A method for tooth profile modification and modeling of logarithmic spiral oscillating tooth transmission
By modifying and modeling the tooth profile of the logarithmic spiral live gear transmission device, and using circular arc curves or high-order polynomial curves to modify the moving teeth and cams, combined with static modeling, the problem of contact impact between the moving teeth and the internal gear ring and cams was solved, thereby improving the transmission performance and manufacturing capabilities.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHENGDU UNIVERSITY OF TECHNOLOGY
- Filing Date
- 2026-06-05
- Publication Date
- 2026-07-03
AI Technical Summary
Existing logarithmic spiral tooth transmission devices suffer from rigid impact during operation, resulting in poor transmission performance and difficulty in meeting manufacturing requirements. In particular, the contact impact between the moving teeth and the internal gear ring, as well as between the cam and the moving teeth, is severe.
The profiles of the moving tooth tip and the cam transition section are modified by using circular arc curves or high-order polynomial curves. The modified tooth profile equation is established, and the meshing function is solved by coordinate transformation matrix. Combined with the static balance model, the sharp point meshing state is eliminated, the conditions for the formation of hydrodynamic oil film are improved, and rigid impact is avoided.
It significantly reduces wear on the tips of moving teeth, improves meshing accuracy and transmission smoothness, enhances load-bearing capacity, provides feasible parameter guidance, and improves transmission performance.
Smart Images
Figure CN122333682A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of mechanical transmission technology, and in particular to a method for tooth profile modification and modeling of logarithmic spiral live gear transmission. Background Technology
[0002] Logarithmic spiral tooth transmission devices have broad application prospects in robotics, high-end CNC equipment, and aerospace. Their logarithmic spiral tooth profile and special transmission structure give them theoretical transmission advantages of high precision and high load capacity.
[0003] Existing technologies have proposed various improvement schemes. For example, CN120991041A discloses a logarithmic spiral tooth transmission device with adjustable backlash, which improves lubrication conditions and transmission accuracy by setting a lubrication component and adjusting the tooth backlash with a double-pitch thread. CN118793747A discloses a precision reduction device for logarithmic spiral tooth transmission, which uses two single-wave cams connected in opposite directions to achieve force balance and higher speed reduction. CN113503353A discloses a logarithmic spiral conjugate tooth profile harmonic reducer, which achieves high load-bearing capacity and low backlash by utilizing the conjugate tooth profile design. Although the above-mentioned existing technologies have optimized the performance of logarithmic spiral tooth transmission devices, they do not consider the meshing state between the moving teeth and the internal gear ring, or the contact between the cam and the moving teeth, resulting in rigid impact problems and making it difficult to form a manufacturable physical product.
[0004] Existing studies on logarithmic spiral tooth transmission devices have not considered the meshing state between the movable tooth and the internal gear ring, as well as the contact between the cam and the movable tooth. This results in severe rigid impact problems during operation, poor transmission smoothness, and difficulty in meeting manufacturing requirements.
[0005] While optimizations to existing logarithmic spiral gear transmission devices (including lubrication, structure, and conjugate tooth profile design) have improved performance to some extent, the structural design process has failed to adequately consider the impact between the contact surfaces of various components, particularly the impact between the cam and the moving teeth, and between the moving teeth and the internal gear ring. This results in poor transmission performance and susceptibility to damage during operation, becoming a key bottleneck hindering the large-scale application of logarithmic spiral gear transmission devices. Summary of the Invention
[0006] To address the impact problem between the contact surfaces of various parts in existing logarithmic spiral gear transmission devices, which leads to poor transmission performance and easy damage, this invention proposes a method for tooth profile modification and modeling of logarithmic spiral gear transmissions to solve the aforementioned problems.
[0007] This application discloses a method for tooth profile modification and modeling of a logarithmic helical live gear transmission, including the following steps: The profile of the movable tooth tip is modified by using circular arc curves or higher-order polynomial curves to obtain the modified profile of the movable tooth. The profiles of the long axis transition section and the short axis transition section of the cam are modified using high-order polynomial curves to obtain the theoretical profile of the modified cam. Based on the motion mapping relationship between the modified moving teeth and the cam, a coordinate transformation matrix is established, and the meshing function during the envelope motion process is derived and solved to obtain the profile of the modified internal gear ring. A static equilibrium model including the cam, cage and all moving teeth was established to analyze the influence of the modification parameters on the contact force under working conditions. Based on the analysis of the profile of the modified moving tooth, the theoretical profile of the modified cam, the profile of the modified internal gear ring, and the modification parameters, a three-dimensional model of the logarithmic spiral moving tooth transmission is established.
[0008] Preferably, a circular arc curve is used to modify the profile of the active tooth tip. By setting the continuity conditions of the values and slopes at the modification segment points, the profile equation of the circular arc modification is established. The equation for the active tooth profile modified by circular arc curve is:
[0009] in, As a reference radius, The radius of the reshaped arc. Polar angle, This refers to the range of removable tooth modification. The helix angle, For the range of moving tooth angles, The x-coordinate of the center of the arc-shaped circle.
[0010] Preferably, a higher-order polynomial curve is used to modify the profile of the active tooth tip, ensuring the continuity of the numerical values, first derivative, and second derivative at the modification segmentation points, and establishing the equation of the active tooth profile modified by the higher-order polynomial curve. The equation for the active tooth profile modified using a higher-order polynomial curve is expressed in polar coordinates as follows:
[0011] in, , , , These are the undetermined coefficients of a higher-order polynomial curve.
[0012] Preferably, the modification of the long axis transition section profile and short axis transition section profile of the cam using high-order polynomial curves includes: Determine the modification range for the short axis and the modification range for the long axis; Establish the minor axis shaping curve and the major axis shaping curve based on five boundary conditions; Solve for the undetermined coefficients of the minor axis shaping curve and the major axis shaping curve respectively; The theoretical profile of the modified cam is obtained based on the undetermined coefficients of the short-axis modification curve and the long-axis modification curve.
[0013] Preferably, the modified theoretical profile of the cam is:
[0014] in, The radius of the base circle, , , , , These are the undetermined coefficients for the minor axis shaping curve. , , , , For the undetermined coefficients of the major axis shaping curve, Let be the rotation angle of the cam. This is the modification segmentation point for the camshaft short shaft. This is the modification segmentation point for the long shaft of the cam. The transmission ratio is... This refers to the number of long or short half-shafts of the cam.
[0015] Preferably, the obtained modified internal gear profile includes: Based on the motion mapping relationship between the modified moving teeth and the cam, a coordinate transformation matrix is established, and the meshing function in the envelope motion process is derived and solved to obtain the modified internal gear profile. The coordinate transformation matrix is as follows:
[0016] in, , ; The profile of the movable tooth follows the rotation angle When the coordinates change, the expression in the static coordinate system is:
[0017] Change and The value is used to obtain the motion trajectory of the moving teeth during the meshing process. and When the value satisfies that the normal direction of the moving tooth at the coordinate point is perpendicular to the direction of the motion velocity, substitute all coordinate points that satisfy the meshing condition into the equation. The expression is used to obtain the modified internal gear profile.
[0018] Preferably, establishing the static equilibrium model includes: Establish the static balance equations for the cam, the cage, and the moving gear. Deformation compatibility equations are established based on the contact deformation relationship between the cam and the movable tooth; The combined static equilibrium equations and deformation compatibility equations form the modified total static equilibrium matrix equation set.
[0019] Preferably, the static equilibrium equation of the cam is:
[0020] in, Indicates the first One movable tooth, The total number of moving teeth. The reaction force provided to each moving tooth The distance from the contact point between the movable tooth and the cam to the center of rotation of the cam. The meshing angle at each contact point of the moving teeth. The friction angle between each moving tooth and the cam. Input torque; The static equilibrium equation of the cage is:
[0021]
[0022] in, Force applied to the outer movable teeth. The force applied to the inner moving teeth, To maintain the radius of the outer circle of the frame, To maintain the radius of the inner circle of the frame, This is the distance from the contact point between the moving tooth and the internal gear ring to the center of rotation of the cam. To maintain the friction angle between the two sides of the frame and the moving teeth, To output torque, For full tooth width, For full tooth height, To maintain the radial distance between the outer side of the frame and the contact point of the movable tooth; The static equilibrium equation for the movable tooth is:
[0023] in, The coefficient matrix, This is the force acting on the internal gear ring.
[0024] Preferably, the modified total static equilibrium matrix equations are as follows:
[0025] In the formula,
[0026]
[0027] , , The coefficient matrix, The angle of cam rotation in the deformation compatibility equation Related variables, For flexible rotation angle, This is the radial distance to the outer contact point of the cage corresponding to the first movable tooth.
[0028] Preferably, establishing the three-dimensional model of the logarithmic spiral gear transmission includes: Based on the modified moving tooth profile equation, the cam theoretical profile equation, and the internal gear ring profile equation, discrete points are calculated respectively. The discrete points are imported into 3D modeling software, fitted into smooth curves, and the moving tooth model, cam model, and internal gear ring model are obtained by mirroring and stretching respectively.
[0029] The beneficial effects of this invention are: (1) The shaping method proposed in this invention can effectively eliminate the sharp point meshing state between the moving tooth and the internal gear ring, significantly reduce the wear of the moving tooth tip, and improve the conditions for the formation of hydrodynamic oil film.
[0030] (2) This invention analyzes the influence of the shaping parameters on the contact force under working conditions through static modeling.
[0031] (3) The present invention can simultaneously avoid sharp points or dotted contour lines on the inner and outer contours of the cam, ensuring the continuity of the contour lines and eliminating the problem of rigid impact.
[0032] (4) The present invention can significantly improve meshing accuracy and transmission smoothness, improve tooth profile contact stress distribution, enhance load-bearing capacity and fatigue resistance, and also provide feasible parameter guidance for actual production. Attached Figure Description
[0033] Figure 1 This is a flowchart illustrating the tooth profile modification process of a logarithmic spiral live gear transmission according to an embodiment of the present invention. Figure 2 This is a schematic diagram illustrating the use of a circular arc curve to modify the profile of the movable tooth in an embodiment of the present invention; Figure 3 This is a schematic diagram illustrating the use of high-order polynomial curves to modify the profile of the movable tooth in an embodiment of the present invention. Figure 4 This is a schematic diagram illustrating the use of high-order polynomial curves to modify the theoretical profile of a cam according to an embodiment of the present invention. Figure 5 This is a schematic diagram of the static balance of the cam according to an embodiment of the present invention; Figure 6 This is a schematic diagram of the static balance of the cage according to an embodiment of the present invention; Figure 7 This is a schematic diagram of the static balance of the movable tooth according to an embodiment of the present invention; Figure 8 This is a schematic diagram of the parameters of the movable teeth in an embodiment of the present invention; Figure 9 This is a schematic diagram of the contact deformation between the cam and the movable tooth in an embodiment of the present invention. Detailed Implementation
[0034] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided with reference to the accompanying drawings and embodiments.
[0035] This application discloses a method for tooth profile modification and modeling of a logarithmic spiral live gear transmission. The modification process is as follows: Figure 1 As shown, firstly, the profile of the movable tooth tip is modified using a high-order polynomial or circular arc curve to obtain the modified movable tooth profile. Then, the profiles of the long axis transition section and the short axis transition section of the cam are modified using a high-order polynomial curve to obtain the theoretical profile of the modified cam. Based on this, according to the motion relationship between the modified movable tooth and the cam, the modified internal gear ring profile is solved through coordinate transformation and the principle of conjugate meshing. Next, through static modeling, the influence of the modification parameters on the contact force under working conditions is analyzed, and a three-dimensional model of the logarithmic spiral movable tooth transmission is established. Finally, a set of methods for profile modification and modeling of the logarithmic spiral movable tooth transmission device that meets the needs of actual processing and manufacturing and performance optimization is formed, providing a necessary foundation for actual production and manufacturing.
[0036] The modified movable tooth profile consists of two segments: the original movable tooth profile without modification and the modified movable tooth tip profile. The movable tooth tip profile can be modified using a circular arc curve or a higher-order polynomial, as shown in Examples 1 and 2, respectively.
[0037] Example 1 In this embodiment, a circular arc curve is used to modify the profile of the active tooth tip. By setting the continuity conditions of the values and slopes at the modification segment points, the equation of the circular arc modified tooth profile is established.
[0038] To modify the profile of the moving tooth tip using a circular arc curve, it is necessary to determine the modification segmentation points and ensure that the modified segment and the original logarithmic spiral segment are continuous in terms of value and first derivative at the segmentation points. This allows for the establishment of the circular arc modified tooth profile equation in polar coordinates. For example... Figure 2The diagram shows a schematic of modifying the profile of a movable tooth using a circular arc curve. The dashed curve segment represents a portion of the movable tooth profile that conforms to the original logarithmic spiral, while the solid curve segment represents the tooth tip profile of the movable tooth after modification using the circular arc. Points A, B, and C correspond to the center of the modification arc, the modification segment point, and the tooth root position of the movable tooth profile in the global coordinate system, respectively.
[0039] Based on the symmetry of the moving teeth along the X-axis, we can first solve... Figure 2 The equation of the tooth profile in the fourth quadrant is then used to establish the complete movable tooth profile based on symmetry. From the coordinate relationships in the diagram, the equation of the circular arc in the fourth quadrant is: (1) (2) in, Let x be the x-coordinate of the center of the arc-shaped circle. The radius of the reshaped arc. As a reference radius, Polar angle, The helix angle, This refers to the range of the moving tooth angle.
[0040] Furthermore, according to equation (2), point B can be represented as:
[0041] in, Let B be the x-coordinate. Let B be the ordinate of point B. This refers to the range of removable tooth shaping.
[0042] To ensure the continuity of the circular arc and the logarithmic spiral at point B, the numerical values and slopes of both curves at that point must be equal. This requires that the coordinates of point B on the circular arc and point B on the logarithmic spiral are equal, and that the first derivatives of the corresponding curves at that point are identical. Solving these equations simultaneously yields the following: and The expression:
[0043] For equation (4), only and Two unknowns, in order to standardize the process of modifying the tooth profile, are specified as follows: Assume the number and solve. After rearranging equation (4), we can... use Represented as:
[0044] To unify the profile of the movable tooth after shaping The representation in polar coordinates will , Substituting into equation (1) and rearranging, we obtain the following equation: (6) in, This refers to the reshaped profile of the movable tooth.
[0045] Two equations were obtained using the quadratic formula. about The expression, and by Figure 2 It can be seen that, along with Increases with increasing. Therefore, the equation for the movable tooth modification segment can be expressed as: (7) In summary, the equation for the active tooth profile modified with a circular arc curve can be expressed in polar coordinates as follows:
[0046] Example 2 In this embodiment, a high-order polynomial curve is used to modify the profile of the active tooth tip. At the modification segmentation point, the numerical values, first derivative, and second derivative are ensured to be continuous, thereby establishing a high-order polynomial profile modification equation.
[0047] like Figure 3 The diagram illustrates the modification of the movable tooth profile using a higher-order polynomial curve. The dashed curve segment represents a portion of the movable tooth profile conforming to the original logarithmic spiral, while the solid curve segment represents the modified movable tooth tip profile using the higher-order polynomial. Points A', B, and C represent the intersection of the tooth profile and the X-axis, the modification segment point, and the position of the movable tooth root, respectively. Similar to modifying the movable tooth tip profile using a circular arc curve, since the tooth profile curve is symmetrical along the X-axis, only the curve expression in the fourth quadrant needs to be solved. At point A', it is necessary to ensure that the tangent direction of the curve is perpendicular to the Y-axis to avoid forming the movable tooth tip point, i.e. .
[0048] At point B, it is necessary to ensure that the values of the polynomial curve and the logarithmic spiral, as well as the first and second derivatives, are the same. Based on these requirements, the higher-order polynomial curve is determined by four constraints. Therefore, an expression for the higher-order polynomial curve can be established using a cubic polynomial, which can be represented in polar coordinates as follows: (9) in, , , , To introduce the undetermined coefficients of the higher-order polynomial curve, This is to unify the expression of logarithmic spirals and higher-order polynomial curves.
[0049] By rearranging and simplifying the above expressions, we can establish a system of equations for solving the undetermined coefficients of the high-order polynomial profile curve of the moving tooth:
[0050] Furthermore, the above equation can be rewritten in matrix form, with... The 4×4 coefficient matrix on the left side of the equation is represented by... This represents the 4×1 coefficient matrix on the right side of the equation, with... Let the matrix of undetermined coefficients represent the profile curve of the movable tooth. Then, the undetermined coefficients of the higher-order polynomial curve can be solved according to the following formula: (11) Finally, the equation for the active tooth profile modified by a high-order polynomial curve can be expressed in polar coordinates as follows:
[0051] Example 3 Based on the active tooth profile modification in Example 1 or Example 2, higher-order polynomial curves are used to modify the profiles of the long axis transition section and the short axis transition section of the cam respectively. At the long and short axes, the radial velocity and acceleration of the active tooth are ensured to be zero. At the segmentation point, the values, first derivative and second derivative of the curve before and after modification are strictly continuous, thereby establishing a higher-order polynomial tooth profile modification equation.
[0052] Cam shaping is used to avoid discontinuities in the inner and outer contours and rigid impacts on the moving teeth. For example... Figure 4 The diagram illustrates the modification of the theoretical cam profile using a high-order polynomial curve. Considering the periodic symmetry of the theoretical cam profile, the theoretical cam profile before and after modification is established in the first quadrant. The red dashed line represents the theoretical cam profile before modification, the blue solid line represents the theoretical cam profile after modification, and the black solid line represents the cam base circle. Let be the rotation angle of the cam. The cam's minor axis modification segment point (i.e., the cam rotation angle corresponding to the segment point between the minor axis modification curve and the non-modified curve). This refers to the camshaft profile trim segment point (i.e., the cam rotation angle corresponding to the segment point between the untrimmed curve and the long axis trim curve), and... , and These correspond to the short axis reshaping range, long axis reshaping range, and no reshaping range, respectively.
[0053] For minor axis contour curves, since the theoretical contour line near the X-axis is concave, therefore The value must be greater than the critical point of the concave curve to avoid the modified curve still exhibiting concavity. According to... Figure 4 The trend of the uncorrected curve shows that the slope at the critical point of the concave curve does not exist. Therefore, the slope of the curve can be used to determine the slope. Determine its critical value .
[0054] Can Represented as:
[0055] in, The transmission ratio is... Let be the radius of the base circle. , These are the equations of the horizontal and vertical coordinates on the theoretical profile of the cam. , They are respectively , Compared to The first derivative.
[0056] In equation (13), if It does not exist, that is Thus, the solution is obtained. ( Minimum value).
[0057] When the moving tooth is in the push stroke phase, the cam pushes it radially outward along the cage; when it is in the return stroke phase, the moving tooth moves in the opposite direction. To reduce the rigid impact on the moving tooth during the instant of reversal between the push and return strokes, it is necessary to ensure that the radial velocity and acceleration of the moving tooth are both zero when it is on the long and short axes of the cam. The moving velocity and acceleration of the moving tooth along the cage can be obtained from the theoretical profile of the cam. Compared to Find the first and second derivatives, that is, the expression along the minor axis ( ) and major axis ( ) place, =0, =0.
[0058] Furthermore, and The location serves as the segmentation point between the modified and unmodified curves along the major and minor axes. It is essential to ensure strict geometric continuity of the curves before and after modification at this location, i.e., at... and At the point where the curves before and after the modification are completely equal in terms of their numerical values, first derivative, and second derivative, the constraint equations for the minor axis modification curve and the major axis modification curve can be listed respectively. Both modification curves are determined by five boundary conditions (the radial velocity and acceleration of the moving teeth at the major and minor axes of the modified curve are both zero, and the numerical values, first derivative, and second derivative are strictly continuous with the unmodified curve at the segmentation point). Therefore, the modification curves are calculated by establishing a fourth-order polynomial. The constraint equation for the minor axis modification curve is shown in equation (14), and the constraint equation for the major axis modification curve is shown in equation (15). (14) (15) in, , , , , These are the undetermined coefficients for the minor axis shaping curve. , , , , For the undetermined coefficients of the major axis shaping curve, This refers to the number of the long or short half-shafts of the cam. In this embodiment, both the number of the long and short half-shafts of the cam are 2. and This is to unify the expression of the unmodified and modified cam curves.
[0059] Furthermore, the constraint equations of the minor axis shaping curve are rearranged and simplified, and a system of equations for solving the undetermined coefficients of the minor axis shaping curve is established:
[0060] Rewrite equation (16) in matrix form, Indicates the left side of the equation coefficient matrix, with Indicates the right side of the equation coefficient matrix, with Let the matrix of undetermined coefficients of the minor axis shaping curve represent the matrix of undetermined coefficients. The undetermined coefficients of the minor axis shaping curve can be solved by the following formula: (17) Similarly, by rearranging and simplifying the constraint equations of the major axis shaping curve, a system of equations for solving the undetermined coefficients of the major axis shaping curve is established:
[0061] Rewrite equation (18) in matrix form, Indicates the left side of the equation coefficient matrix, with Indicates the right side of the equation coefficient matrix, with Let the matrix of undetermined coefficients of the major axis shaping curve represent the matrix of undetermined coefficients. Then, the undetermined coefficients of the major axis shaping curve can be solved by the following formula: (19) Known , , and Then, the undetermined coefficient matrix of the major and minor axis shaping curves can be solved using equations (17) and (19). To facilitate the study of logarithmic spiral gear transmissions, this embodiment specifies... and With the same values, the theoretical profile of the modified cam can be represented by equation (20), while the theoretical profiles of other positions can be solved according to the periodic symmetry relationship.
[0062] (20) Based on the motion mapping relationship between the modified moving teeth and the cam, a coordinate transformation matrix is established, and the meshing function during the envelope motion process is derived and solved to obtain the modified internal gear profile.
[0063] The modification of the moving gear and cam inevitably leads to a change in the profile of the internal gear ring. The transformation matrix after modification is represented as follows: (twenty one) in, , Substituting into equation (21), the transformation matrix can be expressed as: (twenty two) Furthermore, by combining the equation (8) of the movable tooth profile modified by the circular arc curve, the movable tooth profile as a function of the rotation angle can be obtained. When the coordinates change, the expression in the static coordinate system O0-X0Y0 is as follows: ;(twenty three) For equation (23), depending on the shaping method used for the movable teeth, its It can be determined by the active tooth profile equation (8) modified by a circular arc curve or the active tooth profile equation (12) modified by a higher-order polynomial curve. By changing the equation... and Numerical analysis yields the motion trajectory of the moving teeth during meshing, and its envelope is the tooth profile of the internal gear ring that needs to be solved. Therefore, solving for the tooth profile of the internal gear ring can be equivalent to calculating the points of meshing. and .when and The value is determined when the normal direction of the moving tooth at that coordinate point is perpendicular to the direction of the motion velocity. ( Let $\frac{ ... (where is the unit normal vector of the moving tooth profile at that coordinate point), and that coordinate point is the meshing point. All [tooth profiles] satisfying this condition will be [values]. and Substituting into equation (23) will solve for the corresponding internal gear profile.
[0064] Based on the above modification results, a static modeling of the logarithmic spiral tooth transmission tooth profile modification is performed. The modification equation only guarantees geometric continuity; its impact on transmission performance must be quantified through a static model. In this embodiment, a static balance model including the cam, cage, and all moving teeth is established based on the kinematic relationships between the components in the logarithmic spiral tooth transmission system.
[0065] like Figure 5 The diagram shown illustrates the static equilibrium of the cam. This embodiment ignores factors such as inertia and gravity, considering only the contact force between transmission components. The static equilibrium equations for the cam and cage are established based on the static equilibrium conditions. The cam is subjected to an input torque. and the reaction force provided by each moving tooth Due to the symmetrical configuration of the mechanism, the components of force acting on the cam in the X and Y directions are balanced. Therefore, the static equilibrium equation of the cam is as follows: (twenty four) in, Indicates the first One movable tooth, The total number of moving teeth. The distance from the contact point between the movable tooth and the cam to the center of rotation of the cam can be solved by the modified theoretical profile equation (20). The meshing angle at each contact point of the moving teeth. This refers to the friction angle between each moving tooth and the cam.
[0066] like Figure 6 The diagram shows the static equilibrium of the cage, with forces applied to the inner and outer sides by the moving teeth. and The output torque is Based on the static equilibrium conditions, the static equilibrium equations of the cage are established: (25) (26) in, Force applied to the outer movable teeth. The force applied to the inner moving teeth, To maintain the radius of the outer circle of the frame, To maintain the radius of the inner circle of the frame, This is the distance from the contact point between the moving tooth and the internal gear ring to the center of rotation of the cam. To maintain the friction angle between the two sides of the frame and the moving teeth, For full tooth width, For full tooth height, To maintain the radial distance between the outer side of the retainer and the contact point of the movable tooth (i.e., the distance from that contact point to the center of rotation of the cam), its value is determined based on the distance from the contact point between the movable tooth and the internal gear ring to the center of rotation of the cam. Full tooth height and the radius of the outer circle of the cage Confirmed. If If the contact point between the outer side of the cage and the moving tooth is located on the outer circle of the cage, then... ;like If the contact point between the outer side of the retainer and the movable tooth is located on the side of the movable tooth, then... , specifically as formula (26).
[0067] For static balance of moving gears, neglecting the effects of inertia, gravity, and other factors, only considering the contact forces between transmission components, Figure 7 This is a schematic diagram of the static balance of the movable gear. Figure 8 A schematic diagram of the parameters of the movable tooth is shown, illustrating the first... The static equilibrium state of the movable teeth. Subjected to the force of the cam. Force of the internal gear ring and the force of the moving teeth on both sides of the cage , When the reducer is in static equilibrium, the first The static equilibrium equations for the movable teeth are as follows: (27) in, It is a coefficient matrix.
[0068] coefficient matrix As shown below, it is defined as each active tooth edge (No. (X-axis direction of each movable tooth) (No. The forces acting on the moving teeth are balanced along the Y-axis, and the torques exerted by the cam, cage, and internal gear ring on the moving teeth are balanced.
[0069] (28) in, The friction angle between the moving tooth and the internal gear ring.
[0070] Equation (28) is written as an explicit expression of the static equilibrium equation of the movable tooth (29), which represents the contact force between the movable tooth and the internal gear ring (i.e., the force exerted by the internal gear ring). As the objective of the solution, we can obtain Number of static equilibrium equations.
[0071] (29) In summary, the total number of static equations is... There are one unknown quantity to be found. , , , and Total There are several unknowns. This is because the logarithmic helical gear transmission involves multi-tooth meshing and has an over-constrained structural characteristic, thus requiring the establishment of deformation compatibility conditions that conform to its structural characteristics. For example... Figure 9 The diagram shows the contact deformation between the cam and the movable tooth. There is an elastic contact between the cam and the movable tooth roller. Therefore, when an input torque is applied to the cam, contact deformation occurs between the cam and the movable tooth, causing the cam to rotate by a small angle. After (elastic rotation), equilibrium is reached.
[0072] The indentation depth at the contact point between the movable tooth and the cam can be calculated based on the cam's theoretical profile equation; that is, the radial deformation of the movable tooth at contact with the cam. With elastic angle The relationship between them: (30) in, Indicates the first The modified theoretical profile of the cam corresponding to each active tooth.
[0073] The needle rollers in the cam contact the inner contour surface of the cam component, and the theoretical contour line of contact with the bottom surface of the slipper is the outer contour trajectory of the needle roller movement. Therefore, the diameter of the cam needle rollers is used as the offset, and the offset is made inward along the normal direction. The expression of the actual inner contour curve of the cam is as follows: (31) in, Let be the normal vector of the theoretical profile of the cam. The diameter of the needle roller is [value].
[0074] Combine equations (20) and (31), and take into account For a small variable, the radial deformation of the movable tooth is converted into a relationship with the cam rotation angle. Relevant equations: (32) The load and deformation relationship between the cam and the moving tooth can be expressed as: (33) in, This refers to the contact deformation between the bottom of the movable toothed slip shoe and the cam groove needle roller. The contact stiffness coefficient between the cam and the moving tooth is calculated using the following formula: (34) in, Let be the radius of curvature of the theoretical profile of the cam. Let be the radius of curvature of the curved surface of the movable toothed slipper. It can be derived from equation (20). The elastic modulus of the needle rollers within the cam groove. The elastic modulus of the sliding shoe of the cam moving tooth. The Poisson's ratio of the needle rollers in the cam groove. Poisson's ratio for the sliding shoe of the cam movement.
[0075] Based on the above analysis of the contact deformation between the cam and the movable tooth, the relationship between the contact force and the amount of contact deformation can be derived as the relationship between the contact force and the rotational displacement of the cam caused by the deformation. The expression is as follows: (35) Equations (24) and (25) can be written in matrix form: (36) coefficient matrix as follows: (37) (38) coefficient matrix as follows: (39) in, This is the radial distance to the outer contact point of the cage corresponding to the first movable tooth.
[0076] By combining the static equilibrium equations of the movable gear (equation (27), the overall static equilibrium equations of the logarithmic spiral movable gear transmission structure are constructed. The overall static equilibrium matrix equations are as follows: (40) In the formula , The expression is as follows: (41) (42) However, without considering the deformation compatibility conditions of the mechanism, the number of rows in the established logarithmic spiral gear transmission static equilibrium matrix equations is... The number of rows and columns is Since the number of independent equations is less than the number of unknowns, this set of equations is an underdetermined set of equations. The core reason for this is the over-constraint characteristic caused by the multi-tooth meshing of the logarithmic spiral gear transmission. It is impossible to form a closed and solvable set of equations by relying solely on static equilibrium conditions. The above analysis has established deformation coordination equations that match the characteristics of the transmission structure based on the deformation coordination relationship of the transmission system. After organizing them into a standardized matrix form, they are substituted into the total static equilibrium matrix equation set (40) to obtain the modified total static equilibrium matrix equation set as shown in equation (43). In this embodiment, the deformation coordination equation refers to the equation describing the elastic rotation angle of the cam. The relationship between the contact deformation of each moving tooth and the equations that introduce deformation include equations (32), (33), (34) and equation (35) obtained by combining the equations.
[0077] After introducing the deformation compatibility equations, the total equation set has an additional... The system of equations consists of 1 independent equations, with 1 new unknown variable to be solved. The revised total static equilibrium matrix equations have both 100 rows and 100 columns. The unknown quantity to be solved is The column vectors. Thus, the original underdetermined system of equations is transformed into a full-rank square matrix with equal number of rows and columns. The system of equations is fully constrained, and the unknowns can be solved with unique and exact solutions, providing a reliable theoretical model for solving the contact force of the transmission system.
[0078] (43) coefficient matrix in the formula , The angle of cam rotation in the deformation compatibility equation The relevant variables can be derived by combining equations (32), (33), and (34).
[0079] Based on the above static modeling and shaping method of logarithmic spiral tooth transmission, the contact force distribution under different shaping parameters can be obtained, and a three-dimensional model of logarithmic spiral tooth transmission can be accurately established. The tooth profile of a single-sided movable tooth is established, taking a high-order polynomial curve shaping scheme as an example (the modeling method for the circular arc curve shaping scheme is the same as that for the high-order polynomial curve shaping scheme): First, the discrete points of the tooth profile are calculated using MATLAB software. The tooth profile curve consists of two parts: the shaping segment and the logarithmic spiral segment. To ensure fitting accuracy, the number of discrete points for the shaping segment is specified to be no less than 400, and the number of discrete points for the logarithmic spiral segment is no less than 800. Then, the calculated discrete points of the tooth profile are saved as a txt file and imported into the three-dimensional modeling software (UG software is used in this embodiment). Next, the "Curve Fitting" command is used in UG software, and the "Degree and Tolerance" method is used to fit the discrete points into a smooth curve, where the degree of the curve is set to 24 and the tolerance is set to 1×10⁻⁴. -5Finally, the fitted single-sided tooth profile is mirror-symmetric along the x-axis to form a complete movable tooth profile. The established movable tooth profile curve is then stretched along the tooth width direction to form a sheet, which is used to divide the movable tooth blank solid, thus obtaining the movable tooth model.
[0080] The calculation and fitting method for discrete points of the cam profile can refer to the tooth profile calculation and fitting process of the movable tooth. First, the inner and outer profiles of the 1 / 4 cam are obtained, and then the 1 / 4 cam solid is obtained through offset and stretching. Finally, the complete cam model is obtained by using mirror symmetry and Boolean operation.
[0081] The tooth profile curve of a single-sided internal gear ring can be calculated using the above methods. Depending on the modification method used for the movable teeth, there are two types of internal gear ring tooth profile shapes. Here, we take the high-order polynomial curve modification as an example (the modeling method for the circular arc curve modification scheme is the same as that for the high-order polynomial curve modification scheme). First, the tooth profile of the single-sided internal gear ring is calculated, and then the single tooth shape is obtained through mirroring. Subsequently, the complete internal gear ring curve is obtained through circumferential array, and finally, the complete internal gear ring model is obtained through the extrusion command.
[0082] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.
Claims
1. A method of profile modification and modeling of logarithmic spiral oscillating tooth transmission, characterized in that, Includes the following steps: The profile of the movable tooth tip is modified by using circular arc curves or higher-order polynomial curves to obtain the modified profile of the movable tooth. The profiles of the long axis transition section and the short axis transition section of the cam are modified using high-order polynomial curves to obtain the theoretical profile of the modified cam. Based on the motion mapping relationship between the modified moving teeth and the cam, a coordinate transformation matrix is established, and the meshing function during the envelope motion process is derived and solved to obtain the profile of the modified internal gear ring. A static equilibrium model including the cam, cage and all moving teeth was established to analyze the influence of the modification parameters on the contact force under working conditions. Based on the analysis of the profile of the modified moving tooth, the theoretical profile of the modified cam, the profile of the modified internal gear ring, and the modification parameters, a three-dimensional model of the logarithmic spiral moving tooth transmission is established.
2. The method of profiling and modeling of logarithmic spiral oscillating tooth gearing according to claim 1, characterized in that, The tooth profile of the active tooth tip is modified by using a circular arc curve. By setting the continuity conditions of the values and slopes at the modification segment points, the tooth profile equation of the circular arc modification is established. The equation for the active tooth profile modified by circular arc curve is: wherein, is a reference radius, is a radius of the circular arc after modification, is a polar angle, is a range of modification of the movable tooth, is a helix angle, is a range of the angle of the movable tooth, is an abscissa of the center of the circular arc modification.
3. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 1, characterized in that, The tooth profile of the active tooth tip is modified using a higher-order polynomial curve. At the modification segmentation point, the numerical values, first derivative, and second derivative are kept continuous. The equation of the active tooth profile modified by the higher-order polynomial curve is established. The equation for the active tooth profile modified using a higher-order polynomial curve is expressed in polar coordinates as follows: in, , , , These are the undetermined coefficients of a higher-order polynomial curve.
4. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 2 or 3, characterized in that, The process of reshaping the profiles of the long axis transition section and the short axis transition section of the cam using high-order polynomial curves includes: Determine the modification range for the short axis and the modification range for the long axis; Establish the minor axis shaping curve and the major axis shaping curve based on five boundary conditions; Solve for the undetermined coefficients of the minor axis shaping curve and the major axis shaping curve respectively; The theoretical profile of the modified cam is obtained based on the undetermined coefficients of the short-axis modification curve and the long-axis modification curve.
5. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 4, characterized in that, The modified theoretical profile of the cam is as follows: in, The radius of the base circle, , , , , These are the undetermined coefficients for the minor axis shaping curve. , , , , For the undetermined coefficients of the major axis shaping curve, Let be the rotation angle of the cam. This is the modification segmentation point for the camshaft short shaft. This is the trimming and segmentation point for the long shaft of the cam. The transmission ratio is... This refers to the number of long or short half-shafts of the cam.
6. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 5, characterized in that, The obtained modified internal gear profile includes: Based on the motion mapping relationship between the modified moving teeth and the cam, a coordinate transformation matrix is established, and the meshing function in the envelope motion process is derived and solved to obtain the modified internal gear profile. The coordinate transformation matrix is as follows: in, , ; The profile of the movable tooth follows the rotation angle When the coordinates change, the expression in the static coordinate system is: Change and The value is used to obtain the motion trajectory of the moving teeth during the meshing process. and When the value satisfies that the normal direction of the moving tooth at the coordinate point is perpendicular to the direction of the motion velocity, substitute all coordinate points that satisfy the meshing condition into the equation. The expression is used to obtain the modified internal gear profile.
7. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 6, characterized in that, Establishing the static equilibrium model includes: Establish the static balance equations for the cam, the cage, and the moving gear. Deformation compatibility equations are established based on the contact deformation relationship between the cam and the movable tooth; The combined static equilibrium equations and deformation compatibility equations form the modified total static equilibrium matrix equations.
8. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 7, characterized in that, The static equilibrium equation of the cam is: in, Indicates the first One movable tooth, The total number of moving teeth. The reaction force provided to each moving tooth The distance from the contact point between the movable tooth and the cam to the center of rotation of the cam. The meshing angle at each contact point of the moving teeth. The friction angle between each moving tooth and the cam. Input torque; The static equilibrium equation of the cage is: in, Force applied to the outer movable teeth. The force applied to the inner moving teeth, To maintain the radius of the outer circle of the frame, To maintain the radius of the inner circle of the frame, This is the distance from the contact point between the moving tooth and the internal gear ring to the center of rotation of the cam. To maintain the friction angle between the two sides of the frame and the moving teeth, To output torque, For full tooth width, For full tooth height, To maintain the radial distance between the outer side of the frame and the contact point of the movable tooth; The static equilibrium equation for the movable tooth is: in, The coefficient matrix, This is the force acting on the internal gear ring.
9. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 8, characterized in that, The corrected set of total static equilibrium matrix equations is as follows: In the formula, , , The coefficient matrix, The angle of cam rotation in the deformation compatibility equation Related variables, For flexible rotation angle, This is the radial distance to the outer contact point of the cage corresponding to the first movable tooth.
10. The method for tooth profile modification and modeling of logarithmic spiral live gear transmission according to claim 9, characterized in that, The establishment of the three-dimensional model of the logarithmic spiral gear transmission includes: Based on the modified moving tooth profile equation, the cam theoretical profile equation, and the internal gear ring profile equation, discrete points are calculated respectively. The discrete points are imported into 3D modeling software, fitted into smooth curves, and the moving tooth model, cam model, and internal gear ring model are obtained by mirroring and stretching respectively.