CN106845048A - A kind of inside engaged gear axle decelerator Nonlinear dynamic models method for counting friction and backlash - Google Patents

Abstract

The invention discloses a nonlinear dynamic modeling method of an internal meshing gear shaft reducer taking into account friction and tooth side clearance, which belongs to the technical field of mechanical system reliability engineering; the specific steps are as follows: first, a certain internal meshing gear shaft is set The conditions for the modeling of the reducer; then, determine the relative linear displacement of the gear pair along the meshing line, the static transmission error and the nonlinear backlash in turn; further, use the Fourier series to approximate the comprehensive time-varying meshing stiffness of the gear pair, using The Lagrange method establishes the nonlinear dynamic equation of the gear pair, and finally, performs linear transformation and dimensionless processing on the nonlinear dynamic equation. The model of the invention has good universality, and can carry out the numerical simulation of the running state and dynamic characteristics of the internal meshing gear reducer with any parameters, and carry out the output simulation of the vibration signal of the reducer system.

Description

A Nonlinear Dynamics of Internal Gear Shaft Reducer Considering Friction and Backlash learning modeling method

technical field

The invention belongs to the technical field of mechanical system reliability engineering, and relates to a nonlinear dynamic modeling method of an internal meshing gear shaft reducer taking into account friction and tooth side clearance.

Background technique

The internal gear shaft reducer has the advantages of compact structure, stable performance and high efficiency. It is widely used in civil, industrial, aviation and aerospace fields, so the theoretical research on its working mechanism is of great significance. When the gear pair of the reducer is working, even in normal state, due to the influence of tooth surface friction, backlash and time-varying meshing stiffness, the gear vibration shows strong nonlinear characteristics.

In order to achieve the goals of high precision, low noise, low vibration, and improve the controllability of gear transmission, researchers have invested a lot of time and energy on the nonlinear dynamics of gear transmission systems.

The research on the nonlinear dynamics of the internal meshing gear shaft reducer is of great value to the theory and application research of the gear reducer.

Contents of the invention

In order to reveal a theoretical tool for the vibration mechanism of a certain type of internal meshing gear shaft reducer, the present invention establishes a tool that includes tooth surface friction, nonlinear gear, etc. The dynamic model of internal gear shaft reducer with side clearance function and time-varying mesh stiffness, specifically a nonlinear dynamic modeling method of internal gear shaft reducer considering friction and backlash.

Specific steps are as follows:

Step 1. For a certain internal gear shaft reducer, set the conditions for modeling the reducer;

(1) The internal gear and external gear of the reducer are both involute spur gears;

(2) The two gear tooth blanks are regarded as rigid bodies, and the input and output shafts of the reducer are regarded as rigid bodies; the supporting stiffness of the two gear shafts is large enough, and the elastic deformation of the support is not considered;

(3) All components in the reducer are not subjected to axial force, and the vibration vector exists in a plane perpendicular to the axis;

(4) The teeth of the driving wheel (external gear) and the driven wheel (internal gear) are considered as cantilever beams, and there is a relative sliding displacement of the teeth along the meshing line;

(5) The gears in the gear train are installed according to the standard center distance, and the pitch circle of the gear coincides with the index circle;

(6) Parts processing errors and installation errors are not included.

Step 2. For the internal gear shaft reducer, determine the relative linear displacement on the meshing line of the gear pair by using the angular displacement of the gear;

Among them: x is the relative linear displacement of the meshing pair of the internal gear along the meshing line; y is the relative linear displacement of the meshing pair of the external gear along the meshing line; r ₁ is the radius of the pitch circle of the external gear; r ₂ is the pitch of the internal gear degree circle radius; θ ₁ is the torsional angular displacement of the external gear; θ ₂ is the torsional angular displacement of the internal gear; t is the time variable; e(t) is the static transmission error of the gear pair, e(t)=e _a sinω _m t, e _a is the error amplitude, ω _m is the meshing tooth frequency.

Step 3, determining the nonlinear backlash function according to the relative line displacement along the meshing line on the meshing pair of internal gears;

In the formula, b is the tooth backlash constant.

Step 4, using Fourier series to approximate the time-varying mesh stiffness function k _m (t) of the gear pair in a healthy state;

Among them, k _av is the average meshing stiffness of the gear pair, k _a is the amplitude of stiffness change; n takes an integer value.

Step five, using the backlash function of the reducer, combined with the time-varying mesh stiffness function of the gear pair, using the Lagrange method to establish the nonlinear dynamic equation of the gear pair;

The nonlinear dynamic equation of the gear pair includes the input matrix T ₁ and the output matrix T ₂ of the internal meshing gear pair;

Where: I ₁ is the moment of inertia of the external gear; I ₂ is the moment of inertia of the internal gear; is the instantaneous angular acceleration of the external gear; is the instantaneous angular acceleration of the internal gear; c _m is the meshing viscous damping coefficient; l ₁ is the force arm of the meshing friction damping force of the external gear; l ₂ is the moment arm of the meshing frictional damping force of the internal gear; λ is the friction direction coefficient, _ω2 is the nominal angular velocity of the internal gear; is the instantaneous angular velocity of the external gear; is the instantaneous angular velocity of the internal gear; μ is the dynamic friction coefficient of the tooth surface;

Step 6. Perform linear transformation and dimensionless processing on the nonlinear dynamic equation of the gear pair to obtain the dimensionless nonlinear dynamic model of the internal meshing gear pair that finally includes friction and backlash;

τ＝ω _e t; ω _e is the equivalent natural frequency of the gear pair, m _e is the equivalent mass of the gear pair, m ₁ is the mass of the external gear, m ₂ is the mass of the internal gear;

e _a is the amplitude of the comprehensive error change of the gear teeth, which is a constant;

The advantages of the present invention are:

(1) A nonlinear dynamics modeling method of internal meshing gear shaft reducer considering friction and tooth backlash. Based on the Lagrange method, a nonlinear dynamics model of internal meshing gear pair is established. This model has good versatility , can do the dynamic numerical simulation of the internal meshing gear pair under any working condition.

(2) A nonlinear dynamic modeling method for internal meshing gear shaft reducers that takes friction and backlash into account. The internal parameters such as frictional damping effect, nonlinear backlash function, and time-varying meshing stiffness are taken into account in the model Excitation, high accuracy of numerical simulation, high degree of simulation of the vibration signal of the gear reducer system output by the model, and obvious nonlinear dynamic characteristics.

Description of drawings

Fig. 1 is the structural representation of the internal meshing gear shaft speed reducer that the present invention adopts;

Fig. 2 is a schematic diagram of the principle of the nonlinear dynamic model of the internal gear shaft reducer of the present invention;

Fig. 3 is a flow chart of the nonlinear dynamics modeling method of the internal meshing gear shaft reducer of the present invention, which takes friction and backlash into account;

Fig. 4 is the phase trace diagram of the steady state vibration of the internal meshing gear shaft reducer of the present invention;

Fig. 5 is a time domain diagram of the steady state vibration of the internal meshing gear shaft reducer of the present invention;

Fig. 6 is a frequency domain diagram of the steady state vibration of the internal meshing gear shaft reducer of the present invention.

detailed description

The present invention will be further described in detail with reference to the accompanying drawings and embodiments.

The present invention is aimed at a certain type of internal meshing gear shaft reducer, the structure of which is shown in Figure 1. Based on the Lagrange method, a design including tooth surface friction, nonlinear backlash function, time-varying meshing stiffness and meshing viscous damping and other factors is designed. The nonlinear dynamics model of the internal meshing gear shaft reducer, specifically a nonlinear dynamics modeling method of the internal meshing gear shaft reducer taking into account friction and backlash.

The schematic diagram is shown in Figure 2, where N ₁ N ₂ represents the meshing line of the internal meshing gear pair; e(t) is the static transmission error of the gear pair; k _m is the meshing stiffness of the gear pair; c _m is the meshing viscosity damping; α is the meshing angle of the gear pair, which is also the gear pressure angle; r ₁ and r ₂ are the pitch circle radii of the external gear (driving) and internal gear (driven) respectively; ω ₁ and ω ₂ are the external gear and internal gear respectively. The nominal angular velocity of the gear; l ₁ and l ₂ are the force arms of the external gear and the internal gear respectively; y is the meshing point line displacement of the external gear; T ₁ and T ₂ are the driving torque and load torque.

As shown in Figure 3, it includes the following steps:

Step 1. For a certain internal gear shaft reducer, set the conditions for modeling the reducer;

(1) The internal gear and external gear of the reducer are both involute spur gears;

(2) The two gear tooth blanks are regarded as rigid bodies, and the input and output shafts of the reducer are regarded as rigid bodies; the supporting stiffness of the two gear shafts is large enough, and the elastic deformation of the support is not considered;

(3) All components in the reducer are not subjected to axial force, and the vibration vector exists in a plane perpendicular to the axis;

(4) The teeth of the driving wheel (external gear) and the driven wheel (internal gear) are considered as cantilever beams, and there is a relative sliding displacement of the teeth along the meshing line;

(5) The gears in the gear train are installed according to the standard center distance, and the pitch circle of the gear coincides with the index circle;

(6) Parts processing errors and installation errors are not included.

Step 2. For the internal gear shaft reducer, determine the relative linear displacement on the meshing line of the gear pair by using the angular displacement of the gear;

Among them: x is the relative linear displacement of the meshing pair of the internal gear along the meshing line; y is the relative linear displacement of the meshing pair of the external gear along the meshing line; r ₁ is the radius of the pitch circle of the external gear; r ₂ is the pitch of the internal gear degree circle radius; θ ₁ is the torsional angular displacement of the external gear; θ ₂ is the torsional angular displacement of the internal gear; t is the time variable; e(t) is the static transmission error of the gear pair, e(t)=e _a sinω _m t, e _a is the error amplitude, ω _m is the meshing tooth frequency.

Step 3, determining the nonlinear backlash function according to the relative line displacement along the meshing line on the meshing pair of internal gears;

In the formula, b is the tooth backlash constant.

Step 4, using Fourier series to approximate the time-varying mesh stiffness function k _m (t) of the gear pair in a healthy state;

Among them, k _av is the average meshing stiffness of the gear pair, k _a is the amplitude of stiffness change; n takes an integer value.

Step five, using the backlash function of the reducer, combined with the time-varying mesh stiffness function of the gear pair, using the Lagrange method to establish the nonlinear dynamic equation of the gear pair;

The nonlinear dynamic equation of the gear pair includes the input matrix T ₁ and the output matrix T ₂ of the internal meshing gear pair;

Where: I ₁ is the moment of inertia of the external gear; I ₂ is the moment of inertia of the internal gear; is the instantaneous angular acceleration of the external gear; is the instantaneous angular acceleration of the internal gear; c _m is the meshing viscous damping coefficient; km ( _t ) is the time-varying meshing stiffness; f(x) is the backlash function; l ₁ is the meshing frictional damping force of the external gear arm; l ₂ is the force arm of the meshing friction damping force of the internal gear; λ is the coefficient of the friction force direction, _ω2 is the nominal angular velocity of the internal gear; is the instantaneous angular velocity of the external gear; is the instantaneous angular velocity of the internal gear; μ is the dynamic friction coefficient of the tooth surface;

Step 6. Perform linear transformation and dimensionless processing on the nonlinear dynamic equation of the gear pair to obtain the dimensionless nonlinear dynamic model of the internal meshing gear pair that finally includes friction and backlash;

The substitution relations used are:

τ＝ω _e t; ω _e is the equivalent natural frequency of the gear pair, m _e is the equivalent mass of the gear pair, m ₁ is the mass of the external gear, m ₂ is the mass of the internal gear;

e _a is the amplitude of the comprehensive error change of the gear teeth, which is a constant;

Example

Modeling and simulation experiments were carried out on a certain type of internal meshing gear shaft reducer system, the simulation results of pure torsional vibration were obtained, and the phase trace diagram, time domain diagram and frequency domain diagram of the transmission system were obtained; the parameters of the simulation example are listed in the following table Show.

Parameter value table of simulation example of internal gear reducer

The phase trace diagram of the vibration steady-state response of the gear shaft reducer system obtained by numerical simulation is shown in Figure 4, the time domain diagram is shown in Figure 5, and the frequency domain diagram is shown in Figure 6. It can be seen from the figure that when the normal gear vibrates, the time-domain curve has a stable meshing period of 20, and the frequency-domain graph only has the meshing frequency 0.048 and its high-order multiplier, there is no sideband, and the phase trace moves periodically.

Claims (2)

1. A method for modeling nonlinear dynamics of an internal gear shaft reducer that takes into account friction and backlash, is characterized in that the specific steps are as follows: Step 1. For a certain internal gear shaft reducer, set the conditions for modeling the reducer; Step 2. For the internal gear shaft reducer, determine the relative linear displacement on the meshing line of the gear pair by using the angular displacement of the gear; $\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; e</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.$ Among them: x is the relative linear displacement of the meshing pair of the internal gear along the meshing line; y is the relative linear displacement of the meshing pair of the external gear along the meshing line; r ₁ is the radius of the pitch circle of the external gear; r ₂ is the pitch of the internal gear degree circle radius; θ ₁ is the torsional angular displacement of the external gear; θ ₂ is the torsional angular displacement of the internal gear; t is the time variable; e(t) is the static transmission error of the gear pair, e(t)=e _a sinω _m t, e _a is the error amplitude, ω _m is the meshing tooth frequency; Step 3, determining the nonlinear backlash function according to the relative line displacement along the meshing line on the meshing pair of internal gears; $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; b</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& <span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; b</span>}\\ \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; <</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}\end{array}\right.$ In the formula, b is the gear backlash constant; Step 4, using Fourier series to approximate the time-varying mesh stiffness function k _m (t) of the gear pair in a healthy state; ${\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}{\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; \&infin;</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; n\&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span>$ Among them, k _av is the average meshing stiffness of the gear pair, k _a is the amplitude of stiffness change; n takes an integer value; Step five, using the backlash function of the reducer, combined with the time-varying mesh stiffness function of the gear pair, using the Lagrange method to establish the nonlinear dynamic equation of the gear pair; The nonlinear dynamic equation of the gear pair includes the input matrix T ₁ and the output matrix T ₂ of the internal meshing gear pair; $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; \&lambda;</span>}\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&lambda;</span>}\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\end{array}\right.$ Where: I ₁ is the moment of inertia of the external gear; I ₂ is the moment of inertia of the internal gear; is the instantaneous angular acceleration of the external gear; is the instantaneous angular acceleration of the internal gear; c _m is the meshing viscous damping coefficient; l ₁ is the force arm of the meshing friction damping force of the external gear; l ₂ is the moment arm of the meshing frictional damping force of the internal gear; λ is the friction direction coefficient, _ω2 is the nominal angular velocity of the internal gear; is the instantaneous angular velocity of the external gear; is the instantaneous angular velocity of the internal gear; μ is the dynamic friction coefficient of the tooth surface; Step 6. Perform linear transformation and dimensionless processing on the nonlinear dynamic equation of the gear pair to obtain the dimensionless nonlinear dynamic model of the internal meshing gear pair that finally includes friction and backlash; $\left\{\begin{array}{c}\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; \&rsqb;</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; v</span>}}\\ <span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}\mathrm{<span\; class="notranslate">\; \&tau;</span>}\\ \stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; \&rsqb;</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; \&rsqb;</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; v</span>}}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\end{array}\right.$ τ＝ω _e t; ω _e is the equivalent natural frequency of the gear pair, m _e is the equivalent mass of the gear pair, m ₁ is the mass of the external gear, m ₂ is the mass of the internal gear; ${\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}\mathrm{<span\; class="notranslate">\; \&lambda;</span>}<span\; class="notranslate">\; ;</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; ;</span>$ ${\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; v</span>}}}<span\; class="notranslate">\; ;</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; ;</span>$ $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& <span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; <</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.<span\; class="notranslate">\; ,</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ;</span>$ e _a is the amplitude of the comprehensive error change of the gear teeth, which is a constant; $\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\stackrel{<span\; class="notranslate">\; \&CenterDot;\&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; \&zeta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; .</span>$ 2. A nonlinear dynamics modeling method of an internal meshing gear shaft reducer that takes friction and backlash into account as claimed in claim 1, wherein said step 1 is specifically: (1) The internal gear and external gear of the reducer are both involute spur gears; (2) The two gear tooth blanks are regarded as rigid bodies, and the input and output shafts of the reducer are regarded as rigid bodies; the supporting stiffness of the two gear shafts is large enough, and the elastic deformation of the support is not considered; (3) All components in the reducer are not subjected to axial force, and the vibration vector exists in a plane perpendicular to the axis; (4) The teeth of the driving wheel (external gear) and the driven wheel (internal gear) are considered as cantilever beams, and there is a relative sliding displacement of the teeth along the meshing line; (5) The gears in the gear train are installed according to the standard center distance, and the pitch circle of the gear coincides with the index circle; (6) Parts processing errors and installation errors are not included.