A method for accurately measuring and determining the circular cross-section profile and its circumferential harmonic components

Abstract

The present invention discloses a method for accurately measuring and determining a circular cross-section profile and its circumferential harmonic components, so as to directly solve the expression of the circular cross-section profile of a workpiece and determine the different harmonic components of the profile and their locations. The method specifically includes the following steps: establishing a single cross-section measurement mathematical model and determining the parameter relationship; setting the Fourier coefficients A0, A1 and A2 of the circular cross-section profile; and determining the harmonic components of the circular cross-section profile. i 、B i (i=1,2,3,…,N), and determine its relationship with the measured indication; perform Fourier fitting on the measured indication and solve the Fourier coefficients C0, C i 、D i (i=1,2,3,…,N+1); coefficients are aligned to get A0, A i 、B i (i=1,2,3,…,N) expression; solve for the cross-section eccentricity, and substitute the result to solve for A0, A i 、B i (i=1, 2, 3, ..., N) The present invention provides an effective solution to the roundness measurement problem involving multi-system errors.

Description

Technical Field

[0001] The present invention belongs to the field of precision measurement, and in particular relates to a method for accurately measuring and determining a circular cross-section profile and its circumferential harmonic components. Background Art

[0002] Workpiece roundness error is a key indicator of rotating component quality. Unacceptable roundness error can cause abnormal vibration, wear, and impact in the machine. When testing the roundness of shaft-type workpieces, the sensor cannot be perfectly aligned with the workpiece's rotational axis. The distance between the sensor and the workpiece's rotational axis is manually installed, so measurement errors are also present. The workpiece's centroid and rotational center cannot perfectly coincide, resulting in an eccentricity between the workpiece's centroid and rotational center, and an offset between the sensor's measurement direction and the workpiece's rotational axis. Because this eccentricity between the workpiece's centroid and rotational center and the offset between the sensor's measurement direction and the workpiece's rotational axis are very small relative to the workpiece's radius, common testing methods either ignore this offset or initially disregard it, assuming no error in the sensor's installation position. The error caused by this offset is then evaluated after the test. While this method facilitates calculations, it can lead to systematic errors. For roundness measurement problems involving multiple systematic errors, it is necessary to incorporate these unavoidable errors into the calculation process and determine the magnitude of each systematic error through a solution, rather than relying on manual measurement or ignoring them as minor errors.

[0003] In summary, the development of roundness measurement technology involving multi-system errors urgently requires a high-precision and efficient method for accurately measuring and determining the circular cross-sectional profile and its circumferential harmonic components. Summary of the Invention

[0004] The present invention aims to provide a method for accurately measuring and determining the circular cross-sectional profile and its circumferential harmonic components. A workpiece is clamped on a turntable, which rotates the workpiece and drives its rotation. Simultaneously, an encoder connected to the turntable records the workpiece's rotation angle. Using a single-point laser displacement sensor as an example, the sensor emits laser light onto the workpiece surface, collecting information about the workpiece's circular cross-sectional profile in real time and outputting a measurement indication. Based on the displacement sensor's measurement data, an expression for the workpiece's circular cross-sectional profile can be solved, allowing the different harmonic components of the profile (elliptical, triangular, square, etc.) to be determined and their locations.

[0005] The specific operation steps of the present invention are:

[0006] (1) Establish a mathematical model for single-section measurement to determine the relationship between the sensor reading and the workpiece radius, the profile of the workpiece's circular section, the workpiece's eccentricity, the initial eccentricity angle, the angle the workpiece has rotated, the distance between the sensor's measurement direction and the turntable's rotation center, and the distance between the sensor's installation position and the Y-axis, and make the parameters dimensionless;

[0007] (2) Input the measured data z(θ), θ, s, h, and set the Fourier expansion series N and coefficients [A0 A1 B1 ... A] of the profile f(α) of the measured circular section. N B N ] T ;

[0008] (3) Substituting the Fourier fitting formula of f(α) into the theoretical expression of z(θ), we can get z(θ) with respect to [A0 A1 B1 ... A N B N ] T The equation Q z ;

[0009] (4) Fit z(θ) with an N+1-order Fourier series to obtain the coefficients [C0 C1 D1 ... C N+1 D N+1 ] T ;

[0010] (5) Establish [A0 A1 B1 ... A N B N ] T and [C0 C1 D1 ... C N+1 D N+1 ] T The relationship between the equation g c0 、g ci 、g di (i=1,2,3,…,N+1);

[0011] (6) g c0 、g ci 、g di A0, A in (i=1,2,3,…,N+1) i 、B i (i=1,2,3,…,N) perform first-order Taylor expansion, ignore the square terms of e and h and the higher-order terms above the square, and get A0, A i 、B i (i=1,2,3,…,N) expression;

[0012] (7) A N 、B N Substitute the expression of g into c(N+1) 、g d(N+1) In the equation, the two equations are subjected to first-order Taylor expansion to obtain e, Then solve the obtained e, Substitute back to A0, A i 、B iThe expression of (i=1,2,3,…,N) is solved to get A0, A i 、B i (i=1,2,3,…,N);

[0013] (8) By obtaining the 2N+5 parameters s, h, e, A0, A i 、B i (i=1,2,3,…,N), we can get the expressions of the sensor indication z(θ) and the profile f(α) of the circular cross section.

[0014] The workpiece radius in step (1), the angle of the workpiece rotation, the distance between the sensor measurement direction and the turntable rotation center, and the distance between the sensor installation position and the Y-axis need to be determined according to the installation and debugging status of the actual measuring equipment. Before testing, the test bench needs to be calibrated first to obtain the accurate values ​​of the parameters.

[0015] In step (2), z(θ), s, and h are dimensionless values ​​of the sensor reading, the distance between the sensor measurement direction and the turntable rotation center, and the distance between the sensor installation position and the Y axis, respectively. θ is the rotation angle of the workpiece, which can be obtained by encoder recording.

[0016] The theoretical expression of z(θ) in step (3) can be obtained through step (1).

[0017] In step (5), [A0 A1 B1 ... A N B N ] T and [C0 C1 D1 ... C N+1 D N+1 ] T The relationship is obtained by combining steps (3) z(θ) with respect to [A0 A1 B1 ... A N B N ] T The equation is obtained by the N+1-level Fourier fitting formula of z(θ) in step (4).

[0018] In the step (7), e, Back-substituted A0, A i 、B i (i=1,2,3,…,N) is the expression of A0 and A obtained in step (6) i 、B i (i=1,2,3,…,N) expression.

[0019] The advantages of the present invention are:

[0020] (1) This measurement method can directly solve the expression of the circular cross-section profile of the workpiece, intuitively determine the different harmonic components of the profile (ellipse, triangular circle, square circle, etc.) and their locations, and simultaneously obtain the eccentricity data of the measured cross-section, thereby improving the efficiency of roundness measurement. This method can be implemented using a variety of displacement sensors (such as point laser displacement sensors, spectral confocal sensors, etc.), and has strong versatility.

[0021] (2) Compared with the traditional roundness measurement method, this measurement method takes into account the influence of multiple system errors on roundness measurement, incorporates these inevitable errors into the calculation process, and obtains the amount of each system error through the solution method, rather than through manual measurement or simplifying it by ignoring it as a small amount, which greatly improves the accuracy of workpiece roundness detection. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 This is the flow chart of the logic structure of the detection program of the present invention

[0023] Figure 2 Schematic diagram of the measurement process of the present invention

[0024] Figure 3 Schematic diagram of the dimensionless sensor reading of the present invention

[0025] Figure 4 Schematic diagram of the circular cross-section profile of the workpiece measured by the present invention DETAILED DESCRIPTION

[0026] The purpose of the present invention is to provide a method for accurately measuring and determining the circular cross-sectional profile and its circumferential harmonic components. The workpiece is clamped on a turntable, and the rotation of the turntable drives the workpiece to rotate. At the same time, the turntable is connected to an encoder to record the rotation angle of the workpiece. Taking a single-point laser displacement sensor as an example, the sensor emits a laser to the surface of the workpiece, collects the circular cross-sectional profile information of the workpiece in real time, and outputs the measurement indication. Based on the displacement sensor measurement data, the expression of the circular cross-sectional profile of the circular workpiece can be solved, and the different harmonic components of the profile (ellipse, triangular circle, square circle...) and their locations can be determined. In conjunction with the accompanying drawings, the invention is further described below.

[0027] Combine Figure 1 The logical structure shown in FIG. 1 is as follows:

[0028] (1) Establish a mathematical model for single-section measurement to determine the relationship between the sensor reading and the workpiece radius, the profile of the workpiece's circular section, the workpiece's eccentricity, the initial eccentricity angle, the angle the workpiece has rotated, the distance between the sensor's measurement direction and the turntable's rotation center, and the distance between the sensor's installation position and the Y-axis, and make the parameters dimensionless;

[0029] (2) Input the measured data z(θ), θ, s, h, and set the Fourier expansion series N and coefficients [A0 A1 B1 ... A] of the profile f(α) of the measured circular section. N B N ] T ;

[0030] (3) Substituting the Fourier fitting formula of f(α) into the theoretical expression of z(θ), we can get z(θ) with respect to [A0 A1 B1 ... A N B N ] T The equation Q z ;

[0031] (4) Fit z(θ) with an N+1-order Fourier series to obtain the coefficients [C0 C1 D1 ... C N+1 D N+1 ] T .

[0032] (5) Establish [A0 A1 B1 ... A N B N ] T and [C0 C1 D1 ... C N+1 D N+1 ] T The relationship between the equation g c0 、g ci 、g di (i=1,2,3,…,N+1);

[0033] (6) g c0 、g ci 、g di A0, A in (i=1,2,3,…,N+1) i 、B i (i=1,2,3,…,N) perform first-order Taylor expansion, ignore the square terms of e and h and the higher-order terms above the square, and get A0, A i 、B i (i=1,2,3,…,N) expression;

[0034] (7) A N 、B N Substitute the expression of g into c(N+1) 、g d(N+1) In the equation, the two equations are subjected to first-order Taylor expansion to obtain e, Then solve the obtained e, Substitute back to A0, A i 、B i The expression of (i=1,2,3,…,N) is solved to get A0, A i、B i (i=1,2,3,…,N);

[0035] (8) By obtaining the 2N+5 parameters s, h, e, A0, A i 、B i (i=1,2,3,…,N), we can get the expressions of the sensor indication z(θ) and the profile f(α) of the circular cross section.

[0036] The detailed operations, formulas, and symbols of steps (1) to (8) are explained as follows:

[0037] like Figure 2 As shown, the rotation center of the turntable is point O, a plane rectangular coordinate system is established with point O as the origin, a polar coordinate system is established with point O', the centroid of the workpiece, the radius of the workpiece is R, the contour of the measured circular section of the workpiece is represented by F(α), the eccentricity of the workpiece is represented by E, and the initial eccentricity angle is The angle of the workpiece rotation is represented by θ, the distance between the sensor measurement direction and point O is H, the distance between the sensor installation position and the Y axis is S, the sensor indication is Z(θ), and the measured point of the workpiece is point M.

[0038] The coordinates of the measured point are expressed as (x s ,y s ), since the position detected by the sensor is parallel to the X axis, it is y s =H, so:

[0039]

[0040] Among them, α* is the phase angle of the workpiece measurement point, and φ is a small quantity close to 0. Use the two-angle sum and difference formula to sort out equations (1) and (2), and let α * =φ-θ, we get:

[0041]

[0042] Therefore, the sensor measurement theoretical reading is:

[0043]

[0044] Introducing dimensionless coefficients z = Z / R, h = H / R, f = F / R, e = E / R, s = S / R, then:

[0045]

[0046] In engineering practice, R>>(E,H,F), and equations (6) and (7) are expanded into three-variable third-order Taylor series (about e=0,h=0,f=0), and the high-order terms are omitted, resulting in equations (8) and (9):

[0047]

[0048] Expand the circular cross-section profile f(α) into N-level Fourier series:

[0049]

[0050] According to the example, α * =φ-θ is substituted into (10), and after rearrangement, it is substituted into (8) and subjected to first-order Taylor expansion (about e, h, f, φ=0), and assuming f(φ-θ)=0, Get a z(θ) about [A0 A1 B1...A N B N ] T The equation is named Q z (Take N=6 as an example, Q z Satisfies formula (11)).

[0051]

[0052] like Figure 3 The following is a schematic diagram of the sensor data z(θ). The sensor data z(θ) is fitted with an N+1-order Fourier series. A W×(2N+3) matrix F is constructed (in this example, there are W measuring points around the workpiece, so z(θ) has W data points; a vector θ is constructed: θ=[θ1θ2...θ W-1 θ W ] T ):

[0053] F=[1 cos(θ) sin(θ) cos(2θ) sin(2θ) ... cos[(N+1)θ] sin[(N+1)θ]] (12)

[0054] The data measured by the sensor is represented by vector z:

[0055] z=[z1 z2 ... z W-1 z W ] T (13)

[0056] Construct a 2N+3 vector C:

[0057] C=[C0 C1 D1 ... C N+1 D N+1 ] T (14)

[0058] F T FC=F T z (15)

[0059] C is the coefficient of the N+1 Fourier series fitting of the data measured by the sensor, which satisfies:

[0060] C=(F T F) -1 (F T z) (16)

[0061] By solving equation (16), we can obtain the coefficients C0, C1 of the N+1 Fourier series fitting of the sensor measurement data z(θ): i 、D i (I=1,2,3,…,N+1).

[0062] By rearranging the equation Q z , and obtain the coefficients of the Fourier series expansion. These coefficients are fitted with the N+1 level Fourier series of the sensor detection data (Formula 16) coefficients C0, C i 、D i (i=1,2,3,…,N+1) are equal, and we get the following equations, which are named g c0 、g ci 、g di (i=1,2,3,…,N+1):

[0063]

[0064] Where i∈[2,N-1], i∈N+.

[0065] In formulas (17) to (25), A0 and A i 、B i (i=1,2,3,…,N) performs a first-order Taylor expansion (about e=0,h=0). Since e and h are small quantities close to zero, the higher-order terms of e and h are even smaller quantities, so the square terms of e and h and higher-order terms above the square are ignored, and A0 and A are obtained. i 、B i The expression for (i=1,2,3,…,N):

[0066]

[0067]

[0068] Where i∈[2,N-1], i∈N+.

[0069] A will be expanded N 、B N Substitute g c(N+1) 、g d(N+1)In equations (24) and (25), these two equations are subjected to first-order Taylor expansion (about e = 0, h = 0). The following expressions are obtained (assuming the expansion level is N, N∈N+):

[0070]

[0071] From equations (33) and (34), we can get Further solving for e, Then solve the obtained e, Substituting into equations (17) to (25), we can get A0, A i 、B i (i=1,2,3,4,5,6).

[0072] So far, all 2N+5 unknown parameters have been solved: s, h, e, A0, A i 、B i (i=1,2,3,…,N). Substitute these solved parameters back into equation Q z The expression of the sensor indication z(θ) can be obtained.

[0073] The solved s, h, e, Substituting into formula (6), we can get the expression of the workpiece circular cross-section profile f(φ-θ):

[0074]

[0075] The measurement results are as follows: Figure 4 shown.

Claims

1. A method for accurately measuring and determining a circular cross-section profile and its circumferential harmonic components, characterized in that: The specific steps include: (1) Establishing a mathematical model for circular cross-section measurement , determine the relationship between the sensor reading Z (θ) and the workpiece radius R, the profile F of the circular section of the workpiece being measured, the workpiece eccentricity E, the initial eccentricity angle φ, the angle θ rotated by the workpiece, the distance H between the sensor measurement direction and the turntable rotation center, and the distance S between the sensor installation position and the Y axis, For a small quantity close to 0, introduce the dimensionless values ​​z = Z / R, h = H / R, f = F / R, e = E / R, s = S / R; (2) Input the measured data z(θ), θ, s, h, and set the profile of the dimensionless circular section to be measured Fourier expansion series N and coefficients , where z(θ), s, and h are the dimensionless values ​​obtained by dividing Z(θ), S, and H by R, respectively, and α is the phase angle; (3) Substitute the Fourier fitting formula into the theoretical expression of z(θ) , we get z(θ) about The equation Q z ; (4) Fit z(θ) with an N+1-level Fourier series to obtain the coefficients ; (5) Establish and The relationship between the equation g c0 、g ci 、g di , i=1,2,3,…, N+1; (6) g c0 、g ci 、g di ,A0、A in i=1,2,3,…, N+1 i 、B i , i=1,2,3,…,N, perform first-order Taylor expansion, ignore the square terms of e and h and the higher-order terms above the square, and get A0, A i 、B i , expressions where i=1,2,3,…,N; (7) A N 、B N Substitute the expression of g into c(N+1) 、g d(N+1) In the equation, perform first-order Taylor expansion on these two equations to obtain e and φ, and then substitute the obtained e and φ back into A0 and A i 、B i ,i=1,2,3,…,N expression is solved to get A0、A i 、B i , i=1,2,3,…,N; (8) By obtaining the 2N+5 parameters s, h, e, φ, A0, A i 、B i , i=1,2,3,…,N, the dimensionless sensor indication z(θ) and the dimensionless profile of the measured circular section can be obtained expression.

2. The method for accurately measuring and determining a circular cross-sectional profile and its circumferential harmonic components according to claim 1, characterized in that: In step (5), and The equation g is obtained by the relationship c0 、g ci 、g di , i=1,2,3,…, N+1 as follows: (17) (18) (19) (20) (21) (22) (23) (24) (25) Where i∈[2, N-1], i∈N+.

3. The method for accurately measuring and determining a circular cross-sectional profile and its circumferential harmonic components according to claim 1, characterized in that: In step (6), A0 and A i 、B i , the expression for i=1,2,3,…,N is as follows: (26) (27) (28) (29) (30) (31) (32) Where i∈[2, N-1], i∈N+.

4. The method for accurately measuring and determining a circular cross-sectional profile and its circumferential harmonic components according to claim 1, characterized in that: In step (7), A N 、B N Substitute the expression of g into c(N+1) 、g d(N+1) In the equation, the two equations are subjected to first-order Taylor expansion to obtain e and φ as follows: A will be expanded N 、B N Substitute g c(N+1) 、g d(N+1) In the equation, the two equations are expanded by first-order Taylor. With respect to e=0 and h=0, the whole solution is as follows. Assume that the expansion level is N, N∈N+: (33) (34) Ecos(φ) and esin(φ) obtained from equations (33) and (34) can be further solved to obtain e and φ.

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