Sand profile calculation method based on Bezier curve

Abstract

The invention provides a sand contour shape calculation method based on a Bezier curve, and the method specifically comprises the following steps: 1, sequentially building a homogeneous coordinate transformation matrix from a grinding wheel to a workpiece and a tooth surface equation of a helical gear based on a forming gear grinding machine machining principle and a gear meshing principle; 2, assuming to process the internal bevel gear according to the established tooth surface equation, calculating an instantaneous contact line according to the condition that the normal of the tooth surface of the bevel gear intersects with the axis of the grinding wheel, and obtaining the section of the revolution surface of the grinding wheel; and 3, compensating the radius of the grinding wheel, solving an interpolation point through a Bezier curve, writing a calculation program according to related parameters, carrying out MATLAB fitting to preliminarily obtain a corresponding grinding wheel profile, and then carrying out CMM measurement to finally obtain the obtained grinding wheel profile. According to the method, the sand profile shape calculation efficiency can be improved, and a theoretical basis can be provided for follow-up gear shape modification.

Description

Technical Field

[0001] This invention relates to the field of gear processing and grinding technology for CNC forming gear grinding machines. Specifically, it employs a sand profile calculation method based on Bézier curves, which is a method for efficiently obtaining sand profiles for a large number of gears with different parameters. Background Technology

[0002] Currently, gears, as core components of mechanical transmission, have a significant impact on the overall transmission efficiency and lifespan due to their machining accuracy. With the rapid development of advanced equipment such as ships, aircraft, and automobiles, the technical requirements for gears have generally increased. In gear machining, profile grinding is a commonly used high-precision machining method, directly grinding the required gear tooth surface using a profile grinding wheel. This method is highly efficient and precise. Therefore, the design and generation of the sand profile remains a key factor in gear machining. Currently, sand profile design mainly relies on analytical and empirical methods. Analytical methods derive the sand profile using accurate gear geometry formulas, theoretically achieving high accuracy, but they cannot be directly applied when faced with tooth profile modifications. Empirical methods are too flexible, relying primarily on worker experience and lacking theoretical support, making it difficult to maintain stable gear machining. In summary, a sand profile calculation method that guarantees high accuracy and has a wide applicability is needed, which is also one of the current research directions for improving grinding accuracy. Summary of the Invention

[0003] This invention is a sand profile calculation method based on Bézier curves, which can improve the efficiency of sand profile calculation and provide a theoretical basis for subsequent gear modification.

[0004] To achieve the above objectives, the present invention provides the following technical solution: Step 1: Based on the machining principle of the forming gear grinding machine and the gear meshing principle, establish the homogeneous coordinate transformation matrix from the grinding wheel to the workpiece and the tooth surface equation of the helical gear in sequence; Step 2: Assuming the machining of an internal helical gear, based on the tooth surface equation established above, calculate the instantaneous contact line according to the condition that the normal of the helical gear tooth surface intersects with the axis of the grinding wheel, and at the same time obtain the cross-section of the grinding wheel's rotating surface; Step 3: Compensate for the grinding wheel radius, solve for the interpolation point using Bézier curves, fit the points using MATLAB to obtain the initial sand profile, and then measure using CMM to finally obtain the desired sand profile.

[0005] In step one: during the gear grinding process, the grinding wheel coordinate system... S w Around the coordinate axis O w X w Rotation Angle, first alongX w Axis translation x One unit length, then along Z w Axis translation z One unit length, and around Z w Axis rotation β Angle, at this time relative to the workpiece coordinate system S g If they overlap, then we can obtain S wg Homogeneous transformation matrix: , When grinding helical gears, since the grinding wheel axis is located in the normal plane of the gear groove and undergoes helical motion, the non-centrally enveloped surface formed by the grinding wheel profile in space is the involute helical surface of the helical gear. Therefore, the gear tooth surface equation is: , in, r b The radius of the base circle, k=tanα y , α y The pressure angle at any point on the involute spiral surface i This indicates the angle through which the busbar rotates around the axis from its initial position. d b The center half-angle of the base circular groove, β b The helix angle of the base cylinder can represent any point on the tooth tip, tooth root, or tooth flank, providing a basis for the instantaneous contact line.

[0006] In step two: when the workpiece is being machined, the relative position of the tool's rotating surface and the workpiece's helical surface, let's assume the workpiece's coordinate system is... X g Y g Z g Then the coordinate system of the tool is X w Y w Z w Then any point on the tooth surface P ( x 1 , y 1 , z 1 The normal equation of ) is: , And its normal vector is: , Furthermore, the contact condition between the two is obtained by the intersection of the tooth surface normal and the grinding wheel axis: , Point P ( x 1 , y 1 , z 1 Substituting these conditions into the equation, we obtain the conjugate contact conditions between the grinding wheel's machined surface and the workpiece's tooth surface: , Then use [ S w-g ] -1 Homogeneous coordinate transformation, from the workpiece coordinate system to the grinding wheel coordinate system, and then combined with the conjugate contact condition equation, yields the instantaneous contact line equation. Rotating the instantaneous contact line around the grinding wheel axis yields the cross-section of the grinding wheel's rotating surface.

[0007] In step three: During gear machining, the grinding wheel radius is fixed. However, machine tool geometric errors, manufacturing principle errors, etc., can affect the actual profile calculation, necessitating compensation for the grinding wheel radius. This is achieved by calculating the tangent slope at each point on the grinding wheel profile and then increasing the radius along the normal direction. r By calculating the unit length, we can obtain the point after the grinding wheel radius compensation; Next, the interpolation points can be easily solved using the Bézier curve method. The steps are as follows: First, the known points are parameterized, and the control points are obtained by fitting the least squares method. Then, the points on the curve are output by multiple Bézier curves. Finally, the interpolation points are obtained by using equal-parameter interpolation or equal-arc-length interpolation. After obtaining the interpolation points, MATLAB software is used to fit the points and calculate the RMSE. If the RMSE is greater than the threshold, Bézier control points are added; otherwise, a preliminary sand profile is obtained. Then, CMM measurements are performed. If the offset exceeds 0.05 mm, the radius is recompensated; otherwise, the final sand profile is obtained.

[0008] Compared with the prior art, the beneficial effects of the present invention are as follows: Based on the machining principle of profile grinding machines and the gear meshing principle, the homogeneous coordinate transformation matrix from the grinding wheel to the workpiece and the tooth surface equation of the helical gear are established sequentially. According to the established tooth surface equation, assuming the machining of an internal helical gear, the instantaneous contact line is calculated based on the condition that the normal to the helical gear tooth surface intersects the grinding wheel axis. Simultaneously, the cross-section of the grinding wheel's rotation surface is obtained, and the grinding wheel radius is compensated. The interpolation point is solved using Bézier curves. A calculation program is written based on relevant parameters to initially obtain the grinding wheel profile. Finally, the corresponding grinding wheel profile is obtained through CMM measurement. This model achieves a smooth, continuous, and high-precision grinding wheel profile and provides a theoretical basis for subsequently reducing tooth surface errors. Attached Figure Description

[0009] Figure 1 This is a flowchart of the sand contour algorithm.

[0010] Figure 2 This is a schematic diagram of the homogeneous coordinate transformation between the grinding wheel and the workpiece.

[0011] Figure 3 This is a schematic diagram showing the relative positions of the grinding wheel and the workpiece. Detailed Implementation

[0012] To clarify the technical issues, technical solutions, implementation process, and performance demonstration, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. However, the described embodiments are only a part of the embodiments of the present invention and not all of them.

[0013] This embodiment is a method for calculating the profile of sand based on Bézier curves, including the following steps: Step 1: Based on the machining principle of the forming gear grinding machine and the gear meshing principle, establish the homogeneous coordinate transformation matrix from the grinding wheel to the workpiece and the tooth surface equation of the helical gear in sequence; Step 2: Assuming the machining of an internal helical gear, based on the tooth surface equation established above, calculate the instantaneous contact line according to the condition that the normal of the helical gear tooth surface intersects with the axis of the grinding wheel, and at the same time obtain the cross-section of the grinding wheel's rotating surface; Step 3: Compensate for the grinding wheel radius, solve for the interpolation point using Bézier curves, fit the points using MATLAB to obtain the initial sand profile, and then measure using CMM to finally obtain the desired sand profile.

[0014] In step one: during the gear grinding process, the grinding wheel coordinate system... S w Around the coordinate axis O w X w Rotation Angle, first along X w Axis translation xOne unit length, then along Z w Axis translation z One unit length, and around Z w Axis rotation β Angle, at this time relative to the workpiece coordinate system S g If they overlap, then we can obtain S wg Homogeneous transformation matrix: , When grinding helical gears, since the grinding wheel axis is located in the normal plane of the gear groove and undergoes helical motion, the non-centrally enveloped surface formed by the grinding wheel profile in space is the involute helical surface of the helical gear. Therefore, the gear tooth surface equation is: , in, r b The radius of the base circle, k=tanα y , α y The pressure angle at any point on the involute spiral surface i This indicates the angle through which the busbar rotates around the axis from its initial position. d b The center half-angle of the base circular groove, β b The helix angle of the base cylinder. It can represent any point on the tooth tip, tooth root, or tooth flank, providing a basis for the instantaneous contact line.

[0015] In step two: according to Figure 3 As shown, during the machining of the workpiece, the relative positions of the tool's rotating surface and the workpiece's helical surface are such that, without loss of generality, the workpiece's coordinate system is... X g Y g Z g Then the coordinate system of the tool is X w Y w Z w Then any point on the tooth surface P ( x 1 , y 1 , z 1 The normal equation of ) is: , And its normal vector is: , Furthermore, the contact condition between the two is obtained by the intersection of the tooth surface normal and the grinding wheel axis: , Point P ( x 1 , y 1 , z 1 Substituting these conditions into the equation, we obtain the conjugate contact conditions between the grinding wheel's machined surface and the workpiece's tooth surface: , Then use [ S w-g ] -1 Homogeneous coordinate transformation, from the workpiece coordinate system to the grinding wheel coordinate system, and then combined with the conjugate contact condition equation, yields the instantaneous contact line equation. Rotating the instantaneous contact line around the grinding wheel axis yields the cross-section of the grinding wheel's rotating surface.

[0016] In step three: During gear machining, the grinding wheel radius is fixed. However, machine tool geometric errors, manufacturing principle errors, etc., can affect the actual profile calculation, necessitating compensation for the grinding wheel radius. This is achieved by calculating the tangent slope at each point on the grinding wheel profile and then increasing the radius along the normal direction. r By calculating the unit length, we can obtain the point after compensation for the grinding wheel radius.

[0017] Next, the interpolation points can be easily solved using the Bézier curve method. The steps are as follows: First, the known points are parameterized, and the control points are obtained by fitting using the least squares method. Then, the points on the curve are output by using multiple Bézier curves. Finally, the interpolation points are obtained by using equal-parameter interpolation or equal-arc-length interpolation.

[0018] After obtaining the interpolation points, MATLAB software is used to fit the points and calculate the RMSE. If the RMSE is greater than the threshold, Bézier control points are added; otherwise, a preliminary sand profile is obtained. Then, CMM measurements are performed. If the offset exceeds 0.05 mm, the radius is recompensated; otherwise, the final sand profile is obtained.

[0019] The embodiments described above are merely preferred examples to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Various changes and modifications can be made to the present 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 the present invention is defined by the claims.

Claims

1. A method for calculating a grinding wheel profile based on a Bezier curve, characterized by The method comprises the following steps: Step one: based on the forming gear grinding principle and gear meshing principle, the homogeneous coordinate transformation matrix of the grinding wheel to the workpiece, the tooth surface equation of the helical gear are established in turn; Step two: assuming that the internal helical gear is processed, according to the tooth surface equation established above, the instantaneous contact line is calculated according to the condition that the normal of the helical gear tooth surface intersects with the grinding wheel axis, and the grinding wheel rotation surface section shape is also obtained; Step three: the grinding wheel radius is compensated, the interpolation points are solved through the Bezier curve, the obtained points are fitted by using MATLAB, the corresponding grinding wheel profile is preliminarily obtained, then the CMM is measured, and finally the obtained grinding wheel profile is obtained.

2. The method of claim 1, wherein: In step one: when the gear is being processed by the forming grinding, the grinding wheel coordinate system S w Rotating around the coordinate axis O w X w Rotating An angle, first along X w Axis translation x A unit length, then along Z w Axis translation z A unit length, and rotating around Z w Axis rotation β An angle, at this time coinciding with the workpiece coordinate system S g The homogeneous transformation matrix of S wg can be obtained ， When the helical gear is formed by grinding, since the grinding wheel axis is located in the gear groove plane and does spiral motion, the non-ordered envelope surface formed by the grinding wheel profile in space is the involute helical surface of the helical gear, that is, the gear tooth surface equation is: ， wherein r b is the base circle radius, k=tanα y , α y the pressure angle of any point on the involute helicoid, θ denotes the angle through which the generatrix is turned about the axis from the starting position, δ b is the base circle slot center half angle, β b is the base circle helix angle; that is, it can be expressed at any point of the tip, root, flank, and provides the basis for the instantaneous contact line.

3. The method of claim 1, wherein: In the second step: the relative position between the rotary surface of the tool and the helical surface of the workpiece when the workpiece is being processed, let the coordinate system of the workpiece be X g Y g Z g , then the coordinate system of the tool is X w Y w Z w , the normal equation of any point on the tooth surface P x 1 , y 1 , z 1 ​​ ， And its normal vector is: ， Also, the tooth surface normal intersects with the grinding wheel axis to obtain the contact condition of the two: ， The points P ( x 1 , y 1 , z 1 ) are substituted into the conditional formula to obtain the conjugate contact condition of the grinding wheel processing surface and the workpiece tooth surface: ， Then the following equation is obtained: S w-g ] -1 The instantaneous contact line equation is obtained by combining the homogeneous coordinate transformation from the workpiece coordinate system to the grinding wheel coordinate system with the conjugate contact condition equation. The cross-section of the grinding wheel is obtained by revolving the instantaneous contact line around the grinding wheel axis.

4. The method of claim 1, wherein: In step three: During gear machining, the grinding wheel radius is fixed. However, machine tool geometric errors, manufacturing principle errors, etc., can affect the actual profile calculation. Therefore, the grinding wheel radius is compensated. This is done by calculating the tangent slope at each point on the grinding wheel profile and then increasing it along the normal direction. r The unit length is used to obtain the point after grinding wheel radius compensation; Then, the Bezier curve method is used to easily solve the interpolation points, and the steps are as follows: first, the known points are parameterized, the control points are obtained by least square fitting, then the points on the curve are output through the multi-segment Bezier curve, and finally the interpolation points are obtained by using the equal parameter interpolation or equal arc length interpolation; After obtaining the interpolation points, the obtained points are fitted by using MATLAB software, the RMSE is calculated, if it is greater than the threshold value, the Bezier control point is increased, otherwise the preliminary grinding wheel profile is obtained. Then through CMM measurement, if the offset is more than 0.05mm, the radius is compensated again, otherwise the final grinding wheel profile is obtained.

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