Crack spiral bevel gear meshing rigidity calculation method based on meshing theory

Abstract

The invention discloses a method for calculating the meshing rigidity of a crack spiral bevel gear based on a meshing theory. The method comprises the following steps: step (1), establishing a spiral bevel gear space complex tooth surface; (2) analyzing the meshing contact process of the spiral bevel gear; step (3), constructing an analytic calculation model of the time-varying meshing stiffness of the spiral bevel gear in the healthy state; step (4), constructing an analytical calculation model of the time-varying meshing stiffness of the spiral bevel gear in the crack fault state; the method has the beneficial effects that the time-varying meshing stiffness analytical calculation model of the spiral bevel gear in healthy and crack states is constructed, and the technical blank of the time-varying meshing stiffness calculation method of the spiral bevel gear considered from the perspective of meshing theory at present is filled; and an important theoretical basis is provided for dynamic characteristic analysis and fault diagnosis of the gear transmission system.

Description

Technical Field

[0001] The invention relates to the technical field of gear transmission, and in particular to a method for calculating the meshing stiffness of a cracked spiral bevel gear based on meshing theory. Background Art

[0002] Spiral bevel gears are key transmission components in the aerospace industry, and their performance stability and reliability have a direct and significant impact on aircraft operation. However, conventional mesh stiffness calculation methods rely on simplified assumptions and ignore the effects of complex gear geometry and defects such as cracks on stiffness during actual operation. These factors can lead to complex dynamic behavior during gear operation, such as abnormal vibration, noise, and impact, which in turn can affect aircraft stability and safety. Therefore, developing more accurate stiffness calculation methods that consider various factors in actual operation is crucial for improving the performance and reliability of gear systems.

[0003] To address these issues, the present invention proposes a meshing theory-based method for calculating the mesh stiffness of cracked spiral bevel gears. This method, based on the geometric design principles and complex meshing mechanisms of spiral bevel gears, combines the slicing method with the potential energy method to successfully establish an analytical calculation model for the time-varying mesh stiffness of spiral bevel gears in both healthy and cracked fault states. This invention not only fills the technical gap in methods for calculating the time-varying mesh stiffness of spiral bevel gears, but also has significant academic and engineering application value in promoting the development of related engineering technologies, making a positive contribution to technological innovation and progress in the field of gear transmission. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill a gap in related art, this present invention provides a method for calculating the mesh stiffness of cracked spiral bevel gears based on meshing theory. This method, based on the geometric design principles, machining methods, and meshing mechanism of spiral bevel gears, derives gear tooth surface machining equations and accurately establishes the real-space curved tooth surface, laying a theoretical foundation for accurate analysis of the gear meshing transmission process. Furthermore, a tooth contact analysis method is employed to conduct a detailed analysis of the contact trajectory, contact ellipse, and transmission error curve during the gear transmission meshing process. Based on this, a slicing method and a potential energy method are combined to establish analytical calculation models for the time-varying mesh stiffness of spiral bevel gears in healthy and cracked fault states, respectively. The present invention also explores the effects of different crack lengths and the angle between the crack and the centerline on the mesh stiffness, enabling accurate and effective analysis of the time-varying mesh stiffness of spiral bevel gears in both healthy and cracked states. This method fills a gap in the technical field for calculating the time-varying mesh stiffness of spiral bevel gears, promotes the development of related engineering technologies, and possesses significant academic and engineering application value.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a method for calculating the meshing stiffness of a cracked spiral bevel gear based on meshing theory, characterized by comprising the following steps:

[0006] Step (1): Establish the spatial complex tooth surface of the spiral bevel gear;

[0007] The gear tooth surface machining equation considers two parts: the meshing equation of the cutter head surface is derived in the coordinate system, and the tooth surface ∑2 (a) Transition surface with tooth root∑2 (b) Working part; tooth surface ∑2 (a) It is expressed as the following equation:

[0008]

[0009] Where, is the vector function of the working part of the tooth surface in three-parameter form, s g is the cone coordinate of the active spiral bevel gear, θ g is the rotation angle, ψ2 is the general parameter of motion, and the matrix M 2g (ψ2) represents the g The coordinate transformation matrix to the moving coordinate system s2, is a vector function of the working part of the tooth surface in a two-parameter form, is the parametric equation of the working part of the tooth surface;

[0010] By considering the above equation, the tooth surface ∑2 (a) Originally expressed in three-parameter form, it can be simplified to a two-parameter form ∑2 (a) , expressed as the following equation:

[0011]

[0012] Where, is a two-parameter vector function of the working part of the tooth surface, θ2 and ψ2 are general parameters of motion, is the vector function of the working part of the tooth surface in three-parameter form, s g is the cone coordinate of the active spiral bevel gear, θ g is the rotation angle;

[0013] Considering the tooth root transition surface ∑2 (b) , expressed as the following equation:

[0014]

[0015] Where, is the three-parameter vector function of the working part of the tooth root transition surface, λ w is the tooth root transition surface coordinate, θ2 and ψ2 are general parameters of motion, and the matrix M 2g(ψ2) represents the g To the coordinate transformation matrix of S2, is a vector function of the working part of the tooth root transition surface in a two-parameter form, is the parametric equation of the working part of the tooth surface, so the tooth root transition surface ∑2 (b) The two-parameter form of is expressed as the following equation:

[0016]

[0017] Where, is a two-parameter vector function of the working part of the tooth root transition surface, θ2 and ψ2 are general parameters of motion, is the three-parameter vector function of the working part of the tooth root transition surface, λ w is the coordinate of tooth root transition surface;

[0018] Step (2): Analyze the meshing contact process of spiral bevel gears;

[0019] During the meshing process, the driving spiral bevel gear and the driven spiral bevel gear must be continuously tangent, that is, the position vector and the normal vector at any instant must coincide with each other; the equations of the tooth surfaces of the driving spiral bevel gear and the driven spiral bevel gear in the coordinate system are as follows:

[0020]

[0021] Where, is the tooth surface equation of the active spiral bevel gear, s p is the surface coordinate of the active spiral bevel gear, θ p is the rotation angle of the active spiral bevel gear, ψ1 is the rotation angle of the active spiral bevel gear during processing, φ1 is the rotation angle of the active spiral bevel gear, M hb1 M b11 is the coordinate change matrix of the active spiral bevel gear, r1(s p ,θ p ,ψ1) is the three-parameter form of the active spiral bevel gear tooth surface equation, s g is the surface coordinate of the driven spiral bevel gear, θ g is the rotation angle of the driven spiral bevel gear, is the tooth surface equation of the driven spiral bevel gear, ψ2 is the rotation angle of the driven spiral bevel gear during machining, φ2 is the rotation angle of the driven spiral bevel gear, M hb2 M b22 is the coordinate change matrix of the driven spiral bevel gear; r2(s g ,θ g ,ψ2) is the three-parameter form of the driven spiral bevel gear tooth surface equation, equation f 1p (s p ,θ p ,ψ1)=0 and r1(sp ,θ p ,ψ1) uses three coefficient variables to represent the tooth surface of the active spiral bevel gear in the S1 coordinate system; equation f 2g (s g ,θ g ,ψ2)=0 and r2(s g ,θ g ,ψ2) Use three coefficient variables to represent the tooth surface of the driven spiral bevel gear in the S2 coordinate system;

[0022] Position normal vector of driving spiral bevel gear and driven spiral bevel gear The equation in the coordinate system is as follows:

[0023]

[0024] Where n1(s p ,θ p ,ψ1) is the position normal vector equation of the three-parameter form of the active spiral bevel gear, n2(s g ,θ g ,ψ2) is the position normal vector equation of the driven spiral bevel gear in three-parameter form, L h1 、L b22 The matrix representation of the rotation angles φ1 and φ2 of the active spiral bevel gear and the driven spiral bevel gear is as follows:

[0025]

[0026] The equations satisfying the continuous tangency between the driving spiral bevel gear and the driven spiral bevel gear are as follows:

[0027]

[0028] In the formula, the parameter value of the rotation angle φ1 of the active spiral bevel gear is within the range of -π / N1<φ1<π / N1, the tooth surface equations ∑1 and ∑2 are expressed in the coordinate system by three parameter variables, and the equation f 1p (s p ,θ p ,ψ1),f 2g (s g ,θ g ,ψ2) is the meshing equation of tooth surface ∑1 and ∑2; tooth surface vector function Function of tooth surface normal vector The relationship indicates that the position vectors of the tooth surface equations ∑1 and ∑2 at their tangent points coincide with the unit normal vector of the tooth surface;

[0029] The calculation equation of the transmission error function during gear meshing is as follows:

[0030] Δφ2(φ1)=φ2(φ1)-N1 / N2φ1;

[0031] Where φ1 and φ2 are the rotation angles of the driving spiral bevel gear and the driven spiral bevel gear, respectively; N1 and N2 are the surface normal vectors of the driving spiral bevel gear and the driven spiral bevel gear, respectively; Δφ2(φ1) represents the transmission error function variation relationship between the driving spiral bevel gear and the driven spiral bevel gear;

[0032] Step (3): Establish a time-varying meshing stiffness analytical calculation model for healthy spiral bevel gears, analyze the meshing force process of the gear pair, F m is the meshing force, F m Along the spatial coordinate system, it is decomposed into the tangential force F t , radial force F r With axial force F a , the equations for these forces are as follows:

[0033]

[0034] Where β is the helix angle, α is the pressure angle, and δ G is the pitch circle pressure angle;

[0035] Tangential force F t When the gears rotate, the relationship between the meshing force and the load torque T is as follows:

[0036] F m =F t / cos(α)cos(β)=T / r p cos(α)cos(β);

[0037] Where r p The radial distance from the meshing point to the rotation axis during meshing; the gear potential energy mainly consists of two parts: the energy at the tooth root transition curve and the energy from the point of action of the contact force on the tooth surface to the base circle;

[0038] The parameter solution method in the stiffness analysis process of spiral bevel gears is as follows:

[0039]

[0040] Where x k is the distance between any point on the tooth surface and the tooth root, r kb is the base circle radius of the sliced gear, r kf is the tooth root radius, α k Represents the pressure angle at any point on the tooth surface, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α k2 is the base circle half tooth angle, α k3 The angle between the tooth profile at the root circle and the horizontal axis; hk 、h kx are the distances from the meshing contact point and any point on the tooth surface to the center line of the sliced gear, d k is the distance between the contact point and the tooth root, d k1 is the distance between the root circle and the tooth root, d k2 Indicates the distance from the contact point on the tooth surface to the tooth root;

[0041] The equations for calculating the contact area and area moment of inertia of healthy gear teeth are as follows:

[0042]

[0043] Where A xk , I xk A is the contact area and area moment of inertia at the tooth root circle when the action point is located on the involute part; dk1 , I dk1 is the contact area and area moment of inertia at the base circle when the action point is located on the involute part; Δw is the tooth width of each micro segment;

[0044] Under the action of meshing contact force, each slice gear can be regarded as a cantilever beam, and the axial tensile and compressive stiffness k ka , shear stiffness k ks , bending stiffness k kb , Hertz contact stiffness k kh and the gear flexible matrix stiffness k kf The calculation equation is as follows:

[0045]

[0046] Where G is the shear modulus, G = E / 2(1+ν), E is the Young's modulus, ν is the Poisson's ratio, δ kf is the flexibility coefficient, x k is the distance between any point on the tooth surface and the tooth root, r kb is the base circle radius of the sliced gear, α k Represents the pressure angle at any point on the tooth surface, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α k2 is the base circle half tooth angle; h k is the distance from the meshing contact point to the center line of the sliced gear, d k is the distance between the contact point and the tooth root, d k1 is the distance between the root circle and the tooth root; the meshing stiffness k for the kth slice kt The calculation equation is as follows:

[0047]

[0048] Where, subscripts 1 and 2 represent the driving and driven spiral bevel gears, respectively. The calculation equation for the comprehensive meshing stiffness of the gears as they rotate is as follows:

[0049]

[0050] In the formula, the meshing stiffness of a pair of spiral bevel gears is the superposition of the meshing stiffness of n slices at each meshing moment; n is the total number of slices at the meshing moment, m is the number of instantaneous meshing gear pairs, so k single is the single tooth meshing stiffness, k total k is the comprehensive meshing stiffness of the gear pair when it is meshing. kt is the meshing stiffness of the nth slice at each moment;

[0051] Step (4): Establish an analytical calculation model for the time-varying meshing stiffness of spiral bevel gears in a crack fault state; here, the shear and bending stiffnesses under two different conditions are considered. The first condition is that the crack extension direction does not exceed the gear centerline, and the second condition is that the crack extension direction exceeds the gear centerline.

[0052] Case 1: α k1 ≥α ka , h ka ≥h ko

[0053] α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α ka is the angle between the crack direction and the center line, h ka h is the straight-line distance between the extension end point A of the crack model and the gear centerline, ko is the thickness of half of the tooth at the tooth top; based on the transmission process and geometric relationship during meshing, the following relationship exists:

[0054]

[0055] Where, α kr is the angle between the crack termination point on the k-th crack sheet and the tooth root circle, q k1 is the crack depth of the two parts of the outer surface of the k-th crack sheet, v k is the angle between the center line and the crack direction in the k-th layer model, x kl is the distance between the force point on the k-th tooth surface and the tooth root, γ k is the angle between the midpoint of the crack on the k-th crack sheet and the tooth root circle;

[0056] The calculation equations for the contact area and area moment of inertia between the tooth root circle and the gear base circle when there is a crack are as follows:

[0057]

[0058] Where Akaγ , I kaγ are the contact area and area moment of inertia at the tooth root circle, A dkal , I dkal are the contact area and area moment of inertia at the base circle, respectively;

[0059] The calculation equations for the contact area between the point of action of the contact meshing force and the tooth root circle section and its area moment of inertia are as follows:

[0060] A xka =(h kx +h ka )Δw,I xka =1 / 12(h kx +h ka ) 3 Δw;

[0061] Where A xka , I xka are the contact area and area moment of inertia of the cross section from the action point to the tooth root;

[0062] Substituting the parameter relationship after the crack is affected into the calculation formula of the gear stiffness, the shear and bending stiffness when the crack exists are derived, and then the crack stiffness under this condition is solved;

[0063] Case 2: α k1 <α kc , h kc <h ko or h kc ≥h ko

[0064] The length of the crack extension model exceeds the length of the tooth centerline, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α kc h is the pressure angle on the tooth surface when the length of the crack extension model exceeds the length of the tooth centerline, kc h is the distance between the end point C of the crack model and the center line of the sliced gear, ko is the thickness of the half-tooth at the tooth top. By analyzing the transmission process and geometric relationship during meshing, the following relationship is obtained:

[0065]

[0066] Where q k2 is the crack depth of the outer surface of the k-th crack sheet, α kr is the angle between the crack termination point on the k-th crack sheet and the tooth root circle. After deduction, A dkc1 , I dkc1 are the contact area and area moment of inertia of the gear base circle, A kcγ , I kcγare the contact area and area moment of inertia of the tooth root circle, respectively. The calculation equations are as follows:

[0067]

[0068] Where A xkc , I xkc The calculation equations for the contact area and area moment of inertia between the contact meshing force action point and the tooth root circle section are:

[0069] A xkc =(h kx -h kc )Δw,I xkc =1 / 12(h kx -h kc ) 3 Δw;

[0070] Through the above analysis, an analytical calculation model for the time-varying mesh stiffness of spiral bevel gears under crack fault conditions was established. Two situations were considered, one of which was whether the crack extension direction exceeded the gear centerline. The calculation methods for the shear and bending stiffness when cracks existed were derived in detail.

[0071] Compared with the existing technology, the beneficial effects of the present invention are as follows: on the basis of fully considering the complex geometric shape of spiral bevel gears and defects such as cracks, its real space curved tooth surface is accurately established, and the gear transmission meshing process is analyzed in detail by using the gear tooth contact analysis method, covering multiple aspects such as contact trajectory, contact ellipse and transmission error curve. Furthermore, combining the slicing method and the potential energy method, analytical calculation models for the time-varying meshing stiffness of spiral bevel gears in healthy state and crack fault state are established respectively. Through this model, the influence of different crack lengths and the angle between the crack and the center line on the meshing stiffness is deeply explored. The present invention not only fills the technical gap in the calculation method of the time-varying meshing stiffness of spiral bevel gears, but also provides strong theoretical support and reference basis for the vibration reduction, noise reduction, design optimization and performance improvement of spiral bevel gears, which has effectively promoted the development of gear transmission technology in the fields of aerospace and so on, and has significant academic value and engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 This is a flow chart of the calculation method of meshing stiffness of cracked spiral bevel gears based on meshing theory;

[0073] Figure 2 It is the force analysis model of the kth slice of a healthy spiral bevel gear;

[0074] Figure 3 It is the stress analysis model of the kth slice of the cracked spiral bevel gear. DETAILED DESCRIPTION

[0075] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1 — Figure 3 The specific embodiments of the present invention are described in detail.

[0076] Figure 1 This is a flow chart of the method for calculating the meshing stiffness of cracked spiral bevel gears based on meshing theory, which includes the following steps:

[0077] Step (1): Establish the spatial complex tooth surface of the spiral bevel gear;

[0078] The gear tooth surface machining equation considers two parts: the meshing equation of the cutter head surface is derived in the coordinate system, and the tooth surface ∑2 (a) Transition surface with tooth root ∑2 (b) Working part; tooth surface ∑2 (a) It is expressed as the following equation:

[0079]

[0080] Where, is a vector function of the working part of the tooth surface in three-parameter form, s g is the cone coordinate of the active spiral bevel gear, θ g is the rotation angle, ψ2 is the general parameter of motion, and the matrix M 2g (ψ2) represents the g The coordinate transformation matrix to the moving coordinate system s2, is a vector function of the working part of the tooth surface in a two-parameter form, is the parametric equation of the working part of the tooth surface;

[0081] By considering the above equation, the tooth surface ∑2 (a) Originally expressed in three-parameter form, it can be simplified to a two-parameter form ∑2 (a) , expressed as the following equation:

[0082]

[0083] Where, is a two-parameter vector function of the working part of the tooth surface, θ2 and ψ2 are general parameters of motion, is a vector function of the working part of the tooth surface in three-parameter form, s g is the cone coordinate of the active spiral bevel gear, θ g is the rotation angle;

[0084] Considering the tooth root transition surface ∑2 (b) , expressed as the following equation:

[0085]

[0086] Where, is the three-parameter vector function of the working part of the tooth root transition surface, λ w is the tooth root transition surface coordinate, θ2 and ψ2 are general parameters of motion, and the matrix M 2g (ψ2) represents the g To the coordinate transformation matrix of S2, is a vector function of the working part of the tooth root transition surface in a two-parameter form, is the parametric equation of the working part of the tooth surface, so the tooth root transition surface ∑2 (b) The two-parameter form of is expressed as the following equation:

[0087]

[0088] Where, is a two-parameter vector function of the working part of the tooth root transition surface, θ2 and ψ2 are general parameters of motion, is the three-parameter vector function of the working part of the tooth root transition surface, λ w is the coordinate of tooth root transition surface;

[0089] Step (2): Analyze the meshing contact process of spiral bevel gears;

[0090] During the meshing process, the driving spiral bevel gear and the driven spiral bevel gear must be continuously tangent, that is, the position vector and the normal vector at any instant must coincide with each other; the equations of the tooth surfaces of the driving spiral bevel gear and the driven spiral bevel gear in the coordinate system are as follows:

[0091]

[0092] Where, is the tooth surface equation of the active spiral bevel gear, s p is the surface coordinate of the active spiral bevel gear, θ p is the rotation angle of the active spiral bevel gear, ψ1 is the rotation angle of the active spiral bevel gear during processing, φ1 is the rotation angle of the active spiral bevel gear, M hb1 M b11 is the coordinate change matrix of the active spiral bevel gear, r1(s p ,θ p ,ψ1) is the three-parameter form of the active spiral bevel gear tooth surface equation, s g is the surface coordinate of the driven spiral bevel gear, θ g is the rotation angle of the driven spiral bevel gear, is the tooth surface equation of the driven spiral bevel gear, ψ2 is the rotation angle of the driven spiral bevel gear during machining, φ2 is the rotation angle of the driven spiral bevel gear, M hb2 M b22 is the coordinate change matrix of the driven spiral bevel gear; r2(s g ,θ g,ψ2) is the three-parameter form of the driven spiral bevel gear tooth surface equation, equation f 1p (s p ,θ p ,ψ1)=0 and r1(s p ,θ p ,ψ1) uses three coefficient variables to represent the tooth surface of the active spiral bevel gear in the S1 coordinate system; equation f 2g (s g ,θ g ,ψ2)=0 and r2(s g ,θ g ,ψ2) Use three coefficient variables to represent the tooth surface of the driven spiral bevel gear in the S2 coordinate system;

[0093] Position normal vector of driving spiral bevel gear and driven spiral bevel gear The equation in the coordinate system is as follows:

[0094]

[0095] Where n1(s p ,θ p ,ψ1) is the position normal vector equation of the three-parameter form of the active spiral bevel gear, n2(s g ,θ g ,ψ2) is the position normal vector equation of the driven spiral bevel gear in three-parameter form, L h1 、L b22 The matrix representation of the rotation angles φ1 and φ2 of the active spiral bevel gear and the driven spiral bevel gear is as follows:

[0096]

[0097] The equations satisfying the continuous tangency between the driving spiral bevel gear and the driven spiral bevel gear are as follows:

[0098]

[0099] In the formula, the parameter value of the rotation angle φ1 of the active spiral bevel gear is within the range of -π / N1<φ1<π / N1, the tooth surface equations ∑1 and ∑2 are expressed in the coordinate system by three parameter variables, and the equation f 1p (s p ,θ p ,ψ1),f 2g (s g ,θ g ,ψ2) is the meshing equation of tooth surface ∑1 and ∑2; tooth surface vector function Function of tooth surface normal vector The relationship indicates that the position vectors of the tooth surface equations ∑1 and ∑2 at their tangent points coincide with the unit normal vector of the tooth surface;

[0100] The calculation equation of the transmission error function during gear meshing is as follows:

[0101] Δφ2(φ1)=φ2(φ1)-N1 / N2φ1;

[0102] Where φ1 and φ2 are the rotation angles of the driving spiral bevel gear and the driven spiral bevel gear, respectively; N1 and N2 are the surface normal vectors of the driving spiral bevel gear and the driven spiral bevel gear, respectively; Δφ2(φ1) represents the transmission error function variation relationship between the driving spiral bevel gear and the driven spiral bevel gear;

[0103] Step (3): Establish an analytical calculation model for the time-varying meshing stiffness of spiral bevel gears in a healthy state and analyze the meshing force process of the gear pair, such as Figure 2 The figure shows the force analysis model of the kth slice of a healthy spiral bevel gear; F m is the meshing force, F m Along the spatial coordinate system, it is decomposed into the tangential force F t , radial force F r With axial force F a , the equations for these forces are as follows:

[0104]

[0105] Where β is the helix angle, α is the pressure angle, and δ G is the pitch circle pressure angle;

[0106] Tangential force F t When the gears rotate, the relationship between the meshing force and the load torque T is as follows:

[0107] F m =F t / cos(α)cos(β)=T / r p cos(α)cos(β);

[0108] Where r p The radial distance from the meshing point to the rotation axis during meshing; the gear potential energy mainly consists of two parts: the energy at the tooth root transition curve and the energy from the point of action of the contact force on the tooth surface to the base circle;

[0109] The parameter solution method in the stiffness analysis process of spiral bevel gears is as follows:

[0110]

[0111] Where x k is the distance between any point on the tooth surface and the tooth root, r kb is the base circle radius of the sliced gear, r kf is the tooth root radius, αk Represents the pressure angle at any point on the tooth surface, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α k2 is the base circle half tooth angle, α k3 The angle between the tooth profile at the root circle and the horizontal axis; h k 、h kx are the distances from the meshing contact point and any point on the tooth surface to the center line of the sliced gear, d k is the distance between the contact point and the tooth root, d k1 is the distance between the root circle and the tooth root, d k2 Indicates the distance from the contact point on the tooth surface to the tooth root;

[0112] The equations for calculating the contact area and area moment of inertia of healthy gear teeth are as follows:

[0113]

[0114] Where A xk , I xk A is the contact area and area moment of inertia at the tooth root circle when the action point is located on the involute part; dk1 , I dk1 is the contact area and area moment of inertia at the base circle when the action point is located on the involute part; Δw is the tooth width of each micro segment;

[0115] Under the action of meshing contact force, each slice gear can be regarded as a cantilever beam, and the axial tensile and compressive stiffness k ka , shear stiffness k ks , bending stiffness k kb , Hertz contact stiffness k kh and the gear flexible matrix stiffness k kf The calculation equation is as follows:

[0116]

[0117] Where G is the shear modulus, G = E / 2(1+ν), E is the Young's modulus, ν is the Poisson's ratio, δ kf is the flexibility coefficient, x k is the distance between any point on the tooth surface and the tooth root, r kb is the base circle radius of the sliced gear, α k Represents the pressure angle at any point on the tooth surface, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α k2 is the base circle half tooth angle; h k is the distance from the meshing contact point to the center line of the sliced gear, d k is the distance between the contact point and the tooth root, d k1is the distance between the root circle and the tooth root; the meshing stiffness k for the kth slice kt The calculation equation is as follows:

[0118]

[0119] Where, subscripts 1 and 2 represent the driving and driven spiral bevel gears, respectively. The calculation equation for the comprehensive meshing stiffness of the gears as they rotate is as follows:

[0120]

[0121] In the formula, the meshing stiffness of a pair of spiral bevel gears is the superposition of the meshing stiffness of n slices at each meshing moment; n is the total number of slices at the meshing moment, m is the number of instantaneous meshing gear pairs, so k single is the single tooth meshing stiffness, k total k is the comprehensive meshing stiffness of the gear pair when it is meshing. kt is the meshing stiffness of the nth slice at each moment;

[0122] Step (4): Establish an analytical calculation model for the time-varying meshing stiffness of spiral bevel gears in a crack fault state; Figure 3 The figure shows the stress analysis model for the kth slice of a cracked spiral bevel gear. Here, the shear and bending stiffnesses are considered under two different conditions: the first case where the crack extension direction does not exceed the gear centerline, and the second case where the crack extension direction exceeds the gear centerline.

[0123] Case 1: α k1 ≥α ka , h ka ≥h ko

[0124] α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α ka is the angle between the crack direction and the center line, h ka h is the straight-line distance between the extension end point A of the crack model and the gear centerline, ko is the thickness of half of the tooth at the tooth top; based on the transmission process and geometric relationship during meshing, the following relationship exists:

[0125]

[0126] Where, α kr is the angle between the crack termination point on the k-th crack sheet and the tooth root circle, q k1 is the crack depth of the two parts of the outer surface of the k-th crack sheet, v k is the angle between the center line and the crack direction in the k-th layer model, x kl is the distance between the force point on the k-th tooth surface and the tooth root, γ kis the angle between the midpoint of the crack on the k-th crack sheet and the tooth root circle;

[0127] The calculation equations for the contact area and area moment of inertia between the tooth root circle and the gear base circle when cracks occur are as follows:

[0128]

[0129] Where A kaγ , I kaγ are the contact area and area moment of inertia at the tooth root circle, A dkal , I dkal are the contact area and area moment of inertia at the base circle, respectively;

[0130] The calculation equations for the contact area between the point of action of the contact meshing force and the tooth root circle section and its area moment of inertia are as follows:

[0131] A xka =(h kx +h ka )Δw,I xka =1 / 12(h kx +h ka ) 3 Δw;

[0132] Where A xka , I xka are the contact area and area moment of inertia of the cross section from the action point to the tooth root;

[0133] Substituting the parameter relationship after the crack is affected into the calculation formula of the gear stiffness, the shear and bending stiffness when the crack exists are derived, and then the crack stiffness under this condition is solved;

[0134] Case 2: α k1 <α kc , h kc <h ko or h kc ≥h ko

[0135] The length of the crack extension model exceeds the length of the tooth centerline, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α kc h is the pressure angle on the tooth surface when the length of the crack extension model exceeds the length of the tooth centerline, kc h is the distance between the end point C of the crack model and the center line of the sliced gear, ko is the thickness of the half-tooth at the tooth top. By analyzing the transmission process and geometric relationship during meshing, the following relationship is obtained:

[0136]

[0137] Where qk2 is the crack depth of the outer surface of the k-th crack sheet, α kr is the angle between the crack termination point on the k-th crack sheet and the tooth root circle. After deduction, A dkc1 , I dkc1 are the contact area and area moment of inertia of the gear base circle, A kcγ , I kcγ are the contact area and area moment of inertia of the tooth root circle, respectively. The calculation equations are as follows:

[0138]

[0139] Where A xkc , I xkc The calculation equations for the contact area and area moment of inertia between the contact meshing force action point and the tooth root circle section are:

[0140] A xkc =(h kx -h kc )Δw,I xkc =1 / 12(h kx -h kc ) 3 Δw;

[0141] Through the above analysis, an analytical calculation model for the time-varying mesh stiffness of spiral bevel gears under crack fault conditions was established. Two situations were considered, one of which was whether the crack extension direction exceeded the gear centerline. The calculation methods for the shear and bending stiffness when cracks existed were derived in detail.

[0142] Taking the spiral bevel gear transmission system as an example, the dynamic characteristics of the spiral bevel gears in healthy and cracked states were deeply analyzed using the time-varying meshing stiffness calculation method based on meshing theory proposed in the present invention. The study found that under the conditions of different crack lengths and angles between the crack and the centerline, the meshing stiffness of the spiral bevel gear exhibits significant time-varying characteristics, which in turn affects the vibration response of the gear transmission system. Therefore, according to the calculation model of the present invention, the gear design parameters or operating conditions can be adjusted to ensure that the spiral bevel gear transmission system maintains a stable, low-noise motion state during operation, thereby providing a scientific basis for the design and optimization of gear transmission systems in fields such as aerospace.

[0143] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.

Claims

1. A method for calculating meshing stiffness of cracked spiral bevel gears based on meshing theory; characterized in that: The following steps are involved: Step (1): Establish the spatial complex tooth surface of the spiral bevel gear; The gear tooth surface machining equation considers two parts: the meshing equation of the cutter head surface is derived in the coordinate system, and the tooth surface ∑2 (a) Transition surface with tooth root∑2 (b) Working part; tooth surface ∑2 (a) It is expressed as the following equation: Where, is a vector function of the working part of the tooth surface in three-parameter form, s g is the cone coordinate of the active spiral bevel gear, θ g is the rotation angle, ψ2 is the general parameter of motion, and the matrix M 2g (ψ2) represents the g The coordinate transformation matrix to the moving coordinate system s2, is a vector function of the working part of the tooth surface in a two-parameter form, is the parametric equation of the working part of the tooth surface; By considering the above equation, the tooth surface ∑2 (a) Originally expressed in three-parameter form, it can be simplified to a two-parameter form ∑2 (a) , expressed as the following equation: Where, is a two-parameter vector function of the working part of the tooth surface, θ2 and ψ2 are general parameters of motion, is a vector function of the working part of the tooth surface in three-parameter form, s g is the cone coordinate of the active spiral bevel gear, θ g is the rotation angle; Considering the tooth root transition surface ∑2 (b) , expressed as the following equation: Where, is the three-parameter vector function of the working part of the tooth root transition surface, λ w is the tooth root transition surface coordinate, θ2 and ψ2 are general parameters of motion, and the matrix M 2g (ψ2) represents the g To the coordinate transformation matrix of S2, is a vector function of the working part of the tooth root transition surface in a two-parameter form, is the parametric equation of the working part of the tooth surface, so the tooth root transition surface ∑2 (b) The two-parameter form of is expressed as the following equation: Where, is a two-parameter vector function of the working part of the tooth root transition surface, θ2 and ψ2 are general parameters of motion, is the three-parameter vector function of the working part of the tooth root transition surface, λ w is the coordinate of tooth root transition surface; Step (2): Analyze the meshing contact process of spiral bevel gears; During the meshing process, the driving spiral bevel gear and the driven spiral bevel gear must be continuously tangent, that is, the position vector and the normal vector at any instant must coincide with each other; the equations of the tooth surfaces of the driving spiral bevel gear and the driven spiral bevel gear in the coordinate system are as follows: Where, is the tooth surface equation of the active spiral bevel gear, s p is the surface coordinate of the active spiral bevel gear, θ p is the rotation angle of the active spiral bevel gear, ψ1 is the rotation angle of the active spiral bevel gear during processing, φ1 is the rotation angle of the active spiral bevel gear, M hb1 M b11 is the coordinate change matrix of the active spiral bevel gear, r1(s p ,θ p ,ψ1) is the three-parameter form of the active spiral bevel gear tooth surface equation, s g is the surface coordinate of the driven spiral bevel gear, θ g is the rotation angle of the driven spiral bevel gear, is the tooth surface equation of the driven spiral bevel gear, ψ2 is the rotation angle of the driven spiral bevel gear during machining, φ2 is the rotation angle of the driven spiral bevel gear, M hb2 M b22 is the coordinate change matrix of the driven spiral bevel gear; r2(s g ,θ g ,ψ2) is the three-parameter form of the driven spiral bevel gear tooth surface equation, equation f 1p (s p ,θ p ,ψ1)=0 and r1(s p ,θ p ,ψ1) uses three coefficient variables to represent the tooth surface of the active spiral bevel gear in the S1 coordinate system; equation f 2g (s g ,θ g ,ψ2)=0 and r2(s g ,θ g ,ψ2) Use three coefficient variables to represent the tooth surface of the driven spiral bevel gear in the S2 coordinate system; Position normal vector of driving spiral bevel gear and driven spiral bevel gear The equation in the coordinate system is as follows: Where n1(s p ,θ p ,ψ1) is the position normal vector equation of the three-parameter form of the active spiral bevel gear, n2(s g ,θ g ,ψ2) is the position normal vector equation of the driven spiral bevel gear in three-parameter form, L h1 、L b22 The matrix representation of the rotation angles φ1 and φ2 of the active spiral bevel gear and the driven spiral bevel gear is as follows: The equations satisfying the continuous tangency between the driving spiral bevel gear and the driven spiral bevel gear are as follows: In the formula, the parameter value of the rotation angle φ1 of the active spiral bevel gear is within the range of -π / N1<φ1<π / N1, the tooth surface equations ∑1 and ∑2 are expressed in the coordinate system by three parameter variables, and the equation f 1p (s p ,θ p ,ψ1),f 2g (s g ,θ g ,ψ2) is the meshing equation of tooth surface ∑1 and ∑2; tooth surface vector function Function of tooth surface normal vector The relationship indicates that the position vectors of the tooth surface equations ∑1 and ∑2 at their tangent points coincide with the unit normal vector of the tooth surface; The calculation equation of the transmission error function during gear meshing is as follows: Δφ2(φ1)=φ2(φ1)-N1 / N2φ1; Where φ1 and φ2 are the rotation angles of the driving spiral bevel gear and the driven spiral bevel gear, respectively; N1 and N2 are the surface normal vectors of the driving spiral bevel gear and the driven spiral bevel gear, respectively; Δφ2(φ1) represents the transmission error function variation relationship between the driving spiral bevel gear and the driven spiral bevel gear; Step (3): Establish a time-varying meshing stiffness analytical calculation model for healthy spiral bevel gears, analyze the meshing force process of the gear pair, F m is the meshing force, F m Along the spatial coordinate system, it is decomposed into the tangential force F t , radial force F r With axial force F a , the equations for these forces are as follows: Where β is the helix angle, α is the pressure angle, and δ G is the pitch circle pressure angle; Tangential force F t When the gears rotate, the relationship between the meshing force and the load torque T is as follows: F m =F t / cos(α)cos(β)=T / r p cos(α)cos(β); Where r p The radial distance from the meshing point to the rotation axis during meshing; the gear potential energy mainly consists of two parts: the energy at the tooth root transition curve and the energy from the point of action of the contact force on the tooth surface to the base circle; The parameter solution method in the stiffness analysis process of spiral bevel gears is as follows: Where x k is the distance between any point on the tooth surface and the tooth root, r kb is the base circle radius of the sliced gear, r kf is the tooth root radius, α k Represents the pressure angle at any point on the tooth surface, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α k2 is the base circle half tooth angle, α k3 The angle between the tooth profile at the root circle and the horizontal axis; h k 、h kx are the distances from the meshing contact point and any point on the tooth surface to the center line of the sliced gear, d k is the distance between the contact point and the tooth root, d k1 is the distance between the root circle and the tooth root, d k2 Indicates the distance from the contact point on the tooth surface to the tooth root; The equations for calculating the contact area and area moment of inertia of healthy gear teeth are as follows: Where A xk , I xk A is the contact area and area moment of inertia at the tooth root circle when the action point is located on the involute part; dk1 , I dk1 is the contact area and area moment of inertia at the base circle when the action point is located on the involute part; Δw is the tooth width of each micro segment; Under the action of meshing contact force, each slice gear can be regarded as a cantilever beam, and the axial tensile and compressive stiffness k ka , shear stiffness k ks , bending stiffness k kb , Hertz contact stiffness k kh and the gear flexible matrix stiffness k kf The calculation equation is as follows: k kh =πEΔw / 4(1-v 2 ),k kf =F m / d kf Where G is the shear modulus, G = E / 2(1+ν), E is the Young's modulus, ν is the Poisson's ratio, δ kf is the flexibility coefficient, x k is the distance between any point on the tooth surface and the tooth root, r kb is the base circle radius of the sliced gear, α k Represents the pressure angle at any point on the tooth surface, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α k2 is the base circle half tooth angle; h k is the distance from the meshing contact point to the center line of the sliced gear, d k is the distance between the contact point and the tooth root, d k1 is the distance between the root circle and the tooth root; the meshing stiffness k for the kth slice kt The calculation equation is as follows: Where, subscripts 1 and 2 represent the driving and driven spiral bevel gears, respectively. The calculation equation for the comprehensive meshing stiffness of the gears as they rotate is as follows: In the formula, the meshing stiffness of a pair of spiral bevel gears is the superposition of the meshing stiffness of n slices at each meshing moment; n is the total number of slices at the meshing moment, m is the number of instantaneous meshing gear pairs, so k single is the single tooth meshing stiffness, k total k is the comprehensive meshing stiffness of the gear pair when it is meshing. kt is the meshing stiffness of the nth slice at each moment; Step (4): Establish an analytical calculation model for the time-varying meshing stiffness of spiral bevel gears in a crack fault state; here, the shear and bending stiffnesses under two different conditions are considered. The first condition is that the crack extension direction does not exceed the gear centerline, and the second condition is that the crack extension direction exceeds the gear centerline. Case 1: α k1 ≥α ka , h ka ≥h ko α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α ka is the angle between the crack direction and the center line, h ka h is the straight-line distance between the extension end point A of the crack model and the gear centerline, ko is the thickness of half of the tooth at the tooth top; based on the transmission process and geometric relationship during meshing, the following relationship exists: r kb tana ka +h ka / cosα ka =r kb (a k2 +a ka ) α kr =arccos(r kf cosα k3 -q k1 cosv k / r kf ); x kl =r kf cosγ k -r kf cosα kr ,h ka =r kf sinα k3 -q k1 sinv k Where, α kr is the angle between the crack termination point on the k-th crack sheet and the tooth root circle, q k1 is the crack depth of the two parts of the outer surface of the k-th crack sheet, v k is the angle between the center line and the crack direction in the k-th layer model, x kl is the distance between the force point on the k-th tooth surface and the tooth root, γ k is the angle between the midpoint of the crack on the k-th crack sheet and the tooth root circle; The calculation equations for the contact area and area moment of inertia between the tooth root circle and the gear base circle when there is a crack are as follows: Where A kaγ , I kaγ are the contact area and area moment of inertia at the tooth root circle, A dkal , I dkal are the contact area and area moment of inertia at the base circle, respectively; The calculation equations for the contact area between the point of action of the contact meshing force and the tooth root circle section and its area moment of inertia are as follows: A xka =(h kx +h ka )Δw,I xka =1 / 12(h kx +h ka ) 3 Δw; Where A xka , I xka are the contact area and area moment of inertia of the cross section from the action point to the tooth root; Substituting the parameter relationship after the crack is affected into the calculation formula of the gear stiffness, the shear and bending stiffness when the crack exists are derived, and then the crack stiffness under this condition is solved; Case 2: α k1 <α kc , h kc <h ko or h kc ≥h ko The length of the crack extension model exceeds the length of the tooth centerline, α k1 is the angle between the gear tooth surface and the vertical axis at the meshing contact point, α kc h is the pressure angle on the tooth surface when the length of the crack extension model exceeds the length of the tooth centerline, kc h is the distance between the end point C of the crack model and the center line of the sliced gear, ko is the thickness of the half-tooth at the tooth top. By analyzing the transmission process and geometric relationship during meshing, the following relationship is obtained: x kl =r kf cosγ k -(r kf cosα k3 -r kf sinα k3 cotv k +q k2 cosv k ) r kb tanα kc +h kc / cosα kc =r kb (α k2 +α kc ),h kc =q k2 sinv k ; α kr =arccos(r kf cosα k3 -r kf sinα k3 cotv k +q k2 cosv k / r kf ) Where q k2 is the crack depth of the outer surface of the k-th crack sheet, α kr is the angle between the crack termination point on the k-th crack sheet and the tooth root circle. After deduction, A dkc1 , I dkc1 are the contact area and area moment of inertia of the gear base circle, A kcγ , I kcγ are the contact area and area moment of inertia of the tooth root circle, respectively. The calculation equations are as follows: Where A xkc , I xkc The calculation equations for the contact area and area moment of inertia between the contact meshing force action point and the tooth root circle section are: A xkc =(h kx -h kc )Δw,I xkc =1 / 12(h kx -h kc ) 3 Δw; Through the above analysis, an analytical calculation model for the time-varying mesh stiffness of spiral bevel gears under crack fault conditions was established. Two situations were considered, one of which was whether the crack extension direction exceeded the gear centerline. The calculation methods for the shear and bending stiffness when cracks existed were derived in detail.

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