Method for calculating actual contact area of tooth engagement in mixed lubrication state

Abstract

The application provides a mixed lubrication gear meshing actual contact area calculation method, comprising the following steps: S1, measuring the profile height of the gear tooth surface, calculating the static meshing force of the gear tooth according to the external load condition of the gear, and taking the normal load of the mixed lubrication system as the external normal load, which is borne by the solid part and the liquid part; S2, constructing an explicit function relationship between the solid contact force and the maximum contact area of the micro roughness peak; S3, deducing the equivalent thickness expression of the lubricating oil film, and establishing a function relationship between the liquid contact force and the maximum contact area of the roughness peak; S4, solving the normal load borne by the solid part, the liquid part and the maximum contact area of the roughness peak by simultaneously solving the above three equations, and calculating the dimensionless actual contact area of the tooth surface corresponding to different meshing forces under the mixed lubrication state. The application solves the problems of a large number of cyclic iteration calculations and the inability to reflect the dynamic change of the oil film thickness in the gear meshing process in the existing theoretical method.

Description

Technical Field

[0001] This invention relates to the field of gear transmission and engineering lubrication, and specifically to a method for calculating the actual contact area of ​​gear teeth meshing under mixed lubrication conditions. Background Technology

[0002] Gear drives are widely used in aerospace, precision manufacturing, weaponry, and medical devices. Gear drive systems often require lubrication. Under low-speed, heavy-load conditions, the roughness of the gear teeth's meshing surfaces at a microscopic scale easily leads to direct contact between rough peaks, causing localized rupture of the lubricating oil film—a condition known as mixed lubrication. This direct contact of rough peaks results in solid friction, wear, and even early pitting failures. Therefore, accurately and quickly determining the actual contact area of ​​the gear teeth's meshing surfaces is of significant engineering value.

[0003] In mixed lubrication conditions, the meshing force between gear teeth is jointly borne by the solid contact of the rough peaks and the liquid contact of the lubricating oil. Calculating the load borne by the solids of the rough peaks is a prerequisite for determining the actual contact area of ​​the tooth surface. Currently, the theoretical calculation methods for the solid contact load of the rough peaks mainly include: cyclic algorithms and roughness fitting methods. The cyclic algorithm is based on the load distribution principle in mixed lubrication. By iteratively cycling the oil film thickness, it obtains the load distribution ratio coefficient that simultaneously satisfies both the solid and liquid parts. This method requires a large amount of computational space and cannot provide an explicit analytical expression for the equivalent oil film thickness. The roughness fitting method directly fits the lubricating oil film thickness based on statistical parameters characterizing roughness, and then calculates the solid contact force. Although this method reflects to some extent the physical essence that oil molecules can only exist in the gaps between rough peaks in mixed lubrication, it has a static defect and cannot reflect the dynamic changes in the equivalent oil film thickness caused by changes in meshing force during gear transmission. Summary of the Invention

[0004] This invention characterizes tooth surface roughness based on fractal theory. Through mathematical conversion between the tangential contact area and normal deformation of the roughness peak, an analytical formula for the equivalent oil film thickness under mixed lubrication conditions is obtained. Furthermore, the solid contact force of the roughness peak and the actual contact area of ​​the tooth surface are calculated, thus opening up a new method for predicting the actual contact area of ​​the meshing surface of gear teeth.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] A method for calculating the actual contact area of ​​gear teeth meshing under mixed lubrication includes the following steps:

[0007] S1. Measure the profile height of the gear teeth, calculate the fractal dimension D and fractal roughness G, and calculate the static meshing force F of the gear teeth based on the external load conditions. This static force F serves as the external normal load of the hybrid lubrication system, and is controlled by the solid part F. s and liquid part F l By sharing the burden, we can establish equation one: F = F s +F l ;

[0008] S2. Constructing a solid contact force F based on fractal theory, including parameters D and G. s The maximum contact area a with the micro-roughness peak L Explicit functional relation F s =f(a L Thus, equation two is established;

[0009] S3. By mathematically converting the contact area of ​​the micro-roughness peaks to the normal deformation, the expression for the equivalent thickness of the lubricating oil film, h(a), is derived. L ), and establish the liquid contact force F l Maximum contact area a with the rough peak L The analytic expression F between l =g(h(a) L Thus, equation three is established;

[0010] S4. Solve the above three equations simultaneously to find the normal load on the solid part F. s Liquid part F l The maximum contact area a with the rough peak L And based on the maximum contact area a of the rough peak L Calculate the dimensionless actual contact area As(a) of the tooth surface under different meshing forces in mixed lubrication conditions. L ).

[0011] Furthermore, in step S1, a topography measuring instrument is used to measure the profile height z(x) of the tooth surface at different positions x under the sampling length L, and the fractal dimension D and fractal roughness G are calculated by plotting the power spectrum function.

[0012] Furthermore, step S1 specifically includes the following steps:

[0013] For spur gears, the static meshing force F at the gear tooth meshing interface is calculated as follows:

[0014] F = T / R

[0015] In the formula, F is the static meshing force of the gear; T is the driving torque on the driving wheel.

[0016] R—Radius of the base circle of the driving wheel;

[0017] Due to the unavoidable micro-roughness of the gear tooth surface, under low-speed, heavy-load conditions, the oil film is easily incomplete, resulting in a mixed lubrication state. In this case, the load is borne jointly by the solid and liquid components.

[0018] F = F S +F l

[0019] In the formula, F s —Solid contact force;

[0020] F l —Liquid contact force.

[0021] Furthermore, step S2 specifically includes the following steps:

[0022] In hybrid elastohydrodynamic lubrication, the load is shared by both the solid and the fluid. According to fractal theory, the contact force F of the solid part... s for

[0023]

[0024] In the formula, n(a) is the distribution function of the contact area of ​​the rough peak.

[0025] E – Modulus of elasticity;

[0026] a L —The maximum contact area of ​​all microscopic roughness peaks;

[0027] l—base radius of the rough peak;

[0028] l a —The maximum base radius of all rough peaks;

[0029] l s —The minimum base radius of all rough peaks;

[0030] n(l) — Rough peak base radius distribution function, n(l) = Dl a D l -1-D ;

[0031] a — the contact area of ​​a single micro-protrusion;

[0032] f(a) – Elastic contact force of a single micro-convexity; this equation is the second equation established, denoted as:

[0033] F s =f(a L ).

[0034] Furthermore, step S3 specifically includes the following steps:

[0035] In mixed lubrication, the oil film thickness depends on the gap between the roughness peaks on the tooth surface. This gap dynamically changes with the contact deformation of the roughness peaks. Therefore, the expression for the equivalent thickness of the lubricating oil film includes the initial oil film thickness and the equivalent deformation of all roughness peaks involved in actual contact.

[0036] h = h0 - Δh

[0037] In the formula, h is the equivalent thickness of the lubricating oil film;

[0038] h0—Initial oil film thickness affected by rough morphology;

[0039] Δh — the equivalent deformation of all roughness peaks on the tooth surface.

[0040] Furthermore, the initial oil film thickness h0 mainly depends on the initial roughness of the gear meshing surface, which is characterized by the fractal parameters D and G measured above:

[0041]

[0042] In the formula, L represents the tooth width;

[0043] γ — Surface frequency density;

[0044] m — surface frequency parameter;

[0045] x — the different positions of the tooth surface profile line along the sampling length;

[0046] The equivalent deformation Δh of the solid is obtained by summing the deformation of all rough peaks using fractal theory and then dividing by the number of rough peaks.

[0047] Furthermore, the expression for the deformation of a single rough peak is:

[0048]

[0049] In the formula, ω represents the deformation of a single rough peak;

[0050] a L —Maximum contact area of ​​all rough peaks.

[0051] Furthermore, the sum of the rough peak deformations is obtained using fractal theory as follows:

[0052]

[0053] In the formula, l s —The minimum base radius of all rough peaks;

[0054] n(l) — Rough peak base radius distribution function, n(l) = Dl a D l-1-D ;

[0055] l c —The critical base radius between elastic and plastic deformation;

[0056] The total number of rough peaks is obtained by integrating the distribution function:

[0057]

[0058] The average deformation of all roughness peaks on the tooth surface, i.e., the equivalent change in oil film thickness, is obtained as follows:

[0059]

[0060] The difference between the initial oil film thickness h0 and the equivalent change in oil film Δh is the equivalent thickness h(a) of the lubricating oil film at the gear tooth meshing interface. L ).

[0061] Furthermore, the contact force F of the liquid part l Then the equivalent thickness h(a) of the oil film L Related to this can be represented as:

[0062]

[0063] In the formula, L is the contact line length, which is equal to the tooth width here;

[0064] U – Dimensionless relative velocity;

[0065] G m —Material parameters, which are related to elastic modulus and lubricant properties;

[0066] R e —Equivalent radius of contact body

[0067] This equation is the third equation we established, denoted as:

[0068] F l =g(h(aL)).

[0069] Furthermore, step S4 specifically includes the following steps:

[0070] By simultaneously solving equations 1 (S1), 2 (S2), and 3 (S3), a system is established containing three unknowns, F. s F l a L Solve the system of equations to determine the load F borne by the solid. s The load F borne by the liquid l and the maximum contact area a of the tooth surface roughness peak LFinally, the dimensionless actual contact area A corresponding to different meshing forces F under mixed lubrication conditions is obtained. s (a L ):

[0071]

[0072] Compared with the prior art, the present invention has at least the following beneficial effects:

[0073] 1. This invention provides a novel method for calculating the actual contact area of ​​gear meshing surfaces under mixed lubrication. This method directly correlates the gear meshing force with the actual contact area of ​​the tooth surface roughness peaks, solving the problems of existing theoretical methods requiring extensive iterative calculations and failing to reflect the dynamic changes in oil film thickness during gear meshing. 2. The method provided by this invention is based on fractal geometry theory, realizing the mathematical conversion between the tangential contact area and normal deformation of the rough tooth surface, and providing an explicit analytical expression for the equivalent thickness of the oil film under mixed lubrication conditions. 3. Based on the proposed method for calculating the actual contact area of ​​the tooth surface, the actual contact area of ​​the rough tooth surface corresponding to different meshing forces during gear transmission can be predicted, providing basic parameter support for reducing tooth surface wear and effectively avoiding pitting faults. Attached Figure Description

[0074] 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 accompanying 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.

[0075] Figure 1 This is a schematic diagram of the method for calculating the actual contact area of ​​gear teeth meshing under mixed lubrication according to the present invention;

[0076] Figure 2 This is a schematic diagram illustrating the dynamic change of the equivalent oil film thickness with meshing force according to the present invention.

[0077] Figure 3 This is a schematic diagram illustrating the calculation process of the actual contact area of ​​gear teeth meshing under mixed lubrication according to the present invention.

[0078] Figure 4 This is a schematic diagram of the dimensionless actual contact area of ​​gear teeth meshing under mixed lubrication according to the present invention. Detailed Implementation

[0079] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0080] It should be noted that, unless otherwise specified, the experimental methods described in the following embodiments are all conventional methods, and the reagents and materials described are all commercially available unless otherwise specified. In the description of this invention, the terms "lateral", "longitudinal", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", and "outer" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0081] Furthermore, terms such as "horizontal," "vertical," and "sag" do not imply that components must be absolutely horizontal or suspended, but rather that they can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal relative to "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted.

[0082] In the description of this application, it should also be noted that, unless otherwise expressly specified and limited, the terms "set up," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this application based on the specific circumstances.

[0083] This invention provides a novel method for calculating the actual contact area of ​​gear teeth meshing under mixed lubrication, which can quickly and easily predict the actual contact area of ​​the tooth surface during gear meshing transmission based on the external working conditions of the gear.

[0084] First, the fractal parameters D and G, which characterize the tooth surface roughness, are measured. The static meshing force F of the gear teeth is calculated based on the external load on the gear, serving as the external normal load of the hybrid lubrication system. This normal load is controlled by the solid component F. s and liquid part F l The two sides share the responsibility, thus establishing Equation 1; secondly, based on fractal theory, they construct the solid contact force F containing parameters D and G. s The maximum contact area a with the micro-roughness peakL Explicit functional relation F s =f(a L Thus, Equation 2 is established; furthermore, through the mathematical transformation between the contact area of ​​the micro-rough peaks and the deformation, the expression for the equivalent thickness of the lubricating oil film h(a) is derived. L ), and establish the liquid contact force F l Maximum contact area a with the rough peak L The analytic expression F between l =g(h(a) L Thus, equation three is established; finally, the three equations are solved simultaneously to find the three unknowns F. s F l a L And based on the maximum contact area a of the rough peak L Calculate the dimensionless actual contact area A of the tooth surface corresponding to different meshing forces under mixed lubrication conditions. s (a L ).

[0085] This invention considers the self-similar fractal roughness characteristics of tooth surfaces at the microscale, enabling direct and rapid prediction of the actual contact area of ​​tooth surfaces under mixed lubrication conditions based on static meshing force. The process of this invention is as follows: Figure 1 As shown.

[0086] The present invention will be further described in detail below with reference to the accompanying drawings, but this should not be construed as limiting the invention.

[0087] 1. The surface profile height of a standard spur involute gear is measured using an optical microscope (μ-Scan in this embodiment). An autocorrelation function of the tooth surface morphology is established, and a Fourier transform is performed on the autocorrelation function R(τ) to obtain the power spectrum S(ω) of the tooth surface profile height. Subsequently, a double logarithmic coordinate function of S(ω) and spatial frequency ω is established to obtain the slope k of the fitted line. p And the intercept B. Then we have

[0088]

[0089] γ represents the frequency factor of a rough surface. For machined tooth surfaces, γ = 1.5 is generally used.

[0090] From this, the fractal dimension D and fractal roughness G, which characterize the microstructure of the tooth surface, can be calculated: D = 1.4, G = 1 × 10⁻⁶. -10 m.

[0091] In gear transmission, the meshing force of a pair of meshing teeth is obtained by dividing the external torque on the driving gear by the base circle radius. Taking a standard involute spur gear made of carbon steel as an example, the gear module is 2, the pressure angle is 20°, and the elastic modulus E is 2 × 10⁻⁶.11 Pa, base circle radius is 0.05m, rotation speed is 10r / min, and input torque is 5N.m.

[0092] F = T / R = 100N (2)

[0093] Because the surface roughness of the gear teeth is unavoidable, the oil film is incomplete, resulting in mixed lubrication. In this case, the normal load is shared by the solid and liquid components.

[0094] F = F S +F l (3)

[0095] In the formula, F s —Contact force of solid parts;

[0096] F l —Contact force of the liquid portion;

[0097] 2. In hybrid elastohydrodynamic lubrication, the load is borne jointly by the solid and the liquid. According to fractal theory, the contact force F of the solid part is... s for

[0098]

[0099] In the formula, a represents the contact area of ​​a single micro-protrusion, and l s and l a Let E represent the minimum and maximum length scales, respectively, E represent the equivalent elastic modulus, and f(a) represent the elastic contact force of a single micro-protrusion.

[0100]

[0101] n(l) and n(a) represent the distribution functions of the length scale and the contact area, respectively:

[0102]

[0103] Combining equations (4) and (5), the load borne by the solid contact is obtained:

[0104]

[0105] 3. In mixed lubrication, the oil film thickness depends on the gap between the rough peaks of the solid, and the gap changes dynamically with the contact deformation of the solid, such as... Figure 2 As shown, (a) is a schematic diagram before the meshing force is applied, and (b) is a diagram after the meshing force is applied. Therefore, the equivalent oil film thickness is the difference between the initial oil film thickness and the solid deformation:

[0106] h=h0-Δh (7)

[0107] In the formula, h is the oil film thickness;

[0108] h0 — Initial oil film thickness;

[0109] Δh — Solid deformation amount;

[0110] The initial oil film thickness mainly depends on the surface roughness of the gear meshing surface, which is characterized by the fractal parameters D and G measured above:

[0111]

[0112] The solid deformation is obtained by summing the deformation of all rough peaks using fractal theory and then dividing by the total number of rough peaks.

[0113] The expression for the deformation of a single rough peak is:

[0114]

[0115] In the formula, ω represents the deformation of a single rough peak;

[0116] l—base radius of the rough peak;

[0117] l a —The maximum base radius of all rough peaks;

[0118] a L —Maximum contact area of ​​all rough peaks;

[0119] The sum of the rough peak deformations is obtained using fractal theory as follows:

[0120]

[0121] In the formula, l s —The minimum base radius of all rough peaks;

[0122] n(l) — Rough peak base radius distribution function, n(l) = Dl a D l -1-D ;

[0123] l c —The critical base radius between elastic and plastic deformation;

[0124] The total number of rough peaks is obtained by integrating the distribution function:

[0125]

[0126] The average deformation, i.e., the change in oil film thickness, is obtained as follows:

[0127]

[0128] Combining equations (7), (8), and (12), the expression for the equivalent oil film thickness is obtained as follows:

[0129]

[0130] The contact force in the liquid portion is related to the oil film thickness, and is expressed as:

[0131]

[0132] In the formula, L is the contact line length, which is equal to the tooth width here;

[0133] U – Dimensionless relative velocity;

[0134] G m —Material parameters, which are related to the elastic modulus and lubricating oil properties; combining equations (13) and (14), the expression for the load borne by the lubricating oil is obtained as follows:

[0135]

[0136] 4. By combining equation (3) from step 1, equation (6) from step 2, and equation (15) from step 3, the three unknowns F can be solved. s F l a L ,like Figure 3 As shown. The corresponding values ​​of a under different meshing forces F. L Substitute the value into As = a L *D / (2-D)L 2 This allows us to obtain the dimensionless actual contact area of ​​the rough tooth surface under mixed lubrication conditions. The trend of the actual contact area of ​​the tooth surface calculated by this method with the meshing force F is as follows: Figure 4 As shown. Based on the gear parameters in this case, when the meshing force is F = 100N, the dimensionless actual contact area of ​​the tooth surface is A. s =0.01, when the meshing force increases to F = 1000N, the corresponding dimensionless actual contact area of ​​the tooth surface is A. s =0.49.

[0137] The above embodiments are merely illustrative examples of the technical solutions of the present invention. The methods and apparatus involved in the present invention are not limited to those described in the above embodiments, but are defined by the scope of the claims. Any modifications, additions, or equivalent substitutions made by those skilled in the art based on these embodiments are within the scope of protection claimed by the claims of the present invention.

Claims

1. A method for calculating the actual contact area of tooth engagement of a mixed lubrication lower gear, characterized in that: The steps include the following: S1, measure the profile height of the gear tooth surface, calculate the surface fractal dimension D And fractal roughness G , according to the external load condition of the gear, calculate the static meshing force of the gear tooth F , as the external normal load of the mixed lubrication system, the normal load is borne by the solid part F s And liquid part F l Together, recorded as equation one: F=F s + F l ; S2, constructing a solid contact force based on fractal theory D, G F s explicit function relationship with the maximum contact area of the microscopic rough peaks a L ​​ In the formula, - a roughness peak contact area distribution function, - modulus of elasticity; - all micro-roughness peaks maximum contact area; - base radius of the roughness peak; - the maximum base radius of all rough peaks; - the minimum base radius of all rough peaks; - a roughness peak base radius distribution function, ; - the contact area of a single micro-asperity; - elastic contact force of the individual microprotrusions; Equation Two ; S3. By mathematically converting the contact area of ​​the micro-roughness peaks to the normal deformation, the expression for the equivalent thickness of the lubricating oil film is derived. and establish liquid contact force Maximum contact area with rough peak The functional relationship between them: In the formula, — sample length; — dimensionless relative velocity; - material parameters, related to the elastic modulus and the lubricant properties; - contact body equivalent radius; Equation three is written as ; S4, the normal load is solved by simultaneously solving the three equations above for the solid part , the liquid part , and the maximum contact area of the rough peak , and the maximum contact area of the rough peak , the dimensionless actual contact area of the tooth surface corresponding to different meshing forces in the mixed lubrication state is calculated : .

2. The method of claim 1, wherein: In the step S1, a profile measuring instrument is used to measure the profile height of the tooth surface at the sampling length of different positions , and the fractal dimension and fractal roughness are calculated by drawing a power spectrum function . . .

3. The method of claim 1, wherein: The step S1 specifically includes the following steps: For spur gears, the static engagement force of the tooth engagement interface The calculation method is: In the formula, - gear static engagement force; - driving torque on the driving wheel - radius of the base circle of the driving wheel; Due to the inevitable existence of micro-roughness on the tooth surface, under low-speed heavy-load working conditions, it is extremely easy to cause incomplete oil film and mixed lubrication state, at this time the load is shared by solids and liquids, that is In the formulae, — solid contact force; - liquid contact force.

4. The method of claim 1, wherein: The S3 specifically includes the following steps: In mixed lubrication, the oil film thickness depends on the gap between the rough peaks of the tooth surface, and the gap changes dynamically with the contact deformation of the rough peaks, therefore, the expression of the equivalent thickness of the lubricating oil film contains the initial oil film thickness and the equivalent deformation of all rough peaks participating in the actual contact: In the formula, - the equivalent thickness of the lubricating oil film; - initial oil film thickness affected by rough topography; - the equivalent deformation of all roughness peaks of the tooth surface.

5. The method of claim 4, wherein: Initial oil film thickness h 0 depends on the initial roughness of the gear tooth surface, from the measured fractal parameter 、 Characterization: In the formula, - surface frequency density; - a surface frequency parameter; - different positions of the tooth profile line over the sampling length; equivalent deformation of solid The equivalent deformation of solid is then obtained by summing the deformation of all the rough peaks and dividing by the number of rough peaks.

6. The method of claim 4, wherein: The expression of the deformation of a single rough peak is: In the formula, - single roughness peak deformation; - All roughness peak maximum contact area.

7. The method of claim 6, wherein: The sum of the deformation of the rough peaks is obtained by the fractal theory: In the formula, - critical substrate radius between elastic and plastic deformation; The total number of rough peaks is obtained by integrating the distribution function: The average deformation of all rough peaks of the tooth surface, that is, the equivalent change of the oil film thickness, is: Initial oil film thickness The difference between the equivalent change in oil film thickness and the equivalent change in oil film thickness is the equivalent thickness of the lubricating oil film at the tooth meshing interface.

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