Face gear pair time-varying meshing stiffness iterative calculation method

Abstract

The invention discloses an iterative calculation method for time-varying meshing stiffness of a face gear pair. The iterative calculation method is used for accurately calculating a load distribution coefficient and the time-varying meshing stiffness in the meshing process of the face gear pair. The face gear pair is sliced in the radial direction and the axial direction through a slicing method, the slices are calculated and coupled through a potential energy method, the load distribution coefficient is subjected to iterative correction through the time-varying meshing rigidity obtained through calculation in the iterative calculation process, and therefore accurate solving of the time-varying meshing rigidity of the face gear pair is achieved. According to the method, the calculation efficiency and reliability are improved, theoretical support is provided for face gear pair configuration design optimization, and the international related technical blank is filled.

Description

Technical Field

[0001] The present invention relates to the technical field of gear transmission, and in particular to an iterative calculation method for time-varying meshing stiffness of a face gear pair. Background Art

[0002] Face gears have broad application prospects in aerospace transmissions due to their unique advantages, including low axial installation sensitivity, lack of axial forces, and lightweight design. Time-varying mesh stiffness, as the inherent excitation of a gear pair, is an essential parameter for studying the dynamic characteristics of face gear pairs. However, due to the complex development mechanism of face gears, the radial tooth profile changes with tooth diameter, making accurate analytical calculations difficult. Furthermore, traditional finite element methods suffer from inefficiency and computational complexity.

[0003] In order to solve the above problems, the present invention proposes an iterative calculation method for the time-varying meshing stiffness of face gear pairs. This method slices the face gear radially and axially through the slicing method, and uses the potential energy method to calculate and iteratively correct the slices, thereby achieving accurate solution of the time-varying meshing stiffness of the face gear pair, improving the calculation efficiency and reliability, providing theoretical support for the optimization of the configuration design of the face gear pair, filling the gap in related international technologies, promoting the development of engineering technology, and at the same time having great social and economic benefits. Summary of the Invention

[0004] In order to overcome the shortcomings of the existing technology and fill the gaps in the relevant technology, the present invention provides an iterative calculation method for the time-varying meshing stiffness of a face gear pair. The technical solution adopted by the present invention to solve the technical problem is as follows:

[0005] A method for iteratively calculating the time-varying meshing stiffness of a face gear pair, characterized by comprising the following steps:

[0006] Step (1): Set the face gear pair gear shaping parameters, construct the face gear shaping motion coordinate system, and derive the gear shaping position vector r according to the relative position relationship between the face gear and the gear shaping cutter during the face gear shaping motion. f With the normal vector n f ; Get the tooth surface equation r f Perform discrete solution to obtain discrete point cloud data of the face gear tooth surface;

[0007] Step (2): According to the relative position relationship of the face gear pair when meshing, construct the face gear contact analysis coordinate system and obtain the contact coordinate system S after coordinate transformation. a The tooth surface equation of the face gear in r2 (a) and normal vector n2 (a) ;

[0008] Step (3): Calculate the face gear tooth surface Σ f The first fundamental homogeneous ΣfⅠ Homogeneous with the second fundamental Σ fⅡ , solve the principal curvature k of the face gear surface ft1 and k ft2 , the principal curvature k of the cylindrical gear surface pt1 and k pt2 , calculate the normal curvature k of face gears and cylindrical gears according to Euler's formula fn and k pn ;

[0009] k in =k it1 cosβ i1 +k it2 sinβ i2 (i=p,f);

[0010] Among them, β i1 and β i2 (i is p, f) represents the main direction e of the gear pair (cylindrical gear, face gear) it1 and e it2 The angle with a point on the surface;

[0011] Step (4): Calculate the contact deformation δ of any point on the contact ellipse of the mid-face gear of the mating gear pair t :

[0012]

[0013] Among them, E p and E f Respectively represent the elastic modulus of cylindrical gear and face gear; ν p and ν f Represent the Poisson's ratio of cylindrical gear and face gear respectively; K(e t ) and E(e t ) are the complete elliptic integrals of the first and second kinds; e t is the eccentricity of the contact ellipse; a t and b t They represent the length of the major and minor semi-axis of the contact ellipse respectively; ζ and η represent the coordinates of the contact ellipse of the face gear tooth surface; F c is the meshing force on the gear, which is the input torque T of the gear pair in Decision; A t and B t is a second-order parabolic function;

[0014] According to the gear pair's contact ratio ε, the input torque in different meshing zones during the process from meshing to meshing is calculated. In the first calculation, the load distribution coefficient c1 is assumed to be 0.5. According to the input torque, the contact ellipse parameters and the load distribution function F(e) of the face gear pair are calculated. t ):

[0015]

[0016] Step (5): Slice the discrete point cloud of the face gear along the tooth width direction of the face gear. The slice thickness is recorded as Δd. Slice the slices contained in the face gear contact ellipse radially. Slice and calculate from the top to the bottom of the face gear pair tooth profile in sequence. Calculate the tooth meshing stiffness k caused by gear bending respectively. m :

[0017]

[0018] Among them, F m represents the load distribution function F(e t ) the total force exerted on the slice; b s The critical point of the load distribution function is in the contact coordinate system X a OZ a The projected length on the surface; x f (pc) is the load F(e t ) in X a OZ a X of the projected boundary point on the surface a Direction coordinate value; y f (ps) For slices in Y a Coordinate value of direction; x a and y a For X a OZ a About X a and Y a Coordinate value of direction; x f (pfc) and z f (pfc) The contact point is in the contact coordinate system X a OZ a About X a and Z a Coordinate value of α m Any point P in the face gear contact ellipse c Load angle; δ a is the deformation of the gear teeth caused by bending; δ b is the deformation of the gear teeth caused by shear; δ c is the deformation of the gear teeth caused by compression;

[0019] Calculate the matrix coupling stiffness k generated by the gear matrix coupling b :

[0020]

[0021] Among them, L f The contact point P c Distance from tooth bottom; H f is the slicing tooth width;

[0022] Calculate the Hertzian contact stiffness k caused by gear contact c :

[0023]

[0024] The tooth meshing stiffness k m , matrix coupling stiffness k b and Hertz contact stiffness k c By parallel connection, the total meshing stiffness k of the i-th slice in the axial direction and the j-th slice in the radial direction can be obtained. t (ij) :

[0025]

[0026] Determine whether the calculation of all m axial slices is completed. If the axial slice calculation is not completed, calculate the total time-varying meshing stiffness of the next axial slice. If all axial slice calculations are completed, store the data. Determine whether the calculation of all n radial slices is completed. If all radial slice calculations are not completed, calculate the total time-varying meshing stiffness of the next radial slice. If all radial slice calculations are completed, add all the slice couplings to calculate the total meshing stiffness k of the face gear in the face gear pair when a single tooth is engaged and engaged. ft :

[0027]

[0028] Step (6): Modify the load distribution coefficient c1 by the calculated time-varying meshing stiffness. If the modified load distribution coefficient c1 * The convergence condition value Δ=|(c1 * -c1) / c1|<0.001, calculate the total meshing stiffness k of the face gear pair ft Total meshing stiffness k with cylindrical gear pt The meshing stiffness k of the gear pair obtained in parallel st :

[0029]

[0030] The calculation results are calculated as the final load distribution coefficient c1 of the face gear pair and the time-varying meshing stiffness k st If the modified load distribution coefficient does not meet the convergence condition, return to the calculation and calculate the time-varying meshing stiffness k of tooth pair 1. st1 Time-varying mesh stiffness k with tooth pair 2 st2A secondary calculation of the load distribution coefficient can be obtained to obtain the corrected load distribution coefficient c1 * :

[0031]

[0032] The corrected load distribution coefficient c1 * The initial load distribution coefficient c1 is calculated, and the gear is sliced ​​again until the calculation converges to obtain the final structural load distribution coefficient c1 and time-varying meshing stiffness k of the face gear pair. st , output the calculation results and complete the iterative calculation of the time-varying meshing stiffness of the face gear pair.

[0033] Compared with the existing technology, the beneficial effects of the present invention are as follows: the present invention proposes an iterative calculation method for the time-varying meshing stiffness of a face gear pair. Compared with the finite element method that requires a certain technical foundation, the method proposed by the present invention only needs to set the face gear parameters to run, is not limited by the technical level of the operator, and greatly improves the calculation efficiency while maintaining a high calculation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 This is a flow chart of an implementation of an iterative calculation method for time-varying meshing stiffness of a face gear pair proposed by the present invention;

[0035] Figure 2 It is a schematic diagram of the coordinate transformation of the gear shaping process from the profile gear to the face gear of the present invention;

[0036] Figure 3 2. It is a schematic diagram of coordinate transformation of the contact coordinate system of the face gear pair of the present invention;

[0037] Figure 4 Schematic diagram of the elliptical deformation of the tooth surface contact of the face gear pair of the present invention;

[0038] Figure 5 This is a schematic diagram of a tooth section of a face gear of the present invention;

[0039] Figure 6 This is a schematic diagram of the force analysis of the tooth profile slice of the face gear of the present invention;

[0040] Figure 7 It is the iterative calculation result of the time-varying meshing stiffness and load distribution curve of the present invention;

[0041] Figure 8 It is the time-varying meshing stiffness and load distribution curve under multiple cycles of the present invention. DETAILED DESCRIPTION

[0042] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1 — Figure 8 The specific embodiments of the present invention are described in detail.

[0043] Reference Figure 1 , an iterative calculation method for time-varying meshing stiffness of a face gear pair, comprising the following steps:

[0044] Step (1): Set the face gear pair shaping parameters, and the face gear production motion coordinate system is as follows: Figure 2 As shown in the figure, the tooth surface equation of the face gear is calculated based on the coordinate change theory. The face gear shaping motion is the gear shaping motion, and the position vector r of a point on the tooth surface of the gear is s and normal vector n s It can be expressed as:

[0045]

[0046] Among them, r bs Indicates the base circle radius of the gear; θ s Represents the gear tooth profile parameters; θ 0s represents the involute parameter; u s Indicates the tooth parameters of the gear. Figure 2 As shown in the generated coordinate system, the generated gear and the moving coordinate system S s The connection rotates around the axis by an angle Face gear and moving coordinate system S f The connection rotates around the axis by an angle angle With Angle The ratio is the transmission ratio i fs Inversely proportional, transition coordinate system S m 、S n 、S o Connected to the frame. s To the face gear coordinate system S f The coordinate transformation matrix M fs It can be expressed as:

[0047] M fs =M fo M on M nm M ms (3);

[0048]

[0049] Among them, D sf Indicates the distance between the gear shaping cutter coordinate system and the face gear coordinate system.

[0050] The tooth surface equation and position vector of the face gear are obtained by coordinate transformation from the gear equation. fs Indicates M fsRemove the 3×3 matrix of homogeneous coordinates, the tooth surface equation r f With position vector n f It can be expressed as:

[0051]

[0052] In the generating motion, the gear meshing is also subject to the meshing equation, which can be expressed as:

[0053]

[0054] According to the tooth surface equation and meshing equation, the tooth surface of the opposite gear can be f The discrete point cloud data of the face gear tooth surface is obtained by performing discrete solution in the direction.

[0055] Step (2): Figure 3 The contact coordinate system of the face gear pair shown in the figure, in order to meet the meshing requirements of the face gear pair point contact, the cylindrical gear in the contact coordinate system needs to have 1-3 fewer teeth than the shape-forming gear in the shape-forming motion. In the contact coordinate system, the face gear is fixedly connected to the moving coordinate system S2 and rotated around the axis by an angle The cylindrical gear is fixedly connected to the moving coordinate system S1 and rotates around the axis by an angle Transition coordinate system S a 、S b 、S c 、S d Fixedly connected to the frame. Contact analysis coordinate system S a Mid-plane gear tooth surface equation r2 (a) and normal vector n2 (a) It can be expressed as:

[0056]

[0057] M a2 =M ab M bc M cd M d2 (12);

[0058]

[0059] Among them, Δe, ΔB, and Δq are the installation errors in three directions; L a2 M a2 Remove the 3x3 matrix of homogeneous coordinates.

[0060] From the pinion coordinate system S1 to the contact analysis coordinate system S a The tooth surface equation of the face gear after coordinate transformation is r1 a) and normal vector n1 (a) It can be expressed as:

[0061]

[0062] Among them, θ p Represents the tooth profile parameters of cylindrical gears; u p Indicates the tooth parameters of cylindrical gears;

[0063] The meshing of cylindrical gears and face gears in the contact coordinate system can be constrained by the tooth surface equation and normal vector, which can be expressed as:

[0064]

[0065] Step (3): Calculate the contact ellipse of the face gear pair and the face gear tooth surface Σ f The first fundamental homogeneous Σ fⅠ Homogeneous with the second fundamental Σ fⅡ Expressed as:

[0066]

[0067] E, F, G and L, M, N are the first basic homogeneous and second basic homogeneous operators respectively, which can be used to calculate the principal curvature k of the face gear surface. ft1 and k ft2 , which can be expressed as:

[0068]

[0069] The principal curvature k of the cylindrical gear surface pt1 and k pt2 , which can be expressed as:

[0070]

[0071] Among them, r bp represents the radius of the cylinder base circle;

[0072] Normal curvature k of face gears and cylindrical gears fn and k pn , which can be expressed as:

[0073] k in =k it1 cosβ i1 +k it2 sinβ i2 (i=p,f) (26);

[0074] Among them, β i1 and β i2 (i is p, f) represents the main direction e of the gear pair it1 and e it2 The angle with a point on the surface.

[0075] Step (4): Figure 4The figure shows the contact ellipse of the face gear pair. According to Hertz contact theory, the contact deformation of any point on the contact ellipse of the face gear in the paired gear pair can be calculated. t Expressed as:

[0076]

[0077] Among them, E p and E f Respectively represent the elastic modulus of cylindrical gear and face gear; ν p and ν f Represent the Poisson's ratio of cylindrical gear and face gear respectively; K(e t ) and E(e t ) are the complete elliptic integrals of the first and second kinds; e t is the eccentricity of the contact ellipse, which is determined by the length of the major and minor axes of the contact ellipse; ζ and η represent the coordinates of the contact ellipse of the face gear tooth surface; F c is the meshing force on the gear, which is determined by the input torque of the gear pair; A t and B t is a second-order paraboloid function, which can be expressed as:

[0078]

[0079] The meshing force on the gear and the input torque T in It is related to the number of tooth pairs involved in meshing, and the number of tooth pairs involved in meshing is determined by the overlap ε of the gear pair. Taking the overlap between 1 and 2 as an example, when a single tooth is meshing, it goes through the process from meshing in to meshing out. The input torque T in the meshing area of ​​a single tooth is s and the input torque T in the double-tooth meshing area d It can be expressed as:

[0080]

[0081] Where c1 represents the load distribution coefficient of the face gear pair, which needs to be iteratively corrected. The load distribution coefficient is assumed to be 0.5 during the first calculation.

[0082] The major and minor semi-axis a of the contact ellipse on the face gear pair t and b t It can be expressed as:

[0083]

[0084] The load distribution function F(e) at any point on the contact ellipse t ) can be expressed as:

[0085]

[0086] Step (5): Figure 5 As shown, the discrete point cloud of the face gear is sliced ​​along the tooth width direction of the face gear, and the slice thickness is recorded as Δd. The slices contained in the face gear contact ellipse are radially sliced, and the slice calculations are performed from the top to the bottom of the face gear pair tooth profile. Figure 6 The figure shows the force analysis of the slice when meshing, and the meshing stiffness k of the tooth body caused by the bending of the gear. m It can be expressed as:

[0087]

[0088] Among them, F m represents the load distribution function F(e t ) the total force exerted on the slice; b s The critical point of the load distribution function is in the contact coordinate system X a OZ a The projected length on the surface; x f (pc) is the load F(e t ) in X a OZ a X of the projected boundary point on the surface a Direction coordinate value; y f (ps) For slices in Y a Coordinate value of direction; x a and y a For X a OZ a About X a and Y a Coordinate value of direction; x f (pfc) and z f (pfc) The contact point is in the contact coordinate system X a OZ a About X a and Z a Coordinate value of α m Any point P in the face gear contact ellipse c Load angle; δ a is the deformation of the gear teeth caused by bending; δ b is the deformation of the gear teeth caused by shear; δ c is the deformation of the gear teeth caused by compression; h si h is the thickness of the slice tooth profile; a is the distance from the slice to the tooth top; I i is the average moment of inertia of the cross section; L sb L is the length of the lower base of the tooth profile section; st It is the upper base length of the tooth profile section.

[0089] Matrix coupling stiffness k generated by gear matrix coupling b It can be expressed as:

[0090]

[0091] Among them, L f The contact point P c Distance from tooth bottom; H f is the width of the slicing gear.

[0092] Hertzian contact stiffness k caused by gear contact c It can be expressed as:

[0093]

[0094] The total time-varying mesh stiffness k at the i-th slice on the tooth profile and the j-th slice on the tooth direction t (ij) It can be expressed as:

[0095]

[0096] The total time-varying meshing stiffness k of the face gear during the process of single tooth engagement and engagement st It can be expressed as:

[0097]

[0098] Similarly, the total time-varying mesh stiffness k of cylindrical gears is calculated as pt , the meshing stiffness of the cylindrical gear and the face gear is connected in parallel, and the total time-varying meshing stiffness k of the gear pair during the process of single tooth engagement and engagement is st It can be expressed as:

[0099]

[0100] Step (6): The load distribution coefficient is corrected by the calculated time-varying meshing stiffness. The corrected load distribution coefficient c1 * It can be expressed as:

[0101]

[0102] The convergence condition value will affect the convergence accuracy. When the convergence condition value is set to 0.001, it can be expressed as:

[0103]

[0104] The calculation results are calculated as the final load distribution coefficient c1 of the face gear pair and the time-varying meshing stiffness k st If the modified load distribution coefficient does not meet the convergence condition, return to the calculation and calculate the time-varying meshing stiffness k of tooth pair 1. st1 Time-varying mesh stiffness k with tooth pair 2st2 Perform secondary calculation on the load distribution coefficient. * The initial load distribution coefficient c1 is calculated, and the gear is sliced ​​again until the calculation converges to obtain the final structural load distribution coefficient c1 and time-varying meshing stiffness k of the face gear pair. st , output the calculation results and complete the iterative calculation of the time-varying meshing stiffness of the face gear pair.

[0105] The iterative calculation results of the load distribution coefficient are as follows: Figure 7 As shown in the figure, the horizontal axis represents the meshing cycle of a single tooth from meshing in to meshing out, and the vertical axis represents the load distribution coefficient. The iterative calculation results of the time-varying meshing stiffness are shown in Figure 8 The graph shown in Figure 1 represents the meshing cycle of a single tooth from engagement to disengagement. The vertical axis represents the time-varying mesh stiffness. As the number of iterations increases, the curve gradually converges inward, and the calculation accuracy gradually improves. Once the iterations converge, the results can be considered the final output.

[0106] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.

Claims

1. A method for iterative calculation of time-varying meshing stiffness of a face gear pair, characterized by: Step (1): Set the gear shaping processing parameters of the face gear pair, construct the face gear shaping motion coordinate system, and derive the gear shaping processing face gear position vector r according to the relative position relationship between the face gear and the gear shaping cutter during the face gear shaping motion. f With the normal vector n f ; Step (2): According to the relative position relationship of the face gear pair when meshing, the face gear contact analysis coordinate system is constructed to obtain the contact coordinate system S after coordinate transformation a The tooth surface equation of the face gear in r2 (a) and the normal vector n2 (a) ; Step (3): Calculate the principal curvature k of the face gear surface ft1 and k ft2 , the principal curvature k of the cylindrical gear surface pt1 and k pt2 , calculate the normal curvature k of face gears and cylindrical gears according to Euler's formula fn and k pn ; Step (4): Calculate the contact deformation δ of any point on the contact ellipse of the mid-gear of the mating gear pair t : Among them, E p and E f Respectively represent the elastic modulus of cylindrical gear and face gear; ν p and ν f Respectively represent the Poisson's ratio of cylindrical gear and face gear; K(e t ) and E(e t ) is the complete elliptic integral of the first kind and the complete elliptic integral of the second kind; e t is the eccentricity of the contact ellipse; a t and b t They represent the length of the major and minor semi-axis of the contact ellipse respectively; and η represent the coordinates of the face gear tooth contact ellipse; F c is the meshing force on the gear, which is the input torque T of the gear pair in Determined; A t and B t is a second-order parabolic function; According to the gear pair overlap ε, the load distribution coefficient c1 is assumed to be 0.5 in the first calculation, and the contact ellipse parameters and load distribution function F(e) of the face gear pair are calculated according to the input torque. t ); Step (5): Slice the discrete point cloud of the face gear along the tooth width direction of the face gear, and record the slice thickness as Δd. Perform radial slicing on the slices contained in the face gear contact ellipse, and perform slice calculations from the top to the bottom of the face gear pair tooth profile in turn, and calculate the tooth body meshing stiffness k caused by gear bending. m : Among them, F m represents the load distribution function F(e t ) the total force exerted on the slice; b s is the critical point of the load distribution function in the contact coordinate system X a OZ a The length of the projection formed on the surface; is the load F(e t ) in X a OZ a X of the projected boundary point on the surface a Direction coordinate value; For slices in Y a Coordinate value of direction; x a and a For X a OZ a About X a and Y a The coordinate value of the direction; and is the contact point in the contact coordinate system X a OZ a About X a and Z a The coordinate value of m Any point P in the contact ellipse of the face gear c The load angle; δ a is the deformation of the gear teeth caused by bending; b is the deformation of the gear teeth caused by shear; c is the deformation of the gear teeth caused by compression; Calculate the matrix coupling stiffness k generated by the gear matrix coupling b : Among them, L f The contact point P c Distance from tooth bottom; H f is the slicing tooth width; Calculate the Hertzian contact stiffness k caused by gear contact c : The tooth meshing stiffness k m , matrix coupling stiffness k b and Hertz contact stiffness k c By connecting in parallel, the total meshing stiffness k of the i-th slice in the axial direction and the j-th slice in the radial direction can be obtained. t (ij) ; Determine whether the calculation of all m axial slices is completed. If the calculation of the axial slice is not completed, the total time-varying meshing stiffness of the next axial slice is calculated. If the calculation of all axial slices is completed, the data is stored; Determine whether the calculation of all n radial slices is completed. If the calculation of all radial slices is not completed, the total time-varying meshing stiffness of the next radial slice is calculated. If the calculation of all radial slices is completed, all the slice couplings are added to calculate the total meshing stiffness k of the face gear in the face gear pair when a single tooth is meshed in and out. ft : Step (6): The load distribution coefficient c1 is corrected by the calculated time-varying meshing stiffness. If the corrected load distribution coefficient c1 * The convergence condition is satisfied. * -c1) / c1|<0.001, calculate the total meshing stiffness k of the face gear pair ft Total meshing stiffness k with cylindrical gear pt The meshing stiffness k of the gear pair obtained by parallel connection st ; The calculation result is taken as the final load distribution coefficient c1 of the face gear pair and the time-varying meshing stiffness k st ; If the modified load distribution coefficient does not meet the convergence condition, return to the calculation and use the time-varying meshing stiffness k of tooth pair 1 st1 Time-varying mesh stiffness k with tooth pair 2 st2 A secondary calculation of the load distribution coefficient can obtain the corrected load distribution coefficient c1 * : The corrected load distribution coefficient c1 * The initial load distribution coefficient c1 is taken as the initial load distribution coefficient c1, and the gear is sliced ​​and calculated again until the calculation converges. The final structural load distribution coefficient c1 and the time-varying meshing stiffness k of the face gear pair are obtained. st , output the calculation results, and complete the iterative calculation of the time-varying meshing stiffness of the face gear pair.

Images