Straight-tooth double-arc tooth profile spline and modeling method thereof

Abstract

The modeling method comprises the following steps: firstly, analyzing and comparing basic parameters of an involute gear and an involute spline to obtain a design relationship between the involute straight spur gear and the involute spline, and mapping the design relationship to a double-arc gear and the double-arc tooth profile spline needing to be designed to obtain a double-arc tooth profile spline model; analyzing the basic parameters to obtain a basic parameter calculation formula of the double-arc spline; then determining a tooth profile equation; and finally, modeling and assembling the internal and external splines in modeling software. Loads distributed on the tooth surface of the modeled spline are more uniform, abrasion and damage caused by stress concentration are reduced, and therefore the service life of the spline is prolonged.

Description

Technical Field

[0001] The invention belongs to the field of spline tooth profile design and relates to a spline with a straight tooth double arc tooth profile and a modeling method thereof. Background Art

[0002] The involute tooth profile spline has the characteristics of large tooth root arc, small stress concentration, high bending strength of teeth, and can withstand large bending stress. Under the action of load, the involute tooth profile spline has automatic centering, which allows multiple teeth to work at the same time, thereby reducing the load of a single tooth and improving the overall load-bearing capacity. Therefore, involute tooth profile splines are widely used in practical engineering fields such as aerospace, aviation, navigation, and automobiles. However, in some occasions with high precision requirements, the transmission performance will be reduced due to the poor processing accuracy and assembly error of the involute tooth profile. In addition, in view of the complex working conditions of the involute spline pair itself and the complex geometric characteristics of the application occasions, it is always under the dynamic load of the combined action of internal and external excitations during operation. The force and stress distribution on the tooth surface are extremely uneven, which causes serious wear on the tooth surface, and ultimately seriously affects its service life and reliability. Summary of the invention

[0003] The object of the present invention is to provide a spline with a straight tooth double arc tooth profile and a modeling method thereof, wherein the load on the modeled spline tooth surface is more evenly distributed, and the wear and damage caused by stress concentration are reduced, thereby extending the service life of the spline.

[0004] In order to achieve the above object, the present invention provides the following technical solutions:

[0005] A spline modeling method for a straight tooth double arc tooth profile comprises the following steps:

[0006] Step 1: First, analyze and compare the basic parameters of involute gears and involute splines, and obtain the design relationship between involute spur gears and involute splines, which is then mapped to double circular arc gears and the double circular arc tooth profile splines to be designed. By analyzing the basic parameters, the basic parameter calculation formula for double circular arc splines is obtained.

[0007] Step 2: Determine the tooth profile equation;

[0008] Step 3: Model and assemble the internal and external splines in the modeling software.

[0009] Furthermore, the basic parameter calculation formula of the double arc spline described in step 1 is determined by the following steps:

[0010] (1) Determine the pressure angle: According to the mapping relationship between the double circular arc gear and the double circular arc spline to be designed, the pressure angle of the double circular arc spline is set to 24°;

[0011] (2) Determine the basic calculation formula for the major and minor diameters of internal and external splines:

[0012] The calculation formula for the gear geometry of a pair of internally meshing involute spur gears is as follows:

[0013] External gear:

[0014]

[0015] Internal gear:

[0016]

[0017] Where: d and d1 are the pitch circle diameters of the external gear and internal gear respectively; d a1 ,d a2 are the diameters of the tooth top circles of the external gear and the internal gear respectively; d f1 ,d f2 are the root diameters of the external gear and the internal gear respectively; x1 and x2 are the displacement coefficients of the external gear and the internal gear respectively; z1 and z2 are the number of teeth of the external gear and the internal gear respectively; m, c * They are gear module, tooth top height coefficient, and radial top clearance coefficient. is the tooth root circle coefficient;

[0018] When a pair of internally meshed gears are used as a pair of splines to transmit power, the number of teeth on the internal and external splines should be equal, that is, the number of spline teeth z = z1 = z2; in addition, according to the French spline standard, the internal and external spline modification coefficients should be equal, that is, the spline modification coefficient x = x1 = x2, so the geometric dimension calculation formula of the involute spline with a pressure angle of 20° can be obtained as shown in formulas (1) and (2);

[0019] Similarly, for double arc gears, the calculation formula is the same as formula (1) and (2). The difference from involute gears is that c * Different values ​​of x lead to different calculation formulas. In double circular arc gears, Usually 0.65 is taken. Usually 0.3, c * Usually it is taken as 0.25, and the spline displacement coefficient x is taken as 0 when the standard center distance is installed. From this, the calculation formula for the size diameter of the inner and outer splines of the double arc spline is obtained:

[0020]

[0021] Where d a1 ,d a2 are the tooth tip diameters of the external spline (1) and internal spline (2); d f1 ,d f2 are the root diameters of the external spline (1) and the internal spline (2) respectively; z is the number of teeth; m is the module; x is the spline modification coefficient;

[0022] (3) Determine the basic parameter calculation formula of spur gear double arc spline

[0023] By comparing involute splines and involute gears, through double arc gears, when the helix angle is 0, they are installed according to the standard center distance, and the displacement coefficient x is 0, the parameter calculation formula of spur double arc splines is obtained:

[0024] Pitch circle diameter D = mz; tooth pitch p = πm;

[0025] Internal spline major diameter basic size: 24° flat tooth root D ei = m(z+1.8);

[0026] Internal spline minor diameter basic size: 24° flat tooth root D Fimin = m(z-1.1); basic tooth width E = 0.5πm; maximum effective tooth width E Vmax =E max -λ, λ is the comprehensive tolerance; the minimum effective tooth width is E Vmin =0.5πm; Actual maximum tooth width E max =E Vmin +(T+λ), T is the processing tolerance; the minimum value of the actual tooth width is E min =E Vmin +λ;

[0027] External spline major diameter basic size: 24° flat tooth root D ee = m(z+1.1);

[0028] External spline minor diameter basic size: 24° flat tooth root D ie =m(z-1.8); basic tooth thickness S = 0.5πm;

[0029] Maximum effective tooth thickness S Vmax =S+es V ,es V is the upper deviation of effective tooth thickness; the minimum effective tooth thickness S Vmin =S min +λ; Actual maximum tooth thickness Actual minimum tooth thickness S min =S Vmax -(T+λ); Tooth profile allowance C F =0.1m.

[0030] Furthermore, the tooth profile equation in step 2 is determined by the following process:

[0031] Both the involute spline and the involute spur gear have involute tooth profiles, which can be mapped to the double arc gear, that is, the basic tooth profile of the double arc spline is the same as the basic tooth profile of the double arc gear; the double arc gear has a normal double arc and a quasi-end double arc. Take the helix angle as 0° and design the spur double arc spline. At this time, take the basic tooth profile of the double arc gear as the basic tooth profile of the double arc spline. According to the basic tooth profile of the double arc gear of GB / T12759-1991, the calculation formula corresponding to each arc of the double arc spline is as follows:

[0032] The parameter equation of the arc of the convex tooth profile is:

[0033]

[0034] The parametric equation of the connecting arc is:

[0035]

[0036] The arc parameter equation of the concave tooth profile is:

[0037]

[0038] Parameter equation of tooth root arc:

[0039]

[0040] Where: h a - tooth top height; h f - tooth root height; ρ a - radius of the arc of the convex tooth gallery; ρ f - radius of the arc of the concave tooth profile; e a - Displacement of the center of the tooth profile of the convex tooth; e f - displacement of the center of the concave tooth profile; l a -Convex tooth profile center offset: l f - concave tooth profile center offset; δ1- convex tooth process angle; r j - Radius of the connecting arc; r gf - tooth root radius; S ak -Tooth thickness at the contact point of the convex teeth; h ja - Height of the convex tooth connection point; h jf -concave tooth connection point height; δ2-concave tooth process angle.

[0041] Furthermore, the specific process of modeling in step 3 is as follows:

[0042] (1) Determine tolerances and calculate internal and external spline parameters

[0043] When the tolerance grade is 7, when m = 1.5 mm, z = 12, the total tolerance (T + λ) is 0.126 mm, and the cumulative pitch tolerance F p is 0.049, tooth profile tolerance ff is 0.047, take the external spline effective tooth thickness e s The deviation is 0, and the internal and external spline parameters are obtained;

[0044] (2) Use Soildworks software to build part models

[0045] To model the internal and external splines separately, create a sketch on the front view reference plane first. First, use the circle command to draw the pitch circle, major diameter, and minor diameter according to the relevant parameters. Then use the command to create a spline curve, select the equation-driven curve, and input the arc equations of each segment of the double arc spline to obtain the arc contours of each part of the corresponding convex tooth. Then stretch the matrix to get a single tooth. Finally, set the number of teeth to 12 with the center of the pitch circle as the center, and complete the modeling of the internal and external splines through a circular array.

[0046] Furthermore, the specific process of the assembly described in step three is: create a new assembly, add the modeled internal and external splines into the assembly file, use the concentricity constraint in the assembly constraint command to ensure the concentricity of the internal and external splines, and use contact alignment to ensure that the involute stretching surface of the external spline (1) fits the involute stretching surface of the internal spline (2).

[0047] Furthermore, the tooth profile of the spline is in the form of a double arc, and its tooth profile is formed by connecting a convex arc and a concave arc with a common tangent line. The maximum normal contact pressure of the inner spline (2) is 137 MPa, and the maximum normal contact pressure of the outer spline (1) is 112.8 MPa.

[0048] Compared with the prior art, the present invention has the following beneficial effects:

[0049] The present invention proposes a spline with a double circular arc tooth profile. The double circular arc tooth profile is a spatial spiral body, and its tooth profile is formed by connecting a convex circular arc and a concave circular arc with a common tangent line. This design enables the convex tooth profile of the inner spline and the concave tooth profile of the outer spline, as well as the concave tooth profile of the outer spline and the convex tooth profile of the inner spline to simultaneously produce a conjugate "double conjugate" meshing interval during the connection. This double conjugate meshing mode not only increases the number of meshing tooth pairs, but also makes the load distribution on the tooth surface more uniform, improves the service life, and improves the bending fatigue strength of the spline key teeth. Compared with the involute tooth profile, the bending fatigue strength of the double circular arc tooth profile can be increased by 8% to 14%. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 A two-dimensional sketch of the external spline circular arc tooth profile of the present invention;

[0051] Figure 2 It is a three-dimensional part drawing of a single tooth of an external spline of the present invention;

[0052] Figure 3 It is a three-dimensional part drawing of the external spline full teeth of the present invention;

[0053] Figure 4 A two-dimensional sketch of the internal spline circular arc tooth profile of the present invention;

[0054] Figure 5 It is a three-dimensional part drawing of a single tooth of an internal spline of the present invention;

[0055] Figure 6 It is a three-dimensional part drawing of the internal spline full teeth of the present invention;

[0056] Figure 7 It is the internal and external spline assembly diagram of the present invention;

[0057] Figure 8 Meshing the spline tooth surface of the present invention;

[0058] Fig. 9 The load distribution diagram of the external spline of the working surface of the present invention;

[0059] Fig.10 It is the load distribution diagram of the spline in the working surface of the present invention;

[0060] In the figure: 1-external spline; 2-internal spline; 3-external spline convex tooth profile root arc; 4-external spline convex tooth profile connecting arc; 5-external spline convex tooth profile arc; 6-external spline internal spline major diameter; 7-external spline pitch circle; 8-external spline convex tooth profile root connecting arc; 9-internal spline minor diameter; 10-internal spline tooth profile root arc; 11-internal spline major diameter; 12-internal spline tooth profile arc; 13-internal spline tooth profile connecting arc; 14-internal spline pitch circle; 15-internal spline minor diameter. DETAILED DESCRIPTION

[0061] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0062] A spline modeling method for a straight tooth double arc tooth profile described in this embodiment includes the following steps:

[0063] Step 1: First, analyze and compare the basic parameters of involute gears and involute splines, and derive the design relationship between involute spur gears and involute splines. Then, map the relationship to double circular arc gears and the double circular arc splines to be designed. By analyzing the basic parameters, the basic parameter calculation formula of double circular arc splines is obtained.

[0064] (1) Determine the pressure angle: Involute splines are used as couplings, and involute gears are used as transmission parts. The common feature of both is that the tooth profile is involute. The difference is that when the pressure angle is different, the tooth profile shape area intercepted on the involute is different, resulting in different characteristics. The pressure angle of involute splines specified in the national standard is usually 30°, 37.5° and 45°, and the pressure angle of involute spur gears is usually 20°. However, there are 20° involute splines in the NF standard, and this spline is widely used in NGW planetary gear reducers. In "Calculation of Geometric Dimensions of 20° Pressure Angle Spur Involute Splines" by Liu Xiaolong et al., a design method for 20° involute splines with the same gear pressure angle was calculated in comparison with the NF standard. It is pointed out that the 20° pressure angle involute spline with standard teeth is actually an internal meshing gear with equal numbers of teeth on the inner and outer gears. The pressure angle of double circular arc gears in GB / T 12759-1991 standard is 24°. Therefore, by comparing the double circular arc gears with the double circular arc splines to be designed, their geometric characteristics are mapped, and then the pressure angle of the double circular arc splines is set to 24°.

[0065] (2) Determine the basic calculation formula for the major and minor diameters of internal and external splines:

[0066] The calculation formula for the gear geometry of a pair of internally meshing involute spur gears is as follows:

[0067] External gear:

[0068]

[0069] Internal gear:

[0070]

[0071] Where: d and d1 are the pitch circle diameters of the external gear and internal gear respectively; d a1 ,d a2 are the diameters of the tooth top circles of the external gear and the internal gear respectively; d f1 ,d f2 are the root diameters of the external gear and the internal gear respectively; x1 and x2 are the displacement coefficients of the external gear and the internal gear respectively; z1 and z2 are the number of teeth of the external gear and the internal gear respectively; m, c * They are gear module, tooth top height coefficient, and radial top clearance coefficient. is the tooth root circle coefficient.

[0072] When a pair of internally meshing gears are used as a pair of splines to transmit power, the number of teeth on the internal and external splines should be equal, that is, the number of spline teeth z = z1 = z2; in addition, according to the French spline standard, the internal and external spline displacement coefficients should be equal, that is, the spline displacement coefficient x = x1 = x2. In this way, the geometric dimension calculation formula for the involute spline with a pressure angle of 20° can be obtained, such as formulas (1) and (2).

[0073] Similarly, for double arc gears, the calculation formula is the same as formula (1) and (2). The difference from involute gears is that c * Different values ​​of x lead to different calculation formulas. In double circular arc gears, Usually 0.65 is taken. Usually 0.3, c * Usually it is taken as 0.25, and the spline displacement coefficient x is taken as 0 when the standard center distance is installed. From this, the calculation formula for the size diameter of the inner and outer splines of the double arc spline is obtained:

[0074]

[0075] Where d a1 ,d a2 are the tooth tip diameters of the external spline (1) and internal spline (2); d f1 ,d f2 are the root diameters of the external spline (1) and the internal spline (2) respectively; z is the number of teeth; m is the module; and x is the spline modification coefficient.

[0076] (3) Determine the basic parameter calculation formula of the spur double circular arc spline: By comparing the involute spline and the involute gear, the double circular arc gear is installed according to the standard center distance when its helix angle is 0, and the displacement coefficient x is 0, the parameter calculation formula of the spur double circular arc spline is obtained, as shown in Table 1:

[0077] Table 1 Calculation formula for basic parameters of spur double arc spline

[0078]

[0079]

[0080] Step 2: Determine the tooth profile equation

[0081] Both the involute spline and the involute spur gear have involute tooth profiles, which can be mapped to the double arc gear, that is, the basic tooth profile of the double arc spline is the same as the basic tooth profile of the double arc gear. Double arc gears have normal double arcs and quasi-end double arcs. Take the helix angle as 0° and design a spur double arc spline. At this time, the basic tooth profile of the double arc gear is taken as the basic tooth profile of the double arc spline. According to the basic tooth profile of the double arc gear of GB / T12759-1991, the calculation formula corresponding to each arc of the double arc spline is as follows:

[0082] The parameter equation of the arc of the convex tooth profile is:

[0083]

[0084] The parametric equation of the connecting arc is:

[0085]

[0086] The arc parameter equation of the concave tooth profile is:

[0087]

[0088]

[0089] Parameter equation of tooth root arc:

[0090]

[0091] Where: h a - tooth top height; h f - tooth root height; ρ a - radius of the arc of the convex tooth gallery; ρ f - radius of the arc of the concave tooth profile; e a - Displacement of the center of the tooth profile of the convex tooth; e f - displacement of the center of the concave tooth profile; l a -Convex tooth profile center offset: l f - concave tooth profile center offset; δ1- convex tooth process angle; r j - Radius of the connecting arc; r gf - tooth root radius; S ak -Tooth thickness at the contact point of the convex teeth; h ja - Height of the convex tooth connection point; h jf -concave tooth connection point height; δ2-concave tooth process angle.

[0092] Step 3: Model and assemble the internal and external splines in the modeling software

[0093] (1) Determine tolerances and calculate internal and external spline parameters:

[0094] The tolerance of the involute spline is set to the tolerance of the double arc spline. In the involute spline, the total tolerance of the total differential tooth width and tooth thickness is calculated as:

[0095] (T+λ)=40i1±160i2

[0096] i1 is the tolerance based on the pitch circle diameter D, and its tolerance unit i is: when D≤500mm, When D>500mm, i=0.004D+2.1.

[0097] i2 is based on the basic tooth width E or basic tooth thickness S. The tolerance unit i is: or Wherein the unit of D, E and S is mm.

[0098] Comprehensive tolerance λ:

[0099]

[0100] Cumulative tolerance of tooth pitch F p :

[0101]

[0102] Where L is half of the circumference of the graduation circle:

[0103] Tooth profile tolerance f f :

[0104]

[0105] in is the tolerance factor,

[0106] Tooth tolerance F β :

[0107]

[0108] Where g is the spline length;

[0109] The tolerance grade of the involute refers to the tooth width and tooth thickness and related parameters, that is, the tolerance grade of the cumulative pitch error, tooth shape error and tooth direction error. The tolerance grade is divided according to the size of the total tolerance (T+λ), where T is the processing tolerance and λ is the comprehensive tolerance. When the tolerance grade is 7, when m=1.5mm, z=12, it can be obtained from the manual that the total tolerance (T+λ) is 0.126mm, and the cumulative pitch tolerance F p is 0.049, tooth profile tolerance f f is 0.047. Take the external spline effective tooth thickness e s The deviation is 0. Therefore, the parameters of the double arc internal and external splines are obtained, as shown in Table 2:

[0110] Table 2 Internal and external spline parameters

[0111]

[0112] (2) Application software modeling and assembly

[0113] Soildworks software is used to build the part model. First, the internal spline is modeled. A sketch is created on the front view reference plane. The circle command is used to draw the external spline pitch circle 7, the external spline major diameter 6, and the external spline minor diameter 9 in sequence according to the internal spline parameters in Table 2. Then, the spline size relationship expression is established. The command to create a spline curve is used, and the curve driven by the equation is selected. The equations of each arc of the double arc spline are brought in. The calculated parameter equations are brought in in sequence to obtain the external spline convex tooth profile root arc 3, the external spline convex tooth profile connection arc 4, the external spline convex tooth profile arc 5, and the external spline convex tooth profile root connection arc 8, as shown in FIG. Figure 1 Then click Feature, click Boss Stretch Base, and you can extrude an external spline of a tooth, as shown in Figure 2 As shown, then click the circular array command, select the center of the dividing circle as the center point, and select the number of teeth in the array as 12, and you can model a straight tooth double arc external spline, as shown Figure 3 shown.

[0114] Create a new part and then model the external spline. Create a sketch on the front view reference plane. Use the circle command to draw the internal spline pitch circle 14, the internal spline major diameter 11, and the internal spline minor diameter 15 in sequence. Use the command to create a spline curve, select the curve driven by the equation, and substitute the equations of each arc of the double arc spline to obtain the external spline convex tooth profile root arc 10, the internal spline convex tooth profile connection arc 13, the internal spline convex tooth profile arc 12, and the internal spline convex tooth profile arc 13. Figure 4 As shown, then click the boss to stretch the base to extrude an internal spline of a tooth, as shown Figure 5 As shown, then click the circular array command, select the center of the dividing circle as the center point, and select the number of teeth in the array as 12, and you can model the internal spline of the straight tooth double arc, as shown Figure 6 shown.

[0115] Create a new assembly, click Add Part, add the modeled internal and external splines into the assembly file, use the concentricity constraint in the assembly constraint command to ensure the concentricity of the internal and external splines, use contact alignment to ensure that the involute stretch surface of the external spline 1 fits the involute stretch surface of the internal spline 2. The assembly structure is as follows: Figure 7 shown.

[0116] Finite element analysis verification

[0117] During the simulation process, in order to ensure that the simulation results are less than the actual error, the grid needs to be divided evenly and finely, otherwise too many grids will cause the computer to waste too much computing power. By simulating the work of a single tooth, you can efficiently obtain the simulation data rules. In the simulation software ABAQUS, import the spline that has just been assembled. First, in the property module, create the Material-Steel material: an elastic material with a density of 7.81E-09tonne / mm3, a Young's modulus of 210000MPa, and a Poisson's ratio of 0.3. Create a load with a bending moment of 50 (N·m), so the bending moment acting on each pair of teeth is 4.167 (N·m). When dividing the mesh, distinguish the importance of different parts of the model. If the key parts and key nodes need to improve the calculation accuracy, you can choose to refine the mesh, so refine the mesh at the tooth profile, and discretize the mesh in other places, such as Figure 8 shown.

[0118] Then, after meshing the internal and external splines, enter the job module and click Submit to perform finite element analysis. Fig. 9 , 10 As shown, the maximum normal contact pressure of the inner spline 2 is 137MPa, and the maximum normal contact pressure of the outer spline 1 is 112.8MPa. Compared with the involute spline with the same module and number of teeth at a pressure angle of 20°, after calculation by the relevant formula, it can be concluded that the maximum normal contact stress of the outer spline is 158.18MPa, and the maximum normal contact stress of the inner spline is: 129.84MPa. From this comparison, it can be concluded that the maximum normal contact stress of the outer spline with a straight tooth double arc tooth profile is reduced by 13.38%, and the maximum normal contact stress of the inner spline with a straight tooth double arc tooth profile is reduced by 13.12%. Compared with the involute spline, the double arc spline not only improves the durability and reliability of the spline, but also reduces the wear and damage caused by stress concentration, thereby extending the service life of the spline.

Claims

1. A spline modeling method for a straight tooth double arc tooth profile, characterized in that: The following steps are involved: Step 1: First, analyze and compare the basic parameters of involute gears and involute splines, and obtain the design relationship between involute spur gears and involute splines, which is then mapped to double circular arc gears and the double circular arc tooth profile splines to be designed. By analyzing the basic parameters, the basic parameter calculation formula for double circular arc splines is obtained. Step 2: Determine the tooth profile equation; Step 3: Model and assemble the internal and external splines in the modeling software.

2. The spline modeling method of a straight tooth double arc tooth profile according to claim 1, characterized in that: The basic parameter calculation formula of the double arc spline described in step 1 is determined by the following steps: (1) Determine the pressure angle: According to the mapping relationship between the double circular arc gear and the double circular arc spline to be designed, the pressure angle of the double circular arc spline is set to 24°; (2) Determine the basic calculation formula for the major and minor diameters of internal and external splines: The formula for calculating the gear geometry of a pair of internally meshing involute spur gears is as follows: External gear: Internal gear: Where: d and d1 are the pitch circle diameters of the external gear and internal gear respectively; d a1 d a2 are the diameters of the tooth top circles of the external gear and the internal gear respectively; d f1 d f2 are the root diameters of the external gear and the internal gear respectively; x1 and x2 are the displacement coefficients of the external gear and the internal gear respectively; z1 and z2 are the number of teeth of the external gear and the internal gear respectively; m, c * They are gear module, tooth top height coefficient, and radial top clearance coefficient. is the tooth root circle coefficient; When a pair of internally meshed gears are used as a pair of splines to transmit power, the number of teeth on the internal and external splines should be equal, that is, the number of spline teeth z = z1 = z2; in addition, according to the French spline standard, the internal and external spline modification coefficients should be equal, that is, the spline modification coefficient x = x1 = x2, so the geometric dimension calculation formula of the involute spline with a pressure angle of 20° can be obtained as shown in formulas (1) and (2); Similarly, for double arc gears, the calculation formula is the same as formula (1) and (2). The difference from involute gears is that c * Different values ​​of x lead to different calculation formulas. In double circular arc gears, Usually 0.65 is taken. Usually 0.3, c * Usually it is taken as 0.25, and the spline displacement coefficient x is taken as 0 when the standard center distance is installed. From this, the calculation formula for the size diameter of the inner and outer splines of the double arc spline is obtained: Where d a1 d a2 are the tooth tip diameters of the external spline (1) and internal spline (2); d f1 d f2 are the root diameters of the external spline (1) and the internal spline (2) respectively; z is the number of teeth; m is the module; x is the spline modification coefficient; (3) Determine the basic parameter calculation formula of spur gear double arc spline By comparing involute splines and involute gears, through double arc gears, when the helix angle is 0, they are installed according to the standard center distance, and the displacement coefficient x is 0, the parameter calculation formula of spur double arc splines is obtained: Pitch circle diameter D = mz; tooth pitch p = πm; Internal spline major diameter basic size: 24° flat tooth root D ei = m(z+1.8); Internal spline minor diameter basic size: 24° flat tooth root D Fimin = m(z-1.1); basic tooth width E = 0.5πm; maximum effective tooth width E Vmax =E max -λ, λ is the comprehensive tolerance; Minimum effective tooth groove width E Vmin =0.5πm; Actual maximum tooth width E max =E Vmin +(T+λ), T is the processing tolerance; the minimum value of the actual tooth width is E min =E Vmin +λ; External spline major diameter basic size: 24° flat tooth root D ee = m(z+1.1); External spline minor diameter basic size: 24° flat tooth root D ie =m(z-1.8); basic tooth thickness S = 0.5πm; Maximum effective tooth thickness es V is the upper deviation of effective tooth thickness; minimum effective tooth thickness Actual maximum tooth thickness Minimum actual tooth thickness Tooth profile allowance C F =0.1m.

3. The spline modeling method of a straight tooth double arc tooth profile according to claim 2 is characterized in that: The tooth profile equation described in step 2 is determined by the following process: Both the involute spline and the involute spur gear have involute tooth profiles, which can be mapped to the double arc gear, that is, the basic tooth profile of the double arc spline is the same as the basic tooth profile of the double arc gear; the double arc gear has a normal double arc and a quasi-end double arc. Take the helix angle as 0° and design a spur double arc spline. At this time, the basic tooth profile of the double arc gear is taken as the basic tooth profile of the double arc spline. According to the basic tooth profile of the double arc gear of GB / T12759-1991, the calculation formula corresponding to each arc of the double arc spline is as follows: The parameter equation of the arc of the convex tooth profile is: The connecting arc parameter equation is: The arc parameter equation of the concave tooth profile is: Parameter equation of tooth root arc: Where: h a - tooth top height; h f - tooth root height; ρ a - radius of the arc of the convex tooth gallery; ρ f - radius of the arc of the concave tooth profile; e a - Displacement of the center of the tooth profile of the convex tooth; e f - displacement of the center of the concave tooth profile; l a -Convex tooth profile center offset: l f - concave tooth profile center offset; δ1- convex tooth process angle; r j - Radius of the connecting arc; r gf - tooth root radius; S ak -Tooth thickness at the contact point of the convex teeth; h ja - Height of the convex tooth connection point; h jf -concave tooth connection point height; δ2-concave tooth process angle.

4. The spline modeling method of a straight tooth double arc tooth profile according to claim 3 is characterized in that: The specific process of modeling described in step 3 is: (1) Determine tolerances and calculate internal and external spline parameters When the tolerance grade is 7, when m = 1.5 mm, z = 12, the total tolerance (T + λ) is 0.126 mm, and the cumulative pitch tolerance F p is 0.049, tooth profile tolerance f f is 0.047, take the external spline effective tooth thickness e s The deviation is 0, and the internal and external spline parameters are obtained; (2) Use Soildworks software to build part models To model the internal and external splines separately, create a sketch on the front view reference plane first. First, use the circle command to draw the pitch circle, major diameter, and minor diameter according to the relevant parameters. Then use the command to create a spline curve, select the equation-driven curve, and input the arc equations of each segment of the double arc spline to obtain the arc contours of each part of the corresponding convex tooth. Then stretch the matrix to get a single tooth. Finally, set the number of teeth to 12 with the center of the pitch circle as the center, and complete the modeling of the internal and external splines through a circular array.

5. The spline modeling method of a straight tooth double arc tooth profile according to claim 4 is characterized in that: The specific process of assembly described in step 3 is as follows: create a new assembly, add the modeled internal and external splines into the assembly file, use the concentricity constraint in the assembly constraint command to ensure the concentricity of the internal and external splines, and use contact alignment to ensure that the involute stretch surface of the external spline (1) fits the involute stretch surface of the internal spline (2).

6. The spline generated by the modeling method according to claim 5, characterized in that: The tooth profile of the spline is a double arc, and the tooth profile is formed by connecting a convex arc and a concave arc with a common tangent line. The maximum normal contact pressure of the inner spline (2) is 137 MPa, and the maximum normal contact pressure of the outer spline (1) is 112.8 MPa.

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