A Tooth Surface Modification Design Method for Improving the Load-Bearing Capacity of Aero Spiral Bevel Gears

Abstract

The present invention relates to a tooth surface modification design method for improving the load-carrying capacity of an aviation face gear drive. The core lies in changing the relative motion between the pinion and the gear shaper cutter based on a pre-designed high-order transmission error, realizing the high-order tooth surface modification of the pinion along the line of action. At the same time, a second-order parabolic modification is performed on the tooth profile of the gear shaper cutter to achieve the purpose of two-way modification of the pinion. Compared with conventional modification, under two working conditions of having installation errors and without installation errors, the face gear meshing pair with high-order modification designed by the present invention shows better meshing performance. This modification method provides ideas and methods for the tooth surface modification design of non-orthogonal helical face gears.

Description

Technical Field

[0001] The present invention belongs to the technical field of gear modification design, and particularly relates to a tooth surface modification design method for improving the load-bearing capacity of an aviation face gear drive. Background Art

[0002] In practical applications, assembly, manufacturing errors, etc. are inevitable, and it is necessary to perform tooth surface modification design on gears. Currently, the mainstream modification methods on the market mainly include: tooth profile modification, tooth direction modification, three-dimensional modification, etc., removing materials at the tooth tip or tooth root of the tooth to achieve the purpose of vibration reduction, noise reduction, and increased load-bearing. With the development of science and technology, the requirements of modern industry for gear drives are getting higher and higher, and the disadvantages of commonly used tooth profile modification and tooth direction modification methods on the market are gradually emerging. For example: the contact stress on the tooth surface is too large after modification, which exacerbates the wear of the tooth surface and reduces the working life and transmission efficiency of the gear. Therefore, developing a new tooth surface modification design method is the need for the development of modern industry. The high-order modification design method proposed in this patent retains the advantages of conventional modification methods, and at the same time, can reduce the adverse effects brought by modification to the gear. It is a new technology with broad application prospects and provides a new idea for the tooth surface modification design of non-orthogonal helical face gears. Summary of the Invention

[0003] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a tooth surface modification design method for improving the load-bearing capacity of an aviation face gear drive. After performing second-order parabolic modification on the tool, the relative motion relationship between the tool and the pinion is changed according to the pre-designed high-order transmission error, so as to achieve the purpose of high-order two-way modification of the tooth surface of the pinion.

[0004] To achieve the above purpose, the technical solution of the present invention is: a tooth surface modification design method for improving the load-bearing capacity of an aviation face gear drive. First, according to the relative motion relationship between the tool and the face gear during the face gear processing, the tooth surface equation of the non-orthogonal helical face gear is deduced; secondly, second-order parabolic modification is performed on the rack tooth profile, and the relative motion relationship between the tool and the pinion is changed according to the pre-designed high-order transmission error; finally, the meshing equation of the gear pair is established according to the meshing relationship between the cylindrical helical gear and the non-orthogonal helical face gear.

[0005] The specific implementation of this method includes the following steps:

[0006] Step 1: Deduction of the tooth surface equation of the non-orthogonal helical face gear

[0007] According to the machining motion relationship between the tool and the face gear during the face gear processing, a non-orthogonal helical face gear machining coordinate system is established. Figure 2 Among them, S2 and S s are rigidly connected to the tool and the face gear respectively, and the axes Z2 and Z sThe axes Z2 and Z s The angle γ m is the transmission axis angle. Based on the tool tooth surface equation and the machining motion relationship, a tool surface group that changes with the tool rotation angle is formed, and then the surface gear tooth surface ∑2 is obtained by combining the surface group and the meshing equation.

[0008]

[0009] The relative motion relationship between the machined surface gear and the tool is as follows: Figure 2 As shown, the machining coordinate system is Figure 3 As shown, the tool tooth profile coordinate system is Figure 4 :

[0010] Where: Coordinate transformation matrix M 2a It is from S a Coordinate transformation matrix to S2; R as is the tooth profile equation of the insert.

[0011] Step 2: Equation of the second-order parabola modified tooth profile of helical cylindrical gear

[0012] First, the gear cutter tooth profile is modified by a second-order parabola to obtain the second-order parabola modification vector expression of the gear cutter tooth profile. According to the gear meshing principle, the cylindrical gear tooth surface equation is obtained:

[0013]

[0014] Figure 5 The principle of cylindrical gear processing; Figure 6 This is the effect diagram of tooth profile modification;

[0015] Step 3: High-order transmission error correction design

[0016] Figure 7 The coordinate relationship of the cylindrical pinion is given in the figure, where the pinion is rotated and the rack cutter is translated. The high-order modified tooth surface is indirectly controlled by the pre-designed high-order transmission error.

[0017] like Figure 8 As shown in the figure, the transmission error is pre-designed as a sixth-order parabolic function, and its variation trend is controlled by 5 points. There are 7 unknown parameters in the transmission error function, and the 5 pre-designed points on the transmission error function provide 7 equations. The x-coordinates and y-coordinates of the given 5 points are the rotation angle of the pinion and the corresponding transmission error, respectively. Figure 9 is the second-order transmission error of the face gear pair.

[0018] The high-order transmission error can be expressed as:

[0019]

[0020] Where: and are the rotation angle of the pinion and the corresponding geometric transmission error, respectively, and a0~a6 are high-order transmission error coefficients. Based on the five pre-designed points on the transmission error function, the following set of equations can be obtained:

[0021]

[0022] By combining the above equations, the coefficient of the high-order transmission error curve can be solved:

[0023]

[0024] in

[0025]

[0026] Substituting it into the equation, we can get the calculation equation of high-order transmission error:

[0027]

[0028] Step 4: Establish the meshing equation of the face gear pair

[0029] According to the gear meshing principle, the meshing coordinate system is established, and the tooth surface position vectors of the pinion and face gear are and the normal vector Convert to the same coordinate system S f Then, we get the following equation:

[0030]

[0031]

[0032]

[0033]

[0034] Where: [M f1 ] and [M f2 ] are the coordinate transformation matrices of the cylindrical pinion and the face gear, respectively, [L f2 ]、[L f1 ] respectively correspond to [M f2 ]、[M f1 ] is a sub-matrix of .

[0035] According to the gear meshing principle, the contact points on the tooth surface should satisfy the following conditions: in the same coordinate system, the position vectors of the contact points on the two tooth surfaces are the same, and the normal vectors of the points on the two tooth surfaces are the same. That is, the contact points on the tooth surface should satisfy the following relationship:

[0036]

[0037] Where: is the face gear rotation angle; is the rotation angle of cylindrical pinion.

[0038] Give A series of numerical values ​​are used to solve the nonlinear equations. The solved parameters are substituted into the face gear meshing equation to obtain the meshing trajectory on the face gear tooth surface.

[0039] When the two gear teeth exit meshing, the position vector of the pinion tooth top edge and the tooth surface contact point of the face gear are the same, and the tangential vector of the pinion tooth top edge is perpendicular to the tooth surface normal vector of the face gear. Similarly, when the two gear teeth enter meshing, the tooth top edge of the face gear contacts the tooth surface of the cylindrical gear. The following equations can be obtained by transforming the coordinate system:

[0040]

[0041]

[0042] In the formula, ——the tangential vector of the pinion tooth tip edge;

[0043] ——unit normal vector of the face gear tooth surface;

[0044] ——Tangent vector of the face gear tooth tip edge;

[0045] ——Unit normal vector of the pinion tooth surface.

[0046] Combining this equation with the tooth surface position vector equation in the previous article can solve the position of the edge contact point at the root and top of the face gear.

[0047] Compared with the prior art, the present invention has the following beneficial effects: after the method of the present invention performs second-order parabolic shaping on the tool, the relative motion relationship between the tool and the pinion is changed according to the pre-designed high-order transmission error, thereby achieving the purpose of performing high-order bidirectional shaping of the tooth surface of the pinion. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the process of the present invention.

[0049] Figure 2 Schematic diagram of face gear shaping process.

[0050] Figure 3 Establishment of the coordinate system for face gear machining.

[0051] Figure 4is the rack tooth profile coordinate system.

[0052] Figure 5 is the cylindrical gear machining principle.

[0053] Figure 6 is the tooth profile modification effect diagram.

[0054] Figure 7 is the machining coordinate relationship of the cylindrical pinion.

[0055] Figure 8 is the high-order transmission error of the face gear pair.

[0056] Figure 9 is the second-order transmission error of the face gear pair.

[0057] Figure 10 is the unmodified contact pattern and transmission error when Δγ = 0.

[0058] Figure 11 is the contact pattern and transmission error of the second-order parabola modification when Δγ = 0.

[0059] Figure 12 is the contact pattern and transmission error of the high-order modification when Δγ = 0.

[0060] Figure 13 is the unmodified contact pattern and transmission error when Δγ = 1.5'.

[0061] Figure 14 is the contact pattern and transmission error of the second-order parabola modification when Δγ = 1.5'.

[0062] Figure 15 is the contact pattern and transmission error of the high-order modification when Δγ = 1.5'.

[0063] Figure 16 is the unmodified contact pattern and transmission error when Δγ = -1.5'.

[0064] Figure 17 is the contact pattern and transmission error of the second-order parabola modification when Δγ = -1.5'.

[0065] Figure 18 is the contact pattern and transmission error of the high-order modification when Δγ = -1.5'.

[0066] Figure 19 is the traditional second-order parabola modification; (a) Second-order transmission error function curve; (b) Modification diagram.

[0067] Figure 20 is the high-order modification design; (a) High-order transmission error function curve; (b) Modification diagram.

[0068] Figure 21 Comparison of contact stress and bending stress results under standard installation; (a) Contact stress; (b) Bending stress.

[0069] Figure 22 Comparison of contact stress and bending stress results under installation error of Δγ = 1.5'; (a) Contact stress; (b) Bending stress.

[0070] Figure 23 Comparison of contact stress and bending stress results under installation error of Δγ = -1.5'; (a) Contact stress; (b) Bending stress. Specific implementation mode

[0071] The technical solution of the present invention will be specifically described below with reference to the accompanying drawings.

[0072] As Figure 1 shown, a tooth surface modification design method for improving the load-carrying capacity of an aviation face gear drive according to the present invention first derives the tooth surface equation of a non-orthogonal helical face gear based on the relative motion relationship between the tool and the face gear during the face gear machining process; secondly, performs a second-order parabolic modification on the rack tooth profile and changes the relative motion relationship between the tool and the pinion according to the pre-designed high-order transmission error; finally, establishes the meshing equation of the gear pair according to the meshing relationship between the cylindrical helical gear and the non-orthogonal helical face gear.

[0073] The following are specific implementation examples of the present invention.

[0074] The present invention will be further described below with reference to the accompanying drawings by taking a calculation example of a group of gear modifications.

[0075] The design parameters of the gear are shown in Table 1:

[0076] Table 1 Design parameters of non-orthogonal helical face gear

[0077]

[0078] Step 1. TCA

[0079] In order to reflect the design advantages of the high-order two-way modification in this patent, the contact imprints and transmission errors of the unmodified and second-order parabolic modified ones should be compared respectively. Since installation errors are inevitable in actual installation, among various installation errors, the installation error of the shaft intersection angle has the most significant impact on the meshing performance of the face gear. Therefore, for the high-order two-way modification involved in this application, the contact imprints and transmission errors of the unmodified and second-order parabolic modified ones are compared respectively under three conditions of Δγ = 0, Δγ = 1.5', and Δγ = -1.5', see Figures 10 - 18 .

[0080] Step 2. Modification amount calculation

[0081] Based on the designed high-order transmission error function curve, and considering the tool profile modification and the motion relationship between the tool and the gear to be machined, a high-order two-way modification design of the pinion tooth surface is carried out, and the modification amount is compared with that of the traditional second-order parabolic modification; see Figure 19 、 Figure 20 。

[0082] Step 3: Establishment of parametric mesh model

[0083] According to the tooth surface equations of the pinion and face gear, a three-dimensional mesh is formed through MATLAB programming; the tooth surface contact stress is compared with the calculated result of the theoretical Hertz contact stress, as shown in Table 2.

[0084] Table 2 Comparison of finite element results and theoretical Hertz contact stress

[0085]

[0086] Step 4: Finite element analysis

[0087] The models of the unmodified gear pair, second-order modified gear pair, and high-order modified gear pair are respectively imported into the finite element software ABAQUS for simulation analysis. Among various installation errors, the installation error of the shaft intersection angle has the most significant influence on the meshing performance of the face gear, and the greater the transmission ratio, the greater the influence of the shaft intersection angle installation error. Therefore, in this application, under the conditions of different shaft intersection angle installation errors, the simulation results of the static models of the unmodified gear pair, second-order modified gear pair, and high-order modified gear pair are compared, as shown in Tables 3, 4, and 5, and Figures 21 - 23 。

[0088] Table 3 Comparison of results of the maximum contact stress and bending stress of the face gear without installation error

[0089]

[0090] Table 4 Comparison of results of the maximum contact stress and bending stress of the face gear with installation error Δγ = 1.5′

[0091]

[0092] Table 5 Comparison of results of the maximum contact stress and bending stress of the face gear with installation error Δγ = -1.5′

[0093]

[0094] It can be seen from the above data that:

[0095] (1) The introduction of the new high-order modification of the tooth surface can more effectively reduce the maximum contact stress and maximum bending stress of the tooth surface. Compared with the traditional second-order parabolic modification of the tooth surface, the service life of the face gear will be further extended.

[0096] (2) By comparing the maximum contact stress and maximum bending stress of the tooth surface of the example under the installation error condition, the superiority of the new high-order tooth surface modification method is further verified.

[0097] (3) The core of the new high-order tooth surface modification method is the design of the transmission error. Therefore, it has universality and is not limited to face gear transmission, and can be extended to other types of gear transmission.

[0098] The design advantages of the present invention are reflected in this example.

[0099] The above are the preferred embodiments of the present invention. All changes made according to the technical solution of the present invention, when the functional effects produced do not exceed the scope of the technical solution of the present invention, shall fall within the protection scope of the present invention.

Claims

1. A tooth surface modification design method for improving the load-carrying capacity of an aviation face gear drive, characterized in that The tooth surface equation of the non-orthogonal helical gear and the pinion tooth surface equation are derived; the relative motion between the pinion and the gear shaping cutter is changed based on the pre-designed high-order transmission error, and the pinion is subjected to high-order modification along the meshing line. At the same time, the pinion tooth profile is subjected to second-order parabolic modification, so as to achieve the purpose of high-order bidirectional modification of the pinion tooth surface; finally, the meshing equation of the helical cylindrical pinion and the non-orthogonal helical gear is established; the method comprises the following steps: Step 1, derive the tooth surface equation of non-orthogonal helical gear; Step 2: Equation of the second-order parabola-modified tooth profile of helical cylindrical gear; Step 3: High-order transmission error correction design; Step 4: Establish the meshing equation of the face gear pair; Among them, step 1 is specifically as follows: According to the machining motion relationship between the tool and the face gear during the face gear machining process, the non-orthogonal helical face gear machining coordinate system is established; in the moving coordinate system S2 fixed to the face gear and the moving coordinate system S fixed to the tool s In the middle, the axis Z2 and Z s The axes Z2 and Z s The angle γ m is the transmission axis angle; based on the tool tooth surface equation and the machining motion relationship, a tool surface group that changes with the tool rotation angle is formed, and then the surface gear tooth surface ∑2 is obtained by combining the surface group and the meshing equation; Where: coordinate transformation matrix M 2a is the coordinate transformation matrix from S a to S2; R as is the profile equation of the shaper cutter; a cs is the parabola coefficient, u cs , l g are the tooth surface parameters, u 0s is the distance deviating from the node of the basic profile; ψ s is the rotation angle of the shaper cutter, is the normal vector of the shaper cutter, v g2 is the relative velocity between the face gear and the shaper cutter, R2 is the tooth surface position vector of the face gear, and f2 is the meshing equation; Step 2 is as follows: First, the gear cutter tooth profile is modified by a second-order parabola to obtain the second-order parabola modification vector expression of the gear cutter tooth profile. According to the gear meshing principle, the cylindrical gear tooth surface equation is obtained: Where: a ci is the parabola coefficient, u ci , l d are flank parameters, u 0i is the distance deviating from the base profile node; r1 is the position vector of the flank of the cylindrical gear.

2. A tooth surface modification design method for improving the load-carrying capacity of an aerospace face gear drive according to claim 1, characterized in that The method further includes: Step 3: High-order transmission error correction design The processed pinion performs rotational motion, and the rack inserter performs translational motion; the high-order modified tooth surface is indirectly controlled by the pre-designed high-order transmission error; The transmission error is pre-designed as a sixth-order parabolic function, and its changing trend is controlled by 5 points; there are 7 unknown parameters in the transmission error function, and the 5 pre-designed points on the transmission error function provide 7 equations; the x-coordinates and y-coordinates of the given 5 points are the rotation angle of the pinion and the corresponding transmission error respectively; The high-order transmission error is expressed as: In the formula: and are respectively the rotation angle of the pinion and the corresponding geometric transmission error, and a0 to a6 are high-order transmission error coefficients; XY T is the corresponding numerical matrix; Based on the five pre-designed points on the transmission error function, the following set of equations is obtained: where: ε1 - ε4 are respectively the pre-designed high-order transmission errors, T1 and T2 respectively correspond to the rotation angles of the gear pair when engaging and disengaging, T m is the rotation angle at the intermediate position during the meshing process, λ1 and λ2 are respectively the coefficients corresponding to the transmission error curve, and T is the meshing period; Combine the above equations to solve the high-order transmission error curve coefficient: in X = A -1 B Substituting this into the equation, we can get the calculation equation for the high-order transmission error: Step 4: Establish the meshing equation of the face gear pair According to the gear meshing principle, an meshing coordinate system is established, and the tooth surface position vectors of the pinion and the face gear and the normal vectors are respectively transformed into the same coordinate system S f to obtain the following equations: In the formula: are respectively the expressions in the coordinate system S f ; [M f1 and [M f2 are the coordinate transformation matrices of the cylindrical pinion and the face gear respectively, and [L f2 , [L f1 are the sub-matrices corresponding to [M f2 , [M f1 respectively; According to the gear meshing principle, the contact points on the tooth surface should satisfy the following conditions: in the same coordinate system, the position vectors of the contact points on the two tooth surfaces are the same, and the normal vectors of the points on the two tooth surfaces are the same; that is, the contact points on the tooth surface satisfy the following relationship: In the formula: is the rotation angle of the face gear; is the rotation angle of the cylindrical pinion; Given a series of numerical values, solve the non-linear equations, and substitute the solved parameters into the face gear meshing equation to obtain the meshing trajectory on the tooth surface of the face gear; When the two gear teeth exit meshing, the position vector of the pinion tooth top edge and the tooth surface contact point of the face gear are the same, and the tangential vector of the pinion tooth top edge is perpendicular to the tooth surface normal vector of the face gear. Similarly, when the two gear teeth enter meshing, the tooth top edge of the face gear contacts the tooth surface of the cylindrical gear. The following equations are obtained by the conversion of the coordinate system: In the formula, is the tangential vector of the tip edge of the small gear tooth; is the unit normal vector of the face gear tooth surface; is the tangential vector of the tip edge of the face gear; is the unit normal vector of the pinion tooth surface; This equation is solved together with the tooth position vector equation to find the location of the edge contact point at the root and top of the face gear tooth.

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