A Fourier analysis method for tooth surface waviness based on least squares sinusoidal fitting algorithm

Abstract

This invention discloses a tooth surface waviness Fourier analysis method based on a least-squares sine fitting algorithm. Fourier analysis is performed on the tooth surface waviness measured at the gear measurement center, and the spectrum is calculated using the least-squares sine fitting algorithm. The method includes the following steps: Step 1: Acquiring the tooth surface waviness based on the gear measurement center; Step 2: Using a Fourier decomposition algorithm based on the least-squares sine fitting function; Step 3: Determining the order; and Step 4: Calculating the order spectrum. This method addresses the issues of overlap and unequal spacing in gear waviness Fourier analysis and, based on the analysis results, provides guidance on gear processing and gear meshing noise.

Description

Technical Field

[0001] The invention relates to a tooth surface waviness Fourier analysis method based on a least square sine fitting algorithm, and belongs to the technical field of involute cylindrical gear fitting. Background Art

[0002] In the fields of signal processing and mechanical vibration analysis, Fourier analysis is a widely used mathematical tool that can decompose complex signals or functions into a series of simple sine or cosine waves, thereby revealing their frequency characteristics. Fourier analysis is divided into two main forms: Fourier series and Fourier transform. Fourier series is applicable to periodic signals, while Fourier transform is applicable to non-periodic signals. By converting the time domain signal to the frequency domain, Fourier analysis can more clearly display the frequency components of the signal. The order represents the number of vibration events that occur per rotation (360°) of the rotating part, which corresponds to the frequency in conventional vibration analysis. This analysis method is particularly important when dealing with gear noise problems, and it also plays an important role in studying the influence of gear tooth surface errors on gear meshing noise.

[0003] Traditional Fourier transform algorithms are designed for equally spaced data. However, in real-world applications, data points are often not equally spaced, making the direct application of the FFT difficult. Performing a Fourier transform (FFT) on unevenly spaced data requires interpolation to evenly space the data before performing the FFT, significantly increasing the complexity of data processing.

[0004] To address these issues, this paper proposes a Fourier analysis method for tooth surface waviness based on a least-squares sinusoidal fitting algorithm, designed to analyze the periodic components in gear waviness. By performing Fourier analysis on the gear waviness measured at the gear measurement center, a detailed assessment of tooth surface periodic errors can be performed. This method provides accurate data support for gear quality assessment and manufacturing processes. Summary of the Invention

[0005] The purpose of this invention is to propose a Fourier analysis method for tooth surface waviness based on the least squares sine fitting algorithm. The tooth surface waviness measured by the gear measurement center is subjected to Fourier analysis, and the spectrum result is calculated by the least squares sine fitting algorithm.

[0006] The present invention proposes a tooth surface waviness Fourier analysis method based on least squares sinusoidal fitting, which specifically includes the following method steps:

[0007] Step 1: Obtaining the tooth surface waviness based on the gear measurement center

[0008] Tooth surface waviness is the result of the combined influence of geometric features such as tooth surface shape error, surface waviness, and surface roughness. Tooth surface waviness can be divided into two types: tooth profile waviness along the tooth profile direction and helical line waviness along the helix direction. The full tooth surface waviness of a gear is the comprehensive result of superimposing the pitch error along the involute generation line onto the tooth surface waviness based on the gear meshing principle and the generation principle of the involute, referred to as gear waviness. Gear waviness, including tooth profile waviness and helical line waviness, is obtained based on the gear measurement center measurement, providing fitting data for the subsequent Fourier decomposition algorithm based on the least squares fitting sine function.

[0009] Step 2: Fourier decomposition algorithm based on least squares fitting of sine function

[0010] The least squares method is a commonly used data fitting algorithm. Its goal is to find the optimal fitting parameters by minimizing the sum of squared residuals between the observed data and the fitted function. When fitting a sine function, the least squares method can be used to determine parameters such as the amplitude, frequency, and phase of the sine function. The specific steps for sine function fitting are as follows:

[0011] S1. Establish a least squares function fitting model

[0012]

[0013] Where A is the amplitude of the sine function, which represents the maximum amplitude of the fluctuation, and λ is the wavelength. is the phase offset, which determines the translation of the sine wave on the x-axis, and B is the vertical offset, which indicates the center position of the sine wave.

[0014] S2. Define residual

[0015] Given a set of data points (x i ,y i ), (where i = 1, 2, ..., n), the goal of the least squares method is to select parameters A, λ, B, so that the observed value y i and the model predicted value f(x i ) is the smallest residual sum of squares. i Defined as:

[0016]

[0017] S3. Minimize the residual sum of squares

[0018] Minimize the objective function, the residual sum of squares:

[0019]

[0020] S4. Establish the tooth surface lowest square sine fitting algorithm

[0021] According to the least square function fitting model established by formula (1), a sine function is fitted to the tooth surface waviness data, and the fitting model is established as follows:

[0022]

[0023] Where λ k is the wavelength, A k The wavelength is λ k The corresponding amplitude is The wavelength is λ k The initial phase at time x j is the horizontal coordinate of the tooth profile point, y j is the error value of the measured tooth profile point.

[0024] According to formula (4), the result of sine fitting is calculated and the measurement point x is j is arbitrary and does not need to be equally spaced.

[0025] Step 3: Solve for the order

[0026] Solve formula (4) by applying Euler's formula:

[0027] 2·i·sin(x)=exp(i·x)-exp(-i·x)(5)

[0028] In the formula is a complex unit, exp(i·x)—complex Euler exponential function.

[0029] In combination with Euler's formula (5), the fitting formula (4) can be significantly simplified. By minimizing the sum of squared deviations, it can be simplified to maximizing a matrix product that depends only on the unknown wavelength λ k , the simplified result is:

[0030]

[0031] In the formula

[0032] H represents the formation of the Hermite matrix, that is, the vector or matrix needs to be transposed. Maximization of Equation (3) is either performed by iteration or by determining the wavelength grid that maximizes (3). If the wavelength λ max It is determined that the amplitude and initial phase can be determined from the following relations:

[0033]

[0034]

[0035] In the formula

[0036] If only wavelengths that are integer multiples of the measurement distance are considered, and further restricted to a uniform measurement grid, Equation (6) can be simplified to the following expression:

[0037]

[0038] In the formula

[0039] According to the above given λ max , calculate A and values, which are the solutions of the discrete Fourier transform. Therefore, the algorithm introduced can be regarded as an extension of the discrete Fourier transform and is applicable to arbitrary wavelengths and measurement intervals.

[0040] Step 4: Calculate the order spectrum

[0041] The most critical step in the Fourier analysis of tooth surface is to calculate the amplitude of the fitted sine wave function within the selected frequency range. max The fitted sine function is considered the first main order. This main sine wave function is then removed from the waviness curve, and the remaining waviness curve is refitted. The compensating sine wave with the second largest amplitude is considered the second main order. All spectrum results are calculated sequentially. The steps are as follows:

[0042] S1. Determine the wavelength of the fitted sine function

[0043] Assuming that the data of the first m orders are calculated, the wavelength of the corresponding sine function is:

[0044] λ={λ1,λ1,…,λ m}(10)

[0045] S2. Solve the amplitude

[0046] Calculate the corresponding amplitudes at this time:

[0047] A={A1,A2,…,A m} (11)

[0048] S3. Initial phase solution

[0049] The corresponding initial phase is:

[0050]

[0051] S4. Calculate the spectrum

[0052] The spectrum results are:

[0053] Q={(1,A1),(2,A2),…,(m,A m )}(13)

[0054] The present invention proposes a tooth surface waviness Fourier analysis method based on the least squares sine fitting algorithm to solve the problems of overlap and unequal spacing in gear waviness Fourier analysis. According to the analysis results, guiding opinions on gear processing and gear meshing noise are put forward. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 Gear measuring center.

[0056] Figure 2 The tooth profile waviness is the waviness of the tooth surface.

[0057] Figure 3 is the single tooth pitch deviation of the gear.

[0058] Figure 4 It is the waviness of the tooth profile of a single tooth surface.

[0059] Figure 5 It is the waviness of the tooth profile of the entire gear tooth surface.

[0060] Figure 6 This is the result of the spectrum analysis of the tooth profile waviness of a single tooth surface.

[0061] Figure 7 It is the first main order sinusoidal fitting of the gear tooth surface waviness.

[0062] Figure 8 It is the second main order sinusoidal fitting of the gear tooth surface waviness.

[0063] Figure 9 This is the result of the gear tooth surface waviness spectrum analysis. DETAILED DESCRIPTION

[0064] The present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments. However, this should not be construed as limiting the scope of the present invention to the following embodiments. All technologies implemented based on the present invention fall within the scope of the present invention.

[0065] The basic parameters of the selected characteristic gears are shown in Table 1.

[0066] Table 1 Parameters of the gears tested

[0067] parameter Number of teeth Modulus pressure angle Helix angle Rotation Reshaping Tooth width Numerical 21 2.36mm 19° -19.4° Left-handed Middle drum 52mm

[0068] Step 1: Obtaining the tooth surface waviness based on the gear measurement center

[0069] Klingelnberg P26 is selected as the measuring device, such as Figure 1 As shown. The probe diameter is selected to be 1mm to ensure that the full tooth surface waviness of the gear can be measured. In this embodiment, the tooth profile waviness is analyzed, and the spiral line waviness analysis method is the same. The tooth profile waviness is as follows Figure 2As shown. The single pitch deviation is as follows Figure 3 shown.

[0070] Table 2 Evaluation parameters of the tested gears

[0071] Measurement location Diameter of starting point Final measuring point diameter Starting point diameter Final evaluation point diameter Numerical 49.459mm 60.237mm 49.794mm 58.154mm

[0072] The waviness of a single tooth profile includes information such as tooth surface shape error, waviness and roughness, such as Figure 4 As shown. The tooth profile waviness is as follows Figure 5 shown.

[0073] Step 2: Fourier decomposition algorithm based on least squares fitting of sine function

[0074] The Fourier decomposition algorithm based on least squares fitting of the sine function is adopted. The specific steps of sine function fitting are as follows:

[0075] S1. Establish a least squares function fitting model

[0076]

[0077] Where A is the amplitude of the sine function, which represents the maximum amplitude of the fluctuation, and λ is the wavelength. is the phase offset, which determines the translation of the sine wave on the x-axis, and B is the vertical offset, which indicates the center position of the sine wave.

[0078] S2. Define residual

[0079] According to the measured waviness error data point (x i ,y i ), (where i = 1, 2, ..., n), by selecting parameters A, λ, B, so that the observed value y i and the model predicted value f(x i ) is the smallest residual sum of squares. i Defined as:

[0080]

[0081] S3. Minimize the residual sum of squares

[0082] Minimize the objective function, the residual sum of squares:

[0083]

[0084] S4. Establish the tooth surface lowest square sine fitting algorithm

[0085] A sine function is fitted to the measured data of the gear tooth surface, and the fitting model is established as follows:

[0086]

[0087] Where λ k is the wavelength, A k The wavelength is λ k The corresponding amplitude is The wavelength is λ k The initial phase at time x j is the horizontal coordinate of the tooth profile point, y j is the error value of the measured tooth profile point.

[0088] Step 3: Solve for the order

[0089] By applying Euler's formula:

[0090] 2·i·sin(x)=exp(i·x)-exp(-i·x)

[0091] In the formula is a complex unit, exp(i·x)—complex Euler exponential function.

[0092] By minimizing the sum of squared deviations, this can be simplified to maximizing a matrix product that depends only on the unknown wavelength λ k , the simplified result is:

[0093]

[0094] If the wavelength λ max It is determined that the amplitude and initial phase can be determined from the following relations:

[0095]

[0096]

[0097] If we only consider wavelengths that are integer multiples of the measurement distance, and further restrict ourselves to a uniform measurement grid, we can simplify to the following expression:

[0098]

[0099] According to the above given λ max , calculate A and values, which are the solutions of the discrete Fourier transform.

[0100] Step 4: Calculate the order spectrum

[0101] Maximum amplitude A max The fitted sine function is considered the first main order. This main sine wave function is then removed from the waviness curve, and the remaining waviness curve is refitted. The compensating sine wave with the second largest amplitude is considered the second main order. All spectrum results are calculated sequentially. The steps are as follows:

[0102] S1. Determine the wavelength of the fitted sine function

[0103] Assuming that the data of the first order is calculated, the wavelength of the corresponding sine function is:

[0104] λ={λ1,λ1,…,λ m}

[0105] S2. Solve the amplitude

[0106] Calculate the corresponding amplitude at this time

[0107] A={A1,A2,…,A m}

[0108] S3. Solve the initial phase

[0109] The corresponding initial phase is:

[0110]

[0111] S4. Calculate the spectrum

[0112] The spectrum results are:

[0113] Q={(1,A1),(1,A2),…,(m,A m )}

[0114] Figure 4 is the waviness of the tooth profile of a single tooth surface. Figure 4 The single tooth profile waviness is shown in the figure. The sine fitting analysis method used above is used to calculate the single tooth profile waviness spectrum analysis result. Figure 6 shown.

[0115] Figure 5 is the tooth profile waviness. In the Fourier analysis of tooth profile waviness, the maximum amplitude is A max The compensated sine wave of is considered as the first main order, such as Figure 7 As shown. Subsequently, this main sine wave function is removed from the waviness curve, and the remaining waviness curve is fitted and analyzed. The compensating sine wave with the second largest amplitude is considered to be the second main order, as shown in Figure 8 As shown. Plot all the spectrum results in turn, and we get Figure 9 The results of the waviness spectrum analysis of the entire gear tooth surface are shown.

[0116] The above description of the disclosed embodiments is intended to enable one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention is not limited to the embodiments shown herein but is intended to conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A tooth surface waviness Fourier analysis method based on the least squares sine fitting algorithm, characterized in that: The specific steps include: Step 1: Acquire the tooth surface waviness based on the gear measurement center; Step 2: Fourier decomposition algorithm based on least squares fitting of sine function; The specific steps are as follows: S1. Establish a least squares function fitting model; Where A is the amplitude of the sine function, which represents the maximum amplitude of the fluctuation, and λ is the wavelength. is the phase offset, which determines the translation of the sine wave on the x-axis, and B is the vertical offset, which indicates the center position of the sine wave; S2. Define residual; Given a set of data points (x i ,y i ), where i = 1, 2, ..., n, the goal of the least squares method is to select parameters A, λ, B, so that the observed value y i and the model predicted value f(x i ) is the smallest residual sum of squares; the residual R i Defined as: S3. Minimize the residual sum of squares; Minimize the objective function, the residual sum of squares: S4. Establish the tooth surface lowest square sine fitting algorithm; According to the least square function fitting model established by formula (1), a sine function is fitted to the tooth surface waviness data, and the fitting model is established as follows: Where λ k is the wavelength, A k The wavelength is λ k The corresponding amplitude is The wavelength is λ k The initial phase at time x j is the horizontal coordinate of the tooth profile point, y j is the error value of the measured tooth profile point; According to formula (4), the result of sine fitting is calculated and the measurement point x is j It is arbitrary and does not need to be equally spaced; Step 3: Solve the order; Solve formula (4) by applying Euler's formula: 2·i·sin(x)=exp(i·x)-exp(-i·x)(5) In the formula is a complex unit, exp(i·x)—complex Euler exponential function; Combining Euler formula (5) and fitting formula (4); by minimizing the sum of squares of deviations, it is simplified to maximize a matrix product, which depends on the unknown wavelength λ k , the simplified result is: In the formula H represents the formation of the Hermite matrix, that is, the vector or matrix needs to be transposed; the maximization of formula (3) is performed by iteration or by determining the wavelength grid that maximizes (3); if the wavelength λ max Determine the amplitude and initial phase from the following relations: In the formula If only wavelengths that are integer multiples of the measurement distance are considered and the measurement grid is restricted to a uniform one, equation (6) simplifies to the following expression: In the formula According to the given λ max Calculate A and values, which are the solutions of the discrete Fourier transform; Step 4: Calculate the order spectrum; The key to Fourier analysis of tooth surface is to calculate the amplitude of the fitted sine wave function within the selected frequency range; Maximum amplitude A max The fitted sine function is regarded as the first main order; then, the main sine wave function is removed from the waviness curve, and the remaining waviness curve is refitted; the compensating sine wave with the second largest amplitude is regarded as the second main order; all spectrum results are calculated in sequence.

2. The tooth surface waviness Fourier analysis method based on the least squares sine fitting algorithm according to claim 1, characterized in that: Tooth surface waviness is the result of the combined influence of tooth surface shape error, surface waviness and surface roughness geometric characteristics; tooth surface waviness is divided into tooth profile waviness along the tooth profile direction and helix waviness along the helix direction; the full tooth surface waviness of the gear is the comprehensive result of superimposing the pitch error along the generation line direction of the involute on the tooth surface waviness based on the gear meshing principle and the generation principle of the involute, referred to as gear waviness; the gear waviness is obtained based on the gear measurement center measurement, including tooth profile waviness and helix waviness, providing fitting data for the subsequent Fourier decomposition algorithm based on least squares fitting of the sine function.

3. The tooth surface waviness Fourier analysis method based on the least squares sine fitting algorithm according to claim 1, characterized in that: The steps for step 4 are as follows: S1. Determine the wavelength of the fitted sine function; Assuming that the data of the first m orders are calculated, the wavelength of the corresponding sine function is: λ={λ1,λ1,…,λ m }(10) S2. Solve the amplitude; Calculate the corresponding amplitudes at this time: <h2 style=";text-align:left;direction:ltr">A={A1,A2,…,A<h2 style=";text-align:left;direction:ltr"> m <h2 style=";text-align:left;direction:ltr">} (11) S3. Initial phase solution; The corresponding initial phase is: S4. Solve the spectrum; The spectrum results are: Q={(1,A1),(2,A2),…,(m,A m )}(13)。

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