Bevel gear pair tooth surface load distribution calculation method and system considering through cracks

Abstract

The invention specifically discloses a helical gear pair tooth surface load distribution calculation method and system considering through cracks, and the method comprises the following steps: judging the number J of gear tooth pairs participating in meshing at a corresponding moment t, and dispersing the gear tooth pairs into N equant and independent sheet straight gears in the tooth width direction; the total rigidity of the ith normal tooth piece in the bevel gear pair is calculated; the total rigidity of the ith crack tooth piece in the bevel gear pair is calculated; the comprehensive time-varying meshing rigidity of the bevel gear pair is calculated; the inter-tooth load distribution coefficient and the tooth direction load distribution coefficient in the meshing tooth surface of the bevel gear pair are obtained; and calculating the dynamic meshing force (dynamic load) in the gear transmission process, and obtaining the tooth surface load distribution of the helical gear pair under the through crack. By adopting the technical scheme, the dynamic model of the helical gear pair under the through crack is established to calculate the dynamic load of the helical gear pair, and the tooth space distribution coefficient and the tooth direction distribution coefficient of the helical gear pair are multiplied by the dynamic load to obtain the tooth surface dynamic load distribution of the helical gear pair under the through crack.

Description

Technical Field

[0001] The present invention belongs to the technical field of mechanical dynamics and relates to a method and system for calculating tooth surface load distribution of a helical gear pair considering through-cracks. Background Art

[0002] Helical gears are core components of mechanical transmission systems, and their reliability directly impacts equipment life and operational safety. However, gears are susceptible to fatigue cracking under complex alternating loads. Once cracks propagate unstably, they can lead to gear fracture or even catastrophic failure, resulting in significant economic losses and safety incidents.

[0003] During the meshing process of helical gears, the meshing stiffness exhibits time-varying characteristics due to changes in the number of tooth pairs involved in the meshing. When cracks appear at the tooth root or tooth surface, the local stiffness in the cracked area decreases, further reducing the overall meshing stiffness and exacerbating stiffness fluctuations, affecting the smoothness of the transmission. The reduction in meshing stiffness changes the load-sharing characteristics of the gear pair, redistributing the load within the meshing surface. The uneven stiffness caused by cracks may lead to load concentration, causing some tooth surfaces to be subjected to higher stress, accelerating local wear or crack propagation. Uneven load distribution on the meshing surface can significantly affect the initiation and propagation paths of cracks. High-stress areas are prone to becoming crack sources, and fluctuations in dynamic loads can promote crack extension deeper into the tooth root or tooth surface, forming a positive feedback effect. In-depth research on the interaction mechanism among these three factors is of great significance for establishing high-precision fault warning models and optimizing fatigue-resistant designs. It can also provide theoretical support for the intelligent operation and maintenance of major equipment and improve the reliability.

[0004] At present, the research on the stiffness and load distribution of helical gears has been very detailed. However, in practice, crack defects, time-varying meshing stiffness and meshing surface load distribution have a dynamic coupling effect and mutually restrict each other, which is a key factor causing early gear failure: crack propagation will significantly change the stiffness characteristics, and the mutual feedback between time-varying stiffness and load distribution will accelerate crack evolution, forming a vicious cycle.

[0005] The above defects are what those skilled in the art hope to overcome, and therefore a new method is urgently needed to solve the above problems. Summary of the Invention

[0006] The purpose of the present invention is to provide a method and system for calculating the load distribution on the tooth surface of a helical gear pair considering through-cracks, so as to solve the problem that the coupling effect between cracks, stiffness and load distribution is difficult to clarify.

[0007] To achieve the above objectives, the basic solution of the present invention is: a method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks, comprising the following steps:

[0008] S1, discretize the meshing process into a series of different moments and determine the number of tooth pairs J involved in meshing at the corresponding moment t;

[0009] The teeth of the helical gear pair with through-crack fault are discretized into N equally divided and independent thin spur gears along the tooth width direction;

[0010] S2, based on cantilever beam theory and potential energy method, calculate the total stiffness of the i-th normal tooth in the helical gear pair

[0011] S3, assuming that there is a through crack at the root of the driving gear tooth and assuming that the cracked tooth can still be regarded as a cantilever beam model, with the boundary condition that no deformation occurs at the root of the tooth, calculate the total stiffness of the i-th cracked tooth in the helical gear pair

[0012] S4, based on steps S2, S3 and the offset superposition idea, calculate the comprehensive time-varying meshing stiffness K of the helical gear pair, and then calculate the inter-tooth load distribution coefficient LSF within the meshing tooth surface of the helical gear pair tj and tooth load distribution factor LSF ti ;

[0013] S5, establish a 6-DOF helical gear pair bending-axis-torsion coupling dynamic model and calculate the dynamic meshing force F(t) during gear transmission;

[0014] Based on the inter-tooth distribution coefficient LSF in step S4 tj , tooth distribution coefficient LSF ti and dynamic meshing force F(t), and obtain the tooth surface load distribution F of the helical gear pair under the through-crack tji .

[0015] The working principle and beneficial effects of this basic scheme are as follows: this technical scheme calculates the inter-tooth distribution coefficient and tooth direction distribution coefficient of the helical gear pair with time-varying meshing stiffness, calculates the dynamic load of the helical gear pair by establishing a dynamic model of the helical gear pair under a through crack, multiplies the inter-tooth distribution coefficient and tooth direction distribution coefficient of the helical gear pair with the dynamic load, and finally obtains the dynamic load distribution on the tooth surface of the helical gear pair under a through crack.

[0016] Furthermore, the meshing process is discretized into a series of different moments, and the method for determining the number of tooth pairs involved in meshing at the corresponding moment is:

[0017] From t = Q / w p The helical gear pair transmission process is divided into different discrete moments to facilitate the subsequent calculation of the time-varying meshing stiffness K, where Q and w p are the rotation angle and angular velocity of the driving wheel respectively, and t represents the discrete moment;

[0018] Determine the number of tooth pairs J involved in meshing at this moment:

[0019]

[0020] Where, floor is the floor rounding function; Re = rem (Q, 2*π / N1) is the angle that the driving wheel rotates in one meshing cycle, and rem is the remainder function; Q m =(ε-floor(ε))*2*π / N1 is the angle that the gear teeth rotate in the multi-tooth meshing area, ε is the overlap of the helical gear, and N1 is the number of teeth of the driving gear.

[0021] Discrete helical gear pair meshing, determine the number of gear tooth pairs involved in meshing at the corresponding moment, which is beneficial for subsequent use.

[0022] Furthermore, the teeth of the helical gear pair with through-crack fault are discretized into N equally divided and independent thin spur gears along the tooth width direction, specifically:

[0023] Each slice can be regarded as a spur gear. The load position of each slice changes with the offset of the slice. The distance h from the load point of the i-th slice to the center line of the tooth surface is i and the distance d to the tooth root i ,for:

[0024] d i =R b [cosα pi +(α pi +α2)*sinα pi -cosα2]

[0025] h i =R b [(α pi +α2)*cosα pi -sinα pi ]

[0026]

[0027] Among them, R b and α2 are base circle radius and base circle semi-tooth angle respectively; α p1 and α p2 are the meshing angles of the two teeth at both ends of the contact line; α pi is the engagement angle at the meshing point of the i-th slice, w is the tooth width distance from the initial end face of the tooth to the i-th slice, and W is the tooth width.

[0028] The meshing stiffness of a single helical gear tooth can be defined as the contact force required to produce unit deformation along the contact line. This parameter is solved using the slice method theory, the core of which is to discretize the helical gear into a series of micro-element slices along the tooth width direction.

[0029] Furthermore, according to the cantilever beam theory and potential energy method, the total stiffness of the i-th normal tooth in the helical gear pair is calculated as The specific steps are:

[0030] S201, based on cantilever beam theory and potential energy method, the gear tooth stiffness k t Consists of three parts: bending stiffness k b , shear stiffness k s and compressive stiffness k a , that is, the tooth stiffness of the i-th healthy gear for:

[0031]

[0032] in, are the bending stiffness, axial compression stiffness and shear stiffness at the i-th slice respectively:

[0033]

[0034] Among them, h i and d i is the distance from the load action point of the i-th slice to the center line of the tooth surface and to the tooth root; α pi is the meshing angle at the meshing point of the i-th slice; E and G are the elastic modulus and shear modulus of the gear material; I y and A y are the effective section moment of inertia and cross-sectional area at a distance y from the fixed end of the cantilever beam:

[0035]

[0036] Where L is the width of each sliced ​​spur gear; h y is the distance from the point on the tooth profile at the fixed end y of the cantilever beam to the center plane of the tooth;

[0037] S202, matrix stiffness of the i-th healthy tooth segment for:

[0038]

[0039] Among them, u f S is the distance between the intersection of the meshing point load through the gear tooth center line and the tooth root arc; f is the arc length of tooth root; where the coefficients L*, M*, P*, and Q* are obtained from the polynomial function;

[0040] S203, total stiffness of the i-th normal tooth in the helical gear pair The final expression is:

[0041]

[0042] Based on the cantilever beam theory and potential energy method, the core idea is to regard the slice as a non-uniform variable-section cantilever beam and divide the total energy stored in the tooth beam after the meshing force into three parts: bending potential energy, shear potential energy and axial compression potential energy.

[0043] Furthermore, the total stiffness of the i-th cracked tooth in the helical gear pair is calculated as The method is as follows:

[0044] S301, crack depth q at each slice w and crack location l w is constant, the crack depth along the tooth thickness direction is q, the crack position along the tooth width direction is l, and the evolution of the gear crack is:

[0045]

[0046] Where W is the tooth width;

[0047] For the crack depth q: m1 is constant along the tooth width; m2 shows a parabolic trend along the tooth width:

[0048]

[0049] Among them, q i and q e is the crack depth at the initial and final end faces of the cracked tooth; q c is the constant crack depth when the crack is constant along the tooth width direction;

[0050] For crack position l:n1, the crack path is linear; for crack position n2, the crack path is monotonic parabola:

[0051]

[0052] Among them, l i and l e is the distance from the crack position of the initial end face and the final end face of the cracked tooth to the tooth root;

[0053] S302, when cracks exist, the bending stiffness and shear stiffness will be affected. Recalculate the effective section inertia moment and cross-sectional area at a distance y from the tooth root:

[0054] ① When h c ≥h r and d i >l t hour,

[0055]

[0056] ②When hc≥h r and d i ≤l tOr when hc<h r hour,

[0057]

[0058] Among them, h r is the half-tooth top thickness; h y and h c are the distance from the crack initial point to the gear tooth centerline and the distance from the crack end point to the gear tooth centerline respectively; l t l is the distance from the projection point of the crack end point on the involute tooth profile to the tooth root circle; w and l c are the distances from the crack starting point and end point to the tooth root circle respectively; y is the integral variable when calculating the tooth stiffness; I yc 、A yc and are the effective section inertia moment, section area and inertia radius at a distance y from the fixed end of the cantilever beam when the crack exists; d i is the distance from the load action point of the i-th slice to the tooth root; L is the width of each slice spur gear;

[0059] S303, total stiffness of the i-th cracked tooth in the helical gear pair The final expression is:

[0060]

[0061] in, and is the base stiffness and tooth stiffness of the spur gear at the i-th slice; Bending stiffness Axial compression stiffness and shear stiffness It consists of three parts, and the subscript c represents the corresponding stiffness under the condition of through-crack failure.

[0062] Determine the crack evolution path, calculate the effective section inertia moment and cross-sectional area based on the crack, and thus obtain the total stiffness of the i-th cracked tooth in the helical gear pair The calculation is simple.

[0063] Furthermore, in step S4, based on steps S2 and S3 and the offset superposition idea, the comprehensive time-varying meshing stiffness of the helical gear pair is calculated, and then the inter-tooth load distribution coefficient LSF in the meshing tooth surface of the helical gear pair is calculated. tj and tooth load distribution factor LSF ti , the specific steps are as follows:

[0064] S401, based on Hertz contact theory, calculates the gear Hertz contact stiffness k h :

[0065]

[0066] Where E and v are the elastic modulus and Poisson's ratio of the gear material, respectively; L is the width of each sliced ​​spur gear;

[0067] S402, based on the normal tooth plate stiffness and cracked tooth plate stiffness calculated in steps S2 and S3 and the Hertz contact stiffness in S401, the stiffness of the i-th spur gear pair slice of the cracked tooth pair and the non-cracked tooth pair in the helical gear pair Respectively expressed as:

[0068]

[0069]

[0070] in, is the stiffness of the i-th slice straight gear pair, x = c, n corresponds to the crack state and normal state respectively, y = p, g corresponds to the driving wheel and driven wheel respectively;

[0071] Cracked tooth pair K in helical gear pair c_single and normal gear pair K n_single The meshing stiffnesses are:

[0072]

[0073] Then the tooth load distribution coefficient of the gear pair with a through crack is

[0074] S403, based on the determination of the number of meshing gear teeth in step S1, at the tth discrete moment, the total stiffness k of the meshing gear teeth pair at the i-th slice position of the helical gear pair is i for:

[0075]

[0076] in, is the meshing stiffness of the jth normal tooth pair in the meshing gear teeth at the i-th slice; ε is the contact ratio of the helical gear;

[0077] Then the inter-tooth load distribution coefficient of the gear pair with a through-crack is

[0078] S404, the comprehensive time-varying meshing stiffness K of the helical gear pair with a through-crack is:

[0079] K=K c_single +K n1_single +…+K nj_single ,

[0080] or,

[0081]

[0082] Where N represents the number of slices; K nj_single It is the meshing stiffness of the jth normal tooth pair among the gear teeth participating in the meshing at the same time.

[0083] The comprehensive time-varying meshing stiffness K of the helical gear pair with a through-crack is calculated using the relevant parameters obtained above. The operation is simple and easy to use.

[0084] Furthermore, the 6-DOF helical gear pair bending-axis-torsion coupling dynamic model is established as follows:

[0085]

[0086]

[0087] Among them, m1, m2, I1 and I2 are the mass and moment of inertia of the driving and driven wheels respectively; y1, y2, z1, z2, θ z1 and θ z2 are the tangential, axial and torsional vibration displacements of the center points of the driving and driven wheels respectively; k 1y 、k 1z 、k 2y 、k 2z 、c 1y 、c 1z 、c 2y and c 2z are the equivalent bearing support stiffness and equivalent damping in the tangential and axial directions at the center points of the driving and driven wheels, respectively; K is the comprehensive time-varying meshing stiffness between the gear teeth calculated based on step S4, and C is the comprehensive meshing damping between the gear teeth; T1 and T2 are the input torques of the driving and driven wheels, respectively; β is the pitch circle helix angle; R b1 is the base circle radius of the driving wheel.

[0088] The normal dynamic meshing force F(t) of the helical gear pair can be decomposed into the tangential force F along the meshing line of the gear end face. y and the axial force F along the gear axis z , so the dynamic load F(t) of the helical gear pair during the meshing transmission process is:

[0089]

[0090] F n =(F y 2 +F z 2 ) 1 / 2

[0091] The helical gear pair is regarded as a spring with only elasticity but no inertia and a mass block with only inertia but no elasticity. Considering the time-varying nature of the gear meshing stiffness under the influence of a through-crack, a 6-DOF bending-axis-torsion coupled dynamic model is established to obtain the dynamic meshing force during the transmission process.

[0092] Furthermore, based on the inter-tooth distribution coefficient LSF in step S4 tj , tooth distribution coefficient LSF ti and dynamic load F(t), and obtain the tooth surface load distribution F of the helical gear pair under the through-crack tji for:

[0093] F tji =F n LSF ti ·LSF tj

[0094] Among them, F tji represents the load on the i-th slice of the cracked tooth pair at time t.

[0095] Obtaining the load distribution on the meshing interface of the helical gear pair under a through-crack considering the time-varying meshing stiffness factor is helpful to clarify the coupling effect between the crack, stiffness and load distribution.

[0096] The present invention also provides a system for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks, comprising a processing unit, wherein the processing unit executes the method of the present invention to calculate the load distribution on the tooth surface of the helical gear pair.

[0097] This system considers through-cracks and calculates the load distribution on the tooth surface of helical gear pairs, which is beneficial to analyze the coupling effect between cracks, stiffness and load distribution. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] Figure 1 1 is a flow chart of a method for calculating tooth surface load distribution of a helical gear pair considering through-cracks according to the present invention;

[0099] Figure 2 Schematic diagram of the structure of the 6-DOF helical gear pair bending-axis-torsion coupling dynamic model of the helical gear pair considering the through-crack in the tooth surface load distribution calculation method of the present invention;

[0100] Figure 3 Schematic diagram of meshing surface load distribution of the helical gear pair tooth surface load distribution calculation method considering through cracks of the present invention;

[0101] Figure 4 This is a diagram of the comprehensive time-varying meshing stiffness of a helical gear pair under different degrees of fault cracks according to the calculation method of the tooth surface load distribution of the helical gear pair considering the through crack of the present invention;

[0102] Figure 5This is a dynamic meshing force diagram of a helical gear pair under different degrees of fault cracks according to the calculation method of the tooth surface load distribution of a helical gear pair considering a through crack of the present invention. DETAILED DESCRIPTION

[0103] The following describes embodiments of the present invention in detail. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended only to explain the present invention and are not to be construed as limiting the present invention.

[0104] In the description of the present invention, it should be understood that the terms "longitudinal", "transverse", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as limiting the present invention.

[0105] In the description of the present invention, unless otherwise specified and limited, it should be noted that the terms "installed", "connected" and "connected" should be understood in a broad sense. For example, it can be a mechanical connection or an electrical connection, or it can be the internal communication between two components. It can be a direct connection or an indirect connection through an intermediate medium. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to the specific circumstances.

[0106] The present invention discloses a method for calculating the load distribution on the tooth surface of a helical gear pair considering through-cracks. In order to solve the problem that the coupling effect between cracks, stiffness and load distribution is difficult to clarify, a method for calculating the inter-tooth distribution coefficient and tooth direction distribution coefficient of a helical gear pair based on time-varying meshing stiffness is proposed. By establishing a dynamic model of the helical gear pair under through-cracks, the dynamic load of the helical gear pair is calculated, and the inter-tooth distribution coefficient and tooth direction distribution coefficient of the helical gear pair are multiplied by the dynamic load to finally obtain the dynamic load distribution on the tooth surface of the helical gear pair under through-cracks. Figure 1 As shown in Figure 2, the calculation method for the tooth surface load distribution of a helical gear pair considering a through-crack includes the following steps:

[0107] S1, discretize the meshing process into a series of different moments (discretely divide the process of continuous rotation in the meshing of the gear transmission into a series of different small moments), and determine the number of gear tooth pairs J involved in the meshing at the corresponding moment t;

[0108] The teeth of the helical gear pair with through-crack fault are discretized into N equally divided and independent thin spur gears along the tooth width direction;

[0109] S2, Calculation of the stiffness of the normal parallel spur gear slice of the helical gear pair: Calculate the total stiffness of the i-th normal gear slice in the helical gear pair based on the cantilever beam theory and potential energy method.

[0110] S3, Calculation of the stiffness of the cracked parallel spur gear slice of the helical gear pair: Assume that there is a through crack at the root of the driving gear tooth and assume that the cracked tooth can still be regarded as a cantilever beam model. The boundary condition is that there is no deformation at the root of the tooth. Calculate the total stiffness of the i-th cracked tooth slice in the helical gear pair.

[0111] S4, calculation of inter-tooth load distribution coefficient and tooth direction load distribution coefficient: Based on steps S2, S3 and the offset superposition idea, calculate the comprehensive time-varying meshing stiffness of the helical gear pair, and then calculate the inter-tooth load distribution coefficient LSF within the meshing tooth surface of the helical gear pair tj and tooth load distribution factor LSF ti ;

[0112] S5, tooth surface load distribution calculation: establish a 6-DOF helical gear pair bending-axis-torsion coupling dynamic model to calculate the dynamic meshing force (dynamic load) F(t) during gear transmission;

[0113] Based on the inter-tooth distribution coefficient LSF in step S4 tj , tooth distribution coefficient LSF ti and dynamic meshing force F(t), and obtain the tooth surface load distribution F of the helical gear pair under the through-crack tji .

[0114] In a preferred embodiment of the present invention, the meshing process is discretized into a series of different moments, and the method for determining the number of gear tooth pairs involved in meshing at the corresponding moment is:

[0115] Discrete helical gear pair meshing process: t=Q / w p The transmission process of the helical gear pair is divided into different discrete moments and the time-varying meshing stiffness K is calculated, where Q and w p are the rotation angle and angular velocity of the driving wheel respectively, and t represents the discrete moment;

[0116] Determine the number of tooth pairs J involved in meshing at this moment:

[0117]

[0118] Where, floor is the floor rounding function; Re = rem (Q, 2*π / N1) is the angle that the driving wheel rotates in one meshing cycle, and rem is the remainder function; Q m =(ε-floor(ε))*2*π / N1 is the angle that the gear teeth rotate in the multi-tooth meshing area, ε is the overlap of the helical gear, and N1 is the number of teeth of the driving gear.

[0119] In a preferred embodiment of the present invention, the teeth of the helical gear pair with through-crack fault are discretized into N equally divided and independent thin spur gears along the tooth width direction, specifically:

[0120] Decomposing a helical gear pair: The meshing stiffness of a single helical gear tooth can be defined as the contact force required to produce a unit deformation along the contact line. The slice method is used to solve this parameter. The core of this method is to discretize the helical gear into a series of micro-element slices along the tooth width. Each slice can be regarded as an equivalent spur gear. The load position of each slice changes with the offset of the slice. The distance h from the load point of the i-th slice to the centerline of the tooth surface is: i and the distance d to the tooth root i ,for:

[0121] d i =R b [cosα pi +(α pi +α2)*sinα pi -cosα2]

[0122] h i =R b [(α pi +α2)*cosα pi -sinα pi ]

[0123]

[0124] Among them, R b and α2 are base circle radius and base circle semi-tooth angle respectively; α p1 and α p2 are the meshing angles of the two teeth at both ends of the contact line; α pi is the engagement angle at the meshing point of the i-th slice, w is the tooth width distance from the initial end face of the tooth to the i-th slice, and W is the tooth width.

[0125] In a preferred embodiment of the present invention, the total stiffness of the i-th normal tooth in the helical gear pair is calculated based on the cantilever beam theory and potential energy method. The specific steps are:

[0126] S201, based on cantilever beam theory and potential energy method, its core idea is to regard the slice as a non-uniform variable cross-section cantilever beam and divide the total energy stored in the tooth beam after the meshing force into three parts: bending potential energy, shear potential energy and axial compression potential energy. Based on cantilever beam theory and potential energy method, the gear tooth part stiffness k t Consists of three parts: bending stiffness k b , shear stiffness k s and compressive stiffness k a , that is, the tooth stiffness of the i-th healthy gear for:

[0127]

[0128] in, are the bending stiffness, axial compression stiffness and shear stiffness at the i-th slice, respectively;

[0129]

[0130] Among them, h i and d i is the distance from the load action point of the i-th slice to the center line of the tooth surface and to the tooth root; α pi is the meshing angle at the meshing point of the i-th slice; E and G are the elastic modulus and shear modulus of the gear material; I y and A y are the effective section moment of inertia and cross-sectional area at a distance y from the fixed end of the cantilever beam:

[0131]

[0132] Where L is the width of each sliced ​​spur gear; h y is the distance from the point on the tooth profile at the fixed end y of the cantilever beam to the center plane of the gear tooth;

[0133] S202, matrix stiffness of the i-th healthy tooth segment for:

[0134]

[0135] Among them, u f S is the distance between the intersection of the meshing point load through the gear tooth center line and the tooth root arc; f is the arc length of tooth root; L*, M*, P*, Q* are obtained by polynomial functions, specifically:

[0136]

[0137] X i *Represents L*, M*, P*, Q*, h i is the ratio of tooth root radius to hub radius; θ f is the angle between the tooth root and the center axis of the gear tooth;

[0138] When calculating L*, M*, P*, and Q*, the coefficients of the polynomials are selected as follows:

[0139] <![CDATA[A i (×10 -5 )]]> <![CDATA[B i (×10 -3 )]]> <![CDATA[C i (×10 -4 )]]> <![CDATA[D i (×10 -3 )]]> <![CDATA[E i ]]> <![CDATA[F i ]]> L* -5.574 -1.9986 -2.3015 4.7702 0.0271 6.8045 L* 60.111 28.100 -83.431 -9.9256 0.1624 0.9086 P* -50.952 185.50 0.0538 53.300 0.2895 0.9236 Q* -6.2042 9.0899 -4.0964 7.8297 -0.1472 0.6904

[0140] S203, total stiffness of the i-th normal tooth in the helical gear pair The final expression is:

[0141]

[0142] In a preferred embodiment of the present invention, the total stiffness of the i-th cracked tooth in the helical gear pair is calculated as follows: The method is as follows:

[0143] S301, based on step S1, consider the crack depth q at each slice w and crack location l w is constant, the crack depth along the tooth thickness direction is q, the crack position along the tooth width direction is l, and the evolution of the gear crack is:

[0144]

[0145] Where W is the tooth width;

[0146] For the crack depth q: m1 is constant along the tooth width; m2 shows a parabolic trend along the tooth width:

[0147]

[0148] Among them, q i and q e is the crack depth at the initial and final end faces of the cracked tooth; q c is the constant crack depth when the crack is constant along the tooth width direction;

[0149] For crack position l:n1, the crack path is linear; for crack position n2, the crack path is monotonic parabola:

[0150]

[0151] Among them, l i and l e is the distance from the crack position of the initial end face and the final end face of the cracked tooth to the tooth root;

[0152] S302, when cracks exist, the bending stiffness and shear stiffness will be affected. Recalculate the effective section inertia moment and cross-sectional area at a distance y from the tooth root:

[0153] ① When h c ≥h r and d i >l t hour,

[0154]

[0155] ②When hc≥h r and d i ≤l t Or when hc<h r hour,

[0156]

[0157] Among them, h r is the half-tooth top thickness; h y and h c are the distance from the crack initial point to the gear tooth centerline and the distance from the crack end point to the gear tooth centerline respectively; l t l is the distance from the projection point of the crack end point on the involute tooth profile to the tooth root circle; w and l c are the distances from the crack starting point and end point to the tooth root circle respectively; y is the integral variable when calculating the tooth stiffness; I yc 、A yc and are the effective section inertia moment, section area and inertia radius at a distance y from the fixed end of the cantilever beam when the crack exists; d i is the distance from the load action point of the i-th slice to the tooth root; L is the width of each slice spur gear;

[0158] S303, total stiffness of the i-th cracked tooth in the helical gear pair The final expression is:

[0159]

[0160] in, and is the base stiffness and tooth stiffness of the spur gear at the i-th slice; Bending stiffness Axial compression stiffness and shear stiffness It consists of three parts, and the subscript c represents the corresponding stiffness under the condition of through-crack failure.

[0161] In a preferred embodiment of the present invention, in step S4, based on steps S2 and S3 and the offset superposition concept, the comprehensive time-varying meshing stiffness of the helical gear pair is calculated, and then the inter-tooth load distribution coefficient LSF in the meshing tooth surface of the helical gear pair is calculated. tj and tooth load distribution factor LSF ti , the specific steps are as follows:

[0162] S401, based on Hertz contact theory, calculates the gear Hertz contact stiffness k h :

[0163]

[0164] Where E and v are the elastic modulus and Poisson's ratio of the gear material, respectively; L is the width of each sliced ​​spur gear;

[0165] S402, based on the normal tooth plate stiffness and cracked tooth plate stiffness calculated in steps S2 and S3 and the Hertz contact stiffness in S401, the stiffness of the i-th spur gear pair slice of the cracked tooth pair and the non-cracked tooth pair in the helical gear pair Respectively expressed as:

[0166]

[0167]

[0168] in, is the stiffness of the i-th slice straight gear pair, x = c, n corresponds to the crack state and normal state respectively, y = p, g corresponds to the driving wheel and driven wheel respectively;

[0169] Cracked tooth pair K in helical gear pair c_single and normal gear pair K n_single The meshing stiffnesses are:

[0170]

[0171] Then the tooth load distribution coefficient of the gear pair with a through crack is

[0172] S403, based on the determination of the number of meshing gear teeth in step S1, at the tth discrete moment, the total stiffness k of the meshing gear teeth pair at the i-th slice position of the helical gear pair is i for:

[0173]

[0174] in, is the meshing stiffness of the jth normal tooth pair in the meshing gear teeth at the i-th slice; ε is the contact ratio of the helical gear;

[0175] Then the inter-tooth load distribution coefficient of the gear pair with a through-crack is

[0176] S404, the comprehensive time-varying meshing stiffness K of the helical gear pair with a through-crack is:

[0177] K=K c_single +K n1_single +…+K nj_single ,

[0178] or,

[0179]

[0180] Where N represents the number of slices; K nj_singleIt is the meshing stiffness of the jth normal tooth pair among the gear teeth participating in the meshing at the same time.

[0181] In a preferred embodiment of the present invention, Figure 2 As shown in the figure, dynamic modeling is performed to obtain the dynamic meshing force during the transmission process: the helical gear pair is regarded as a spring with only elasticity but no inertia and a mass block with only inertia but no elasticity. The time-varying nature of the gear meshing stiffness under the influence of the through-crack is considered, and a 6-DOF bending-axis-torsion coupled dynamic model of the helical gear pair is established. The Lagrange equation is used to derive the undamped motion equation of the helical gear pair, namely:

[0182]

[0183]

[0184] Among them, m1, m2, I1 and I2 are the mass and moment of inertia of the driving and driven wheels respectively; y1, y2, z1, z2, θ z1 and θ z2 are the tangential, axial and torsional vibration displacements of the center points of the driving and driven wheels respectively; k 1y 、k 1z 、k 2y 、k 2z 、c 1y 、c 1z 、c 2y and c 2z are the equivalent bearing support stiffness and equivalent damping in the tangential and axial directions at the center points of the driving and driven wheels, respectively; K is the comprehensive time-varying meshing stiffness between the gear teeth calculated based on step S4, and C is the comprehensive meshing damping between the gear teeth; T1 and T2 are the input torques of the driving and driven wheels, respectively; β is the pitch circle helix angle; R b1 is the base circle radius of the driving wheel.

[0185] The normal dynamic meshing force F(t) of the helical gear pair can be decomposed into the tangential force F along the meshing line of the gear end face. y and the axial force F along the gear axis z , so the dynamic load F(t) of the helical gear pair during the meshing transmission process is:

[0186]

[0187] F n =(F y 2 +F z 2 ) 1 / 2

[0188] In a preferred embodiment of the present invention, based on the inter-tooth distribution coefficient LSF of step S4 tj , tooth distribution coefficient LSFti and dynamic meshing force F(t), and obtain the tooth surface load distribution F of the helical gear pair under the through-crack tji for:

[0189] F tji =F n ·LSF ti ·LSF tj

[0190] Among them, F tji represents the load of the i-th slice of the cracked tooth pair at time t, as Figure 3 As shown; Figure 4 and Figure 5 They are the comprehensive time-varying meshing stiffness diagram and the dynamic meshing force diagram of the helical gear pair under different degrees of fault cracks.

[0191] The present invention also provides a system for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks, comprising a processing unit, which executes the method of the present invention to calculate the load distribution on the tooth surface of the helical gear pair.

[0192] Throughout this specification, reference to terms such as "one embodiment," "some embodiments," "examples," "specific examples," or "some examples" means that a specific feature, structure, material, or characteristic described in conjunction with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0193] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to the embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the claims and their equivalents.

Claims

1. A method for calculating the load distribution on the tooth surface of a helical gear pair considering through-cracks, characterized in that: The steps include: S1, discretize the meshing process into a series of different moments and determine the number of tooth pairs J involved in meshing at the corresponding moment t; The teeth of the helical gear pair with through-crack fault are discretized into N equally divided and independent thin spur gears along the tooth width direction; S2, based on cantilever beam theory and potential energy method, calculate the total stiffness of the i-th normal tooth in the helical gear pair S3, assuming that there is a through crack at the root of the driving gear tooth and assuming that the cracked tooth can still be regarded as a cantilever beam model, with the boundary condition that no deformation occurs at the root of the tooth, calculate the total stiffness of the i-th cracked tooth in the helical gear pair S4, based on steps S2, S3 and the offset superposition idea, calculate the comprehensive time-varying meshing stiffness K of the helical gear pair, and then calculate the inter-tooth load distribution coefficient LSF within the meshing tooth surface of the helical gear pair tj and tooth load distribution factor LSF ti ; S5, establish a 6-DOF helical gear pair bending-axis-torsion coupling dynamic model and calculate the dynamic meshing force F(t) during gear transmission; Based on the inter-tooth distribution coefficient LSF in step S4 tj , tooth distribution coefficient LSF ti and dynamic meshing force F(t), and obtain the tooth surface load distribution F of the helical gear pair under the through-crack tji .

2. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 1, characterized in that: The meshing process is discretized into a series of different moments. The method for determining the number of tooth pairs involved in meshing at the corresponding moment is: From t = Q / w p The helical gear pair transmission process is divided into different discrete moments, where Q and w p are the rotation angle and angular velocity of the driving wheel respectively, and t represents the discrete moment; Determine the number of tooth pairs J involved in meshing at this moment: Where, floor is the floor rounding function; Re = rem (Q, 2*π / N1) is the angle that the driving wheel rotates in one meshing cycle, and rem is the remainder function; Q m =(ε-floor(ε))*2*π / N1 is the angle that the gear teeth rotate in the multi-tooth meshing area, ε is the overlap of the helical gear, and N1 is the number of teeth of the driving gear.

3. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 1, wherein: The teeth of the helical gear pair with through-crack fault are discretized into N equally divided and independent thin spur gears along the tooth width direction, specifically: Each slice can be regarded as a spur gear. The load position of each slice changes with the offset of the slice. The distance h from the load point of the i-th slice to the center line of the tooth surface is i and the distance d to the tooth root i ,for: d i =R b [things] pi +(a pi +α2)*sinα pi -cosα2] h i =R b [(a pi +α2)*cosα pi -sina pi ] Among them, R b and α2 are base circle radius and base circle semi-tooth angle respectively; α p1 and α p2 are the meshing angles of the two teeth at both ends of the contact line; α pi is the engagement angle at the meshing point of the i-th slice, w is the tooth width distance from the initial end face of the tooth to the i-th slice, and W is the tooth width.

4. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 1, wherein: According to the cantilever beam theory and potential energy method, calculate the total stiffness of the i-th normal tooth in the helical gear pair The specific steps are: S201, based on cantilever beam theory and potential energy method, the gear tooth stiffness k t Consists of three parts: bending stiffness k b , shear stiffness k s and compressive stiffness k a , that is, the tooth stiffness of the i-th healthy gear for: in, are the bending stiffness, axial compression stiffness and shear stiffness at the i-th slice respectively: Among them, h i and d i are the distances from the load action point of the i-th slice to the center line of the tooth surface and to the tooth root respectively; α pi is the meshing angle at the meshing point of the i-th slice; E and G are the elastic modulus and shear modulus of the gear material; I y and A y are the effective section moment of inertia and cross-sectional area at a distance y from the fixed end of the cantilever beam: <h2 style=";text-align:left;direction:ltr"> <h2 style=";text-align:left;direction:ltr"> A<h2 style=";text-align:left;direction:ltr"> y <h2 style=";text-align:left;direction:ltr"> (2 hours)<h2 style=";text-align:left;direction:ltr"> y <h2 style=";text-align:left;direction:ltr"> )L Where L is the width of each sliced ​​spur gear; h y is the distance from the point on the tooth profile at the fixed end y of the cantilever beam to the center plane of the gear tooth; S202, matrix stiffness of the i-th healthy tooth segment for: Among them, u f S is the distance between the intersection of the meshing point load through the gear tooth center line and the tooth root arc; f is the arc length of tooth root; where the coefficients L*, M*, P*, and Q* are obtained from the polynomial function; S203, total stiffness of the i-th normal tooth in the helical gear pair The final expression is:

5. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 1, wherein: Calculate the total stiffness of the i-th cracked tooth in a helical gear pair The method is as follows: S301, crack depth q at each slice w and crack location l w is constant, the crack depth along the tooth thickness direction is q, the crack position along the tooth width direction is l, and the evolution of the gear crack is: Where W is the tooth width; For the crack depth q: m1 is constant along the tooth width; m2 shows a parabolic trend along the tooth width: Among them, q i and q e is the crack depth at the initial and final end faces of the cracked tooth; q c is the constant crack depth when the crack is constant along the tooth width direction; For crack position l:n1, the crack path is linear; for crack position n2, the crack path is monotonic parabola: Among them, l i and l e is the distance from the crack position of the initial end face and the final end face of the cracked tooth to the tooth root; S302, when cracks exist, the bending stiffness and shear stiffness will be affected. Recalculate the effective section inertia moment and cross-sectional area at a distance y from the tooth root: ① When h c ≥h r and d i >l t hour, ②When h c ≥h r and d i ≤l t or when h c <h r hour, A yc =H yc L=(h y +h c )L Among them, h r is the half-tooth top thickness; h y and h c are the distance from the crack initial point to the gear tooth centerline and the distance from the crack end point to the gear tooth centerline respectively; l t l is the distance from the projection point of the crack end point on the involute tooth profile to the tooth root circle; w and l c are the distances from the crack starting point and end point to the tooth root circle respectively; y is the integral variable when calculating the tooth stiffness; I yc 、A yc and are the effective section inertia moment, section area and inertia radius at a distance y from the fixed end of the cantilever beam when the crack exists; d i is the distance from the load action point of the i-th slice to the tooth root; L is the width of each slice spur gear; S303, total stiffness of the i-th cracked tooth in the helical gear pair The final expression is: in, and is the base stiffness and tooth stiffness of the spur gear at the i-th slice; Bending stiffness Axial compression stiffness and shear stiffness It consists of three parts, and the subscript c represents the corresponding stiffness under the condition of through-crack failure.

6. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 1, wherein: In step S4, based on steps S2 and S3 and the offset superposition concept, the comprehensive time-varying meshing stiffness of the helical gear pair is calculated, and then the inter-tooth load distribution coefficient LSF within the meshing tooth surface of the helical gear pair is calculated. tj and tooth load distribution factor LSF ti , the specific steps are as follows: S401, based on Hertz contact theory, calculates the gear Hertz contact stiffness k h : Where E and v are the elastic modulus and Poisson's ratio of the gear material, respectively; L is the width of each sliced ​​spur gear; S402, based on the normal tooth plate stiffness and cracked tooth plate stiffness calculated in steps S2 and S3 and the Hertz contact stiffness in S401, the stiffness of the i-th spur gear pair slice of the cracked tooth pair and the non-cracked tooth pair in the helical gear pair Respectively expressed as: in, is the stiffness of the i-th slice straight gear pair, x = c, n corresponds to the crack state and normal state respectively, y = p, g corresponds to the driving wheel and driven wheel respectively; Cracked tooth pair K in helical gear pair c_single and normal gear pair K n_single The meshing stiffnesses are: Then the tooth load distribution coefficient of the gear pair with a through crack is S403, based on the determination of the number of meshing gear teeth in step S1, at the tth discrete moment, the total stiffness k of the meshing gear teeth pair at the i-th slice position of the helical gear pair is i for: in, is the meshing stiffness of the jth normal tooth pair in the meshing gear teeth at the i-th slice; ε is the contact ratio of the helical gear; Then the inter-tooth load distribution coefficient of the gear pair with a through-crack is S404, the comprehensive time-varying meshing stiffness K of the helical gear pair with a through-crack is: or, Where N represents the number of slices; K nj_single It is the meshing stiffness of the jth normal tooth pair among the gear teeth participating in the meshing at the same time.

7. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 1, wherein: The bending-axis-torsion coupling dynamic model of the 6-DOF helical gear pair is established as follows: Among them, m1, m2, I1 and I2 are the mass and moment of inertia of the driving and driven wheels respectively; y1, y2, z1, z2, θ z1 and θ z2 are the tangential, axial and torsional vibration displacements of the center points of the driving and driven wheels respectively; k 1y 、k 2y 、k 1z 、k 2z 、c 1y 、c 1z 、c 2y and c 2z are the equivalent bearing support stiffness and equivalent damping in the tangential and axial directions at the center points of the driving and driven wheels, respectively; K is the comprehensive time-varying meshing stiffness between the gear teeth calculated based on step S4, and C is the comprehensive meshing damping between the gear teeth; T1 and T2 are the input torques of the driving and driven wheels, respectively; β is the pitch circle helix angle; R b1 is the base circle radius of the driving wheel; The normal dynamic meshing force F(t) of the helical gear pair can be decomposed into the tangential force F along the meshing line of the gear end face. y and the axial force F along the gear axis z , so the dynamic load F(t) of the helical gear pair during the meshing transmission process is: F n =(F y 2 +F z 2 ) 1 / 2 。 8. The method for calculating the load distribution on the tooth surface of a helical gear pair considering through cracks according to claim 7, wherein: Based on the inter-tooth distribution coefficient LSF in step S4 tj , tooth distribution coefficient LSF ti and dynamic meshing force F(t), and obtain the tooth surface load distribution F of the helical gear pair under the through-crack tji for: F tji =F(t)·LSF ti ·LSF tj Among them, F tji represents the load on the i-th slice of the cracked tooth pair at time t.

9. A system for calculating tooth surface load distribution of helical gear pairs considering through-cracks, characterized in that: The invention comprises a processing unit, wherein the processing unit executes the method according to any one of claims 1 to 8 to calculate the load distribution on the tooth surface of the helical gear pair.

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