are the moments of inertia of the planet carrier, ring gear, sun gear and planet gear, m p is the mass of the planetary gear, r r ,r s ,r p are the base circle radii of the ring gear, sun gear and planet gear respectively, r c is the equivalent radius of the planet carrier, u c ,u r ,u s are the displacements of the planet carrier, ring gear, and sun gear, respectively, and u n is the displacement of the nth planetary gear, k sn ,k rn are the meshing stiffness of the nth planetary gear, sun gear and inner gear ring, respectively, s ,α r are the meshing angles between the sun gear and the planetary gear, and between the inner ring gear and the planetary gear; Step S32: Calculate the time-varying meshing stiffness, and the calculation formula is: Where, T m is the meshing period, γ sn is the phase γ between the nth planetary gear and the first planetary gear s1 =0,γ sr is the phase difference between the internal and external meshing, when the number of planetary gear teeth is odd or even respectively, γ sr Take 1 / 2 and 0 respectively, ε r ,ε s They are the internal and external meshing overlap respectively; Step S33: Calculate the meshing phase, and the calculation formula is: c sn =c rn =ψ n Z r / 2π Among them, γ sn is the external meshing phase of the nth planetary gear, γ rn is the internal meshing phase of the nth planetary gear, ψ n is the circumferential angle between the nth planetary gear and planetary gear 1, the planet carrier rotates counterclockwise as + and clockwise as -; Step S34: Calculate the dynamic transmission error, the calculation formula is: Where, e(t) is the static transmission error of the gear, ω f and ψ f are the gear rotation frequency and initial phase, ω m and are meshing frequency and initial meshing phase respectively; for planetary gears, the internal and external meshing errors can be expressed as: In the formula, e rn (t), e sn (t) are the internal and external meshing transmission errors of the gears, E sn 、E rn are the amplitudes of the total cumulative deviations of the internal and external meshing tooth pitches, respectively; are the amplitudes of the tangential deviation of the inner and outer meshing single tooth pairs respectively; ω s ,ω m are the sun gear rotation frequency and meshing frequency respectively; is the initial phase of internal and external meshing; ψ s It is the initial phase of the sun's rotation; is the internal and external phase difference; Step S4: Analyze the dynamic characteristics of the three-stage planetary gear transmission system based on the Runger-Kutta method; the specific method is as follows: The fourth-order Runger-Kutta algorithm is: For the kinetic equation: Convert it to the following form: make I is the unit matrix of the same order as M, and the above formula can be written in matrix form: make but Rewrite the above formula as: The dynamic response of the planetary gear system is analyzed using numerical integration methods.

2. A computer-readable storage medium, characterized in that: include: The computer-readable storage medium stores a computer program, wherein the computer program, when executed by a processor, implements the steps of a method for analyzing dynamic characteristics of a multi-stage planetary gear structure as claimed in claim 1.

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