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 centers exceed their detection capabilities. In such cases, traditional universal tool microscopes paired with tooth shape coordinate points are used to assess tooth profile accuracy. With technological advancements, new designs are increasingly applied. When machining height-modified straight bevel gears, accurately determining whether the processed tooth profile meets standards requires providing reliable tooth shape coordinate data for inspection. This paper addresses these challenges by proposing a calculation method for tooth shape coordinate points of height-modified straight bevel gears and verifying its correctness through CAD-assisted drawing, filling a gap in small-module height-modified bevel gear detection and ensuring product quality.
The tooth profile accuracy is a critical indicator of 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 involves comparing the actual tooth profile with the theoretical one. The actual 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 entails determining the theoretical positions of these points in this coordinate system. Once theoretical positions are established, using a universal tool microscope involves dividing the middle portion from the tooth tip to the root equally along the Y-axis into several segments based on accuracy requirements. At each division point from the tip downward, the X-values of points on both tooth flanks are measured in this coordinate system, recorded as absolute values, and compared with theoretical values. The difference is calculated for all Y-axis division points. The tooth profile accuracy for the left flank is the maximum 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. However, due to design complexity and manufacturing difficulties, the back cone angle concept is introduced to simplify design and production. The spherical surface is approximated to a conical surface, 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 developed surface are approximated as part of a virtual straight cylindrical gear. The number of teeth required to complete this virtual cylindrical gear is denoted as \( z_y \), the virtual tooth count. This simplification allows straight bevel gear tooth shape calculations to be treated as straight gear calculations, substituting the tooth count \( z \) with \( z_y \). 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-modified straight bevel gears, the calculation of tooth shape detection points translates into that of a modified straight gear with the original module and virtual tooth count as primary parameters. When calculating tooth shape coordinates for a modified gear, the modified tooth profile is considered as that of a non-modified gear, and the module \( m’ \) and tooth count \( z’ \) forming this profile are computed. Substituting these into the calculation formula yields the required coordinate points. In height-modified straight gears, the base circle remains unchanged, so the involute shape generated from it also remains constant. The changes pertain to the 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 derived, with the fixed chordal tooth thickness serving as this connection point.
The fixed chordal tooth thickness \( S \) of a modified gear varies with the modification coefficient \( \xi \). Treating the modified \( S \) as the non-modified fixed chordal tooth thickness determined by \( m’ \) and \( z’ \) establishes the following correspondence. The fixed chordal tooth thickness of a modified gear is:
$$ S = \left( \frac{\pi}{2} \cos^2 \alpha_f \pm \xi \sin 2\alpha_f \right) m $$
The fixed chordal tooth thickness of a gear with parameters \( m’ \) and \( z’ \) is:
$$ S = \frac{\pi m’}{2} \cos^2 \alpha_f $$
From these two 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, \( z \) is the number of teeth, and \( \alpha_f \) is the pressure angle.
Since the pitch circle position remains unchanged when calculating based on \( m’ \) and \( z’ \), the coordinate origin should be set at a distance \( m’ \) from the pitch circle, resulting in a difference \( \Delta h \) from the theoretical tip of the modified gear. The calculation for \( \Delta h \) is:
$$ \Delta h = h_e – m’ = m(f \pm \xi) – m’ $$
where \( h_e \) is the addendum of the modified gear.
To calculate the tooth shape coordinates, the drop height from the tooth tip is determined by \( h_x = \Delta h + P_x m’ \), where \( P_x \) is the drop height coefficient. The corresponding chordal tooth thickness at each height is calculated using \( m’ \) and \( z’ \), or found in reference tables and multiplied by \( m’ \). For small-module gears, typically 3 to 5 points are selected along the tooth height for measurement.
To verify the correctness of this method for height-modified straight bevel gears, CAD software is used to draw a tooth of the gear after developing the back cone surface. The chordal tooth thickness at different drop 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 count \( z_y \):
$$ z_y = \frac{15}{\cos 30.96^\circ} \approx 17.49 $$
Next, compute \( m’ \) and \( z’ \):
$$ m’ = 0.6 \times \left( 1 + \frac{4 \times 0.3 \times \tan 20^\circ}{\pi} \right) \approx 0.6834 $$
$$ z’ = \frac{17.49}{1 + \frac{4 \times 0.3 \times \tan 20^\circ}{\pi}} \approx 15.687 $$
Then, determine \( \Delta h \):
$$ \Delta h = 0.6 \times (1 + 0.3) – 0.6834 = 0.0966 $$
Select three points with drop height coefficients \( P_1 = 0.1 \), \( P_2 = 0.5 \), and \( P_3 = 1 \) to define the tooth profile. The drop heights from the tooth tip are:
$$ h_1 = 0.0966 + 0.1 \times 0.6834 = 0.1649 $$
$$ h_2 = 0.0966 + 0.5 \times 0.6834 = 0.4383 $$
$$ h_3 = 0.0966 + 1 \times 0.6834 = 0.78 $$
The corresponding half chordal tooth thickness values are calculated or looked up and multiplied by \( m’ \). The calculated half chordal tooth thicknesses are approximately 0.2628 mm, 0.4049 mm, and 0.5285 mm for \( h_1 \), \( h_2 \), and \( h_3 \), respectively.
In CAD, draw the base circle, pitch circle, and tip circle for a straight gear with \( m = 0.6 \), \( z_y = 17.49 \), 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. Using half the chordal tooth thickness of the modified straight bevel gear at the pitch circle (1.0716 / 2) as the offset distance, offset the X-axis bilaterally to intersect 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. Draw three parallel lines at intervals \( h_1 \), \( h_2 \), and \( h_3 \), and measure the distances from these lines to the intersections with the left and right tooth profiles. The measured half chordal tooth thicknesses are 0.26 mm, 0.40 mm, and 0.53 mm, respectively.
The comparison between calculated and measured values is summarized in the table below. Excluding drawing precision errors, the coordinate values align closely, with negligible differences in practical measurement, confirming the correctness of the calculation method for tooth shape coordinate points of height-modified straight bevel gears.
| Interval | Drop Height (mm) | Calculated Half Chordal Thickness (mm) | Measured Half Chordal 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 calculation of tooth shape detection points is vital for ensuring the performance of straight bevel gears in applications such as aerospace and military equipment, where lightweight and compact designs are paramount. The method outlined here leverages virtual tooth counts and modification transformations to adapt straight gear calculations to straight bevel gears, providing a reliable approach for quality control. By integrating CAD verification, potential errors in manual calculations are minimized, enhancing the reliability of gear inspection processes. This approach not only supports the production of high-precision straight bevel gears but also contributes to advancements in gear manufacturing technology, particularly for small-module components requiring high accuracy. Future work could explore automation of these calculations in software tools to further streamline the inspection of straight bevel gears.
In summary, calculating tooth shape coordinate points for height-modified straight bevel gears involves determining the virtual tooth count \( z_y \), then deriving the calculation module \( m’ \) and tooth count \( z’ \) using the original module and \( z_y \). Substituting these into formulas along with drop height coefficients \( P \) yields the coordinate values needed for tooth shape measurement. This method effectively addresses the limitations of conventional gear testing centers for straight bevel gears and small-module gears, ensuring accurate quality assessment and supporting the production of reliable straight bevel gears for critical applications.