A method for analyzing dynamics of planetary gears considering flexibility of ring gear

Abstract

The application discloses a kind of considering flexibility of ring gear planetary gear dynamics analysis method, comprising the following steps: the flexible body model of free flexible ring gear is established;Determine the elastic support boundary condition;Establish rigid-flexible coupling dynamics model;The vibration response of gear system is calculated.The application component considers the dynamics model of planetary gear system of "real rotation", accurately depicts the signal modulation phenomenon caused by through effect, and has good physical interpretability.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of gear transmission, and particularly relates to a planetary gear dynamics analysis method considering flexibility of a gear ring. BACKGROUND

[0002] The planetary gear transmission system is a typical transmission component in a wind turbine, a helicopter main reducer and an automobile gearbox, and research on dynamics characteristics of the planetary gear transmission system is of great significance for fault initiation mechanism, fault evolution characteristics and gear monitoring and diagnosis. In aviation gear transmission, due to the requirement of light weight, components such as gear rings in the gear system often have great flexibility. Therefore, it is necessary to establish a dynamics model of the planetary gear system considering flexibility of the gear ring.

[0003] At present, in the field of planetary gear dynamics, mainstream modeling methods can be divided into lumped parameter models and flexible body dynamics models. The lumped parameter model has a simple modeling process and high calculation efficiency, and is therefore most widely used. However, the lumped parameter model ignores the flexibility of components in the planetary gear system, and cannot consider the influence of flexibility of the gear ring on dynamics characteristics of the system. The flexible body dynamics model is generally implemented by using the finite element method, and although the flexibility of the gear ring can be considered, the flexible body dynamics model has the disadvantage of low calculation efficiency.

[0004] Unlike the fixed shaft gear transmission system, the planetary carrier and the planet wheel revolve around the sun wheel in the planetary gear transmission system, and the revolution of the planet wheel causes the planet wheel through effect. Sensors are generally arranged on fixed components such as the gear ring or the gearbox. The planet wheel through effect causes the length of the transmission path between the excitation source (the gear meshing position) and the measuring point to be time-varying, resulting in amplitude and phase modulation of the signal on the gear ring. In dealing with the planet wheel through effect, the traditional model often adopts the following two ideas: one is to use a phenomenological model based on a window function to describe the planet wheel through effect; and the other is to use a traditional dynamics model of a "pseudo-rotation" strategy. The phenomenological model based on the window function lacks physical interpretability, and the traditional dynamics model based on the "pseudo-rotation" strategy approximates the planet wheel through effect by using a time-varying azimuth angle, without truly updating the position of the planet wheel, so that the planet wheel through effect cannot be truly considered.

[0005] Therefore, the person skilled in the art is committed to providing a planetary gear dynamics analysis method considering flexibility of a gear ring to solve the problems that the planet wheel through effect cannot be truly considered and physical interpretability is lacking in the prior art. SUMMARY

[0006] In view of the defects in the prior art, the technical problem to be solved by the present application is how to provide a planetary gear dynamics analysis method capable of solving the problem that the planet wheel through effect cannot be truly considered in the prior art.

[0007] To achieve the above object, the application provides a planet gear dynamics analysis method considering the flexibility of a gear ring, comprising the following steps:

[0008] Step 1, establishing a flexible body model of a free flexible gear ring;

[0009] Step 2, determining elastic support boundary conditions;

[0010] Step 3, establishing a rigid-flexible coupling dynamics model;

[0011] Step 4, calculating the vibration response of the gear system.

[0012] Further, the step 1 specifically comprises:

[0013] Step 1.1, obtaining the constitutive relation of the flexible gear ring;

[0014] Step 1.2, establishing a displacement discrete format.

[0015] Further, the step 1.1 specifically comprises:

[0016] A three-dimensional vibration displacement of the flexible gear ring is established using six generalized coordinates:

[0017]

[0018] According to Hamilton's variational principle, the variational functional of the flexible gear ring is:

[0019] Π ring = ∫(T ring -U ring +W ext )dt = ∫(T ring -U in -U out +W ext )dt (2)

[0020] In the formula, T ring and U ring are the potential energy and kinetic energy of the flexible gear ring respectively, and W ext represents the work done by external force;

[0021] The strain energy of the flexible gear ring is decomposed into an in-plane component U in and an out-of-plane component U out , and the relationship between displacement and strain is:

[0022]

[0023] In the formula, R is the neutral surface radius of the flexible gear ring;

[0024] The in-plane component and the out-of-plane component of the strain energy are:

[0025]

[0026] where E and G are the elastic modulus and shear modulus, respectively, A is the cross-sectional area of the flexible gear ring, I x and I y are the moments of inertia about the x and y axes, respectively, and κ is the shear correction factor.

[0027] Considering the warping effect of the cross section, the Saint-Venant torsion constant is calculated as follows:

[0028]

[0029] where t and h are the thickness and height of the flexible gear ring, respectively.

[0030] Further, the step 1.2 specifically includes:

[0031] The displacement admissible function is constructed using trigonometric series, and the vibration displacement is expanded as:

[0032]

[0033] where U x ,Φ y ,Φ z ,U z ,U y and Φ x are the displacement admissible functions corresponding to the displacement coordinates u x , u z ,u y and , respectively, N t is the modal stage order, and c and s are the cosine and sine coefficient terms, respectively.

[0034] Substituting equation (6) into equation (4), the stiffness matrix expression is obtained as:

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047] The kinetic energy of the flexible ring gear is expressed as:

[0048]

[0049] where I z is the moment of inertia about the z-axis, and p is the density of the flexible ring gear;

[0050] Substituting equation (6) into equation (19), the mass matrix expression is obtained:

[0051]

[0052] Further, the step 2 specifically comprises:

[0053] The stiffness matrix of the elastic support boundary is:

[0054]

[0055] where the stiffness matrix in different directions is expressed as:

[0056]

[0057] where the superscripts c and d represent the centralized and distributed constraints, respectively, k c i is the centralized support stiffness in different directions, k d i is the distributed support stiffness per unit length, N s is the number of centralized support springs, and s and e are the starting and ending angles of the support circular arc, respectively.

[0058] Further, the step 3 specifically comprises:

[0059] Step 3.1, determining the rotation strategy of the planetary gear position;

[0060] Step 3.2, constructing the continuous-discrete coupling matrix of the components;

[0061] Step 3.3, constructing the total stiffness matrix of the components and the total mass matrix.

[0062] Further, the rotation strategy of step 3.1 is a real rotation strategy of updating the instantaneous position of the planetary gear in real time.

[0063] Further, the step 3.2 specifically comprises:

[0064] The continuous-discrete coupling matrix is composed of K DD ,K DC and K CC three sub-matrices.

[0065] K DD is a discrete-discrete sub-matrix:

[0066]

[0067] K ppi (j,k)=A i (j,k)(i=1~N p ,j=1~6,k=1~6) (24)

[0068] In the formula, N p is the number of planetary gears, and A i characterizes the meshing relationship between the planetary gears and the gear rings:

[0069]

[0070] In the formula, k rpi is the i-th planetary gear-gear ring meshing tooth pair.

[0071] The meshing projection vector of the planetary gear-gear ring is:

[0072] P rpi =[-sinψ rpi cosβ b ,cosψ rpi cosβ b ,sinβ b ,-r bp sinψ rpi sinβ b ,r bp cosψ rpi sinβ b ,-r bp cosβ b ,-sinα t cosβ b ,cosα t cosβ b ,-sinβ b ] (26)

[0073] In the formula, α t is the end face pressure angle, and β bis the base circle helix angle, r bp is the base circle radius of the planetary gear, ψ rpi is the angle between the meshing plane of the ith planetary gear-ring gear meshing tooth pair and the y-axis:

[0074]

[0075] where, and are the initial and instantaneous meshing azimuth angles of the ith planetary gear-ring gear meshing tooth pair, ω c is the angular velocity of the planet carrier;

[0076] K DC is the discrete-continuous sub-matrix:

[0077]

[0078] where, K DCi is the discrete-continuous sub-matrix corresponding to the ith planetary gear-ring gear meshing tooth pair:

[0079]

[0080] K CC is the continuous-continuous sub-matrix:

[0081]

[0082] Further, the step 3.3 specifically comprises:

[0083] The overall stiffness matrix of the system is:

[0084]

[0085] where, K lumped (t) represents the time-varying stiffness matrix of the lumped parameter sub-model;

[0086] The overall mass matrix is:

[0087] M = diag(M lumped , M ring ) (32)

[0088] where, M lumped is the mass matrix of the lumped parameter sub-model;

[0089] The time-varying damping matrix of the system is constructed using Rayleigh damping:

[0090] C(t) = a D M + b D K(t) (33)

[0091] The proportional damping coefficient in the formula is estimated by modal damping ratio:

[0092]

[0093]

[0094] In the formula, ω1 and ω2 are the circular frequencies of the 1st and 2nd order elastomer modes, respectively.

[0095] Further, the step 4 specifically comprises: mobile elastic coupling boundary connection between the flexible gear ring and the planetary gear, and numerical iterative solution of the dynamic differential equation is realized by using the Newmark-β method.

[0096] The present application has at least the following beneficial technical effects:

[0097] The planetary gear dynamics analysis method considering the flexibility of the gear ring provided by the present application has a "real rotation" planetary gear system dynamics model, accurately depicts the signal modulation phenomenon caused by the through effect, and has good physical interpretability. The semi-analytical modeling method greatly reduces the model dimension, and greatly improves the calculation efficiency. The present application can establish a planetary gear-gear ring meshing gear pair at any position of the gear ring, and can more accurately reflect the process of planetary gear revolution.

[0098] The concept, specific structure and technical effects of the present application will be further described below in combination with the drawings, so as to fully understand the purpose, features and effects of the present application. BRIEF DESCRIPTION OF DRAWINGS

[0099] Figure 1 is the flexible gear ring model of the embodiment of the present application, wherein a is a flexible gear ring schematic diagram, and b is a local coordinate system definition;

[0100] Figure 2 is the flexible gear ring model of the embodiment of the present application and two typical elastic boundaries;

[0101] Figure 3 is the dynamics model of the planetary gear considering the flexibility of the gear ring, wherein a is a lumped parameter submodel, b is a flexible body submodel, and c is an overall dynamics model of the planetary gear system;

[0102] Figure 4 is the flexible gear ring with mobile elastic coupling boundary of the embodiment of the present application;

[0103] Figure 5 is the continuous-discrete coupling matrix of the embodiment of the present application;

[0104] Figure 6 is the analysis flowchart of the embodiment of the present application;

[0105] Figure 7 are the vibration modes of the ring gear obtained by the simulation and experimental modal analysis of the embodiments of the present application, wherein a-c are the first 6 orders of vibration modes obtained by the experimental modal analysis, and d-f are the 6 orders of vibration modes obtained by the semi-analytical flexible gear model;

[0106] Figure 8 are the time-domain waveforms of the radial acceleration signals of the ring gear, wherein a is the experimental result, and b is the simulation result;

[0107] Figure 9 are the frequency spectrums of the radial acceleration signals of the ring gear, wherein a is the experimental result, and b is the simulation result;

[0108] Figure 10 are the order spectrums near the 3rd harmonic of the meshing frequency, wherein a is the experimental result, b is the result of the embodiments of the present application, and c is the result of the traditional model;

[0109] Figure 11 are the influences of the relative positions between the measuring points and the bolt holes, wherein a is the definition of the angular positions of the measuring points, and b is the maximum radial acceleration response under different measuring point positions. DETAILED DESCRIPTION

[0110] The preferred embodiments of the present application are described below to make the technical content of the present application more clear and convenient to understand. The present application can be embodied in many different forms of embodiments, and the protection scope of the present application is not limited to the embodiments mentioned herein.

[0111] In the drawings, the same components have the same reference numerals, and components with similar structures or functions are denoted by similar reference numerals. The size and thickness of each component shown in the drawings are arbitrarily shown, and the present application does not limit the size and thickness of each component. In order to make the drawings clearer, the thickness of some components is appropriately exaggerated in some places in the drawings.

[0112] The present application provides a planet gear dynamics analysis method considering the flexibility of the ring gear, and the specific analysis process of the embodiments of the present application is as follows.

[0113] Step 1, establishing a flexible body model of a free flexible ring gear.

[0114] Step 1.1, obtaining the constitutive relation of the flexible ring gear.

[0115] Based on the thick ring elasticity theory, a semi-analytical dynamics model is established to describe the three-dimensional elastic deformation of the ring gear. Unlike most traditional flexible ring gear models, the embodiments of the present application can simultaneously consider the in-plane and out-of-plane vibrations of the flexible ring gear. In addition, the semi-analytical dynamics model based on the thick ring theory also considers the shear effect, rotational inertia and warping effect. Six generalized coordinates are used to describe the three-dimensional vibration displacement of the flexible ring gear:

[0116]

[0117] In the above equation, the first three terms and the last three terms correspond to the out-of-plane and in-plane vibration displacements, respectively. The local coordinate system is defined as shown in Figure 1 .

[0118] According to the Hamilton's variational principle, the variational functional of the flexible gear ring can be expressed as

[0119]

[0120] where T ring and U ring are the potential energy and kinetic energy of the flexible gear ring, respectively, W ext represents the work done by external forces, and the strain energy of the flexible gear ring can be decomposed into an in-plane component U in and an out-of-plane component U out , and the relationship between displacement and strain can be expressed as

[0121]

[0122] where R is the neutral surface radius of the flexible gear ring. The in-plane component and the out-of-plane component of the strain energy can be expressed as

[0123]

[0124] where E and G are the elastic modulus and shear modulus, respectively, A is the cross-sectional area of the flexible gear ring, I x and I y are the moments of inertia about the x and y axes, respectively, and κ is the shear correction factor (κ = 5 / 6 for a rectangular cross-section).

[0125] Considering the warping effect of the cross-section, the Saint-Venant torsion constant can be calculated using the following equation:

[0126]

[0127] where t and h are the thickness and height of the flexible gear ring, respectively.

[0128] Step 1.2, Establishing the displacement discretization format.

[0129] Given the symmetry of the flexible gear ring, a trigonometric series is chosen to construct the displacement admissible function. The vibration displacement can be expanded as follows:

[0130]

[0131] where U x ,Φ y ,Φ z , Uz y x are displacement admissible functions corresponding to displacement coordinates u x z y N t is the modal phase order, c and s are the coefficient terms of cosine and sine respectively.

[0132] Substituting equation (6) into equation (4), the stiffness matrix expression can be obtained as follows:

[0133]

[0134]

[0135]

[0136]

[0137]

[0138]

[0139]

[0140]

[0141]

[0142]

[0143]

[0144]

[0145] The kinetic energy of the flexible gear ring can be expressed as follows:

[0146]

[0147] In the formula, I z is the moment of inertia about the z-axis, and p is the density of the flexible gear ring. Substituting equation (6) into equation (19), the mass matrix expression can be obtained as follows:

[0148]

[0149] Step 2, determine the elastic support boundary condition.

[0150] ​​​​​​The derivation of the stiffness matrix and mass matrix of the flexible ring above is based on the smooth ring. For the ring gear component in the planetary gear transmission system, there are two typical boundary conditions as follows:

[0151] (i) Concentrated constraint: for the ring gear fixed by bolts / pins, the bolt connection on the ring gear can be simulated as a concentrated elastic support boundary (concentrated spring).

[0152] (ii) Distributed constraint: for the ring gear constrained by the matching surface, a distributed elastic constraint can be used to simulate the supporting arc.

[0153] The above concentrated and distributed elastic support constraints can be simulated as additional concentrated elements on the smooth ring, as shown in Figure 2 .

[0154] The stiffness matrix of the elastic support boundary is:

[0155]

[0156] In the formula, the stiffness matrix in different directions can be expressed as:

[0157]

[0158] In the formula, the superscripts c and d represent the concentrated and distributed constraints respectively, k c i is the concentrated support stiffness in different directions k d i is the distributed support stiffness per unit length, N s is the number of concentrated support springs, θ s and θ e are the starting and ending angles of the support arc respectively.

[0159] As shown in Figure 3 , the overall dynamic model of the planetary gear transmission system is composed of two sub-models: a lumped parameter sub-model and a flexible body sub-model; the lumped parameter sub-model includes the sun gear, the planet gears and the planet carrier, and the flexibility of the ring gear is considered through a semi-analytical flexible body model.

[0160] Step 3, establish a rigid-flexible coupling dynamic model.

[0161] Step 3.1, determine the rotation strategy of the planet wheel position

[0162] In the traditional model, in order to simplify modeling, a rotating coordinate system following the rotation of the planet carrier is generally adopted, and the traditional model adopts a "pseudo-rotation" strategy to consider the through effect of the planet wheel; in other words, the position of the planet wheel remains unchanged, and the revolution of the planet carrier and the planet wheel is realized by changing the azimuth angle of gear meshing. The embodiment adopts a fixed coordinate system fixed to the gear ring to describe the vibration of the planetary gear system, and realizes a "true rotation" strategy by updating the instantaneous position of the planetary gear in real time. After each integral step Δt, the position of the planetary gear should be updated.

[0163] In the traditional discrete body dynamics model (finite element model and lumped parameter model), the meshing spring between the planet wheel and the gear ring cannot be connected to any position on the gear ring, therefore, the discrete body dynamics model cannot realize the "true rotation" strategy. By utilizing the unique advantages of the semi-analytical continuum model, in the embodiment, the meshing spring between the planet wheel and the gear ring can be connected to any position on the gear ring. The time-varying planet wheel-gear ring meshing interface can be realized by moving the elastic coupling boundary, as shown in Figure 4 .

[0164] Step 3.2, construct a continuous-discrete coupling matrix.

[0165] The moving elastic coupling boundary between the lumped parameter submodel and the flexible body submodel is a typical continuous-discrete coupling boundary, as shown in Figure 5 , the continuous-discrete coupling matrix is composed of K DD , K DC and K CC three submatrices.

[0166] K DD is a discrete-discrete submatrix:

[0167]

[0168] K ppi (j,k)=A i (j,k)(i=1~N p ,j=1~6,k=1~6) (24)

[0169] In the formula, N p is the number of planet wheels, and A i characterizes the meshing relationship between the planet wheel and the gear ring:

[0170]

[0171] In the formula, k rpi is the i th planet wheel-gear ring meshing tooth pair, and the meshing projection vector of the planet wheel-gear ring is:

[0172] P rpi= [-sin ψ rpi cos β b , cos ψ rpi cos β b , sin β b , -r bp sin ψ rpi sin β b , r bp cos ψ rpi sin β b , -r bp cos β b , -sin α t cos β b , cos α t cos β b , -sin β b ] (26)

[0173] where α t is the face pressure angle, β b is the base circle helix angle, r bp is the base circle radius of the planet, and ψ rpi is the angle between the meshing plane of the ith planet-ring pair and the y-axis:

[0174]

[0175] where θ and θ are the initial and instantaneous meshing azimuth angles of the ith planet-ring pair, and ω c is the angular velocity of the carrier.

[0176] K DC is the discrete-continuous sub-matrix:

[0177]

[0178] where K DCi is the discrete-continuous sub-matrix corresponding to the ith planet-ring pair is:

[0179]

[0180] K CC is the continuous-continuous sub-matrix:

[0181]

[0182] Step 3.3, construct the overall stiffness matrix and the overall mass matrix.

[0183] The overall stiffness matrix of the system can be expressed as:

[0184]

[0185] where K lumped (t) represents the time-varying stiffness matrix of the lumped parameter sub-model. Since the azimuthal angle of the planetary gear is time-varying, the stiffness matrix of the lumped parameter sub-model is also time-varying.

[0186] The global mass matrix can be expressed as:

[0187] M = diag(M lumped ,M ring ) (32)

[0188] where M lumped is the mass matrix of the lumped parameter sub-model. The mass and stiffness matrices of the lumped parameter sub-model are similar to the traditional planetary gear dynamics model.

[0189] Rayleigh damping is used to construct the time-varying damping matrix of the system:

[0190] C(t) = a D M + β D K(t) (33)

[0191] The proportional damping coefficients in the equation are estimated by modal damping ratios:

[0192]

[0193]

[0194] where ω1and ω2are the circular frequencies of the 1st and 2nd elastic body modes, respectively. The modal damping ratios are assumed to be: ξ1= ξ2= 0.02.

[0195] Step 4 Calculate the vibration response of the gear system

[0196] The time-varying mesh stiffness, time-varying mesh damping and the unloaded transmission error are used as parametric excitations of the system. The parametric excitations of the sun-planet and planet-ring gear pairs are introduced into the dynamics model to calculate the vibration response of the system. The moving elastic coupling boundary is used to connect the flexible ring gear and the planet gear. The Newmark-β method is used to realize the numerical iterative solution of the dynamics differential equations. After each integration step Δt, the instantaneous position of the planetary gear is updated. The flow chart of the model is shown in Figure 6 .

[0197] The experimental modal analysis is used to verify the effectiveness of the modal characteristics of the present embodiment. The frequency response function of the system is obtained by the method of fixing the force hammer and moving the measuring points under the free boundary condition. The natural frequency of the ring gear of the present embodiment is compared with the finite element method, the traditional method (Wang method) and the experimental results, and the results are shown in Table 1. As shown in Table 1, the error between the natural frequency obtained by the finite element method and the experiment is less than 2.5%, which has high precision. However, the finite element method depends on the fine solid element finite element grid, so it will lead to a large dimension of the overall model. For the traditional semi-analytical method, the Wang method adopts the plane assumption and only considers the in-plane degrees of freedom, so only the in-plane natural frequency is obtained. In addition, the Wang method greatly overestimates the in-plane vibration natural frequency, because the Wang method does not consider the shear effect, the moment of inertia and the warping effect. Through the above comparison, it can be seen that the analytical model of the present embodiment can obtain accurate calculation results (the maximum error is less than 3%) without sacrificing the solving efficiency.

[0198] Table 1. Ring gear free modal frequencies

[0199]

[0200] The ring gear mode shapes obtained by simulation and experiment are shown in Figure 7 As shown in Figure 7 , the first six order mode shapes of simulation and experiment are in good agreement, which proves the effectiveness of the semi-analytical flexible ring gear model of the present embodiment.

[0201] The response test of the planetary gear transmission system is carried out to verify the effectiveness of the vibration response calculated by the model of the present embodiment. The radial acceleration response of the No. 2 measuring point on the ring gear is shown in Figure 8 , and the time span shown is two planetary carrier periods. As shown in Figure 8 , the vibration amplitudes of the simulation and experimental signals are relatively close. Due to the through effect of the planetary gear, the simulation signal presents obvious amplitude modulation phenomenon. Due to the mixed non-periodic noise component in the measured signal, the amplitude modulation phenomenon is not obvious.

[0202] The frequency spectrum of the radial acceleration signal of the No. 2 measuring point on the ring gear is shown in Figure 9 , and as shown in Figure 9 , the frequency spectrum structures of the simulation and experimental signals are in good agreement, and the amplitudes near the third harmonic of the meshing frequency are very high. This is caused by the fact that the 10th order natural frequency (f n10 = 1780.4 Hz) of the planetary gear system is close to the 3rd order meshing frequency (3x f m = 1776.3 Hz) of the gear.

[0203] The order spectrum near the third harmonic of the meshing frequency is shown in Figure 10As shown. Comparing the order spectrum characteristics obtained from the traditional model ("pseudo-rotation" strategy), this embodiment ("real rotation" strategy), and experiments, it can be found that although both simulation models and experiments show obvious modulation phenomena, their spectral characteristics differ significantly. For the order spectrum obtained from the traditional dynamic model, the amplitude of the third harmonic of the meshing frequency is the largest, while the amplitude of the sideband frequencies generated by the adjustment is lower than the amplitude of the meshing frequency and its harmonics. However, for this embodiment, the amplitudes of the following two frequency components are significantly greater than the amplitudes of the meshing frequency and its harmonics:

[0204]

[0205] In the formula, f m and f c These are the meshing frequency and the planet carrier frequency, respectively.

[0206] Induction reveals that the amplitude reaches its maximum when the order of the planetary carrier is an integer multiple of the number of planetary gears. This pattern is consistent with... Figure 10 The experimental results in a are consistent with those in model a. Therefore, this embodiment can obtain more accurate sideband modulation characteristics compared to the traditional model. Compared with the "pseudo-rotation" strategy used in the traditional model, the "real rotation" strategy in this embodiment can more realistically reflect the physical nature of the planetary gear passing effect.

[0207] This study investigates the influence of the relative positional relationship between the measuring point and the gear ring bolt hole on the radial acceleration signal. The angular position of the measuring point is defined as follows: Figure 11 As shown in a, θ is defined directly above the bolt hole. sensor =0; the position between the two bolt holes, θ sensor =π / 16. The maximum radial acceleration response at different measuring point locations is as follows: Figure 11 As shown in b. (By) Figure 11 It can be seen that the position (θ) between the two bolt holes sensor =π / 16), the vibration response is most obvious and the signal-to-noise ratio is best. Therefore, the sensor arranged on the gear ring should be selected at the middle position between the two bolt holes as much as possible.

[0208] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, those skilled in the art can obtain the following results based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology.

Claims

1. A method for planetary gear dynamics analysis considering flexibility of a ring gear, characterized in that, The method comprises the following steps: Step 1, establishing a flexible body model of the free flexible gear ring, specifically comprising: Step 1.1, obtaining the constitutive relation of the flexible gear ring, specifically comprising: Three-dimensional vibration displacement of the flexible gear ring is established using six generalized coordinates: (1) According to Hamilton's variational principle, the variational functional of the flexible gear ring is: (2) wherein T ring and U ring are the potential and kinetic energy of the flexible ring gear, respectively, W ext is the work done by external forces; The strain energy of the flexible ring gear is decomposed into in-plane and out-of-plane components U in and out-of-plane components U out The relationship between displacement and strain is (3) In the formula, R is the neutral face radius of the flexible ring gear The in-plane component and the out-of-plane component of the strain energy are: (4) wherein E and G are the elastic modulus and the shear modulus, respectively, A is the cross-sectional area of the flexible ring gear, I x and I y are the moments of inertia about the x and y axes, Kappa is the shear correction factor; Considering the warping effect of the section, the calculation method of the Saint-Venant torsional constant is: (5) wherein t and h respectively the thickness and height of the flexible ring gear Step 1.2, establishing a displacement discretization format; Step 2, determining the elastic support boundary condition; Step 3, establishing a rigid-flexible coupling dynamic model; Step 4, calculating the vibration response of the gear system.

2. The planet gear dynamics analysis method considering flexibility of ring gear according to claim 1, wherein, The step 1.2 specifically comprises: The displacement admissible function is constructed by selecting a trigonometric series, and the vibration displacement is expanded as: (6) wherein U x , Phi y , Phi z , U z , U y and Phi x are displacement admissible functions corresponding to displacement coordinates u x , y , z , u z , u y and x , N t is the modal phase order, c and s are the coefficient terms of cosine and sine, respectively; Substituting formula (6) into formula (4), the stiffness matrix expression is obtained: (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) The kinetic energy of the flexible gear ring is represented as: (19) In the formula, I z To bypass z Moment of inertia of the axis, Rho The density of the flexible gear ring; Substituting formula (6) into formula (19), the mass matrix expression is obtained: (20)。 3. The planet gear dynamics analysis method considering flexibility of ring gear according to claim 1, wherein, The step 2 specifically comprises: The stiffness matrix of the elastic support boundary is: Wherein, the stiffness matrix in different directions is represented as: where the superscripts c and d represent the concentrated and distributed constraints, respectively, k c i Kc is the concentrated support stiffness for different directions, k d i Kd is the distributed support stiffness per unit length, N s Nc is the number of concentrated support springs, Theta s and Theta e and are the start and end angles of the support arc, respectively.

4. The planet gear dynamics analysis method considering flexibility of ring gear according to claim 1, wherein, The step 3 specifically comprises: Step 3.1, determining the rotation strategy of the position of the planetary gear; Step 3.2, constructing the continuous-discrete coupling matrix of the component; Step 3.3, constructing the total stiffness matrix and the total mass matrix of the component.

5. The planet gear dynamics analysis method considering flexibility of ring gear according to claim 4, wherein, The rotation strategy of the step 3.1 is a real rotation strategy of real-time updating the instantaneous position of the planetary gear.

6. The planet gear dynamics analysis method considering flexibility of ring gear according to claim 5, wherein, The step 3.2 specifically comprises: The continuous-discrete coupling matrix is composed of K DD , K DC and K CC three sub-matrices; K DD is a discrete-discrete submatrix: (24) wherein N p the number of planetary gears, A i characterizes the meshing relationship between the planetary gears and the ring gear: (25) In the formula, k rpi is the first i planetary gear-race meshing tooth pair; The meshing projection vector of the planetary gear-gear ring is: (26) wherein α t is the face pressure angle, β b is the base circle helix angle, r bp is the base circle radius of the planetary gear, Psi rpi is the angle between the meshing plane of the i th planetary gear-ring gear meshing tooth pair and the axis of rotation, y ​ (27) wherein i0 and i (θp0, θp) are the initial and instantaneous meshing azimuth angles of the mth planetary gear-raceway meshing pair, respectively, t i Omega c is the angular velocity of the planet carrier;​​ K DC is a discrete-continuous submatrix: (28) In the formula, K DCi For the first i Discrete-continuous sub-matrix corresponding to the mth planetary gear-race meshing tooth pair: (29) K CC is a continuous-continuous submatrix: (30)。 7. The planet gear dynamics analysis method considering flexibility of the ring gear according to claim 6, wherein The step 3.3 specifically comprises: The total stiffness matrix of the system is: (31) wherein K lumped t represents the time-varying stiffness matrix of the lumped parameter sub-model​ The total mass matrix is: (32) In the formula, M lumped is the mass matrix of the lumped parameter submodel; The time-varying damping matrix of the system is constructed by using Rayleigh damping: (33) The proportional damping coefficient in the formula is estimated by using the modal damping ratio: (34) (35) wherein Omega 1 and Omega 2 are the circular frequencies of the 1st and 2nd order elastic body modes, respectively.

8. The planet gear dynamics analysis method considering flexibility of ring gear according to claim 1, wherein, The step 4 specifically comprises: mobile elastic coupling boundary connection between the flexible gear ring and the planetary gear, and numerical iterative solution of the kinetic differential equation is realized by Newmark- β method.

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