Two-stage planetary transmission system dynamics modeling method under extreme working condition

Abstract

The invention discloses a dynamic modeling method for a two-stage planetary transmission system under an extreme working condition. The dynamic modeling method comprises the following steps: step (1), establishing a time-varying meshing stiffness model of coupling friction; (2) establishing a dynamic model of the gear transmission system by taking the time-varying meshing stiffness as the basis and comprehensively considering the internal intertooth friction effect and relative vibration displacement characteristics of the system; (3) establishing a nonlinear dynamic oscillatory differential equation set of the two-stage planetary gear transmission system; (4) parameters are selected to solve nonlinear response characteristics of the system; the method has the beneficial effects that the nonlinear dynamic response characteristics of the two-stage planetary gear transmission system under the extreme working condition can be solved, and the research blank of the two-stage planetary gear transmission system under the extreme working condition at present is filled.

Description

Technical Field

[0001] The present invention relates to the technical field of gear dynamics, and in particular to a dynamic modeling method for a two-stage planetary transmission system under extreme working conditions. Background Art

[0002] The two-stage planetary gear transmission system is a mechanical transmission mechanism with a wide range of uses. It plays a key role in advanced manufacturing industries such as automotive transmissions, wind turbines, and aerospace equipment. The system is known for its high transmission efficiency, high power density, and strong impact resistance, and is indispensable in the field of high-end manufacturing. By analyzing the nonlinear response characteristics of the system and obtaining the laws of the system response, the stable state of the system under operation can be found, providing theoretical support for improving system performance, reducing noise, and increasing the service life of the structure. However, with its in-depth application in the field of high-end manufacturing, the two-stage planetary gear transmission system will inevitably face more complex and extreme working conditions. Theoretical research on this transmission system only focuses on conventional input conditions and external environments, and there is still a gap in the nonlinear dynamic analysis under certain specific extreme working conditions.

[0003] In order to solve the above problems, the present invention proposes a dynamic modeling method for a two-stage planetary transmission system under extreme working conditions. This method can effectively and accurately study the nonlinear dynamic characteristics of the system under such conditions; it fills the technical gap in the dynamic analysis and nonlinear characteristic calculation of the two-stage planetary gear transmission system under certain specific extreme working conditions, promotes the development of engineering technology, and can also generate greater social and economic benefits. Summary of the Invention

[0004] In order to overcome the shortcomings of the existing technology and fill the gaps in related technologies, the present invention provides a dynamic modeling method for a two-stage planetary transmission system under extreme working conditions. This method establishes a two-stage planetary gear transmission system and constructs a nonlinear dynamic model, taking into account multiple factors such as support stiffness, coupled friction time-varying meshing stiffness, tooth side clearance, comprehensive transmission error, and inter-tooth friction. The differential equations of the transmission system are derived, and the Runge-Kutta integration method is used to solve the differential dynamic equations of the system to obtain the nonlinear response characteristics of the system under the influence of external excitation and internal factors.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a dynamic modeling method for a two-stage planetary transmission system under extreme working conditions, characterized by comprising the following steps:

[0006] Step (1): Establish a time-varying meshing stiffness model of coupling friction based on the potential energy method: The two-stage planetary gear transmission system has two meshing states: internal meshing and external meshing; the meshing between the planetary gear and the inner ring gear is internal meshing; in the internal meshing state, the potential energy method is used to solve the axial compressive stiffness k of the coupling friction.a , bending stiffness k b , shear stiffness k s The expressions are:

[0007]

[0008] Among them, α is the meshing angle of the gear, α2 is the planetary gear half-tooth angle, and the calculation formula is: α1 is the force component F y The angle between the force and the resultant force F is calculated as -inv(α0), μ p is the friction coefficient of the planetary gear, N p 、N r are the number of teeth of the planetary gear and the inner ring gear respectively, φ is the half tooth angle of the inner ring gear, and the calculation formula is R rb is the base circle radius of the inner gear ring, R rr is the root circle radius of the inner ring gear;

[0009] When meshing, the meshing between the sun gear and the planet gear is external meshing; in the external meshing state, the potential energy method is used to solve the axial compressive stiffness k of the coupling friction. a , bending stiffness k b , shear stiffness k s The expressions are:

[0010]

[0011]

[0012] Among them, the calculation formula of α2 is the same as that of internal meshing, and the calculation formula of angle α1 is:

[0013]

[0014] Among them, μ s is the friction coefficient of the sun gear, N s 、N p are the number of teeth of the sun gear and planet gear respectively;

[0015] The overall expression for meshing stiffness k(t) solved by potential energy method is:

[0016]

[0017] Among them, k h is the Hertzian contact stiffness of the gear, which is expressed as:

[0018]

[0019] Where E is the elastic modulus of the gear material used, L is the width of the gear teeth, ν is the Poisson's ratio of the gear material used; k bn 、k sn 、k an 、k fn (n=1, 2) represent the bending stiffness, shear stiffness, axial compressive stiffness and base stiffness of the driving wheel and the driven wheel respectively;

[0020] Step (2): Couple the analysis of meshing force fluctuation, damping dissipation characteristics and tooth surface friction excitation to establish the vibration displacement control equation with time-varying parameters:

[0021] The gear pair vibration displacement expression is:

[0022]

[0023] Among them, δ sp ,δ pr They represent the vibration displacements generated by the sun gear-planet gear pair and the planet gear-ring gear pair, respectively, and x s 、y s 、u s Represents the vibration displacement and torsional displacement of the sun gear in the x and y directions respectively. p 、y p 、u p Represents the vibration displacement and torsional displacement of the planetary gear in the x and y directions respectively. r 、y r 、u r Represent the vibration displacement and torsional displacement of the inner gear ring in the x and y directions, ψ spi Indicates the sun gear-ith planet gear pair relative to the sun gear coordinate system X S The rotation angle of the axis is expressed as ψ spi =2π(i-1) / N-α(i=1,2,3); ψpri represents the position of the i-th planetary gear ring gear pair relative to X S The rotation angle of the axis is expressed as ψ pri =2π(i-1) / N+α; α is the gear pressure angle, e sp 、e pr They represent the transmission errors between the sun gear-planet gear pair and the planet gear-ring gear pair respectively;

[0024] The gear-planet carrier vibration displacement expression is:

[0025]

[0026] Among them, δ cpx , δ cpy , δ cpuThey represent the vibration displacement and torsional displacement in the x and y directions under the interaction between the planet gear and the planet carrier, respectively. c 、y c 、u c Represents the vibration displacement and torsional displacement of the planet carrier in the x and y directions respectively, p 、y p Represent the vibration displacement of the planetary gear in the x and y directions respectively;

[0027] The transmission error is mainly caused by external excitation, according to the meshing frequency ω m Perform a first-order Fourier series expansion:

[0028] where e m is the error constant, e A represents the error fluctuation amplitude, is the initial phase angle, ω m is the meshing frequency;

[0029] The expression of meshing force between teeth is:

[0030]

[0031] in are the time-varying meshing stiffness of the first-stage sun gear-planet gear pair, the planet gear-ring gear pair, the second-stage sun gear-planet gear pair, and the planet gear-ring gear pair, respectively. They are the meshing damping of the first-stage sun gear-planet gear pair, planet gear-internal gear pair, the second-stage sun gear-planet gear pair, and planet gear-internal gear pair, respectively. are the relative vibration displacements of the first-stage sun gear-planet gear pair, the planet gear-ring gear pair, the second-stage sun gear-planet gear pair, and the planet gear-ring gear pair; f(δ) is the tooth side clearance function, expressed as:

[0032]

[0033] Where b is half of the tooth side clearance;

[0034] The expression of meshing damping is:

[0035]

[0036] in, Respectively represent the meshing damping ratios of the first-stage sun gear-planet gear pair, the planet gear-internal gear pair, the second-stage sun gear-planet gear pair, and the planet gear-internal gear pair, m e is the equivalent mass of the gear pair, and the expression is I s , Ip , I r Represent the rotational inertia of the sun gear, planet gear, and inner ring gear, r s 、r p 、r r Represent the base circle radius of the sun gear, planet gear and inner ring gear respectively;

[0037] Friction between teeth: The expression is F f =λμF; where μ is the friction coefficient, F is the meshing force, and λ is the directional coefficient, which is related to the direction of the friction force and can be {-1, 0, 1};

[0038] Step (3): Establish the nonlinear dynamic equations of the two-stage planetary gear transmission system:

[0039] Taking into account the time-varying meshing stiffness of coupling friction, meshing damping dissipation, and internal tooth surface friction effects, the vibration differential equation of the two-stage planetary gear transmission system can be derived through dimensionless transformation:

[0040] The vibration differential equation of the first-stage sun gear is:

[0041]

[0042]

[0043] in, They are the lateral and longitudinal vibration displacements and torsional displacements of the first layer sun gear, is the friction coefficient of the first-stage sun gear, is the friction main force arm of the sun gear-planet gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage sun gear respectively, are the lateral and longitudinal support damping and torsional damping of the first stage sun gear respectively, Respectively represent the meshing stiffness and meshing damping between the first-stage sun gear and the planetary gear, It represents the relative vibration displacement between the first-stage sun gear and the i-th planet gear, t in is the torque output from the input motor to the transmission system through the input shaft, is the tooth backlash function of the relative vibration displacement between the first-stage sun gear and the i-th planet gear;

[0044] The vibration differential equation of the first-stage planetary gear is:

[0045]

[0046] in, are the lateral and longitudinal vibration displacements and torsional displacements of the i-th planetary gear in the first layer, are the friction coefficients of the first-stage inner gear ring, is the friction main force arm of the planetary gear-ring gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage planetary gear respectively, They are the lateral and longitudinal support damping and torsional damping of the first stage planetary gear respectively. It represents the relative vibration displacement between the i-th planetary gear and the inner ring gear. is the tooth backlash function of the relative vibration displacement generated by the i-th planetary gear and the first-stage internal gear ring;

[0047] The vibration differential equation of the first-stage planet carrier is:

[0048]

[0049] in, is the lateral, longitudinal vibration displacement and torsional displacement of the first layer planet carrier, is the angular velocity of the first stage planet carrier, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage planet carrier, They are the lateral and longitudinal support damping and torsional damping of the first stage planetary gear respectively. are the lateral and longitudinal vibration displacements and torsional displacements generated by the first-stage planet carrier and the i-th planet gear, respectively. They represent the inner diameter of the first-stage planet carrier and the base radius of the second-stage sun gear, represents the torsional displacement of the second-stage sun gear, represents the interstage coupling stiffness between the first and second stages;

[0050] The vibration differential equation of the first-stage internal gear ring is:

[0051]

[0052] in, is the lateral, longitudinal vibration displacement and torsional displacement of the first layer inner gear ring, are the lateral and longitudinal support stiffness and torsional stiffness of the first-stage inner gear ring, They are the lateral and longitudinal support damping and torsional damping of the first-stage inner gear ring respectively;

[0053] The vibration differential equation of the second-stage sun gear is:

[0054]

[0055] in, is the lateral, longitudinal vibration displacement and torsional displacement of the second sun gear, is the friction coefficient of the second-stage sun gear, is the friction main force arm of the second-stage sun gear-planet gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage sun gear respectively, are the lateral and longitudinal support damping and torsional damping of the second-stage sun gear respectively, Respectively represent the meshing stiffness and meshing damping between the second-stage sun gear and the planetary gear, It represents the relative vibration displacement between the second-stage sun gear and the j-th planet gear. is the tooth backlash function of the relative vibration displacement between the second-stage sun gear and the j-th planet gear;

[0056] The vibration differential equation of the second-stage planetary gear is:

[0057]

[0058] in, are the lateral and longitudinal vibration displacements and torsional displacements of the j-th planetary gear in the second layer, are the friction coefficients between the second-stage sun gear and the inner ring gear, It is the friction main force arm of the second-stage planetary gear-ring gear pair. are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage planetary gear respectively, are the lateral and longitudinal support damping and torsional damping of the second-stage planetary gear respectively, Respectively represent the meshing stiffness and meshing damping between the second-stage planetary gear and the inner ring gear, It represents the relative vibration displacement between the jth planetary gear and the inner ring gear of the second stage. is the backlash function of the relative vibration displacement between the jth planetary gear and the second-stage internal gear ring;

[0059] The vibration differential equation of the second-stage planet carrier is:

[0060]

[0061] in, is the lateral, longitudinal vibration displacement and torsional displacement of the second-layer planetary carrier, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage planetary carrier, are the lateral and longitudinal support damping and torsional damping of the second-stage planetary gear respectively, are the lateral and longitudinal vibration displacements and torsional displacements generated by the first-stage planet carrier and the j-th planet gear respectively;

[0062] The vibration differential equation of the second-stage internal gear ring is:

[0063]

[0064] in, is the lateral, longitudinal vibration displacement and torsional displacement of the second layer inner gear ring, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage inner gear ring, They are the lateral and longitudinal support damping and torsional damping of the second-stage inner gear ring respectively;

[0065] Step (4): Select reasonable parameters to calculate the nonlinear response characteristics of the system; solve the obtained vibration differential equations through numerical methods to obtain the dynamic response of the system under the initial conditions.

[0066] Compared with the prior art, the present invention has the following beneficial effects: taking into account multiple factors such as support stiffness and support damping, time-varying meshing stiffness, tooth side clearance, comprehensive transmission error, and inter-tooth friction, a nonlinear dynamic calculation model of a two-stage planetary gear transmission system under extreme input conditions is established, and a set of differential equations of the two-stage planetary gear transmission system under these conditions is derived. Then, the four-step Runge-Kutta method is used to solve the system differential dynamic equations, and the nonlinear response characteristics of the system under the influence of external excitation and internal factors are obtained. This fills the technical gap in the calculation of nonlinear characteristics of dynamic analysis of two-stage planetary gear transmission systems under certain specific extreme working conditions, and provides a reference for vibration reduction, noise reduction and performance improvement of two-stage planetary gear transmissions. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 It is a flow chart of nonlinear dynamic calculation of two-stage planetary gear transmission system;

[0068] Figure 2 This is a schematic diagram of the gear tooth equivalent calculation of the time-varying meshing stiffness of coupled friction using the potential energy method;

[0069] Figure 3 It is a three-dimensional dynamic model diagram of a two-stage planetary gear transmission system;

[0070] Figure 4 It is the bifurcation diagram of the input and output ends of the two-stage planetary gear transmission system. DETAILED DESCRIPTION

[0071] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1-Figure 4 The specific embodiments of the present invention are described in detail.

[0072] like Figure 1 The figure shows a flow chart of the nonlinear dynamics calculation method of a two-stage planetary gear transmission system, which includes the following steps:

[0073] Step (1): Determine the gear parameters of the two-stage planetary gear transmission system and add a high-torque, low-speed motor to simulate extreme working conditions such as heavy trucks and mining machinery; the gear parameters and motor parameters are shown in Table 1 and Table 2 respectively:

[0074] Table 1 Gear parameters of two-stage planetary gear transmission system

[0075]

[0076] Table 2 Main parameters of switched reluctance motor

[0077]

[0078] Step (2): Establish a time-varying meshing stiffness model of coupled friction based on the potential energy method: The two-stage planetary gear transmission system has two meshing states: internal meshing and external meshing. Figure 2 A schematic diagram of the calculation method for time-varying meshing stiffness is shown; in the internal meshing state, the potential energy method is used to solve the axial compressive stiffness k of the coupled friction. a , bending stiffness k b , shear stiffness k s The expressions are:

[0079]

[0080] Among them, α is the meshing angle of the gear, α2 is the planetary gear half-tooth angle, and the calculation formula is: α1 is the force component F y The angle between the force and the resultant force F is calculated as -inv(α0), μ p is the friction coefficient of the planetary gear, N p 、N r are the number of teeth of the planetary gear and the inner ring gear respectively, φ is the half tooth angle of the inner ring gear, and the calculation formula is R rb is the base circle radius of the inner gear ring, R rr is the root circle radius of the inner ring gear;

[0081] When meshing, the meshing between the sun gear and the planet gear is external meshing; in the external meshing state, the potential energy method is used to solve the axial compressive stiffness k of the coupling friction. a , bending stiffness k b , shear stiffness k s The expressions are:

[0082]

[0083]

[0084] Among them, the calculation formula of α2 is the same as that of internal meshing, and the calculation formula of angle α1 is:

[0085]

[0086] Among them, μ s is the friction coefficient of the sun gear, N s 、N p are the number of teeth of the sun gear and planet gear respectively;

[0087] The overall expression for meshing stiffness k(t) solved by potential energy method is:

[0088]

[0089] Among them, k h is the Hertzian contact stiffness of the gear, expressed as

[0090]

[0091] Where E is the elastic modulus of the gear material used, L is the width of the gear teeth, ν is the Poisson's ratio of the gear material used; k bn 、k sn 、k an 、k fn (n=1, 2) represent the bending stiffness, shear stiffness, axial compressive stiffness and base stiffness of the driving wheel and the driven wheel respectively;

[0092] Step (3): Couple the analysis of meshing force fluctuation, damping dissipation characteristics and tooth surface friction excitation to establish the vibration displacement control equation with time-varying parameters:

[0093] The gear pair vibration displacement expression is:

[0094]

[0095] Among them, δ sp ,δ pr They represent the vibration displacements generated by the sun gear-planet gear pair and the planet gear-ring gear pair, respectively, and x s 、y s 、u s Represents the vibration displacement and torsional displacement of the sun gear in the x and y directions respectively. p 、y p 、u p Represents the vibration displacement and torsional displacement of the planetary gear in the x and y directions respectively. r 、y r 、u r Represent the vibration displacement and torsional displacement of the inner gear ring in the x and y directions, ψ spi Indicates the sun gear-ith planet gear pair relative to the sun gear coordinate system X S The rotation angle of the axis is expressed as ψ spi=2π(i-1) / N-α(i=1,2,3); ψpri represents the position of the i-th planetary gear ring gear pair relative to X S The rotation angle of the axis is expressed as ψ pri =2π(i-1) / N+α; α is the gear pressure angle; e sp 、e pr They represent the transmission errors between the sun gear-planet gear pair and the planet gear-ring gear pair respectively;

[0096] The gear-planet carrier vibration displacement expression is:

[0097]

[0098] Among them, δ cpx , δ cpy , δ cpu They represent the vibration displacement and torsional displacement in the x and y directions under the interaction between the planet gear and the planet carrier, respectively. c 、y c 、u c Represents the vibration displacement and torsional displacement of the planet carrier in the x and y directions respectively, p 、y p Represent the vibration displacement of the planetary gear in the x and y directions respectively;

[0099] The transmission error is mainly caused by external excitation, according to the meshing frequency ω m Perform a first-order Fourier series expansion:

[0100] where e m is the error constant, e A represents the error fluctuation amplitude, is the initial phase angle, ω m is the meshing frequency; the meshing force between teeth is expressed as:

[0101]

[0102] in are the time-varying meshing stiffness of the first-stage sun gear-planet gear pair, the planet gear-ring gear pair, the second-stage sun gear-planet gear pair, and the planet gear-ring gear pair, respectively. They are the meshing damping of the first-stage sun gear-planet gear pair, planet gear-internal gear pair, the second-stage sun gear-planet gear pair, and planet gear-internal gear pair, respectively. are the relative vibration displacements of the first-stage sun gear-planet gear pair, the planet gear-ring gear pair, the second-stage sun gear-planet gear pair, and the planet gear-ring gear pair; f(δ) is the tooth side clearance function, expressed as:

[0103]

[0104] Where b is half of the tooth side clearance;

[0105] The expression of meshing damping is:

[0106]

[0107] in, Respectively represent the meshing damping ratios of the first-stage sun gear-planet gear pair, the planet gear-internal gear pair, the second-stage sun gear-planet gear pair, and the planet gear-internal gear pair, m e is the equivalent mass of the gear pair, and the expression is I s , I p , I r Represent the rotational inertia of the sun gear, planet gear, and inner ring gear, r s 、r p 、r r Represent the base circle radius of the sun gear, planet gear and inner ring gear respectively;

[0108] Friction between teeth: The expression is F f =λμF; where μ is the friction coefficient, F is the meshing force, and λ is the directional coefficient, which is related to the direction of the friction force and can be {-1, 0, 1};

[0109] Step (3): Establish the nonlinear dynamic equations of the two-stage planetary gear transmission system:

[0110] Figure 3 The figure below is a schematic diagram of the support stiffness and damping, meshing stiffness and damping of the transmission system. Taking into account the time-varying meshing stiffness of the coupled friction, the meshing damping dissipation mechanism, and the internal tooth surface friction effect, the vibration differential equation of the two-stage planetary gear transmission system can be derived through dimensionless transformation:

[0111] The vibration differential equation of the first-stage sun gear is:

[0112]

[0113]

[0114] in, They are the lateral and longitudinal vibration displacements and torsional displacements of the first layer sun gear, is the friction coefficient of the first-stage sun gear, is the friction main force arm of the sun gear-planet gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage sun gear respectively, are the lateral and longitudinal support damping and torsional damping of the first stage sun gear respectively, Respectively represent the meshing stiffness and meshing damping between the first-stage sun gear and the planetary gear, It represents the relative vibration displacement between the first-stage sun gear and the i-th planet gear, t in is the torque output from the input motor to the transmission system through the input shaft, is the tooth backlash function of the relative vibration displacement between the first-stage sun gear and the i-th planet gear;

[0115] The vibration differential equation of the first-stage planetary gear is:

[0116]

[0117] in, are the lateral and longitudinal vibration displacements and torsional displacements of the i-th planetary gear in the first layer, are the friction coefficients of the first-stage inner gear ring, is the friction main force arm of the planetary gear-ring gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage planetary gear respectively, They are the lateral and longitudinal support damping and torsional damping of the first stage planetary gear respectively. It represents the relative vibration displacement between the i-th planetary gear and the inner ring gear. is the tooth backlash function of the relative vibration displacement generated by the i-th planetary gear and the first-stage internal gear ring;

[0118] The vibration differential equation of the first-stage planet carrier is:

[0119]

[0120] in, is the lateral, longitudinal vibration displacement and torsional displacement of the first layer planet carrier, is the angular velocity of the first-stage planet carrier, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage planet carrier, They are the lateral and longitudinal support damping and torsional damping of the first stage planetary gear respectively. are the lateral and longitudinal vibration displacements and torsional displacements generated by the first-stage planet carrier and the i-th planet gear, respectively. They represent the inner diameter of the first-stage planet carrier and the base radius of the second-stage sun gear, represents the torsional displacement of the second-stage sun gear, represents the interstage coupling stiffness between the first and second stages;

[0121] The vibration differential equation of the first-stage internal gear ring is:

[0122]

[0123] in, is the lateral, longitudinal vibration displacement and torsional displacement of the first layer inner gear ring, are the lateral and longitudinal support stiffness and torsional stiffness of the first-stage inner gear ring, They are the lateral and longitudinal support damping and torsional damping of the first-stage inner gear ring respectively;

[0124] The vibration differential equation of the second-stage sun gear is:

[0125]

[0126] in, is the lateral, longitudinal vibration displacement and torsional displacement of the second sun gear, is the friction coefficient of the second-stage sun gear, is the friction main force arm of the second-stage sun gear-planet gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage sun gear respectively, are the lateral and longitudinal support damping and torsional damping of the second-stage sun gear respectively, Respectively represent the meshing stiffness and meshing damping between the second-stage sun gear and the planetary gear, It represents the relative vibration displacement between the second-stage sun gear and the j-th planet gear. is the tooth backlash function of the relative vibration displacement between the second-stage sun gear and the j-th planet gear;

[0127] The vibration differential equation of the second-stage planetary gear is:

[0128]

[0129] Among them, are the lateral and longitudinal vibration displacements and torsional displacements of the j-th planetary gear in the second layer, are the friction coefficients between the second-stage sun gear and the inner ring gear, It is the friction main force arm of the second-stage planetary gear-ring gear pair. are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage planetary gear respectively, are the lateral and longitudinal support damping and torsional damping of the second-stage planetary gear respectively, Respectively represent the meshing stiffness and meshing damping between the second-stage planetary gear and the inner ring gear, It represents the relative vibration displacement between the jth planetary gear and the inner ring gear of the second stage. is the backlash function of the relative vibration displacement between the jth planetary gear and the second-stage internal gear ring;

[0130] The vibration differential equation of the second-stage planet carrier is:

[0131]

[0132]

[0133] in, is the lateral, longitudinal vibration displacement and torsional displacement of the second-layer planetary carrier, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage planetary carrier, are the lateral and longitudinal support damping and torsional damping of the second-stage planetary gear respectively, are the lateral and longitudinal vibration displacements and torsional displacements generated by the first-stage planet carrier and the j-th planet gear respectively;

[0134] The vibration differential equation of the second-stage internal gear ring is:

[0135]

[0136] in, is the lateral, longitudinal vibration displacement and torsional displacement of the second layer inner gear ring, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage inner gear ring, They are the lateral and longitudinal support damping and torsional damping of the second-stage inner gear ring respectively;

[0137] Step (4): Solve the nonlinear response characteristics of the system using the Runge-Kutta method;

[0138] Figure 4 The figure shows the bifurcation diagram of the dynamic response of the transmission system output as the external excitation changes. It can be seen that as the external excitation frequency changes, the system's dynamic response also changes irregularly, with a recurring phenomenon of single-cycle, multi-cycle, and chaotic responses, confirming the theme of nonlinear dynamics. This reveals the evolution of the vibration response of the planetary gear transmission system under varying input speeds, providing strong theoretical support for the efficient optimization of planetary transmission system parameters and the improvement of planetary transmission system stability.

[0139] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.

Claims

1. A dynamic modeling method for a two-stage planetary transmission system under extreme working conditions, characterized by: Step (1): Establish a time-varying meshing stiffness model of coupling friction based on the potential energy method: The two-stage planetary gear transmission system has two meshing states: internal meshing and external meshing; the meshing between the planetary gear and the inner ring gear is internal meshing; in the internal meshing state, the potential energy method is used to solve the axial compressive stiffness k of the coupling friction. a , bending stiffness k b , shear stiffness k s The expressions are: Among them, E is the Young's modulus of the gear material, L is the tooth width, α is the meshing angle of the gear, α2 is the semi-tooth angle of the planetary gear; α1 is the component force F y The angle with the resultant force F, φ is the half tooth angle of the inner ring gear, μ p is the friction coefficient of the planetary gear; When meshing, the meshing between the sun gear and the planet gear is external meshing; in the external meshing state, the potential energy method is used to solve the axial compressive stiffness k of the coupling friction. a , bending stiffness k b , shear stiffness k s The expressions are: Among them, μ s is the friction coefficient of the sun gear; The overall expression for meshing stiffness k(t) solved by potential energy method is: Among them, k h is the Hertzian contact stiffness of the gear, k bn 、k sn 、k an 、k fn (n=1, 2) represent the bending stiffness, shear stiffness, axial compressive stiffness and base stiffness of the driving wheel and the driven wheel respectively; Step (2): Couple the analysis of meshing force fluctuation, damping dissipation characteristics and tooth surface friction excitation to establish the vibration displacement control equation with time-varying parameters: The gear pair vibration displacement expression is: Among them, δ sp ,δ pr They represent the vibration displacements generated by the sun gear-planet gear pair and the planet gear-ring gear pair, respectively, and x s 、y s 、u s Represents the vibration displacement and torsional displacement of the sun gear in the x and y directions respectively. p 、y p 、u p Represents the vibration displacement and torsional displacement of the planetary gear in the x and y directions respectively. r 、y r 、u r Represent the vibration displacement and torsional displacement of the inner gear ring in the x and y directions, ψ spi Indicates the sun gear-ith planet gear pair relative to the sun gear coordinate system X S The rotation angle of the axis is expressed as ψ spi =2π(i-1) / N-α(i=1,2,3); ψpri represents the position of the i-th planetary gear ring gear pair relative to X S The rotation angle of the axis is expressed as ψ pri =2π(i-1) / N+α, α is the gear pressure angle, e sp 、e pr They represent the transmission errors between the sun gear-planet gear pair and the planet gear-ring gear pair respectively; The gear-planet carrier vibration displacement expression is: Among them, δ cpx , δ cpy , δ cpu They represent the vibration displacement and torsional displacement in the x and y directions under the interaction between the planet gear and the planet carrier, respectively. c 、y c 、u c Represents the vibration displacement and torsional displacement of the planet carrier in the x and y directions respectively, p 、y p Represent the vibration displacement of the planetary gear in the x and y directions respectively; Step (3): Establish the nonlinear dynamic equations of the two-stage planetary gear transmission system: Taking into account the time-varying meshing stiffness of coupling friction, meshing damping dissipation mechanism, and internal tooth surface friction effect, the vibration differential equation of the two-stage planetary gear transmission system can be derived through dimensionless transformation: The vibration differential equation of the first-stage sun gear is: in, They are the lateral and longitudinal vibration displacements and torsional displacements of the first layer sun gear, is the friction coefficient of the first-stage sun gear, is the friction main force arm of the sun gear-planet gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage sun gear respectively, are the lateral and longitudinal support damping and torsional damping of the first stage sun gear respectively, Respectively represent the meshing stiffness and meshing damping between the first-stage sun gear and the planetary gear, It represents the relative vibration displacement between the first-stage sun gear and the i-th planet gear, t in is the torque output from the input motor to the transmission system through the input shaft, is the tooth backlash function of the relative vibration displacement between the first-stage sun gear and the i-th planet gear; The vibration differential equation of the first-stage planetary gear is: in, are the lateral and longitudinal vibration displacements and torsional displacements of the i-th planetary gear in the first layer, are the friction coefficients of the first-stage inner gear ring, is the friction main force arm of the planetary gear-ring gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage planetary gear respectively, They are the lateral and longitudinal support damping and torsional damping of the first stage planetary gear respectively. It represents the relative vibration displacement between the i-th planetary gear and the inner ring gear. is the tooth backlash function of the relative vibration displacement generated by the i-th planetary gear and the first-stage internal gear ring; The vibration differential equation of the first-stage planet carrier is: in, is the lateral, longitudinal vibration displacement and torsional displacement of the first layer planet carrier, is the angular velocity of the first-stage planet carrier, are the lateral and longitudinal support stiffness and torsional stiffness of the first stage planet carrier, They are the lateral and longitudinal support damping and torsional damping of the first stage planetary gear respectively. are the lateral and longitudinal vibration displacements and torsional displacements generated by the first-stage planet carrier and the i-th planet gear, respectively. They represent the inner diameter of the first-stage planet carrier and the base radius of the second-stage sun gear, represents the torsional displacement of the second-stage sun gear, represents the interstage coupling stiffness between the first and second stages; The vibration differential equation of the first-stage internal gear ring is: in, is the lateral, longitudinal vibration displacement and torsional displacement of the first layer inner gear ring, are the lateral and longitudinal support stiffness and torsional stiffness of the first-stage inner gear ring, They are the lateral and longitudinal support damping and torsional damping of the first-stage inner gear ring respectively; The vibration differential equation of the second-stage sun gear is: in, is the lateral, longitudinal vibration displacement and torsional displacement of the second sun gear, is the friction coefficient of the second-stage sun gear, is the friction main force arm of the second-stage sun gear-planet gear pair, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage sun gear respectively, are the lateral and longitudinal support damping and torsional damping of the second-stage sun gear respectively, Respectively represent the meshing stiffness and meshing damping between the second-stage sun gear and the planetary gear, It represents the relative vibration displacement between the second-stage sun gear and the j-th planet gear. is the tooth backlash function of the relative vibration displacement between the second-stage sun gear and the j-th planet gear; The vibration differential equation of the second-stage planetary gear is: in, are the lateral and longitudinal vibration displacements and torsional displacements of the j-th planetary gear in the second layer, are the friction coefficients between the second-stage sun gear and the inner ring gear, It is the friction main force arm of the second-stage planetary gear-ring gear pair. are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage planetary gear respectively, are the lateral and longitudinal support damping and torsional damping of the second-stage planetary gear respectively, Respectively represent the meshing stiffness and meshing damping between the second-stage planetary gear and the inner ring gear, It represents the relative vibration displacement between the jth planetary gear and the inner ring gear of the second stage. is the backlash function of the relative vibration displacement between the jth planetary gear and the second-stage internal gear ring; The vibration differential equation of the second-stage planet carrier is: in, is the lateral, longitudinal vibration displacement and torsional displacement of the second-layer planetary carrier, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage planetary carrier, are the lateral and longitudinal support damping and torsional damping of the second-stage planetary gear respectively, are the lateral and longitudinal vibration displacements and torsional displacements generated by the first-stage planet carrier and the j-th planet gear respectively; The vibration differential equation of the second-stage internal gear ring is: in, is the lateral, longitudinal vibration displacement and torsional displacement of the second layer inner gear ring, are the lateral and longitudinal support stiffness and torsional stiffness of the second-stage inner gear ring, They are the lateral and longitudinal support damping and torsional damping of the second-stage inner gear ring respectively; Step (4): Select reasonable parameters to calculate the nonlinear response characteristics of the system; solve the obtained vibration differential equations through numerical methods to obtain the dynamic response of the system under the initial conditions.

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