Semi-analytic geometric contact analysis method suitable for rough tooth surface of modified gear

Abstract

The invention relates to the field of gear contact analysis, and discloses a semi-analytic geometric contact analysis method suitable for a modified gear rough tooth surface, which comprises the following steps of: firstly, modeling a modified tooth surface and a rough tooth surface; solving a contact point of an ideal tooth surface of the modified helical gear pair; searching a point pair with maximum geometric interference in a deviation curved surface convex region of a meshing point neighborhood; the actual meshing position and the rotation angle of the rough tooth surface are obtained, and then the no-load transmission error is obtained; the problems that in the prior art, singular points exist on the tooth surface due to addition of tooth surface microscopic features, a traditional TCA method is not suitable any more, meshing points cannot be solved through the gear meshing principle, and the solving precision and the solving efficiency are low when a tooth surface searching method is directly adopted are solved.

Description

Technical Field

[0001] The present invention relates to the field of gear contact analysis, and particularly to a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear. Background Art

[0002] As a core component of mechanical transmission, gears play a key role in the power transmission systems of fields such as automobiles, aviation, and ships. Their contact performance directly affects the transmission efficiency, vibration and noise, and service life. Research shows that the tooth surface modification technology can effectively suppress the edge contact effect and reduce the contact stress concentration phenomenon by topologically optimizing the tooth surface geometry. However, inevitable tooth surface deviations (such as surface roughness) in the actual machining process will cause a multi-scale coupling effect, resulting in the deviation of the actual contact path from the theoretical contact trajectory, and significantly aggravating the dynamic impact load under high-speed and heavy-load conditions. The traditional geometric contact analysis (Tooth Contact Analysis, TCA) method is based on the assumption of an ideal smooth tooth surface and is difficult to characterize the local stress singularity and contact non-linearity caused by the rough tooth surface. Therefore, developing a contact analysis method that integrates macroscopic modification characteristics and microscopic roughness effects has important engineering value for improving the service performance of gear pairs under extreme conditions.

[0003] Regarding the influence of tooth surface deviation on the contact characteristics of gears, most current studies simulate the tooth surface contact path and the size of the contact area through the geometric contact analysis (TCA) model.

[0004] For the above method, the inventor believes that due to the addition of microscopic features of the tooth surface, there are singular points on the tooth surface, and the traditional TCA method is no longer applicable. It is impossible to solve the meshing point through the gear meshing principle, and the solution accuracy and solution efficiency of directly using the tooth surface search method are relatively low. Summary of the Invention

[0005] The present invention aims to provide a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear, which solves the problems in the prior art that the addition of microscopic features of the tooth surface causes singular points on the tooth surface, the traditional TCA method is no longer applicable, it is impossible to solve the meshing point through the gear meshing principle, and the solution accuracy and solution efficiency of directly using the tooth surface search method are relatively low.

[0006] To achieve the above object, the present invention provides the following method:

[0007] A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by the present invention is:

[0008] S1: Establish the tooth surface of a standard involute helical gear. Superimpose a modified tooth surface on the tooth surface of the standard involute helical gear to construct a modified tooth surface. When constructing the modified tooth surface, simultaneously superimpose modifications in the tooth direction and the tooth profile direction, using parabolic modification to establish a rough tooth surface of the modified gear. Superimpose an end face tooth profile deviation on the modified tooth surface to obtain the modified tooth surface and the normal vector.

[0009] S2: Establish a coordinate system for the meshing relationship of the gear pair. Based on the gear tangency principle, derive independent equations. Given one of the parameters, solve for all unknowns to obtain the ideal tooth surface contact path of the modified gear.

[0010] S3: Establish a local mesh within the ideal tooth surface contact path of the modified gear. Use the local search method to determine the pair of points with the maximum geometric interference in the local area of the rough tooth surface of the modified gear at the same rotation angle. Find the matching point and the initial separation distance on the driven member. When the initial separation distance is less than zero, it proves that the workpiece is in an interference state at this time.

[0011] S4: At the same rotation angle of the ideal tooth surface meshing, when the rough tooth surface of the modified gear is in an interference state, at this time, apply a small reverse rotation angle Δφg to the driving wheel w , so that the imaginary geometric interference between the rough tooth surfaces is eliminated, and obtain the small reverse rotation angle Δφg w After that, use the least squares method to find the optimal rotation angle Δφgw’ in the vicinity of the Δφg w neighborhood, so that the rough tooth surfaces are exactly tangent at the main contact point M'. After obtaining the actual meshing position and rotation angle of the rough tooth surface of the modified gear, the no-load transmission error of the helical gear pair can be further calculated.

[0012] Preferably, the step of establishing the tooth surface of a standard involute helical gear, superimposing a modified tooth surface on the tooth surface of the standard involute helical gear to construct a modified tooth surface, simultaneously superimposing modifications in the tooth direction and the tooth profile direction, using parabolic modification to establish a rough tooth surface of the modified gear, and superimposing an end face tooth profile deviation on the modified tooth surface to obtain the modified tooth surface and the normal vector includes: establishing the tooth surface r g (u c , θ c ) of a standard involute helical gear, and superimposing a modified surface on the tooth surface of the standard involute helical gear to obtain a modified tooth surface r g (u c , θ c ). Wherein the tooth surface r g (u c , θ c ) of the standard involute helical gear and the normal vector n g (u c , θ c), Modified gear tooth surface r g1 (u c , θ c ) and the normal vector n g 1(u c , θ c ) are shown as follows:

[0013]

[0014] r g1 (u c , θ c ) = r g (u c , θ c ) + c a u c 2 n g (u c , θ c ) + c b θ c 2 n g (u c , θ c ) + c c n g (u c , θ c );

[0015]

[0016] Among them, r b is the base circle radius of the gear, and θ0 is the developed angle of the tooth width on the base circle. u c and θ c are the tooth surface parameters of the helical gear, u c is the tooth direction parameter, and θ c is the tooth profile parameter. c a is the tooth direction modification parameter, and c b , c c are the tooth profile modification parameters.

[0017] Preferably, the step S1 further includes: superimposing an end face tooth profile deviation on the modified tooth surface, and the direction of the tooth profile deviation is the normal vector direction of the tooth surface of the standard involute helical gear, and considering the tooth surface roughness as a two-dimensional deviation; the relationship between the actual points of the tooth surface considering roughness and the actual points of the modified gear tooth surface is expressed by the following formula:

[0018]

[0019] Among them, r gi is the position vector of the tooth surface including the tooth surface deviation, and n gibis the normal vector of the standard tooth surface end face profile without modification, δ ei is the total normal deviation of the end face profile, expressed as a two-dimensional deviation with the tooth profile and tooth direction parameters as variables.

[0020] Preferably, for the step of establishing the coordinate system of the gear pair meshing relationship, based on the gear tangency principle, deriving independent equations, giving one of the parameters, and solving for all unknowns to obtain the ideal tooth surface contact path of the modified gear, it includes: establishing the gear pair meshing coordinate system Sf, the origin of which coincides with the follower coordinate system Sg of the driving gear, and Sp′, Sp″, Sp″′ are the auxiliary coordinate systems of the driven gear. Considering the gear pair installation error, all of them are merged on the driven gear. Based on the gear pair continuous tangency principle, the contact points of the ideal tooth surface of the modified helical gear pair are analytically solved through the following meshing equations:

[0021]

[0022] In the formula, u ci and θ ci (i = g, p) are the tooth direction and tooth profile parameters of the driving and driven gears. r g1 f and r p1 f are the position vectors of the ideal tooth surfaces of the gear and the pinion respectively in the coordinate system of the gear pair meshing relationship, and n g1 f and n p1 f are the unit normal vectors of the ideal tooth surfaces of the gear and the pinion respectively in the meshing coordinate system.

[0023] Preferably, the continued derivation of the contact points of the ideal tooth surface of the modified helical gear pair by analytically solving the meshing equation is as follows:

[0024]

[0025] In the formula, M fg , M fp″′ , M p″′p″ , M p″p′ , M p′p is the coordinate transformation matrix, which can be expressed according to the geometric relationship as:

[0026]

[0027] where θ px , θ py , ΔE is the installation error of the driven gear, representing the angular deviation of the x-axis, the angular deviation of the y-axis, and the center distance deviation respectively. E is the center distance between the driving and driven gears. Ψ g and Ψ pThey are the rotations of the driving and driven wheels respectively; five independent equations can be derived according to the meshing equation, which contain six unknowns in total. When the rotation of the driven wheel Ψ p is given, these five equations can be solved to obtain the ideal tooth surface contact path of the modified gear, which can be expressed as:

[0028]

[0029] Preferably, the step of establishing a local mesh within the ideal tooth surface contact path of the modified gear, determining the pair of points with the maximum geometric interference in the local area of the rough tooth surface of the modified gear at the same rotation angle using the local search method, finding the matching point and the initial separation distance on the driven member, and proving that the workpiece is in an interference state when the initial separation distance is less than zero includes:

[0030] The introduction of tooth surface roughness significantly changes the local contact state of the tooth surface, thereby causing changes in the transmission error. A local tooth surface mesh is established around the theoretical tooth surface contact point, and the pair of points with the maximum geometric interference in the local area of the rough tooth surface at the same rotation angle is determined using the local search method. The matching point and the initial separation distance of any point on the driving tooth surface on the driven gear are solved through the following non-linear equations. The formula is:

[0031]

[0032] The above formula can derive three independent equations containing three unknowns l r , u cp , θ cp . Solving this system of equations can find a unique matching point on the driven tooth surface. l r is the initial separation distance of this pair of matching points. When the l r is positive, this pair of points is in a separated state; when the l r equals 0, it indicates just contact; when the l r is negative, it indicates that this pair of matching points is in an interference state.

[0033] Preferably, at the same rotation angle of the ideal tooth surface meshing, when the rough tooth surface of the modified gear is in an interference state, a small reverse rotation angle Δφg w is applied to the driving wheel at this time, so that the imaginary geometric interference between the rough tooth surfaces is eliminated, and the small reverse rotation angle Δφg w is obtained. After that, the best rotation angle Δφgw’ in the neighborhood of the Δφg w is found through the least squares method, so that the rough tooth surfaces are exactly tangent at the main contact point M'. The steps include: when the driving and driven wheels are at the same rotation angle, interference will occur at the meshing point. A reverse rotation angle is applied to the driving wheel to eliminate the imaginary geometric interference between the rough tooth surfaces, and the tooth surface deviation is much smaller than r g, the angle Δφ rotated by the driving wheel g w is simplified, and Δφ g w is approximately M g ’ is translated by l along the direction of the unit normal vector n of the theoretical tangent point M rmin . Then the tangency condition of the rough surface can be approximately expressed as:

[0034]

[0035] After obtaining the angle rotated by the driving wheel, then slowly and slightly adjust the angle Δφ rotated by the driving wheel g w , so that the rough surface is exactly tangent at the main contact point M′, that is:

[0036] (r pm - r gm )·n gm = 0;

[0037] In the formula, rgm and rpm are the position vectors of the actual meshing points of the rough tooth surfaces of the driving and driven gears respectively, and n gm is the unit normal vector of the actual meshing point of the driving wheel. In order to obtain the accurate rotation angle value Δφ g w , set the above tangency condition for inspection as f(φ g ) = 0, then there is:

[0038]

[0039] Since after adding the rotation angle, the interference degree of the driving and driven wheels decreases as the rotation angle of the driving wheel increases. From the description of step S3 for l r , it can be known that in the neighborhood of Δφ gw , f(φ g ) is monotonically changing, that is, f’(φ g ) is always greater than or always less than 0. Then the bisection method can be used to quickly obtain a relatively accurate φ g , so that f(φ g ) is close to 0, and the accuracy is 10^-9.

[0040] Preferably, after step S4, it further includes: setting the global iteration number counter s = 1, the maximum iteration number s max = 100, the contact accuracy threshold k = 10^-10, and defining the initial rotation angle range Δφg max(s＝1) and Δφ gmin(s＝1) = 0, corresponding to the driving wheel rotation angle formula:

[0041]

[0042] where Δφ g w is the initial approximate rotation angle value, which is determined according to step S3;

[0043] Perform the initial interference state determination; calculate the initial interval endpoint function values:

[0044]

[0045] If then the initial interval is valid, and proceed to the subsequent steps; otherwise, expand by an exponential step size until the opposite sign condition is satisfied.

[0046] Preferably, the subsequent steps further include: calculating the current iteration midpoint rotation angle:

[0047]

[0048] Solve for the current midpoint function value

[0049] Perform the contact state determination and interval update, interference determination: If the tooth surfaces of the driving and driven wheels are separated, the rotation angle needs to be reduced:

[0050]

[0051] If the tooth surfaces of the driving and driven wheels are separated, the rotation angle needs to be increased:

[0052]

[0053] The steps for verifying the convergence condition are:

[0054] If the following conditions are met, terminate the iteration:

[0055]

[0056] Otherwise, let s = s + 1 and recalculate the rotation angle of the point in the current iteration.

[0057] Preferably, the steps after terminating the iteration are: after obtaining the actual meshing position of the rough tooth surface of the modified gear and the rotation angle of the point, calculate the no-load transmission error of the helical gear pair as:

[0058]

[0059] In the formula, z g and z p are the number of teeth of the driving and driven wheels respectively.

[0060] The beneficial effects of the present invention are embodied in that when analyzing the contact situation of the rough tooth surface of a modified gear set with singular points, the technical solution of the present application can simultaneously consider the influence of the modified tooth surface and the microscopic tooth surface deviation. Combining the principle of continuous tooth surface tangency and the local tooth surface search method, it is faster and more applicable when analyzing the contact characteristics of the rough tooth surface of the modified gear. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following will briefly introduce the drawings required for use in the description of the specific embodiments or the prior art. In all the drawings, similar elements or parts are generally denoted by similar reference numerals. In the drawings, the elements or parts do not necessarily draw to actual scale.

[0062] Figure 1 It is a schematic flowchart of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention;

[0063] Figure 2 It is a schematic diagram of a modified tooth surface model of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention;

[0064] Figure 3 It is a modified tooth surface diagram including tooth surface deviation of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention;

[0065] Figure 4 It is a diagram of the gear contact coordinate system of the driving and driven wheels of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention;

[0066] Figure 5 It is a schematic diagram for calculating the contact distance and the initial separation distance of the ideal tooth surface distance of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention;

[0067] Figure 6 It is a flowchart for obtaining the actual meshing position and rotation angle of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention;

[0068] Figure 7 It is a schematic diagram of the actual meshing position of the rough tooth surface of a semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0069] In order to enable those skilled in the art to better understand the solution of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below 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.

[0070] The terms "first", "second", etc. in the specification and claims of the present invention and the above-mentioned drawings are used to distinguish different objects, rather than to describe a specific order. In addition, the terms "comprising" and "having" and any variations thereof are intended to cover non-exclusive inclusion. For example, a process, method, device, product or end that includes a series of steps or units is not limited to the listed steps or units, but may optionally further include steps or units not listed, or may optionally further include other steps or units inherent to these processes, methods, products or ends.

[0071] Referring to "embodiments" herein means that a specific feature, structure or characteristic described in connection with the embodiments can be included in at least one embodiment of the present invention. The phrase appears in various places in the specification and does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment mutually exclusive with other embodiments. Those skilled in the art will explicitly and implicitly understand that the embodiments described herein can be combined with other embodiments.

[0072] The traditional Tooth Contact Analysis (TCA) method is based on the assumption of an ideal smooth tooth surface and is difficult to characterize the local stress singularity and contact nonlinearity caused by a rough tooth surface. Therefore, developing a contact analysis method that integrates macroscopic modification features and the influence of microscopic roughness has important engineering value for improving the service performance of gear pairs under extreme working conditions.

[0073] Regarding the influence of tooth surface deviation on the contact characteristics of gears, most current studies simulate the tooth surface contact path and the size of the contact area through a geometric contact analysis (TCA) model.

[0074] For the above method, the inventor believes that due to the addition of microscopic features of the tooth surface, there are singular points on the tooth surface, and the traditional TCA method is no longer applicable. It is impossible to solve the meshing point through the gear meshing principle, and the solution accuracy and solution efficiency of directly using the tooth surface search method are relatively low.

[0075] The present invention aims to provide a semi - analytical geometric contact analysis method applicable to the rough tooth surface of modified gears, which solves the problems in the prior art that the addition of microscopic features of the tooth surface leads to singular points on the tooth surface, the traditional TCA method is no longer applicable, it is impossible to solve the meshing point through the gear meshing principle, and the solution accuracy and efficiency of directly using the tooth surface search method are relatively low.

[0076] Figure 2 It is a schematic diagram of the established modified tooth surface model. Through Figure 2 it can be seen that on the basis of the theoretical tooth surface, the tooth profile and tooth direction are modified for the modified tooth surface;

[0077] Figure 3 It is a modified tooth surface diagram including tooth surface deviation. From Figure 3 it can be seen that there are obvious tooth surface deviations on the tooth surface, resulting in singular points on the tooth surface, and the traditional TCA method is no longer applicable. In this paper, a semi - analytical geometric basic analysis method is adopted;

[0078] Figure 4 It is a diagram of the contact coordinate system of the driving and driven gears. Using Figure 4 the relative position relationship of the coordinate systems and the continuous tangency principle of the gear pair, the ideal contact path of the modified rough gear can be solved;

[0079] Figure 5 It is a schematic diagram for calculating the contact distance and the initial separation distance of the ideal tooth surface, showing the matching point and the initial separation distance on the moving tooth surface. M gi and M pi represent points on the driving and driven gears, and M gi M pi represents the position vector between the meshing points;

[0080] Figure 6 It is a flowchart for obtaining the actual meshing position and rotation angle, showing the program flow for solving the actual position and rotation angle;

[0081] Figure 7 It is a schematic diagram of the actual meshing position of the rough tooth surface. In Figure a, it is the meshing of the ideal tooth surface, which will cause interference; Figure b is a schematic diagram of the actual meshing of the rough tooth surface. After adjusting the rotation angle Δφ g w the actual meshing point of the driving and driven gear tooth surfaces is point M'.

[0082] As Figures 1-7 shown, the specific implementation manner of the present invention provides a semi - analytical geometric contact analysis method applicable to the rough tooth surface of modified gears, including the following steps:

[0083] S1: Establish the tooth surface of a standard involute helical gear. Superimpose a modified tooth surface on the tooth surface of the standard involute helical gear to construct a modified tooth surface. When constructing the modified tooth surface, simultaneously superimpose modifications in the tooth direction and the tooth profile direction, using a parabolic modification to establish the rough tooth surface of the modified gear. Superimpose an end face tooth profile deviation on the modified tooth surface to obtain the modified tooth surface and the normal vector.

[0084] In the embodiment of the present invention, establish the tooth surface r g (u c , θ c ) of a standard involute helical gear. Superimpose a modified surface on the tooth surface of the standard involute helical gear to obtain the modified tooth surface r g (u c , θ c ). Among them, the tooth surface r g (u c , θ c ) of the standard involute helical gear and the normal vector n g (u c , θ c ), the tooth surface r g1 (u c , θ c ) of the modified gear and the normal vector n g 1(u c , θ c ) are shown as follows:

[0085]

[0086] r g1 (u c , θ c ) = r g (u c , θ c ) + c a u c 2 n g (u c , θ c ) + c b θ c 2 n g (u c , θ c ) + c c n g (u c , θ c );

[0087]

[0088] Among them, r b is the base circle radius of the gear, and θ0 is the developed angle of the tooth width on the base circle. uc With θ c is the helical gear tooth surface parameter, u c is the tooth direction parameter, θ c is the tooth profile parameter. c a is the tooth direction modification parameter, c b , c c is the tooth profile modification parameter; a profile deviation of the end face is superimposed on the modified tooth surface, and the direction of the profile deviation is the normal vector direction of the tooth surface of the standard involute helical gear. The tooth surface roughness is considered as a two-dimensional deviation; the relationship between the actual points on the tooth surface considering roughness and the actual points on the tooth surface of the modified gear is expressed by the following formula:

[0089]

[0090] where, r gi is the position vector of the tooth surface with tooth surface deviation, n gib is the normal vector of the end face tooth profile of the standard tooth surface without modification, δ ei is the total normal deviation of the end face tooth profile, expressed as a two-dimensional deviation with tooth profile and tooth direction parameters as variables.

[0091] S2: Establish a coordinate system for the meshing relationship of the gear pair. Based on the gear tangency principle, derive independent equations, specify one of the parameters, solve for all unknowns, and obtain the contact path of the ideal tooth surface of the modified gear.

[0092] In the embodiment of the present invention, a meshing coordinate system Sf of the gear pair is established, and the origin of the coordinate system coincides with the follower coordinate system Sg of the driving wheel. Sp′, Sp″, Sp″′ are the auxiliary coordinate systems of the driven wheel. Considering the installation error of the gear pair, it is merged on the driven wheel. Based on the continuous tangency principle of the gear pair, the contact points of the ideal tooth surface of the modified helical gear pair are analytically solved through the following meshing equations:

[0093]

[0094] In the formula, u ci and θ ci (i = g, p) are the tooth direction and tooth profile parameters of the driving and driven wheels. r g1 f and r p1 f are the position vectors of the ideal tooth surfaces of the gear and the pinion in the coordinate system of the gear pair meshing relationship, n g1 f and n p1 f are the unit normal vectors of the ideal tooth surfaces of the gear and the pinion in the meshing coordinate system respectively; continue to derive the contact points of the ideal tooth surface of the modified helical gear pair by analytically solving the meshing equations as follows:

[0095]

[0096] Wherein, M fg , M fp″′ , M p″′p″ , M p″p′ , M p′p is a coordinate transformation matrix, which can be expressed according to geometric relationships as:

[0097]

[0098]

[0099] where θ px , θ py , ΔE is the installation error of the driven gear, representing the deviation of the x-axis intersection angle, the deviation of the y-axis intersection angle, and the center distance deviation respectively. E is the center distance between the driving and driven gears. Ψ g and Ψ p are the rotation angles of the driving and driven gears respectively; five independent equations can be derived according to the meshing equation, which altogether contain six unknowns. When the rotation angle Ψ p of the driven gear is given, these five equations can be solved to obtain the ideal tooth surface contact path of the modified gear, which can be expressed as:

[0100]

[0101] S3: Establish a local mesh within the ideal tooth surface contact path of the modified gear, use the local search method to determine the pair of points with the maximum geometric interference in the local area of the rough tooth surface of the modified gear at the same rotation angle, find the matching point and the initial separation distance on the driven member. When the initial separation distance is less than zero, it proves that the workpiece is in an interference state at this time.

[0102] In the embodiment of the present invention, introducing tooth surface roughness significantly changes the local contact state of the tooth surface, thereby causing a change in the transmission error. A local tooth surface mesh is established around the theoretical tooth surface contact point, and the local search method is used to determine the pair of points with the maximum geometric interference in the local area of the rough tooth surface at the same rotation angle. The matching point and the initial separation distance of any point on the driving tooth surface on the driven gear are solved through the following non-linear equations. The formula is:

[0103]

[0104] The above formula can be derived to obtain three independent equations containing three unknowns l r , u cp , θ cp . Solving this system of equations can find a unique matching point on the driven tooth surface. l r is the initial separation distance of this matching point pair. When l r is positive, this point pair is in a separated state; when l r is equal to 0, it indicates just contact; when l rWhen it is negative, it indicates that the matching point pair is in an interference state.

[0105] S4: Under the same rotation angle as the ideal tooth surface, the rough tooth surface of the modified gear is in an interference state. At this time, a small reverse rotation angle Δφg is applied to the driving wheel. w , so that the imaginary geometric interference between the rough tooth surfaces is eliminated, and a small angle Δφg of reverse rotation is obtained w Then, the least square method is used to find the w The optimal rotation angle Δφgw' near the neighborhood makes the rough tooth surfaces exactly tangent at the main contact point M'; after obtaining the actual meshing position and rotation angle of the rough tooth surface of the modified gear, the no-load transmission error of the helical gear pair can be further calculated.

[0106] In the embodiment of the present invention, when the driving and driven wheels are at the same rotation angle, interference will occur at the meshing point, and a reverse rotation angle will be applied to the driving wheel to eliminate the imaginary geometric interference between the rough tooth surfaces. The tooth surface deviation is much smaller than r g , rotate the driving wheel by an angle Δφ g w To simplify, Δφ g w Approximately M g 'Translate l along the direction of the unit normal vector n of the theoretical tangent point M rmin The tangency condition of the rough surface can be approximately expressed as:

[0107]

[0108] After obtaining the rotation angle of the driving wheel, slowly adjust the rotation angle of the driving wheel in a small range Δφ g w , so that the rough surfaces are exactly tangent at the main contact point M′, that is:

[0109] (r pm -r gm )·n gm =0;

[0110] Where rgm and rpm are the position vectors of the actual meshing points of the rough tooth surfaces of the master and slave gears, respectively, and n gm is the unit normal vector of the actual meshing point of the driving wheel. In order to obtain the accurate rotation angle value Δφ g w , set the tangency condition used for the test above to f(φ g )=0, then:

[0111]

[0112] Since after adding the corner, the interference degree between the driving and driven wheels decreases as the rotation angle of the driving wheel increases. From the description of step S3 for l r it can be known that within the neighborhood of Δφ gw , f(φ g ) is monotonically changing, that is, f’(φ g ) is always greater than or always less than 0. Then the dichotomy method can be used to quickly obtain a relatively accurate φ g such that f(φ g ) is close to 0 with a precision of 10^-9. After step S4, it further includes: setting the global iteration count counter s = 1, the maximum iteration count s max = 100, the contact precision threshold k = 10^-10, defining the initial corner range Δφg max(s＝1) and Δφ gmin(s＝1) = 0, corresponding to the driving wheel corner formula:

[0113]

[0114] where Δφ g w is the initial approximate corner value, and the initial approximate corner value is determined according to step S3;

[0115] Perform the initial interference state determination; calculate the function values of the initial interval endpoints:

[0116]

[0117] If then the initial interval is valid and proceed to the subsequent steps; otherwise, expand by an exponential step size until the opposite sign condition is satisfied;

[0118] The subsequent steps further include: calculating the current iteration midpoint corner:

[0119]

[0120] Solving the current midpoint function value [[ID=**50**]]

[0121] Perform the contact state determination and interval update, interference determination: If the tooth surfaces of the driving and driven wheels are separated, the corner needs to be reduced:

[0122]

[0123] If the tooth surfaces of the driving and driven wheels are separated, the corner needs to be increased:

[0124]

[0125] The steps for convergence condition verification are:

[0126] Terminate the iteration if the following conditions are met:

[0127]

[0128] Otherwise, let s = s + 1 and recalculate the point rotation angle at the current iteration;

[0129] The steps after terminating the iteration are as follows: After obtaining the actual meshing position and point rotation angle of the rough tooth surface of the modified gear, calculate the no-load transmission error of the helical gear pair as:

[0130]

[0131] In the formula, z g and z p are the number of teeth of the driving and driven wheels respectively.

[0132] The beneficial effects of the present invention are as follows: When analyzing the contact situation of the rough tooth surface of the modified gear set with singularities, the technical solution of the present application can simultaneously consider the influence of the modified tooth surface and the microscopic tooth surface deviation. Combining the principle of continuous tooth surface tangency and the local tooth surface search method, it is faster and more applicable when analyzing the contact characteristics of the rough tooth surface of the modified gear.

[0133] The above are only embodiments of the present invention. Specific technical solutions or common knowledge such as well-known characteristics are not described in detail here. It should be noted that for those skilled in the art, without departing from the solution of the present invention, several modifications and improvements can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effects of the present invention and the practicality of the patent. The protection scope required by this application should be based on the content of its claims, and the specific implementation manners described in the specification can be used to interpret the content of the claims.

Claims

1. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear, characterized in that The method includes the following steps: S1: Establish the tooth surface of a standard involute helical gear, and superimpose a modified tooth surface on the tooth surface of the standard involute helical gear to construct a modified tooth surface. When constructing the modified tooth surface, modify it simultaneously in the tooth direction and the tooth profile direction, use parabolic modification, establish the rough tooth surface of the modified gear, and superimpose an end face tooth profile deviation on the modified tooth surface to obtain the modified tooth surface and the normal vector; S2: Establish the coordinate system of the gear pair meshing relationship. Based on the gear tangency principle, derive independent equations, give one of the parameters, solve for all unknowns, and obtain the ideal tooth surface contact path of the modified gear; S3: Establish a local grid within the ideal tooth surface contact path of the modified gear. Use the local search method to determine the pair of points with the maximum geometric interference in the local area of the rough tooth surface of the modified gear at the same rotation angle. Find the matching point and the initial separation distance on the driven member. When the initial separation distance is less than zero, it proves that the workpiece is in an interference state at this time; S4: At the same rotation angle of the ideal tooth surface engagement, the rough tooth surface of the modified gear is in an interference state. At this time, apply a small reverse rotation angle Δφg to the driving wheel w , so that the imaginary geometric interference between the rough tooth surfaces is eliminated, and the small reverse rotation angle Δφg is obtained w After that, use the least squares method to find the optimal rotation angle Δφgw’ near the neighborhood of the Δφg w so that the rough tooth surfaces are exactly tangent at the main contact point M'; after obtaining the actual meshing position and rotation angle of the rough tooth surface of the modified gear, the no-load transmission error of the helical gear pair can be further calculated.

2. The semi-analytical geometric contact analysis method for a modified gear with a rough tooth surface according to claim 1, characterized in that The steps of establishing the tooth surface of a standard involute helical gear, superimposing a modified tooth surface on the tooth surface of the standard involute helical gear to construct a modified tooth surface, modifying it simultaneously in the tooth direction and the tooth profile direction when constructing the modified tooth surface, using parabolic modification, establishing the rough tooth surface of the modified gear, and superimposing an end face tooth profile deviation on the modified tooth surface to obtain the modified tooth surface and the normal vector include: Establish the tooth surface r of a standard involute helical gear g (u c , θ c ), a modified surface is superimposed on the tooth surface of the standard involute helical gear to obtain a modified tooth surface r g (u c , θ c ). Among them, the tooth surface r g (u c , θ c ) and the normal vector n g (u c , θ c ) of the standard involute helical gear, the tooth surface r g1 (u c , θ c ) and the normal vector n g1 (u c , θ c ) of the modified gear are shown as follows: r g1 (u c , θ c ) = r g (u c , θ c ) + c a u c 2 n g (u c , θ c ) + c b θ c 2 n g (u c , θ c ) + c c n g (u c , θ c ); where r b is the base circle radius of the gear, and θ0 is the developed angle of the tooth width on the base circle. u c and θ c are helical gear tooth surface parameters, u c is the tooth direction parameter, and θ c is the tooth profile parameter. c a is the tooth direction modification parameter, c b , c c is the tooth profile modification parameter.

3. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear according to claim 1, characterized in that The step S1 further includes: Superimpose an end face tooth profile deviation on the modified tooth surface. The direction of the tooth profile deviation is the normal vector direction of the tooth surface of the standard involute helical gear, and consider the tooth surface roughness as a two-dimensional deviation; The relationship between the actual points of the tooth surface considering roughness and the actual points of the tooth surface of the modified gear is expressed by the following formula: where r gi is the position vector of the tooth surface with tooth surface deviation, n gib is the normal vector of the standard tooth surface end profile without modification, δ ei is the total normal deviation of the end profile, expressed as a two-dimensional deviation with profile and helix parameters as variables.

4. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear according to claim 1, characterized in that The steps of establishing the coordinate system of the gear pair meshing relationship, based on the gear tangency principle, deriving independent equations, giving one of the parameters, solving for all unknowns, and obtaining the ideal tooth surface contact path of the modified gear include: Establish the gear pair meshing coordinate system Sf, the origin of which coincides with the follower coordinate system Sg of the driving wheel. Sp′, Sp″, Sp″′ are the auxiliary coordinate systems of the driven wheel. Considering the gear pair installation error, all of them are merged on the driven wheel. Based on the continuous tangency principle of the gear pair, solve the contact points of the ideal tooth surface of the modified helical gear pair by the following meshing equations: where, u ci and θ ci (i = g, p) are the tooth direction and tooth profile parameters of the driving and driven gears. r g1 f and r p1 f are the position vectors of the ideal tooth surfaces of the gear and pinion, respectively, in the coordinate system of the gear pair meshing relationship, and n g1 f and n p1 f are the unit normal vectors of the ideal tooth surfaces of the gear and pinion, respectively, in the meshing coordinate system.

5. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of modified gears according to claim 1, characterized in that Continue to derive the solution of the contact points of the ideal tooth surface of the modified helical gear pair for the meshing equations: where M fg , M fp″′ , M p″′p″ , M p″p′ , M p′p is a coordinate transformation matrix and can be expressed according to geometric relationships as: where θ px , θ py , ΔE is the installation error of the driven wheel, representing the angular deviation of the x-axis intersection angle, the angular deviation of the y-axis intersection angle, and the center distance deviation respectively. E is the center distance between the driving and driven wheels. Ψ g and Ψ p are the rotation angles of the driving and driven wheels respectively; five independent equations can be derived from the meshing equation, which altogether contain six unknowns. When the rotation angle Ψ p of the driven wheel is given, these five equations can be solved, thereby obtaining the ideal tooth surface contact path of the modified gear, which can be expressed as:

6. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear, characterized in that, The steps of establishing a local grid within the ideal tooth surface contact path of the modified gear, using the local search method to determine the pair of points with the maximum geometric interference in the local area of the rough tooth surface of the modified gear at the same rotation angle, finding the matching point and the initial separation distance on the driven member, and when the initial separation distance is less than zero, it proves that the workpiece is in an interference state at this time include: Introduce the tooth surface roughness to significantly change the local contact state of the tooth surface, thereby causing changes in the transmission error. Establish a local grid of the tooth surface around the theoretical tooth surface contact point, use the local search method to determine the pair of points with the maximum geometric interference in the local area of the rough tooth surface at the same rotation angle, and solve the matching point and the initial separation distance of any point on the driving tooth surface on the driven gear through the following non-linear equation. The formula is: The above equation can be derived to obtain three independent equations containing three unknowns l r , u cp , θ cp . Solving this system of equations can find a unique matching point on the driven tooth surface. l r is the initial separation distance of this pair of matching points. When the said l r is positive, this pair of points is in a separated state; when the said l r equals 0, it indicates just contact; when the said l r is negative, it indicates that this pair of matching points is in an interference state.

7. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear according to claim 1, characterized in that, Under the same rotation angle of the ideal tooth surface meshing, the rough tooth surface of the modified gear is in an interference state. At this time, a small reverse rotation angle Δφg is applied to the driving wheel w , so that the imaginary geometric interference between the rough tooth surfaces is eliminated, and the small reverse rotation angle Δφg is obtained w After that, the best rotation angle Δφgw’ in the neighborhood of the Δφg w is found by the least square method, so that the rough tooth surfaces are exactly tangent at the main contact point M'. The steps include: If the driving and driven wheels are at the same rotation angle, interference will occur at the meshing point. Apply a reverse rotation angle to the driving wheel to eliminate the imaginary geometric interference between the rough tooth surfaces, and the tooth surface deviation is much smaller than r g Simplify the rotation angle Δφ g w of the driving wheel. Δφ g w is approximately equal to M g translates along the direction of the unit normal vector n of the theoretical tangent point M by l rmin . Then the tangency condition of the rough surface can be approximately expressed as: After obtaining the rotation angle of the driving wheel, then slowly and slightly adjust the rotation angle Δφ of the driving wheel g w , so that the rough surface is exactly tangent at the main contact point M′, that is: (r pm -r gm )·n gm ＝0； wherein, rgm and rpm are respectively the position vectors of the actual meshing points of the rough tooth surfaces of the driving and driven gears, and n gm is the unit normal vector of the actual meshing point of the driving gear. In order to obtain an accurate rotation angle value Δφ g w , the above tangency condition for inspection is set as f(φ g ) = 0, then we have: Since after adding the corner, the interference degree between the driving and driven wheels decreases as the rotation angle of the driving wheel increases. From the description of step S3 for l r , it can be known that within the neighborhood of Δφ gw , f(φ g ) changes monotonically, that is, f’(φ g ) is always greater than or always less than 0. Then the dichotomy method can be used to quickly obtain a relatively accurate φ g such that f(φ g ) is close to 0 with an accuracy of 10^-9.

8. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear according to claim 7, characterized in that After step S4, it further includes: Set the global iteration counter s = 1, the maximum number of iterations s max = 100, the contact precision threshold k = 10^-10, and define the initial rotation angle range Δφg max(s＝1) and Δφ gmin(s＝1) = 0, corresponding to the driving wheel rotation angle formula: where Δφ g w is the initial approximate rotation angle value, which is determined according to step S3; Perform the initial interference state determination; Calculate the function values of the endpoints of the initial interval: If then the initial interval is valid and proceed to the subsequent steps; Otherwise, expand by exponential steps until the condition of opposite signs is satisfied.

9. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear according to claim 8, characterized in that The subsequent steps also include: Calculate the midpoint rotation angle of the current iteration: Solve the function value of the current midpoint Perform the contact state determination and interval update Interference determination: If the tooth surfaces of the driving and driven wheels are separated, the rotation angle needs to be reduced: If The tooth surfaces of the driving and driven wheels are separated, and the rotation angle needs to be increased: The steps for verifying the convergence condition are: If the following conditions are met, terminate the iteration: Otherwise, let s = s + 1 and recalculate the rotation angle of the point in the current iteration.

10. A semi-analytical geometric contact analysis method applicable to the rough tooth surface of a modified gear according to claim 9, characterized in that, The steps after terminating the iteration are: After obtaining the actual meshing position of the rough tooth surface of the modified gear and the rotation angle of the point, calculate the no-load transmission error of the helical gear pair as: where z g and z p are the number of teeth of the driving and driven wheels respectively.

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