Helical gear nonlinear dynamic modeling method based on three-dimensional anisotropy

Abstract

The invention discloses a helical gear nonlinear dynamic modeling method based on three-dimensional anisotropy, and belongs to the technical field of dynamic simulation of gear transmission systems. Comprising the following steps: establishing a three-dimensional anisotropic tooth surface microstructure model based on a W-M fractal theory; correcting a gear backlash nonlinear displacement function and a bearing clearance nonlinear displacement function; establishing a nonlinear dynamic model of the six-degree-of-freedom helical gear transmission system containing a gear backlash, a bearing clearance and friction; performing linear change and dimensionless on the nonlinear kinetic equation of the gear pair to obtain a dimensionless nonlinear kinetic model considering the backlash and the bearing clearance based on the three-dimensional anisotropy; replacing the non-linear displacement function of the gear backlash and the non-linear displacement function of the bearing clearance in the step 2 into a dimensionless non-linear dynamic model; and calculation is carried out in MATLAB simulation software. According to the method, the gear backlash is used as a variable for modeling, and the modeling precision is improved.

Description

Technical Field

[0001] The invention relates to the technical field of dynamics simulation of gear transmission systems, and in particular to a nonlinear dynamics modeling method for helical gears based on three-dimensional anisotropy. Background Art

[0002] Gear transmission has the advantages of high transmission efficiency, accurate transmission ratio, compact structure and long life, and is widely used in mechanical equipment and power transmission devices. As an important component of mechanical transmission, the dynamic characteristics of gears directly affect the performance and reliability of mechanical equipment.

[0003] The influence of friction is often ignored in traditional dynamic models, resulting in inaccurate transmission analysis. The microscopic morphology of the tooth surface affects the nonlinear dynamic characteristics of the gear system. The Chinese patent CN117574580A "A method for dynamic modeling of small-module gears with shaft-hole clearance friction" focuses on the clearance between the shaft and the hole and its response for modeling. The clearance model is only a traditional three-stage model, and the tooth side clearance is a constant, which cannot be modeled dynamically and finely.

[0004] Therefore, a nonlinear dynamic modeling method for helical gears based on three-dimensional anisotropic rough tooth surface is needed, in which the tooth side clearance is modeled as a variable to make the model more refined. Summary of the invention

[0005] The purpose of the present invention is to propose a nonlinear dynamic modeling method for helical gears based on three-dimensional anisotropy, and the specific steps are as follows:

[0006] Step 1: Establish a three-dimensional anisotropic tooth surface micro-morphology model based on WM fractal theory;

[0007] Step 2, correcting the nonlinear displacement function of the tooth side clearance and the nonlinear displacement function of the bearing clearance;

[0008] The nonlinear displacement function of the tooth side clearance is:

[0009]

[0010] Where b m To consider the time-varying backlash function of the anisotropic microstructure, the expression is:

[0011]

[0012] Where b0 is the initial tooth side clearance and b0 is considered as a constant; D x1 , G x1 is the fractal factor of the large gear, D x2 , G x2is the fractal factor of the pinion, Z1(t) is the time-varying backlash of the driving wheel, and Z2(t) is the time-varying backlash of the driven wheel; n is the frequency index; γ is a constant that satisfies the probability distribution; γ n is the discrete spectrum; L is the sampling length;

[0013] The nonlinear displacement function of the bearing clearance is:

[0014]

[0015] Where b i To consider the time-varying bearing clearance function of anisotropic micromorphology, the expression is:

[0016]

[0017] Where b0 is the initial bearing clearance and b0 is considered as a constant; D x1 , G x1 is the fractal factor of the bearing in the x direction, D y1 , G y1 is the fractal factor of the bearing in the y direction; n is the frequency index; γ is a constant that satisfies the probability distribution; γ n is the discrete spectrum; L is the sampling length;

[0018] Step 3, establishing a nonlinear dynamic model of the six-degree-of-freedom helical gear transmission system including tooth side clearance, bearing clearance and friction;

[0019] Step 4, linearly transform and dimensionlessly transform the nonlinear dynamic equation of the gear pair to obtain a dimensionless nonlinear dynamic model that considers the tooth side clearance and bearing clearance based on three-dimensional anisotropy;

[0020] Step 5, replacing the nonlinear displacement function of the tooth side clearance and the nonlinear displacement function of the bearing clearance in step 2 into the dimensionless nonlinear dynamic model;

[0021] Step 6: Perform calculations in MATLAB simulation software.

[0022] The three-dimensional anisotropic tooth surface micromorphology model in step 1 is:

[0023]

[0024] Where Z(x,y) is the surface profile height; D x and G x is the fractal dimension and characteristic scale coefficient in the x direction; D y and G y is the fractal dimension and characteristic scale coefficient in the y direction; n is the frequency index; γ is a constant that satisfies the probability distribution; γ n is the discrete spectrum; L is the sampling length.

[0025] The nonlinear dynamic model of the six-degree-of-freedom helical gear transmission system in step 3 is:

[0026]

[0027] Where, T i , i=1,2 is the constant external torque applied to the gear; R1, R2 is the base circle radius of the driving wheel and the driven wheel; m i ,I i , i=1,2 is the mass and moment of inertia of the gear; F i is the radial load; k xi , c xi are the equivalent bearing stiffness and damping of the gear in the X direction; k yi , c yi are the equivalent bearing stiffness and damping of the gear in the Y direction; k m , c m are the meshing stiffness and meshing damping of the gear pair respectively; e(t) is the static transmission error; β is the helical angle of the helical gear; F mi is the dynamic meshing force of gears, F mi =c mi δ(t)+k mi f(δ(t)), i=1,2; F fi is the friction force of the helical gear, F fi =μ(t)F mi , μ(t) is the time-varying friction coefficient.

[0028] The dimensionless nonlinear dynamic model in step 4 is:

[0029]

[0030] In the formula,

[0031] ω=ω e / ω n , b c is the nominal displacement scale, ω n is the time scale, m e is the equivalent mass, is the dimensionless displacement, is the dimensionless meshing damping, is the dimensionless bearing damping, is the dimensionless meshing stiffness, is the dimensionless bearing stiffness.

[0032] The beneficial effects of the present invention are:

[0033] The method of the present invention uses the tooth side clearance as a variable for modeling, thereby improving the modeling accuracy, effectively characterizing the three-dimensional rough surface of the high-precision gear, and being conducive to accurately analyzing the dynamic characteristics of the helical gear. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 It is a flow chart of the nonlinear dynamic modeling method of helical gears based on three-dimensional anisotropy of the present invention;

[0035] Figure 2 Yes D x >D y , G x =G y Anisotropic three-dimensional rough surface map when ;

[0036] Figure 3 Yes D x <D y , G x =G y Anisotropic three-dimensional rough surface map when ;

[0037] Figure 4 Yes D x =D y , G x =G y Anisotropic three-dimensional rough surface map when ;

[0038] Figure 5 is the fractal clearance diagram of the gear pair;

[0039] Figure 6 It is the dynamic model diagram of the helical gear meshing pair;

[0040] Figure 7 Yes D x =1.1,D y =Time chart at 1.5 hours;

[0041] Figure 8 Yes D x =1.5,D y =Time chart at 1.5 hours;

[0042] Fig. 9 Yes D x =1.5,D y =Time chart when 1.55. DETAILED DESCRIPTION

[0043] The present invention proposes a nonlinear dynamic modeling method for helical gears based on three-dimensional anisotropy, which will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0044] Figure 1 The flowchart of the nonlinear dynamic modeling method of helical gears based on three-dimensional anisotropy of the present invention comprises the following steps:

[0045] Step 1: Construct a two-dimensional contour curve based on WM fractal theory:

[0046]

[0047] Where Z(x) is the height of the surface profile; D is the fractal dimension; G is the characteristic scale coefficient; n is the frequency index; γ is a constant that satisfies the probability distribution (usually γ = 1.5); γ n is the discrete spectrum; n l is the minimum cutoff frequency of the surface profile; L is the sampling length.

[0048] In order to get close to the actual tooth surface morphology, the two-dimensional contour curve is superimposed to obtain a rough surface model with three-dimensional anisotropic tooth surface micromorphology:

[0049]

[0050] Where Z(x,y) is the surface profile height; D x and G x is the fractal dimension and characteristic scale coefficient in the x direction; D y and G y is the fractal dimension and characteristic scale coefficient in the y direction; n is the frequency index; γ is a constant that satisfies the probability distribution (generally γ=1.5); γ n is the discrete spectrum; L is the sampling length.

[0051] Step 2: The nonlinear displacement function based on the modified tooth side clearance is:

[0052]

[0053] Where b m To consider the time-varying backlash function of the anisotropic microstructure, the expression is:

[0054]

[0055] Where b0 is the initial tooth side clearance and b0 is considered as a constant; D x1 , G x1 is the fractal factor of the large gear, D x2 , G x2 is the fractal factor of the pinion, Z1(t) is the time-varying backlash of the driving wheel, and Z2(t) is the time-varying backlash of the driven wheel; n is the frequency index; γ is a constant that satisfies the probability distribution (generally γ=1.5); γ n is the discrete spectrum; L is the sampling length;

[0056] The nonlinear displacement function based on the modified bearing clearance is:

[0057]

[0058] Where b i To consider the time-varying bearing clearance function of anisotropic micromorphology, the expression is:

[0059]

[0060] Where b0 is the initial bearing clearance and b0 is considered as a constant; D x1 , G x1 is the fractal factor of the bearing in the x direction, D y1 , G y1 is the fractal factor of the bearing in the y direction; n is the frequency index; γ is a constant that satisfies the probability distribution (generally γ=1.5); γ n is the discrete spectrum; L is the sampling length.

[0061] Step 3: Due to the existence of tooth surface roughness, friction cannot be ignored. The nonlinear dynamic model of the six-degree-of-freedom helical gear transmission system including tooth side clearance, bearing clearance and friction is established using the concentrated mass method:

[0062]

[0063] Where Ti (i = 1, 2) is the constant external torque applied to the gear; R1, R2 are the base circle radii of the driving wheel and the driven wheel; mi, Ii (i = 1, 2) are the mass and moment of inertia of the gear; F i is the radial load; k xi , c xi are the equivalent bearing stiffness and damping of the gear in the X direction; k yi , c yi are the equivalent bearing stiffness and damping of the gear in the Y direction; k m , c m are the meshing stiffness and meshing damping of the gear pair respectively; e(t) is the static transmission error; β is the helical angle of the helical gear;

[0064] Dynamic meshing force of gears F mi for:

[0065] F mi =c mi δ(t)+k mi f(δ(t))(i=1,2)

[0066] In the formula, δ=(R1θ1-R2θ2+(x1-x2)sinψ+(y1-y2)cosψ-e(t))cosβ

[0067] Helical gear friction F fi for:

[0068] F fi =μ(t)F mi

[0069] μ(t) is the time-varying friction coefficient. The friction coefficient based on non-Newtonian, thermal-elastic-hydrodynamic lubrication is:

[0070]

[0071] In the formula, b i is a constant factor that depends on the type of lubrication, b i (i=1,2,...,9)=-8.92,1.03,1.04,-0.35,2.81,-0.10,0.75,-0.39,0.62,S is the root mean square value of the comprehensive roughness of the tooth surface; η is the absolute viscosity of the lubricating oil; R ij (t) is the comprehensive curvature radius of the jth contact line of the i-th meshing pair; P hij (t) is the maximum Hertzian contact pressure of the jth segment; SR ij (t) is the slide-roll ratio of the jth contact line.

[0072] Step 4: In order to improve the calculation accuracy and eliminate the unit limitation, the parameters of the dynamic equation are dimensionless. Introduce the displacement nominal scale b c and time scale ω n ,

[0073] The dimensionless nonlinear dynamic model considering the tooth side clearance and bearing clearance based on three-dimensional anisotropy is:

[0074]

[0075] In the formula,

[0076] ω=ω e / ω n , m e is the equivalent mass, is the dimensionless displacement, is the dimensionless meshing damping, is the dimensionless bearing damping, is the dimensionless meshing stiffness, is the dimensionless bearing stiffness.

[0077] Step 5, replacing the nonlinear displacement function of the tooth side clearance and the nonlinear displacement function of the bearing clearance based on the three-dimensional anisotropic micromorphology in step 2 into the overall dimensionless dynamic equation of the system;

[0078] Step 6: Calculate in MATLAB simulation software.

[0079] In order to verify the accuracy and effectiveness of the nonlinear dynamic modeling method of helical gears based on three-dimensional anisotropy of the present invention, the following simulation experiment was carried out.

[0080] Most of the previous dynamic models are based on two-dimensional fractal tooth surface, ignoring the fractal characteristics in the tooth width direction. Based on this, a three-dimensional anisotropic tooth surface micromorphology is established, such as Figure 2 , Figure 3 , Figure 4 shown. Figure 2 D x >D y , G x =G y From the anisotropic three-dimensional rough surface diagram, it can be seen that the fineness in the x direction is higher than that in the y direction. Figure 3 D x <D y , G x =G y From the anisotropic three-dimensional rough surface diagram, it can be seen that the fineness in the x direction is lower than that in the y direction. Figure 4 D x =D y , G x =G y From the anisotropic three-dimensional rough surface diagram, it can be seen that the fineness in the x direction is equal to that in the y direction, and the surface is isotropic. This shows that considering the meshing surface as a simple two-dimensional fractal function cannot accurately describe the gear surface morphology.

[0081] like Figure 5 The figure shows the microscopic rough surface of the meshing gear. Z1 and Z2 are the heights of the micro-convex bodies on the contact surface, and b0 is the initial tooth side clearance. Due to the uneven tooth surface and complex structure, the tooth side clearance is regarded as a constant, and the accuracy of the model is not high. In actual application scenarios, tooth side clearance and bearing clearance exist at the same time. Traditional modeling regards them as constants, which will also affect the modeling accuracy. Figure 6 This is the dynamic model diagram of the helical gear meshing pair. The nonlinear displacement function of the tooth side clearance and the nonlinear displacement function of the bearing clearance based on the three-dimensional anisotropic micromorphology are replaced into the overall dimensionless dynamic equation of the system for simulation. The simulation results are shown in Figure 7 , Figure 8 and Fig. 9 shown.

[0082] like Figure 7As shown, when D x =1.1, D y =1.5, the dynamic transmission error amplitude in the figure is 71.0069; Figure 8 As shown, when D x =1.5, D y =1.5, the dynamic transmission error amplitude in the figure is 4.5785; Fig. 9 As shown, when D x =1.5, D y =1.55, the dynamic transmission error amplitude is 3.5604 in the figure. It can be seen that with the increase of fractal dimension D, the dynamic error amplitude decreases, and the system presents a better stable state, thus improving the modeling accuracy.

Claims

1. A nonlinear dynamic modeling method for helical gears based on three-dimensional anisotropy, characterized in that: The specific steps are as follows: Step 1: Establish a three-dimensional anisotropic tooth surface micro-morphology model based on WM fractal theory; Step 2, correcting the nonlinear displacement function of the tooth side clearance and the nonlinear displacement function of the bearing clearance; The nonlinear displacement function of the tooth side clearance is: Where b m To consider the time-varying backlash function of the anisotropic microstructure, the expression is: Where b0 is the initial tooth side clearance and b0 is considered as a constant; D x1 , G x1 is the fractal factor of the large gear, D x2 , G x2 is the fractal factor of the pinion, Z1(t) is the time-varying backlash of the driving wheel, and Z2(t) is the time-varying backlash of the driven wheel; n is the frequency index; γ is a constant that satisfies the probability distribution; γ n is the discrete spectrum; L is the sampling length; The nonlinear displacement function of the bearing clearance is: Where b i To consider the time-varying bearing clearance function of anisotropic micromorphology, the expression is: Where b0 is the initial bearing clearance and b0 is considered as a constant; D x1 , G x1 is the fractal factor of the bearing in the x direction, D y1 , G y1 is the fractal factor of the bearing in the y direction; n is the frequency index; γ is a constant that satisfies the probability distribution; γ n is the discrete spectrum; L is the sampling length; Step 3, establishing a nonlinear dynamic model of the six-degree-of-freedom helical gear transmission system including tooth side clearance, bearing clearance and friction; Step 4, linearly transform and dimensionlessly transform the nonlinear dynamic equation of the gear pair to obtain a dimensionless nonlinear dynamic model that considers the tooth side clearance and bearing clearance based on three-dimensional anisotropy; Step 5, replacing the nonlinear displacement function of the tooth side clearance and the nonlinear displacement function of the bearing clearance in step 2 into the dimensionless nonlinear dynamic model; Step 6: Perform calculations in MATLAB simulation software.

2. The nonlinear dynamic modeling method of helical gears based on three-dimensional anisotropy according to claim 1 is characterized in that: The three-dimensional anisotropic tooth surface micro-morphology model in step 1 is: Where Z(x,y) is the surface profile height; D x and G x is the fractal dimension and characteristic scale coefficient in the x direction; D y and G y is the fractal dimension and characteristic scale coefficient in the y direction; n is the frequency index; γ is a constant that satisfies the probability distribution; γ n is the discrete spectrum; L is the sampling length.

3. The nonlinear dynamic modeling method of helical gears based on three-dimensional anisotropy according to claim 1 is characterized in that: The nonlinear dynamic model of the six-degree-of-freedom helical gear transmission system in step 3 is: Where, T i , i=1,2 is the constant external torque applied to the gear; R1, R2 is the base circle radius of the driving wheel and the driven wheel; m i ,I i , i=1,2 is the mass and moment of inertia of the gear; F i is the radial load; k xi , c xi are the equivalent bearing stiffness and damping of the gear in the X direction; k yi , c yi are the equivalent bearing stiffness and damping of the gear in the Y direction; k m , c m are the meshing stiffness and meshing damping of the gear pair respectively; e(t) is the static transmission error; β is the helical angle of the helical gear; F mi is the dynamic meshing force of gears, F mi =c mi δ(t)+k mi f(δ(t)), i=1,2; F fi is the friction force of the helical gear, F fi =μ(t)F mi , μ(t) is the time-varying friction coefficient.

4. The nonlinear dynamic modeling method of helical gears based on three-dimensional anisotropy according to claim 1 or 3, characterized in that: The dimensionless nonlinear dynamic model in step 4 is: In the formula, ω=ω e / ω n , i=1,2, b c is the nominal displacement scale, ω n is the time scale, m e is the equivalent mass, is the dimensionless displacement, is the dimensionless meshing damping, is the dimensionless bearing damping, is the dimensionless meshing stiffness, is the dimensionless bearing stiffness.

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