Design method of parallel axis bidirectional pure rolling line gear pair

Abstract

This invention proposes a design method for a parallel-axis, bidirectional, purely rolling linear gear pair. First, a meshing coordinate system is established and the coordinate transformation matrix is ​​derived. An extended linear gear meshing equation is constructed to solve for the relative velocity and unit common normal vector at the meshing point. Second, the contact line equation is solved based on the coordinate transformation matrix and the meshing equation, and contact lines are arranged on both sides of the gear tooth to achieve bidirectional transmission. Combining a preset normal tooth profile shape and tooth radius, the equations of the center guide lines corresponding to the four contact lines are calculated. Then, based on the criterion that the relative velocity of the driving and driven gears is zero during meshing, a pure rolling condition equation is established. Finally, based on the contact line equation, the center guide line equation, and the pure rolling condition, combined with the preset normal tooth profile, a scanning method is used to construct a single linear gear tooth and array it to generate a three-dimensional solid model. The linear gear designed in this invention can achieve bidirectional, continuous, and smooth transmission.

Description

Technical Field

[0001] This invention relates to the field of gear design technology, and in particular to a design method for a parallel-axis bidirectional pure rolling line gear pair. Background Technology

[0002] Linear gears are a novel type of gear mechanism based on the theory of spatial curve meshing. Compared to traditional gears, linear gears have significant advantages such as compact structure, small size, light weight, ability to achieve large transmission ratios, minimum number of teeth of 1, and flexible design. They are particularly suitable for applications with strict requirements on space and weight, such as micro-mechanisms, robot joints, and medical devices.

[0003] With the continuous development of spur gear technology, its spatial curve meshing theory has undergone significant evolution. Early spur gear spatial curve meshing equations ensured continuous contact of the conjugate curves during meshing by constraining the dot product of the relative velocities of the driving and driven gears at the contact point with the principal normal vector of the driving contact line to be zero. Subsequently, this meshing equation was extended to intersecting shafts at arbitrary angles, staggered shafts, and variable transmission ratio conditions, establishing a unified theoretical framework applicable to various transmission forms. Based on this, researchers proposed a direct modeling method based on the "center guide line" (tooth profile center trajectory line), as well as modeling strategies using different tooth profile forms such as circular arcs and elliptical arcs. To meet the needs of bidirectional transmission, novel spur gear structures with double-sided contact curves have emerged in existing technologies, achieving backlash-free transmission in both forward and reverse directions through conjugate curve matching. In the field of pure rolling transmission, existing technologies have also been able to achieve zero slip ratio of parallel or intersecting shaft gears under ideal conditions by optimizing design parameters. However, the following shortcomings still exist between existing research on spur gears and practical engineering applications:

[0004] First, in existing studies, the normal direction of the meshing equations is fixed as the principal normal vector direction of the contact point on the contact line. While this constraint simplifies the equation form, it also severely limits the degree of freedom in design. When it is necessary to optimize the transmission performance of gears by adjusting the shape of the contact line, this fixed normal direction cannot meet the flexible and varied structural design requirements.

[0005] Secondly, existing bidirectional linear gear design methods often involve complex geometric parameters and stringent constraints, requiring traditional linear gear modeling to first solve complex tooth surface equations and then construct a solid model through scanning or surface fitting. This process is cumbersome and places extremely high demands on designers, making it highly unfavorable for rapid iteration and application in engineering practice.

[0006] Third, existing research on pure rolling line gears is mostly limited to unidirectional transmission conditions. For bidirectional rolling line gears that require pure rolling meshing in both directions, there is a lack of systematic theoretical support and structural implementation schemes.

[0007] Therefore, it is necessary to further improve and refine the existing technology to overcome these shortcomings, and this invention is made based on this situation. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and provide a design method for a parallel shaft bidirectional pure rolling linear gear pair that enables the constructed linear gear pair to achieve continuous and stable transmission.

[0009] This invention is achieved through the following technical solution:

[0010] To solve the above technical problems, the present invention provides a design method for a parallel-axis bidirectional pure rolling line gear pair, comprising the following steps:

[0011] S1: Establish the meshing coordinate system of the gear pair on the parallel shaft external meshing line, and derive the coordinate transformation matrix between each coordinate system;

[0012] S2: Construct an extended linear gear meshing equation containing angular parameters, and solve for the relative velocity and unit common normal vector of the two gears at the meshing point by analyzing the extended linear gear meshing equation;

[0013] S3: First, based on the coordinate transformation matrix and the extended linear gear meshing equation, solve the contact line equations of the driving and driven gears in the linear gear pair. To achieve bidirectional transmission, a contact line is arranged on each side of the tooth surface of each tooth of the two gears, namely the first driving contact line, the second driving contact line, the first driven contact line, and the second driven contact line. Then, based on the preset normal tooth profile shape and tooth profile radius, and combined with the contact line equations, calculate the center guide line equations corresponding to the four contact lines by offsetting a set distance along a specific vector within the contact line normal plane. These equations are the equations of the first driving center guide line, the second driving center guide line, the first driven center guide line, and the second driven center guide line. The specific vector lies within the normal plane spanned by the principal normal and secondary normal of the corresponding contact line.

[0014] S4: Based on the criterion that the relative speed of the driving and driven gears is zero during meshing, an equation for the pure rolling condition of the spur gear pair is established, so that the relative speed of the driving and driven gears at the meshing point is always zero.

[0015] S5: Based on the contact line equation, the center guide line equation, and the pure rolling condition, and combined with the preset normal tooth profile, a single tooth of the linear gear is constructed using the scanning method, and then an array is used to generate a three-dimensional solid model of the linear gear pair.

[0016] To further address the technical problem this invention aims to solve, the specific process of deriving the coordinate transformation matrix in step S1 of the parallel-axis bidirectional pure rolling line gear pair design method provided by this invention includes: given two fixed Cartesian coordinate systems... , and two rotating Cartesian coordinate systems fixed to the driving wheel and driven wheel respectively. , Based on the angular velocities, rotation angles, and distance between the central axes of the driving and driven wheels, the coordinate system is derived. and Coordinate transformation matrix between :

[0017]

[0018] In the formula, and α represents the angles rotated by the driving wheel and the driven wheel, respectively, and 'a' represents the distance between the central axes of the driving wheel and the driven wheel.

[0019] To further address the technical problem addressed by this invention, the specific expression for the extended line gear meshing equation in step S2 of the parallel shaft bidirectional pure rolling line gear pair design method provided by this invention is as follows:

[0020]

[0021] in, For the contact point in the coordinate system The relative velocity in and Coordinate systems and The unit common normal vector of the two corresponding contact lines at the contact point. and coordinate system The principal and secondary normals of the lower contact line; by introducing angle parameters. Release the direction of the common normal to the principal normal vector of the contact line. With Sub-Hyaksha Within Zhang Cheng's complete normal plane; the angle parameters The value range is set to .

[0022] To further address the technical problem addressed by this invention, the method for solving the contact line equation in step S3 of the parallel shaft bidirectional pure rolling line gear pair design method provided by this invention includes: giving the first active contact line on one side of a single tooth of the driving gear in the coordinate system... The parametric equations in the equations; under forward rotation, the first driven contact line in conjugate meshing is calculated from the first active contact line through coordinate transformation; the first active contact line is rotated by an angle... The second active contact line is obtained on the other side of the same single tooth; for the corresponding second driven contact line under the reverse working condition, it is obtained by coordinate transformation of the second active contact line to obtain the third driven contact line, and then the third driven contact line is rotated around the axis by an angle. The angle obtained, or directly derived from the rotation of the first driven contact line around the axis. The result is obtained; where the third driven contact line is a driven contact line on the adjacent tooth surface, with a rotation angle of... , In the formula The number of teeth on the driving gear. This is the transmission ratio.

[0023] To further address the technical problem addressed by this invention, the parallel-axis bidirectional pure rolling line gear pair design method provided by this invention uses a cylindrical helix with equal pitch in the coordinate system. The parametric equation in the equation is expressed as:

[0024]

[0025] In the formula, Where is the helix radius. The pitch coefficient, Let be the independent variable of a cylindrical helix with equal pitch. The variable value at the start of engagement. This represents the variable value at the end of engagement.

[0026] To further address the technical problem addressed by this invention, the specific mathematical model for solving the center guide line equation in step S3 of the parallel shaft bidirectional pure rolling line gear pair design method provided by this invention is as follows:

[0027]

[0028] In the formula, The center guide line position vector, This is the position vector corresponding to the contact line. This is the offset distance. Take the radius of the arc of the normal tooth profile; Let be the offset vector, and

[0029]

[0030] Where i and j are either 1 or 2; , These are the principal normal and secondary normal of the corresponding contact line, respectively. The angle between the offset direction and the principal normal vector; the corresponding active center guide line is offset equidistantly along the offset vector direction. The corresponding driven center guide line is offset equidistantly in the opposite direction of the offset vector. .

[0031] To further address the technical problem addressed by this invention, in the parallel-axis bidirectional pure rolling line gear pair design method provided by this invention, step S4 involves setting the relative speed of the driving and driven gears... The pure rolling condition equation when the contact line is a cylindrical helix is ​​derived as follows:

[0032]

[0033] In the formula, The center distance, Where is the helix radius. This is the transmission ratio.

[0034] In order to further solve the technical problem to be solved by the present invention, in the parallel shaft bidirectional pure rolling line gear pair design method provided by the present invention, the preset normal tooth profile cross-sectional shape in step S5 is designed as any one of circular arc, elliptical arc, involute or cycloid.

[0035] To further address the technical problem addressed by this invention, in the parallel-axis bidirectional pure rolling line gear pair design method provided by this invention, when the preset normal tooth profile cross-sectional shape is an arc, the specific process of constructing the gear pair using the scanning method in step S5 is as follows:

[0036] Create a normal reference plane through one endpoint of the corresponding contact line, and sketch a fan-shaped tooth profile on the normal plane, so that the center of the arc segment is located at the endpoint of the corresponding center guide line, and the endpoint of the corresponding contact line is located in the middle of the arc segment.

[0037] Using the contact line as the scanning path and the central guide line as the guide line to control the torsion amplitude of the tooth surface, the scanning generates one side of the tooth surface of a single tooth.

[0038] Repeat the above steps to generate the other side of the tooth surface, construct a single line tooth entity, and generate a complete tooth crown after circumferential array.

[0039] To further address the technical problems addressed by this invention, the parallel-axis bidirectional pure rolling linear gear pair design method provided by this invention further includes step S6: performing forward and reverse kinematic simulation on the generated three-dimensional solid model of the linear gear pair; extracting the instantaneous transmission ratio of the driving and driven gears; and determining the continuity and stability of the linear gear pair transmission based on whether the relative standard deviation of the instantaneous transmission ratio is less than a preset threshold.

[0040] Compared with the prior art, the present invention has the following advantages:

[0041] 1. This invention proposes a generalized spatial curve meshing equation that includes angular parameters, breaking the limitation of fixed common normal direction in traditional equations. This theoretical innovation significantly expands the design space, providing essential theoretical degrees of freedom for subsequent active control of transmission errors and optimization of tooth surface contact performance.

[0042] 2. This invention establishes a completely new direct modeling system that eliminates the need to solve extremely complex implicit equations for the tooth surface. By calculating the contact line and center guide line and performing direct scanning in conjunction with the tooth profile, this method bypasses tedious and complex derivation steps, significantly simplifying the design process and opening up new avenues for the rapid design, development, and engineering verification of linear gears.

[0043] 3. This invention not only enables rapid design and 3D modeling of parallel-axis bidirectional pure rolling line gear pairs, but also directly utilizes a kinematic simulation module for dynamic verification. Analysis of the simulated transmission ratio and instantaneous transmission ratio successfully confirms that the gear pair possesses the capability for continuous and stable transmission, thus definitively demonstrating the accuracy and feasibility of the entire design method. Attached Figure Description

[0044] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings, wherein:

[0045] Figure 1 This is a flowchart of the present invention;

[0046] Figure 2 This is a schematic diagram of the parallel shaft external meshing line gear meshing coordinate system of the present invention;

[0047] Figure 3 This is a schematic diagram of the linear gear meshing curve of the present invention;

[0048] Figure 4 This is a schematic diagram of the center guide line for modeling linear gears according to the present invention;

[0049] Figure 5 This is a schematic diagram of the process of modeling the driving wheel using the scanning method in this invention;

[0050] Figure 6 This is a schematic diagram of a three-dimensional model of the parallel-shaft external meshing line gear pair of the present invention;

[0051] Figure 7 This is a schematic diagram of the driving gear speed curve obtained from the motion simulation of the linear gear pair of the present invention;

[0052] Figure 8 This is a schematic diagram of the instantaneous transmission ratio obtained from the motion simulation of the linear gear pair of the present invention. Detailed Implementation

[0053] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0054] like Figures 1 to 8 The diagram illustrates a design method for a parallel-axis, two-way, purely rolling line gear pair, comprising the following steps:

[0055] S1: Establish the meshing coordinate system of the gear pair on the parallel shaft external meshing line, and derive the coordinate transformation matrix between each coordinate system;

[0056] S2: Construct an extended linear gear meshing equation containing angular parameters, and solve for the relative velocity and unit common normal vector of the two gears at the meshing point by analyzing the extended linear gear meshing equation;

[0057] S3: First, based on the coordinate transformation matrix and the extended linear gear meshing equation, solve the contact line equations of the driving and driven gears in the linear gear pair. To achieve bidirectional transmission, a contact line is arranged on each side of the tooth surface of each tooth of the two gears, namely the first driving contact line, the second driving contact line, the first driven contact line, and the second driven contact line. Then, based on the preset normal tooth profile shape and tooth profile radius, and combined with the contact line equations, calculate the center guide line equations corresponding to the four contact lines by offsetting a set distance along a specific vector within the contact line normal plane. These equations are the equations of the first driving center guide line, the second driving center guide line, the first driven center guide line, and the second driven center guide line. The specific vector lies within the normal plane spanned by the principal normal and secondary normal of the corresponding contact line.

[0058] S4: Based on the criterion that the relative speed of the driving and driven gears is zero during meshing, an equation for the pure rolling condition of the spur gear pair is established, so that the relative speed of the driving and driven gears at the meshing point is always zero.

[0059] S5: Based on the contact line equation, the center guide line equation, and the pure rolling condition, and combined with the preset normal tooth profile, a single tooth of the linear gear is constructed using the scanning method, and then an array is used to generate a three-dimensional solid model of the linear gear pair.

[0060] like Figure 1 The diagram shown is a flowchart of the present invention. The details of each step are described below.

[0061] Step S1: Establish the meshing coordinate system and derive the coordinate transformation matrix.

[0062] Parallel shaft external meshing line gear meshing coordinate system as follows Figure 1 As shown. In four coordinate systems in space, two fixed Cartesian coordinate systems are given. , and two rotating Cartesian coordinate systems , The coordinate system Fixed to the driving wheel, coordinate system Fixed to the driven wheel. At the initial engagement position, the coordinate system... and Relative to the coordinate system and Coincidence. The driving wheel moves at an angular velocity Around the coordinate system of The shaft rotates, and the driven wheel moves at an angular velocity Around the coordinate system of The axis rotates, for a certain period of time The angles through which the rotation is performed are respectively and 'a' is the coordinate system. and of The distance between the shafts is the distance between the central axes of the driving gear and the driven gear, and b is the width of the gear. and This represents two meshing conjugate curves. For active contact line, The superscript indicates the coordinate system for the driven contact line. Indicates active contact line and driven contact line The point of engagement at a certain moment. According to the definition of transmission ratio, the transmission ratio... With angular velocity and corner The relationship is shown in equation (1).

[0063] = / = / (1)

[0064] coordinate system and Coordinate transformation matrix between As shown in equation (2):

[0065] (2)

[0066] coordinate system and Coordinate transformation matrix between As shown in equation (3):

[0067] (3)

[0068] coordinate system and Coordinate transformation matrix between As shown in equation (4):

[0069] (4)

[0070] The coordinate system can be obtained from equations (2)-(4). and Transformation matrix between As shown in equation (5).

[0071] (5)

[0072] Step S2: Extend the traditional meshing equation of the linear gear and analyze the extended meshing equation.

[0073] The traditional meshing equation for linear gears is extended to the form shown in equation (6):

[0074] (6)

[0075] In the formula, For the contact point in the coordinate system The relative velocity in and Coordinate systems and The unit common normal vector of the two contact lines at the contact point. can be Obtained by coordinate transformation. and coordinate system The principal and secondary normals of the lower contact line. This is an angle parameter with a value between [0, π / 2]. The angle parameter is introduced... Release the direction of the public normal to the direction of the public normal. and Within Zhang Cheng's complete normal plane, thus providing a basis for subsequent optimization. This provides the necessary theoretical freedom to achieve proactive design of transmission errors and optimization of contact performance.

[0076] Let coordinate system Lower active contact line The parametric equation is shown in equation (7).

[0077] (7)

[0078] Then the unit tangent vector, principal normal vector and secondary normal vector of the active contact line can be obtained, as shown in equation (8).

[0079] (8)

[0080] The coordinate system can be obtained from equations (2), (6) and (8). Lower common normal vector .

[0081] Figure 2 In the middle, let the meshing point In coordinate system The coordinates are , The position vector of the active contact line. Let be the position vector of the driven contact line. ,but

[0082] (9)

[0083] meshing point Speed ​​of the driving wheel and the speed of the driven wheel during its motion Its expression is shown in equation (10).

[0084] (10)

[0085] We can obtain the results from equations (9) and (10). and As shown in equation (11)

[0086] (11)

[0087] Then the relative velocity of the two gears at the meshing point can be obtained. As shown in equation (12).

[0088] (12)

[0089] Combining formulas (2) to (6) and formula (12), we can obtain and Based on the relationship, the equations of the contact line and the center guide line can be derived.

[0090] Step S3: Solve for the equations of the contact lines of the primary and secondary wheels and the corresponding center guide lines based on S1 and S2.

[0091] (1) Solve the contact line equation

[0092] The transmission of a spur gear pair is achieved through the meshing of two contact lines located on the driving and driven spur gears, respectively. To achieve bidirectional transmission in both forward and reverse directions, each individual tooth needs to have a contact line distributed on both sides of its tooth surface. That is, a single tooth of the driving gear has a first driving contact line and a second driving contact line, and a single tooth of the driven gear has a corresponding first driven contact line and a second driven contact line. The first driving contact line is generally given.

[0093] exist Figure 3 In this context, assuming that the first driving contact line and the first driven contact line are conjugate meshing when the spur gear pair rotates forward, the first driven contact line can be calculated from the first driving contact line through coordinate transformation. The second driving contact line is obtained by rotating the first driving contact line by an angle. Therefore, for the second driven contact line, when the spur gear pair reverses, the line directly conjugately meshing with the second driving contact line is not the second driven contact line, but a third driven contact line on the adjacent tooth surface. The third driven contact line can be calculated from the second driving contact line through coordinate transformation. Because the spur teeth are evenly distributed on the gear body, the second driven contact line used for modeling can be obtained by rotating the third driven contact line around the driven gear axis by an angle of 2. It can also be obtained by the first driven contact line rotating around the wheel shaft. We obtain the rotation angle here. and The value is determined by the number of teeth on the driving gear. And the transmission ratio is determined, where , .

[0094] If the first active contact line is given in the coordinate system The equation in the equation is shown in equation (13):

[0095] (13)

[0096] In equation (13), Let be the independent variable of a cylindrical helix with equal pitch, where The variable value at the start of engagement. This represents the variable value at the end of engagement.

[0097] According to equations (5) and (13), the first driven contact line corresponding to the first active contact line can be obtained in the coordinate system. The parametric equation in the equation is shown in equation (14):

[0098] (14)

[0099] Rotate the first active contact line by an angle Obtain the second active contact line in the coordinate system The parametric equations in the equation are shown in equation (15):

[0100] (15)

[0101] Similarly in the coordinate system In the middle, the first driven contact wire winds around Axis rotation The parametric equation of the second driven contact line can be obtained as shown in equation (16):

[0102] (16)

[0103] (2) Solve for the equation of the central leading line

[0104] To create an accurate geometric model of linear gear teeth using the scanning method, and to ensure no interference during meshing, the normal tooth profile of the linear gear teeth must be scanned along the contact line during modeling. An auxiliary line is also needed to guide the torsional amplitude of the generated tooth surface. The tooth profile of the linear gear can be designed as any cross-sectional shape, such as a circular arc, elliptical arc, involute, or cycloid. This paper selects a circular arc tooth profile for analysis.

[0105] The scanning guide line can be viewed as the trajectory formed by the center of the circular arc tooth profile, so it is simply called the center guide line. Its offset diagram is shown below. Figure 4 As shown.

[0106] Figure 4 In the middle, the center guide line It can be made by contact wire Along the vector The direction is offset by a certain distance get, Let be the radius of the aforementioned circular arc tooth profile. Specifically, the active center guide line originates from the active contact line along the vector... Equidistant offset We obtain that the driven centerline is derived from the driven contact line along the vector. equidistant offset in the opposite direction The result is shown in equation (17). In the equation, the vector... , , These are the unit tangent vector, principal normal, and secondary normal of the contact line. It is a vector and The included angle.

[0107] (17)

[0108] More specifically, the first active center guiding line The first active contact line shown in equation (13) follows the vector directional offset distance Obtain, second active center guide line From equation (15), the second active contact line along the vector directional offset distance Obtain, first driven center guide line From equation (14), the first driven contact line along the vector offset distance in the opposite direction Obtain, second driven center guide line From equation (16), the second driven contact line along the vector offset distance in the opposite direction get.

[0109] (3) Specific design examples

[0110] Given the parametric equations of the first active contact line in the coordinate system As shown in equation (18).

[0111] (18)

[0112] In the formula, The radius of the cylindrical helix; The pitch coefficient, Let be the independent variable of a cylindrical helix with equal pitch, where The variable value at the start of engagement. The values ​​represent the end-point values ​​of the meshing. Specific design parameters for the linear gear are shown in Table 1.

[0113] Table 1 Design parameters of parallel shaft external meshing line gear pairs

[0114]

[0115] The first active contact line in the coordinate system can be obtained from equations (2) and (18). parametric equations in As shown in equation (19).

[0116] (19)

[0117] The first active contact line in the coordinate system can be obtained from equations (2), (6), (8) and (19). The unit common normal vector at the contact point As shown in equation (20).

[0118] (20)

[0119] By combining equations (6), (12), (19), and (20), the meshing equation is obtained:

[0120] (twenty one)

[0121] Therefore, we obtain

[0122] (twenty two)

[0123] Substituting the parameters of equation (22) and Table 1 into equations (13) to (16), the first active contact line, the first passive contact line, the second active contact line and the second passive contact line are obtained as shown in equations (23) to (26).

[0124] (twenty three)

[0125] (twenty four)

[0126] (25)

[0127] (26)

[0128] Similarly, by substituting the parameters in Table 1 into Equation (17), the center guide lines corresponding to the four contact lines can be obtained, as shown in Equations (27) to (30).

[0129] (27)

[0130] (28)

[0131] (29)

[0132] (30)

[0133] Step S4: Derive the pure rolling condition (that is, establish the equation for the pure rolling condition).

[0134] Substituting equations (19) and (22) into equation (12), we get

[0135] (31)

[0136] make The pure rolling condition for a linear gear pair with a contact line that is a cylindrical helix is ​​shown in equation (32):

[0137] (32)

[0138] In Table 1, a = 40 mm, m = 10 mm, i 12 =3, substituting into equation (32), the equation holds true.

[0139] Step S5: Based on steps S3 and S4 and the tooth surface profile, construct a three-dimensional model of the linear gear using the sweep method.

[0140] Based on the contact line and center guide line equations obtained from the above solution, a 3D solid model of the spur gear is created in 3D software using the sweep method. Taking the driving gear as an example, the modeling process is as follows: Figure 5 As shown.

[0141] a) Create the tooth surface containing one side of the meshing line of a single tooth using the first active contact line and the first active center guide line: Create a normal reference plane for the contact line through one endpoint of the first active contact line. This normal plane is also the normal plane at the endpoint of the first active center guide line. Sketch an arc sector on the normal plane. The central angle of the sector depends on the actual situation; in this paper, it is taken as 80°. The center of the arc segment is at the endpoint of the first active center guide line, and the endpoint of the first active contact line is located at the middle position of the arc segment. The radius of the arc segment is the aforementioned offset distance. .

[0142] b) Using the same method, create the other side of the tooth surface of a single tooth through the second active contact line and the second active center guide line, and obtain a single tooth according to the tooth width requirements.

[0143] c) The aforementioned single tooth in the circular array generates all the line teeth.

[0144] d) Create a solid at the center of the linear gear and create the gear hole as needed.

[0145] Similarly, the three-dimensional model of the driven gear can be obtained, and the three-dimensional model of the entire spur gear pair is as follows: Figure 6 As shown.

[0146] Step S6: Simulate the linear gear transmission using 3D software, solve for the driven gear speed and instantaneous transmission ratio, and plot the speed curve and instantaneous transmission ratio curve.

[0147] The three-dimensional model of the above-designed linear gear pair was used to perform forward and reverse kinematic simulation analysis using CAD software. Given a constant rotational speed of 600 deg / s for the driving gear, the contact between the linear gears was set to solid contact. The starting position of forward rotation was obtained through point contact at the end face positions, and the ending position of forward rotation was directly used as the starting position of reverse rotation.

[0148] The simulation yielded the rotational speeds and instantaneous transmission ratios of the driving and driven wheels in both forward and reverse directions as follows: Figure 7 and Figure 8As shown. During both forward and reverse rotation, the speed of the drive wheel is constantly set to 600 dec / s. In forward rotation, the average transmission ratio is 3.0014, the relative error of the average transmission ratio is 0.0467%, the absolute standard deviation of the instantaneous transmission ratio is 0.0546, and the relative standard deviation of the instantaneous transmission ratio is 1.82%. In reverse rotation, the average transmission ratio is 3.0013, the relative error of the average transmission ratio is 0.0433%, the absolute standard deviation of the instantaneous transmission ratio is 0.0471, and the relative standard deviation of the instantaneous transmission ratio is 1.57%.

[0149] Motion simulation results show that the spur gear pair can continuously and accurately output the designed transmission ratio in both forward and reverse rotation. The relative standard deviation for both forward and reverse rotation is less than 2%, indicating that the spur gear pair transmission is relatively smooth, verifying the correctness and feasibility of the method of this invention. The parallel shaft external meshing spur gear pair designed by the method of this invention can achieve continuous and stable transmission under both forward and reverse operating conditions. The method of this invention provides a new approach for the rapid design, modeling, and verification of spur gears, and provides a technical foundation for subsequent optimization of spur gear design.

[0150] This invention provides a design method for a parallel-axis, bidirectional, purely rolling linear gear pair. It selects an equidistant cylindrical helix as the spatial conjugate meshing curve and directly constructs the gear tooth surface using a circular arc tooth profile, combined with the contact line and center guide line. The constructed linear gear pair achieves continuous and smooth transmission. The linear gear pair designed in this invention can provide a theoretical basis and technical support for the engineering application of linear gears in mechanical systems requiring bidirectional transmission (such as reciprocating mechanisms and precision control systems).

[0151] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively describe all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A design method for a parallel-axis bidirectional pure rolling line gear pair, characterized in that, Includes the following steps: S1: Establish the meshing coordinate system of the gear pair on the parallel shaft external meshing line, and derive the coordinate transformation matrix between each coordinate system; S2: Construct an extended linear gear meshing equation containing angular parameters, and solve for the relative velocity and unit common normal vector of the two gears at the meshing point by analyzing the extended linear gear meshing equation; S3: First, based on the coordinate transformation matrix and the extended linear gear meshing equation, solve the contact line equations of the driving and driven gears in the linear gear pair. To achieve bidirectional transmission, a contact line is arranged on each side of the tooth surface of each tooth of the two gears, namely the first driving contact line, the second driving contact line, the first driven contact line, and the second driven contact line. Then, based on the preset normal tooth profile shape and tooth profile radius, and combined with the contact line equations, calculate the center guide line equations corresponding to the four contact lines by offsetting a set distance along a specific vector within the contact line normal plane. These equations are the equations of the first driving center guide line, the second driving center guide line, the first driven center guide line, and the second driven center guide line. The specific vector lies within the normal plane spanned by the principal normal and secondary normal of the corresponding contact line. S4: Based on the criterion that the relative speed of the driving and driven gears is zero during meshing, an equation for the pure rolling condition of the spur gear pair is established, so that the relative speed of the driving and driven gears at the meshing point is always zero. S5: Based on the contact line equation, the center guide line equation, and the pure rolling condition, and combined with the preset normal tooth profile, a single tooth of the linear gear is constructed using the scanning method, and then an array is used to generate a three-dimensional solid model of the linear gear pair.

2. The design method for a parallel-axis bidirectional pure rolling line gear pair according to claim 1, characterized in that, The specific process of deriving the coordinate transformation matrix in step S1 includes: given two fixed Cartesian coordinate systems , and two rotating Cartesian coordinate systems fixed to the driving wheel and driven wheel respectively. , Based on the angular velocities, rotation angles, and distance between the central axes of the driving and driven wheels, the coordinate system is derived. and Coordinate transformation matrix between : ； In the formula, and α represents the angles rotated by the driving wheel and the driven wheel, respectively, and 'a' represents the distance between the central axes of the driving wheel and the driven wheel.

3. The design method for a parallel-shaft bidirectional pure rolling line gear pair according to claim 1, characterized in that, The specific expression for the extended line gear meshing equation in step S2 is as follows: ； in, For the contact point in the coordinate system The relative velocity in and Coordinate systems and The unit common normal vector of the two corresponding contact lines at the contact point. and coordinate system The principal and secondary normals of the lower contact line; by introducing angle parameters. Release the direction of the common normal to the principal normal vector of the contact line. With Sub-Hyaksha Within Zhang Cheng's complete normal plane; the angle parameters The value range is set to .

4. The design method for a parallel-axis bidirectional pure rolling line gear pair according to claim 1, characterized in that, The method for solving the contact line equation in step S3 includes: given the first active contact line on one side of a single tooth of the driving gear in the coordinate system. The parametric equations in the equations; under forward rotation, the first driven contact line in conjugate meshing is calculated from the first active contact line through coordinate transformation; the first active contact line is rotated by an angle... The second active contact line is obtained on the other side of the same single tooth; for the corresponding second driven contact line under the reverse working condition, it is obtained by coordinate transformation of the second active contact line to obtain the third driven contact line, and then the third driven contact line is rotated around the axis by an angle. The angle obtained, or directly derived from the rotation of the first driven contact line around the axis. The result is obtained; where the third driven contact line is a driven contact line on the adjacent tooth surface, with a rotation angle of... , In the formula The number of teeth on the driving gear. This is the transmission ratio.

5. The design method for a parallel-axis bidirectional pure rolling line gear pair according to claim 4, characterized in that, The first active contact line adopts a cylindrical helix with equal pitch, which is in the coordinate system The parametric equation in the equation is expressed as: ； In the formula, Where is the helix radius. The pitch coefficient, Let be the independent variable of a cylindrical helix with equal pitch. The variable value at the start of engagement. This represents the variable value at the end of engagement.

6. The design method for a parallel-shaft bidirectional pure rolling line gear pair according to claim 1, characterized in that, The specific mathematical model for solving the equation of the central guiding line in step S3 is as follows: ； In the formula, The center guide line position vector, This is the position vector corresponding to the contact line. This is the offset distance. Take the radius of the arc of the normal tooth profile; Let be the offset vector, and ； Where i and j are either 1 or 2; , These are the principal normal and secondary normal of the corresponding contact line, respectively. The angle between the offset direction and the principal normal vector; the corresponding active center guide line is offset equidistantly along the offset vector direction. The corresponding driven center guide line is offset equidistantly in the opposite direction of the offset vector. .

7. The design method for a parallel-axis bidirectional pure rolling line gear pair according to claim 5, characterized in that, In step S4, the relative speed of the driving and driven wheels is set. The pure rolling condition equation when the contact line is a cylindrical helix is ​​derived as follows: ； In the formula, The center distance, Where is the helix radius. This is the transmission ratio.

8. The design method for a parallel-axis bidirectional pure rolling line gear pair according to claim 1, characterized in that, In step S5, the preset normal tooth profile cross-sectional shape is designed as any one of a circular arc, an elliptical arc, an involute, or a cycloid.

9. The design method for a parallel-axis bidirectional pure rolling line gear pair according to claim 8, characterized in that, When the preset normal tooth profile cross-sectional shape is a circular arc, the specific process of constructing it using the scanning method in step S5 is as follows: Create a normal reference plane through one endpoint of the corresponding contact line, and sketch a fan-shaped tooth profile on the normal plane, so that the center of the arc segment is located at the endpoint of the corresponding center guide line, and the endpoint of the corresponding contact line is located in the middle of the arc segment. Using the contact line as the scanning path and the central guide line as the guide line to control the torsion amplitude of the tooth surface, the scanning generates one side of the tooth surface of a single tooth. Repeat the above steps to generate the other side of the tooth surface, construct a single line tooth entity, and generate a complete tooth crown after circumferential array.

10. The design method for a parallel-shaft bidirectional pure rolling line gear pair according to any one of claims 1 to 9, characterized in that, The method also includes step S6: performing forward and reverse kinematic simulation on the generated three-dimensional solid model of the spur gear pair; extracting the instantaneous transmission ratio of the driving and driven gears, and determining the continuity and stability of the spur gear pair transmission based on whether the relative standard deviation of the instantaneous transmission ratio is less than a preset threshold.

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