Face gear transmission dynamics modeling method considering time-varying temperature

Abstract

The invention discloses a face gear transmission dynamics modeling method considering time-varying temperature, and belongs to the technical field of gears. The method comprises the following steps: 1) fitting elastic modulus and Poisson's ratio functions of a material at different temperatures, inputting the elastic modulus and Poisson's ratio functions, friction heat flow and a convective heat transfer coefficient into a finite element model, and solving thermal time-varying meshing stiffness and thermal meshing damping; 2) establishing a meshing unit containing a time-varying thermal parameter, a static transfer error, a meshing clearance and a friction force, a shaft beam unit containing a time-varying temperature and a gyroscopic effect, a rotor unit and a bearing support unit; (3) assembling all the units into a face gear transmission system-level multi-node dynamic model based on a finite element method, and deducing a motion differential equation of the face gear transmission system-level multi-node dynamic model; and 4) solving the characteristic value equation and the motion differential equation, and analyzing the dynamic characteristics of the system under the working conditions of normal temperature and time-varying temperature. According to the method, the temperature serves as a time-varying parameter to be dynamically input into a kinetic model, the system working state can be reflected more truly, and a theoretical basis is provided for face gear optimization design.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of gears, and particularly relates to a face gear transmission dynamics modeling method considering time-varying temperature. BACKGROUND

[0002] The face gear becomes the core component of the high-speed heavy-load extreme working condition of the aerospace equipment and high-precision transmission system due to the advantages of high bearing capacity, speed adaptability, power density and installation tolerance. In the continuous high-speed heavy-load operation, the meshing temperature presents regular time-varying fluctuations due to the friction heat of the tooth surface. The time-varying temperature changes the inherent properties of the material, such as the elastic modulus and Poisson's ratio, causes the dynamic evolution of the contact deformation, triggers the drift of the time-varying meshing stiffness and damping and other dynamic parameters, and finally significantly amplifies the system vibration response, thereby restricting the reliability and application range in extreme environments. It has core theoretical value and engineering significance to construct a dynamics model driven by the time-varying temperature to suppress the system vibration and noise.

[0003] Under the high-speed heavy-load working condition, the friction heat is diffused to the wheel body, gear shaft and bearing and other structures through heat conduction / convection, thereby changing the thermal-mechanical coupling characteristics of the system. The existing dynamics model is mostly based on the constant temperature or constant temperature assumption, and the temperature is not dynamically coupled to the motion differential equation as a time-varying parameter. Moreover, the existing method is limited to the steady-state / transient temperature field of a single working condition, ignores the time-varying characteristics of the temperature caused by the actual variable speed and variable load, and has insufficient prediction accuracy. The method of the patent breaks through the limitation of the traditional temperature field analysis, and initiates a face gear transmission dynamics modeling method considering time-varying temperature. Through the construction of the dynamic coupling mechanism of the thermal-mechanical field, the regulation effect of the temperature on the material properties and the contact behavior is mapped in real time, the correlation between the time-varying temperature and the dynamic characteristics (vibration amplitude, spectral characteristics and the like) of the system is accurately quantified, theoretical support is provided for the design of the high-reliability face gear transmission system, and the development of the aerospace transmission technology towards high efficiency and light weight is promoted. SUMMARY

[0004] The purpose of the embodiment of the application is to provide a face gear transmission dynamics modeling method considering time-varying temperature, which takes the face gear as the research object, and solves the limitation of the vibration analysis method under the traditional constant temperature or constant temperature condition by establishing a flexible multi-node dynamics model considering time-varying temperature.

[0005] In order to solve the above technical problems, the application is implemented as follows:

[0006] The embodiment of the application provides a face gear transmission dynamics modeling method considering time-varying temperature, which comprises the following steps:

[0007] In step S1, the least square method is used to perform function fitting on the experimental data of the elastic modulus and Poisson's ratio of the material at different temperatures.

[0008] Step S2, calculating the friction heat flux and the convection heat transfer coefficient of the tooth end surface, meshing surface, tooth top surface and non-working surface;

[0009] Step S3, inputting the elastic modulus, Poisson's ratio, frictional heat flux and convection heat transfer coefficient of each surface under time-varying temperature into the face gear transmission finite element simulation model;

[0010] Step S4, solving the time-varying meshing stiffness of the gear transmission under time-varying temperature based on the finite element simulation model, and calculating the thermal meshing damping under time-varying temperature;

[0011] Step S5, respectively establishing a face gear meshing unit including thermal time-varying meshing stiffness, thermal meshing damping, static transmission error, meshing clearance, and friction, a shaft beam unit including time-varying temperature and gyroscopic effect, a rigid rotor unit including gyroscopic effect, and a bearing support unit;

[0012] Step S6, solving the dynamic equations of each unit, and assembling the units into a multi-node dynamic model of the face gear transmission system with time-varying temperature based on the finite element method, and deriving the system dynamic equations;

[0013] Step S7: Based on the differential equation of motion of the dynamic model, a corresponding eigenvalue equation is obtained, the eigenvalue equation and the differential equation of motion are solved, and the inherent characteristics and vibration characteristics of the system under the traditional normal temperature working condition and the time-varying temperature working condition are compared.

[0014] Optionally, in step S1, the least squares method is used to fit the elastic modulus of the material under time-varying temperature to a function formula as follows:

[0015] E(T)=208.5-0.057T-1.48×10 -4 T 2 (1.29)

[0016] The fitting function formula for the Poisson's ratio of materials under time-varying temperature is:

[0017] υ(T)=0.289+4.0×10 -5 T(1.30)

[0018] Optionally, in step S2, the calculation method of the three parameters of the gear relative sliding speed, the tooth surface contact stress and the tooth surface friction coefficient in the friction heat flow includes:

[0019] The relative sliding speed of the gears when meshing is calculated using equations (1.31) and (1.32):

[0020]

[0021] Among them, d s is the position of the meshing point on the meshing line; v (p,g)is the absolute velocity of the contact point of the driving wheel and the driven wheel along the contact tangent direction, d (p,g) is the pitch circle diameter of the driving wheel and the driven wheel, a n is the gear pressure angle, n (p,g) is the speed of the driving wheel and the driven wheel;

[0022] The average contact pressure of the gear meshing surface is calculated by formula (1.33):

[0023]

[0024] Among them, F s is the normal load on the tooth surface per unit tooth width, E0 is the comprehensive elastic modulus, b is the gear tooth width, ρ e is the equivalent curvature radius of the gear;

[0025] The tooth surface friction coefficient is calculated by formula (1.34):

[0026]

[0027] Among them, μ(v s ) is the friction coefficient μ with the relative motion speed v s The function of change, r z The value is 200 to establish a more accurate friction model;

[0028] Friction heat flow of driving wheel:

[0029] q p =βγ1μ(v s )P ca v s (1.35)

[0030] Friction heat flow of driven wheel:

[0031] q g =(1-β)γ1μ(v s )P ca v s (1.36)

[0032]

[0033] Among them, β is the friction heat flow distribution factor, γ1 is the heat energy conversion coefficient, μ(v s ) is the friction coefficient, λ (p,g) is the thermal conductivity of the driving wheel and the driven wheel, ρ (p,g) is the material density of the driving wheel and the driven wheel, c (p,g) is the specific heat capacity of the driving wheel and the driven wheel.

[0034] Alternatively, in step S2, the gear pair convective heat transfer is divided into four cases of tooth end face, meshing face, tooth top face and non-working face, and the calculation method includes:

[0035] The convective heat transfer phenomenon of the tooth end face is simplified as the convective heat transfer analysis of a rotating disc; the lubricating oil has three forms of laminar flow, transitional laminar flow and turbulent flow with the rotation of the disc, and the three forms correspond to the corresponding Reynolds value range;

[0036]

[0037] wherein ω is the angular velocity, r k is the arbitrary radius of the disc surface, v l is the kinematic viscosity of the fluid medium;

[0038] When the numerical range R e ≤ 2 × 10 5 , the lubricating oil on the tooth end face is in laminar flow, and the convective heat transfer coefficient in the laminar flow state can be calculated according to formula (1.39):

[0039] h s = 0.308 λ l (m c + 2) 0.5 (ρ l v l c l / λ l ) 0.5 (ω / v l ) 0.5 (1.39)

[0040] When the numerical range 2 × 10 5 < R e < 2.5 × 10 5 , the lubricating oil on the tooth end face is in transitional laminar flow, and the convective heat transfer coefficient in the transitional laminar flow state can be calculated according to formula (1.40):

[0041]

[0042] When the numerical range R e > 2.0 × 10 5 , the lubricating oil on the tooth end face is in turbulent flow, and the convective heat transfer coefficient in the turbulent flow state can be calculated according to formula (1.41):

[0043]

[0044] wherein λ l is the thermal conductivity of the fluid medium, m c is a general exponential constant, and the value is 2, and ρ lis the density of the fluid medium, c l is the specific heat capacity of the fluid medium;

[0045] Calculation of meshing surface heat transfer coefficient:

[0046]

[0047] The maximum value of the tooth end face heat transfer coefficient is taken for the tooth top face heat transfer:

[0048] h d =max(h s )(1.43)

[0049] Non-working surface flow heat transfer coefficient:

[0050]

[0051] Optionally, in step S4, the thermal time-varying meshing stiffness under time-varying temperature is calculated as follows:

[0052]

[0053] Thermal mesh damping calculation:

[0054]

[0055] Among them, F n is the normal contact force, r (p,g) is the base circle radius of the gear, θ (p,g) are the gear rotation angles, ζ is the damping ratio, which is 0.025, m (p,g) Indicates the mass of the gear.

[0056] Optionally, in step S5, the calculation method of the beam unit including time-varying temperature and gyroscopic effect includes:

[0057] Calculation of shear modulus of gear material:

[0058]

[0059] Calculation of shear correction factor for gear materials:

[0060]

[0061] Where ε is the ratio of the inner and outer radii of the gear shaft;

[0062] The total potential energy expression of the beam unit under time-varying temperature is:

[0063]

[0064] Where Θ(T) is the shear modulus of the material, Λs is the polar moment of inertia of the section, Λ is the diameter moment of inertia of the section, μd (T) is a shear correction factor, x' and y' are the curvature components of the beam along the x and y axes, θx, θy are the axis section angles, θ z is the rate of change of the torsion angle.

[0065] Optionally, in step S6, the dynamic equations of the face gear meshing unit, the shaft beam unit, the rotor unit, and the bearing support unit under time-varying temperature and the system dynamic equation include:

[0066] The dynamic equation of the face gear meshing unit under time-varying temperature:

[0067]

[0068] wherein M m , C m , K m and F m are the mass matrix, the damping matrix, the stiffness matrix and the external force matrix of the beam unit, q m is the generalized coordinate vector of the face gear pair;

[0069] The dynamic equation of the shaft beam unit under time-varying temperature:

[0070]

[0071] wherein M , G , C and K are the mass matrix, the gyro matrix, the damping matrix and the stiffness matrix of the beam unit,

[0072] is the generalized coordinate vector of the shaft;

[0073] The dynamic equation of the rotor unit:

[0074] The bearing support unit of the vibration analysis model in the present application is general, and thus a corresponding dynamic equation of the bearing support unit is not needed to be established, and a conventional dynamic parameter is directly substituted;

[0075] The motion equation of the face gear transmission system:

[0076]

[0077] wherein M is the system mass matrix, C is the system damping matrix, G is the system gyro matrix, K is the system stiffness matrix, q is the generalized coordinate vector of the face gear transmission, f(q) is the nonlinear function of the face gear transmission containing the meshing gap, and Q is the external load vector.

[0078] Optionally, in step S7, the motion differential equation is converted into a state equation:

[0079]

[0080] State variable equation:

[0081]

[0082] State coefficient matrix:

[0083]

[0084] Eigenvalue calculation of the state equation:

[0085]

[0086] Among them, ω i is the natural frequency, ξ i is the damping ratio.

[0087] A computer system comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor executes the steps of any one of the methods when executing the computer program.

[0088] Compared with the prior art, this application has the following beneficial effects:

[0089] 1. This application proposes a face gear transmission dynamics modeling method that considers time-varying temperature, and establishes a numerical calculation model for the gear transmission elastic modulus, Poisson's ratio, thermal time-varying meshing stiffness, and thermal meshing damping under time-varying temperature conditions.

[0090] 2. Based on the finite element node method, a system-level dynamic vibration analysis model of the face gear meshing unit, shaft beam unit, gear rotor unit, and bearing support unit was constructed, taking into account the time-varying temperature. Based on the vibration analysis model, the natural frequencies under normal temperature and time-varying temperature conditions were solved;

[0091] 3. In an optional solution, the vibration response of the flexible face gear transmission system, such as vibration displacement and dynamic transmission error, can also be analyzed based on the vibration analysis model. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following briefly introduces the drawings required for describing the embodiments. Obviously, the drawings described below are only some embodiments of the present application. Those skilled in the art can also derive other drawings based on these drawings without inventive work, among which:

[0093] Figure 1 Flowchart provided for the embodiment of this application;

[0094] Figure 2Schematic diagram of setting thermal boundary conditions on different surfaces of the face gear transmission provided in an embodiment of the present application;

[0095] Figure 3 A finite element simulation model of a gear transmission with time-varying temperature provided in an embodiment of the present application;

[0096] Figure 4 A schematic diagram of the calculation results of the thermal time-varying mesh stiffness of the face gear provided in an embodiment of the present application;

[0097] Figure 5 A schematic diagram of a gear meshing unit with a time-varying temperature according to an embodiment of the present application;

[0098] Figure 6 Schematic diagram of the gear shaft beam unit with time-varying temperature provided in an embodiment of the present application;

[0099] Figure 7 A multi-node dynamic model of a gear transmission with time-varying temperature provided in an embodiment of the present application;

[0100] Figure 8 Typical inherent characteristics of the system provided by the embodiments of this application;

[0101] Figure 9 Typical vibration displacement phase diagram and Poincare screenshot of the system provided in the embodiment of this application;

[0102] Figure 10 This is a frequency sweep diagram of the dynamic transmission error of a typical face gear of the system provided in the embodiment of the present application. DETAILED DESCRIPTION

[0103] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are part of the embodiments of this application, not all of them. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0104] The terms "first," "second," and the like in the specification and claims of this application are used to distinguish similar objects, and are not used to describe a specific order or precedence. It should be understood that the terms used in this manner are interchangeable where appropriate, so that the embodiments of this application can be implemented in an order other than that illustrated or described herein, and that the objects distinguished by "first," "second," and the like are generally of the same type, and do not limit the number of objects; for example, the first object can be one or more. In addition, the term "and / or" in the specification and claims refers to at least one of the connected objects, and the character " / " generally indicates that the objects connected are in an "or" relationship.

[0105] The embodiment of the present application provides a face gear transmission dynamics modeling method considering time-varying temperature, comprising the following steps:

[0106] 1) The least squares method is used to fit the elastic modulus and Poisson's ratio experimental data of the material at different temperatures, and then the friction heat flux and the convective heat transfer coefficient of the meshing surface, tooth top surface, tooth end surface and non-working surface are calculated; 2) The elastic modulus, Poisson's ratio, friction heat flux and convective heat transfer coefficient of each surface under time-varying temperature are input into the finite element simulation model of the face gear transmission to solve the thermal time-varying meshing stiffness of the gear transmission under time-varying temperature, and calculate the thermal meshing damping under time-varying temperature; 3) The face gear meshing unit containing thermal time-varying meshing stiffness, thermal meshing damping, static transmission error, meshing clearance, and friction force, the shaft beam unit containing time-varying temperature and gyroscopic effect, the rigid rotor unit containing gyroscopic effect, and the bearing support unit are respectively established; 4) Based on the finite element method, the various units are assembled into a multi-node dynamic model of the face gear transmission system with time-varying temperature, and its dynamic equations and eigenvalue equations are derived and solved, and the inherent characteristics and vibration characteristics of the system under traditional normal temperature conditions and time-varying temperature conditions are compared. Among them:

[0107] Step 1) Based on the elastic modulus and Poisson's ratio data measured by the material tensile test, the least squares method is used to fit the function and establish a continuous function relationship with temperature change; according to the geometric structure, material properties and specific working parameters of the face gear transmission, the friction heat flow caused by tooth surface friction is calculated, and based on the principle of heat transfer, the friction heat flow caused by tooth surface friction is calculated. Figure 2 Convective heat transfer coefficient of the meshing surface, tooth top surface, tooth end surface and non-working surface of face gear transmission;

[0108] Step 2) The temperature-related elastic modulus, Poisson's ratio, friction heat flow and convection heat transfer coefficient of the face gear transmission meshing surface, tooth top surface, tooth end surface and non-working surface in step 1 are applied as thermal boundary conditions to the Figure 3 The finite element simulation model of the face gear transmission is used. After executing the finite element simulation, the angular displacement of the driving wheel and the driven wheel is output; then based on the output angular displacement and the finite element contact analysis results, the solution is obtained. Figure 4 The thermal meshing damping of the gear transmission is further calculated based on the time-varying temperature.

[0109] Step 3) Based on the physical parameters in steps 1 and 2, a key unit model is systematically constructed to characterize the core dynamic behavior of the face gear transmission system, specifically including: Figure 5 Face gear meshing unit model, which integrates thermal time-varying mesh stiffness, thermal mesh damping, static transmission error, meshing clearance and tooth surface friction; Figure 6The shaft beam unit includes the combined effects of the thermal deformation of the shaft structure under time-varying temperature and the gyroscopic torque effect generated by high-speed rotation conditions; the rigid rotor unit model considers the mass distribution of the rotor and the gyroscopic effect generated by high-speed rotation conditions; the stiffness and damping characteristics of the bearing support unit;

[0110] Step 4) Based on the established unit models, the units are systematically integrated and assembled in generalized coordinates based on the finite element node method to construct Figure 7 A flexible multi-node dynamic model of a face gear transmission system with time-varying temperature. The two gear rotors in this model correspond to nodes 5 and 10, which are affected by the rotor and the beam. The bearing nodes are numbered 1, 3, 7, and 11. Based on the Lagrange method, the nonlinear dynamic equations and eigenvalue equations corresponding to this multi-degree-of-freedom system are derived. Then, the eigenvalue analysis method is used to solve the inherent characteristics of the system, and the dynamic equations are solved by the Newmark-β method to obtain the dynamic response of the system. Finally, for example, Figure 8 The inherent characteristics of the system under traditional normal temperature and time-varying temperature conditions, and Figure 9 、 Figure 10 Vibration characteristics of the system under two working conditions.

[0111] Based on the above analysis, see Figure 1 The embodiment of the present application provides a face gear transmission dynamic modeling method considering time-varying temperature, comprising the following steps:

[0112] Step S1, using the least squares method to perform function fitting on the elastic modulus and Poisson's ratio experimental data of the material at different temperatures;

[0113] Step S2, calculating the friction heat flux and the convection heat transfer coefficient of the meshing surface, tooth top surface, tooth end surface and non-working surface;

[0114] Step S3, inputting the elastic modulus, Poisson's ratio, frictional heat flux and convection heat transfer coefficient of each surface under time-varying temperature into the face gear transmission finite element simulation model;

[0115] Step S4, solving the time-varying meshing stiffness of the gear transmission under time-varying temperature based on the finite element simulation model, and calculating the thermal meshing damping under time-varying temperature;

[0116] Step S5, respectively establishing a face gear meshing unit including thermal time-varying meshing stiffness, thermal meshing damping, static transmission error, meshing clearance, and friction under time-varying temperature, an axis beam unit including time-varying temperature and gyroscopic effect, a rigid rotor unit including gyroscopic effect, and a bearing support unit;

[0117] Step S6, solving the dynamic equations of each unit, and assembling each unit into a multi-node dynamic model of the face gear transmission system with time-varying temperature based on the finite element method, and deriving the system dynamic equations;

[0118] Step S7: Based on the differential equation of motion of the dynamic model, a corresponding eigenvalue equation is obtained, the eigenvalue equation and the differential equation of motion are solved, and the inherent characteristics and vibration characteristics of the system under the traditional normal temperature working condition and the time-varying temperature working condition are compared.

[0119] In step S1, based on the data of the tensile test and combined with the least squares method, the elastic modulus of the material under time-varying temperature is fitted with the following function formula:

[0120] E(T)=208.5-0.057T-1.48×10 -4 T 2 (1.59)

[0121] The fitting function formula for the Poisson's ratio of materials under time-varying temperature is:

[0122] υ(T)=0.289+4.0×10 -5 T (1.60)

[0123] In step S2, the calculation method of friction heat flow and convection heat transfer coefficient includes:

[0124] Friction heat flow of driving wheel:

[0125] q p =βγ1μP ca v s (1.61)

[0126] Friction heat flow of driven wheel:

[0127] q g =(1-β)γ1μP ca v s (1.62)

[0128]

[0129] The convective heat transfer phenomenon on the tooth end face is simplified to the convective heat transfer analysis of a rotating disk. The lubricating oil exhibits three forms as the disk rotates: laminar flow, transitional laminar flow, and turbulent flow. Each of these forms corresponds to a corresponding Reynolds value range.

[0130] When the value range R e ≤2×10 5 When , the lubricating oil flow on the tooth end face is laminar flow, and the convective heat transfer coefficient under laminar flow is calculated as:

[0131] h s=0.308λ l (m c +2) 0.5 (ρ l v l c l / λ l ) 0.5 (ω / v l ) 0.5 (1.64)

[0132] When the value range is 2×10 5 <R e <2.5×10 5 When , the lubricating oil flow on the tooth end face is transition laminar flow. Calculate the convective heat transfer coefficient under transition laminar flow state:

[0133]

[0134] When the value range R e >2.0×10 5 When , the lubricating oil flow on the tooth end face is turbulent. Calculate the convective heat transfer coefficient under turbulent state:

[0135]

[0136] Among them, λ l is the thermal conductivity of the fluid medium, m c is a general exponential constant, with a value of 2, ρ l is the density of the fluid medium, c l is the specific heat capacity of the fluid medium;

[0137] Calculation of meshing surface heat transfer coefficient:

[0138]

[0139] The maximum value of the tooth end face heat transfer coefficient is taken for the tooth top face heat transfer:

[0140] h d =max(h s )(1.68)

[0141] Non-working surface flow heat transfer coefficient:

[0142]

[0143] In step S3, the calculated elastic modulus, Poisson's ratio, frictional heat flux and convection heat transfer coefficient of each surface are input into Figure 3 Finite element simulation model of face gear transmission.

[0144] In step S4, the finite element simulation model outputs the rotation angle values ​​of the driving wheel and the driven wheel to solveFigure 4 Thermal time-varying mesh stiffness of gear transmission under time-varying temperature:

[0145]

[0146] Then calculate the thermal meshing damping of the gear transmission under time-varying temperature:

[0147]

[0148] In step S5, establish Figure 5 The face gear meshing unit considering thermal time-varying meshing stiffness, thermal meshing damping, static transmission error, meshing clearance and friction is designed. Figure 6 Axis-beam unit with time-varying temperature and gyroscopic effect, rigid rotor unit with gyroscopic effect, and bearing support unit.

[0149] In step S6, the differential equations of motion of the multi-node dynamic model of the face gear transmission system with time-varying temperature include:

[0150] The dynamic equation of the gear meshing unit under time-varying temperature is:

[0151]

[0152] Among them, M m , C m , K m and F m are the mass matrix, damping matrix, stiffness matrix and external force matrix of the beam element, respectively. m is the generalized coordinate vector of the face gear pair;

[0153] Dynamic equations of the beam element under time-varying temperature:

[0154]

[0155] in, and are the mass matrix, gyro matrix, damping matrix, and stiffness matrix of the beam element respectively. is the axis generalized coordinate vector;

[0156] Rotor unit dynamic equations:

[0157]

[0158] The bearing support unit of the vibration analysis model in this application is general, so there is no need to establish the corresponding bearing support unit dynamic equation, and conventional dynamic parameters can be directly substituted;

[0159] Motion equation of face gear transmission system:

[0160]

[0161] Among them, M is the system mass matrix, C is the system damping matrix, G is the system gyro matrix, K is the system stiffness matrix, q is the generalized coordinate vector of the face gear transmission, f(q) is the nonlinear function of the face gear transmission with meshing clearance, and Q is the external load vector.

[0162] In step S7, the differential equation of motion is converted into a state equation:

[0163]

[0164] State variable equation:

[0165]

[0166] State coefficient matrix:

[0167]

[0168]

[0169] Eigenvalue calculation of the state equation:

[0170]

[0171] Among them, ω i is the natural frequency, ξ i is the damping ratio.

[0172] Compared with the prior art, this application has the following beneficial effects:

[0173] 1. This application accurately characterizes the evolution of key parameters such as elastic modulus and Poisson's ratio with temperature under time-varying temperature conditions, and innovatively establishes a numerical calculation model for thermal time-varying mesh stiffness and thermal mesh damping;

[0174] 2. Based on the finite element nodal method, this application constructs a complete and highly accurate multi-node vibration analysis model for the system. This model innovatively integrates the gear meshing unit, shaft beam unit, gear rotor unit, and bearing support unit under time-varying temperature. By comparing the system's natural frequency and vibration response under normal temperature and time-varying temperature conditions, the influence of time-varying temperature on the system's dynamic characteristics is revealed.

[0175] 3. In an optional implementation scheme, the multi-node vibration analysis model of the system established in this application can further solve the vibration displacement and dynamic transmission error of key nodes, so as to comprehensively evaluate the dynamic performance and vibration noise level of the system under complex time-varying temperature conditions;

[0176] 4. Compared with the traditional system vibration analysis method at room temperature, the face gear transmission dynamic modeling method considering time-varying temperature introduced in this application is more innovative and closer to the actual working conditions of engineering applications, providing new ideas and methods for vibration reduction, vibration suppression and optimization of face gear transmission.

[0177] It should be noted that, in this document, the terms "comprises," "includes," or any other variations thereof are intended to encompass non-exclusive inclusion, such that a process, method, article, or apparatus comprising a series of elements includes not only those elements but also other elements not explicitly listed, or elements inherent to such process, method, article, or apparatus. In the absence of further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of other identical elements in the process, method, article, or apparatus comprising the element.

[0178] Furthermore, it should be noted that the scope of the methods and systems herein is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in reverse order depending on the functions involved. For example, the methods described may be performed in an order different from that described, and various steps may be added, omitted, or combined. Furthermore, features described with reference to certain examples may be combined in other examples.

[0179] The embodiments of the present application are described above in conjunction with the accompanying drawings, but the present application is not limited to the above-mentioned specific implementation methods. The above-mentioned specific implementation methods are merely illustrative and not restrictive. Under the guidance of this application, ordinary technicians in this field can also make many forms without departing from the purpose of this application and the scope of protection of the claims, all of which are within the protection of this application.

Claims

1. A face gear transmission dynamics modeling method considering time-varying temperature, characterized in that: The following steps are involved: Step S1, using the least squares method to perform function fitting on the elastic modulus and Poisson's ratio experimental data of the material at different temperatures; Step S2, calculating the friction heat flux and the convection heat transfer coefficient of the tooth end surface, meshing surface, tooth top surface and non-working surface; Step S3, inputting the elastic modulus, Poisson's ratio, frictional heat flux and convection heat transfer coefficient of each surface under time-varying temperature into the face gear transmission finite element simulation model; Step S4, solving the time-varying meshing stiffness of the gear transmission under time-varying temperature based on the finite element simulation model, and calculating the thermal meshing damping under time-varying temperature; Step S5, respectively establishing a face gear meshing unit including thermal time-varying meshing stiffness, thermal meshing damping, static transmission error, meshing clearance, and friction, a shaft beam unit including time-varying temperature and gyroscopic effect, a rigid rotor unit including gyroscopic effect, and a bearing support unit; Step S6, solving the dynamic equations of each unit, and assembling each unit into a multi-node dynamic model of a face gear transmission system with time-varying temperature based on the finite element method, and deriving its dynamic equations; Step S7: Based on the differential equation of motion of the dynamic model, a corresponding eigenvalue equation is obtained, the eigenvalue equation and the differential equation of motion are solved, and the inherent characteristics and vibration characteristics of the system under the traditional normal temperature working condition and the time-varying temperature working condition are compared.

2. The method according to claim 1, characterized in that In step S1, the elastic modulus and Poisson's ratio function fitting calculation method under time-varying temperature includes: Based on the material tensile test data, the elastic modulus function under time-varying temperature is fitted: E(T)=208.5-0.057T-1.48×10 -4 T 2 (1.1) Where T is the time-varying temperature; Based on the material tensile test data, the Poisson's ratio function is fitted at time-varying temperature: υ(T)=0.289+4.0×10 -5 T(1.2)。 3. The method according to claim 2, characterized in that In step S2, the calculation method of friction heat flow includes: The friction heat flow is mainly affected by three aspects: the relative sliding speed of the gear, the contact stress of the tooth surface and the friction coefficient of the tooth surface; The relative sliding speed of the gears is calculated by equations (1.3) and (1.4): Among them, d s is the position of the meshing point on the meshing line; v (p,g) is the absolute velocity of the contact point of the driving wheel and the driven wheel along the contact tangent direction, d (p,g) is the pitch circle diameter of the driving wheel and the driven wheel, a n is the gear pressure angle, n (p,g) is the speed of the driving wheel and the driven wheel; The tooth surface contact stress is calculated by formula (1.5): Among them, F s is the normal load on the tooth surface per unit tooth width, E0 is the comprehensive elastic modulus, b is the gear tooth width, ρ e is the equivalent curvature radius of the gear; The tooth surface friction coefficient is calculated by formula (1.6): Among them, μ(v s ) is the friction coefficient μ with the relative motion speed v s The function of change, r z The value is 200 to establish a more accurate friction model; Friction heat flow of driving wheel: q p =βγ1μ(v s )P ca v s (1.7) Friction heat flow of driven wheel: q g =(1-β)γ1μ(v s )P ca v s (1.8) Among them, β is the friction heat flow distribution factor, γ1 is the heat energy conversion coefficient, μ(v s ) is the friction coefficient, λ (p,g) is the thermal conductivity of the driving wheel and the driven wheel, ρ (p,g) is the material density of the driving wheel and the driven wheel, c (p,g) is the specific heat capacity of the driving wheel and the driven wheel.

4. The method according to claim 3, characterized in that In step S2, the method for calculating the convective heat transfer coefficient of the tooth end surface, meshing surface, tooth top surface, and non-working surface includes: The tooth end surface heat transfer coefficient is calculated by formula (1.10) and formula (1.11): Among them, λ l is the thermal conductivity of the fluid medium, m c is a general exponential constant, with a value of 2, ρ l is the density of the fluid medium, c l is the specific heat capacity of the fluid medium; ω is the angular velocity, r k is an arbitrary radius of the disk surface, v l is the kinematic viscosity of the fluid medium; Calculation of meshing surface heat transfer coefficient: The maximum value of the tooth end face heat transfer coefficient is taken for the tooth top face heat transfer: h d =max(h s )(1.13) Non-working surface flow heat transfer coefficient:

5. The method according to claim 4, characterized in that Step S4, the calculation expressions for thermal time-varying meshing stiffness and thermal meshing damping include: Calculation of thermal time-varying mesh stiffness: Thermal mesh damping calculation: Among them, F n is the normal contact force, r (p,g) is the base circle radius of the gear, θ (p,g) are the gear rotation angles, ζ is the damping ratio, which is 0.025, m (p,g) Indicates the mass of the gear.

6. The method according to claim 5, characterized in that Step S5, the calculation method of the beam unit including time-varying temperature and gyroscopic effect includes: Calculation of shear modulus of gear material: Calculation of shear correction factor for gear materials: Where ε is the ratio of the inner and outer radii of the gear shaft; Calculation of total potential energy of beam element: Where Λs is the polar moment of inertia of the section, Λ is the diametrical moment of inertia of the section, x' and y' are the curvature components of the beam along the x and y axes, θx and θy are the axial section rotation angles, and θ′ z is the rate of change of the torsion angle.

7. The method according to claim 6, characterized in that Step S6, the kinetic equations of each unit include: Dynamic equations of the beam element under time-varying temperature: in, and are the mass matrix, gyro matrix, damping matrix, and stiffness matrix of the beam element respectively. is the axis generalized coordinate vector; Rotor unit dynamic equations: Dynamic equation of face gear meshing unit: Among them, M m , C m , K m and F m are the mass matrix, damping matrix, stiffness matrix and external force matrix of the beam element, respectively. m is the generalized coordinate vector of the face gear pair; The bearing support unit of the vibration analysis model in this application is general, so there is no need to establish the corresponding bearing support unit dynamic equation, and conventional dynamic parameters can be directly substituted; Optionally, in step S6, the differential equation of motion of the face gear transmission system is: Among them, M is the system mass matrix, C is the system damping matrix, G is the system gyro matrix, K is the system stiffness matrix, q is the generalized coordinate vector of the face gear transmission, f(q) is the nonlinear function of the face gear transmission with meshing clearance, and Q is the external load vector.

8. The method according to claim 7, characterized in that Step S7, convert the differential equation of motion into the state equation: State variable equation: State coefficient matrix: Eigenvalue calculation of the state equation: Among them, ω i is the natural frequency, ξ i is the damping ratio.

9. The method according to claim 8, characterized in that The Lagrangian method is used to solve the system's differential equations of motion, and the inherent characteristics and vibration characteristics of the system under normal temperature and time-varying temperature conditions are obtained and analyzed, where the vibration characteristics include vibration displacement and dynamic transmission error.

10. A computer system comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When executing the computer program, the processor performs the steps of the method according to any one of claims 1 to 9.

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