Method for calculating time-varying meshing stiffness of solid lubrication coating gear

Abstract

The invention relates to the technical field of machinery, and discloses a solid lubrication coating gear time-varying meshing stiffness calculation method, which comprises the following steps: S1, introducing a frequency response function; s2, introducing an equivalent elastic modulus E *; s3, calculating the time-varying meshing stiffness of the gear on the basis of the equivalent elastic modulus defined by the expansion Hertz theory in the S2 in combination with a potential energy method; and S4, replacing the equivalent elastic modulus E * defined by the expansion Hertz theory in the S2 into the part E in the stiffness calculation formula in the S3, and finally obtaining a time-varying meshing stiffness calculation model of the coated gear. The expansion Hertz theory is applied to gear time-varying meshing stiffness calculation, an equivalent elastic modulus capable of being applied to solid lubricating coating object contact is defined by deducing the expansion Hertz theory, then the equivalent elastic modulus is applied to gear time-varying meshing stiffness calculation, and a time-varying meshing stiffness calculation model of the gear with the solid lubricating coating is completed.

Description

Technical Field

[0001] The present invention relates to the field of mechanical technology, and specifically to a calculation method for the time-varying meshing stiffness of a solid lubricating coating gear. Background Technique

[0002] A Chinese patent with the publication number CN118260937A discloses a calculation method for the meshing stiffness of a planetary gear considering elastohydrodynamic lubrication; it includes the following steps: S1. Use orthogonal design to determine the parameter combinations of the training set required to generate the PCE surrogate model; S2. Establish an elastohydrodynamic lubrication model, substitute the parameters of the training set into the model and solve for the pressure and film thickness curves; S3. Correct the contact deformation of the solved elastohydrodynamic lubrication model; S4. Extract the minimum oil film thickness and the corrected contact deformation from the solution results of the elastohydrodynamic lubrication model, construct a training set, and use it to replace the solution of the elastohydrodynamic lubrication model; S5. Call the PCE surrogate model to solve for the minimum oil film thickness and contact deformation, so as to solve the LTCA model under elastohydrodynamic lubrication; S6. Calculate the meshing stiffness according to the LTCA results. This method corrects the problem that the traditional series method calculates the meshing stiffness of the planet gear-ring inaccurately, and makes up for the gap in the calculation method of the meshing stiffness of planetary gears under elastohydrodynamic lubrication;

[0003] However, the above solution only considers the calculation of the time-varying meshing stiffness of planetary gears, and does not consider the calculation of the time-varying meshing stiffness of planetary gear mechanisms with solid lubricating coatings commonly used in the aerospace field; in view of this, we propose a calculation method for the time-varying meshing stiffness of solid lubricating coating gears. Summary of the Invention

[0004] The purpose of the present invention is to provide a calculation method for the time-varying meshing stiffness of a solid lubricating coating gear to solve the problems raised in the above background technique.

[0005] To achieve the above purpose, the present invention provides the following technical solution: A calculation method for the time-varying meshing stiffness of a solid lubricating coating gear includes the following steps:

[0006] S1. Introduce the frequency response function, which is the fundamental solution of the problem in the frequency domain;

[0007] S2. To combine the material responses of the coating and the substrate, introduce the equivalent elastic modulus E * ;

[0008] S3. Calculate the time-varying meshing stiffness of the gear by combining the equivalent elastic modulus defined by the extended Hertz theory in S2 with the potential energy method;

[0009] S4. The equivalent elastic modulus E defined by the extended Hertz theory in S2 *Replace the part of E in the stiffness calculation formula in S3, and finally obtain the time-varying meshing stiffness calculation model of the coated gear.

[0010] Optionally, the S1 includes: For the FRF of the normal displacement caused by pressure on the surface of a three-dimensional object with a lubricating coating, the following expression is given:

[0011]

[0012] where e = exp(-2Ft), F is the contact force, t is the dimensionless coating thickness, p c ,p s are the Poisson's ratios of the coating material and the substrate material respectively, and G c ,G s are the shear moduli of the coating material and the substrate material respectively.

[0013] Optionally, the S2 includes: The combined response of the coating and the substrate can be regarded as the response of an equivalent half-plane, and the response of the equivalent half-plane can be described by the equivalent elastic modulus E * to describe.

[0014] Optionally, the S2 further includes: Convert the shear modulus G c ,G s in the FRF expression to the equivalent elastic modulus E c * ,E s * : The following formula can be obtained:

[0015]

[0016] By comparing the expression of FRF with the limit cases of uncoated and infinitely thick coatings, an extended Hertz theory is established to define the equivalent elastic modulus:

[0017]

[0018] where F is the meshing force along the meshing line at the meshing point.

[0019] Optionally, the S3 includes: The potential energy method calculates according to the principles of mechanics of materials and elasticity, etc., and simplifies the gear tooth to a cantilever beam fixed on the root circle of the tooth, and combines the deformation and energy conversion of the gear under force;

[0020] During the gear meshing process, due to the force between the teeth, the teeth of the gear will undergo a certain degree of elastic deformation, thus storing the corresponding potential energy; at the same time, the deformation of the tooth matrix will also generate additional potential energy U f; During the tooth contact process, contact potential energy U is also generated due to the contact action. h ; Different forms of potential energy together constitute the total potential energy distribution in the gear system;

[0021]

[0022] In the formula, the meshing force can be further decomposed into a radial component F y and a tangential component F x ; I y represents the moment of inertia of the tooth cross-section at a certain position y from the base circle, and C f is a correction coefficient, with a value of 1.2 for a rectangular cross-section; A x represents the cross-sectional area, h is the distance from the meshing point to the base circle, and d is the distance from the meshing point to the tooth symmetry line; E and G respectively represent the elastic modulus and shear modulus of the material.

[0023] Optionally, the potential energy includes: bending potential energy U caused by bending deformation b , shear potential energy U caused by shear deformation s and axial compression potential energy U formed by axial compression a .

[0024] Optionally, the S3 further includes: stiffness K generated due to the deformation of the tooth matrix f which can be calculated by the following formula:

[0025]

[0026] In the formula, α represents the pressure angle, L is the tooth width, u f is the distance from the tooth root along the tooth center line to the point of action of the meshing force, and S f represents the tooth root thickness; under the assumption that the smooth strain condition holds, L * , M * , P * , Q * can be obtained by fitting with a polynomial function; in addition, X * represents L * , M * , P * , Q * , A, B, C, D, E, F, and the values of u f , S f , θ f , h f are calculated by the subsequent formulas; among them, is the dimensionless tool nose radius, and r i represents the radius of the gear shaft hole:

[0027]

[0028] The contact potential energy U generated by the tooth-to-tooth meshing h can be expressed based on the Hertz contact theory; meanwhile, the contact potential energy U h and the contact stiffness k h The relationship between them can be described by the following formula; where p represents the Poisson's ratio, and L represents the effective contact tooth width;

[0029]

[0030] Based on the potential energy method, the total meshing stiffness when a pair of gears mesh can be obtained, and the specific expression is as follows:

[0031]

[0032] If there are n pairs of gears meshing simultaneously, the meshing stiffness of the gears can be calculated by the second formula.

[0033] Compared with the prior art, the present invention provides a method for calculating the time-varying meshing stiffness of a solid lubricant coating gear, which has the following beneficial effects:

[0034] 1. The method for calculating the time-varying meshing stiffness of the solid lubricant coating gear extends the application of the Hertz theory to the calculation of the time-varying meshing stiffness of gears. By deriving the extended Hertz theory, an equivalent elastic modulus applicable to the contact of solid lubricant coating objects is defined, and then it is applied to the calculation of the time-varying meshing stiffness of gears to complete the time-varying meshing stiffness calculation model of the gear with a solid lubricant coating.

[0035] 2. The method for calculating the time-varying meshing stiffness of the solid lubricant coating gear can analyze the dynamic response of the mechanism, and can make targeted improvements to the mechanism to obtain better transmission performance to meet the requirements of aerospace missions. In addition, the influence law of the coating properties on the lubrication conditions and transmission performance of the transmission mechanism can be analyzed through the time-varying meshing stiffness calculation model of the coated gear of the present invention, providing a theoretical support for selecting lubricant coating materials with better lubrication performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is a schematic flow chart of the present invention;

[0037] Figure 2 is a schematic diagram of the tooth profile parameters of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0038] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0039] As Figure 1 - Figure 2 shown, the present invention provides a technical solution: The present invention provides a calculation method for the time-varying meshing stiffness of a solid lubricant coating gear, including the following steps:

[0040] S1. Introduce the frequency response function, which is the fundamental solution of the problem in the frequency domain. For the FRF of the normal displacement caused by pressure on the surface of a three-dimensional object with a lubricant coating, the following expression is given:

[0041]

[0042] where e = exp(-2Ft), F is the contact force, t is the dimensionless coating thickness, p c , p s are the Poisson's ratios of the coating material and the substrate material respectively, and G c , G s are the shear moduli of the coating material and the substrate material respectively.

[0043] S2. In order to combine the material responses of the coating and the substrate, an equivalent elastic modulus E * is introduced. The definition of the equivalent modulus is based on the following idea: The combined response of the coating and the substrate can be regarded as the response of an equivalent half-plane, and the response of the equivalent half-plane can be described by the equivalent elastic modulus E * .

[0044] Convert the shear moduli G c , G s in the FRF expression to the equivalent elastic moduli E c * , E s * : The following formula can be obtained:

[0045]

[0046] By comparing the expression of the FRF with the limiting cases of no coating and infinitely thick coating, an extended Hertz theory is established to define the equivalent elastic modulus:

[0047]

[0048] Among them, F is the meshing force at the meshing point along the meshing line direction;

[0049] S3. Calculate the time-varying meshing stiffness of the gear by combining the equivalent elastic modulus defined by the extended Hertz theory in S2 with the potential energy method. The potential energy method is calculated based on principles such as mechanics of materials and elasticity of materials, and simplifies the gear teeth into a cantilever beam fixed on the root circle of the tooth, and combines the deformation and energy conversion of the gear under force.

[0050] During the gear meshing process, due to the force between the teeth, the teeth of the gear will undergo a certain degree of elastic deformation, thus storing corresponding potential energy. The potential energy includes the bending potential energy U caused by bending deformation b and the shear potential energy U caused by shear deformation s as well as the axial compression potential energy U formed by axial compression a . At the same time, the deformation of the tooth matrix will also generate additional potential energy U f . In addition, during the tooth contact process, contact potential energy U will also be generated due to the contact action h . Different forms of potential energy together constitute the total potential energy distribution in the gear system.

[0051]

[0052] In the formula, F represents the meshing force at the meshing point along the meshing line direction, and this force can be further decomposed into a radial component F y and a tangential component F x . Among them, I y represents the moment of inertia of the tooth cross-section at a certain position y from the base circle, and C f is a correction coefficient, usually taking a value of 1.2 for a rectangular cross-section. In addition, A x represents the area of the cross-section, h is the distance from the meshing point to the base circle, and d is the distance from the meshing point to the tooth symmetry line. At the same time, E and G respectively represent the elastic modulus and shear modulus of the material.

[0053] The stiffness K generated due to the deformation of the tooth matrix f can be calculated by the following formula:

[0054]

[0055] In the formula, α represents the pressure angle, L is the tooth width, u f is the distance from the tooth root along the tooth center line to the point of action of the meshing force, and S f represents the tooth root thickness. Under the assumption that the smooth strain condition holds, L * , M * , P * , Q * can be obtained by fitting with a polynomial function. In addition, X* Representing L * , M * , P * , Q * , A, B, C, D, E, F can take values referring to the following table, u f , S f , θ f , h f are calculated from the subsequent formulas. Among them, is the dimensionless tool nose radius, and r i represents the radius of the gear shaft hole.

[0056]

[0057]

[0058]

[0059] Finally, the contact potential energy U h generated by the tooth engagement can be expressed based on the Hertz contact theory. At the same time, the relationship between the contact potential energy U h and the contact stiffness k h can be described by the following formula. Among them, p represents the Poisson's ratio, and L represents the effective contact tooth width.

[0060]

[0061] Up to this point, based on the potential energy method, the total meshing stiffness when a pair of gears are meshing can be obtained, and the specific expression is as follows. If there are multiple pairs of gears meshing simultaneously, the meshing stiffness of the gears can be calculated by the second formula.

[0062]

[0063] S4. Substitute the equivalent elastic modulus E * defined by the extended Hertz theory in S2 into the part of E in the stiffness calculation formula in S3, and finally obtain the time-varying meshing stiffness calculation model of the coated gear.

[0064] The above has generally described the present invention in detail. However, based on the present invention, some modifications or improvements can be made, which are obvious to those of ordinary skill in the technical field. Therefore, the modifications or improvements made without departing from the spirit of the present invention are within the protection scope of the present invention.

Claims

1. A method for calculating the time-varying meshing stiffness of a solid lubricating coating gear, characterized in that: The steps include: S1. Introduce the frequency response function, which is the basic solution to the problem in the frequency domain; S2. In order to combine the material response of the coating and the substrate, the equivalent elastic modulus E is introduced. * ; S3, the equivalent elastic modulus defined based on the extended Hertz theory of S2 is combined with the potential energy method to calculate the time-varying mesh stiffness of the gear; S4, the equivalent elastic modulus E defined by the extended Hertzian theory in S2 * The E part in the stiffness calculation formula in S3 is replaced, and finally the time-varying meshing stiffness calculation model of the coated gear is obtained.

2. The method for calculating the time-varying meshing stiffness of a solid lubricating coating gear according to claim 1, characterized in that: S1 includes: For the FRF of the normal displacement of the surface of a three-dimensional object with a lubricating coating due to pressure, the following expression is given: in F is the contact force, t is the dimensionless coating thickness, p c ,p s are the Poisson’s ratios of the coating material and the substrate material, respectively, G c ,G s are the shear moduli of the coating material and the substrate material, respectively.

3. The method for calculating the time-varying meshing stiffness of a solid lubricating coating gear according to claim 1, characterized in that: The S2 includes: the combined response of the coating and the substrate can be regarded as the response of an equivalent half-plane, and the response of the equivalent half-plane can be expressed by the equivalent elastic modulus E * to describe.

4. The method for calculating the time-varying meshing stiffness of a solid lubricating coating gear according to claim 1, characterized in that: S2 further includes: replacing the shear modulus G in the FRF expression c ,G s Convert to the equivalent elastic modulus E of coating and substrate c * ,E s * : The following formula can be obtained: By comparing the expression of FRF with the limiting cases of no coating and infinitely thick coating, the extended Hertz theory is established to define the equivalent elastic modulus: Where F is the meshing force along the meshing line at the meshing point.

5. The method for calculating the time-varying meshing stiffness of a solid lubricating coating gear according to claim 4, characterized in that: S3 includes: the potential energy method is based on the principles of material mechanics and elastic mechanics, and the gear teeth are simplified into cantilever beams fixed on the root circle, and the calculation is performed in combination with the deformation and energy conversion of the gear when subjected to force; During the gear meshing process, due to the force between the gear teeth, the gear teeth will undergo a certain degree of elastic deformation, thereby storing the corresponding potential energy; at the same time, the deformation of the gear tooth matrix will also generate additional potential energy U f ; During the contact process of the gear teeth, contact potential energy U is generated due to the contact action h ; Potential energy in different forms together constitutes the total potential energy distribution in the gear system; In the formula, the meshing force can be further decomposed into the radial component F y and the tangential component F x ;I y represents the moment of inertia of the gear tooth section at a certain position y from the base circle, and C f is the correction factor, which is 1.2 for rectangular cross-sections; A x represents the area of ​​the cross section, h is the distance between the meshing point and the base circle, and d is the distance between the meshing point and the tooth symmetry line; E and G represent the elastic modulus and shear modulus of the material respectively.

6. The method for calculating the time-varying meshing stiffness of a solid lubricating coating gear according to claim 5, characterized in that: The potential energy includes: bending potential energy U caused by bending deformation b , shear potential energy U caused by shear deformation s And the axial compression potential energy U formed by axial compression a .

7. The method for calculating the time-varying meshing stiffness of a solid lubricating coating gear according to claim 1, characterized in that: The S3 further includes: the stiffness K generated by the deformation of the gear tooth matrix f It can be calculated by the following formula: In the formula, α represents the pressure angle, L is the tooth width, and u f is the distance from the tooth root along the tooth centerline to the point where the meshing force acts, and S f represents the tooth root thickness; assuming smooth strain conditions, L * 、M * , P * , Q * Its value can be obtained by fitting a polynomial function; in addition, X * Represents L * 、M * , P * , Q * The values ​​of A, B, C, D, E, and F can be found in the following table, and u f , S f ,θ f 、h f Calculated by the following formula; where, is the dimensionless tool corner radius, r i Indicates the radius of the gear shaft hole: The contact potential energy U generated by the meshing of teeth h It can be expressed based on Hertz contact theory; at the same time, the contact potential energy U h and contact stiffness k h The relationship between can be described by the following formula; where p represents Poisson's ratio and L represents the effective contact tooth width; Based on the potential energy method, the total meshing stiffness of a pair of gears when meshing can be obtained. The specific expression is as follows: If there are pairs of gears involved in meshing at the same time, the meshing stiffness of the gears can be calculated using the second formula.

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