Method for obtaining comprehensive meshing stiffness of spiral bevel gear

Abstract

This invention relates to a method for obtaining the comprehensive meshing stiffness of a spiral bevel gear. The method specifically comprises: S1: Based on multi-tooth pair and meshing state analysis, and combining the given gear displacement angle and the rotation angle of the target tooth pair at the theoretical meshing position, obtaining the single tooth pairs that generate contact; S2: Based on the penetration amount between single tooth pairs and the meshing stiffness between single tooth pairs, obtaining the distributed load of all tooth pairs that generate contact; S3: Based on the deformation amount between single tooth pairs and the distributed load, obtaining the comprehensive meshing stiffness under multi-tooth meshing conditions. This invention incorporates the penetration amount between tooth pairs, more closely approximating the actual situation, thus reducing the error compared to reality.

Description

Technical Field

[0001] This invention relates to the field of gear design technology, and specifically to contact analysis of spiral bevel gears. Background Technology

[0002] In actual transmission processes, due to the high overlap ratio of spiral bevel gears, they are usually in a state of alternating single-tooth and multi-tooth meshing. The overall stiffness of the gear in multi-tooth meshing cannot be directly derived from the single-tooth meshing stiffness. Currently, the most common method for calculating the overall stiffness is to consider the influence of the overlap ratio in multi-tooth meshing and obtain it by superimposing the single-tooth stiffness. However, in actual multi-tooth meshing, the load is distributed among the meshing tooth pairs, and the load distribution among each tooth pair changes with the rotation angle. Consequently, the stiffness also changes with the load, resulting in a smaller single-tooth meshing stiffness when considering multi-tooth meshing compared to a purely single-tooth meshing stiffness. Therefore, simply calculating the stiffness by superimposing the theoretical overlap ratio does not satisfy the deformation compatibility principle of multi-tooth meshing, leading to significant deviations in the calculation results. Summary of the Invention

[0003] The purpose of this invention is to provide a method for obtaining the comprehensive meshing stiffness of a spiral bevel gear, so as to solve the problem that the existing technology does not meet the deformation coordination principle of multi-tooth meshing, which leads to large deviations in the calculation results.

[0004] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0005] A method for obtaining the comprehensive meshing stiffness of a spiral bevel gear, the method specifically comprising:

[0006] S1: Based on the analysis of multiple tooth pairs and meshing state, and combined with the given gear displacement angle and the rotation angle of the target tooth pair at the theoretical meshing position, obtain the single tooth pair that produces contact;

[0007] S2: Based on the penetration amount between single tooth pairs and the meshing stiffness between single tooth pairs, obtain the distributed load of all tooth pairs that generate contact;

[0008] S3: Based on the deformation and load distribution between single tooth pairs, obtain the comprehensive meshing stiffness under multi-tooth meshing conditions.

[0009] Furthermore, S1 specifically involves selecting a tooth pair that satisfies the following formula;

[0010] ;

[0011] in, ; This represents the displacement angle of a given gear; Indicates transmission error, where ; This represents the rotation angle of a given target tooth pair at the theoretical meshing position; This represents the angle between the normal vector and the direction of the circumferential tangent vector at the point of engagement. Indicates the radius of the circle where the meshing point is located;

[0012] in, , and The following relationship exists:

[0013] ,

[0014] The number of teeth on the large or small wheel. This represents the rotation angle of the preceding tooth pair for a given target tooth pair. This indicates the rotation angle of the subsequent tooth pair of the given target tooth pair.

[0015] Furthermore, S2 specifically refers to:

[0016] S21: Judgment Is it greater than 0? If it is greater than 0, proceed to S22; otherwise, let... =0, proceed to S24;

[0017] S22: Given the initial load assignment and iteration step size ;

[0018] S23: Based on S22 Obtain the single-tooth meshing stiffness Then judge Does the following expression satisfy the condition? If it does, proceed to S24; otherwise, let... = + Repeat S22;

[0019] ;

[0020] Indicates the single-tooth meshing stiffness. Indicates the penetration depth;

[0021] S24: Output ;

[0022] S25: Determine the output If the following formula is satisfied, proceed to step S3; otherwise, reselect the gear displacement angle and the angle of the target tooth pair at the theoretical meshing position, and repeat step S1 until the output is achieved. The following expression is satisfied.

[0023] ;

[0024] in To select the torque for either the large or small wheel;

[0025] Furthermore, the method for obtaining the comprehensive meshing stiffness under the multi-tooth meshing state is shown in the following formula:

[0026] ;

[0027] in: Indicates the first The amount of deformation of the meshing gear teeth. .

[0028] The beneficial effects of this invention are:

[0029] This invention starts with the analysis of multi-tooth pair meshing. Given the displacement angle of the gear and the rotation angle of the target tooth pair at the theoretical meshing position, it combines the penetration amount of the target tooth pair, the preceding tooth pair, and the following tooth pair to obtain the distributed load generated by gear meshing. Based on the distributed load and stiffness between different tooth pairs, the comprehensive meshing stiffness is obtained. Compared with the calculation of simply superimposed stiffness, this invention combines the penetration amount between tooth pairs, which is closer to the actual situation, thus reducing the error with reality. Attached Figure Description

[0030] Figure 1 This is a flowchart of the present invention;

[0031] Figure 2 Model of a spiral bevel gear meshing pair;

[0032] Figure 3 The transmission error curve and the large wheel rotation angle variation curve are shown.

[0033] Figure 4 The curves show the relationship between the meshing stiffness of a single tooth and the rotation angle of the large wheel.

[0034] Figure 5 To solve the load distribution process for multi-tooth pairs;

[0035] Figure 6 The process for solving the stiffness of the distributed load. Detailed Implementation

[0036] The following description, with reference to the accompanying drawings and preferred embodiments, illustrates the implementation of the technical solution of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be understood that the preferred embodiments are only for illustrating the present invention and not for limiting the scope of protection of the present invention.

[0037] It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Therefore, the drawings only show the components related to the present invention and are not drawn according to the actual number, shape and size of the components in the actual implementation. In the actual implementation, the form, quantity and proportion of each component can be arbitrarily changed, and the layout of the components may also be more complex.

[0038] This embodiment proposes a method for obtaining the comprehensive meshing stiffness of a spiral bevel gear, such as... Figure 1 As shown, it includes the following steps:

[0039] S1: Based on multi-tooth pair and meshing state analysis, combined with the given gear displacement angle. The angle of rotation of the target teeth at the theoretical meshing position , to obtain the single tooth pair that makes contact.

[0040] Spiral bevel gears typically mesh with alternating single and multiple teeth. Considering a single tooth pair as a rigid body, relative penetration can occur during meshing. In an ideal meshing state, there is no meshing clearance; this clearance is actually the transmission error. Gears without transmission error will experience an increase in transmission error during actual transmission due to the gradual increase in load. However, if a reasonable transmission error value is preset during tooth surface design, the amplitude of the transmission error will show a decreasing trend followed by an increasing trend during load analysis due to the influence of tooth deformation. This is beneficial for reducing the amplitude of the transmission error under rated load when designing steady-state loads, resulting in superior transmission performance. Therefore, in practical design, given a transmission error... To ensure smoother transmission of the gear pair, when the first When the teeth are just meshed, it is necessary to overcome transmission errors. This means that the smaller wheel needs to rotate an additional angle in space. It can be obtained through spatial geometric relationships. and The relationship is given by equation (1).

[0041] Equation (1)

[0042] In the formula, The radius of the circle where the meshing point is located; Let be the angle between the normal vector and the circumferential tangent vector at the meshing point, which can be calculated using equation (1). At this point, since the is at the ... right, The transmission error value on the teeth is greater than There are still gaps , If the two pairs of teeth are not engaged, then the two pairs of teeth are not meshing. As the smaller wheel continues to rotate, due to the load on the larger wheel, the larger wheel, which is in the meshing position, will actually bend and deform due to the push of the smaller wheel. At this point, the deformation is converted into the normal penetration amount of the larger and smaller wheels. and Based on the continuous deformation condition, it can be known that when When the gear teeth penetrate, , The gear teeth will also follow This produces a corresponding displacement on the gear teeth. When the generated displacement is greater than... When considering the theoretical transmission error of the gear teeth, the following occurs: , When two teeth mesh simultaneously, the force that should act on the teeth should be applied to the pair of teeth. Total load on Will follow The burden is gradually distributed as the penetration of the tooth pair increases.

[0043] Based on the above theoretical multi-tooth meshing situation, a multi-tooth meshing model of spiral bevel gears is established, such as... Figure 2 As shown, contact pair 2 is rotated to the initial meshing position and used as the target tooth pair for calculating the complete meshing cycle. Simultaneously, a functional relationship is established between the transmission error value, the single-tooth meshing stiffness value, and the rotation angle of the target tooth pair, yielding the transmission error value at any angle during the rotation cycle. With single tooth meshing stiffness value Turn the corner The rotation angles of the preceding and following tooth pairs are obtained by adding or subtracting one rotation cycle respectively. and Equation (2) can be obtained.

[0044] Equation (2)

[0045] In the formula: This refers to the number of teeth on the large gear. , By expressing the transmission error value as a function, the transmission error functions of the front and rear teeth of the target tooth pair can be obtained. , Single-tooth meshing stiffness value function , The center of rotation of the large gear tooth surface is the coordinate system. Coordinate system of the rotation center of the pinion tooth surface .

[0046] Based on the above meshing pair model, the pinion rotation angle is given. Meanwhile, it is assumed that the deformation is small enough and does not affect the change of the conjugate contact trace. At this point, the engagement angles of the three pairs of gear teeth are respectively... , , According to Figure 2 The theoretical model under load. Based on the above meshing conditions, it can be seen that when contact pair 1 rotates... When it occurs on the two curved surfaces The penetration amount, while contact pair 2 just makes contact when it rotates by the same angle, the two tooth surfaces do not penetrate, and contact pair 3 does not make contact under this condition. The judgment condition for this state is Equation (3):

[0047] Equation (3)

[0048] In the formula: Choose 1, 2, or 3. When When this formula holds true, it means that at the rotation angle Below this, the distance the tooth surface moves in the normal direction is greater than or equal to the normal clearance between the two curved surfaces. At this time, the two curved surfaces are in a state of penetration or just in contact but there is no contact force. This can be extrapolated to multiple contact pairs.

[0049] Specifically, first, the displacement angle of the gear teeth is given. Angle of rotation at the theoretical meshing position of the target tooth pair Because when the target tooth pair begins to mesh, its preceding tooth pair has already meshed, the maximum normal load can be calculated from the meshing position of the preceding tooth pair. The maximum penetration of the preceding tooth pair is ,Will The rotation angle, transformed into the coordinate system, provides the initial value for iteration. Subsequent iterations use the previously converged rotation angle as the initial value. The current meshing angle of the front and rear gears is calculated using the phase difference between the gear rotation angles. , The conjugate contact points of the current three tooth pairs are solved based on the conjugate difference surface theory. , , The radius of the conjugate contact point in the small wheel coordinate system space is calculated using equation (4) and Hertzian point contact theory. , , And the angular relationship between the surface normal vector and the circumferential tangent vector. , , Through the transmission error curve ( Figure 3 and Figure 4 The transmission error of the three pairs of gear teeth at their respective meshing angles can be obtained. , , In this embodiment Figure 3 and Figure 4 The relationship represented is existing technology. Finally, the relationship is determined by equation (3). The number of tooth pairs that come into contact.

[0050] Equation (4)

[0051] Tangential direction of the instantaneous meshing circle at the meshing point That is, the direction vector of the tangential force. It is the normal vector of the instantaneous engagement point, and also the direction vector of the normal load at the engagement point position. . for and The angular relationship in space, from which we can obtain... Let the driving torque of the small wheel at this time be... ,

[0052] S2: Based on the penetration amount and meshing stiffness between individual tooth pairs, the distributed load of all tooth pairs that come into contact is obtained. Specifically, as follows... Figure 5 As shown:

[0053] S21: Judgment Is it greater than 0? If it is greater than 0, proceed to S22; otherwise, let... =0, proceed to S24;

[0054] S22: Given the initial load assignment and iteration step size The load acting on each tooth pair decreases with the increase of meshing tooth pairs and the increase of penetration of each tooth pair. Therefore, two load distribution methods are considered under the limit conditions: 1) the gear teeth are in a single meshing state, at which time the tooth pair bears all the load brought by the torque; 2) when the front gear is in contact but no obvious penetration occurs, the critical state is considered, then the load borne by the tooth is considered to be not 0, but very small. Combining the above two limit conditions, the initial load range of the deformation compatibility equation can be defined as follows: .

[0055] S23: Based on S22 Obtain the single-tooth meshing stiffness Obtain the single-tooth meshing stiffness Methods such as Figure 6 As shown, specifically: solving the rotation angle based on the conjugate difference surface equation. The conjugate contact point is used to calculate the curvature characteristics of the point using differential geometry principles. Substitute the Hertzian correction formula to calculate the major axis of the contact ellipse under load. Contact short shaft With maximum contact stress Determine whether the current load contact ellipse is a semi-ellipse.

[0056] Semi-ellipse and full ellipse are defined as follows: taking the large gear as an example, in the tooth tip direction, it should not exceed the tooth tip line of the large gear; in the tooth root direction, it should not exceed the theoretical projection line position of the tooth tip line of the small gear in the meshing state with the large gear; in the tooth length direction, it should not exceed the boundary of the large end or small end. In this paper, elliptical contact states that exceed these constraints are collectively referred to as "semi-ellipse", and complete elliptical contact states are collectively referred to as "full ellipse".

[0057] When the contact length is semi-elliptical, the contact state is corrected using the contact equilibrium condition to obtain the corrected contact length. short axis Contact stress Divide the semiellipse into A two-dimensional tooth profile boundary element model is used to obtain the local contact load. In the above formula If the contact ellipse is a full ellipse, then the maximum contact stress value in the above formula remains the same. Similarly, calculate the local contact load. Local contact load Substituting these values ​​into the boundary element model and the Hertzian line contact model respectively, we can obtain the bending deformation. With contact deformation The meshing stiffness of the plate can be obtained by combining the series model of unit gear meshing and the parallel model of single-tooth meshing stiffness calculation. Stiffness of single tooth meshing .

[0058] Alternatively, the meshing stiffness of a single tooth can be obtained through finite element analysis. .

[0059] Then determine Does it satisfy equation (5)? ​​If it does, proceed to S24; otherwise, let... = + Repeat S22;

[0060] Equation (5)

[0061] Indicates the single-tooth meshing stiffness. Indicates the penetration depth;

[0062] S24: Output ;

[0063] S25: Determine the output If equation (6) is satisfied, proceed to step S3; otherwise, reselect the displacement angle. The angle of rotation of the target teeth at the theoretical meshing position Repeat S1 until the output is complete. Satisfying equation (6):

[0064] Equation (6)

[0065] in To select the torque for either the large or small wheel;

[0066] in, Let be the angle between the surface normal vector and the circumferential tangent vector at the meshing point of the i-th pair of teeth; Let be the radius of the circle where the meshing point of the i-th pair of teeth is located.

[0067] S3: Based on the deformation and load distribution between single tooth pairs, the comprehensive meshing stiffness under multi-tooth meshing state is obtained by equation (7).

[0068] Equation (7)

[0069] in, .

[0070] The above embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention.

Claims

1. A method for obtaining the comprehensive meshing stiffness of a spiral bevel gear, characterized in that: The method is specifically as follows: S1: Based on the analysis of multiple tooth pairs and meshing state, and combined with the given gear displacement angle and the rotation angle of the target tooth pair at the theoretical meshing position, obtain the single tooth pair that produces contact. S2: Based on the penetration amount between single tooth pairs and the meshing stiffness between single tooth pairs, obtain the distributed load of all tooth pairs that generate contact; S3: Based on the deformation and load distribution between single tooth pairs, obtain the comprehensive meshing stiffness under multi-tooth meshing conditions; Specifically, S1 involves selecting a tooth pair that satisfies the following formula; in, ; This represents the displacement angle of a given gear; Indicates transmission error, where ; This represents the rotation angle of a given target tooth pair at the theoretical meshing position; This represents the angle between the normal vector and the direction of the circumferential tangent vector at the point of engagement. Indicates the radius of the circle where the meshing point is located; in, , and The following relationship exists: The number of teeth on the large or small wheel. This represents the rotation angle of the preceding tooth pair for a given target tooth pair. This indicates the rotation angle of the subsequent tooth pair of the given target tooth pair; Specifically, S2 is: S21: Judgment Is it greater than 0? If it is greater than 0, proceed to S22; otherwise, let... =0, proceed to S24; S22: Given the initial load assignment and iteration step size ; S23: Based on S22 Obtain the single-tooth meshing stiffness Then judge Does the following expression satisfy the condition? If it does, proceed to S24; otherwise, let... = + Repeat S22; Indicates the single-tooth meshing stiffness. Indicates the penetration depth; S24: Output ; S25: Determine the output If the following formula is satisfied, proceed to step S3; otherwise, reselect the gear displacement angle and the angle of the target tooth pair at the theoretical meshing position, and repeat step S1 until the output is achieved. Satisfy the following formula: ； in To select the torque for either the large or small wheel; in, Let be the angle between the surface normal vector and the circumferential tangent vector at the meshing point of the i-th pair of teeth; Let be the radius of the circle where the meshing point of the i-th pair of teeth is located.

2. The method for obtaining the comprehensive meshing stiffness of a spiral bevel gear according to claim 1, characterized in that: The method for obtaining the overall meshing stiffness under the multi-tooth meshing state is shown in the following formula: ； in: Indicates the first The amount of deformation of the meshing gear teeth. .

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