Method for checking contact strength and bending strength of asymmetric helical cylindrical gear pair
Through the analytical calculation method of asymmetric helical cylindrical gear pair, the maximum contact stress and maximum bending stress are calculated, which solves the problems of long and high cost of finite element simulation calculation time in the prior art, and realizes the effectiveness of gear strength verification.
Patent Information
- Application Number
- CN202510178431.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-18
- Publication Date
- 2025-06-27
AI Technical Summary
The stress calculation of existing asymmetric helical cylindrical gears mainly relies on finite element simulation methods, and there are problems such as strict grid cell size requirements, long calculation time, and not suitable for engineering applications.
A method of analytical calculation of the maximum contact stress and maximum root bending stress of asymmetric helical cylindrical gear pair is proposed. The gear strength verification is achieved by defining basic parameters and calculating the maximum line load, maximum contact stress and maximum bending stress.
It effectively avoids the high cost and time-consuming problems of finite element simulation and experimental methods, provides a theoretical basis for parameter design and strength verification of asymmetric helical cylindrical gears, and realizes strength verification.
Smart Images
Figure CN120217574A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of gear drive strength calculation and analysis, and specifically to a method for checking the contact strength and bending strength of an asymmetric helical cylindrical gear pair. Background Art
[0002] Involute gear drives are widely used in various fields such as vehicle transportation, aerospace, and construction machinery. With the development of global energy conservation, emission reduction, and carbon neutrality requirements, the demand for the strength of gear components in new energy construction machinery and high-power high-speed precision electric drive reducers for vehicles is increasing. Asymmetric helical cylindrical gears have different pressure angles on both sides. By increasing the pressure angle on the driving side and decreasing the pressure angle on the sliding side, not only can the load-carrying capacity be increased, but also better transmission smoothness can be achieved.
[0003] Strength check is an important step in the design process of cylindrical gear pairs. To avoid failure modes such as pitting, scoring, wear, and tooth breakage, it is necessary to accurately calculate the maximum contact stress on the tooth surface and the bending stress at the tooth root. The existing stress calculations for asymmetric helical cylindrical gears mainly use the finite element simulation method. However, the finite element method has strict requirements for the size of grid elements. A denser grid requires a long calculation time and is not suitable for engineering applications, while a sparse grid cannot calculate the true gear stress.
[0004] To solve the above problems, the present invention proposes an analytical calculation method for the maximum contact stress and tooth root bending stress of an asymmetric helical cylindrical gear pair, providing a theoretical basis for the parameter design and strength check of asymmetric helical cylindrical gears, effectively avoiding the problems of high cost and long time consumption in finite element simulation and experimental methods, and realizing strength check. Summary of the Invention
[0005] The purpose of the present invention is to provide a method for checking the contact strength and bending strength of an asymmetric helical cylindrical gear pair, avoiding the deficiencies of the prior art. By proposing an analytical calculation method for the maximum contact stress and maximum tooth root bending stress of an asymmetric helical cylindrical gear pair, the stress data required for gear strength check is obtained, solving the problems existing in the prior art.
[0006] To achieve the above purpose, the technical solution adopted by the present invention is a method for checking the contact strength and bending strength of an asymmetric helical cylindrical gear pair, including the following steps: The first step is to define parameters. Determine the basic parameters of the asymmetric helical cylindrical gear pair, including the number of teeth z, normal module m n , normal pressure angle α on the driving side d , normal pressure angle α on the sliding side c , pitch circle helix angle β, normal modification coefficient x, tooth tip height h a, Total tooth height h, installation center distance a1, tooth width B Step 2: Calculate the maximum line load Based on the basic parameters of the asymmetric helical cylindrical gear pair, calculate the minimum total contact line length on the driving side of the gear pair L min , combined with the input torque of the pinion M , calculate the maximum line load ; The minimum total contact line length on the driving side of the gear pair L min is: (1), In the above formula, is the face contact ratio, is the axial contact ratio, is the fractional part of is the fractional part of is the base pitch in the transverse plane, is the base helix angle; For the given input torque M of the pinion, the maximum line load is: (2), In the above formula, F n is the normal force of the gear pair, is the input torque of the pinion, K is the load factor, is the normal pressure angle at the pitch circle on the driving side, is the normal module, is the minimum total contact line length, z 1 is the number of teeth of the pinion; Step 3: Calculate the maximum contact stress Based on the maximum line load calculated in Step 2 , and combined with the comprehensive curvature radius of the tooth profile in the normal section at the pitch circle, calculate the maximum contact stress according to the Hertz contact stress calculation formula The comprehensive curvature radius of the tooth profile in the normal section at the pitch circle is: (3), In the above formula, is the pitch circle radius of the pinion, α t is the transverse pressure angle at the pitch circle, u is the transmission ratio, is the base helix angle; The equivalent elastic modulus is: (4), In the above formula, is the Poisson's ratio of the pinion material, is the Poisson's ratio of the gear material, is the Poisson's ratio of the pinion material, is the Poisson's ratio of the gear material; From the Hertz contact stress calculation formula, the maximum contact stress is: (5), In the above formula, is the maximum line load, is the composite curvature radius, is the equivalent elastic modulus; Fourth step, solve the maximum bending stress, First, solve the radius at the highest point of single-tooth or double-tooth meshing of the end face profile, then solve the coordinates and normal vectors at the highest point of single-tooth or double-tooth meshing of the normal profile, and then establish the theoretical bending stress analytical formula. Finally, perform iteration on the theoretical bending stress analytical formula to solve the maximum bending stress; Fifth step, check the contact strength and bending strength, By comparing the calculated maximum contact stress with the allowable contact stress of the gear, and comparing the calculated maximum bending stress with the allowable bending stress of the gear, the contact strength and bending strength of the gear are checked.
[0007] Furthermore, the solution of the radius at the highest point of single-tooth or double-tooth meshing of the end face profile described in the fourth step is specifically as follows: According to the basic parameters of the asymmetric helical cylindrical gear pair, calculate the face contact ratio of the driving side. If the face contact ratio is greater than 2, solve the radius at the highest point of double-tooth meshing of the end face. If the face contact ratio is greater than 1 and less than 2, then solve the radius at the highest point of single-tooth meshing of the end face; When the face contact ratio is greater than 1 and less than 2, the radius at the highest point of single-tooth meshing of the end face profile is: (6), When the face contact ratio is greater than 2, the radius at the highest point of double-tooth meshing of the end face profile is: (7), In the above formula, is the addendum circle radius of the gear, is the base circle radius of the gear, is the face contact ratio, is the face pitch.
[0008] Furthermore, the solution of the coordinates and normal vectors at the highest point of single-tooth or double-tooth meshing of the normal profile in the fourth step is specifically as follows: Solve the normal profile equation and the normal vector equation. According to the radius of the highest point of single-tooth or double-tooth meshing on the end face obtained in the fourth step, substitute this radius into the normal profile equation and the normal vector equation to obtain the coordinates and normal vector of the highest point of single-tooth or double-tooth meshing on the normal profile; The tooth surface of an asymmetric helical cylindrical gear is in the coordinate system Ss(x s ,y s ,z s ), where the z s axis is the gear axis direction, and the end face profile is located on the plane formed by the x s axis and the y s axis. In the coordinate system Ss, the tooth surface equation r s and the normal vector n s are: (8), In the above formula, x s , y s , z s are the coordinate values of the tooth surface point in the coordinate system Ss, , , are the normal vector components at the tooth surface point in the coordinate system Ss; The normal profile is located on the plane formed by the x n ,y n ,z n in the coordinate system Sn(x n axis and the y n axis. The tooth surface equation r n and the normal vector n n are expressed in the coordinate system Sn as: (9), In the above formula, x n , y n , z n are the coordinate values of the tooth surface point in the coordinate system Sn, , , are the normal vector components at the tooth surface point in the coordinate system Sn; is the coordinate transformation matrix from the coordinate system Ss to the coordinate system Sn, which is: (10), Let z in formula (9) be nThe coordinates are 0, and the normal profile equation and normal vector equation are obtained as follows: (11), Substitute the radius at the highest point of single-tooth or double-tooth meshing of the end face profile into the normal profile equation and normal vector equation, and the coordinates and the normal vector of the highest point C of single-tooth or double-tooth meshing of the normal profile can be obtained by solving the system of equations: (12), In the above formula, and are the coordinate position components, and are the normal vector components. The C in the upper brackets represents that these parameters belong to the highest point C of single-tooth or double-tooth meshing of the normal profile, and the subscript n represents that these vectors are located in the coordinate system Sn.
[0009] Furthermore, in the fourth step, the analytical formula for the theoretical bending stress is established. Specifically, Apply the maximum line load at the highest point C of single-tooth or double-tooth meshing of the normal profile, and the loading direction is the normal vector at the position of the highest point of single-tooth or double-tooth meshing . Take a circle that intersects the transition arcs of the normal profiles on the driving side and the sliding side at two points D i and C i . Similarly, substitute the radius of this circle into the system of equations in the fifth step to obtain the coordinates of the two intersection points D i and C i . Take a perpendicular line at a point m i on the line connecting these two points. This perpendicular line intersects the loading direction of the maximum line load at a point O; The analytical formula for the theoretical bending stress is: (13), In the above formula, is the maximum line load, is the loading angle, is the thickness of the tooth root section, is the force arm, is the helix angle influence coefficient. These parameters are calculated as follows: The loading angle is: (14), The thickness of the tooth root section is: (15), The force arm is: (16), In the above formula, , , is the tooth thickness on the driving side of the indexing line of the rack cutter, is the tooth thickness of the indexing line of the rack cutter, and the calculation is as follows: (17), (18), In the formula, is the addendum height of the cutter tooth of the rack, is the addendum arc radius of the rack cutter; the parameter calculation is as follows: (19), (20), In the formula, is the total height of the gear tooth, is the addendum height of the gear tooth, is the normal modification coefficient.
[0010] Furthermore, in the fourth step, the analytical formula of the theoretical bending stress is iterated to solve the maximum bending stress, specifically: Take the iteration termination circle radius as the root circle radius, and the iteration start circle radius as the root circle radius plus 0.25m n , and evenly divide the arc between the iteration start circle and the termination circle into n - 1 arcs. By calculating and iterating the theoretical bending stress formula (13) at each arc position, the maximum bending stress obtained is: (21), In the above formula, each parameter is the parameter at the position of the maximum bending stress, and the meaning of each parameter is the same as above, is the correction coefficient of the maximum theoretical bending stress: (22), In the above formula, is the root fillet radius of curvature at the point of the maximum theoretical bending stress.
[0011] The beneficial effect of the present invention is: The present invention provides a method for checking the contact strength and bending strength of an asymmetric helical cylindrical gear pair. Through the proposed analytical calculation method for the maximum contact stress and the maximum tooth root bending stress of the asymmetric helical cylindrical gear pair, the stress data required for gear strength checking is obtained, providing a theoretical basis for the parameter design and strength checking of the asymmetric helical cylindrical gear, effectively avoiding the problems of high cost and long time consumption in the finite element simulation and experimental methods, and realizing strength checking. BRIEF DESCRIPTION OF THE DRAWINGS
[0012] Figure 1 is the flowchart of the present invention; Figure 2 is the normal tooth profile of the asymmetric helical cylindrical gear in the embodiment of the present invention; Figure 3 is the bending stress calculation model diagram in the embodiment of the present invention; Figure 4 is the rack cutter normal section profile in the embodiment of the present invention; Figure 5 is the tooth root bending stress iteration model in the embodiment of the present invention. Specific embodiments
[0013] The principles and features of the present invention will be described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.
[0014] To achieve the above object, the present invention provides the following specific embodiments: As Figure 1 shown, a method for checking the contact strength and bending strength of an asymmetric helical cylindrical gear pair includes the following steps: The first step is to define parameters, Determine the basic parameters of the asymmetric helical cylindrical gear pair, including the number of teeth z, the normal module m n , the driving side normal pressure angle α d , the sliding side normal pressure angle α c , the pitch circle helix angle β, the normal modification coefficient x, the addendum height h a , the total tooth height h, the installation center distance a1, the tooth width B, and the specific values are shown in Table 1; Table 1 Basic parameters of the asymmetric gear pair
[0015] The second step is to calculate the maximum line load, According to the basic parameters of the asymmetric helical cylindrical gear pair, calculate the minimum contact line total length on the driving side of the gear pair L min , and combine with the input torque M of the pinion to calculate the maximum line load ; The minimum contact line total length on the driving side of the gear pair L min is: (1) In the above formula, is the face contact ratio, is the axial contact ratio, is the fractional part of is the fractional part of is the face base pitch, is the base helix angle; For a given pinion input torque \(M = 344\ Nm\), the maximum line load is: (2) In the above formula, F n is the normal contact force of the gear pair, is the pinion input torque, K is the load factor, is the normal pressure angle on the driving side, is the normal module, is the minimum total contact line length, z 1 is the number of teeth of the pinion; Step 3: Calculate the maximum contact stress, Based on the maximum line load calculated in Step 2, and using the Hertz contact stress calculation formula, combined with the comprehensive curvature radius of the tooth profile in the normal section at the pitch circle, calculate the maximum contact stress, The comprehensive curvature radius of the tooth profile in the normal section at the pitch circle is: (3) In the above formula, is the pitch circle radius of the pinion, α t is the transverse pressure angle at the pitch circle, u is the transmission ratio, is the base helix angle; The equivalent elastic modulus is: (4) In the above formula, is the Poisson's ratio of the pinion material, is the Poisson's ratio of the gear material, is the Poisson's ratio of the pinion material, is the Poisson's ratio of the gear material; From the Hertz contact stress calculation formula, the maximum contact stress is: (5) Step 4: Solve for the radius at the highest point of single or double tooth contact on the transverse tooth profile, Based on the basic parameters of the asymmetric helical cylindrical gear pair, calculate the transverse contact ratio on the driving side. If the transverse contact ratio is greater than 2, solve for the radius at the highest point of double tooth contact on the transverse tooth profile. If the transverse contact ratio is greater than 1 and less than 2, then solve for the radius at the highest point of single tooth contact on the transverse tooth profile; When the transverse contact ratio is greater than 1 and less than 2, the radius at the highest point of single tooth contact on the transverse tooth profile is: (6) When the transverse contact ratio is greater than 2, the radius at the highest point of double tooth contact on the transverse tooth profile is: (7) In the above formula, is the addendum circle radius of the gear, is the base circle radius of the gear, is the face contact ratio, is the face pitch; Step 5: Solve the coordinates and normal vectors of the highest point of single or double tooth meshing on the normal tooth profile. Solve the normal tooth profile equation and the normal vector equation. According to the radius of the highest point of single or double tooth meshing on the face obtained in Step 4, substitute this radius into the normal tooth profile equation and the normal vector equation to obtain the coordinates and normal vectors of the highest point of single or double tooth meshing on the normal tooth profile. The tooth surface of an asymmetric helical cylindrical gear in the coordinate system Ss(x s , y s , z s ) is shown in Figure 2 . Among them, the z s axis is the gear axis direction, the face tooth profile is located on the plane formed by the x s axis and the y s axis. In the coordinate system Ss, the tooth surface equation r s and the normal vector n s are: (8) In the above formula, x s , y s , z s are the coordinate values of the tooth surface point in the coordinate system Ss, , , are the normal vector components at the tooth surface point in the coordinate system Ss; The normal tooth profile is located on the plane formed by the x n , y n , z n ) in the coordinate system Sn. As shown in n axis and the y n axis. The tooth surface equation Figure 2 and the normal vector r n are expressed in the coordinate system Sn as: n n In the above formula, (9) In the above formula, x n , y n , z n are the coordinate values of the tooth surface point in the coordinate system Sn, , , is the normal vector component at the tooth surface point in the coordinate system Sn; is the coordinate transformation matrix from the coordinate system Ss to the coordinate system Sn, is: (10) Let the z n coordinate in Equation (9) be 0, and the normal section tooth profile equation and the normal vector equation are obtained as: (11) Substitute the radius of the highest point of single or double tooth meshing of the end face tooth profile into Equation (11), and the coordinates of the highest point C of single or double tooth meshing of the normal section tooth profile and the normal vector can be obtained from the following system of equations: (12) In the above formula, , are the coordinate position components, , are the normal vector components. The C in the upper brackets represents that these parameters belong to the highest point C of single or double tooth meshing of the normal section tooth profile, and the subscript n represents that these vectors are located in the coordinate system Sn; Sixth step, establish the analytical formula of the theoretical bending stress, Apply the maximum line load at the highest point C of single or double tooth meshing of the normal section tooth profile, and the loading direction is the normal vector at the position of the highest point of single or double tooth meshing of the tooth profile , take a circle that intersects the transition arcs of the normal tooth profiles on the driving side and the sliding side at two points D i and C i , similarly substitute the radius of this circle into Equation (12) to obtain the coordinates of the two intersection points D i and C i , take a perpendicular line at a point m i on the line connecting these two points, and this perpendicular line intersects the loading direction of the maximum line load at a point O, as Figure 3 shown.
[0016] The analytical formula of the theoretical bending stress is: (13) In the above formula, is the maximum line load, is the loading angle, is the tooth root section thickness, is the acting force arm, is the helix angle influence coefficient, as Figure 3 shown, and these parameters are calculated as follows: Loading angle is: (14) Tooth root section thickness is: (15) Acting force arm is: (16) In the above formula, , , is the tooth thickness on the driving side of the pitch line of the rack cutter, is the tooth thickness of the pitch line of the rack cutter, as Figure 4 shown, the calculation is as follows: (17) (18) In the formula, is the addendum height of the cutter tooth of the rack, is the addendum arc radius of the rack cutter; the parameter calculation is as follows: (19) (20) In the formula, is the total tooth height of the gear, is the addendum height of the gear tooth, is the normal modification coefficient; Step 7, iterative solution of the theoretical bending stress analytical formula to obtain the maximum bending stress, Take the iteration termination circle radius as the root circle radius, and the iteration start circle radius as the root circle radius plus 0.25m n , and evenly divide the arc between the iteration start circle and the termination circle into n - 1 arcs, as Figure 5 shown, by calculating and iterating the theoretical bending stress formula (13) at each arc position, the actual maximum bending stress obtained is: (21) In the above formula, each parameter is the parameter at the position of the maximum bending stress, and the meaning of each parameter is the same as the parameter in formula (13), is the correction coefficient of the maximum theoretical bending stress: (22) In the above formula, is the curvature radius of the tooth root arc at the point of the maximum theoretical bending stress.
[0017] Step 8, check the contact strength and bending strength, By comparing the calculated maximum contact stress with the allowable contact stress of the gear and the calculated maximum bending stress with the allowable bending stress of the gear, the contact strength and bending strength of the gear are checked. The above example takes the calculation of the maximum bending stress of the pinion as an example. The input torque is 344 Nm. After calculation, the maximum contact stress of the pinion is 1202.90 Mpa, and the maximum bending stress is 384.02 Mpa.
[0018] The results obtained by using the present invention can provide a basic basis for the strength check and parameter optimization design of the asymmetric helical cylindrical gear.
[0019] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.
Claims
1. A method for checking the contact strength and bending strength of an asymmetric helical cylindrical gear pair, characterized in that: The following steps are involved: The first step is to define the parameters. Determine the basic parameters of asymmetric helical cylindrical gear pairs, including the number of teeth z and the normal module m n , driving side normal pressure angle α d , normal pressure angle α of sliding side c , pitch circle helix angle β, normal displacement coefficient x, tooth top height h a , tooth height h, installation center distance a1, tooth width B, The second step is to calculate the maximum line load, According to the basic parameters of the asymmetric helical cylindrical gear pair, calculate the minimum total length of the contact line on the driving side of the gear pair L min , combined with the input torque of the pinion M , calculate the maximum line load ; Minimum total contact line length on the driving side of the gear pair L min for: (1)， In the above formula, is the end face overlap, is the axial overlap, for The decimal part of for The decimal part of is the end face base circle pitch, is the base circle helix angle; For a given pinion input torque M , maximum line load for: (2)， In the above formula, F n is the normal meshing force of the gear pair, is the pinion input torque, K is the load factor, is the normal pressure angle on the driving side, is the normal modulus, is the minimum total length of the contact line, z 1 is the number of small gear teeth; The third step is to calculate the maximum contact stress. The maximum line load calculated in step 2 Based on the Hertz contact stress calculation formula, combined with the comprehensive curvature radius of the normal section tooth profile at the pitch circle, the maximum contact stress is calculated. The comprehensive curvature radius of the normal section tooth profile at the pitch circle is: (3)， In the above formula, is the pinion pitch radius, α t is the end pressure angle at the pitch circle, u is the transmission ratio, is the base circle helix angle; The equivalent elastic model quantity is: (4)， In the above formula, is the Poisson's ratio of the pinion material, is the Poisson's ratio of the gear material, is the Poisson's ratio of the pinion material, is the Poisson's ratio of the gear material; According to the Hertz contact stress calculation formula, the maximum contact stress is: (5)， In the above formula, is the maximum line load, is the integrated curvature radius, is the equivalent elastic modulus; The fourth step is to solve the maximum bending stress. First, the radius of the highest point of the end face tooth profile with a single tooth or double teeth is solved, and then the coordinates and normal vector of the highest point of the normal face tooth profile with a single tooth or double teeth are solved, and then the theoretical bending stress analytical formula is established, and finally the theoretical bending stress analytical formula is iterated to solve the maximum bending stress; The fifth step is to check the contact strength and bending strength. By comparing the calculated maximum contact stress with the allowable contact stress of the gear, and comparing the calculated maximum bending stress with the allowable bending stress of the gear, the contact strength and bending strength verification of the gear is completed.
2. A method for checking contact strength and bending strength of an asymmetric helical cylindrical gear pair as claimed in claim 1, characterized in that: The solution for the radius of the highest point of engagement of a single or double tooth on the end face tooth profile described in step 4 is as follows: According to the basic parameters of the asymmetric helical cylindrical gear pair, calculate the end face overlap of the driving side. If the end face overlap is greater than 2, solve the radius of the highest point of the double-tooth meshing of the end face. If the end face overlap is greater than 1 but less than 2, solve the radius of the highest point of the single-tooth meshing of the end face. The end face overlap is greater than 1 and less than 2, and the radius of the highest point of the end face tooth profile single tooth meshing is: (6)， The end face overlap is greater than 2, and the radius of the highest point of the double tooth meshing of the end face tooth profile is: (7)， In the above formula, is the radius of the gear tooth tip circle, is the radius of the gear base circle, is the end face overlap, is the end face tooth pitch.
3. A method for checking contact strength and bending strength of an asymmetric helical cylindrical gear pair as claimed in claim 1, characterized in that: In the fourth step, the coordinates and normal vector of the highest point of the normal tooth profile single or double tooth meshing are solved, specifically: Solve the normal tooth profile equation and the normal vector equation. According to the radius of the highest point of the end face single tooth or double tooth meshing solved in the fourth step, substitute the radius into the normal tooth profile equation and the normal vector equation to obtain the coordinates and normal vector of the highest point of the normal tooth profile single tooth or double tooth meshing; The tooth surface of the asymmetric helical cylindrical gear in the coordinate system Ss(x s ,y s ,z s ), where z s The axis is the gear axis direction, and the end face tooth profile is located at x s Axis, y s On the plane formed by the axis, the tooth surface equation in the coordinate system Ss is r s and the law vector n s for: (8)， In the above formula, x s , y s , z s is the coordinate value of the tooth surface point in the coordinate system Ss, , , is the normal vector component at the tooth surface point in the coordinate system Ss; The normal tooth profile is located in the coordinate system Sn(x n ,y n ,z n ) n Axis, y n On the plane formed by the axis, the tooth surface equation is r n and the law vector n n In the coordinate system Sn, it is expressed as: (9)， In the above formula, x n , y n , z n is the coordinate value of the tooth surface point in the coordinate system Sn, , , is the normal vector component at the tooth surface point in the coordinate system Sn; is the coordinate transformation matrix from coordinate system Ss to coordinate system Sn, for: (10)， Imperative form (9) z n The coordinate is 0, and the normal tooth profile equation and normal vector equation are: (11)， The radius of the highest point of engagement of single or double teeth on the end face tooth profile Substitute the normal tooth profile equation and normal vector equation into the coordinates of the highest point C of the normal tooth profile single or double tooth meshing and the law vector It can be obtained by solving the equations: (12)， In the above formula, , is the coordinate position component, , is the normal vector component. The C in the upper brackets indicates that these parameters belong to the highest point C of the normal tooth profile single or double tooth meshing. n Represents that these vectors are located in the coordinate system Sn.
4. A method for checking contact strength and bending strength of an asymmetric helical cylindrical gear pair as claimed in claim 1, characterized in that: In the fourth step, the analytical formula of theoretical bending stress is established, which is: The maximum linear load is applied at the highest point C of the normal tooth profile with a single tooth or double teeth meshing. The loading direction is the normal vector of the highest point of the tooth profile with a single tooth or double teeth meshing. , take a circle that intersects the normal tooth profile transition arcs on the driving side and the sliding side at two points D i and C i Similarly, substitute the radius of the circle into the equations solved in step 5 to obtain the intersection point D i and C i The coordinates of these two points are taken from a point m on the line connecting these two points. i The perpendicular line intersects the loading direction of the maximum linear load at a point O; The analytical formula for the theoretical bending stress is: (13)， In the above formula, is the maximum line load, is the loading angle, is the tooth root section thickness, is the force arm, is the helix angle influence factor, and these parameters are calculated as follows: Loading Angle for: (14)， Root section thickness for: (15)， Working arm for: (16)， In the above formula, , , is the tooth thickness on the drive side of the rack cutter indexing line, is the tooth thickness of the rack cutter indexing line, calculated as follows: (17)， (18)， In the formula, is the tool tooth top height of the rack, is the radius of the rack tool tooth top arc; the parameter calculation is as follows: (19)， (20)， In the formula, is the full height of the gear teeth, is the gear tooth top height, is the normal displacement coefficient.
5. A method for checking contact strength and bending strength of an asymmetric helical cylindrical gear pair as claimed in any one of claims 1 to 4, characterized in that: In the fourth step, the theoretical bending stress analytical formula is iterated to solve the maximum bending stress, which is: The radius of the iteration end circle is the root circle radius, and the radius of the iteration start circle is the root circle radius plus 0.25m. n , the area between the iteration start circle and the end circle is uniformly divided into n-1 arcs, and the maximum bending stress is obtained by iteratively calculating the theoretical bending stress formula (13) at each arc position: (21)， In the above formula, each parameter is the parameter at the position of maximum bending stress, and the meaning of each parameter is the same as above. is the maximum theoretical bending stress correction factor: (22)， In the above formula, is the radius of curvature of the tooth root arc at the point of maximum theoretical bending stress.