Helical gears are critical components in mechanical transmission systems, and their spiral angle accuracy significantly impacts transmission smoothness, noise levels, and service life. Traditional methods for analyzing the spiral angle of helical gears, such as geometric projection-based measurements and contact probe-based detection, often fail to capture subtle variations in the spiral line and dynamic coupling effects between parameters. This leads to inaccuracies in calculating the tangent relationship between the pitch circle spiral angle and the tooth tip circle spiral angle, resulting in significant measurement uncertainties and errors in spiral angle deviation. To address these limitations, I propose a laser interference detection algorithm that deeply analyzes and derives the parametric equations of the spiral line, considers the complex characteristics and parameter interactions of helical gears, and obtains precise tangent values. By using a laser interferometer to measure relevant parameters and combining them with the tangent values, the actual spiral angle is calculated and compared with the standard value to determine the deviation. This approach ensures high-precision detection of spiral angle deviations in helical gears.
The tooth line of helical gears in processing machine tools consists of a series of complex parameterized spiral lines wrapped around a cylindrical surface. By placing the spiral line in a spatial Cartesian coordinate system, its parametric equations can be expressed as follows:
$$ x = r \times \cos\left(\theta + \theta_0 + \frac{(n-1) \times 2\pi}{c}\right) $$
$$ y = r \times \sin\left(\theta + \theta_0 + \frac{(n-1) \times 2\pi}{c}\right) $$
$$ z = \pm b \times \theta = \pm r \times \theta \times \cot(\beta) = \pm \frac{r \times \theta}{\tan(\beta)} $$
where \( r \) is the radius of the helical gear, \( c \) is the number of teeth, \( b \) is the thickness, \( \theta \) is the angular parameter ranging from \( -\frac{b}{r} \) to 0 radians, \( n \) is the number of spiral lines, \( \pi \) is the circular constant, \( \beta \) is the spiral angle of the helical gear (pitch circle), and \( \theta_0 \) is the initial angle. In the end-face coordinate system, the rotation angle of each spiral line of the helical gear is \( \frac{2\pi n}{c} \). When both \( r \) and \( b \) are negative, the helical gear rotates leftward; otherwise, it rotates rightward. The projection of the spiral line in the xoz plane approximates a straight line, but detailed analysis of the parametric equations reveals the functional relationship between \( x \) and \( z \):
$$ x = r \times \cos\left(\theta_0 + \frac{z \times \tan(\beta)}{r} + \frac{2\pi n}{c}\right) $$
Assuming the helical gear axis intersects the upper end face at point \( (x_0, z_0) \), the expressions for \( x \), \( y \), and \( z \) are derived as:
$$ z = b \times \theta $$
$$ x = r \times \cos\left(\theta_0 + \frac{(z – z_0) \times \tan(\beta)}{r} + \frac{2\pi n}{c}\right) $$
$$ y = r \times \sin\left(\theta_0 + \frac{(z – z_0) \times \tan(\beta)}{r} + \frac{2\pi n}{c}\right) $$
Typically, the tooth tip circle diameter of helical gears is twice the sum of the pitch circle diameter and the module. The tangent values of the pitch circle spiral angle \( \beta \) and the tooth tip circle spiral angle \( \beta_y \) are given by:
$$ \tan(\beta) = \frac{P_z}{\pi d} $$
$$ \tan(\beta_y) = \frac{P_z}{\pi d_y} $$
where \( d \) is the pitch circle diameter, \( d_y \) is the tooth tip circle diameter, and \( P_z \) is the lead of the spiral surface. Through in-depth derivation of the spiral line parametric equations, the tangent relationship between the pitch circle and tooth tip circle spiral angles is established, overcoming the computational challenges of traditional methods.
Although the tangent values provide a foundation for calculating the spiral angle, they alone are insufficient to determine the actual spiral angle due to ideal model assumptions and practical factors like machine tool accuracy and tool wear. Therefore, I employ a laser interferometer to measure relevant parameters such as module, number of teeth, rotation angle, and tooth width. The laser interferometer uses polarized beam frequencies \( f_1 \) and \( f_2 \), and the emitted beam is converted into a reference signal of \( f_1 – f_2 \). The interference mirror separates the beam, directing \( f_1 \) and \( f_2 \) to reflector 1 and reflector 2, respectively. After reflection, the beams are recombined and reflected back to the laser interferometer, forming a signal of \( f_1 \pm \Delta f \) after A/D conversion. Comparing this with the reference signal yields a \( \pm \Delta f \) pulse signal. The frequency of the reflected beam from the helical gear after laser interference is expressed as:
$$ f \pm \Delta f = f \times \frac{a – v}{a + v} \approx f \times \left(1 – \frac{2v}{a}\right) $$
where \( a \) is the speed of light, and \( v \) is the velocity of the moving component. Based on the movement distance \( l \) of the laser interferometer, the laser interference phase equation is derived as:
$$ s(t) = s_0(t) + \frac{\lambda}{2\pi} \times \arctan\left(\frac{C \times \sin(s_F(t) + \alpha)}{1 + C \times \cos(s_F(t) + \alpha)}\right) $$
where \( \lambda \) is the wavelength of the laser reflected by the helical gear, \( s_0(t) \) is the laser external cavity phase without optical feedback, \( s_F(t) \) is the phase with optical feedback, \( C \) is the optical feedback coefficient, and \( \alpha \) is the linewidth broadening factor of the helical gear. Setting the sampling frequency of the laser interferometer to 50 kHz, I measure and collect signals such as module, number of teeth, rotation angle, and tooth width of the helical gear using the phase equation \( s_0(t) \). These are input into MATLAB for spiral angle calculation, and the deviation is obtained by comparing with the standard value.
To compute the spiral angle, each tooth of the helical gear is connected by straight lines to form a convex polygon. The laser interferometer refracts this polygon onto a reflector, yielding the optical feedback coefficient \( C \), which is fitted using the least squares method to obtain a circular curve of radius \( R_k \). Assuming the center of this circle is \( (x’, y’) \) and the radius of the maximum inscribed circle is \( R_j \), the module \( m \) of the helical gear is calculated as:
$$ m = \frac{R_k – R_j}{M} $$
where \( M \) is the standard module value from the module table, set to 2.25. The number of teeth \( c \) is determined by refracting a circle with center \( (x’_1, y’_1) \) and radius \( \frac{R_k + R_j}{2} \) onto the helical gear image using the laser interferometer. After processing and binarization, the tooth profile is obtained, and the actual number of teeth is identified. The rotation angle \( \gamma \) is the minimum angle rotated about the gear center on the upper end face until the tangent values of the pitch circle spiral angle and tooth tip circle spiral angle are \( \tan(\beta) \) and \( \tan(\beta_y) \), respectively. It is computed using the cosine theorem:
$$ \gamma = \arccos\left(\frac{s_1^2 + s_2^2 – s_3^2}{2 s_1 s_2}\right) $$
where \( s_1 \) is the distance between the upper end face center coordinates and the circle center coordinates, and \( s_2 \) is the distance between the lower end face center coordinates and the circle center coordinates. The tooth width \( h \) is measured with a vernier caliper. Using \( m \), \( c \), \( \gamma \), and \( h \), the actual spiral angle \( \beta_{\text{actual}} \) of the helical gear is calculated as:
$$ \tan(\beta_{\text{actual}}) = \frac{c \times m \times \gamma \times \pi}{360 \times h} $$
The deviation value \( \Delta \beta \) is then obtained by comparing with the standard spiral angle \( \beta_{\text{std}} \):
$$ \Delta \beta = \beta_{\text{actual}} – \beta_{\text{std}} $$
To validate the proposed algorithm, I conducted comparative experiments with traditional methods, such as the “roll-grinding” process calculation method and parameter visual detection method. The experimental setup included helical gears with a tooth tip circle diameter of 32.00 mm, pitch circle diameter of 35.66 mm, 25 teeth, module of 1.25 mm, and material of 45 steel. Measurement devices included a laser interferometer (Renishaw XL-80, accuracy ±0.1 μm), a high-precision rotary table (accuracy ±1″), and an industrial camera (resolution 5 megapixels) for parameter visual detection. The experimental procedure involved data collection under simulated operational conditions, with artificial deviations introduced at specific intervals to test detection accuracy.
The results demonstrated the effectiveness of the proposed method. For instance, in 10 detection experiments, the spiral angle deviation values measured by the proposed method were consistently small and stable, as summarized in the following table:
| Experiment Number | Actual Spiral Angle \( \beta_{\text{actual}} \) (°) | Standard Spiral Angle \( \beta_{\text{std}} \) (°) | Deviation Value \( \Delta \beta \) (°) |
|---|---|---|---|
| 1 | 19.41 | 19.32 | 0.09 |
| 2 | 19.40 | 19.32 | 0.08 |
| 3 | 19.42 | 19.32 | 0.10 |
| 4 | 19.43 | 19.32 | 0.11 |
| 5 | 19.42 | 19.32 | 0.10 |
| 6 | 19.44 | 19.32 | 0.12 |
| 7 | 19.45 | 19.32 | 0.13 |
| 8 | 19.40 | 19.32 | 0.08 |
| 9 | 19.41 | 19.32 | 0.09 |
| 10 | 19.42 | 19.32 | 0.10 |
The relative error \( E_r \) was calculated to compare the performance of the three algorithms over 60 experiments, using the formula:
$$ E_r = \frac{|l – l_0|}{l_0} \times 100\% $$
where \( l_0 \) is the actual deviation value and \( l \) is the algorithm-detected value. The proposed method exhibited the smallest relative errors, followed by the parameter visual detection method, and the “roll-grinding” process calculation method had the largest errors. This confirms that the proposed method achieves precise detection of helical gear spiral angle deviations with minimal relative error.
Additionally, in 50 detection experiments with artificial deviations introduced at specific intervals (e.g., experiments 10, 15, 20, 25, 35, 40, 45), the proposed method consistently identified and quantified spiral angle deviations with high stability and accuracy. In contrast, the “roll-grinding” process calculation method and parameter visual detection method showed varying degrees of detection bias under complex conditions, indicating lower robustness. To evaluate stability in complex environments, 10 spiral angle deviation detection experiments were conducted under simulated uncertain factors like temperature changes and vibrations. The fluctuation ranges of deviation values for the three methods were as follows: the proposed method within ±1.5°, the “roll-grinding” process calculation method within ±3.5°, and the parameter visual detection method within ±5.5°. This demonstrates the superior stability of the proposed method for helical gears in practical applications.
In conclusion, the laser interference detection algorithm for spiral angle deviation of helical gears in processing machine tools offers a high-precision and stable solution. By deriving the parametric equations of the spiral line and integrating laser interferometer measurements, it accurately calculates the spiral angle deviation, with a maximum relative error not exceeding 0.13° and deviation fluctuations within ±1.5°. This method effectively addresses the limitations of traditional approaches and ensures the machining accuracy of helical gears, making it suitable for industrial applications where precision is paramount. The repeated emphasis on helical gears throughout this study underscores their importance in mechanical systems and the need for reliable detection methods.