A high load carrying cylindrical gear design method based on curvilinear meshing line

Abstract

The application provides a high-bearing cylindrical gear design method based on a curved meshing line, comprising the following steps: step 1, tooth profile solving based on a curved meshing line; step 2, calculation of an undercut critical value; step 3, derivation of a rack cutter equation; step 4, tooth profile design based on a parabolic meshing line; step 5, three-dimensional finite element grid modeling of a new gear parameterization; and step 6, error sensitivity analysis. The application of the technical scheme can realize that the bearing capacity of the new gear designed based on the curved meshing line is significantly improved compared with the involute gear.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of gear tooth profile design, and particularly relates to a high-load cylindrical gear design method based on a curved meshing line. BACKGROUND

[0002] With the vigorous development of modern industry, the demand for high-power and high-load gear transmission systems of mechanical equipment is gradually increasing. Although involute gears have superior performance, they also gradually expose the shortcomings of low tooth surface contact strength and limited bearing capacity with the progress of science and technology. In some large mechanical equipment, tooth surface pitting and even galling occur in the gear transmission system, which seriously affects the performance and reliability. As known, the tooth profile shape is the fundamental reason for determining the performance of gear transmission. Therefore, in order to meet the demand of modern industry for high-load and high-power gear transmission systems, new non-involute tooth profiles need to be explored to design new gear transmission systems. SUMMARY

[0003] In view of this, the purpose of the present application is to provide a high-load cylindrical gear design method based on a curved meshing line, which can significantly improve the bearing capacity of the new gear designed based on the curved meshing line compared with the involute gear.

[0004] To achieve the above purpose, the present application adopts the following technical solution: a high-load cylindrical gear design method based on a curved meshing line, comprising the following steps:

[0005] Step 1: tooth profile solving based on a curved meshing line;

[0006] Step 2: calculation of undercut critical value;

[0007] Step 3: derivation of rack cutter equation;

[0008] Step 4: tooth profile design based on a parabolic meshing line;

[0009] Step 5: parameterized three-dimensional finite element mesh modeling of the new gear;

[0010] Step 6: error sensitivity analysis.

[0011] In a preferred embodiment, the step 1 is specifically: a coordinate system S f is a fixed coordinate system, and the coordinate system origin is located at the node; the coordinate systems S1 and S2 are dynamic coordinate systems, and the coordinate system origins are located at the centers of the two gears; the locus formed by the contact points in the fixed coordinate system S f during the transmission process of the gear pair is the meshing line; according to the gear meshing principle, the locus formed by transforming the meshing points represented in the fixed coordinate system into the dynamic coordinate systems S1 and S2 respectively satisfies the conjugate profile of the preset meshing line; x f and y frespectively, the coordinate components of the X axis and the Y axis of the fixed coordinate system S f The coordinate components of the X axis and the Y axis of the fixed coordinate system S f The coordinate components of the X axis and the Y axis of the fixed coordinate system S

[0012]

[0013] The coordinate components of the X axis and the Y axis of the fixed coordinate system S

[0014]

[0015]

[0016] wherein: Φ1, Φ2 are the angular displacements of the two gears in the meshing process respectively; r1, r2 are the pitch circle radii of the two gears respectively;

[0017] u t is the parameter of the preset meshing line; [L] i,f is the coordinate transformation matrix from S f to S i , i = 1, 2;

[0018] According to the meshing principle of the plane gear, the common normal line of the spur gear at the meshing point must pass through the instantaneous center O f ; in S f , the unit normal vector is represented as:

[0019]

[0020] The unit normal vector is represented in S1:

[0021]

[0022] According to the meshing principle of the gear, the common normal line of the two conjugate tooth profiles at the contact point position is perpendicular to the relative velocity direction:

[0023]

[0024] f1 is the meshing equation, which means that the common normal line of the conjugate tooth surface at the contact point position is perpendicular to the relative velocity at the point;

[0025] is the position vector of the common normal vector of the two tooth surfaces in the coordinate system S1; x′ f , y′ f are the first-order derivative functions of x f and y f about u t ;

[0026] After the meshing equation is arranged, the relationship between the angular displacement of the first gear and the parameter u t of the preset meshing line is obtained:

[0027]

[0028] According to the two-gear transmission ratio relationship, the relationship between the second gear angular displacement and the preset meshing line parameter u can be obtained: t

[0029]

[0030] In a preferred embodiment, the step 2 is specifically: calculating the new gear undercut critical parameter;

[0031] The tooth surface sliding speed is used as a condition to derive the new gear undercut critical value:

[0032]

[0033]

[0034] The relative velocity vector at the tooth profile contact point position;

[0035] The equation is obtained by sorting to get the first gear undercut critical value:

[0036] (1+m 12 )x f x′ f +[(1+m 12 )y f +(r1+r2)]y′ f =0

[0037] m 12 is the transmission ratio of the two gears;

[0038] Similarly, the second gear undercut critical value is:

[0039] (1+m 12 )x f x′ f +{y f +m 12 [y f -(r1+r2)]}y′ f =0.

[0040] In a preferred embodiment, the step 3 is specifically: presetting the meshing line P1P2, establishing the coordinate system S C1 is a moving coordinate system fixed with the rack cutter, the coordinate axis X C1 is coincident with the pitch line of the rack cutter; in the initial position, the moving coordinate system S C1 is coincident with the fixed coordinate system S f ​The rack cutter tooth profile intersects with the meshing line P1P2 at point D0, and with the meshing line P1P2 at point D as the rack cutter moves;

[0041]

[0042] According to the gear meshing theory, the tangent line at point D of the rack cutter tooth profile must be perpendicular to the line segment O f -D, thus we have:

[0043]

[0044]

[0045] x c1 and y c1 are the coordinate components of the X axis and Y axis of the rack cutter equation in the fixed coordinate system S c1

[0046] Thus the rack cutter equation for machining the first gear is obtained:

[0047]

[0048] Wherein:

[0049] α is a preset parabola parameter, and Δ is an integral constant determined by the initial position of the rack cutter;

[0050] According to the Camus theorem, when two rack cutters capable of being mutually embedded are used to perform gear generating according to the conjugate machining principle, the machined gears are also mutually conjugate; therefore, after the rack tooth profile of the first gear is obtained, the derivation of the rack tooth profile equation of the second gear is completed by using the spatial embedding relationship of the two rack cutters.

[0051] In a preferred embodiment, the step 4 is specifically that: the parabola vertex in the first quadrant is located at O f (0,0), the focus is located at FF(0, p1 / 2), and α is a preset parabola parameter;

[0052] In the coordinate system S f , the position vector of the preset meshing line is represented as:

[0053]

[0054] p1 is the first quadrant preset parabola focal distance; in the triangle O f MF, there are the following geometric relationships:

[0055]

[0056] ​

[0057]

[0058] The relationship between the parameter a and u is obtained as follows: t

[0059]

[0060] The equation of the parabolic meshing line in the first quadrant is:

[0061]

[0062] Wherein:

[0063]

[0064] The equation of the meshing line in the third quadrant is:

[0065]

[0066] p2 is the preset parabolic focal distance in the third quadrant; wherein:

[0067] Substitute the meshing line equation obtained above into the undercut relationship formula derived in step 2:

[0068] (1+m 12 )x f x′ f +{y f +m 12 [y f -(r1+r2)]}y′ f =0;

[0069] The undercut critical condition is obtained as follows:

[0070]

[0071] According to the gear meshing theory, the gear coincidence degree is ensured to be greater than 1; when the moving point M is located on the meshing line in the first quadrant, the angle β should be greater than the angle corresponding to one fourth of the gear teeth, so the range of the parameter β is expressed as:

[0072]

[0073] In the triangle O2MF, there are the following geometric relationships:

[0074]

[0075]

[0076] Therefore, the following is obtained:

[0077] ​

[0078] The condition that k1 does not produce root cutting is obtained from the above derivation:

[0079]

[0080] Similarly, the condition that k2 does not produce root cutting is:

[0081]

[0082] In a preferred embodiment, the step 5 is specifically: based on MATLAB, a gear simulation machining program is written; firstly, the instantaneous cutting point coordinates of the rack cutter from entering engagement to exiting engagement are solved by using gear engagement principle; then, the plane node coordinates of the double-sided tooth profile of the single gear tooth end surface are determined by rotation projection transformation; secondly, the plane node coordinates of the gear tooth base body are obtained by rotation projection transformation; finally, all the space node coordinates of the single tooth are calculated by using the coordinate transformation in the direction of the tooth width;

[0083] The calculated node coordinate file is imported into the finite element software ABAQUS;

[0084] The tooth modification curve is composed of two second-order parabolas and a straight line, wherein y1 and y2 are the maximum modification amounts; y3 is the unmodified length in the tooth width direction; the unmodified gear tooth surface and the modification amount surface are meshed; the tooth modification mesh model is established based on the superposition method of the original tooth surface and the tooth modification amount surface; the modification amount of the corresponding mesh node of the tooth surface is calculated according to the tooth modification curve equation; the point P(x, y) is the mesh space node coordinate of the tooth surface, and the modification value δ of the node P(x, y) is determined according to the two-dimensional modification curve F ; the space node coordinates of the gear are re-solved according to the modification amount at the tooth surface node; finally, the calculated modification gear node coordinate file is imported into ABAQUS, and the rapid three-dimensional modeling of the modification gear is realized.

[0085] In a preferred embodiment, the step 6 is specifically: installation error Δγ is added in the meshing coordinate system of the gear; the coordinate system Sg is the gear coordinate system without installation error, and the coordinate system SF is the gear coordinate system with installation error; the rotation transformation equation from the coordinate system SF to the coordinate system Sg is:

[0086]

[0087] Compared with the prior art, the present application has the following beneficial effects:

[0088] The application provides a high-load new type cylindrical gear design method based on a curved meshing line, and carries out meshing characteristic analysis of the new type gear by taking into account installation error and tooth modification by using a load contact analysis technology. First, according to the knowledge of differential geometry and the gear meshing theory, the tooth profile design of the high-load cylindrical gear based on the parabolic meshing line is completed, and the rack tooth profile equation for machining the new type gear is calculated; the accurate three-dimensional grid space node coordinates of the new type gear are solved according to the relative motion relationship between the rack cutter and the machined gear, and the parameterized three-dimensional finite element grid modeling of the new type gear is completed; the tooth modification finite element grid model of the new type gear is established in the form of superposition of the original tooth surface and the tooth modification amount. The finite element analysis result of the new type gear shows that, compared with the involute gear, the new type gear designed based on the parabolic meshing line has higher load capacity, and has high application potential in multi-stage gear transmission and planetary gear transmission. BRIEF DESCRIPTION OF DRAWINGS

[0089] Figure 1 The gear meshing coordinate system of the preferred embodiment of the application;

[0090] Figure 2 The derivation of the rack cutter tooth profile equation based on the meshing line of the preferred embodiment of the application;

[0091] Figure 3 The parabolic meshing line in the first quadrant of the preferred embodiment of the application;

[0092] Figure 4 The parabolic meshing line in the third quadrant of the preferred embodiment of the application;

[0093] Figure 5 The gear tooth surface grid division of the preferred embodiment of the application;

[0094] Figure 6 The modification amount curved surface grid division of the preferred embodiment of the application;

[0095] Figure 7 The gear coordinate system considering installation error Δγ of the preferred embodiment of the application;

[0096] Figure 8 The new type gear designed based on different meshing lines of the preferred embodiment of the application;

[0097] Figure 9 The influence of the parabolic parameter k1 on the tooth profile of the new type gear of the preferred embodiment of the application;

[0098] Figure 10 The influence of the parabolic parameter k2 on the tooth profile of the new type gear of the preferred embodiment of the application;

[0099] Figure 11Comparison of contact stress between new type gear and involute gear for preferred embodiment of the present application;

[0100] Figure 12 Comparison of bending compressive stress between new type gear and involute gear for preferred embodiment of the present application;

[0101] Figure 13 Comparison of bending tensile stress between new type gear and involute gear for preferred embodiment of the present application;

[0102] Figure 14 Tooth surface load distribution of new type gear under different modification parameters for preferred embodiment of the present application;

[0103] Figure 15 Comparison of contact stress between new type gear under different modification amount for preferred embodiment of the present application;

[0104] Figure 16 Comparison of bending compressive stress between new type gear under different modification amount for preferred embodiment of the present application;

[0105] Figure 17 Comparison of bending tensile stress between new type gear under different modification amount for preferred embodiment of the present application;

[0106] Figure 18 Tooth surface load distribution of unmodified involute gear under error working condition for preferred embodiment of the present application;

[0107] Figure 19 Tooth surface load distribution of unmodified new type gear under error working condition for preferred embodiment of the present application;

[0108] Figure 20 Tooth surface load distribution of modified involute gear under error working condition for preferred embodiment of the present application;

[0109] Figure 21 Tooth surface load distribution of modified new type gear under error working condition for preferred embodiment of the present application;

[0110] Figure 22 Contact stress of unmodified involute gear under error working condition for preferred embodiment of the present application;

[0111] Figure 23 Contact stress of unmodified new type gear under error working condition for preferred embodiment of the present application;

[0112] Figure 24 Contact stress of tooth direction modified involute gear under error working condition for preferred embodiment of the present application;

[0113] Figure 25 Contact stress of tooth direction modified new type gear under error working condition for preferred embodiment of the present application;

[0114] Figure 26 Bending compressive stress of the unmodified involute gear under error condition for the preferred embodiment of the present application;

[0115] Figure 27 Bending compressive stress of the unmodified new type gear under error condition for the preferred embodiment of the present application;

[0116] Figure 28 Bending compressive stress of the modified in tooth direction involute gear under error condition for the preferred embodiment of the present application;

[0117] Figure 29 Bending compressive stress of the modified in tooth direction new type gear under error condition for the preferred embodiment of the present application;

[0118] Figure 30 Bending tensile stress of the unmodified involute gear under error condition for the preferred embodiment of the present application;

[0119] Figure 31 Bending tensile stress of the unmodified new type gear under error condition for the preferred embodiment of the present application;

[0120] Figure 32 Bending tensile stress of the modified in tooth direction involute gear under error condition for the preferred embodiment of the present application;

[0121] Figure 33 Bending tensile stress of the modified in tooth direction new type gear under error condition for the preferred embodiment of the present application. DETAILED DESCRIPTION

[0122] The present application will be further described below in conjunction with the accompanying drawings and embodiments.

[0123] It should be noted that the following detailed description is exemplary in nature and is intended to provide further description of the application. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.

[0124] It is to be understood that the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. It should also be noted that, as used herein, the term “and / or” includes any and all combinations of one or more of the associated listed items.

[0125] A high load cylindrical gear design method based on a curved meshing line, with reference to Figures 1 to 33 , comprising the following steps:

[0126] Step 1: Tooth profile solving based on curvilinear meshing line

[0127] As Figure 1 shown, coordinate system S f is a fixed coordinate system, and the coordinate system origin is located at the node; coordinate systems S1 and S2 are dynamic coordinate systems, and the coordinate system origins are located at the centers of the two gears. The trajectory formed by the contact points in the fixed coordinate system S f during the gear transmission process is the meshing line. According to the gear meshing principle, the trajectory formed by the meshing points represented in the fixed coordinate system is the conjugate profile that satisfies the preset meshing line.

[0128] The coordinate expression of the position vector of the preset meshing line in S f is:

[0129]

[0130] The position vector expression of the conjugate tooth profile is:

[0131]

[0132]

[0133] Wherein:

[0134] u t is the parameter of the preset meshing line

[0135] [L] i,f is the coordinate transformation matrix from S f to S i , i = 1, 2

[0136] According to the meshing principle of spur gears, the common normal line of the spur gears at the meshing point must pass through the instantaneous center O f . In S f , the unit normal vector is expressed as:

[0137]

[0138] Express the unit normal vector in S1:

[0139]

[0140] According to the gear meshing principle, the common normal line of the two conjugate tooth profiles at the contact point position is perpendicular to the relative velocity direction:

[0141]

[0142] After rearranging the meshing equation, the relationship between the angular displacement of the first gear 1 and the preset meshing line parameter u t can be obtained:

[0143]

[0144] Based on the transmission ratio between the two gears, the angular displacement of the second gear and the preset meshing line parameter u can be obtained. t The relationship between them:

[0145]

[0146] Step 2: Calculation of Critical Root Cutting Value

[0147] During gear manufacturing, improper design parameters can lead to undercutting of the gear teeth, resulting in decreased bending strength and reduced contact ratio, thus affecting the smoothness of gear transmission. Therefore, it is necessary to calculate the critical parameters for undercutting of new gears.

[0148] According to gear meshing theory, when the gear tooth profile curve slides relative to each other at a singular point, undercutting may occur at the singular point.

[0149] Therefore, the critical value for undercut of a new type of gear can be derived using the tooth surface sliding speed as a condition:

[0150]

[0151]

[0152] By rearranging the equations, we can obtain the critical value for the first gear to be cut at one root:

[0153] (1+m 12 )x f x′ f +[(1+m 12 )y f +(r1+r2)]y′ f =0

[0154] Similarly, the critical value for the two tangents of the second gear is:

[0155] (1+m 12 )x f x′ f +{y f +m 12 [y f -(r1+r2)]}y′ f =0

[0156] Step 3: Derivation of the equation for the rack cutter

[0157] To complete the machining and manufacturing of the new type of gear, it is necessary to derive the tooth profile equation of the rack cutter.

[0158] like Figure 2As shown, curves P1P2 are the preset meshing lines, and the coordinate system is S. C1 It is a moving coordinate system fixed to the rack cutter, with the X-axis as the coordinate axis. C1 Align with the pitch line of the rack cutter. At the initial position, move coordinate system S. C1 With fixed coordinate system S f The tooth profile of the rack cutter intersects the line of action P1P2 at point D0. As the rack cutter moves, the rack cutter and the line of action P1P2 intersect at point D.

[0159]

[0160] According to gear meshing theory, the tangent at point D on the tooth profile of the rack cutter must be perpendicular to line segment O. f -D, from which we can obtain:

[0161]

[0162]

[0163] This yields the equation for the rack cutter used to machine the first gear 1:

[0164]

[0165] in:

[0166] Δ is the integration constant, determined by the initial position of the rack cutter.

[0167] According to Camus' theorem, when two interlocking rack cutters are used to generate a gear according to the principle of conjugate machining, the resulting gears will also mesh conjugately. Therefore, after obtaining the rack tooth profile of the first gear 1, the equation of the rack tooth profile of the second gear 2 can be derived using the spatial interlocking relationship of the two rack cutters.

[0168] Step 4: Tooth profile design based on parabolic meshing line

[0169] like Figure 3 As shown, the vertex of the parabola in the first quadrant is located at O. f (0,0), with focus at F(0,p1 / 2), and α is the preset parabola parameter.

[0170] Coordinate system S f Below, the position vector of the preset engagement line is represented as:

[0171]

[0172] In triangle O f In MF, the following geometric relations exist:

[0173]

[0174]

[0175]

[0176] The relationship between the parameter a and u is obtained as follows: t

[0177]

[0178] The equation of the parabolic meshing line in the first quadrant is:

[0179]

[0180] Wherein:

[0181]

[0182] The equation of the meshing line in the third quadrant is:

[0183]

[0184] Wherein:

[0185] Substitute the meshing line equation obtained above into the undercutting relationship derived in step 2:

[0186] (1+m 12 )x f x′ f +{y f +m 12 [y f -(r1+r2)]}y′ f =0

[0187] The critical condition of undercutting is obtained as follows:

[0188]

[0189] According to the gear meshing theory, to ensure smooth transmission, the gear addendum must be greater than 1. When the moving point M is located on the meshing line in the first quadrant, the angle β should be greater than the angle corresponding to one-fourth of the gear teeth, so the range of the parameter β can be expressed as:

[0190]

[0191] In the triangle O2MF, there are the following geometric relationships:

[0192]

[0193]

[0194] ​Thus we get:

[0195]

[0196] From the above derivation, the condition that k1 does not produce undercutting must be met:

[0197]

[0198] Similarly, the condition that k2 does not produce undercutting must be met:

[0199]

[0200] Step 5: Parametric 3D finite element mesh modeling of new gear

[0201] A gear simulation machining program is written based on MATLAB. First, the coordinates of the instantaneous cutting points of the rack cutter from entering engagement to exiting engagement are solved using the gear meshing principle; then, the plane node coordinates of the double-sided tooth profile of the single tooth face are determined through rotation projection transformation; secondly, the plane node coordinates of the gear body are obtained through rotation projection transformation; finally, all spatial node coordinates of a single tooth are calculated using coordinate transformation along the tooth width direction; as shown in Figures 5-6 .

[0202] The calculated node coordinate file is imported into the finite element software ABAQUS to realize fast 3D modeling of the new gear.

[0203] The tooth modification curve is composed of two second-order parabolas and a straight line, where y1 and y2 are the maximum modification amounts; y3 is the unmodified length in the tooth width direction. The unmodified gear tooth surface and the modification amount surface are meshed. Based on the method of superimposing the original tooth surface and the tooth modification amount surface, a tooth modification mesh model is established, and the modification amount of the corresponding mesh nodes of the tooth surface is calculated according to the tooth modification curve equation. Point P(x, y) is the spatial node coordinate of the tooth surface mesh, and the modification value δ F of node P(x, y) can be determined according to the two-dimensional modification curve. According to the modification amount at the tooth surface node, the spatial node coordinates of the gear are recalculated. Finally, the calculated modification gear node coordinate file is imported into ABAQUS to realize fast 3D modeling of the modification gear.

[0204] Step 6: Error sensitivity analysis

[0205] Installation error Δγ is added to the gear engagement coordinate system to analyze the effect of installation error on the performance of the proposed new gear. Coordinate system S g is the gear coordinate system without installation error, and coordinate system S F is the gear coordinate system with installation error. The rotation transformation equation from coordinate system S F to coordinate system S g :

[0206]

[0207] The design parameters of the gear are shown in Table 1. The Young's modulus of the gear material is 2.01 x 105MPa; the Poisson's ratio is 0.29. The effect of friction between the two contacting tooth surfaces is not considered. A torque of 134 Nm is applied at the reference point 1 of the pinion. 5

[0208] Table 1 Design parameters of the gear pair

[0209]

[0210] To verify the effectiveness of the proposed design method of high load carrying cylindrical gears based on the curved meshing line, the gear transmission with parabolic meshing line is taken as an example to illustrate the proposed design method. The contact stress and bending stress of the new gear transmission are calculated by using the finite element method, and are compared with those of the involute gear.

[0211] The design parameters of the gear are shown in Table 2. The Young's modulus of the gear material is 2.01 x 105MPa; the Poisson's ratio is 0.29. The effect of friction between the two contacting tooth surfaces is not considered. A torque of 134 Nm is applied at the reference point 1 of the pinion.

[0212] Table 2 Design parameters of the gear pair

[0213]

[0214]

[0215] Step 1: Tooth profile design of the new gear

[0216] According to the design parameters of the new gear in Table 2, the parameter ranges of the parabolic line without undercut are 0.2≤k1<0.473 and 0.2≤k2<0.454, respectively. In order to study the influence of the parameters of the parabolic meshing line on the tooth profile of the new gear, the parabolic lines with different parameters are designed, and the corresponding tooth profiles of the new gear are given, as shown in Figs. 2-5. Figure 8

[0217] Figure 9 The influence of the parameter k1 on the tooth profile of the new gear is reflected. As shown in the figure, the addendum tooth profile of the pinion and the dedendum tooth profile of the gear are determined by the meshing line located in the first quadrant. The addendum thickness of the pinion increases with the increase of the first quadrant parabolic coefficient k1, and the dedendum thickness of the gear decreases with the increase of the coefficient k1.

[0218] As shown in Figs. 6-9, the influence of the parameter k2 on the tooth profile of the new gear is reflected. As shown in the figure, the addendum tooth profile of the pinion and the dedendum tooth profile of the gear are determined by the meshing line located in the first quadrant. The addendum thickness of the pinion increases with the increase of the first quadrant parabolic coefficient k2, and the dedendum thickness of the gear decreases with the increase of the coefficient k2. Figure 10 ​​The influence of the parameter k2 on the new gear tooth profile is shown in the figure. It can be seen that the dedendum of the pinion and the addendum of the gear are affected by the meshing line in the third quadrant. With the increase of the parabolic coefficient k2, the dedendum of the pinion decreases, and the addendum of the gear increases.

[0219] Figures 11-13 The stress comparison between the new gear and the involute gear is given. As shown in the figure, the contact stress and the bending stress of the new gear are much smaller than those of the involute gear with the same parameters; the maximum contact stress and the bending stress of the involute gear both appear near the node, while the maximum bending stress of the new gear appears at the tooth exit meshing position.

[0220] The maximum stress values of the new gear and the involute gear in the meshing period are given as shown in Table 3. It can be seen that compared with the involute gear, the maximum contact stress of the new gear is reduced by 16.10%, the maximum bending compressive stress is reduced by 6.75%, and the maximum bending tensile stress is reduced by 8.49%.

[0221] Table 3 Maximum stress comparison of the new gear and the involute gear (MPa)

[0222]

[0223]

[0224] Step two: tooth direction modification of the new gear

[0225] Gear assembly errors are difficult to avoid in actual use. Tooth direction modification is the most economical and effective technique to reduce the influence of errors. The influence of modification on the meshing performance of the new gear is analyzed comprehensively below. Only the pinion tooth surface is modified here, and the gear is not treated. y1 and y2 are the maximum modification amounts at both ends of the tooth width, and y3 is the non-modification length in the tooth width direction.

[0226] In order to further study the influence of tooth direction modification on the load capacity of the new gear, Figure 14 The tooth surface load distribution of the new gear at the node position when meshing is given under different modification amounts. It can be seen that due to the edge effect, the stress is concentrated at both ends of the tooth width of the unmodified gear, while the load is concentrated at the midpoint of the tooth width of the gear after tooth direction modification, which is beneficial to improve the meshing performance of the gear under error conditions.

[0227] Table 4 and Figures 15-17 The stress comparison of the new gear under different modification amounts is given. It can be seen that the contact stress of the new gear increases with the increase of the modification amount, the maximum contact stress is located near the node, the bending compressive stress and the bending tensile stress have the same trend as the unmodified gear with the change of the pinion angle, and the maximum bending stress occurs at the tooth exit meshing position.

[0228] Table 4 Maximum stress (MPa) of new gear with different modification

[0229]

[0230] Step three: Error sensitivity analysis of new gear

[0231] Error sensitivity analysis of new gear is carried out below and compared with involute gear. The tooth modification parameters yi(i = 1, 2 and 3) of new gear and involute gear are 8 μm, 8 μm and 8 mm respectively.

[0232] Figures 18-21 The tooth surface load distribution of unmodified and modified new gear with installation error is given and compared with the same parameter involute gear. As shown in the figure, with the increase of error angle, the tooth surface load distribution of gear is more and more serious, which is extremely unfavorable for gear transmission. By comparing the situation before and after tooth modification, it can be found that the load distribution has been obviously improved after tooth modification.

[0233] Figures 22-33 The contact stress, bending compressive stress and bending tensile stress of new gear and involute gear with installation error are given respectively. From table 5 and table 6, it can be seen that the contact stress and bending stress of new gear are significantly lower than those of involute gear, and the load capacity of new gear is significantly improved compared with involute gear.

[0234] Table 5 Maximum stress (MPa) of unmodified new gear and involute gear with different installation error

[0235]

[0236] Table 6 Maximum stress (MPa) of modified new gear and involute gear with different installation error

[0237]

[0238] From the above data, it can be seen that:

[0239] (1) The new gear is designed based on parabolic meshing line, and the shape of tooth profile can be controlled by adjusting the parabolic parameters.

[0240] (2) The tooth surface contact fatigue strength and tooth root bending fatigue strength of new gear are significantly improved compared with involute gear. Under the given load condition, the tooth surface contact stress of new gear is reduced by 16.10% compared with involute gear. New gear has higher load capacity and lower error sensitivity than involute gear. Whether modified or not and whether installation error exists or not, the stress of new gear is significantly lower than that of involute gear according to the finite element analysis results.

[0241] The design advantages of the present invention are embodied in this example.

Claims

1. A method of designing a high load carrying cylindrical gear based on a curvilinear line of action, characterized by The method comprises the following steps: Step 1: tooth profile solving based on curved meshing line; Step 2: calculation of undercut critical value; Step 3: derivation of rack cutter equation; Step 4: tooth profile design based on parabolic meshing line; Step 5: three-dimensional finite element mesh modeling of gear parameterization; Step 6: error sensitivity analysis; The step 2 is specifically: calculating the gear undercut critical parameter; The gear undercut critical value is derived by using the sliding velocity of the tooth surface as a condition: is the relative velocity vector at the point of tooth profile contact; The first gear undercut critical value is obtained by arranging the equation: (1 + m 12 )x f x′ f + [(1 + m 12 )y f + (r1 + r2)]y′ f = 0 m 12 is the ratio of the two gear transmissions; Similarly, the second gear undercut critical value is: (1 + m 12 )x f x′ f +{y f +m 12 [y f -(r1+r2)]}y′ f = 0; The step 3 is specifically: presetting the engagement line P1P2, establishing a coordinate system S C1 The moving coordinate system is fixed with the rack cutter, and the coordinate axis X C1 The moving coordinate system is fixed with the rack cutter, and the coordinate axis X C1 The moving coordinate system is fixed with the rack cutter, and the coordinate axis X f The moving coordinate system is fixed with the rack cutter, and the coordinate axis X The tooth profile of the rack cutter intersects with the engagement line P1P2 at the D0 point, and the rack cutter intersects with the engagement line P1P2 at the D point with the movement of the rack cutter. According to the gear engagement theory, the tangent of the rack cutter tooth profile point D must be perpendicular to the line segment O f - Thus, we have: x c1 and y c1 are the coordinate components of the X-axis and Y-axis of the fixed coordinate system S c1 lower rack cutter equation Thus, the rack cutter equation for machining the first gear is obtained: Wherein: Alpha is a preset parabolic parameter, and delta is an integral constant, which is determined by the initial position of the rack cutter; According to the Camus theorem, when two rack cutters capable of being mutually embedded are used to perform gear generation according to the conjugate machining principle, the machined gears are also mutually conjugate; therefore, after the rack tooth profile of the first gear is obtained, the derivation of the rack tooth profile equation of the second gear is completed by using the spatial embedding relationship of the two rack cutters.

2. A high load carrying cylindrical gear design method based on curvilinear line of action as claimed in claim 1 wherein, The step 1 is specifically: coordinate system S f is a fixed coordinate system, and the coordinate system origin is located at the node; the coordinate systems S1 and S2 are dynamic coordinate systems, and the coordinate system origins are located at the centers of the two gears; the locus formed by the contact points in the fixed coordinate system S f during the gear pair transmission process is the meshing line; according to the gear meshing principle, the locus formed by the meshing points represented in the fixed coordinate system and transformed into the dynamic coordinate systems S1 and S2 respectively is the conjugate profile satisfying the preset meshing line; x f and y f are the coordinate components of the preset meshing line in the X axis and the Y axis in the fixed coordinate system S f , and the coordinate representation of the position vector of the preset meshing line in S f is: The position vector of the conjugate tooth profile is: Wherein: Phi1 and phi2 are the angular displacements of the two gears in the meshing process, respectively; r1 and r2 are the pitch circle radii of the two gears, respectively; u t is a parameter of the preset engagement line; [L] i,f is S f to S i coordinate transformation matrix of i = 1, 2; According to the principle of the engagement of the spur gears, the common normal line at the engagement point of the spur gears passes through the instantaneous center O f ; at S f , the unit normal vector is represented as: The unit normal vector is expressed in S1: According to the gear meshing principle, the common normal line of the two conjugate tooth profiles at the contact point position is perpendicular to the relative velocity direction: F1 is the meshing equation, which means that the common normal line of the conjugate tooth surface at the contact point position is perpendicular to the relative velocity at the contact point; is the position vector of the common normal vector of the two tooth surfaces in the coordinate system S1; x' f , y' f are the first-order derivative functions of x f and y f with respect to u t ; The relationship between the first gear angular displacement and the preset meshing line parameter u is obtained by arranging the meshing equation t ​ According to the two-gear transmission ratio relationship, the relationship between the angular displacement of the second gear and the preset meshing line parameter u is obtained: t between the angular displacement of the second gear and the preset meshing line parameter u is obtained:

3. A high load carrying cylindrical gear design method based on curvilinear line of action as claimed in claim 1 wherein, The step 4 is specifically that the parabola vertex of the first quadrant is located at O f (0,0), the focus is located at F F(0,p1 / 2), and α is a preset parabola parameter. Coordinate system S f The position vector of the preset engagement line is represented as follows. p1 is a first quadrant preset parabolic focal distance; in triangle O f In the MF, there are the following geometric relationships: From this the relation of the parameter a to u t is obtained: The equation of the parabolic meshing line in the first quadrant is: Wherein: Similarly, the meshing line equation in the third quadrant is: p2 is a third quadrant preset parabolic focal distance; wherein: The meshing line equation obtained above is substituted into the undercut relationship derived in step 2: (1 + m 12 )x f x′ f +{y f +m 12 [y f -(r1+r2)]}y′ f = 0; The undercut critical condition is obtained: According to the gear engagement theory, the gear coincidence degree is ensured to be greater than 1; when the moving point M is located on the engagement line in the first quadrant, the angle β should be greater than the angle corresponding to the quarter tooth of the gear, and thus the range of the parameter β is represented as: In triangle O2MF, there are the following geometric relationships: Thus, we obtain: The conditions that k1 does not produce undercut are obtained by the above derivation: Similarly, the conditions that k2 does not produce undercut are obtained:

4. A high load carrying cylindrical gear design method based on curvilinear line of action as claimed in claim 1 wherein, The step 5 is specifically: writing a gear simulation machining program based on MATLAB; first, the coordinates of the instantaneous cutting points of the rack cutter from entering meshing to exiting meshing are solved by using the gear meshing principle; then, the plane node coordinates of the double-sided tooth profile of the gear tooth body are obtained by rotation projection transformation; secondly, the plane node coordinates of the gear tooth body are obtained by rotation projection transformation; finally, all the space node coordinates of a single tooth are calculated by using the coordinate transformation along the tooth width direction; The calculated node coordinate file is imported into the finite element software ABAQUS; The tooth direction modification curve is composed of two second-order parabolas and a straight line, wherein y1 and y2 are maximum modification amounts; y3 is an unmodified length in the tooth width direction; the unmodified gear tooth surface and the modification amount surface are meshed; a tooth direction modification mesh model is established based on the superposition method of the original tooth surface and the tooth direction modification amount surface, the modification amount of the corresponding mesh node of the tooth surface is calculated according to the tooth direction modification curve equation; point P(x, y) is a tooth surface mesh space node coordinate, and the modification value δ of node P(x, y) is determined according to the two-dimensional modification curve F ; the space node coordinates of the gear are re-solved according to the modification amount at the tooth surface node; finally, the calculated modification gear node coordinate file is imported into ABAQUS, and the rapid three-dimensional modeling of the modification gear is realized.

5. A high load carrying cylindrical gear design method based on curvilinear line of action as claimed in claim 1 wherein, The step 6 is specifically: adding installation error Dgamma in the gear meshing coordinate system; the coordinate system Sg is the gear coordinate system without installation error, and the coordinate system SF is the gear coordinate system with installation error; the rotation transformation equation from the coordinate system SF to the coordinate system Sg is:

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