Method for detecting tooth surface of plane secondary enveloping ring surface worm

Abstract

The invention discloses a method for detecting a tooth surface of a planar double-enveloping worm, which comprises the following steps of: S1, establishing a theoretical mathematical equation of the tooth surface of the worm according to a gear meshing principle and basic parameters of the planar double-enveloping worm; s2, the actual coordinate points of the tooth surface of the worm are subjected to data acquisition and represented by y, z and theta, y is in the radial direction of the worm, z is in the axial direction of the worm, theta is the axial rotation angle around the worm and is increased by a certain step length, and coordinate transformation is conducted on the measurement points; and S3, calculating a corresponding theoretical point based on an axial and radial equality principle, matching and comparing the measurement point with the theoretical point to obtain a tooth surface deviation and draw a tooth surface error topological graph, and completing the tooth surface detection of the planar secondary enveloping ring surface worm. The method can fundamentally solve the detection problem of the tooth surface precision of the toroidal worm, and has the characteristics of wide application value, low cost and high precision.

Description

Technical Field

[0001] The invention relates to worm detection, in particular to a method for detecting a plane secondary enveloping toroidal worm tooth surface, and belongs to the technical field of precision testing mechanical transmission. Background Art

[0002] The plane double enveloping toroidal worm is a precision mechanical transmission, which consists of a double enveloping worm wheel 2 and a toroidal worm 1. Its structure is as follows: Figure 1 As shown. Different from the traditional worm gear enveloping worm pair, the tooth surface of the plane secondary enveloping worm gear is a secondary surface enveloping the spiral surface formed by the first enveloping motion with the worm gear through conjugate motion, rather than a rotating surface. This transmission mechanism has the advantages of high load capacity, low wear, smooth transmission and long life.

[0003] With the development of science and technology, higher and higher requirements are put forward for the accuracy of the plane double enveloping toroidal worm transmission system, and a higher precision plane double enveloping toroidal worm tooth surface detection method is needed. The helical error of the toroidal worm is mainly composed of local error, periodic error (including long period error and short period error) and progressive error, so the helical error of the toroidal worm is a comprehensive error. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the purpose of the present invention is to provide a planar secondary enveloping toroidal worm tooth surface detection method, which can achieve high-precision detection of the toroidal worm tooth surface and determine whether the tooth surface processing accuracy meets the requirements.

[0005] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0006] A method for detecting the tooth surface of a planar secondary enveloping toroidal worm, the steps are as follows:

[0007] S1. According to the gear meshing principle and the basic parameters of the planar secondary enveloping toroidal worm, the theoretical mathematical equation of the tooth surface of the planar secondary enveloping toroidal worm is established;

[0008] S2. Collect data on the actual coordinate points of the plane secondary enveloping torus worm tooth surface to obtain a number of measurement points represented by polar coordinates (y, z, θ), where y is along the radial direction of the worm, z is along the axial direction of the worm, and θ is the angle of rotation around the axial direction of the worm. θ increases in a certain step size, and the measurement points are transformed in coordinates;

[0009] S3. Based on the principle of axial and radial equality of the measuring point and the theoretical point, the theoretical point corresponding to each measuring point is calculated, the measuring point is matched and compared with the theoretical point, the tooth surface deviation of each measuring point is obtained and the tooth surface deviation topology diagram is drawn to complete the planar secondary enveloping toroidal worm tooth surface detection.

[0010] Furthermore, in step S2, the method for collecting data on the actual coordinate points of the planar secondary enveloping toroidal worm tooth surface is to use a spherical probe to scan the axial profile to obtain the polar coordinates (y, z, θ) of the actual tooth surface measurement points, and convert the polar coordinates into the three-dimensional coordinate system of the toroidal worm tooth surface through coordinate transformation to obtain the three-dimensional coordinates (x, y, z).

[0011] Furthermore, the axial profile of the worm is scanned as follows:

[0012] 2.1) The plane secondary enveloping toroidal worm is radially positioned by the upper and lower tips of the measuring center through the center holes at both ends;

[0013] 2.2) Place the spherical probe on the uppermost tooth surface of the worm, measure the corresponding Z value, and use the distance conversion of this reference relative to the throat center during the machining process to perform axial positioning and determine the center value z0 of the plane secondary enveloping toroidal worm;

[0014] 2.3) Place the spherical probe in the YOZ plane, i.e. the X-axis coordinate is 0 and lock the X-axis;

[0015] 2.4) Lock the C-axis coordinate of the rotation center, use the profile sweep function to scan, and move downward along the Z axis to sweep the axial tooth profile of the toroidal worm;

[0016] 2.5) The C axis rotates a certain angle θ c After locking, repeat step 2.4) to complete another axial tooth profile measurement;

[0017] 2.6) Repeat step 2.5) to measure a series of axial tooth profiles until the tooth surface measurement of the entire toroidal worm is completed.

[0018] Furthermore, in step S3, the method for calculating the corresponding theoretical point based on the principle of axial and radial equality is to use the theoretical mathematical equation of the planar secondary enveloping toroidal worm tooth surface in S1 to obtain the coordinate value of the theoretical point corresponding to the planar secondary enveloping toroidal worm tooth surface by measuring the axial coordinate value and radial coordinate value of the point.

[0019] Furthermore, in S2, the radius of the spherical probe is automatically compensated during data collection, and the data obtained by the spherical probe is the actual data of the measuring points of the toroidal worm tooth surface.

[0020] Furthermore, a set of data obtained by measuring a certain angle of rotation of the C-axis is selected as a reference, and the corresponding rotation angle is selected and coordinate transformation is performed, that is, polar coordinates are converted to obtain the three-dimensional coordinates of the toroidal worm tooth surface in the measurement coordinate system.

[0021] Furthermore, in S3, the measuring point is matched with the theoretical point by circumferential rotation and transformed into a right triangle, the tooth surface deviation of the measuring point is calculated, and the full tooth surface deviation topology map is obtained through the tooth surface deviations of all measuring points.

[0022] Compared with the prior art, the present invention has the following beneficial effects:

[0023] The present invention provides a method for detecting the tooth surface of a plane secondary enveloping toroidal worm. A spherical probe is used to collect coordinate data of a series of spiral lines from the top to the bottom of the plane secondary enveloping toroidal worm. The coordinate values ​​of the corresponding points on the tooth surface calculated by the theoretical worm mathematical equation are compared to obtain the absolute error of the tooth surface of the worm being measured. The spiral line error Δf can be calculated by measuring the spiral line on the indexing annular surface. k1 , and then the tooth surface machining accuracy is obtained to determine whether the tooth surface machining accuracy meets the requirements. By changing the position of the contact in the direction of the worm tooth height, the spiral lines on different annular surfaces can be obtained, and the coordinates of multiple spiral lines are used to describe the entire tooth surface of the worm being tested, so as to calculate other error items. This method is also suitable for the detection of various toroidal worm tooth surfaces such as conical envelope toroidal worms and drum toroidal worms. It can fundamentally solve the detection problem of toroidal worm tooth surface accuracy, and has the characteristics of wide application value, low cost and high precision. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be further described in detail below with reference to the accompanying drawings, in which:

[0025] Figure 1 It is a schematic diagram of a plane secondary enveloping toroidal worm gear transmission mechanism;

[0026] Figure 2 It is a topological diagram of the tooth surface error of the plane secondary enveloping toroidal worm of the present invention;

[0027] Figure 3 This is a data collection diagram of the plane secondary enveloping toroidal worm tooth surface of the present invention;

[0028] Figure 4 It is a geometric relationship diagram between the measured points and the theoretical points of the present invention;

[0029] Figure 5 This is a flow chart of measurement and data processing of the present invention. DETAILED DESCRIPTION

[0030] The embodiments of the present invention are described in further detail below.

[0031] See also Figure 1-Figure 5 The present invention provides a method for detecting the tooth surface of a plane secondary enveloping toroidal worm, comprising the following specific implementation steps:

[0032] S1. First, based on the basic parameters of the planar secondary enveloping toroidal worm and worm wheel and the principle of spatial gear meshing, the theoretical mathematical equation of the planar secondary enveloping toroidal worm tooth surface is established:

[0033] The worm gear tooth surface equation during the first enveloping process is:

[0034]

[0035] The worm gear tooth surface equation during the second enveloping process is:

[0036]

[0037] Among them, X0, Y0, and Z0 are the coordinates of the tool tooth surface equation during the first enveloping process; X1, Y1, and Z1 are the coordinates of the worm tooth surface equation during the first enveloping process; S1 (O1-X1, Y1, Z1)) and S2 (O2-X2, Y2, Z2) are the motion coordinate systems fixed to the worm wheel during the first and second enveloping processes, respectively. The rotation angle of the parent plane The worm rotation angle i 10 is the transmission ratio, a is the transmission pair center distance, r b is the main base circle radius, u and v are the tooth surface parameters, θ1 is the tool gear rotation angle (θ1=θ2i 10 ), θ2 is the worm gear rotation angle, β is the parent plane inclination angle, is the working half angle, τ is the pitch angle, z k is the number of surrounding teeth.

[0038] S2. Referring to the inspection standards and methods of bevel gears, an axial profile scan is performed from the top to the root of the plane secondary enveloping toroidal worm, and data is collected on the actual coordinate points of the worm tooth surface to obtain a number of measurement points represented by polar coordinates (y, z, θ), where y is along the radial direction of the worm, z is along the axial direction of the worm, and θ is the angle of rotation around the axial direction of the worm, and θ increases in a certain step size. The coordinates of the measurement points are transformed, and the polar coordinates are converted into the three-dimensional coordinate system of the toroidal worm tooth surface to obtain the three-dimensional coordinates (x, y, z);

[0039] S3. Based on the principle of axial and radial equality of the measurement point and the theoretical point, the theoretical point corresponding to each measurement point is calculated, the measurement point is matched and compared with the theoretical point, the distance between the actual measurement point and the theoretical point is expressed as the deviation value at the point, the tooth surface deviation of each measurement point is obtained and the tooth surface deviation topology diagram is drawn to complete the plane secondary enveloping toroidal worm tooth surface detection. The present invention shows the gap between the actual value and the theoretical value in the form of a topological diagram. When the actual tooth surface is within the tooth body of the theoretical tooth surface, the deviation is negative; otherwise, the deviation is positive. Figure 2This is the topological diagram of the tooth surface error of the plane secondary enveloping toroidal worm.

[0040] The tooth surface accuracy is described by the maximum deviation and average deviation values, and the calculation principle is:

[0041]

[0042] Specifically, in S2, the data collection of the toroidal worm gear tooth surface is obtained by the following method, which can also be referred to at the same time Figure 3 :

[0043] 2.1) The toroidal worm is radially positioned through the center holes at both ends and the upper and lower tips of the gear measuring center.

[0044] 2.2) The spherical probe is located at Figure 3 At point ①, the corresponding Z value is measured, and the distance conversion relative to the throat center during the machining process is used to perform axial positioning and determine the center value z0 of the plane secondary enveloping toroidal worm.

[0045] 2.3) Place the spherical probe in the YOZ plane, with the X-axis coordinate at 0 and lock the X-axis.

[0046] 2.4) Lock the C-axis coordinate, use the profile sweep function to scan, and move along the -Z direction to sweep the axial tooth profile of the toroidal worm. The schematic diagram of the probe moving during the sweep process is shown in Figure 3 As shown in ②.

[0047] 2.5) The C axis rotates a certain angle θ c After locking, repeat step 2.4) to complete another axial tooth profile measurement. If you need to improve the accuracy of tooth surface measurement, the angle θ c A smaller value can be used.

[0048] 2.6) Repeat step 2.5) to measure a series of axial tooth profiles until the tooth surface measurement of the entire toroidal worm is completed.

[0049] Since the gear measurement center uses a three-dimensional probe to measure, the probe radius has been compensated, so the data obtained by the probe is the actual data of the toroidal worm gear tooth surface. Since the probe data is expressed in y, z and θ, it is polar coordinate data, so it needs to be converted into three-dimensional coordinates corresponding to the coordinates of the theoretical point. When converting, select a set of data as the reference, and select its corresponding rotation angle θ c By performing coordinate transformation, the three-dimensional coordinate value (x c ,y c , z c). This coordinate system and the theoretical coordinate system of the toroidal worm gear tooth surface have been radially and axially positioned during the measurement process through steps 2.1) and 2.2), and its circumferential positioning is determined by least square matching of the above actual measurement data and theoretical data.

[0050] Assume that the circumferential matching angle between the measured coordinate system and the theoretical coordinate system is Then the coordinate value of the measured data in the theoretical coordinate system is:

[0051]

[0052] Through the coordinate values ​​of the measured points in the theoretical coordinate system, the torus spiral line 5 to which the point 4 belongs is calculated, and the radius of the torus arc is:

[0053]

[0054] The geometric relationship between the measured points and the theoretical points on the toroidal worm tooth surface is as follows: Figure 4 As shown in the figure. ij is the measured point, Q ij is the normal theoretical point corresponding to the theoretical torus spiral line, S ij is the theoretical torus helix line with P ij The corresponding isointerface theoretical point, λ is S ij The lead angle of a point on its torus helix, line segment P ij Q ij The length of the tooth surface deviation is Δf ij .

[0055] The tooth surface deviation Δf of the above toroidal worm ij It can be solved by the following steps:

[0056] 1. Since the measured point P ij and S on the torus ij The points are on the same plane, so their Z-axis coordinates are equal, that is, (z 2c ) ij =(z c' )S ij ;

[0057] 2. Given the radius of the annular arc, determine the tooth surface parameter u 2 =z 2 c 2 +z 2 ;

[0058] 3. At the worm gear angle In the range of values, select one value, through the meshing equation in the following formula

[0059]

[0060] Solve for the corresponding θ, and use x1=u,y1=rcosθ, Determine z2 and use the function f = z2-z 2c Solve the corresponding f value at this time; i 12 is the transmission ratio of the tool worm and the worm wheel in the second enveloping.

[0061] 4. The value increases in a certain step size, and the corresponding θ and f values ​​are calculated according to step 3. The two values ​​when the value f changes positively or negatively are calculated. value;

[0062] 5. The value is a variable, f is the objective function, and the two Use the binary search method to find the value corresponding to f = 0 in the range of values. and θ value;

[0063] 6. Substitute the u value and the value obtained in step 5 Substitute the values ​​of and θ into the coordinates X1, Y1, and Z1 of the worm gear tooth surface equation during the first enveloping process to obtain the coordinate point S ij (x 2′ ,y 2′ , z 2′ );

[0064] 7. Line Segment

[0065] 8. Triangle P in the microscopic field ij Q ij S ij Considered as a right triangle, ∠P ij Q ij S ij The torus helix is ​​in S ij From differential geometry, we know that the tangent line of the intersection of two surfaces in space at a certain point is perpendicular to the normal lines of the two intersecting surfaces at that point.

[0066] Through coordinate transformation, the expression of the normal vector of the worm gear tooth surface in the fixed coordinate system S2 of the plane secondary enveloping toroidal worm is:

[0067]

[0068] i1, j1, k1 are the components of the normal vector of the worm gear tooth surface in three directions, which are unit vectors; n 1x 、n 1y 、n 1z is the magnitude of the components of the normal vector n1 in the X, Y, and Z directions;

[0069] Through differential geometry, the normal vector of the plane secondary enveloping toroidal worm tooth surface in the fixed coordinate system S2 is expressed as:

[0070]

[0071] i2, j2, k2 are the components of the normal vector of the worm tooth surface in three directions, which are unit vectors; n 2x 、n 2y 、n 2z is the magnitude of the components of the normal vector n2 in the X, Y, and Z directions;

[0072] Torus spiral line S ij The tangent vector at the point is:

[0073] t p =n1×n2=t px i2+t py j2+t pz k2

[0074] t px ,t py ,t pz is the tangent vector t p The magnitude of the components in the X, Y, and Z directions;

[0075] In the above formula, n1 and n2 are the worm wheel and the annular surface enveloping the worm in S ij The normal vector at .

[0076] According to the definition of lead angle and geometric relationship, λ can be obtained as:

[0077]

[0078] 9. Calculate line segment P ij Q ij The length of the tooth surface deviation is:

[0079]

[0080] In the above formula, k is the discrimination coefficient. When the measuring point P ij When inside the entity, k = -1; when the measuring point P ij Outside the entity, k=1.

[0081] Circumferential matching angle As a variable, the goal is to minimize the deviation between the measured surface and the theoretical surface of the plane quadratic enveloping toroidal worm, and the evaluation function is constructed based on the least squares method:

[0082] The present invention uses the dichotomy method and the stepwise approximation method to measure and process the tooth surface data. The flow chart is as follows: Figure 5As shown, where h is the circumferential matching angle is the transformation step size, and ε is the calculation accuracy.

[0083] The present invention uses a spherical probe to collect a series of coordinate data of the spiral line of the plane secondary enveloping toroidal worm from top to bottom, and compares the coordinate values ​​of the corresponding points on the tooth surface calculated by the theoretical worm mathematical equation to obtain the absolute error of the measured worm tooth surface. The spiral line error Δf can be calculated by completing the measurement of the spiral line on the indexing annular surface. u , and then the tooth surface machining accuracy is obtained to determine whether the tooth surface machining accuracy meets the requirements. This method is also suitable for the detection of various toroidal worm tooth surfaces such as conical envelope toroidal worms and drum toroidal worms. It can fundamentally solve the detection problem of toroidal worm tooth surface accuracy and has the characteristics of wide application value, low cost and high accuracy.

[0084] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described with reference to the preferred embodiments of the present invention, it should be understood by those skilled in the art that various changes may be made in form and details without departing from the spirit and scope of the present invention as defined in the appended claims.

Claims

1. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm, characterized in that: Here are the steps: S1. According to the gear meshing principle and the basic parameters of the planar secondary enveloping toroidal worm, the theoretical mathematical equation of the tooth surface of the planar secondary enveloping toroidal worm is established; S2. Collect data on the actual coordinate points of the plane secondary enveloping torus worm tooth surface to obtain a number of measurement points represented by polar coordinates (y, z, θ), where y is along the radial direction of the worm, z is along the axial direction of the worm, and θ is the angle of rotation around the axial direction of the worm. θ increases in a certain step size, and the measurement points are transformed in coordinates; S3. Based on the principle of axial and radial equality of the measuring point and the theoretical point, the theoretical point corresponding to each measuring point is calculated, the measuring point is matched and compared with the theoretical point, the tooth surface deviation of each measuring point is obtained and the tooth surface deviation topology diagram is drawn to complete the planar secondary enveloping toroidal worm tooth surface detection.

2. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 1, characterized in that: In S2, the method for collecting data on the actual coordinate points of the planar secondary enveloping toroidal worm tooth surface is to use a spherical probe to scan the axial profile to obtain the polar coordinates (y, z, θ) of the actual tooth surface measurement points, and convert the polar coordinates into the three-dimensional coordinate system of the toroidal worm tooth surface through coordinate transformation to obtain the three-dimensional coordinates (x, y, z).

3. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 2, characterized in that: Scan the axial profile of the worm as follows: 2.1) The plane secondary enveloping toroidal worm is radially positioned by the upper and lower tips of the measuring center through the center holes at both ends; 2.2) Place the spherical probe on the uppermost tooth surface of the worm, measure the corresponding Z value, and use the distance conversion of this reference relative to the throat center during the machining process to perform axial positioning and determine the center value z0 of the plane secondary enveloping toroidal worm; 2.3) Place the spherical probe in the YOZ plane, i.e. the X-axis coordinate is 0 and lock the X-axis; 2.4) Lock the C-axis coordinate of the rotation center, use the profile sweep function to scan, and move downward along the Z axis to sweep the axial tooth profile of the toroidal worm; 2.5) After the C-axis rotates a certain angle, lock it and repeat step 2.4) to complete another axial tooth profile measurement; 2.6) Repeat step 2.5) to measure a series of axial tooth profiles until the tooth surface measurement of the entire toroidal worm is completed.

4. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 1, characterized in that: In S3, the method for calculating the corresponding theoretical point based on the principle of axial and radial equality is to use the theoretical mathematical equation of the planar secondary enveloping toroidal worm tooth surface in S1, and to obtain the coordinate value of the theoretical point corresponding to the planar secondary enveloping toroidal worm tooth surface by measuring the axial coordinate value and radial coordinate value of the point.

5. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 4, characterized in that: According to the theoretical mathematical equation, the normal vectors n1 and n2 of each theoretical point on the toroidal worm are calculated, and the corresponding lead angle λ is obtained.

6. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 3, characterized in that: In S2, the radius of the spherical probe is automatically compensated during data collection, and the data obtained by the spherical probe is the actual data of the measuring points on the toroidal worm tooth surface.

7. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 3, characterized in that: A set of data obtained by measuring a certain angle of rotation of the C-axis is selected as a reference, and the corresponding rotation angle is selected and coordinate transformation is performed, that is, polar coordinates are converted to obtain the three-dimensional coordinates of the toroidal worm tooth surface in the measurement coordinate system.

8. A method for detecting the tooth surface of a planar secondary enveloping toroidal worm according to claim 1, characterized in that: In S3, the measuring point is matched with the theoretical point by circumferential rotation and transformed into a right triangle, the tooth surface deviation of the measuring point is calculated, and the full tooth surface deviation topology map is obtained through the tooth surface deviations of all measuring points.

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