A new algorithm for solving contact path of non-orthogonal non-symmetrical helical gear

Abstract

The application discloses a new algorithm for solving contact trace of a non-orthogonal non-symmetrical helical gear, and the specific steps are as follows: (1) establishing a non-symmetrical helical gear and a non-orthogonal non-symmetrical helical gear meshing transmission coordinate system containing installation error; (2) deducing a contact trajectory equation group considering the installation error; (3) combining the fourth formula and the fifth formula in the equation group to obtain the components of a unit normal vector of a contact point of the face gear along the x-axis and the y-axis; (4) combining the first formula, the second formula in the equation group and the coordinate axis components at the contact point to obtain the relationship between the normal vector components and the coordinate components at the contact point of the face gear tooth surface; and (5) combining the meshing equations of the tooth surfaces to obtain the tooth surface contact trace of the non-orthogonal non-symmetrical helical face gear. The method can be used for contact characteristic analysis and evaluation, can reduce the nonlinear dimension of the contact trace equation group, saves the calculation time, improves the solving speed and has high solving precision.

Description

Technical Field

[0001] This invention relates to the field of gear systems in aerospace engineering, and in particular to a novel algorithm for solving the contact trace of non-orthogonal and asymmetric helical gears. Background Technology

[0002] Face gear transmission is a novel type of gear transmission, and it has been successfully applied in helicopter main drives abroad. Conventional helical tooth contact analysis (TCA) involves finding points on two conjugate tooth surfaces that have the same instantaneous position coordinates and normal vectors. This method requires solving high-dimensional nonlinear equations, making the solution process extremely complex and demanding in terms of initial value selection. For novel non-orthogonal and asymmetric helical tooth gear transmissions, the method for solving the contact trace is even more challenging, and no reports have been published to date.

[0003] To address the aforementioned issues, this invention proposes a novel algorithm for solving the contact traces of non-orthogonal asymmetric helical gears. This method can solve the contact traces of non-orthogonal asymmetric helical gears, enabling the analysis and evaluation of contact characteristics of novel high-performance non-orthogonal asymmetric helical gears. Furthermore, it can transform a TCA equation system containing 6 equations and 7 unknowns into a simplified TCA equation system containing 3 equations and 4 unknowns, significantly reducing the nonlinear complexity of the TCA equation system, saving computation time, and achieving higher accuracy. This fills a gap in related international technologies, promotes the development of engineering technology in the field of high-speed, heavy-load, and precision transmission, and can generate significant social and economic benefits. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill the gaps in related technologies, this invention provides a novel algorithm for solving the contact traces of non-orthogonal and asymmetric helical gears. The technical solution adopted by this invention to solve its technical problem is as follows: A novel algorithm for solving the contact traces of non-orthogonal and asymmetric helical gears, characterized in that:

[0005] Step (1): The expressions for the position vector and unit normal vector of the helical gear are:

[0006] ,

[0007] ;

[0008] Step (2): In the same coordinate system, the two tooth surfaces have the same coordinates and unit normal vector at the meshing point. Transform the position vectors and unit normal vectors of the helical gear and the face gear to the auxiliary coordinate system S. f In the diagram, the point that satisfies the contact condition is denoted as r. f (k) = r f (2) n f (k)= n f (2) Expressed mathematically: In the same coordinate system, the two tooth surfaces have the same coordinates and unit normal vector at the meshing point; an auxiliary coordinate system S is chosen. f For reference, the above conditions can be expressed as:

[0009] ;

[0010] In the formula:

[0011] ,

[0012] ,

[0013] ,

[0014] ;

[0015] Substituting the values, we obtain the contact trace equations and solve the system of equations:

[0016] ;

[0017] Step (3): Solve equations 4 and 5 simultaneously using the contact trace equations to obtain the components n of the unit normal vector at the contact point of the face gear along the x-axis and y-axis. 2x and n 2y It should satisfy:

[0018] ;

[0019] Step (4): Solve the first and second equations of the contact trace equation system and the coordinate axis components at the contact point to obtain the relationship between the normal vector component and the coordinate components at the contact point of the face gear tooth surface; the relationship between the normal vector component and the coordinate components at the contact point of the face gear tooth surface is:

[0020] ,

[0021] ;

[0022] Simplifying, we get:

[0023] ,

[0024] ;

[0025] Step (5): The simplified TCA equation system has 3 equations and 4 unknowns, all of which are surface gear tooth surface parameters. Select the tooth surface... The corner Using the rotation angle as input, the tooth surface parameters at the contact point are obtained by iterative solution. , , Then, substituting into the equation r2(u, l) of the non-orthogonal and asymmetric helical gear tooth surface,... S , From this, the contact point on the tooth surface of the face gear can be obtained; the simplified equation for solving the contact trace is:

[0026] .

[0027] Compared with existing technologies, the beneficial effects of this invention are: transforming the contact trace solution equation set containing 6 equations and 7 unknowns into a simplified contact trace solution equation set containing 3 equations and 4 unknowns, greatly reducing the nonlinear complexity of the contact trace solution equation set and improving the accuracy of stability calculation and determination for complex engineering systems. Attached Figure Description

[0028] Figure 1 Flowchart of the algorithm for contact trace of non-orthogonal and asymmetric helical gear tooth surface;

[0029] Figure 2 Diagram of non-orthogonal, asymmetric helical gear meshing transmission;

[0030] Figure 3 It is the gear hobbing cutter to the auxiliary coordinate system S a The coordinate system transformation matrix;

[0031] Figure 4 It is the coordinate system transformation matrix that introduces bias error;

[0032] Figure 5 It is the coordinate system transformation matrix that introduces the axis intersection angle error;

[0033] Figure 6 It is the auxiliary coordinate system S p The coordinate transformation matrix of the face gear;

[0034] Figure 7 The contact traces of non-orthogonal, asymmetric helical gears;

[0035] Figure 8 The transmission error curve of a non-orthogonal, asymmetric helical gear. Detailed Implementation

[0036] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 8 The specific embodiments of the present invention will be described in detail below.

[0037] Step (1): Select one side of the tooth surface of a non-orthogonal, asymmetric helical gear, i.e., the active side tooth surface, as the meshing object, and analyze the contact trajectory of the gear. According to the point contact theory of helical gears, except for the number of teeth, the other main parameters of the helical gear are the same as those of the gear shaper. Therefore, the expression of the position vector of the helical gear is similar to that of the gear shaper. The expressions for its position vector and unit normal vector are:

[0038] ,

[0039] ;

[0040] Step (2): According to the gear meshing principle, the condition for two tooth surfaces to mesh correctly is that, in the same coordinate system, the two meshing tooth surfaces are tangent to each other at any time and do not interfere with each other. This can be expressed mathematically as follows: In the same coordinate system, the two tooth surfaces have the same coordinates and unit normal vector at the meshing point. Choose an auxiliary coordinate system S. f For reference, the above conditions can be expressed as:

[0041] ;

[0042] In the formula:

[0043] ,

[0044] ,

[0045] ,

[0046] ;

[0047] Substituting, we get:

[0048] ;

[0049] Since the normal vector at the meshing point is a unit vector, only two of the three equations in the above system of equations (4, 5, and 6) are independent. That is, there are a total of 6 independent equations and 7 unknowns in the system of equations.

[0050] Step (3): Solving equations 4 and 5 simultaneously in the TCA equation system, we obtain the components of the unit normal vector at the contact point of the face gear along the x-axis and y-axis directions, which satisfy:

[0051] ;

[0052] Step (4): Solve the first and second equations in the system of equations and the coordinate axis components at the contact point to obtain the relationship between the normal vector component and the coordinate components at the contact point of the gear tooth surface:

[0053] ,

[0054] ;

[0055] Simplifying, we get:

[0056] ,

[0057] ;

[0058] Step (5): Combining the meshing equation with the tooth surface, the simplified TCA equation is obtained as follows:

[0059] ;

[0060] The simplified TCA equation system consists of 3 equations and 4 unknowns, all of which are surface parameters of the gear teeth. The tooth surface is selected as the unknown. The corner Using the rotation angle as input, the tooth surface parameters at the contact point are obtained by iterative solution. , , Then, substituting into the equation r2(u, l) of the non-orthogonal and asymmetric helical gear tooth surface,... S , The contact point of the gear tooth surface can be obtained from the above.

[0061] Step (6): Substituting equations 3 and 4 into the simplified TCA equations, the rotation angle of the asymmetric helical gear is then calculated. Represented as:

[0062] ;

[0063] Figure 2 This is a diagram of a non-orthogonal, asymmetric helical gear meshing transmission. Based on the meshing principle and coordinate transformation, the tooth surface equation is obtained. The tooth surface equation is solved by programming. The generated mathematical model is discretized. Through uniform sampling, the coordinate values ​​of the tooth surface points of the non-orthogonal, asymmetric helical gear are obtained. A point cloud file is generated and saved as a ".txt" document. It is then imported into 3D modeling software to obtain a complete solid model. Figure 3 It is the gear hobbing cutter to the auxiliary coordinate system S a The coordinate system transformation matrix, Figure 4 It is the coordinate system transformation matrix that introduces bias error. Figure 5 It is the coordinate system transformation matrix that introduces the axial intersection error. Figure 6 It is the auxiliary coordinate system S p The coordinate transformation matrix of the face gear; Figure 7To solve the TCA equations using a new algorithm, we can see that the tooth surface contact trace of the non-orthogonal and asymmetric helical gear is an oblique line. As the pinion rotates, the contact point on the tooth surface moves from the tooth tip to the tooth root. Figure 8 The transmission error curve of a non-orthogonal and asymmetric helical gear is parabolic, which helps to absorb the vibration generated when the tooth surfaces are in contact, resulting in a relatively smooth transmission process.

[0064] The above description is merely a preferred embodiment of the invention and does not constitute any limitation on the invention. Any modifications, alterations, or equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the protection scope of the invention.

Claims

1. A novel algorithm for solving the contact trace of non-orthogonal and asymmetric helical gears, characterized in that: Step (1): The expressions for the position vector and unit normal vector of the helical gear are: ， ； Step (2): In the same coordinate system, the two tooth surfaces have the same coordinates and unit normal vector at the meshing point. Transform the position vectors and unit normal vectors of the helical gear and the face gear to the auxiliary coordinate system S. f In the diagram, the point that satisfies the contact condition is denoted as r. f (k) = r f (2) n f (k) = n f (2) Expressed mathematically: In the same coordinate system, the two tooth surfaces have the same coordinates and unit normal vector at the meshing point; an auxiliary coordinate system S is chosen. f For reference, the above conditions can be expressed as: ； In the formula: ， ， ， ； Substituting the values, we obtain the contact trace equations and solve the system of equations: ； Step (3): Solve equations 4 and 5 simultaneously using the contact trace equations to obtain the components n of the unit normal vector at the contact point of the face gear along the x-axis and y-axis. 2x and n 2y It should satisfy: ； Step (4): Solve the first and second equations of the contact trace equation system and the coordinate axis components at the contact point to obtain the relationship between the normal vector component and the coordinate components at the contact point of the face gear tooth surface; the relationship between the normal vector component and the coordinate components at the contact point of the face gear tooth surface is: ， ； Simplifying, we get: ， ； Step (5): The simplified TCA equation system has 3 equations and 4 unknowns, all of which are surface gear tooth surface parameters. Select the tooth surface... The corner Using the rotation angle as input, the tooth surface parameters at the contact point are obtained by iterative solution. , , Then, substituting into the equation r2(u, l) of the non-orthogonal and asymmetric helical gear tooth surface,... S , From this, the contact point on the tooth surface of the face gear can be obtained; the simplified equation for solving the contact trace is: 。

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