A method for obtaining transmission error of error tooth contact analysis

Abstract

The application discloses a kind of error tooth surface contact analysis transmission error acquisition methods, this method includes the following steps, (1) establish error tooth surface mathematical model: establish gear pair servo coordinate system, generate mathematical model containing error tooth surface, automatically divide tooth surface initial grid, introduce contact coordinate system, obtain the gear equation of master and driven wheel under contact coordinate system;(2) solve gear pair transmission error: using the way of local iterative tooth surface search, determine the initial meshing position of gear pair, solve the transmission error of gear pair.The gear pair transmission error acquisition method of the application does not need to solve complex tooth surface engagement equation, can effectively solve error tooth surface singular point problem, and can determine error tooth surface initial meshing point by tooth surface multi-scale iterative search algorithm, and accurately solve transmission error, and can be applied to any type of gear and any scale error, with strong robustness, stability and universality.

Description

Technical Field

[0001] The present invention belongs to the technical field of gear transmission, and in particular relates to a method for obtaining transmission error through error tooth surface contact analysis. Technical Background

[0002] Gears are commonly used transmission devices in machinery, widely used in applications such as automotive transmissions, aircraft landing gear, and marine engines. In these applications, controlling the efficiency, vibration, noise, and reliability of gear products is crucial. Static transmission error is a key indicator in contact analysis and a primary driver of vibration and noise in gear systems. Tooth surface deviation in high-speed gears directly affects gear transmission error.

[0003] Currently, the classic and most widely used method for calculating gear transmission errors is the tooth geometric contact method proposed by Professor Litvin. Its core is based on gear geometry and rigid body dynamics. Based on the continuous resection condition between the tooth surfaces, the meshing equations are established and solved to obtain the instantaneous contact position and no-load transmission error. This method has since been widely used to analyze the contact characteristics of various types of gears. However, due to the highly nonlinear nature of the meshing equations, they are very difficult to solve. Furthermore, the error tooth surfaces contain multiple singular points, which makes geometric contact analysis fraught with convergence issues.

[0004] To address the difficulty of solving meshing equations, Professor Tang Jinyuan's team proposed a contact analysis method based on slicing theory. This method slices the gear into a series of thin slices and proposes a contact judgment strategy to determine whether the slices at both ends of the contact curve are in contact. This method is suitable for ideal tooth surfaces where the meshing equations are difficult to solve. However, when applied to error tooth surfaces, the number of slices must be significantly increased to ensure calculation accuracy, resulting in low computational efficiency. Summary of the Invention

[0005] In order to avoid the shortcomings of the prior art, the present invention proposes a method for obtaining transmission errors through error tooth surface contact analysis to solve the problem that the current traditional contact analysis method cannot be applied to error tooth surfaces.

[0006] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0007] A method for obtaining transmission error by error tooth surface contact analysis, the method comprising the following steps:

[0008] Step A: Establish the follow-up coordinate system S g and S p , are rigidly connected to the driving wheel and the driven wheel respectively; the coordinate system S g (O g -x g ,y g ,z g) is the driving wheel coordinate system, z g The axis coincides with the driving wheel axis, and the coordinate system S p (O p -x p ,y p ,z p ) is the driven wheel coordinate system, z p The shaft coincides with the driven wheel axis.

[0009] Step B: Based on the tooth profile / tooth guide deviation data obtained by the gear measuring machine, the deviation data is fitted using the Fourier series to obtain the normal deviation surface w(u,l), which is superimposed on the ideal tooth surface to obtain the mathematical model of the error tooth surface of the driving and driven gears. The error tooth surface of the driving and driven gears can be expressed in the corresponding follower coordinate system as follows:

[0010]

[0011] Where R g w and R p w are the position vectors of the error tooth surfaces of the driving wheel and the driven wheel in the corresponding follower coordinate system, R1 and R2 are the position vectors of the ideal tooth surfaces of the driving wheel and the driven wheel, w1 and w2 are the normal deviation surfaces of the driving wheel and the driven wheel, and u and l are the tooth surface parameters.

[0012] Step C: Initial meshing of tooth surface. Initial meshing of the full error tooth surface is performed based on the tooth surface parameters u and l to preliminarily search for the contact area on the entire tooth surface. To improve the calculation efficiency, the number of tooth surface meshes can be small. The nodes of the master and slave gear tooth surfaces after initial meshing are R in their corresponding follower coordinate systems. g w (u gi ,l gi ) and R g w (u gj ,l gj ).

[0013] Step D: Assemble the mathematical models of the master and slave wheels in the same coordinate system and introduce the contact coordinate system S f (O f -x f ,y f ,z f ), all tooth surface point coordinates and unit normal vectors are expressed in it, and the origin of the coordinate system is the same as the driving wheel following coordinate system S g Origin coincides with O g , and z f Axis and follower coordinate system S g z g The axes coincide, and the driving wheel revolves around zg The axis rotates counterclockwise, with an angle of Ψ g , the initial moment and the coordinate system S g With S f Coincident. All installation errors are attributed to the pinion, and the moving coordinate system S p Origin O p With S f Origin O f The distance is E′, x p Axis and x f The axis intersection angle is θ px ,y p Axis and y f The axis intersection angle is θ py , the driven wheel revolves around z p The axis rotates clockwise, with an angle of Ψ p .

[0014] The driving wheel follows the coordinate system S1 to the contact coordinate system S f The transformation matrix is ​​as follows:

[0015]

[0016] The driven wheel follows the coordinate system S2 to the contact coordinate system S f The transformation matrix is ​​as follows:

[0017]

[0018] Any node of the driving wheel in the contact coordinate system S f The tooth surface expression is as follows:

[0019]

[0020] Any node of the driven wheel is in the contact coordinate system S f The tooth surface expression is as follows:

[0021]

[0022] Step E: Determine the initial meshing position and use a local iterative tooth surface search method. The basic procedure of the specific iterative algorithm is as follows:

[0023] Step 1: Set the number of iterations to h, set the contact accuracy to κ = 10^-10; maintain the driven wheel angle Ψ p If the driving wheel is rotated counterclockwise, the angle of the driving wheel will be g is the following formula:

[0024] ψ g =m gp ψ p +Δψ g

[0025] where m gp is the ideal transmission ratio of the gear pair, ΔΨ g is the additional rotation angle of the master and driven gear surface, which needs to be determined iteratively, the iteration number is s, and the maximum value of the additional rotation angle at the initial time is set as ΔΨ gmax (s＝1) , the minimum value is ΔΨ gmin (s＝1) , and the master gear at any time is ΔΨ gk (s) which can be expressed as the following formula:

[0026]

[0027] Step 2: set the current k (s) = 0;

[0028] Step 3: search the full tooth surface through the initial mesh of the master and driven gears. Let the iteration number h = 0, and for any node M pj in the master gear, calculate the nearest point M gi on the tooth surface of the master gear, and the normal distance function of the tooth surface mesh point is as follows:

[0029]

[0030] where λ ij is the normal distance function of the surface mesh node of the master and driven gears, is the normal vector of the surface node i of the driven gear.

[0031] Step 4: find the minimum normal distance λ min (h＝0) based on the initial mesh in the full tooth surface search. gmin (h＝0) and M pmin (h＝0) are the master contact point pairs with the nearest distance obtained by the initial mesh full tooth surface search.

[0032] Step 5: generate a local network around the master contact point based on the error tooth surface parameters u and l, and the value range of the tooth surface parameters is as follows:

[0033]

[0034] where u g (h+1) , l g (h+1) , u p (h+1) , l p (h+1) are the value ranges of the tooth surface parameters of the master and driven gears in the h+1 iteration; ugmin-1 (h) ,l gmin-1 (h) ,u pmin-1 (h) ,l pmin-1 (h) u is the tooth surface parameter value of the node before the main contact point found in the hth iteration; gmin+1 (h) ,l gmin+1 (h) ,u pmin+1 (h) ,l pmin+1 (h) The tooth surface parameter value of the node after the main contact point found in the hth iteration;

[0035] According to step D, the local tooth surface nodes divided above are represented in the contact coordinate system S f Down.

[0036] Step 6: Find the minimum normal distance of the local tooth surface mesh in the h+1th iteration. The calculation method refers to step 2.

[0037] Step 7: Determine whether the master and slave tooth surfaces are in contact. If the minimum normal distance λ min (h+1) >0, it means the tooth surfaces of the driving and driven wheels are separated, so the driving wheel angle is increased, and k (s) =k (s) +1; if the minimum normal distance λ min (h+1) <0, indicating interference between the master and slave gear tooth surfaces, let k end =k (s) , execute Step 8.

[0038] Step 8: Determine whether the master and slave tooth surfaces are in contact without interference. If the minimum normal distance |λ is satisfied min |<κ, end the iteration, and record the driving wheel angle Ψ at that moment g ti , driven wheel angle Ψ p ti ; If the minimum normal distance |λ min |>κ, the condition is not met and the driving wheel rotation angle range needs to be further narrowed. Let s=s+1 and update the driving wheel rotation angle range to the following formula:

[0039]

[0040] Repeat Step 2-Step 7 until the minimum normal distance |λ is satisfied. min |<κ, that is, the master and slave tooth surfaces are just in contact without interference, and the iteration ends.

[0041] Step F: Solve the error tooth surface transmission error by the following formula:

[0042] TE=ψ g -m gp ψ p

[0043] By executing steps E to F at different times, the temporal variation pattern of the transmission error of the error-containing gear pair can be obtained.

[0044] The beneficial effects of the present invention are as follows: The gear pair transmission error acquisition method of the present invention does not require solving complex tooth surface meshing equations, can effectively solve the error tooth surface singularity problem, and can determine the initial meshing point of the error tooth surface through the tooth surface multi-scale iterative search algorithm, and accurately solve the transmission error. It can be applied to any type of gear and any scale of error, and has strong robustness, stability and versatility. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is the mathematical model diagram of error tooth surface;

[0046] Figure 2 is the initial mesh map of the tooth surface;

[0047] Figure 3 To establish the contact coordinate system diagram;

[0048] Figure 4 Schematic diagram of tooth surface distance calculation method;

[0049] Figure 5 is the local iterative grid map;

[0050] Figure 6 This is an example diagram of the transmission error change law. DETAILED DESCRIPTION

[0051] The present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments.

[0052] A method for obtaining transmission error by error tooth surface contact analysis, the method comprising the following steps:

[0053] Table 1 shows the basic design parameters of the error helical gear pair, and the mathematical expressions of the master and slave tooth surfaces are generated based on these design parameters;

[0054] Table 1 Basic design parameters of helical gear pairs

[0055]

[0056] Step A: Establish the follow-up coordinate system S g and S p , which are rigidly connected to the driving wheel and the driven wheel respectively; coordinate system Sg (O g -x g ,y g ,z g ) is the driving wheel coordinate system, z g The axis coincides with the driving wheel axis, and the coordinate system S p (O p -x p ,y p ,z p ) is the driven wheel coordinate system, z p The shaft coincides with the driven wheel axis.

[0057] Step B: Based on the deviation data of tooth profile / tooth direction obtained by the gear measuring machine (such as P40), the deviation data is fitted according to the Fourier series to obtain the normal deviation surface w(u,l), which is superimposed on the ideal tooth surface. Here, the ideal tooth surface is input according to the corrugation deviation in Table 1 to obtain the error tooth surface mathematical model of the driving wheel and the driven wheel, as shown in the figure. Figure 1 As shown. The error tooth surface of the driving wheel and the driven wheel can be expressed as follows in the corresponding follower coordinate system:

[0058]

[0059] Where R g w and R p w are the position vectors of the error tooth surfaces of the driving wheel and the driven wheel in the corresponding follower coordinate system, R1 and R2 are the position vectors of the ideal tooth surfaces of the driving wheel and the driven wheel, w1 and w2 are the normal deviation surfaces of the driving wheel and the driven wheel, and u and l are the tooth surface parameters.

[0060] Step C: Initial meshing of tooth surface. Based on the tooth surface parameters u and l, the initial meshing of the full error tooth surface is performed to preliminarily search the contact area of ​​the entire tooth surface. To improve the calculation efficiency, the number of tooth surface meshes can be small, such as Figure 2 As shown. After the initial mesh division, the nodes of the master and slave gear tooth surfaces are R in their corresponding follower coordinate systems. g w (u gi ,l gi ) and R g w (u gj ,l gj ).

[0061] Step D: Assemble the mathematical models of the master and slave wheels in the same coordinate system and introduce the contact coordinate system S f (O f -x f ,y f ,z f ),like Figure 3 All tooth surface point coordinates and unit normal vectors are shown in it, and the origin of the coordinate system and the driving wheel follower coordinate system S g Origin coincides with O g , and z f Axis and follower coordinate system S g z g The axes coincide, and the driving wheel revolves around z g The axis rotates counterclockwise, with an angle of Ψ g , the initial moment and the coordinate system S g With S f Overlap. Set the installation error θ px ,θ py , ΔE is merged into the pinion, x p Axis and x f The axis intersection angle is θ px ,y p Axis and y f The axis intersection angle is θ py , the driven wheel revolves around z p The axis rotates clockwise, with an angle of Ψ p . Moving coordinate system S p Origin O p With S f Origin O f The distance between the driving and driven wheels is E', which is called the center distance between the driving and driven wheels. The calculation method is as follows:

[0062]

[0063] Where ΔE is the center distance error.

[0064] Driving wheel following coordinate system S g To contact coordinate system S f The transformation matrix is ​​as follows:

[0065]

[0066] The driven wheel following coordinate system S p To contact coordinate system S f The transformation matrix is ​​as follows:

[0067]

[0068] Any node of the driving wheel in the contact coordinate system S f The tooth surface expression is as follows:

[0069]

[0070] Any node of the driven wheel is in the contact coordinate system S f The tooth surface expression is as follows:

[0071]

[0072] Step E: Determine the initial meshing position and use a local iterative tooth surface search method. The basic procedure of the specific iterative algorithm is as follows:

[0073] Step 1: Set the number of iterations to h, set the contact accuracy to κ = 10^-10; maintain the driven wheel angle Ψ p If the driving wheel is rotated counterclockwise, the angle of the driving wheel will be g is the following formula:

[0074] ψ g =m gp ψ p +Δψ g

[0075] Where m gp is the ideal transmission ratio of the gear pair, ΔΨ g In order to make the master and slave tooth surfaces contact the additional rotation angle, it is necessary to iterate and determine it continuously. The number of iterations is s, and the maximum value of the additional rotation angle at the initial moment is set to ΔΨ gmax (s＝1) , minimum value ΔΨ gmin (s＝1) is 0, and the driving wheel is ΔΨ at any time gk (s) It can be expressed as the following formula:

[0076]

[0077] Step 2: Set the current k (s) =0;

[0078] Step 3: Search the entire tooth surface through the initial mesh of the driving and driven wheels. Let the number of iterations h = 0, for any node M in the driving wheel pj , calculate the nearest point M on the corresponding driving gear tooth surface gi ,like Figure 4 As shown in the figure, the normal distance function of the tooth surface grid point is as follows:

[0079]

[0080] Where λ ij is the normal distance function of the mesh nodes on the surface of the master and driven wheels, n f g is the normal vector from the gear surface to node i.

[0081] Step 4: Find the minimum normal distance λ on the entire tooth surface based on the initial mesh min (h＝0) ,M gmin (h＝0) With M pmin(h＝0) The pair of main contact points with the closest distance obtained by searching the entire tooth surface of the initial mesh.

[0082] Step 5: Generate a local network around the main contact point of the master and driven wheels based on the error tooth surface parameters u and l, such as Figure 5 As shown; the tooth surface parameter value range is shown in the following formula:

[0083]

[0084] Where u g (h+1) ,l g (h+1) ,u p (h+1) ,l p (h+1) is the range of the master and slave gear tooth surface parameters in the h+1th iteration; u gmin-1 (h) ,l gmin-1 (h) ,u pmin-1 (h) ,l pmin-1 (h) u is the tooth surface parameter value of the node before the main contact point found in the hth iteration; gmin+1 (h) ,l gmin+1 (h) ,u pmin+1 (h) ,l pmin+1 (h) The tooth surface parameters of the node after the main contact point found in the hth iteration are taken.

[0085] According to step D, the local tooth surface nodes divided above are represented in the contact coordinate system S f Down.

[0086] Step 6: Find the minimum normal distance of the local tooth surface mesh in the h+1th iteration. The calculation method refers to step 2.

[0087] Step 7: Determine whether the master and slave tooth surfaces are in contact. If the minimum normal distance λ min (h+1) >0, it means the tooth surfaces of the driving and driven wheels are separated, so the driving wheel angle is increased, and k (s) =k (s) +1; if the minimum normal distance λ min (h+1) <0, indicating interference between the master and slave gear tooth surfaces, let k end =k (s) , execute Step 8.

[0088] Step 8: Determine whether the master and slave tooth surfaces are in contact without interference. If the minimum normal distance |λ is satisfied min |<κ, end the iteration, and record the driving wheel angle Ψ at that moment g ti , driven wheel angle Ψ p ti ; If the minimum normal distance |λ min |>κ, the condition is not met and the driving wheel rotation angle range needs to be further narrowed. Let s=s+1 and update the driving wheel rotation angle range to the following formula:

[0089]

[0090] Repeat Step 2-Step 7 until the minimum normal distance |λ is satisfied min |<κ, that is, the master and slave tooth surfaces are just in contact without interference, and the iteration ends.

[0091] Step F: Solve the error tooth surface transmission error by the following formula:

[0092] TE=ψ g -m gp ψ p

[0093] By executing steps E to F at different times, the time variation law of the transmission error of the error-containing gear pair in this example can be obtained, as shown in FIG. Figure 6 shown.

[0094] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A method for obtaining transmission error by error tooth surface contact analysis, characterized in that: The method comprises the following steps: (1) Establishing a mathematical model of the error tooth surface Establish a follower coordinate system for the gear pair, generate a mathematical model of the tooth surface with errors, automatically divide the initial mesh of the tooth surface, introduce a contact coordinate system, and obtain the tooth surface equations of the master and driven gears in the contact coordinate system; (2) Solving the transmission error of the gear pair The local iterative tooth surface search method is used to determine the initial meshing position of the gear pair and solve the transmission error of the gear pair; To determine the initial meshing position, a local iterative tooth surface search method is used. The basic procedure of the specific iterative algorithm is as follows: Step 1: Let the number of iterations be h and the contact accuracy be κ = 10 -10 ; Maintain the driven wheel angle Ψ p If the driving wheel is rotated counterclockwise, the angle of the driving wheel will be g is the following formula: ψ g =m gp ψ p +Dψ g Where m gp is the ideal transmission ratio of the gear pair, ΔΨ g In order to make the additional rotation angle of the driving wheel when the master and driven tooth surfaces are in contact, it is necessary to iterate and determine it continuously. The number of iterations is s, and the maximum value of the additional rotation angle of the driving wheel at the initial moment is set to ΔΨ gmax (s＝1) , minimum value ΔΨ gmin (s＝1) is 0, and the additional turning angle of the driving wheel at any time is ΔΨ gk (s) It can be expressed as the following formula: Step 2: Set the current k (s) =0; Step 3: Search the entire tooth surface through the initial mesh of the driving and driven wheels, and set the number of iterations h = 0. For any node M in the driving wheel, pj , calculate the nearest point M on the corresponding driving gear tooth surface gi , the normal distance function of the tooth surface grid point is as follows: Where λ ij is the normal distance function of the mesh nodes on the master and slave surfaces, is the normal vector from node i on the gear surface; Step 4: Find the minimum normal distance λ on the entire tooth surface based on the initial mesh min (h＝0) ,M gmin (h＝0) With M pmin (h＝0) The main contact point pair with the closest distance obtained by searching the entire tooth surface of the initial mesh; Step 5: Generate a local network around the main contact point of the master and slave wheels based on the error tooth surface parameters u and l. The tooth surface parameter value range is shown in the following formula: Where u g (h+1) ,l g (h+1) ,u p (h+1) ,l p (h+1) is the range of the master and slave gear tooth surface parameters in the h+1th iteration; u gmin-1 (h) ,l gmin-1 (h) ,u pmin-1 (h) ,l pmin-1 (h) u is the tooth surface parameter value of the node before the main contact point found in the hth iteration; gmin+1 (h) ,l gmin+1 (h) ,u pmin+1 (h) ,l pmin+1 (h) The tooth surface parameter value of the node after the main contact point found in the hth iteration; Step 6: Represent the local tooth surface nodes divided above in the contact coordinate system S f Next; find the minimum normal distance of the local tooth surface mesh in the h+1th iteration. The calculation method refers to step 2; Step 7: Determine whether the master and slave tooth surfaces are in contact. If the minimum normal distance λ min (h+1) >0, it means the tooth surfaces of the driving and driven wheels are separated, so the driving wheel angle is increased, and k (s) =k (s) +1; if the minimum normal distance λ min (h+1) <0, indicating interference between the master and slave gear tooth surfaces, let k end =k (s) , execute Step 8; Step 8: Determine whether the master and slave tooth surfaces are in contact without interference. If the minimum normal distance |λ is satisfied min |<κ, end the iteration, and record the driving wheel angle Ψ at that moment g ti , driven wheel angle Ψ p ti ; If the minimum normal distance |λ min |>κ, the condition is not met and the driving wheel rotation angle range needs to be further narrowed. Let s=s+1 and update the driving wheel rotation angle range to the following formula: Repeat Step 2-Step 7 until the minimum normal distance |λ is satisfied. min |<κ, that is, the master and slave tooth surfaces are just in contact without interference, and the iteration ends.

2. The method for obtaining transmission error by error tooth surface contact analysis according to claim 1, characterized in that: Establish the mathematical model of the error tooth surface. The specific steps are as follows: Establish the follow-up coordinate system S g and S p , are rigidly connected to the driving wheel and the driven wheel respectively; the coordinate system S g (O g -x g ,y g ,z g ) is the driving wheel coordinate system, z g The axis coincides with the driving wheel axis, and the coordinate system S p (O p -x p ,y p ,z p ) is the driven wheel coordinate system, z p The shaft coincides with the driven wheel axis; Based on the deviation data of the tooth profile / tooth guide obtained by the gear measuring machine; the deviation data is fitted according to the Fourier series to obtain the normal deviation surface w(u,l), which is superimposed on the ideal tooth surface to obtain the mathematical model of the error tooth surface of the driving wheel and the driven wheel; the error tooth surface of the driving wheel and the driven wheel can be expressed as the following formula in the corresponding follower coordinate system: Where R g w and R p w are the position vectors of the error tooth surfaces of the driving wheel and the driven wheel in the corresponding follower coordinate system, R g and R p are the position vectors of the ideal tooth surfaces of the driving wheel and the driven wheel, w g and w p is the normal deviation surface of the driving wheel and the driven wheel, u and l are the tooth surface parameters.

3. The method for obtaining transmission error by error tooth surface contact analysis according to claim 2, characterized in that: The full error tooth surface is automatically meshed initially based on the tooth surface parameters u and l to preliminarily search for the contact area on the entire tooth surface. The number of tooth surface meshes is relatively small.

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