Method for optimizing meshing rigidity of involute spur gear for modifying shape based on dynamic transmission error

Abstract

The invention discloses an involute spur gear meshing stiffness optimization method for modification based on dynamic transmission errors, which is characterized in that the involute tooth profile is accurately modeled, the meshing stiffness is calculated by adopting a Weber energy method, and the impact generated by single-tooth and double-tooth meshing alternation is reduced through tooth profile modification, so that the transmission errors are reduced, and the involute spur gear meshing stiffness is optimized. The technology is applied to an idle gear system, and the meshing rigidity is iteratively optimized by establishing an idle gear nonlinear kinetic equation and analyzing dynamic transmission errors responded by the system until the requirements for transmission precision and design strength are met.

Description

Technical Field

[0001] The invention belongs to the technical field of gear structure optimization, and in particular relates to a method for optimizing the meshing stiffness of an involute spur gear by performing modification based on dynamic transmission error. Background Art

[0002] Due to its compact structure, high transmission efficiency, high speed, and multi-stage transmission gear system, it is widely used in the field of industrial machinery, including automobile gearboxes, aircraft engines, and precision CNC machining machine tools. However, five-axis high-end CNC machine tools and automobile reducers have higher requirements for the stability and accuracy of gear transmission, and the existing gears are difficult to achieve the required accuracy due to processing errors, installation errors, and randomness of product dimensions.

[0003] In response to the above problems, tooth modification or tooth profile modification is generally performed to improve the load-bearing capacity of the gear and reduce the vibration and impact of the gear, but analysis and optimization from the perspective of dynamic transmission error are rarely performed. Secondly, some parameters of tooth profile modification are derived from empirical parameters, and the correlation between modification parameters and dynamic response is not given.

[0004] Therefore, based on the analysis of dynamic transmission error, this paper optimizes the tooth profile modification parameters (including modification amount, modification length and modification method), and then obtains the dynamic meshing stiffness related to the system response, and obtains the effective meshing stiffness value with the minimum dynamic transmission error. Summary of the invention

[0005] The purpose of the present invention is to design a method for optimizing the meshing stiffness of involute spur gears based on dynamic transmission error modification, so as to improve the stability and accuracy of the transmission system.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A method for optimizing meshing stiffness of involute spur gears by modifying the shape based on dynamic transmission error, characterized in that: the method is as follows:

[0008] S1: Establish an accurate model of the involute tooth profile: Take the center of the gear as the origin and the center line of the gear as the y-axis to establish the coordinate system XOY. The tooth profile curve is mainly composed of the tooth root arc line AB, the tooth root transition curve BC, the involute tooth profile curve CD, and the tooth top arc curve DE. Establish the tooth root arc line AB segment equation, the tooth root transition curve BC segment equation, the involute tooth profile CD segment equation, and the tooth top arc curve DE segment equation respectively to obtain the meshing stiffness;

[0009] S2: Based on elastic mechanics and Weber energy method, the meshing stiffness of different meshing positions is solved, and the engineering simulation software ANSYS is used for analysis and verification to obtain the meshing stiffness at different meshing points. The gear meshing process is the process from the root of the tooth to the top of the tooth. Different meshing positions refer to the process from the meshing start point to the meshing end point of the gear. Each meshing position contains three parts of deformation.

[0010] S3: Determine the modification method, adopt the short tooth top modification, express the three modification types of linear modification, quadratic curve modification and exponential modification into a general form, and find the optimal modification parameters through comparative analysis of different values;

[0011] S4: Establish the nonlinear dynamic equation of the idler system, expand the meshing stiffness in S2 by Fourier series and substitute it into the nonlinear dynamic equation for response analysis. According to the final dynamic response error, optimize the meshing stiffness and obtain a stable and reliable transmission structure.

[0012] Furthermore, the equation for establishing the tooth root arc line AB segment of S1 is as follows:

[0013]

[0014] Among them, r f is the tooth root circle radius, δ1∈[b / r,π / z], b is the machining rack parameter, z is the number of teeth, and r is the pitch circle radius;

[0015] The method to establish the BC segment equation of the tooth root transition curve is as follows:

[0016] The part of the tool tooth profile forms an involute tooth profile, the part of the tool tooth profile is a straight line and forms the root circle of the gear, and the part of the tool tooth profile is an arc with a radius of r0, which forms the transition curve of the gear;

[0017] The main parameters of tool tooth profile are as follows:

[0018]

[0019] Among them, m is the module, α is the pitch circle pressure angle, h a ＊ is the tooth addendum coefficient, c ＊ is the head clearance coefficient;

[0020] A coordinate system is fixed to the rack tool and the processed gear respectively, and a fixed coordinate system X is given at the same time f O f Y f ; When the processing gear rotates After the angle, according to the coordinate transformation law, the expression of point M in the coordinate system X2O2Y2 (coordinate system 2) is obtained;

[0021] In the coordinate system X1O1Y1 (coordinate system 1), the homogeneous coordinates of point M are:

[0022] (x 1M ,y 1M ,z 1M ,1) T =(r0cosα′,-a1-r0sinα′,0,1) T

[0023] When the rack tool performs machining motion, coordinate system 1 is relative to the fixed coordinate system X f O f Y f The transformation is as follows:

[0024]

[0025] The coordinate system 2 of the gear revolves around z f Rotated Angle, its transformation matrix is:

[0026]

[0027] Then the transformation matrix of coordinate system 1 relative to coordinate system 2 is:

[0028]

[0029] The expression of point M in coordinate system 2 is:

[0030]

[0031] right 2 Further simplification of the M expression yields:

[0032]

[0033] In the above formula, α′∈[α′ min ,π / 2], when there is no root cutting in the gear, the tooth root transition curve is tangent to the tooth profile involute, at this time α′ min =α =π / 9; if there is undercut, α′ min Less than α, the value at this time is determined by the intersection H of the tooth profile involute equation and the tooth root transition curve equation; r is the gear machining pitch circle radius, a1 = a-χm, varies with α′, and

[0034]

[0035] Establish the CD segment equation of the involute tooth profile:

[0036] The time-varying meshing point K of the involute is expressed by the following equation:

[0037] OK=OP+PK

[0038] in,

[0039]

[0040] PK=r b tanα k [cos(π-β k )sin(π-β k )]

[0041] Then the equation of point K is:

[0042]

[0043] in

[0044]

[0045] but

[0046]

[0047] Among them, r b is the base circle radius, γ=tanα k , α k is the pressure angle at the meshing point, and α k ∈[α H ,arcos(r b / r a )],α H is the pressure angle corresponding to the intersection point H of the transition curve and the involute. The coordinates of point H are given by the following formula:

[0048]

[0049] α H =arccos(r b / r H )

[0050] The expression of δ2 is:

[0051]

[0052] Establishment of the DE segment equation of the tooth top arc curve:

[0053]

[0054] In the formula, δ3∈[0,β-(θ αa -θ α )],and

[0055] θ α =tanα-α.

[0056] Among them, Z is the number of teeth of the gear, α a is the tooth tip circle pressure angle, and α is the pitch circle pressure angle.

[0057] Further, the S2 comprises the following steps:

[0058] The total load deformation of the gear teeth q Tj The contact point consists of the following three parts: 1) Bending, shearing and axial compression deformation of the involute part of the gear tooth, which is regarded as a cantilever beam Bj ; 2) Transition fillet and base deformation q connected to the tooth Fj ; 3) Local contact deformation q C ;

[0059] Assuming the gear is a non-uniform cantilever beam, the equivalent length of the cantilever beam is L e is the distance from the base point M to the top circle. Let the thickness of the infinitesimal element i be δ i The cross-sectional area is The moment of inertia is L ij represents the distance between the microelement i and the load point j in the X-axis direction, β j It is W j The angle between the Y axis and j is the normal meshing force at point j;

[0060] 1) Bending, shearing and axial compression deformation of the involute gear part:

[0061] For the involute tooth profile part regarded as an equivalent cantilever beam, its deformation is divided into three parts: F x Compression deformation y1 under the action of F y and bending deformation y2 under the action of M′; F y Shear deformation y3 under action;

[0062] The component of force along the x-axis:

[0063] F x =W j sinβ j

[0064] The component of force along the y-axis:

[0065] F y =W j cosβ j

[0066] Equivalent bending moment:

[0067]

[0068] The deformations of the three parts are superimposed to obtain the total deformation of a single microelement:

[0069]

[0070] In the formula, This value can improve the calculation accuracy of the small unit i; h j is the tooth thickness of the gear at the meshing point j; μ is the Poisson's ratio; E is the elastic modulus; β j is the angle between the load and the Y axis. j and β j The specific calculation is given as follows.

[0071] h j =2r j [h p / (2r p )+inv(α)-inv(θ j )]

[0072] Among them, θ j =arcos(r b / r j );r p is the pitch circle radius, h p is the pitch circle tooth thickness, and the involute function is expressed as: inv(f)=tan(f)-f.

[0073] The angle between the load direction and the X-axis is

[0074] β j =θ j –ψ j

[0075] Among them, ψ j =h p / (2r p )+inv(α)-inv(θ j )

[0076] The total deformation of the involute part of the gear tooth at the meshing point j is

[0077]

[0078] Transition fillets and base deformation:

[0079] The calculation formula of the infinitesimal deformation of the transition fillet is consistent with that of the involute part of the gear, so the superposition formula is:

[0080]

[0081] Cornell's study on the deformation of the substrate used O'Donnell's derived formula:

[0082] For plane stress:

[0083]

[0084] For plane strain:

[0085]

[0086] In the formula,

[0087] L f =X j -X M -Y j tanβ j ,h f =2Y M

[0088] The total deformation of the transition curve part and the base part is

[0089] q Fj =q FBj +q FFj

[0090] 3) Local contact deformation:

[0091] For the local contact deformation of gears, the Weber method is used, that is,

[0092]

[0093] Where b is the width of the semi-contact area, and is the height of the contact deformation zone;

[0094]

[0095] In the formula, R c1 and R c2 is the equivalent radius of the Hertz contact, and its calculation formula is

[0096]

[0097]

[0098] In the formula, R p1 and R p2 is the pitch circle radius, R b1 is the base circle radius of gear 1, θ is the pitch circle pressure angle, and r1 is the radius corresponding to the pressure angle of gear 1 at the meshing point;

[0099] The total deformation of the meshing teeth at point j is:

[0100] q Tj =(q Bj1 +q Bj2 )+(q Fj1 +q Fj2 )+q Cj

[0101] In the formula, the subscripts 1 and 2 are the numbers of the driving gear and the driven gear, representing the deformation amounts of the corresponding gears. Furthermore, the meshing stiffness at point j can be obtained:

[0102] K j =W j / q Tj

[0103] Based on the meshing stiffness calculation method, the meshing stiffness at different meshing points is calculated.

[0104] Furthermore, in S3, both the large and small gears are subjected to tip relief. The relief amounts of the large and small gears are respectively equal to the comprehensive deformation amounts of the tooth pair at the highest point HPSTC and the lowest point LPSTC in the single-tooth meshing area of the small gear;

[0105] According to the relief length, the tooth profile relief can be divided into long relief and short relief

[0106]

[0107] In the formula, L max is the relief length corresponding to the highest point of single-tooth meshing. When ΔL < 1, it is called short relief. When ΔL > 1, it is called long relief.

[0108] Furthermore, the three relief types of linear relief, quadratic curve relief, and exponential relief are expressed in the following general form:

[0109]

[0110] In the formula, a and b are coefficients related to the relief type, and they need to satisfy the relief amount boundary condition, that is, a + b = 1;

[0111] When a = 1 and b = 0, the relief type is linear relief. When 1 < n < 2 and a = 0, the relief type is exponential relief; when n = 2 and a ≠ 1, the relief type is quadratic curve relief. Among them, when a = 0 and b = 1, the relief type is also called parabolic relief;

[0112] Considering the same relief amount condition, different relief curves or relief lengths are adopted to study the influence of the meshing stiffness on the dynamic transmission error. Through the comparative analysis of different values, the optimal relief curve and relief length are found.

[0113] Furthermore, the nonlinear dynamic model in S4 is as follows:

[0114]

[0115] Among them, M is the mass matrix, C is the damping matrix, and K is the time-varying meshing stiffness matrix;

[0116] According to the result calculated in step S2, the meshing stiffness is expanded by Fourier series, and the first three order terms are taken, namely:

[0117]

[0118] Where i = 1, 2. Parameter Ω m is the meshing base ratio of the gear pair, and

[0119] Ω m =2πn1z1 / 60

[0120] k im , a (i) mj , b (i) mj are the coefficients of Fourier expansion respectively, n1 is the gear speed, z1 is the number of teeth of gear 1;

[0121] Substitute the expanded meshing stiffness into the dynamic equation, combine with the modification parameters in step S3, perform MATLAB calculation simulation on the dynamic transmission error of the system, and obtain the optimal modification parameters through continuous iterative optimization, thereby obtaining the required optimal meshing stiffness value.

[0122] The following beneficial effects can be obtained through the above technical solution:

[0123] The present invention accurately models the involute tooth profile, calculates the meshing stiffness using the Weber energy method, reduces the impact caused by the alternating meshing of single and double teeth through tooth profile modification, reduces the transmission error, and applies this technology to the idler system. By establishing the nonlinear dynamic equation of the idler, the dynamic transmission error of the system response is analyzed, and the meshing stiffness is iteratively optimized until the transmission accuracy and design strength requirements are met.

[0124] The method of the present invention can fully improve the stability and accuracy of gear transmission in high-end CNC machine tools or precision reducers, can reduce the noise generated by vibration during the transmission process, and extend the service life of the gears. Taking the power transmission system of the servo tool holder of a CNC machine tool as an example, by using this optimization method, the stability of the cutting tool working process can be guaranteed, the processing quality of the product can be improved, and the chatter phenomenon of the processing process can be effectively suppressed. Compared with the prior art, the optimization method in the present invention is more reliable, the entire design process is standardized, and the defects of the empirical method can be avoided. Moreover, this method is based on the dynamics point of view and is more in line with the actual working conditions than the analysis of static transmission errors. BRIEF DESCRIPTION OF THE DRAWINGS

[0125] Figure 1 A geometric model diagram of an involute tooth profile optimized for an embodiment of the present invention;

[0126] Figure 2 A geometric model diagram of a rack-type tool with double rounded corners at the top according to an embodiment of the present invention;

[0127] Figure 3 A schematic diagram of coordinate transformation of a rack and a gear when a hob cuts gears in an embodiment of the present invention;

[0128] Figure 4 A schematic diagram of deriving relevant parameters from the involute equation in an embodiment of the present invention;

[0129] Figure 5 Schematic diagram of the calculation model of gear tooth deformation in an embodiment of the present invention;

[0130] Figure 6 It is a flow chart of meshing stiffness calculation in an embodiment of the present invention;

[0131] Figure 7 This is a diagram showing the meshing stiffness calculation results based on Weber theory and ANSYS analysis in an embodiment of the present invention;

[0132] Figure 8 It is a schematic diagram of a shaping curve in an embodiment of the present invention;

[0133] Fig. 9 It is a schematic diagram of the training parameters along the meshing line direction in an embodiment of the present invention;

[0134] Fig.10 A diagram showing a dynamic model of an idler system in an embodiment of the present invention;

[0135] Fig.11 A diagram of a dynamic design scheme in an embodiment of the present invention;

[0136] Fig.12 is a dynamic response error diagram under different meshing stiffnesses in an embodiment of the present invention;

[0137] Fig.13 Graph showing the dynamic response error under different torques in an embodiment of the present invention. DETAILED DESCRIPTION

[0138] The present invention will be further described below in conjunction with the accompanying drawings:

[0139] like Figure 1-13 As shown, a method for optimizing meshing stiffness of involute spur gears based on dynamic transmission error modification comprises the following steps:

[0140] S1: Use the coordinate transformation method to establish an accurate model of the involute tooth profile and solve the meshing stiffness;

[0141] The coordinate system XOY is established with the center of the gear as the origin and the center line of the gear as the y-axis. The tooth profile curve is mainly composed of the tooth root arc line AB, the tooth root transition curve BC, the involute tooth profile curve CD, and the tooth top arc curve DE ( Figure 1 The present invention adopts the hobbing method to cut teeth and generate a single tooth geometric model of an involute gear.

[0142] Establishment of the equation of the AB segment of the tooth root arc line:

[0143]

[0144] Among them, r f is the root circle radius, δ1∈[b / r,π / z], b is the machining rack parameter, z is the number of teeth, and r is the pitch circle radius.

[0145] Establishment of the BC segment equation of tooth root transition curve:

[0146] The tooth root transition curve is the focus of involute modeling. The transition area of ​​the plane gear is a curve that connects the working part (involute part) of the gear tooth profile with the tooth root circle. The shape of the tooth root transition curve is determined by two factors: the relative motion relationship between the tool and the gear blank (processing method); the blade shape of the tool top. This paper takes the rack-type tool with double fillet at the top commonly used in hobbing gear cutting as an example for analysis.

[0147] Figure 2 As shown, part 1 of the tool tooth profile forms an involute tooth profile, part 3 of the tool tooth profile is a straight line and forms the root circle of the gear, and part 2 of the tool tooth profile is an arc with a radius of r0, which forms a transition curve of the gear.

[0148] The main parameters of tool tooth profile are as follows:

[0149]

[0150] Among them, m is the module, α is the pitch circle pressure angle, h a ＊is the tooth addendum coefficient, c ＊ is the head clearance coefficient.

[0151] Figure 3 As shown in the figure, the rack tool and the processed gear are respectively connected to a coordinate system, and a fixed coordinate system X is given. f O f Y f When the machining gear turns After calculating the angle, according to the coordinate transformation law, we get the expression of point M in the coordinate system X2O2Y2 (coordinate system 2).

[0152] In the coordinate system X1O1Y1 (coordinate system 1), the homogeneous coordinates of point M are:

[0153] (x 1M ,y 1M ,z 1M ,1) T =(r0cosα′,-a1-r0sinα′,0,1) T

[0154] When the rack tool performs machining motion, coordinate system 1 is relative to the fixed coordinate system X f O f Y f The transformation is as follows:

[0155]

[0156] The coordinate system 2 of the gear revolves around z f Rotated Angle, its transformation matrix is:

[0157]

[0158] Then the transformation matrix of coordinate system 1 relative to coordinate system 2 is:

[0159]

[0160] The expression of point M in coordinate system 2 is:

[0161]

[0162] right 2 Further simplification of the M expression yields:

[0163]

[0164] In the above formula, α′∈[α′ min ,π / 2], when there is no root cutting in the gear, the tooth root transition curve is tangent to the tooth profile involute, at this time α′ min=α =π / 9; if there is undercut, α′ min Less than α, the value at this time is determined by the intersection H of the tooth profile involute equation and the tooth root transition curve equation. r is the gear machining pitch radius, a1 = a-χm, varies with α′, and

[0165]

[0166] Establishment of the CD segment equation of the involute tooth profile:

[0167] Figure 4 As shown, the time-varying meshing point K of the involute is expressed by the following equation:

[0168] OK=OP+PK

[0169] in,

[0170]

[0171] PK=r b tanα k [cos(π-β k )sin(π-β k )]

[0172] Then the equation of point K is:

[0173]

[0174] in

[0175]

[0176] but

[0177]

[0178] Among them, r b is the base circle radius, γ=tanα k , α k is the pressure angle at the meshing point, and α k ∈[α H ,arcos(r b / r a )],α H is the pressure angle corresponding to the intersection point H of the transition curve and the involute. The coordinates of point H are given by the following formula:

[0179]

[0180] The expression of δ2 is:

[0181]

[0182] Establishment of the DE segment equation of the tooth top arc curve:

[0183]

[0184] In the formula, δ3∈[0,β-(θ αa -θ α )],and

[0185] θ α =tanα-α.

[0186] Among them, Z is the number of teeth of the gear, α a is the tooth tip circle pressure angle, and α is the pitch circle pressure angle.

[0187] S2: Solve the meshing stiffness of different meshing positions based on elastic mechanics and Weber energy method, and use ANSYS for analysis and verification ( Figure 7 shown);

[0188] The total load deformation of the gear teeth q Tj The contact point consists of the following three parts: 1) Bending, shearing and axial compression deformation of the involute part of the gear tooth, which is regarded as a cantilever beam Bj ; 2) Transition fillet and base deformation q connected to the tooth Fj ; 3) Local contact deformation q Cj .

[0189] Assuming the gear is a non-uniform cantilever beam, the equivalent length of the cantilever beam is L e is the distance from the base point M to the top circle. Let the thickness of the infinitesimal element i be δ i The cross-sectional area is The moment of inertia is L ij represents the distance between the microelement i and the load point j in the X-axis direction, β j It is W j The angle between the Y axis and j is the normal meshing force at point j, as Figure 5 shown.

[0190] Bending, shearing and axial compression deformation of involute gear parts:

[0191] For the involute tooth profile part regarded as an equivalent cantilever beam, its deformation is mainly divided into three parts: F x Compression deformation y1 under the action of F y and bending deformation y2 under the action of M′; F y Shear deformation y3 under the action.

[0192] The component of force along the x-axis:

[0193] Fx =W j sinβ j

[0194] The component of force along the y-axis:

[0195] F y =W j cosβ j

[0196] Equivalent bending moment:

[0197]

[0198] The deformations of the three parts are superimposed to obtain the total deformation of a single microelement:

[0199]

[0200] In the formula, This value can improve the calculation accuracy of the small unit i; h j is the tooth thickness of the gear at the meshing point j; μ is the Poisson's ratio; E is the elastic modulus; β j is the angle between the load and the Y axis. j and β j The specific calculation is given as follows.

[0201] h j =2r j [h p / (2r p )+inv(α)-inv(θ j )]

[0202] Among them, θ j =arcos(r b / r j );r p is the pitch circle radius, h p is the pitch circle tooth thickness, and the involute function is expressed as: inv(f)=tan(f)-f.

[0203] The angle between the load direction and the X-axis is

[0204] β j =θ j –ψ j

[0205] Among them, ψ j =h p / (2r p )+inv(α)-inv(θ j )

[0206] The total deformation of the involute part of the gear tooth at the meshing point j is

[0207]

[0208] Transition fillets and base deformation:

[0209] The calculation formula of the infinitesimal deformation of the transition fillet is consistent with that of the involute part of the gear, so the superposition formula is:

[0210]

[0211] Cornell's study on the deformation of the substrate used O'Donnell's derived formula:

[0212] For plane stress:

[0213]

[0214] For plane strain:

[0215]

[0216] In the formula,

[0217] L f =X j -X M -Y j tanβ j ,h f =2Y M

[0218] The total deformation of the transition curve part and the base part is

[0219] q Fj =q FBj +q FFj

[0220] Local contact deformation:

[0221] For the local contact deformation of gears, Hertz theory is usually used. The present invention adopts the method given by Weber, that is,

[0222]

[0223] Where b is the width of the semi-contact area, and is the height of the contact deformation zone, μ1 and μ2 are the Poisson's ratios of the corresponding gears, and E1 and E2 are the elastic moduli of the corresponding gears.

[0224]

[0225] In the formula, R c1 and R c2is the equivalent radius of the Hertz contact, and its calculation formula is

[0226]

[0227] In the formula, R p1 and R p2 is the pitch circle radius, R b1 is the base circle radius of gear 1, θ is the pitch circle pressure angle, and r1 is the radius corresponding to the pressure angle of gear 1 at the meshing point.

[0228] The total deformation of the meshing teeth at point j is:

[0229] q Tj =(q Bj1 +q Bj2 )+(q Fj1 +q Fj2 )+q Cj

[0230] In the formula, the subscripts 1 and 2 are the numbers of the driving wheel and the driven wheel, representing the deformation of the corresponding gear. Then the meshing stiffness of point j can be obtained:

[0231] K j =W j / q Tj

[0232] Given meshing stiffness calculation flow chart ( Figure 6 As shown), the meshing stiffness at different meshing points can be calculated.

[0233] S3: Determine the modification method. The present invention adopts the tooth top short modification and performs result analysis by changing the modification parameters.

[0234] The effect of modifying the tooth root of the gear is the same as that of modifying the tooth top of the paired gear, so the present invention adopts a solution of modifying the tooth top of both the large and small gears. The modification amount of the large and small gears is equal to the comprehensive deformation of the tooth pair at the highest point (HPSTC) and the lowest point (LPSTC) of the single tooth meshing area of ​​the small gear.

[0235] According to the modification length, the tooth profile modification can be divided into long modification and short modification.

[0236]

[0237] Where, L max It is the modification length corresponding to the highest point of single tooth meshing. When ΔL<1, it is called short modification, and when ΔL>1, it is called long modification.

[0238] The transmission of the involute spur gear needs to ensure the smoothness of the transmission, so the overlap should be as large as possible. For this reason, the present invention adopts a short modification.

[0239] In the research on tooth profile modification and optimal design based on the dynamic characteristics of a gear transmission system, the two most commonly used modification methods are linear modification and parabolic modification. The research of this invention considers three modification types: linear modification, conic curve modification, and exponential modification, and expresses them in a general form.

[0240]

[0241] In the formula, a and b are coefficients related to the modification type, and they need to satisfy the modification amount boundary condition, that is, a + b = 1.

[0242] When a = 1 and b = 0, the modification type is linear modification; when 1 < n < 2 and a = 0, the modification type is exponential modification; when n = 2 and a ≠ 1, the modification type is conic curve modification, where when a = 0 and b = 1, the modification type is also called parabolic modification. Based on the study of the above modification parameters, considering under the condition of the same modification amount, using different modification curves or modification lengths, the influence of the meshing stiffness on the dynamic transmission error is studied, and through the comparative analysis of different values, the optimal modification curve and modification length are found.

[0243] S4: Establish the non-linear dynamic equation of the idler gear system ( Fig.10 as shown), substitute the obtained meshing stiffness into it for response analysis, and optimize the meshing stiffness based on the final dynamic response error to obtain a stable and reliable transmission structure;

[0244] Table 1 gives the basic gear parameters of the idler gear system and establishes a non-linear dynamic model

[0245]

[0246] Among them, M is the mass matrix, C is the damping matrix, and K is the time-varying meshing stiffness matrix. Expand the meshing stiffness in Fourier series according to the results of step S2, and take the first three terms, that is

[0247]

[0248] In the formula, i = 1, 2. The parameter Ω m is the meshing fundamental frequency of the gear pair, and

[0249] Ω m = 2πn1z1 / 60

[0250] k im a (i) mj b (i) mj are the coefficients of the Fourier expansion respectively, n1 is the gear rotation speed, and z1 is the number of teeth of gear 1.

[0251] Substitute the expanded meshing stiffness into the dynamic equation, write program code, combine the shaping parameters in step S3, perform MATLAB calculation simulation on the dynamic transmission error of the system, and obtain the optimal shaping parameters through continuous iterative optimization, thereby obtaining the required optimal meshing stiffness value.

[0252] Through accurate modeling of the involute tooth profile, the meshing stiffness is calculated using the Weber energy method, and the impact caused by the alternating meshing of single and double teeth is reduced by tooth profile modification, thereby reducing the transmission error. This technology is applied to the idler system, and the nonlinear dynamic equation of the idler is established to analyze the dynamic transmission error of the system response, and the meshing stiffness is iteratively optimized until the transmission accuracy and design strength requirements are met. The specific implementation steps are as follows:

[0253] Establishing an accurate involute model is a prerequisite for accurately calculating meshing stiffness. It is necessary to obtain the expression of the four-segment involute curve by using the mathematical theory of spatial coordinate transformation. The specific structure is referenced Figure 1 For the calculation of meshing stiffness, the present invention adopts two methods, MATALB and ANSYS finite element analysis, for comparison. Given the basic parameters of the gear, according to Figure 6 The calculation is performed according to the process shown.

[0254] Table 1 Basic parameters of meshing gear pairs

[0255]

[0256] The calculation result of the Japan Society of Mechanical Engineers based on the Ishikawa formula is 17.2N / (mm·μm); the approximate value of the ISO draft formula based on the Weber-Banaschek formula is 18.91N / (mm·μm); the average value of the ANSYS calculation result is 18.74N / (mm·μm); the average value obtained by the algorithm of the embodiment of the present invention is 19.78N / (mm·μm). The errors of the embodiment of the present invention and other methods are 13.04%, 4.4%, and 5.26% respectively; because the Ishikawa formula regards the gear as a combination of a trapezoid and a rectangle, its tooth thickness is smaller than that of the accurate modeling, the calculated deformation is larger, and the stiffness value is smaller, so the error is larger. The other two methods have smaller errors than the method used in this article, so the stiffness calculation is more accurate.

[0257] The embodiment of the present invention adopts a short shaping method for shaping the tooth tops of large and small gears. The shaping curve adopts three methods: linear shaping, quadratic curve shaping and exponential shaping. The value range of the shaping amount can be obtained by calculating the meshing stiffness. The embodiment of the present invention takes [40,45] (μm), and the shaping length is normalized. The highest point of the single tooth meshing area (HPSTC) to the tooth top is 1, and the shaping lengths of 0.9, 0.8, 0.7, and 0.6 are selected for analysis respectively. The calculation results are brought into the dynamic model. According to Fig.11 Dynamic design scheme, calculate the dynamic transmission error under different meshing stiffness ( Fig.12 As shown), considering the design strength problem, the dynamic transmission error under different torques is analyzed ( Fig.13 As shown). Through analysis, it can be seen that under the premise of ensuring the overlap, the meshing stiffness peak is small, which is beneficial to the stability of the system. Therefore, under the premise of ensuring that the gear has sufficient bending strength, the gear uses a standard gear or a gear pair with zero displacement. Through theoretical analysis, it can be seen that optimizing the meshing stiffness can control the unilateral impact and bilateral impact during the system transmission process, reduce vibration impact, and avoid excessive damage to the gear.

[0258] The above are all preferred embodiments of the present invention. For ordinary technicians in this technical field, without departing from the principle of the present invention, various equivalent modifications to the present invention belong to the protection scope of the claims attached to this application.

Claims

1. A method for optimizing meshing stiffness of involute spur gears based on dynamic transmission error modification, characterized in that: The method is as follows: S1: Establish an accurate model of the involute tooth profile: Take the center of the gear as the origin and the center line of the gear as the y-axis to establish the coordinate system XOY. The tooth profile curve is mainly composed of the tooth root arc line AB, the tooth root transition curve BC, the involute tooth profile curve CD, and the tooth top arc curve DE. Establish the tooth root arc line AB segment equation, the tooth root transition curve BC segment equation, the involute tooth profile CD segment equation, and the tooth top arc curve DE segment equation respectively to obtain the involute tooth profile model; S2: The meshing stiffness at different meshing positions is solved based on elastic mechanics and Weber energy method, and the engineering simulation software ANSYS is used for analysis and verification to obtain the meshing stiffness at different meshing points; S3: Determine the modification method, adopt the short tooth top modification, express the three modification types of linear modification, quadratic curve modification and exponential modification into a general form, and find the optimal modification parameters through comparative analysis of different values; S4: Establish the nonlinear dynamic equation of the idler system, expand the meshing stiffness in S2 by Fourier series and substitute it into the nonlinear dynamic equation for response analysis. According to the final dynamic response error, optimize the meshing stiffness and obtain a stable and reliable transmission structure.

2. The method for optimizing meshing stiffness of involute spur gears by modifying the shape based on dynamic transmission error according to claim 1, characterized in that: The equation for establishing the tooth root arc line AB segment of S1 is as follows: Among them, r f is the tooth root circle radius, δ1∈[b / r,π / z], b is the machining rack parameter, z is the number of teeth, and r is the pitch circle radius; The method to establish the BC segment equation of the tooth root transition curve is as follows: The part of the tool tooth profile forms an involute tooth profile, the part of the tool tooth profile is a straight line and forms the root circle of the gear, and the part of the tool tooth profile is an arc with a radius of r0, which forms the transition curve of the gear; The main parameters of tool tooth profile are as follows: Among them, m is the module, α is the pitch circle pressure angle, h a ＊ is the tooth addendum coefficient, c ＊ is the head clearance coefficient; A coordinate system is fixed to the rack tool and the processed gear respectively, and a fixed coordinate system X is given at the same time f O f Y f ; When the processing gear rotates After the angle, according to the coordinate transformation law, we can get the expression of point M in the coordinate system X2O2Y2; In the coordinate system X1O1Y1, the homogeneous coordinates of point M are: (x 1M ,y 1M ,z 1M ,1) T =(r0cosα′,-a1-r0sinα′,0,1) T When the rack tool performs machining motion, the coordinate system X1O1Y1 is relative to the fixed coordinate system X f O f Y f The transformation is as follows: The coordinate system X2O2Y2 of the gear revolves around z f Rotated Angle, its transformation matrix is: Then the transformation matrix of coordinate system X1O1Y1 relative to coordinate system X2O2Y2 is: The expression of point M in the coordinate system X2O2Y2 is: right 2 Further simplification of the M expression yields: In the above formula, α′∈[α′ min ,π / 2], when there is no root cutting in the gear, the tooth root transition curve is tangent to the tooth profile involute, at this time α′ min =α =π / 9; if there is undercut, α′ min Less than α, the value at this time is determined by the intersection H of the tooth profile involute equation and the tooth root transition curve equation; r is the gear machining pitch circle radius, a1 = a-χm, varies with α′, and Establish the CD segment equation of the involute tooth profile: The time-varying meshing point K of the involute is expressed by the following equation: OK=OP+PK in, PK=r b tana k [cos(π-β k )sin(π-β k )] Then the equation of point K is: in but Among them, r b is the base circle radius, γ=tanα k , α k is the pressure angle at the meshing point, and α k ∈[α H ,arcos(r b / r a )],α H is the pressure angle corresponding to the intersection point H of the transition curve and the involute. The coordinates of point H are given by the following formula: The expression of δ2 is: Establishment of the DE segment equation of the tooth top arc curve: In the formula, and i α =tanα-α; Among them, Z is the number of teeth of the gear, α a is the tooth tip circle pressure angle, and α is the pitch circle pressure angle.

3. The method for optimizing meshing stiffness of involute spur gears by modifying the shape based on dynamic transmission error according to claim 1 or 2, characterized in that: The gear meshing process is the process of change from tooth root to tooth top. Different meshing positions refer to the process from the starting point to the end point of meshing. Each meshing position contains three parts of deformation. The S2 comprises the following steps: The total load deformation of the gear teeth q Tj The contact point consists of the following three parts: 1) Bending, shearing and axial compression deformation of the involute part of the gear tooth, which is regarded as a cantilever beam Bj ; 2) Transition fillet and base deformation q connected to the tooth Fj ; 3) Local contact deformation q C ; Assuming the gear is a non-uniform cantilever beam, the equivalent length of the cantilever beam is L e is the distance from the base point M to the top circle; let the thickness of the infinitesimal element i be δ i The cross-sectional area is The moment of inertia is L ij represents the distance between the microelement i and the load point j in the X-axis direction, β j It is W j The angle between the Y axis and j is the normal meshing force at point j; 1) Bending, shearing and axial compression deformation of the involute gear part: For the involute tooth profile part regarded as an equivalent cantilever beam, its deformation is divided into three parts: F x The compression deformation y1 under the action of F y and bending deformation y2 under the action of M′; F y Shear deformation y3 under action; The component of force along the x-axis: F x =W j sinβ j The component of force along the y-axis: F y =W j cosβ j Equivalent bending moment: The deformations of the three parts are superimposed to obtain the total deformation of a single microelement: In the formula, The value of h can improve the calculation accuracy of small unit i; j is the tooth thickness of the gear at the meshing point j; μ is the Poisson's ratio; E is the elastic modulus; β j is the angle between the load and the Y axis, h j and β j The specific calculation is given as follows; h j =2r j [h p / (2r p )+inv(α)-inv(θ j )] Among them, θ j =arcos(r b / r j );r p is the pitch circle radius, h p is the pitch circle tooth thickness, and the involute function is expressed as: inv(f) = tan(f)-f; The angle between the load direction and the X-axis is b j =θ j –ψ j among them, ps j =h p / (2r p )+inv(α)-inv(θ j ) The total deformation of the involute part of the gear tooth at the meshing point j is 2) Transition fillet and substrate deformation: The calculation formula of the infinitesimal deformation of the transition fillet is consistent with that of the involute part of the gear, so the superposition formula is: For the plane stress formula: For the plane strain formulation: In the formula, L f =X j -X M -Y j andβ j h f [2Y M The total deformation of the transition curve part and the base part is q Fj =q FBj +q FFj 3) Local contact deformation: For the local contact deformation of gears, the Weber method is used, that is, Where b is the width of the semi-contact area, and is the height of the contact deformation zone, μ1 and μ2 are the Poisson's ratios of the corresponding gears, and E1 and E2 are the elastic moduli of the corresponding gears; In the formula, R c1 and R c2 is the equivalent radius of the Hertz contact, and its calculation formula is In the formula, R p1 and R p2 is the pitch circle radius, R b1 is the base circle radius of gear 1, θ is the pitch circle pressure angle, and r1 is the radius corresponding to the pressure angle of gear 1 at the meshing point; The total deformation of the meshing teeth at point j is: q Tj =(q Bj1 +q Bj2 )+(q Fj1 +q Fj2 )+q Cj In the formula, subscripts 1 and 2 are the numbers of the driving wheel and the driven wheel, and the meshing stiffness of point j can be obtained: K j =W j / q Tj Based on the meshing stiffness calculation method, the calculation of the meshing stiffness at different meshing points is realized.

4. The method for optimizing meshing stiffness of involute spur gears by modifying the shape based on dynamic transmission error according to claim 1, characterized in that: In S3, tooth tip modification is performed on both the large and small gears, and the modification amounts of the large and small gears are respectively equal to the comprehensive deformation amounts of the tooth pair at the highest point HPSTC and the lowest point LPSTC in the single tooth meshing area of the small gear; According to the modification length, the tooth profile modification can be divided into long modification and short modification Where, L max It is the modification length corresponding to the highest point of single tooth meshing. When ΔL<1, it is called short modification, and when ΔL>1, it is called long modification.

5. The method for optimizing meshing stiffness of involute spur gears by modifying the shape based on dynamic transmission error according to claim 1 or 4, characterized in that: The three modification types of linear modification, quadratic curve modification and exponential modification are expressed in the following general form: In the formula, a and b are coefficients related to the modification type, and they need to satisfy the modification amount boundary condition, that is, a + b = 1; When a = 1 and b = 0, the modification type is linear modification. When 1 < n < 2 and a = 0, the modification type is exponential modification; when n = 2 and a ≠ 1, the modification type is quadratic curve modification. Among them, when a = 0 and b = 1, the modification type is also called parabola modification; Considering that under the condition of the same modification amount, different modification curves or modification lengths are adopted to study the influence of the meshing stiffness on the dynamic transmission error. Through the comparative analysis of different values, the optimal modification curve and modification length are found.

6. The method for optimizing meshing stiffness of involute spur gears by modifying the shape based on dynamic transmission error according to claim 1, characterized in that: The non-linear dynamic model in S4 is as follows: Where, M is the mass matrix, C is the damping matrix, and K is the time-varying meshing stiffness matrix; According to the result calculated in step S2, the meshing stiffness is expanded by Fourier series, and the first three terms are taken, that is In the formula, i = 1, 2, parameter Ω m is the meshing base ratio of the gear pair, and Oh m =2πn1z1 / 60 k im , a (i) mj , b (i) mj are the coefficients of Fourier expansion respectively, n1 is the gear speed, z1 is the number of teeth of gear 1; Substitute the expanded meshing stiffness into the dynamic equation, combine the modification parameters in step S3, and perform MATLAB calculation and simulation on the dynamic transmission error of the system. The optimal modification parameters are obtained through continuous iterative optimization, so as to obtain the required optimal meshing stiffness value.

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