Meshing efficiency calculation method based on fractal reconstruction tooth surface model

Abstract

The invention discloses a meshing efficiency calculation method based on a fractal reconstruction tooth surface model, and belongs to the technical field of gear meshing efficiency calculation. According to the method, an anisotropic rough surface model is constructed through actually measured tooth surface data, and the local roughness of discrete points is extracted; calculating a tooth surface sliding-rolling speed, normal load distribution and a time-varying friction coefficient in combination with a tooth profile geometric equation and motion contact analysis; and sliding and rolling friction power losses are calculated based on the friction acting principle, and finally the average meshing efficiency is obtained. The method provided by the invention overcomes the defect that a constant is used for representing the roughness of the whole tooth surface in a traditional method, and realizes accurate evaluation of the meshing efficiency under different roughness characteristics.

Description

Technical Field

[0001] This invention belongs to the field of gear meshing efficiency calculation technology, specifically relating to a method for calculating meshing efficiency based on a fractal reconstructed tooth surface model. Background Technology

[0002] Gears, as core transmission components, are widely used in automobiles, aerospace, and robotics. Their meshing efficiency directly affects the system's energy consumption, thermal management, and vibration and noise performance. Gear surface roughness is a key factor influencing gear friction loss, contact fatigue, and transmission efficiency. Traditional research methods have the following limitations: Insufficient characterization of tooth surface roughness: Existing technologies mostly use a single constant to characterize the roughness of the entire tooth surface, which cannot reflect the local roughness differences at various points on the actual tooth surface. Although isotropic rough surface models have been developed based on fractal theory, and anisotropic fractal contact models have been further proposed, more accurate reconstruction methods are still needed for the anisotropic characteristics of actual tooth surfaces such as those produced by gear grinding.

[0003] Limitations of meshing efficiency models: Current research on gear meshing efficiency mainly relies on elastohydrodynamic lubrication theory and finite element analysis to calculate the friction coefficient, and focuses on the influence of macroscopic parameters (tooth profile, speed, torque). These models do not fully incorporate tooth surface roughness data, making it difficult to quantify the dynamic impact of roughness on meshing power loss.

[0004] Limitations of friction coefficient calculation: Traditional methods use theoretical constants or simplified models to calculate time-varying friction coefficients, neglecting the spatial distribution characteristics of roughness at discrete points on the tooth surface, resulting in insufficient meshing power loss (especially sliding friction loss) and inaccurate efficiency prediction.

[0005] In summary, there is an urgent need for a meshing efficiency evaluation method that integrates measured tooth surface roughness data with anisotropic fractal reconstruction and local friction coefficient calculation, so as to accurately quantify the impact of roughness on gear transmission performance. Summary of the Invention

[0006] This invention provides a method for calculating meshing efficiency based on a fractal reconstructed tooth surface model. It calculates sliding and rolling friction power losses based on the principle of frictional work, using the roughness of each point on the reconstructed anisotropic surface model. Based on the tooth profile geometry equation and kinematic contact analysis, it calculates the tooth profile meshing path, tooth surface sliding-rolling speed, normal load distribution, and time-varying friction coefficient. This provides a method for calculating gear meshing power loss and meshing efficiency, effectively solving at least one technical problem mentioned in the background art.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows: A method for calculating meshing efficiency based on a fractal reconstructed tooth surface model includes the following steps: Step S1: Construct a fractal reconstruction anisotropic rough surface model based on the measured tooth surface data, and calculate the root mean square roughness of discrete points. ; Step S2: Based on the rough surface model, calculate the tooth surface motion parameters and friction parameters, including the tooth profile meshing path, tooth surface sliding-rolling speed, normal load distribution, and time-varying friction coefficient. Step S3: Calculate the sliding friction power loss according to the principle of work done by friction. and rolling friction power loss The instantaneous meshing power loss is obtained by superposition. ; Step S4, based on input power and instantaneous engagement power loss Calculate instantaneous meshing efficiency η The average meshing efficiency of the gear pair is obtained by integration. .

[0008] Optionally, in step S1, the specific process of constructing the fractal reconstruction anisotropic rough surface model includes: Tooth surface height data is measured using a white light interferometer, with sampling along the machining direction X and the vertical direction Y. The fractal dimensions Dx and Dy in the X and Y directions are calculated using the power spectral density function method, and satisfy the following formula: , ; in, β for logS(ω) The slope of the linear regression; γ For scale parameters; D s G is the fractal dimension; G is the characteristic scale coefficient. ω Spatial frequency; Power spectral density; Generate rough surface models based on anisotropic 3D rough surface modeling formulas: ; in, Z(x) i ), Z (y i ） Indicates along X and Y Rough surfaces derived from directional profile curve arrays; A i Transformation operator A No. i The element with the largest absolute value in the row; Bj Represents the coefficient vector B No. j One element; n It is classified as a micro-convex body. The root mean square roughness of each discrete point is calculated based on the reconstructed surface. : ; in, The total number of discrete points. Single point height, The average height.

[0009] Optionally, in step S2, the tooth surface sliding speed and scrolling speed The calculation formula is: ; ; in, , For the gear angular velocity, The distance from the contact point to the meshing node The distance.

[0010] Optionally, in step S2, the time-varying coefficient of friction The calculation formula is: ; ; in, P h Hertzian stress; SR The slip-roll ratio; V e This refers to the suction speed; ν 0 represents dynamic viscosity; R The radius of curvature is the composite radius of curvature. b 1~ b 9 is the regression coefficient.

[0011] Optionally, in step S3, the sliding friction power loss P S and rolling friction power loss P R The calculation formula is: ; ; in, , This is the heat reduction factor. G , U , W For dimensionless parameters, For the combined radius of curvature, is the viscosity-pressure coefficient.

[0012] Optionally, in step S4, the average meshing efficiency The calculation process includes: Instantaneous power loss during the meshing cycle Integrating along the line of engagement yields the average meshing power loss. : ; in The base circle tooth pitch; B 1 B 2 represents the actual line of engagement; Based on input power Calculate the average meshing efficiency : .

[0013] Optionally, the method is applicable to cylindrical gear pairs, and the rough surface model is constructed based on measured data of the gear teeth produced by grinding.

[0014] The advantages of this invention compared to the prior art are as follows: 1. This invention extracts the fractal dimension of the parallel / perpendicular machining directions (X / Y directions) from measured tooth surface data (measured with a white light interferometer) (as shown in the embodiment). Dx =1.6289, Dy =1.9617), and an anisotropic three-dimensional rough surface model is constructed by combining the improved WM fractal function. Compared with the traditional isotropic model, this method more realistically restores the actual texture features of the ground tooth surface, laying the foundation for accurate analysis of the influence of local roughness on meshing behavior.

[0015] 2. This invention is based on the root mean square roughness of each discrete point on the tooth surface. Rq By combining an empirical friction model under lubrication conditions, the time-varying friction coefficient at each contact point during meshing is dynamically calculated. This overcomes the shortcomings of traditional methods that use a single constant to represent the roughness of the entire tooth surface, significantly improving the accuracy of friction coefficient calculation.

[0016] 3. The present invention calculates sliding / rolling friction losses separately: Sliding friction power loss and rolling friction power loss under elastohydrodynamic lubrication are derived separately based on the principle of frictional work, and then superimposed to obtain the total instantaneous power loss; through tooth profile geometry equations and kinematic contact analysis, the meshing path, sliding and rolling speeds, and normal load distribution are calculated in real time, and the instantaneous meshing efficiency and periodic average efficiency are dynamically output by combining the time-varying friction coefficient. Examples show that this method can quantify the meshing efficiency (such as average efficiency) under different roughness characteristics. =0.9815), providing a reliable basis for gear design.

[0017] 4. This invention provides a complete implementation process (modeling → parameter calculation → efficiency evaluation), supporting closed-loop applications from measured data to efficiency simulation. An example using ground spur gears verifies the feasibility of this method in real-world industrial scenarios.

[0018] In summary, this invention, through fractal reconstruction of anisotropic tooth surface models and dynamic calculation of local friction coefficients, achieves for the first time a refined assessment of the impact of spatial distribution of tooth surface roughness on meshing efficiency. Compared with prior art, its advantages are: the model is more closely aligned with engineering realities (anisotropic surfaces); it overcomes the limitations of a "single roughness constant," improving calculation accuracy; and it provides theoretical support for energy consumption optimization of gear transmission systems. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort, wherein: Figure 1 A diagram illustrating the gear sampling process provided by this invention; Figure 2 A schematic diagram illustrating the principle of external meshing of involute tooth profiles provided by this invention; Figure 3 This is one of the measurement tooth surface roughness topography images provided by the present invention; Figure 4 This is the second measurement image of tooth surface roughness provided by the present invention; Figure 5 Logarithmic curves of the X and Y direction power spectral density function method provided by this invention; Figure 6 A linear fitting curve of the X and Y direction power spectral density function method provided by the present invention; Figure 7 The fractal reconstruction anisotropic tooth surface model diagram provided by this invention; Figure 8 The root mean square roughness diagram of the gear model provided by this invention; Figure 9 The time-varying friction coefficient diagram obtained by the existing method provided by this invention; Figure 10 The time-varying friction coefficient diagram obtained by this method is provided by the present invention; Figure 11 The sliding friction work loss diagram is obtained by the existing method provided by this invention; Figure 12The sliding friction loss diagram obtained by the method provided by this invention. Detailed Implementation

[0020] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0021] It should be noted that all directional indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present invention are only used to explain the relative positional relationship and movement of each component in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indication will also change accordingly.

[0022] Furthermore, in this invention, descriptions involving "first," "second," etc., are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0023] In this invention, unless otherwise explicitly specified and limited, the terms "connection," "fixed," etc., should be interpreted broadly. For example, "fixed" can mean a fixed connection, a detachable connection, or an integral part; it can mean a mechanical connection; it can mean a direct connection or an indirect connection through an intermediate medium; it can mean the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0024] Furthermore, the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are feasible for those skilled in the art. If the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0025] This invention provides a method for calculating meshing efficiency based on a fractal reconstructed tooth surface model, characterized by the following steps: Step S1: Construct a fractal reconstruction anisotropic rough surface model based on the measured tooth surface data, and calculate the root mean square roughness of discrete points. ; Step S2: Based on the rough surface model, calculate the tooth surface motion parameters and friction parameters, including the tooth profile meshing path, tooth surface sliding-rolling speed, normal load distribution, and time-varying friction coefficient. Step S3: Calculate the sliding friction power loss according to the principle of work done by friction. and rolling friction power loss The instantaneous meshing power loss is obtained by superposition. ; Step S4, based on input power and instantaneous engagement power loss Calculate instantaneous meshing efficiency η The average meshing efficiency of the gear pair is obtained by integration. .

[0026] In step S1, the specific process of constructing the fractal reconstruction anisotropic rough surface model includes: See Figure 1 As shown, tooth surface height data is measured using a white light interferometer, and samples are taken along the machining direction X and the vertical direction Y to filter out macroscopic shape errors such as tilt and curvature. The fractal dimensions Dx and Dy in the X and Y directions are calculated using the power spectral density function method (PSD) and satisfy the following formula: , ; in, β for logS(ω) The slope of the linear regression; γ For scale parameters; D s G is the fractal dimension; G is the characteristic scale coefficient. ω Spatial frequency; Power spectral density; Based on the two-dimensional WM function curve, and according to the modeling formula for anisotropic three-dimensional rough surfaces, and combined with the fractal dimensions Dx and Dy, a rough surface model is generated: ; in, Z(x) i ), Z (y i ） Indicates along X and Y Rough surfaces derived from directional profile curve arrays; A i Transformation operator A No. i The element with the largest absolute value in the row; B j Represents the coefficient vectorB No. j One element; n It is classified as a micro-convex body. The root mean square roughness of each discrete point is calculated based on the reconstructed rough surface. : ; in, The total number of discrete points. Single point height, The average height.

[0027] In step S2, combined Figure 2 As shown, P For meshing nodes, K This is the instantaneous contact point between the two tooth surfaces. N 1 N 2 represents the theoretical line of engagement. B 1 B 2 represents the actual line of engagement. Let the angular velocities of the two gears be respectively... ω 1. ω 2, then the two tooth surfaces are at K Point tooth surface sliding speed for: ; The two tooth surfaces are K Scrolling speed of the point for: ; in, The distance from the contact point to the meshing node The distance.

[0028] The normal load on the tooth surface is calculated based on the relationship between the meshing stiffness of a single tooth pair and the overall stiffness of the gear pair: ; ; in, K lsf For load distribution factor; F n,max The maximum normal load; k single For single-tooth meshing stiffness; k total This represents the overall stiffness of the gear pair.

[0029] The root mean square roughness at each discrete point obtained by substituting the tooth surface friction coefficient under fully lubricated conditions into the above formula. R q The time-varying friction coefficient was calculated. It can be expressed by the following formula: ; ; in, P h Hertzian stress; SR The slip-roll ratio; V e This refers to the suction speed; ν 0 represents dynamic viscosity; R The radius of curvature is the composite radius of curvature. b 1~ b 9 is the regression coefficient.

[0030] In step S3, sliding friction power loss accounts for the majority of meshing power loss. This is due to the energy loss caused by the sliding between the tooth surfaces at the meshing point, resulting from the different velocities of the contacting tooth surfaces. The sliding friction power loss can be calculated from the power loss principle. P S It can be expressed by the following formula: ; in, f The coefficient of sliding friction; F n This is the normal load.

[0031] When gears are in elastohydrodynamic lubrication, an elastic hydrodynamic oil film forms between the meshing tooth profiles. Due to the uneven distribution of the oil film pressure, rolling friction resistance torque is generated, resulting in rolling friction power loss. The following rolling friction power loss is obtained. P R Calculation formula: ; in, , This is the heat reduction factor. G , U , W For dimensionless parameters, For the combined radius of curvature, is the viscosity-pressure coefficient.

[0032] After calculating the instantaneous sliding and rolling friction power losses separately using the above formula, the sum of these values ​​yields the instantaneous gear meshing power loss. P loss = P S + P R .

[0033] In step S4, the instantaneous gear meshing efficiency is obtained from the definition of transmission efficiency. η for: .

[0034] Calculate the average meshing power loss during the gear meshing cycle. P loss,ave That is, to obtain the average meshing efficiency of the gear pair. η ave .like Figure 1 As shown, for the tooth profile of a cylindrical gear, its meshing period is the distance the contact point travels along the meshing line by one base circle pitch. p b The time when |B2D|=|CB1|, according to the gear meshing principle, the distance the contact point moves along the meshing line is proportional to the gear rotation angle and time. When gear tooth 2 is in B 2 D During segmental meshing, gear tooth 1 is in DB One stage of meshing, therefore the average meshing efficiency The calculation process includes: Instantaneous power loss during the meshing cycle Integrating along the line of engagement yields the average meshing power loss. : ; in The base circle tooth pitch; B 1 B 2 represents the actual line of engagement; Based on input power Calculate the average meshing efficiency : .

[0035] The method is applicable to cylindrical gear pairs, and the rough surface model is constructed based on measured data of the gear teeth produced by grinding.

[0036] The following detailed description of the meshing efficiency calculation method based on the fractal reconstructed tooth surface model provided by the present invention will be based on a specific embodiment 1.

[0037] Example 1 Based on the parameter data of a pair of coarse spur gears processed by grinding in Table 1, the fractal tooth surface model was reconstructed using this method.

[0038] Table 1 Design parameters of gear embodiment The surface morphology of the specimen was measured using a white light interferometer, and height information at various points on the tooth surface was collected, for example... Figure 3 and Figure 4 As shown.

[0039] To obtain X and Y The height data at each point in both directions, based on the linear fitting results, are as follows: Figure 5 and Figure 6 As shown, calculate the fractal dimension of a point. D x =1.6289, D y =1.9617.

[0040] Based on the obtained fractal dimension D x and D y In MATLAB, calculate and generate anisotropic tooth surface models, such as... Figure 7 As shown.

[0041] extract Figure 5 Root mean square roughness of each point in the reconstructed tooth surface model R q ,like Figure 8 As shown.

[0042] Based on the model parameters in Table 1, the time-varying friction coefficient of the tooth surface was calculated. The results of calculating the time-varying friction coefficient using the method provided in this invention are as follows: Figure 10 As shown, comparison Figure 9 The root mean square roughness is 0.825. μm The coefficient of friction obtained at that time.

[0043] Time-varying sliding friction power loss, such as Figure 11 and Figure 12 As shown.

[0044] The average meshing efficiency of the gear is calculated based on the method provided in this invention. P loss,ave =0.9815.

[0045] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0046] Furthermore, it should be noted that the scope of the methods and systems in the embodiments of the present invention is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in the reverse order, depending on the functions involved. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. In addition, features described with reference to certain examples may be combined in other examples.

[0047] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of the present invention.

Claims

1. A method for calculating meshing efficiency based on a fractal reconstructed tooth surface model, characterized in that, Includes the following steps: Step S1: Construct a fractal reconstruction anisotropic rough surface model based on the measured tooth surface data, and calculate the root mean square roughness of discrete points. ; Step S2: Based on the reconstructed rough surface model, calculate the motion parameters and friction parameters of the tooth surface, including the tooth profile meshing path, the sliding-rolling speed of the tooth surface, the normal load distribution, and the time-varying friction coefficient. Step S3: Calculate the sliding friction power loss according to the principle of work done by friction. and rolling friction power loss The instantaneous meshing power loss is obtained by superposition. ; Step S4, based on input power and instantaneous engagement power loss Calculate instantaneous meshing efficiency η The average meshing efficiency of the gear pair is obtained by integration. .

2. The method according to claim 1, characterized in that, In step S1, the specific process of constructing the fractal reconstruction anisotropic rough surface model includes: Tooth surface height data is measured using a white light interferometer, with sampling along the machining direction X and the vertical direction Y. The fractal dimensions Dx and Dy in the X and Y directions are calculated using the power spectral density function method, and satisfy the following formula: ， ； in, β for logS(ω) The slope of the linear regression; γ For scale parameters; D s G is the fractal dimension; G is the characteristic scale coefficient. ω Spatial frequency; Power spectral density; Generate rough surface models based on anisotropic 3D rough surface modeling formulas: ； in, Z(x) i ), Z (y i ） Indicates along X and Y Rough surfaces derived from directional profile curve arrays; A i Transformation operator A No. i The element with the largest absolute value in the row; B j Represents the coefficient vector B No. j One element; n It is classified as a micro-convex body. The root mean square roughness of each discrete point is calculated based on the reconstructed surface. : ； in, The total number of discrete points. Single point height, The average height.

3. The method according to claim 1, characterized in that, In step S2, the tooth surface sliding speed and scrolling speed The calculation formula is: ； ； in, , For the gear angular velocity, The distance from the contact point to the meshing node The distance.

4. The method according to claim 1, characterized in that, In step S2, the time-varying coefficient of friction The calculation formula is: ； ； in, P h Hertzian stress; SR The slip-roll ratio; V e This refers to the suction speed; ν 0 represents dynamic viscosity; R The radius of curvature is the composite radius of curvature. b 1~ b 9 is the regression coefficient.

5. The method according to claim 1, characterized in that, In step S3, the sliding friction power loss P S and rolling friction power loss P R The calculation formula is: ； ； in, , This is the heat reduction factor. G , U , W For dimensionless parameters, For the combined radius of curvature, is the viscosity-pressure coefficient.

6. The method according to claim 1, characterized in that, In step S4, the average meshing efficiency The calculation process includes: Instantaneous power loss during the meshing cycle Integrating along the line of engagement yields the average meshing power loss. : ； in The base circle tooth pitch; B 1 B 2 represents the actual line of engagement; Based on input power Calculate the average meshing efficiency : 。 7. The method according to any one of claims 1-6, characterized in that, The method is applicable to cylindrical gear pairs, and the rough surface model is constructed based on measured data of the gear teeth produced by grinding.

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