In the machining of small module gears, adjusting machine parameters according to different part specifications is essential. After confirming gear accuracy through specialized testing equipment and personnel, the machine can be locked for batch processing. For cylindrical gears with a module above 0.3, gear testing centers quickly and accurately output results. However, for straight bevel gears and cylindrical gears with a module below 0.3, these exceed the capabilities of gear testing centers. In such cases, traditional universal tool microscopes combined with tooth shape coordinate points are used to detect tooth profile accuracy. With technological advancements, new designs are increasingly applied in practice. For machining height modification straight bevel gears, accurately determining whether the processed tooth profile meets standards requires providing reliable tooth shape coordinate point information for inspection. This paper addresses these challenges by proposing a calculation method for tooth shape coordinate points of height modification straight bevel gears and verifying its correctness through CAD-assisted drawings, filling a gap in small module height modification bevel gear detection and ensuring product quality.
The tooth profile accuracy is a critical indicator for assessing gear qualification, directly reflecting the smoothness of gear meshing. Various methods exist for detecting tooth profiles, including advanced gear testing centers, projection magnification comparison, and universal tool microscopes. The underlying principle remains consistent: comparing the actual tooth profile with the theoretical one. The actual tooth profile is fitted from points on the tooth contour in a coordinate system where the gear rotation center is the origin, and the line connecting the midpoint of one tooth’s tip to the origin is the Y-axis. Calculating tooth shape detection points involves determining the theoretical positions of these points in this coordinate system. Once theoretical positions are established, using a universal tool microscope for tooth profile detection involves dividing the middle portion from the tooth tip to the root along the Y-axis into equal segments based on accuracy requirements. Starting from the tooth tip, at each division point, the X-values of points on both tooth flanks in this coordinate system are measured, recorded as absolute values, and compared with theoretical values. The differences are noted, and after calculating differences for all Y-axis division points, the tooth profile accuracy for the left flank is the maximum difference minus the minimum difference, and similarly for the right flank. The highest value between the two is taken as the final tooth profile error.
For straight bevel gears, the tooth profile is theoretically a spherical involute. To simplify design and manufacturing, the concept of the back cone angle is introduced, approximating the spherical surface as a conical surface and transforming the spherical involute into an involute within the back cone plane, commonly referred to as the large-end tooth profile. To facilitate tooth shape calculation, the back cone surface is developed into a plane, and the gear teeth on this surface are approximated as part of a virtual straight cylindrical gear. The number of teeth required to complete this virtual gear is denoted as the virtual tooth number \( z_y \), thus simplifying straight bevel gear tooth shape calculation to that of a straight gear. By substituting the virtual tooth number \( z_y \) into the straight gear tooth shape calculation formula, the required values can be derived. The formula for \( z_y \) is:
$$ z_y = \frac{z}{\cos \phi_f} $$
where \( \phi_f \) is the pitch cone angle and \( z \) is the number of teeth. For height modification straight bevel gears, the tooth shape calculation involves treating the modified tooth profile as that of a non-modified gear and computing the forming module \( m’ \) and tooth number \( z’ \) to determine the coordinate points. Since the base circle remains unchanged for height-modified straight gears, the involute shape generated from this base circle also remains the same, with changes only in addendum, dedendum, and tooth thickness at the pitch circle. Thus, by finding the relationship between the modified and non-modified tooth profiles, the modified tooth shape points can be calculated, with the fixed chord tooth thickness serving as the connection point.
The fixed chord tooth thickness \( S \) for a modified gear varies with the modification coefficient \( \xi \). By considering \( S \) as the non-modified fixed chord tooth thickness determined by \( m’ \) and \( z’ \), the corresponding relationship can be established as follows. The fixed chord tooth thickness for a modified gear is:
$$ S = \left( \frac{\pi}{2} \cos^2 \alpha_f \pm \xi \sin 2\alpha_f \right) m $$
The fixed chord tooth thickness for a gear with parameters \( m’ \) and \( z’ \) is:
$$ S = \frac{\pi m’}{2} \cos^2 \alpha_f $$
From these equations, we derive:
$$ m’ = m \left( 1 \pm \frac{4\xi \tan \alpha_f}{\pi} \right) $$
$$ z’ = \frac{d_f}{m’} = \frac{z}{1 \pm \frac{4\xi \tan \alpha_f}{\pi}} $$
where \( d_f \) is the pitch diameter and \( \alpha_f \) is the pressure angle. Since the pitch circle position remains unchanged when calculating with \( m’ \) and \( z’ \), the coordinate origin should be adjusted by a distance \( \Delta h \) from the pitch circle, differing from the modified gear’s theoretical addendum \( h_e \). The calculation for \( \Delta h \) is:
$$ \Delta h = h_e – m’ = m(f \pm \xi) – m’ $$
where \( f \) is the addendum coefficient. For tooth shape detection, points are selected at specific descent heights from the tooth tip. The descent height \( h_x \) is given by:
$$ h_x = \Delta h + P_x m’ $$
where \( P_x \) is the descent coefficient. The corresponding half chord tooth thickness can be calculated using \( m’ \) and \( z’ \) or looked up in involute gear tooth shape coordinate tables and multiplied by \( m’ \).
To verify the correctness of this calculation method for height modification straight bevel gears, CAD software is used to draw a tooth of the straight bevel gear after developing the back cone surface. The chord tooth thickness at different descent heights is measured and compared with calculated data. Consider a straight bevel gear with module \( m = 0.6 \), teeth \( z = 15 \), pressure angle \( \alpha_f = 20^\circ \), pitch cone angle \( \phi_f = 30.96^\circ \), and modification coefficient \( \xi = 0.3 \). First, calculate the virtual tooth number \( z_y \):
$$ z_y = \frac{15}{\cos 30.96^\circ} \approx 17.49 $$
Then compute \( m’ \) and \( z’ \):
$$ m’ = 0.6 \left( 1 + \frac{4 \times 0.3 \times \tan 20^\circ}{\pi} \right) \approx 0.6834 $$
$$ z’ = \frac{17.49}{1 + \frac{4 \times 0.3 \times \tan 20^\circ}{\pi}} \approx 15.687 $$
Next, determine \( \Delta h \):
$$ \Delta h = 0.6 \times (1 + 0.3) – 0.6834 = 0.0966 $$
Select three points with descent coefficients \( P_1 = 0.1 \), \( P_2 = 0.5 \), and \( P_3 = 1 \) to define the tooth shape. The descent heights from the tooth tip are:
$$ h_1 = 0.0966 + 0.1 \times 0.6834 \approx 0.1649 $$
$$ h_2 = 0.0966 + 0.5 \times 0.6834 \approx 0.4383 $$
$$ h_3 = 0.0966 + 1 \times 0.6834 \approx 0.78 $$
The corresponding half chord tooth thickness values are calculated or looked up and multiplied by \( m’ \). The calculated half chord tooth thicknesses are approximately 0.2628 mm, 0.4049 mm, and 0.5285 mm, respectively.
In CAD, draw the base circle, pitch circle, and tip circle for a straight gear with \( m = 0.6 \) mm, \( z_y = 17.49 \), and modification coefficient 0.3. Plot the base circle involute curve \( a \) intersecting the pitch circle at point A. Mirror curve \( a \) across the X-axis to get curve \( b \) intersecting the pitch circle at point B. Offset the X-axis bilaterally by half the pitch chord tooth thickness of this modified straight bevel gear (1.0716 / 2), intersecting the pitch circle at points C and D. Rotate curve \( a \) from point A to point D around the center, and curve \( b \) from point B to point C. After removing unnecessary lines, draw a perpendicular to the X-axis at the midpoint of the tip arc, and create three parallel lines at intervals \( h_1 \), \( h_2 \), and \( h_3 \). Measure the distances between the intersection points of these lines with the left and right tooth profiles (half the distance between intersection points is recorded). The measured half chord tooth thicknesses are 0.26 mm, 0.40 mm, and 0.53 mm, respectively.
The results are summarized in the table below. Excluding drawing accuracy errors, the coordinate values align closely, with negligible errors in practical measurement, confirming the correctness of the calculation method for tooth shape coordinate points of height modification straight bevel gears.
| Interval | Descent Height (mm) | Calculated Half Chord Tooth Thickness (mm) | Measured Half Chord Tooth Thickness (mm) |
|---|---|---|---|
| h1 | 0.1649 | 0.2628 | 0.26 |
| h2 | 0.4383 | 0.4049 | 0.40 |
| h3 | 0.78 | 0.5285 | 0.53 |
The accurate detection of tooth profiles in straight bevel gears is crucial for ensuring meshing performance and longevity. For height modification straight bevel gears, the calculation of tooth shape coordinate points involves determining the virtual tooth number \( z_y \), then computing the forming module \( m’ \) and tooth number \( z’ \) using the original module and \( z_y \). By substituting these into formulas and incorporating descent height coefficients \( P \), the coordinate point values needed for tooth shape measurement can be obtained. This method is reliable for inspection tasks, particularly for small module straight bevel gears where traditional gear testing centers are inadequate.
In practical applications, the straight bevel gear’s tooth profile must be meticulously verified to prevent inefficiencies in power transmission. The use of universal tool microscopes, combined with calculated coordinate points, provides a cost-effective solution for small-scale production or prototyping. Moreover, the integration of CAD software for validation enhances confidence in the calculated points, reducing the risk of errors in mass production. The straight bevel gear’s geometry, especially under height modification, requires precise calculations to account for changes in tooth thickness and addendum. The derived formulas facilitate this by transforming the complex straight bevel gear into a manageable virtual cylindrical gear model.
Further considerations include the impact of manufacturing tolerances on tooth profile accuracy. For instance, deviations in the pitch cone angle or module can affect the virtual tooth number and subsequent calculations. Regular calibration of measurement instruments is essential to maintain accuracy. Additionally, the straight bevel gear’s alignment during inspection must be perfect to avoid skewed readings. The proposed method simplifies this by relying on a standardized coordinate system, making it accessible for technicians without advanced gear testing equipment.
In summary, the calculation of tooth shape detection points for height modification straight bevel gears involves a systematic approach that leverages virtual gear parameters and descent height adjustments. The verification through CAD drawings confirms the method’s validity, ensuring that inspected gears meet required standards. This approach not only fills a gap in small module straight bevel gear detection but also promotes quality assurance in industries relying on precision gears, such as aerospace and military equipment. As technology evolves, this method can be integrated with digital twin simulations for real-time monitoring and adjustment during machining, further enhancing the straight bevel gear’s performance and reliability.