A calculation method for the length of the meshing contact line of helical gears based on three-dimensional space

Abstract

The present invention relates to the technical field of machine gears, and discloses a method for calculating the length of the meshing contact line of helical gears in three-dimensional space, including the following steps: obtaining the two-dimensional coordinates of the limit points; dividing the contact line; constructing a three-dimensional space geometric relationship diagram; obtaining each part of the contact line; and obtaining the total length of the contact line. By using spatial analytic geometry and combining with the transient finite element simulation results of helical gear meshing, the present invention establishes a mathematical model, conducts research on the calculation of the length of the meshing contact line of helical gears in three-dimensional space, and transforms the calculation of the length of the meshing contact line of helical gears into a spatial analytic geometry mathematical problem, realizing accurate calculation according to the three-dimensional space characteristics of the meshing contact line of helical gears.

Description

Technical Field

[0001] The present invention relates to the technical field of machine gears, and particularly relates to a method for calculating the length of the meshing contact line of helical gears in three-dimensional space. Background Art

[0002] Gear transmission is formed by the meshing of two gears respectively installed on the driving shaft and the driven shaft. Gear transmission is the most widely used transmission form in the mechanical power transmission system. As basic components in gear transmission, there are quite a variety of gears. Among them, helical gears are widely used due to their characteristics such as smooth transmission, large contact ratio, and strong load-bearing capacity.

[0003] Currently, due to the wide application of helical gears in actual engineering, the research on helical gears is broader and deeper. Among them, accurately calculating the contact load of helical gears and further carrying out basic research such as elastohydrodynamic lubrication analysis and Hertz contact strength check of helical gear meshing is particularly important. However, only by accurately calculating the length of the meshing contact line of helical gears can these basic researches be accurately carried out.

[0004] At present, the calculation of the length of the meshing contact line of helical gears mainly uses classical theoretical methods. When calculating, it is necessary to consider the influence of parameters such as the width of the helical gear, contact ratio, and helix angle on the contact line length. However, the contact line during the meshing of helical gears has strong three-dimensional spatial characteristics. It is almost impossible to imagine the spatial relationship between the meshing teeth of helical gears by abstract thinking, and thus it is impossible to accurately calculate it according to the three-dimensional spatial characteristics of the meshing contact line of helical gears. Summary of the Invention

[0005] The present invention provides a method for calculating the length of the meshing contact line of helical gears in three-dimensional space to solve the problem that it is difficult to imagine the spatial relationship between the meshing teeth of helical gears by abstract thinking, and thus it is difficult to accurately calculate it according to the three-dimensional spatial characteristics of the meshing contact line of helical gears.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A method for calculating the length of the meshing contact line of helical gears in three-dimensional space, characterized by comprising the following steps:

[0008] 1) Obtain the two-dimensional coordinates of the limit point: Taking the center of the driving wheel end face as the coordinate origin, establish a two-dimensional coordinate system, and solve the coordinates of the actual meshing limit point on the driving wheel end face according to the straight-line equation of the meshing line on the helical gear end face;

[0009] 2) Divide the contact line: Divide the contact line during the meshing process of the helical gear into a part with a constant contact line length and a part with a gradually changing contact line length according to the schematic diagram of the change of the meshing contact line of the helical gear;

[0010] 3) Construct a three-dimensional spatial geometric relationship diagram: Based on the parameter information of the helical gear pair, establish a transient finite element simulation model of helical gear meshing and submit it for calculation to obtain the transient diagram of the tooth contact stress patch of the driving gear during helical gear meshing, which is used to construct a three-dimensional spatial geometric relationship diagram showing the contact line during helical gear meshing;

[0011] 4) Determine each part of the contact line: Construct a three-dimensional rectangular coordinate system, and then based on the spatial characteristics of the meshing contact line in the three-dimensional rectangular coordinate system, establish and solve the three-dimensional spatial geometric equation of the vanishing point of the contact line at the tooth surface and tooth root positions, and propose the calculation principle and method of the coordinate values of this point, and then solve the lengths of the variable and invariant parts of the helical gear meshing contact line;

[0012] 5) Determine the total length of the contact line: Through MATLAB programming, divide the meshing time period of each tooth participating in a complete meshing process into equal time sub-periods, calculate the contact line length of each tooth at each time equal point, and sum up the contact lines of the teeth at each time equal point to obtain the total length of the helical gear meshing contact line.

[0013] As a preferred technical solution of the present invention, the straight line equation of the helical gear end face meshing line is a straight line equation established according to the parameter information of the helical gear pair. The parameter information of the helical gear pair consists of the basic parameter information of the helical gear pair, the length of the helical gear end face meshing line calculated according to the helical gear end face meshing principle diagram and the basic parameter information of the helical gear pair, and other geometric parameters of the helical gear. Through rigorous basic data calculation and the establishment of relevant equations, it is beneficial to obtain more accurate data results.

[0014] As a preferred technical solution of the present invention, the three-dimensional rectangular coordinate system is a three-dimensional rectangular coordinate system based on the two-dimensional coordinate system of the driving gear end face on the transient diagram of the tooth contact stress patch of the driving gear, with the axis of the driving gear as its Z axis. By constructing a three-dimensional rectangular coordinate system on the gear body, it is beneficial to more intuitively understand the relationship between data and spatial positions.

[0015] As a preferred technical solution of the present invention, the number of segments of the equal time sub-period ranges from 80 to 120. Through a large number of practices, it is found that this value range is more convenient and accurate for calculation and analysis, which is beneficial to efficient and accurate data analysis and calculation.

[0016] As a preferred technical solution of the present invention, the summation is carried out in the order of each tooth entering meshing successively.

[0017] The present invention has the following beneficial effects:

[0018] The present invention is applicable to a method for calculating the length of the meshing contact line of helical gears in three-dimensional space. By using spatial analytic geometry and combining with the transient finite element simulation results of helical gear meshing, a mathematical model is established to study the calculation of the length of the meshing contact line of helical gears in three-dimensional space, and the calculation of the length of the meshing contact line of helical gears is transformed into a mathematical problem of spatial analytic geometry, thus solving the problem that it is difficult to imagine the spatial relationship between the meshing teeth of helical gears by abstract thinking and it is difficult to accurately calculate it according to the three-dimensional spatial characteristics of the meshing contact line of helical gears in the classical theoretical method. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0020] Figure 1 is the flow chart of the present invention;

[0021] Figure 2 is the schematic diagram of the calculation principle of the meshing contact line of helical gears;

[0022] Figure 3 is the schematic diagram of the meshing principle of the helical gear end face;

[0023] Figure 4 is the schematic diagram of the end face X-Y coordinate system of the helical gear;

[0024] Figure 5 is the schematic diagram of the change of the meshing contact line of helical gears;

[0025] Figure 6 is the transient diagram of the change of the contact stress spot of the teeth of the driving gear in helical gear meshing over time;

[0026] Figure 7 is B′ 1 Schematic diagram of the geometric parameters for calculating the Z coordinate of point (the part with unchanged length calculation of the meshing contact line of helical gears);

[0027] Figure 8 is the schematic diagram of the principle of the difference in the developed angles of the addendum circle and the root forming circle;

[0028] Figure 9 is B′ 1 Schematic diagram of the geometric parameters for calculating the Z coordinate of point (the part with gradually changing length calculation of the meshing contact line of helical gears);

[0029] Figure 10 is the schematic diagram of the change curve of the meshing contact line for each tooth of the helical gear to mesh once;

[0030] Figure 11 Schematic diagram of the contact lines of multiple teeth of a helical gear and their superposition diagram. Specific embodiments

[0031] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0032] As Figures 1-11 shown, this embodiment provides a method for calculating the length of the meshing contact line of a helical gear based on three-dimensional space. Taking the parameters of the helical gear pair provided in Table 1 as an example, the length of the meshing contact line of the helical gear is calculated. The basic parameters of the helical gear pair not involved in the table are defaulted to standard values, such as the addendum coefficient, clearance coefficient and other basic parameters are defaulted to standard values, and the modification is not considered. The specific steps are as follows:

[0033] Table 1 Parameters of helical gear pair

[0034]

[0035] 1) Calculate the two-dimensional coordinates of the limit points:

[0036] As Figure 3 shown is the schematic diagram of the helical gear end face meshing principle. N 1 N 2 is the theoretical meshing line, B 1 B 2 is the actual meshing line. When the gear pair participates in meshing, the driving gear enters meshing from the tooth root part (point B 1 ) and exits meshing from the tooth tip part (point B 2 ). Among them, the addendum circle radius of the driving gear is r a1 , the root forming circle radius is r f1 , the base circle radii are r b1 , the end face pressure angle of the pitch circle is α t , the end face pressure angle of the addendum circle is α t1 ; the base circle radius of the driven gear is r b2 , and the end face pressure angle of the addendum circle is α t2 .

[0037] According to the geometric relationship of the helical gear end face meshing in the figure, the calculation formula for the length of the helical gear end face meshing line is derived as:

[0038]

[0039]

[0040]

[0041]

[0042] In the formula:

[0043] r 1 is the pitch circle radius of the driving gear (mm);

[0044] r 2 is the pitch circle radius of the driven gear (mm);

[0045] r b1 is the base circle radius of the driving gear (mm);

[0046] r b2 is the base circle radius of the driven gear (mm);

[0047] α t1 is the addendum circle face pressure angle of the driving gear (°);

[0048] α t2 is the addendum circle face pressure angle of the driven gear (°);

[0049] Calculate other geometric parameters of the helical gear according to the basic parameter information of the helical gear pair, such as the normal pressure angle, pitch circle radius, etc. Formulas (5)-(10) give the calculation of other geometric parameters of the driving gear, and the calculation formulas for other geometric parameters of the driven gear can be deduced by analogy.

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056] In the formula:

[0057] α n is the normal pressure angle (°);

[0058] p s is the helix lead (mm);

[0059] β 2 is the helix angle of the root forming circle of the driving gear (°).

[0060] Substitute the parameters of the helical gear pair in Table 1 into Equations (1)-(4) to obtain the helical gear end face engagement line and The length of is:

[0061]

[0062] Furthermore, substitute the parameters of the helical gear pair in Table 1 into Equations (5)-(10) to obtain the other basic parameters of the helical gear as follows:

[0063] α t = 24.933°; r 1 = 84.455 mm; r 2 = 82.491 mm; r b1 = 76.584 mm; r b2 = 74.803 mm;

[0064] r f1 = 81.529 mm; r f2 = 79.568 mm; β 2 = 26.191°; p s = 1041.453 mm.

[0065] As Figure 4 shown is the end face X-Y coordinate system of the helical gear. In the figure, a two-dimensional X-Y coordinate system is established with the center of the active wheel end face as the coordinate origin. Substitute the parameters of the helical gear pair in Table 1 into Equation (13) to obtain the straight line equation of the engagement line: y = -0.46x + 80.53

[0066] Substitute the parameters of the helical gear pair in Table 1 into Equations (14)-(20) to obtain the X-Y coordinates of points B 1 、B 2 as follows (unit: mm).

[0067] B 1 (6.9, 77.32); B 2 (-6.92, 83.74).

[0068] 2) Divide the contact line:

[0069] Taking the center of the active wheel end face during helical gear meshing as the coordinate origin, establish a two-dimensional X-Y coordinate system (see Figure 4 ), and according to the helical gear end face meshing principle and the basic parameter information of the helical gear pair, establish the straight line equation of the helical gear end face engagement line . Among them, the calculation formulas for the slope K and intercept b of the straight line are:

[0070] K = -tan(α t ) (11)

[0071]

[0072] Therefore, the end face engagement line has a straight-line equation of:

[0073] y = -tan(α t )x + r 1 (13)

[0074] Based on the engagement line equation, the actual engagement limit points B 1 and B 2 of the driving gear end face are solved, and the calculation formulas are:

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081]

[0082] Furthermore, the X and Y coordinates of points B 1 and B 2 are obtained as: B 1 (x 1 , y 1 ); B 2 (x 2 , y 2 ).

[0083] As Figure 5 shown is the schematic diagram of the change of the helical gear engagement contact line. The tooth width is B, and enclose the area as the actual engagement area. N 1 N 2 is the front end face of the helical gear, and N′ 1 N′ 2 is the rear end face of the helical gear. The slant line in the actual engagement area is the change process of the contact line. When the helical gear is engaged, the teeth enter the engagement area from point B 1 and exit the engagement area from point B′ 2 . When the teeth enter the engagement, the length of the contact line changes from short to long (i.e., the area ① in the figure). At this time, the instantaneous length of the contact line is represented by L 2It is indicated that the length of the contact line remains unchanged for a certain period in the middle (i.e., the area ② in the figure), and at this time, the contact line length is represented by L 1 It is indicated that finally the contact line length gradually shortens until it becomes zero (i.e., the area ③ in the figure), and at this time, the instantaneous contact line length is represented by L 2 Furthermore, the contact line in the helical gear meshing process can be divided into two parts: the first part is the part where the contact line length remains unchanged; the second part is the part where the contact line length is gradually changing. The calculation of the contact line length for these two parts will be carried out separately later.

[0084] 3) Construct a three-dimensional space geometric relationship diagram:

[0085] According to the parameter information of the helical gear pair, establish a transient finite element simulation model of helical gear meshing and submit it for calculation to obtain the transient diagram of the tooth contact stress spot of the driving gear in helical gear meshing (see Figure 6 ). It can be seen from the figure that the stress spots are distributed in a diagonal state on the tooth surface, which is the contact line trace of helical gear meshing.

[0086] 4) Find each part of the contact line:

[0087] On the transient diagram of the tooth contact stress spot of the driving gear (see Figure 6 ), based on the two-dimensional X-Y coordinate system of the driving gear end face, construct a three-dimensional rectangular coordinate system, with the axis of the driving gear as its Z axis (see Figure 6 ). At the same time, three auxiliary planes are established, namely the end face meshing line (the green inclined plane in the figure), the end face meshing point B 1 (the light blue vertical plane in the figure), and the plane formed by stretching B 2 (the white vertical plane in the figure) along the gear axis. Figure 6 In it, the Z coordinate of point B 2 is 0, so the three-dimensional coordinates of point B 2 are: B 2 (-6.92, 83.74, 0). is parallel to the Z axis, so the X-Y coordinates of point B′ 1 are the same as the X-Y coordinates of point B 1 . Therefore, only the Z coordinate of point B′ 1 needs to be solved.

[0088] As Figure 7 shown is the schematic diagram of the geometric parameters for calculating the Z coordinate of point B′ 1 (calculation of the contact line length of the invariant part of helical gear meshing), point B 1 is the position where the tooth root of the tooth enters meshing at the end face of the tooth, point A is the position on the previous tooth on the same arc as point B 1 ; B 1The point ′ is the position where the tooth root of the tooth where point A is located enters meshing at a certain moment. Therefore, the straight line is the set of points where the tooth root of this tooth enters meshing, The included angle between and 2 is the helix angle β of the tooth root forming circle of the helical gear; α 1 is the included angle between the meshing point B 1 B 2 on the end face; α2 is the difference in the developed angles between the addendum circle and the tooth root forming circle; α tf is the end face pressure angle of the tooth root forming circle. Since Figure 7 in is parallel to the Z-axis, so is perpendicular to the gear end face. Therefore, ΔAB 1 B′ 1 is a right triangle, and ∠AB′ 1 B 1 is the helix angle β of the tooth root forming circle. Therefore, the formula for calculating the Z coordinate of point B′ 2 is: 1 Since

[0089]

[0090] points A and B Figure 7 in 1 are on the arc of the tooth root forming circle, so ΔAOB 1 is an isosceles triangle. Therefore, The calculation formula of

[0091]

[0092] To calculate the α angle in formula (22), the schematic diagram for calculating the difference in the developed angles α Figure 8 between the addendum circle and the tooth root forming circle as shown in 2 is introduced. The calculation formula of α 2 is:

[0093]

[0094]

[0095] α 2 = θ a - θ f (25)

[0096] In the formula:

[0097] is the addendum circle meshing angle (°);

[0098] is the tooth root forming circle meshing angle (°);

[0099] θ a is the addendum position expansion angle (°);

[0100] θ f is the dedendum position expansion angle (°).

[0101] In addition, referring to ΔB Figure 7 in 1 ON 1 the formula for the pressure angle α tf of the dedendum forming circle end face is:

[0102]

[0103] The angle α 1 B 2 between the end face engagement points B 1 is calculated by the formula:

[0104]

[0105] Therefore, the formula for calculating the α angle in formula (21) is:

[0106] α = α 1 + α 2 (28)

[0107] Furthermore, by jointly solving equations (21)-(28), the Z coordinate of point B 1 ′ is obtained. Therefore, the formula for the length L 1 of the invariant part of the helical gear meshing contact line is:

[0108]

[0109] Substituting the parameters of the helical gear pair in Table 1 into equations (21)-(28) for calculation, the numerical value z of the Z coordinate of point B′ 1 is: z = 32.92 mm; furthermore, the three-dimensional coordinates of point B′ 1 are: B′ 1 (6.9, 77.32, 32.92) (unit: mm).

[0110] Therefore, substituting the three-dimensional coordinates of point B 2 and point B′ 1 into formula (29), the length L 1 of the invariant part of the helical gear meshing contact line can be obtained as:

[0111] L 1 = 36.28 mm.

[0112] As Figure 9 shown for B′ 1Schematic diagram of geometric parameters for point Z coordinate calculation (calculation of the length of the meshing contact line of the tapered part of the helical gear), tooth root B of the tooth 1 Point is B' at a certain instant during the movement of point B along the axial direction of the gear 1 Point. In the figure, point K is an arbitrary point on the face meshing line. When point K moves from the tooth root to the tooth tip (from point B 1 point to point B 2 point), the length of the meshing contact line of the tooth gradually increases from zero. This is Figure 5 L in 2 . Calculate the length L of the contact line in the tapered part 2 The principle of calculating the length L of the contact line in the tapered part is the same as that of calculating the length L 1 of the constant part of the contact line. That is, for each determined point K, the X-Y coordinates of point K on the face meshing line can be calculated according to formulas (14)-(20) and combined with Figure 2 , and according to formulas (21)-(28) and combined with the geometric relationship of helical gear meshing, the tooth root position B' 1 corresponding to each point K during the gradual change of the contact line length is obtained, and the z coordinate of point 2 . Therefore, the calculation formula for the length L

[0113]

[0114]

[0115]

[0116]

[0117] α K2 = θ k - θ f (34)

[0118] α K = α K1 + α K2 (35)

[0119]

[0120]

[0121]

[0122] Therefore, substituting the parameters of the helical gear pair in Table 1 into formulas (30)-(38) to calculate the length L 2 of the meshing tapered part of the helical gear

[0123] 5) Calculate the total length of the contact line: By programming in MATLAB, the meshing time period of each gear tooth participating in a complete meshing process is divided into equal time periods, the contact line length of each time point of the gear tooth is calculated, and the contact line of the gear tooth is superimposed and summed at each time point to obtain the total length of the meshing contact line of the helical gear. After repeated experiments, it was found that the meshing time period is divided into 100 parts, which is the best solution. Therefore, this embodiment divides the meshing time period into 100 parts; in this embodiment, the contact line of the gear teeth is superimposed and summed at each time point according to the order in which each gear tooth enters the meshing.

[0124] like Figure 10 The figure shows the contact line change curve for each tooth of the helical gear when it meshes once.

[0125] like Figure 11 The figure shows the contact lines of multiple teeth of helical gears and their superposition diagrams. Curve a in the figure reflects the change process of the contact line when a tooth is fully meshed; curve b reflects the change process of the contact line when a tooth is out of meshing; curve c reflects the change of the contact line when a tooth is in meshing and is about to maintain a stable state; the meanings of curves d, e, f, and g are similar. Therefore, the total contact line length of the helical gear meshing is obtained by adding up the lengths of the contact lines of multiple teeth at the same time, and its change law with the rotation of the gear is shown in curve h in the figure.

[0126] Depend on Figure 11 It can be seen that the total length of the helical gear meshing contact line is time-varying, and the variation is affected by the overlap of the helical gears, specifically showing an alternating meshing form of four-tooth meshing and three-tooth meshing. For example, the area (1) in the figure is a four-tooth meshing area, in which there are 4 gear teeth participating in the meshing, and the gear teeth are numbered abcd; the area (2) in the figure is a three-tooth meshing area, in which there are 3 gear teeth participating in the meshing, and the gear teeth are numbered acd. The meshing conditions in the remaining areas are similar.

[0127] Through the implementation of the above steps, it is possible to accurately calculate the helical gear meshing contact line based on its three-dimensional spatial characteristics.

[0128] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A method for calculating the length of the meshing contact line of helical gears in three-dimensional space, characterized in that, it includes the following steps: 1) Obtain the two-dimensional coordinates of the limit point: Taking the center of the active wheel end face as the coordinate origin, establish a two-dimensional coordinate system, and solve the coordinates of the actual meshing limit point on the active wheel end face according to the straight-line equation of the helical gear end face meshing line; 2) Divide the contact line: According to the schematic diagram of the change of the helical gear meshing contact line, divide the contact line in the helical gear meshing process into a part with a constant contact line length and a part with a gradually changing contact line length; 3) Construct a three-dimensional space geometric relationship diagram: According to the parameter information of the helical gear pair, establish a transient finite element simulation model of helical gear meshing and submit it for calculation to obtain the transient diagram of the tooth contact stress spot of the active wheel in helical gear meshing, which is used to construct a three-dimensional space geometric relationship diagram showing the contact line when helical gears mesh; 4) Obtain each part of the contact line: Construct a three-dimensional rectangular coordinate system, and then based on the spatial characteristics of the meshing contact line in the three-dimensional rectangular coordinate system, establish and solve the three-dimensional space geometric equation of the vanishing point of the contact line at the tooth surface and tooth root positions, and propose the calculation principle and method of the coordinate value of this point, and then solve the lengths of the variable and invariant parts of the helical gear meshing contact line; 5) Obtain the total length of the contact line: Through MATLAB programming, evenly divide the meshing time period of each tooth participating in a complete meshing process into equal time periods, calculate the contact line length of each tooth at each time equal point, and sum up the contact lines of the teeth at each time equal point to obtain the total length of the helical gear meshing contact line, In step 4, the calculation principle and method of the vanishing point of the contact line at the tooth surface and tooth root positions include: On the transient diagram of the tooth contact stress spot of the driving gear, a three-dimensional rectangular coordinate system is constructed based on the two-dimensional X-Y coordinate system of the end face of the driving gear, with the axis of the driving gear as its Z-axis. At the same time, three auxiliary planes are established, namely the end face meshing line End face meshing point B 1 , B 2 The plane formed by stretching along the axial direction of the gear, and the three-dimensional coordinates of point B are obtained. Also, since 2 is parallel to the Z-axis, the X-Y coordinates of point B ′ are equal to the X-Y coordinates of point B 1 ; 1 point. B 1 The point is the position where the tooth root of the tooth enters meshing at the end face. Assume that point A is the position on the previous tooth on the same circular arc as point B 1 ; B 1 ' is the position where the tooth root of the tooth where point A is located enters meshing at a certain moment. Therefore, the straight line is the set of points where the tooth root of this tooth enters meshing, and The included angle between them is the helix angle β of the tooth root forming circle of the helical gear 2 ; α 1 is the included angle between the meshing point B 1 B 2 on the end face; α 2 is the difference in the developed angle between the addendum circle and the tooth root forming circle; α tf is the end face pressure angle of the tooth root forming circle; Since is parallel to the Z-axis, so is perpendicular to the gear end face. Therefore, ΔAB 1 B 1 ' is a right triangle, and ∠AB 1 'B 1 is the helix angle β of the tooth root forming circle 2 . Therefore, the formula for calculating the Z coordinate of point B 1 ' is: Since point A and point B 1 are on the arc of the root forming circle of the tooth, so ΔAOB 1 is an isosceles triangle. Therefore, The calculation formula is as follows: To calculate α in the above formula, the principle of calculating the difference in involute angles α between the addendum circle and the root circle of the formed gear is introduced. 2 For α 2 the calculation formula is as follows: α 2 = θ a - θ f In the formula: is the addendum circle engagement angle, in degrees; is the circular contact angle for root forming, in degrees; θ a is the angular displacement at the tooth tip, in degrees; θ f is the angular spread at the tooth root position, in degrees; In addition, the calculation formula for the pressure angle α of the root forming circular end face tf is as follows: N 1 B 1 is a line segment, and r b is the base circle radius; End face meshing point B 1 B 2 The included angle α 1 The calculation formula is as follows: N 1 B 2 is a line segment; Therefore, the calculation formula for the α angle is: α＝α 1 +α 2 Solve the above equation jointly to obtain B 1 Z coordinate of point '.

2. A method for calculating the length of the meshing contact line of helical gears in three-dimensional space according to claim 1, characterized in that, the straight-line equation of the helical gear end face meshing line is a straight-line equation established according to the parameter information of the helical gear pair, and the parameter information of the helical gear pair is composed of the basic parameter information of the helical gear pair, the length of the helical gear end face meshing line calculated according to the helical gear end face meshing principle diagram and the basic parameter information of the helical gear pair, and other geometric parameters of the helical gear.

3. A method for calculating the length of the meshing contact line of helical gears in three-dimensional space according to claim 1, characterized in that, the three-dimensional rectangular coordinate system is a three-dimensional rectangular coordinate system with the axis of the active wheel as its Z axis based on the two-dimensional coordinate system of the active wheel end face on the transient diagram of the tooth contact stress spot of the active wheel.

4. A method for calculating the length of the meshing contact line of helical gears in three-dimensional space according to claim 1, characterized in that, the number of segments of the equal time period ranges from 80 to 120.

5. A method for calculating the length of the meshing contact line of helical gears in three-dimensional space according to claim 1, characterized in that, the summation by superposition is carried out in the order of each tooth entering meshing successively.

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