A tooth surface coupling load contact analysis method for herringbone planetary gear system

Abstract

The present invention relates to a tooth surface coupling load contact analysis method for a herringbone gear planetary gear system, which considers the installation error of the system, the meshing phase of each inner (outer) gear and its left and right tooth surfaces and the coupling relationship of their forces, establishes a tooth surface geometric contact analysis model of a herringbone gear planetary gear system (DHPGs), obtains the initial contact clearance of the discrete points of the meshing tooth pair contact line of the inner (outer) gear at the meshing position; establishes a finite element model of each gear, obtains the tooth surface node flexibility coefficient, and interpolates to further obtain the flexibility coefficient of the discrete points of the contact line of multiple tooth pairs at the meshing position, establishes a tooth surface load contact analysis equation based on the principles of deformation coordination and tooth pair force balance, and obtains the gear deformation and tooth surface load after loading by solving the nonlinear programming method, thereby laying a theoretical foundation for the tooth surface design and static and dynamic analysis of high-performance DHPGs. The method integrates geometric analysis and mechanical analysis, requires only one finite element calculation, improves the calculation efficiency, and provides a key technology for the transmission performance analysis of herringbone gear planetary systems and power distribution and convergence systems in important application occasions.

Description

Technical Field

[0001] The invention belongs to the technical field of aviation and marine gears, and in particular relates to a tooth surface coupling load contact analysis method for a herringbone planetary gear system. Background Art

[0002] Herringbone gears overcome the disadvantage of helical gears that they generate large axial forces. They can use larger helix angles and tooth widths to obtain greater overlap, making herringbone planetary gear transmission systems (DHPGs) have the advantages of smaller axial forces, higher power density, and smoother transmission. They have gradually replaced spur and helical gears and become the first choice for high-speed, heavy-load power distribution and converging gear transmission systems such as aviation and navigation. Load-bearing contact analysis (LTCA) technology is an important analysis method for numerical simulation of the meshing process of gear teeth under heavy loads. It is a bridge between geometric design and mechanical analysis, and plays a critical role in the research of various types of gears. The static load distribution and load transmission error of the tooth surface obtained by LTCA simulation are the main indicators for measuring the comprehensive meshing performance of the tooth surface. At present, the LTCA method is mainly applied to the meshing of a single pair of gear pairs. Since there is a coupling relationship between the force coupling and the tooth surface contact clearance between the inner and outer gear pairs and the left and right tooth surfaces of the gear pairs in DHPGs, the traditional single pair of gear pair LTCA method is not suitable for the meshing of multiple herringbone planetary gears. Although commercial finite element software is a very effective analysis tool, the mesh quality of the finite element model, the geometric accuracy of the tooth surface nodes, and the assembly accuracy of the gear pair have become key factors affecting the accuracy and efficiency of LTCA solutions, and are not suitable for promotion in engineering applications. There is currently a lack of efficient numerical methods for tooth surface coupled load contact analysis (DPLTCA) for DHPGs at home and abroad. In order to solve the above technical problems, a new technical solution is proposed. Summary of the invention

[0003] Aiming at the coupling characteristics of the forces between the internal and external gear pairs and their left and right tooth surfaces in DHPGs, a numerical method of tooth surface coupling load-bearing contact analysis (DPLTCA) is provided. This method combines the finite element method to closely integrate the precise geometric characteristics and mechanical characteristics of each gear pair and its left and right tooth surfaces. Only one finite element calculation is needed to obtain the load distribution on the entire tooth surface of each gear pair in the system, the load transmission error, meshing stiffness, and load-sharing coefficient, etc. This provides a key technology for the transmission performance analysis of herringbone planetary systems and power distribution and convergence systems in important application scenarios.

[0004] The present invention is achieved through the following technical solutions:

[0005] A tooth surface coupling load contact analysis method for a herringbone planetary gear system comprises the following steps:

[0006] Step 1, setting the basic parameters of the gears in the herringbone planetary gear transmission system;

[0007] Step 2: There are multiple meshing gear pairs in the herringbone planetary gear transmission system, and each gear pair has differences in geometric characteristics caused by phase difference and installation error. In order to reflect the differences in geometric characteristics, the tooth surface contact analysis equations of each meshing gear pair are transformed into a unified fixed coordinate system, and the transmission error calculation of each gear pair uses the same initial rotation angle. According to the TCA solution method of a single pair of gear pairs, the relative initial clearance of each gear pair and its left and right tooth surfaces in the system are determined respectively, that is, the geometric characteristics of each gear pair and its left and right tooth surfaces are obtained;

[0008] Step 3: Construct a finite element mesh model of a herringbone gear by numerical method. The mechanical properties of discrete points at the meshing position of the gear are represented by a matrix. The boundary conditions are that the two sides and the lower edge of the gear body are fixed. The flexibility coefficients of N×N mesh nodes of the gear are calculated by the finite element numerical method. Combined with these mesh node flexibility coefficients, the flexibility coefficients of discrete points of the instantaneous contact line of the tooth surface are obtained by binary interpolation. Further, the flexibility coefficient matrix of discrete points of the contact line of the meshing tooth pair at the meshing position of the system is obtained by synthesis, that is, the mechanical properties of the tooth surface are obtained.

[0009] Step 4: According to the geometric and mechanical characteristics of the herringbone planetary gear transmission system, and in accordance with the deformation coordination, meshing force balance and non-embedded conditions, the tooth surface load contact analysis equation of the herringbone gear planetary system is established. The tooth surface load contact analysis equation is expressed in terms of the normal flexibility matrix F k and its initial clearance w k , normal meshing force P are known input quantities, and the mathematical programming method is used to solve the normal load deformation Z of the gear teeth after loading, the load p of the discrete points of the contact line and the normal contact clearance d of the discrete points of the contact line after loading, and then the transmission performance of the herringbone gear system is analyzed.

[0010] Preferably, the basic parameters of the gear in step 1 include the number of teeth, module, pressure angle, helix angle, tooth width, tooth addendum height, tooth root height and tool fillet radius, and the installation error of the sun gear and the ring gear relative to the reference coordinate system.

[0011] Preferably, the method for determining the geometric characteristics of each gear pair and its left and right tooth surfaces by the tooth surface geometric contact analysis method in step 2 is as follows:

[0012] According to the tooth surface geometric contact analysis equation of the herringbone gear planetary system, the geometric transmission error and the normal clearance of the contact point of the planetary gear relative to the sun gear (ring gear) are obtained. The geometric transmission error is converted into the inter-tooth clearance of the contact point and superimposed with the normal clearance of the contact point to obtain the initial tooth surface clearance of discrete points on the instantaneous contact line of each gear pair. The initial tooth surface clearance is used as the geometric characteristic of the tooth surface contact.

[0013] Preferably, when there is a phase difference in the meshing of the herringbone star gear and its left and right tooth surfaces, the relative phase difference after alignment is further obtained based on the phase difference between the left and right tooth surfaces of the sun gear and the planetary gear, and the phase difference is converted into a normal displacement and superimposed on the initial gap on one side to obtain the geometric characteristics of the system gear pair in which the star gear and its left and right tooth surfaces contain a phase difference.

[0014] Preferably, the tooth surface geometric contact analysis equation of the herringbone gear planetary system is as follows:

[0015]

[0016]

[0017] Among them, R i 、n i are the tooth surface position vector and unit normal vector respectively, i=s, p, r represents the sun gear, planetary gear and ring gear; is the meshing angle of each gear; M fi is the transformation matrix from the gear motion coordinate system to the installation reference coordinate system, L fi is the 3×3 submatrix above; R fi 、N fi is the position vector and unit normal vector of the tooth surface in the reference coordinate system.

[0018] Preferably, the expression of the geometric transmission error is as follows:

[0019]

[0020] In the formula, Z s and Z p are the number of teeth on the sun gear and planet gear respectively, are the initial rotation angles of the sun gear and the planet gear respectively.

[0021] Preferably, the expression of the initial tooth surface clearance is as follows:

[0022] w=[w 11 L a1 ,w 12 L a2 ,L,w ij L ab ], where w ij =δ ij +c ij

[0023] Where w ij is the initial tooth surface clearance at the discrete points of the contact line, i = 1, ..., a, a is the number of discrete points of the contact line, j = 1, ..., b, b is the number of contacting tooth pairs meshing at the same time, c is the normal clearance of the contact point, δ ij is the gap between teeth at the contact point.

[0024] Preferably, the flexibility coefficient of discrete points of the instantaneous contact line of the tooth surface in step 3 is expressed as follows:

[0025]

[0026] Among them, F kn The normal flexibility matrix interpolation of the nth meshing position needs to be obtained through the following step-by-step synthesis method. Assume that the flexibility matrix of the n discrete points of the long axis of the instantaneous contact ellipse of the left and right tooth surfaces of the herringbone gear pair obtained by gear interpolation is

[0027]

[0028] Assume that there are M on the left and right ends of the meshing position 1 、M 2 If the gear teeth are meshed simultaneously, the normal flexibility matrix of the left and right meshing teeth is F pL 、F pR , then the normal flexibility matrix of the meshing teeth at the left and right ends is F t ;

[0029]

[0030] Assume that there are K planetary gears meshing with the ring gear at a certain meshing position, and the normal flexibility matrix interpolation is F k ;

[0031]

[0032] Preferably, the tooth surface load contact analysis equation of the herringbone gear planetary system in step 4, the internal meshing is expressed as follows:

[0033]

[0034] Convert Z into displacement on the meshing line, which is expressed in the form of angle, that is, the load transmission error. The load distribution coefficient on the contact line of the gear pair is the ratio of the sum of the loads of the discrete points on the contact line after loading to the total load. The load distribution coefficient of the left or right tooth surface of the planetary gear at the meshing position is as follows:

[0035]

[0036] The left and right tooth surfaces of the planetary gear are regarded as the comprehensive meshing stiffness of the inner (outer) gear pair m of a parallel spring herringbone gear, and the expression is as follows:

[0037]

[0038] Compared with the prior art, the present invention has the following beneficial technical effects:

[0039] The present invention relates to a method for analyzing the coupled load-bearing contact of the tooth surfaces of a double-helical planetary gear system, which takes into account the installation errors of the system, the meshing phases of each internal (external) gear and its left and right tooth surfaces, and the coupling relationship of the forces, establishes a tooth surface geometric contact analysis model for the double-helical planetary gear system (DHPGs), obtains the initial contact clearance of the discrete points of the contact line of the simultaneously meshing tooth pairs of the internal (external) gears at the meshing position; establishes finite element models of each gear, obtains the tooth surface node flexibility coefficients, and further interpolates to obtain the flexibility coefficients of the discrete points of the multi-tooth pair contact line at the meshing position, establishes a tooth surface load-bearing contact analysis equation according to the principles of deformation coordination and tooth pair force balance, and solves through a nonlinear programming method to obtain the tooth deformation and tooth surface load after loading, laying a theoretical foundation for the tooth surface design and static and dynamic analysis of high-performance DHPGs. This method integrates geometric analysis and mechanical analysis, only requires one finite element calculation, improves the calculation efficiency, and provides key technologies for the transmission performance analysis of double-helical gear systems and power split and confluence systems in important application scenarios. Description of the Drawings

[0040] Figure 1 It is a 3-tooth finite element model of the double-helical gear ring, planet gear, and sun gear of the present invention;

[0041] Figure 2 It is the initial contact clearance of the tooth surfaces of the gear pair of the present invention (tooth space clearance, normal clearance of the contact line);

[0042] Figure 3a It is the installation coordinate system (left end face direction) of the double-helical planetary gear system (DHPGs) of the present invention;

[0043] Figure 3b It is the installation coordinate system of a certain external meshing pair in the DHPGs of the present invention;

[0044] Figure 4 It is the tooth surface contact position sequence of the double-helical gear of the present invention;

[0045] Figure 5 It is the tooth surface mesh and contact line of a helical gear at one end of the double-helical gear of the present invention;

[0046] Figure 6 It is the tooth surface load-bearing contact analysis (DPLTCA) model of the double-helical planetary gear system of the present invention;

[0047] Figure 7 It is the DPLTCA process of the present invention;

[0048] Figure 8a It is the TCA simulation of the external meshing pair in the DHPGs of the present invention;

[0049] Figure 8b It is the tooth surface load distribution coefficient of the external meshing planet gear in the DHPGs of the present invention;

[0050] Figure 8c The load distribution on the tooth surface of the externally meshing planetary gears in the DHPGs of the present invention;

[0051] Figure 8d is the load-sharing coefficient of the outer meshing planetary gears in the DHPGs of the present invention;

[0052] Figure 8e is the tooth surface load coefficient of the outer meshing planetary gear under multiple load conditions in the DHPGs of the present invention;

[0053] Figure 8f It is the transmission error of the external meshing multi-load bearing in the DHPGs of the present invention;

[0054] Figure 8g is the comprehensive meshing stiffness of each gear pair in the external meshing of the DHPGs of the present invention;

[0055] Figure 9a It is the simulation of the internal meshing TCA in DHPGs of the present invention;

[0056] Figure 9b is the load distribution coefficient of the tooth surface of the internal meshing planetary gear in the DHPGs of the present invention;

[0057] Fig.9c The load distribution on the tooth surface of the internally meshing planetary gear in the DHPGs of the present invention; DETAILED DESCRIPTION

[0058] The present invention will be further described in detail below in conjunction with the accompanying drawings, which are intended to explain the present invention rather than to limit it.

[0059] See also Figure 1 -9. A tooth surface coupling load contact analysis method for a herringbone planetary gear system, comprising the following steps:

[0060] Step 1: Set the basic parameters of gears in the herringbone planetary gear transmission system (DHPGs).

[0061] The basic parameters of the gear are shown in Table 1, including: number of teeth, module, pressure angle, helix angle, tooth width, tooth top height, tooth root height and tool fillet radius. The installation error, namely the axis angle error and center distance error, is simplified to the error of the sun gear and ring gear relative to the reference coordinate system. It is assumed that the planetary gears have no installation error.

[0062] Step 2: Transform the TCA equations of each meshing gear pair into a unified fixed coordinate system, and at the same time make the transmission error of each gear pair use the same initial rotation angle. Use the tooth surface geometric contact analysis method to determine the geometric characteristics of each gear pair and its left and right tooth surfaces in the system, that is, the relative initial clearance.

[0063] The geometric characteristics of DHPGs are the meshing positions of each gear pair in the system under no load and the initial contact clearance of the tooth surface at the meshing position. The initial clearance consists of two parts: the inter-tooth clearance and the normal clearance of the tooth surface (see Figure 2 ); It is obtained through the tooth contact analysis (TCA) method. Since there are multiple meshing conditions of gear pairs in the system, and each gear pair has differences in geometric characteristics caused by phase difference and installation error, the TCA method of DPHGs is different from that of a single pair of gears. When TCA analysis is performed on each individual gear pair in the system, this difference in geometric characteristics will disappear, resulting in distortion of subsequent analysis results. The principle of TCA is that two tooth surfaces in continuous tangential contact have common contact points and common normals at any time in the same coordinate system. Therefore, when performing TCA analysis on DHPGs, the specific steps are as follows:

[0064] S2.1. Transform the TCA equations of each internal (external) meshing gear pair into a unified fixed coordinate system;

[0065] Coordinate system such as Figure 3a As shown, O f -X f Y f Z f is a unified fixed coordinate system with the planet carrier rotation center as the origin, and Y f The axis passes through the center of the first planetary gear, and the origin is at the midpoint of the large gear tooth groove; the planetary gear reference coordinate system O fpi -X fpi Y fpi Z fpi (i=1,2...N,N is the number of planetary gears) parallel to it, the planetary gears are evenly distributed, and the reference coordinate system of the sun gear and the ring gear is O fs -X fs Y fs Z fs , O fr -X fr Y fr Z fr ;O pi -X pi Y pi Z pi , O s -X s Y s Z s , O r -X r Y r Z r It is the moving coordinate system of the planetary gear, sun gear and ring gear. The origin coincides with their respective reference coordinate systems and rotates around the z-axis. The meshing coordinate system of a certain external meshing gear pair is shown in Figure 3b ,θ p1 ,θ s, E are the rotation angles of the planetary gear and sun gear and the installation center distance, γ s , ΔE s They are the installation errors of the axis angle and the center distance respectively. To distinguish the right tooth surface coordinate system, the "'" in the subscript indicates the right tooth surface coordinate system and the establishment of the internal meshing reference.

[0066] S2.2. The geometric transmission error obtained by TCA analysis of a single gear pair is the value of the actual rotation angle of the driven gear deviating from the theoretical rotation angle. In order to reflect the difference in geometric characteristics, the transmission error calculation of each gear pair must use the same initial rotation angle, so as to reflect the relative initial clearance of the contact tooth surfaces of each gear pair and the left and right tooth surfaces under the installation error of the system;

[0067] S2.3. After unifying the meshing reference coordinate system (fixed coordinate system) and initial rotation angle of each internal (external) gear pair and its left and right tooth surfaces in DHPGs, the geometric characteristics of each gear pair and its left and right tooth surfaces in the system can be calculated separately according to the TCA method of a single pair of gear pairs.

[0068] When the herringbone gear star wheel and its left and right tooth surface meshing phase difference are not considered, the calculation method of the geometric characteristics is as follows:

[0069] The tooth surface contact analysis (DPTCA) equation of the herringbone gear planetary system is:

[0070]

[0071]

[0072] Where: R i 、n i are the tooth surface position vector and unit normal vector respectively, i=s, p, r represents the sun gear, planetary gear and ring gear; is the meshing angle of each gear; M fi is the transformation matrix from the gear motion coordinate system to the installation reference coordinate system, L fi is the 3×3 submatrix above; R fi 、N fi is the position vector and unit normal vector of the tooth surface in the reference coordinate system; the above formula results in five independent scalar equations, taking a series of As the input, solve the remaining 5 unknown quantities to obtain all the contact points and rotation angles of the meshing positions of the two tooth surfaces; TCA analysis obtains the geometric transmission error of the planetary gear relative to the sun gear (gear ring) and the normal clearance of the contact point:

[0073]

[0074] In the formula, Z s and Z p are the number of teeth on the sun gear (ring gear) and planet gear, They are the initial rotation angles of the sun gear (ring gear) and the planet gear respectively;

[0075] Therefore, the geometric transmission error is converted into the gap between the contact points and superimposed with the normal gap at the contact points. The initial tooth surface gap of the discrete points on the contact line of each external (internal) gear pair at a certain moment is expressed by the vector w as follows:

[0076] w=[w 11 L a1 ,w 12 L a2 ,L,w ij L ab ], where w ij =δ ij +c ij (4)

[0077] Where w ij is the initial tooth surface clearance at the discrete points of the contact line, i = 1, ..., a, a is the number of discrete points of the contact line, j = 1, ..., b, b is the number of contacting tooth pairs meshing at the same time, c is the normal clearance of the contact point, δ ij is the gap between teeth at the contact point.

[0078] When considering the herringbone gear star wheel and its left and right tooth surface meshing phase difference;

[0079] The relative phase difference after alignment is further obtained based on the phase difference between the left and right tooth surfaces of the sun gear (ring gear) and the planetary gear. The phase difference can be converted into a normal displacement and superimposed on the initial gap on one side, thus completing the influence of the phase difference between the left and right tooth surfaces on the geometric characteristics.

[0080] The DPHGs geometric characteristic calculation of the herringbone star gear and its left and right tooth surface meshing phase difference. The geometric characteristic data of a certain meshing position of the system specifically includes the size of the initial contact gap of each gear pair at a certain instantaneous meshing position and the position of each discrete contact point. The installation error and phase difference will lead to differences in geometric characteristics. In order to simplify the calculation in the above TCA method, the output geometric characteristic data has not yet considered the left and right tooth surface phase difference. In addition, the output geometric characteristic data of a single gear pair does not reflect the phase difference of the simultaneously meshing tooth pairs between the gear pairs, so further improvement is needed. Taking the external gear pair as an example, the phase difference of each external meshing pair is:

[0081] Δt=dec(Z s (i-1) / N)T (5)

[0082] In the formula, dec(A) represents the decimal part of A, T is the meshing period, and N is the number of planetary gears. The geometric characteristic data is processed according to the phase difference relationship, that is, the influence of the phase difference of the planetary gear on the geometric characteristics is completed. The relative phase difference after alignment is further obtained according to the phase difference between the left and right tooth surfaces of the sun gear (ring gear) and the planetary gear. The phase difference can be converted into a normal displacement and superimposed on the initial gap on one side, that is, the influence of the phase difference between the left and right tooth surfaces on the geometric characteristics is completed.

[0083] Taking the parameters in Table 1 as an example, one meshing cycle is divided into 5 equal parts, with a total of 14 contact positions. Figure 4 As shown in Table 2, when the number of planetary gears is N=4, each gear pair has no phase difference, that is, at time 0, each planetary gear contacts at tooth surface positions "1", "6" and "11", and the tooth pair contact sequence is shown in Table 2. When the number of planetary gears is N=5, each gear pair has a phase difference, and at time 0, planetary gear 2 contacts at tooth surface positions "4", "9" and "12", and so on. The phase difference between adjacent planetary gears is 0.2T, and the tooth pair contact sequence is shown in Table 3.

[0084] Step 3: Calculate the normal flexibility F, i.e., the mechanical properties, of DPHGs based on the finite element mesh model of the herringbone gears.

[0085] Construct a finite element mesh model of a herringbone gear. The tooth surface finite element mesh model is the basis for calculating the finite element flexibility coefficient. A complete herringbone gear can be regarded as a mirror image of a helical gear plus a tooth groove. The helical gear can be divided into: tooth surface, tooth root, wheel body, tooth groove on the left side of the wheel body, and tooth groove on the right side of the wheel body; first, according to the principle of gear meshing, the tool tooth profile is expressed in the gear coordinate system to obtain the helical gear tooth surface. According to the number of nodes in the radial and axial parts of a single tooth, the node coordinates of the tooth surface and tooth root transition profile on both sides of a single helical gear are determined. Secondly, the contour of the intermediate wheel body is determined according to the boundary points of the tooth root transition profile, and the contour of the intermediate wheel body is rotated around the axis by the angle occupied by one tooth to obtain the tooth groove contour on both sides. Then, after having these boundary contours, according to the number of nodes in the tooth thickness direction, the entity part between the contours is "filled" with discrete points through coordinate rotation transformation, so that the entity of a single helical gear is constructed with discrete points; the node coordinates of the other half of the helical gear are determined by the "mirror" method; the tooth groove nodes are determined by "stretching" the inner end face wheel body of one end of the helical gear. At this point, all nodes of a single herringbone gear have been determined, and the nodes of other teeth are "arrayed" by rotating the coordinate transformation around the axis. According to the requirements of finite element mesh generation, the nodes are renumbered, and through three-layer cyclic control, the outer layer controls the radial direction from the tooth top to the wheel body, the second outer layer controls the tooth thickness direction from one side to the other, and the innermost layer controls the axial direction from one end face to the other. All nodes are reordered tooth by tooth, and then the units and unit nodes are renumbered in the order of 8-node hexahedral linear unit nodes. The applicant has made a detailed introduction in the invention patent of "A method for automatic modeling and assembly of finite element meshes of planetary transmission herringbone gears", which will not be repeated here. Figure 1 It is a finite element model of three gears: ring gear, sun gear and planet gear.

[0086] The DPLTCA model conducts load-bearing research on the initial clearance w and normal flexibility F of the tooth surface at a certain position. The above steps obtain the geometric characteristic data of a certain meshing position. For further analysis, it is also necessary to calculate the mechanical characteristic data of the discrete points of the contact line of these meshing teeth, that is, the flexibility coefficient. For the convenience of calculation, the mechanical characteristics of the discrete points of the meshing position are represented by a matrix. The boundary conditions are that the two sides and the lower edge of the gear wheel body are fixed, and the flexibility coefficient f of the N×N grid nodes of the active (passive) wheel is obtained by finite element calculation. ij , combined with these mesh node flexibility coefficients, and then through binary interpolation to obtain the discrete points of the tooth surface instantaneous contact line ( Figure 5 middle is a grid node, represents the flexibility coefficient of the interpolation point); the normal flexibility matrix of the inner (outer) gear pair is obtained by superimposing the normal flexibility matrices of each planetary gear and the ring gear (sun gear); the calculation steps of the flexibility coefficient of the discrete points of the meshing position and the meshing tooth contact line are as follows:

[0087] S3.1. Assume that the flexibility matrix of n discrete points on the long axis of the instantaneous contact ellipse of the left and right tooth surfaces of the herringbone gear pair is obtained by gear interpolation:

[0088]

[0089] S3.2, Assume that the left and right ends of the meshing position have M 1 、M 2 If the gear teeth are meshed simultaneously, the normal flexibility matrix of the left and right meshing teeth is F pL 、F pR

[0090]

[0091] S3.3. The meshing characteristics of the left and right tooth surfaces of the herringbone gear need to be considered at the same time. The normal flexibility matrix of the meshing teeth on the left and right ends is F t

[0092]

[0093] S3.4. In addition, the normal flexibility of the contact point of the herringbone gear tooth surface needs to consider the number of planetary gears at the meshing position. Assuming that there are K planetary gears and the ring gear (planetary gear and sun gear) meshing at a certain meshing position, the normal flexibility matrix interpolation is F k

[0094]

[0095] S3.5, Assuming that there are 5 meshing positions in one cycle, the normal flexibility matrix is ​​F

[0096]

[0097] Step 4: According to the geometric and mechanical properties, and in accordance with the deformation coordination, meshing force balance, and non-embedded conditions, the tooth surface load contact analysis equation of the herringbone gear planetary system is established. The equation is based on F k 、w k , P are known input quantities. The mathematical programming method is used to solve the normal displacement Z after loading, the discrete load p, and the normal contact clearance d of the discrete points on the tooth surface after loading. The transmission performance of the herringbone gear system is analyzed.

[0098] The tooth surface coupling load contact analysis model (DPLTCA) of the herringbone planetary gear system. Figure 6 As shown in the figure, the contact force of each gear is expanded along the meshing surface. Before the gear teeth are deformed, it is assumed that the internal meshing teeth pair that mesh simultaneously at a certain instant are pair I ab and II ab And the external meshing tooth pair is IIIab and IV ab , where subscript a=1,2 represents planetary gear 1 and planetary gear 2, and subscript b=L,R represents left and right gear teeth. The initial clearance of the tooth surface at this meshing position is: w k =δ k +b k k=I ab , II ab , III ab , IV ab , where w k =[w 1 ,w 2 ,…,w i ,…,w n ] T ; n is the number of discrete points on the major axis of the instantaneous contact ellipse. Taking internal meshing as an example, under the action of load P, the gear teeth only undergo elastic deformation. Assuming that the friction on the tooth surface is ignored, the instantaneous contact line I along the tooth surface ab ,II ab The normal load acting simultaneously produces only a normal displacement Z r ,

[0099] Then ① the tooth surface contact clearance d and Z after loading r The sum should be equal to the initial clearance of the tooth surface before loading;

[0100] ② The sum of the normal loads p acting simultaneously along the tooth surface contact line should be equal to the normal external load P;

[0101] ③ In addition, if a discrete point on the contact line does not bear load, there must still be tooth surface clearance at that point, otherwise there must be no tooth surface clearance at that point. The above conditions are expressed by the mathematical equation as follows:

[0102]

[0103] Step 5: DPLTCA simulation results. The known quantity in the above formula is F k 、w k , P, the mathematical programming method is used to solve the results of the normal displacement Z after loading, the discrete load p, and the normal contact clearance d of the discrete points on the tooth surface after loading; the process of the herringbone planetary gear system load contact analysis (DPLTCA) is shown in Figure 7 Convert Z into displacement on the meshing line and express it in the form of angle, which is the load transmission error. The load distribution coefficient on the contact line of the inner (outer) gear pair is the ratio of the sum of the loads of the discrete points on the contact line after loading to the total load; the load distribution coefficient of a left or right tooth surface of a planetary gear at a certain meshing position is

[0104]

[0105] The comprehensive meshing stiffness of the left and right tooth surfaces as a parallel spring herringbone gear inner (outer) gear pair m is:

[0106]

[0107] Simulation analysis.

[0108] Table 1 Taking planetary gear transmission as an example, the sun gear input torque is 3500N·m and the installation error is ΔE r =ΔE r =0.003mm,γ s =-γ r =0.5′. The number of planetary gears is N=4. Here, the main consideration is the situation where there is no meshing phase difference between the gear pairs, that is, the inner and outer gear pairs are meshed at the same time without installation error. In addition, the floating characteristics of the components and the tooth surface modification are not considered. The DPLTCA simulation results are analyzed as follows.

[0109] Table 1 Basic parameters of planetary transmission gear pairs

[0110]

[0111] Table 2 Contact sequence of internal (external) meshing multiple teeth (no phase difference)

[0112]

[0113] Table 3 Contact sequence of external meshing multiple teeth (with phase difference)

[0114]

[0115]

[0116] Taking the external meshing gear pair as an example, TCA simulation shows that the installation error causes the relative initial clearance of each gear pair to increase, and the center distance error causes the pitch circle position to change, resulting in the radial change of the gear pair contact trace ( Figure 8a ); The initial clearance of the standard tooth surface gear pair is the tooth clearance. Since the tooth clearance of the left tooth surface is planetary gear 1 < planetary gear 4 < planetary gear 2 < planetary gear 3, and the tooth clearance of the right tooth surface is planetary gear 4 < planetary gear 3 < planetary gear 1 < planetary gear 2, the smaller the tooth surface clearance, the more load it bears. Therefore, the tooth surface load distribution ( Figure 8b )、Maximum load distribution factor( Figure 8c ) changes gradually accordingly, that is, planetary gear 1>planetary gear 4>planetary gear 2>planetary gear 3 in the left tooth surface, and planetary gear 4>planetary gear 3>planetary gear 1>planetary gear 2 in the right tooth surface; the sum of the tooth surface loads of all gear pairs in the same meshing position is equal to the total load, that is, the sum of the load distribution coefficients at the contact positions of the left tooth surfaces 1, 6, 11 and the right tooth surfaces 1, 6, 11 is equal to 1.

[0117] The load-sharing coefficient reflects the macroscopic load distribution between the planetary gears. During the meshing process, the load borne by the tooth surface also changes gradually with the corresponding inter-tooth gap size. The sum of the left or right load-sharing coefficients of the planetary gears at a certain meshing moment is equal to 4. As the load increases, the load borne by each planetary gear gradually becomes equal, that is, the load-sharing coefficient gradually approaches 1 ( Figure 8d , Figure 8e ). It can be seen that when there is no floating of the components, in addition to causing uneven load distribution between the planetary gears, it is also easy to cause uneven loads on the left and right tooth surfaces of each gear.

[0118] Figure 8f It reflects the changes in the comprehensive deformation of each gear pair with the change of load torque, and can also reflect the load change trend of each gear. When the system torque is <100N.m, the left tooth surface of each external meshing planetary gear is not loaded, and it is mainly loaded by the right tooth surface; when the load torque gradually increases to 1000N.m, the left tooth surface of the planetary gear gradually loads, and all gears are fully loaded at 1000N.m. For the right tooth surface of the planetary gear, when the load is <100N.m, it is mainly carried by planetary gear 1 and planetary gear 4. As the torque increases to 600N.m, planetary gear 1 and planetary gear 2 are loaded successively, until the load increases to 2000N.m, the initial clearance of the right tooth surface of planetary gear 4 is completely eliminated, and all planetary gears are fully loaded.

[0119] The meshing stiffness of the left and right tooth surfaces of the herringbone gears depends on two parallel springs. Therefore, the size of the comprehensive meshing stiffness of each gear pair is consistent with the size of the tooth surface load capacity. That is, in the comprehensive meshing stiffness, planetary gear 1> planetary gear 4> planetary gear 2> planetary gear 3 ( Figure 8g ).

[0120] The same analysis method can be used for internal gear pairs, mainly providing the tooth surface TCA simulation, contact line load distribution coefficient and tooth surface load distribution simulation. Since the inter-tooth clearance of the left tooth surface is planetary gear 1 < planetary gear 4 < planetary gear 2 = planetary gear 3, and the inter-tooth clearance of the right tooth surface is planetary gear 2 < planetary gear 3 < planetary gear 1 < planetary gear 4 ( Figure 9a ), so the maximum load distribution factor ( Figure 9b ) and tooth surface load distribution ( Fig.9c ) is the corresponding and opposite changes gradually.

[0121] The present invention can complete the loading contact analysis of the herringbone planetary gear system without the need for modeling, assembly, pre-processing, etc. with the help of commercial finite element software. This method takes into account the accurate geometric characteristics of the tooth surface of the system gear pair under the condition of installation error and the coupling characteristics of the force between the gear pairs, and closely integrates the geometric analysis of the tooth surface of each gear pair in the system with the mechanical analysis. By programming and inputting the basic parameters and installation errors of the planetary gear system, only one finite element calculation is required to obtain the tooth surface deformation, meshing stiffness, load distribution, and load-sharing coefficient after loading, providing key data for the further modification design, load-sharing analysis, and dynamic analysis of the high-performance herringbone planetary transmission system, and developing related numerical simulation software to improve product design efficiency.

[0122] The above contents are only for explaining the technical idea of ​​the present invention and cannot be used to limit the protection scope of the present invention. Any changes made on the basis of the technical solution in accordance with the technical idea proposed by the present invention shall fall within the protection scope of the claims of the present invention.

Claims

1. A tooth surface coupling load contact analysis method for a herringbone planetary gear system. It is characterized in that The following steps are involved: Step 1, setting the basic parameters of the gears in the herringbone planetary gear transmission system; Step 2: There are multiple meshing gear pairs in the herringbone planetary gear transmission system, and each gear pair has differences in geometric characteristics caused by phase difference and installation error. In order to reflect the differences in geometric characteristics, the tooth surface contact analysis equations of each meshing gear pair are transformed into a unified fixed coordinate system, and the transmission error calculation of each gear pair uses the same initial rotation angle. According to the TCA solution method of a single pair of gear pairs, the relative initial clearance of each gear pair and its left and right tooth surfaces in the system are determined respectively, that is, the geometric characteristics of each gear pair and its left and right tooth surfaces are obtained; The method of using the tooth surface geometric contact analysis method to determine the geometric characteristics of each gear pair and its left and right tooth surfaces is as follows: According to the tooth surface geometric contact analysis equation of the herringbone gear planetary system, the geometric transmission error and the normal clearance of the contact point of the planetary gear relative to the sun gear ring are obtained, and the geometric transmission error is converted into the contact point inter-tooth clearance and superimposed with the contact point normal clearance to obtain the initial tooth surface clearance of the discrete points on the instantaneous contact line of each gear pair, and the initial tooth surface clearance is used as the geometric characteristic of the tooth surface contact; The tooth surface geometric contact analysis equation of the herringbone gear planetary system is as follows: Among them, R i 、n i are the tooth surface position vector and unit normal vector respectively, i=s, p, r represents the sun gear, planetary gear and ring gear; is the meshing angle of each gear; M fi is the transformation matrix from the gear motion coordinate system to the installation reference coordinate system, L fi is the 3×3 submatrix above; R fi 、N fi is the position vector and unit normal vector of the tooth surface in the reference coordinate system; The expression of the initial tooth surface clearance is as follows: w = [w 11 …w a1 ,w 12 …w a2 ,…,w ij …w ab , where w ij = δ ij + c ij Where w ij is the initial tooth surface clearance at the discrete points of the contact line, i = 1, ..., a, a is the number of discrete points of the contact line, j = 1, ..., b, b is the number of contacting tooth pairs meshing at the same time, c is the normal clearance of the contact point, δ ij is the gap between teeth at the contact point; When there is a phase difference between the herringbone gear star wheel and its left and right tooth surfaces, the relative phase difference after alignment is further obtained according to the phase difference between the left and right tooth surfaces of the sun gear and the planetary gear. The phase difference is converted into a normal displacement and superimposed on the initial gap on one side to obtain the geometric characteristics of the system gear pair containing the phase difference between the star wheel and its left and right tooth surfaces. Step 3: Construct a finite element mesh model of a herringbone gear by numerical method. The mechanical properties of discrete points at the meshing position of the gear are represented by a matrix. The boundary conditions are that the two sides and the lower edge of the gear body are fixed. The flexibility coefficients of N×N mesh nodes of the gear are calculated by the finite element numerical method. Combined with these mesh node flexibility coefficients, the flexibility coefficients of discrete points of the instantaneous contact line of the tooth surface are obtained by binary interpolation. Further, the flexibility coefficient matrix of discrete points of the contact line of the meshing tooth pair at the meshing position of the system is obtained by synthesis, that is, the mechanical properties of the tooth surface are obtained. The flexibility coefficient of the discrete points of the instantaneous contact line of the tooth surface is expressed as follows: Among them, F kn The normal flexibility matrix interpolation of the nth meshing position needs to be obtained through the following step-by-step synthesis method. Assume that the flexibility matrix of the n discrete points of the long axis of the instantaneous contact ellipse of the left and right tooth surfaces of the herringbone gear pair obtained by gear interpolation is Assume that there are M on the left and right ends of the meshing position 1 、M 2 If the gear teeth are meshed simultaneously, the normal flexibility matrix of the left and right meshing teeth is F pL 、F pR， Then the normal flexibility matrix of the meshing teeth on the left and right ends is F t ; Assume that there are K planetary gears meshing with the ring gear at a certain meshing position, and the normal flexibility matrix interpolation is F k ; Step 4: According to the geometric and mechanical characteristics of the herringbone planetary gear transmission system, and in accordance with the deformation coordination, meshing force balance and non-embedded conditions, the tooth surface load contact analysis equation of the herringbone gear planetary system is established. The tooth surface load contact analysis equation is expressed in terms of the normal flexibility matrix F k and its initial clearance w k , normal meshing force P are known input quantities, and the normal load deformation Z of the gear teeth after loading, the load p of the discrete points of the contact line and the normal contact clearance d of the discrete points of the contact line after loading are solved by mathematical programming method, and then the transmission performance of the herringbone gear system is analyzed; The tooth surface load contact analysis equation of the herringbone gear planetary system and the internal meshing are expressed as follows: Convert Z into displacement on the meshing line, which is expressed in the form of angle, that is, the load transmission error. The load distribution coefficient on the contact line of the gear pair is the ratio of the sum of the loads of the discrete points on the contact line after loading to the total load. The load distribution coefficient of the left or right tooth surface of the planetary gear at the meshing position is as follows: The left and right tooth surfaces of the planetary gear are regarded as the comprehensive meshing stiffness of the internal and external gear pair m of a parallel spring herringbone gear. The expression is as follows:

2. A tooth surface coupling load contact analysis method for a herringbone planetary gear system according to claim 1, It is characterized in that The basic parameters of the gears described in step 1 include the number of teeth, module, pressure angle, helix angle, tooth width, tooth top height, tooth root height and tool fillet radius, and the installation error of the sun gear and ring gear relative to the reference coordinate system.

3. The tooth surface coupling load contact analysis method of a herringbone planetary gear system according to claim 1, It is characterized in that The expression of the geometric transmission error is as follows: In the formula, Z s and Z p are the number of teeth on the sun gear and planet gear respectively, are the initial rotation angles of the sun gear and the planet gear respectively.

Images