An improved method of considering lateral and angular stiffness of a spline coupling with parallel misalignment

Abstract

The application aims to provide an improved method for considering the lateral stiffness and the corner stiffness of the spline coupling with misalignment, comprising the following steps: establishing a schematic diagram of the load and deformation of the misaligned spline coupling, deducing the load and deformation balance equation of the misaligned spline engagement state according to the deformation geometry of the misaligned spline; deducing the single-tooth engagement stiffness according to the energy method, correcting the gear tooth engagement stiffness; bringing the single-tooth engagement stiffness of the spline obtained in step (2) into the engagement force balance equation obtained in step (1), and calculating the lateral stiffness and the corner stiffness of the misaligned spline coupling. Through comparison with the finite element results, the calculation results of the application are more efficient, and the vacancy of the calculation method of the lateral stiffness and the corner stiffness of the spline coupling under the misalignment condition at the present stage is made up.

Description

TECHNICAL FIELD

[0001] The present application relates to a spline coupling stiffness analytical method, in particular to a spline coupling lateral and angular stiffness calculation method. BACKGROUND

[0002] Due to the large number of transmission components of high-speed tracked vehicles, the transmission system is more likely to fail under alternating and impact loads, and the tooth surface is prematurely worn due to the tooth-to-tooth sliding caused by vibration. Therefore, it is crucial to ensure the stability and reliability of the transmission system. As an indispensable part of the transmission system, the performance of the spline coupling is directly related to the reliable operation of the high-speed tracked vehicle. Since the spline coupling is often in a complex working condition, the spline is always in a slightly misaligned state, which will cause the spline coupling to bend and bring additional load to the spline shaft. Therefore, it is of great significance to study the stiffness characteristics of the misaligned spline coupling.

[0003] From previous studies, it can be seen that most of the work has focused on the meshing force and load distribution of the misaligned spline coupling. The research on the stiffness of the spline coupling is mostly carried out in the healthy state, and the research on the stiffness characteristics under misalignment is very few. Some scholars use energy method to calculate the gear mesh stiffness, and then deduce the spline stiffness expression. However, when calculating the gear mesh stiffness, the base and gear cantilever beam overlap at the tooth root, which leads to repeated calculation of energy in the traditional method. SUMMARY

[0004] The purpose of the present application is to provide an improved method for considering the lateral and angular stiffness of the spline coupling considering parallel misalignment, which can solve the problems existing in the existing spline misalignment fault characterization model and dynamic analysis technology.

[0005] The purpose of the present application is achieved in that:

[0006] The improved method for considering the lateral and angular stiffness of the spline coupling considering parallel misalignment according to the present application is characterized in that:

[0007] (1) a schematic diagram of the load and deformation of the misaligned spline coupling is established, and the balance equation of the load and deformation of the misaligned spline meshing state is derived according to the deformation geometric relationship of the misaligned spline;

[0008] (2) the single-tooth meshing stiffness is derived according to the energy method, wherein the tooth base stiffness is considered, and the gear mesh stiffness is corrected by removing the repeated calculation of the gear and base energy;

[0009] (3) According to the spline single-tooth meshing stiffness obtained in step (2), the meshing force balance equation obtained in step (1) is brought in to calculate the lateral stiffness and rotation angle stiffness of the misaligned spline coupling.

[0010] The present application can also include:

[0011] 1. In step (1), the process of deriving the balance equation of the load and deformation of the misaligned spline meshing state is as follows:

[0012] a. When the spline is misaligned, assuming that the misalignment amounts in the x direction and y direction are x0 and y0 respectively, the radial misalignment amount e is:

[0013]

[0014] Let the angle between the radial misalignment amount e and the x axis be θ, then:

[0015]

[0016] b. In the Cartesian coordinate system oxy, assuming that the spline has N teeth in total, the number of the tooth above the x axis is 1, and the numbers of the teeth in clockwise order are 2, 3, 4, … i, the angle between the center line of the i th tooth and the y axis is as follows:

[0017]

[0018] At this time, the equivalent meshing distance L of each tooth of the spline is: i

[0019]

[0020] The meshing stiffness k of each single tooth of the misaligned spline coupling is: gi represented as:

[0021] k gi =f(L i );

[0022] c. According to the slicing method to obtain the relationship between the external load and deformation of the misaligned spline coupling, first, the spline coupling is divided into independent slices, and the thickness of each slice is dz, then the force of each slice is determined from the meshing tooth shape, and the offset δ(z) of each slice when the outer spline shaft is deformed is obtained by integration:

[0023] δ(z)=(z-z a )θ x ;

[0024] d. Assuming that the pitch circle radius of the initial meshing position when the spline is not misaligned is r m ​; when there is no misalignment e, when the external spline slice moves under the action of external load by displacement δ, the pitch circle radius r of each tooth mi As follows:

[0025]

[0026] When there is misalignment e, and the slice has displacement δ, the meshing circle radius r of each tooth mi As follows:

[0027]

[0028] e. Load displacement δ and misalignment e caused by spline will cause the normal penetration of each tooth, the normal penetration amount Δ caused by displacement δ of the i-th tooth b1i For:

[0029] Δ b1i = δ cos α i

[0030] Where is the angle between the normal line at the meshing position of the i-th tooth and the y direction, β i is the angle between the tooth center line and the meshing line;

[0031] The normal penetration amount Δ caused by misalignment e b2i For:

[0032]

[0033] Assuming that the torsional displacement of the external spline shaft under the action of lateral force and torque is Then the normal penetration amount Δ of the meshing point of the i-th tooth caused by torque displacement is ti For:

[0034]

[0035] The total normal penetration expression Δ i For:

[0036] Δ i = Δ b1i + Δ b2i + Δ ti ;

[0037] f. For the i-th tooth pair, the normal force F generated by the meshing tooth i The expression is as follows:

[0038] dF i = H(Δ i ) k gi Δ i dz

[0039] Where k giH(Δ i ) is the unit step function defined in equation,

[0040]

[0041] g. By combining the forces of all the teeth and integrating along the axial direction, the relationship between the tooth deformation and the external load is obtained as follows:

[0042]

[0043] where n, N, x j and Δx j represent the number of slices along the axial direction, the total number of teeth, the axial coordinate of slice j, and the length of slice j, respectively.

[0044] 2. In step (2), the step of calculating the single-tooth meshing stiffness includes:

[0045] A. The single-tooth meshing stiffness k g is expressed as:

[0046]

[0047] B. The tooth base stiffness of the external spline is:

[0048]

[0049] C. The transition curve CD of the rack-type cutter machining the gear, which considers the tooth tip chamfer, is the equidistant curve of the long-width involute, and its expression is:

[0050]

[0051] D. The single-tooth bending, shearing, and axial compression stiffness of the spline is obtained through two representations of deformation energy in elasticity and material mechanics, and the modified external spline stiffness expression is:

[0052]

[0053]

[0054]

[0055] Part of the stiffness is integrated with angular displacement, and the modified external spline stiffness is simplified as:

[0056]

[0057]

[0058]

[0059] E. For internal splines, all teeth have an involute geometry, and their base circle is smaller than their addendum circle. The bending, shear, and axial stiffness of the internal spline are calculated using the potential energy method. The expression for the internal spline stiffness is:

[0060]

[0061]

[0062]

[0063] 3. In step (3), lateral stiffness and angular stiffness represent the ability to resist lateral and angular deformation, and their calculation methods are as follows:

[0064]

[0065]

[0066] Where δ y It is the deflection of the coupling center of the external spline end face, F y It is the deflection δ y The corresponding shear force in the corresponding direction, θ z M is the angle at the coupling center of the external spline end face. x Because of θ x Bending moment in the corresponding direction;

[0067] When spline misalignment occurs, the stiffness in the x and y directions is no longer the same. Ignoring cross stiffness, the relationship between the stiffness in the two directions is as follows:

[0068] k x (θ)=k y (θ+90°)

[0069] k θx (θ)=k θy (θ+90°).

[0070] The advantages of this invention are as follows: Based on contact theory and the slicing method, this invention proposes an improved method for considering the stiffness of splines with parallel misalignment. Based on spline contact theory, it proposes a balance equation for the meshing force of misaligned splines. In the calculation of meshing stiffness, the tooth base stiffness is considered, and the tooth stiffness is corrected by removing redundant energy calculations. This solves the problems in existing spline coupling meshing characteristic analysis and static analysis techniques, such as the lack of research on stiffness under misalignment faults. Attached Figure Description

[0071] Figure 1 This is a schematic diagram of a spline coupling.

[0072] Figure 2a is the engagement state before spline deformation (spline is completely centered), Figure 2b is the engagement state before spline deformation (spline is not centered) ;

[0073] Figure 3a is the simplified model of spline shaft (force diagram of outer spline shaft), Figure 3b is the simplified model of spline shaft (deformation diagram of spline shaft) ;

[0074] Figure 4a is the engagement state after spline deformation (spline is completely centered), Figure 4b is the engagement state after spline deformation (spline is not centered) ;

[0075] Figure 5a is the normal penetration after spline deformation (normal penetration caused by displacement), Figure 5b is the normal penetration after spline deformation (normal penetration caused by misalignment), Figure 5c is the normal penetration after spline deformation (normal penetration caused by torsion), Figure 5d is the normal penetration after spline deformation (spline tooth engagement parameter definition) ;

[0076] Figure 6a is the outer spline cantilever beam model before and after correction (before correction), Figure 6b is the outer spline cantilever beam model before and after correction (after correction) ;

[0077] Figure 7 is the inner spline cantilever beam model;

[0078] Figure 8 is the flowchart for solving spline stiffness;

[0079] Figure 9a is the schematic diagram of different engagement stiffness calculation models (the present application), Figure 9b is the schematic diagram of different engagement stiffness calculation models (traditional model) ;

[0080] Figure 10 is the entity finite element model of spline coupling;

[0081] Figure 11a is the spline load loading schematic diagram (shear force loading scheme schematic diagram), Figure 11b is the spline load loading schematic diagram (bending moment loading scheme schematic diagram) ;

[0082] Figure 12a is the comparison of stiffness of the model and the finite element model of the present application (lateral stiffness), Figure 12b is the comparison of stiffness of the model and the finite element model of the present application (rotation angle stiffness), T = 1500 N·m;

[0083] Figure 13a Comparison of lateral and angular stiffness (lateral stiffness) of the model of the present application and the finite element model under the condition of e=0.005 mm, θ=0°, Figure 13b Comparison of lateral and angular stiffness (angular stiffness) of the model of the present application and the finite element model under the condition of e=0.005 mm, θ=0°.

[0084] Figure 14a Comparison of lateral and angular stiffness (lateral stiffness) of the model of the present application and the finite element model under the condition of e=0.01 mm, θ=0°, Figure 14b Comparison of lateral and angular stiffness (angular stiffness) of the model of the present application and the finite element model under the condition of e=0.01 mm, θ=0°. DETAILED DESCRIPTION

[0085] The present application will be described in more detail below with examples and accompanying drawings:

[0086] In combination Figures 1-14b , the improved method for considering the lateral and angular stiffness of the spline coupling with misalignment of the present application comprises the following steps:

[0087] Step 1, a schematic diagram of the load and deformation of the misaligned spline coupling is established, and the balance equation of the load and deformation of the misaligned spline engagement state is derived according to the deformation geometric relationship of the misaligned spline;

[0088] S1.1 In the healthy state of the spline, the engagement distance of each tooth is the root circle of the inner spline to the top circle of the outer spline, as shown in Figure 2a , at this time the equivalent engagement distance is recorded as L0, when the spline exists misalignment, assuming that the misalignment amount in the x direction and y direction is x0 and y0 respectively, then the radial misalignment amount e is:

[0089]

[0090] Let the angle between the radial misalignment amount e and the x axis be θ, then:

[0091]

[0092] S1.2 When the spline appears misalignment, the engagement of the inner and outer splines is not uniform, some teeth are engaged tightly and some teeth are engaged loosely. Since the centers of the inner and outer splines do not coincide, the engagement distance L of each tooth of the inner and outer splines is not the same. When the misalignment direction θ is positive, the tooth engagement is the tightest, and vice versa, when the misalignment direction θ is reversed, the tooth engagement is the loosest, as shown in Figure 2b In the Cartesian coordinate system oxy, assuming that the spline has N teeth in total, the number of the tooth above the x axis is recorded as 1, and clockwise in turn as 2, 3, 4, … i, the angle between the center line of the i th tooth and the y axis is :

[0093]

[0094] The equivalent meshing distance L of each tooth of the spline at this time i is:

[0095]

[0096] Therefore, the meshing stiffness k of each single tooth of the misaligned spline coupling gi can be expressed as:

[0097] k gi = f(L i ) (5)

[0098] S1.3 The schematic diagram of the spline load and deformation is shown in Figures 3a-3b . According to the slicing method, the relationship between the external load and deformation of the misaligned spline coupling is obtained. First, the spline coupling is divided into multiple independent slices, and the thickness of each slice is dz. Then the force of each slice is determined from the angle of the meshing tooth profile, and the overall load of the misaligned spline can be obtained by integration. When the spline shaft is deformed, the offset δ(z) of the slice is expressed as:

[0099] δ(z) = (z-z a )θ x (6)

[0100] S1.4 As shown in Figure 4a , the pitch circle radius of the initial meshing position of the spline when the spline is not misaligned is r m ; when there is no misalignment e, when the outer spline slice moves under the action of the external load, the pitch circle radius r mi of each tooth is as follows:

[0101]

[0102] As shown in Figure 4b , when there is misalignment e, and the slice has displacement δ, the meshing circle radius r mi of each tooth is as follows:

[0103]

[0104] S1.5 In addition, the displacement δ and the misalignment e caused by the load to the spline will cause the normal penetration of each tooth. As shown in Figure 5a , the normal penetration Δ b1i of the i-th tooth caused by the displacement δ is:

[0105] Δ b1i = δcosα i (9)

[0106] where is the angle between the normal at the i-th tooth engagement point and the y-direction, β i is the angle between the tooth center line and the line of action.

[0107] As Figure 5b shown, the normal penetration Δ b2i due to misalignment e is given by:

[0108]

[0109] In addition to the lateral displacement δ and misalignment e that can cause normal penetration, the torque experienced by the external spline can also cause normal penetration. As Figure 5c shown, assuming the torsional displacement of the external spline shaft under lateral force and torque is then the normal penetration Δ ti at the i-th tooth engagement point due to the torque displacement is given by:

[0110]

[0111] Therefore, the total normal penetration expression Δ i is given by:

[0112] Δ i = Δ b1i + Δ b2i + Δ ti (12)

[0113] S1.6 For the i-th tooth pair, the normal force F i generated by the engaging teeth is given by:

[0114] dF i = H(Δ i )k gi Δ i dz (13)

[0115] where k gi is the single tooth engagement stiffness per unit length of the i-th tooth pair in the misaligned spline. H(Δ i ) is the unit step function defined in equation (14) and is given by equation (15). It must be noted that k gi and Δ i are generally different for different splines in the misaligned case.

[0116]

[0117] S1.7 Therefore, by combining the forces of all the tooth segments and integrating along the axial direction, the relationship between the tooth deformation and the external load can be obtained as follows:

[0118]

[0119] In the formula, n, N, x j and Δx j These represent the number of slices along the spline axis, the total number of teeth, the axial coordinate of slice j, and the length of slice j, respectively.

[0120] Step 2 derives the single-tooth meshing stiffness based on the energy method, taking into account the tooth base stiffness, and corrects the gear tooth meshing stiffness for the first time by removing redundant energy calculations;

[0121] S2.1 Single Tooth Meshing Stiffness k g The expression is:

[0122]

[0123] The tooth base stiffness of the S2.2 external spline is:

[0124]

[0125] S2.3 For gear machining with a rack-type cutter considering tooth tip chamfering, the transition curve CD is an equidistant curve of a long involute, and its expression is:

[0126]

[0127] The bending, shearing, and axial compressive stiffness of a single spline tooth in S2.4 is obtained through two methods: elasticity and material mechanics, representing the deformation energy. The modified expression for the external spline stiffness is as follows:

[0128]

[0129]

[0130]

[0131] U b1 U s1 U a1 , and These are the external spline gear tooth region ABDGF (after correction), ABDF (before correction), and the sector region, respectively. The bending, shearing and axial compressive energy, F a =Fsinβ,F b =Fcosβ;x β It is the distance between the meshing point and the center line of the gear teeth; y β G is the horizontal distance between the meshing point and the origin; G = E / 2(1+v) is the shear modulus; υ is Poisson's ratio; y1, y2 and y3 represent the transition curve, involute and circular arc, respectively. The horizontal coordinate of any point on the y-axis;C y D I represents the horizontal coordinates of the start and end points of the transition curve; y1 A y1 I y2 A y2 I y3 A y3 Transition curves, involutes, and arcs The moment of inertia and cross-sectional area of ​​the cross section at any position on the surface; M1 = F b (y β -y1)-F a x β M2 = F b (y β -y2)-F a x β and M3 = F b (y β -y3)-F a x β These represent the meshing forces for the transition curve, involute, and circular arc, respectively. The torque generated at any point on it. Other geometric parameters are as follows: Figures 6a-6b As shown.

[0132] For ease of spline stiffness integration calculation, some stiffness is integrated using angular displacement. The corrected external spline stiffness is simplified as follows:

[0133]

[0134]

[0135]

[0136] For internal splines, S2.5, all teeth have an involute geometry, and their base circles are smaller than their addendum circles. This paper calculates the bending, shear, and axial stiffness of internal splines using the potential energy method. The cantilever beam model of an internal spline is shown below. Figure 7 As shown, the expression for the stiffness of the internal spline is:

[0137]

[0138]

[0139]

[0140] Step 3: Based on the spline single-tooth meshing stiffness obtained in Step S2, substitute it into the meshing force balance equation obtained in S1 to calculate the lateral stiffness and angular stiffness of the misaligned spline coupling. Their calculation methods are as follows:

[0141]

[0142]

[0143] Where δ y It is the deflection of the coupling center of the external spline end face, F y It is the deflection δ y The corresponding shear force in the corresponding direction. θ z M is the angle at the coupling center of the external spline end face. x Because of θ x The bending moment in the corresponding direction. This paper proposes a method for calculating the lateral stiffness and angular stiffness in the case of spline misalignment, the flowchart of which is shown below. Figure 8 .

[0144] It is worth noting that when spline misalignment occurs, the stiffness in the x and y directions is no longer the same, which differs from a healthy spline. Ignoring cross stiffness, the relationship between the stiffness in the two directions is as follows:

[0145] k x (θ)=k y (θ+90°) (30)

[0146] k θx (θ)=k θy (θ+90°) (31)

[0147] Example

[0148] Based on the results of the misaligned spline stiffness analysis model in step 3, experimental verification was conducted, using the traditional model, the Y model, and the finite element model for comparison. Figures 9a-9b As shown, the traditional model does not consider the combined stiffness k of the gear body. tf However, considering the tooth base stiffness k f The model only considers the stiffness of the gear teeth.

[0149] The spline parameters used here are shown in Table 1. Unless otherwise specified, subsequent spline parameters are the same as those used in this analysis. A finite element model of the spline coupling is created in ANSYS software, as follows: Figure 10 As shown. The entire structure was simulated using Solid185 elements. The contact process of the meshing teeth was simulated using three-dimensional contact surface Conta173 elements and target surface Targe170 elements. Contact elements were only set on the driving side of the meshing teeth, and it was assumed that the meshing teeth on other sides did not contact, which is consistent with the assumptions in the analysis model. To improve the accuracy of the calculation results, the mesh of the meshing teeth was refined. Figures 11a-11bAs shown, one end of the inner spline is fixed, the other end of the outer spline is subjected to an external load, the surface of one end of the outer spline and the center point are rigidly bound by a Mass21 unit to apply torque, shear force and bending moment in the calculation of lateral stiffness and rotational stiffness.

[0150] Table 1 Spline coupling structure parameters

[0151]

[0152] First, the y-direction stiffness under misalignment e = 0 mm and torque T = 1500 N·m is calculated by the model of the application, the traditional model, the Yu model and the finite element model respectively, and the results are shown in Figures 12a-12b The change rule of spline stiffness can be roughly divided into three stages: stage one, the spline lateral stiffness is basically stable, because the radial force at the initial stage of loading mainly acts on the outer spline, the deformation caused by the radial force and the existing initial gap offset, resulting in the number of spline meshing teeth and the contact area remaining unchanged, and finally the stiffness almost tends to be constant; stage two, the spline lateral stiffness decreases sharply with the increase of lateral displacement, because the number of contact teeth decreases significantly with the increase of lateral displacement, the contact area decreases sharply, and thus the stiffness decreases rapidly; stage three, the spline lateral stiffness decreases slowly and remains stable again with the increase of lateral displacement, the main reason is that the number of contact teeth decreases slowly, only the contact pressure and contact area change slightly; therefore, the lateral stiffness in this stage decreases slightly again.

[0153] In Figures 12a-12b comparison with the finite element model, the maximum calculation error of the stiffness of different models is in the first stage. The maximum calculation error of the Yu model is about 45%, that of the traditional model is about 12%, and that of the model of the application is about 0.4%. The calculation results show that the consistency of the spline stiffness obtained by the method of the application and the finite element method is the best. The calculation time of the proposed model is 1.26 seconds, the Yu model needs 4.06 seconds, and the finite element model needs about 6.5 hours. This shows that the model proposed in this paper is correct and effective.

[0154] In order to verify the effectiveness of the analytical model under different misalignment conditions, the model verification is carried out under the conditions of misalignment amount e = 0.005 mm and e = 0.01 mm (θ = 0°) as follows.

[0155] As shown in Figures 13a-13b and Figures 14a-14bIt can be seen that the results of the misalignment spline analysis model are in good agreement with the finite element model, and it is worth noting that in the first stage, the model of the application and the finite element model have some differences, the main reason is that the ANSYS software will consider the factors such as the gap of contact and the friction coefficient, while the model of the application does not consider the influence of these factors, and because the first stage is greatly affected by the contact, so only the stiffness difference of the first stage is large.

Claims

1. An improved method for considering the lateral and angular stiffness of a misaligned spline coupling, characterized by: (1) establishing a schematic diagram of the load and deformation of the misaligned spline coupling, and deriving a balance equation of the load and deformation of the misaligned spline in the meshing state according to the geometric relationship of the deformation of the misaligned spline; (2) deriving the single-tooth meshing stiffness according to the energy method, wherein the tooth base stiffness is considered, and the tooth meshing stiffness is corrected by removing the repeated calculation of the energy of the tooth and the base; In step (2), the step of calculating the single-tooth meshing stiffness includes: A. Single tooth engagement stiffness k g The expression is: B. The tooth base stiffness of the external spline is: C. The transition curve CD of the rack-type cutter machining the gear considering the tooth tip chamfer is the equidistant curve of the long-width involute, and the expression is: D. The bending, shear and axial compression stiffness of the spline single tooth is obtained by two expressions of deformation energy in elasticity and material mechanics, wherein the corrected external spline stiffness expression is: Part of the stiffness is integrated by angular displacement, and the corrected external spline stiffness is simplified as: E. For the internal spline, the geometric shape of all teeth is involute, and the base circle is smaller than the tooth tip circle, and the bending, shear and axial stiffness of the internal spline is calculated by the potential energy method, wherein the internal spline stiffness expression is: (3) According to the single-tooth meshing stiffness of the spline obtained in step (2), the meshing force balance equation obtained in step (1) is brought in to calculate the lateral stiffness and angular stiffness of the misaligned spline coupling.

2. The improved method of considering lateral and angular stiffness of a splined shaft coupling for parallel misalignment as recited in claim 1 wherein: In step (1), the process of deriving the balance equation of the load and deformation of the misaligned spline in the meshing state is: a. When the spline exists misalignment, assuming that the misalignment amounts in the x direction and the y direction are x0 and y0 respectively, the radial misalignment amount e is: Let the angle between the radial misalignment amount e and the x axis be θ, then: b.In the Cartesian coordinate system oxy, assume that the spline has a total of N teeth, and the number of the tooth directly above the x-axis is 1, and the numbers of the teeth in clockwise order are 2, 3, 4, … i, the angle between the center line of the ith tooth and the y-axis is as follows: At this time, the equivalent meshing distance L of each tooth of the spline i is: The engagement stiffness k of each single tooth of the misaligned spline coupling gi is expressed as: c. According to the slicing method to obtain the relationship between the external load and the deformation of the misaligned spline coupling, first divide the spline coupling into independent slices, and the thickness is dz, then determine the single slice force from the meshing tooth shape, and then obtain the overall load of the misaligned spline by integration, when the deformation of the external spline shaft occurs, the offset amount δ(z) of the slice is: δ(z) = (z - z a )θ x ; d. Assume the pitch circle radius of the initial meshing position when the splines are not misaligned is r m ; when misalignment e is not present, when the outer spline slice has moved by displacement δ under the action of the external load, the pitch circle radius r mi is as follows: When there is misalignment e, and the slice has displacement δ, the engagement circle radius r of each tooth mi As follows: e. The displacement δ and misalignment e caused by the load to the spline both cause a normal penetration of each tooth, the amount of normal penetration Δ caused by the displacement δ of the ith tooth is b1 i is: Δ b1i = δ cos α i wherein is the angle between the normal to the i-th tooth engagement and the y-direction, β i is the angle between the tooth center line and the engagement line; Normal penetration amount Δ due to misalignment e b2 i is: Assuming the torsional displacement of the external spline shaft under the action of lateral force and torque is then the normal penetration amount Δni of the i th tooth engagement point due to the torque displacement is t i is: Total normal penetration expression Δ i is: Δ i = Δ b1i + Δ b2i + Δ ti ; f.For the ith pair of teeth, the normal force F generated by the meshing teeth i The expression is as follows: dF i = H(Δ i )k gi Δ i dz where k gi is the single tooth engagement stiffness per unit length of the ith tooth pair in the case of spline misalignment, H(A i ) is the unit step function defined in the equation, g. By combining the forces of all tooth slices and integrating along the axial direction, the relationship between the tooth deformation and the external load is obtained as follows: In the formula, n, N, x j and Δx j respectively represent the number of slices into which the spline is divided along the axial direction, the total number of teeth, the axial coordinate of the slice j, and the length of the slice j.

3. The improved method of considering lateral and angular stiffness of a spline coupling for misalignment in parallel as claimed in claim 1 wherein: In step (3), the lateral stiffness and the angular stiffness represent the ability to resist lateral and angular deformation, and their calculation methods are as follows: where δ y is the deflection of the spline end face coupling center, F y is the force in the corresponding direction, θ y is the corresponding shear force in the respective direction, θ z is the rotation angle of the spline end face coupling center, M x is the bending moment in the respective direction due to θ x in the respective direction; When the spline produces misalignment, the stiffness in the x direction and the y direction is no longer the same, and under the condition of ignoring the cross stiffness, the relationship between the stiffness in the two directions is as follows: k x (θ) = k y (θ + 90°) k θx (θ) = k θy (θ + 90°).

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