Herringbone gears are widely used in marine propulsion, aerospace, and other large power equipment owing to their high load-carrying capacity and small axial thrust. However, the centering of tooth flanks on the two sides of a herringbone gear relative to the central symmetry plane directly affects the uniformity of contact load distribution, which can cause vibration and noise during transmission. Since the contact lines on the same flank of the two oppositely inclined gear parts are often not symmetric after machining, eccentric load and axial vibration inevitably occur during meshing, reducing the stability and reliability of the transmission. From the viewpoint of service performance, this paper studies the measurement and evaluation of the symmetry error of herringbone gear flanks. The contact line of the tooth flank is chosen as the measurable physical quantity because it reflects the meshing state. Based on the definition of helical gear contact lines, the definitions of contact lines on the same flank of herringbone gears and their symmetrical degree error are studied. An evaluation coordinate system is established, and the mathematical model of the contact lines is derived. Simulated discrete data of theoretical contact lines and actual contact lines with superimposed errors are used to fit the actual contact line and obtain the symmetry error. The proposed evaluation method is then implemented on a CNC gear measuring machine.
In order to transform the theoretical model into an industrial measurement procedure, the measurement coordinate system is formed by applying the evaluation coordinate system to the gear measuring machine. The axial measurement reference is determined by measuring the end faces of the gear. In this coordinate system, two contact lines with opposite helix directions on each tooth are measured tooth-by-tooth. A dedicated software package is developed using C++ to control the probe and rotary table motions, acquire the measured data, and compute the symmetry error of the contact lines. Experiments are carried out on two herringbone gears with different modules. The measurement results are compared with those obtained by the fast cross-section method. The comparison shows that the contact-line method provides higher precision and better repeatability, although its measurement time is longer. The uncertainty of the measurement is also analyzed, demonstrating that the proposed method is reliable and effective for high-precision evaluation of herringbone gear symmetry.
1. Introduction
Herringbone gears, also known as double-helical gears, can be considered as two helical gears of opposite helix angle that are rigidly connected. This construction eliminates the axial thrust generated by a single helical gear while retaining the advantages of large overlap ratio and smooth meshing. Consequently, herringbone gears are widely used in high-power transmission systems such as marine gearboxes and aero-engine drives. Nevertheless, the manufacturing process of herringbone gears is more complex than that of spur or helical gears. In particular, the alignment of the two helical halves is difficult to control precisely. Even small deviations in the angular position or axial position between the left-hand and right-hand teeth will produce an asymmetry of the tooth flanks relative to the central plane. This asymmetry directly affects the load distribution between the two halves and gives rise to axial vibration and noise during operation.
Many researchers have investigated the dynamics of herringbone gear systems and pointed out that the symmetry error of the two sides is one of the main excitation sources. For example, theoretical and experimental studies have demonstrated that the axial displacement of the pinion is closely related to the intersection point displacement of the contact lines. If the two contact lines on the same flank do not intersect exactly on the symmetry plane, the gear pair will experience periodic axial excitation. However, the existing measurement standards for herringbone gears still follow the accuracy evaluation system of helical gears, i.e., the two helical halves are separately evaluated. Such an approach cannot reflect the actual symmetry performance of the complete gear. In machining practice, some manufactures use a dial indicator to check the common normal of the tooth space after rough cutting, or measure the phase angle between the two halves on a gear hobbing machine. These methods are indirect and often have limited accuracy.
For this reason, the present research aims at developing a direct, high-precision method for measuring and evaluating the symmetry of herringbone gears based on the tooth flank contact lines. The contact line of a gear pair is the instantaneous contact trajectory during meshing, and it is directly related to the transmission performance. For a herringbone gear, two contact lines on the same flank (one on the left-hand side and one on the right-hand side) intersect at a point. The distance from this intersection point to the central symmetry plane represents the degree of symmetry of that tooth. This quantity can be measured on a gear measuring machine and serves as a performance-oriented symmetry error.
In this paper, we first present the definitions of contact lines and symmetry errors for herringbone gears. Then we establish the mathematical model of the contact lines and develop a least-squares fitting algorithm to compute the symmetry error. After that, we design the measurement principle on a CNC gear measuring machine and implement the measurement software. Finally, we verify the method by measuring two herringbone gears with different modules and compare the results with the fast cross-section method. The measurement uncertainty is analyzed to evaluate the reliability of the proposed method.
2. Definitions of Contact Lines and Symmetry Error
2.1 Definition of the contact line
For a helical gear, the contact line is defined as the instantaneous contact line on the tooth flank when a pair of ideal, non-deformed gear teeth are meshing. It is a straight line located in the tangent plane of the base cylinder, and its angle with the base cylinder generatrix is the base helix angle βb. Since a herringbone gear is composed of two helical gears with opposite helix directions, the same definition is applicable to each half. The contact line is a straight segment on the flank, and every position along the face width corresponds to one point on the line.
For a herringbone gear, the left-hand and right-hand contact lines on the same flank are inclined in opposite directions. If the gear is perfectly manufactured, these two lines intersect exactly on the central symmetry plane. The intersection point is denoted as A in Fig. 2.1 (not shown here; the schematic is given in the original paper). In a gear pair, the intersection points of the pinion and the wheel should coincide axially during meshing. Any deviation will cause axial float or vibration.
2.2 Symmetry error of a single tooth
The symmetry error of a single tooth, denoted as fAi, is defined as the axial distance between the actual intersection point of the two contact lines on tooth i and the central symmetry plane. If the intersection point is above the symmetry plane (positive Z direction), the error is positive; otherwise, it is negative. This error directly reflects the degree of misalignment of that tooth with respect to the central plane.
2.3 Other evaluation indices
In addition to the single-tooth error, we define three other indices to describe the symmetry quality of the whole gear:
- Adjacent tooth difference fAu: the maximum absolute difference between the symmetry errors of neighboring teeth. This quantity reflects short-period variation that can excite vibration.
- Total symmetry error FA: the algebraic difference between the maximum and minimum single-tooth errors over all teeth. This reflects long-period variation.
- Mean symmetry error fAs: the arithmetic mean of all single-tooth errors. This indicates the overall offset of the actual gear with respect to the nominal central plane.
3. Mathematical Model of Contact Lines
3.1 Coordinate system and contact line equations
To evaluate the symmetry error, we establish a Cartesian coordinate system T’RZ’ as shown in Fig. 3.1 (not reproduced). The T’ axis lies in the tangent plane of the base cylinder and is perpendicular to the gear axis; the Z’ axis is parallel to the gear axis; the R axis is perpendicular to the tangent plane. The origin O’ is located at the intersection of the line connecting the two end-points of the left-hand and right-hand contact lines and the central symmetry plane. After translating the coordinate system along the R axis to the gear center, we obtain the actual measurement coordinate system TRZ, where Z is the gear axis.
In the T’O’Z’ plane, the left-hand contact line can be expressed as a linear function:
$$Z’ = k_L T’ + b_L \tag{1}$$
where kL is the slope and bL is the intercept on the Z’ axis. Similarly, the right-hand contact line is:
$$Z’ = k_R T’ + b_R \tag{2}$$
The slopes are given by:
$$k_L = \tan\left(90^\circ – \beta_b\right) \tag{3}$$
$$k_R = -\tan\left(90^\circ – \beta_b\right) \tag{4}$$
where βb is the base helix angle, computed from the normal pressure angle αn, the helix angle β, and the normal module mn:
$$\beta_b = \tan^{-1}\left\{\tan\beta \cdot \cos\left[\tan^{-1}\left(\frac{\tan\alpha_n}{\cos\beta}\right)\right]\right\} \tag{5}$$
By measuring the coordinates of any two points on each contact line, we can determine the coefficients k and b, thereby obtaining the equation of the actual contact line.
3.2 Evaluation range
Considering the actual tooth flank modifications and the effective contact width, the evaluation range is selected as the middle 60% of the face width of each helical half. Let B be the width of one helical half. The evaluation range is from −0.3B to +0.3B relative to the center line of that half. This range avoids edge contacts and edge modifications in practice.
3.3 Extraction of discrete measurement points
As shown in Fig. 3.3 (schematic), the left-hand contact line passes through the point M at the lower end of the left-hand flank. The coordinates of M are (0, bt/2) where bt is the width of the central gap (withdrawal groove). Substituting M into the left-hand line equation yields the theoretical contact line. Similarly, due to symmetry, the right-hand contact line can be obtained. Along the theoretical contact lines, a set of n equally spaced points is generated. These are the theoretical measurement points PLj (j = 1, 2, …, n) for the left-hand side and PRj for the right-hand side.
Because of manufacturing errors, the actual measurement points will deviate from the theoretical points in a direction perpendicular to the theoretical contact line. To simulate actual measurements, we add a random error δn to each theoretical measurement point along the normal direction. The resulting points are denoted as GLj and GRj. In our simulation, the random errors are taken from the actual error values measured by the cross-section method on the same gear at corresponding positions.
3.4 Least-squares fitting
The least-squares method is used to fit the measured points into straight lines. For the left-hand side, the objective function is:
$$\varphi(b_L,k_L)=\sum_{j=1}^{n}\left[Z_j-\left(b_L+k_L T_j\right)\right]^2 \tag{6}$$
By setting the partial derivatives with respect to bL and kL to zero, we obtain:
$$b_L=\frac{\sum T_j^2 \sum Z_j – \sum T_j \sum T_j Z_j}{n\sum T_j^2 – \left(\sum T_j\right)^2} \tag{7}$$
$$k_L=\frac{n\sum T_j Z_j – \sum T_j \sum Z_j}{n\sum T_j^2 – \left(\sum T_j\right)^2} \tag{8}$$
Similarly, the coefficients bR and kR for the right-hand side are obtained. The fitted lines are denoted as l1 and l2 for the theoretical data, and l3 and l4 for the actual data.
3.5 Computation of symmetry error
Extending the two actual contact lines l3 and l4 to their intersection point A( T, Z ), the symmetry error for the tooth is simply the Z-coordinate of this point:
$$f_A = Z \tag{9}$$
If the intersection point lies on the positive Z axis, fA is positive; otherwise, it is negative. The same calculation is repeated for every tooth of the gear.
3.6 Numerical validation
To validate the proposed evaluation method, we used an actual herringbone gear with the following basic parameters:
| Parameter | Value |
|---|---|
| Normal module mn / mm | 2.2 |
| Number of teeth z | 27 |
| Normal pressure angle αn / ° | 22 |
| Helix angle β / ° | 30 |
| Face width of one half B / mm | 30 |
| Gap width bt / mm | 20 |
We selected two teeth separated by 120° (tooth #1 and tooth #10). On each tooth, 41 theoretical measurement points were generated along the contact lines. The random errors were taken from the actual deviations obtained by the cross-section method. After fitting, the theoretical contact lines have zero symmetry error, as expected. For the actual contact lines, the symmetry errors were calculated to be -0.0275 mm for tooth #1 and -0.0256 mm for tooth #10. The cross-section method gave -0.0278 mm and -0.0258 mm, respectively. The maximum difference is only 0.0003 mm, which confirms that the contact-line evaluation method agrees well with the established cross-section method.
4. Measurement Principle on a CNC Gear Measuring Machine
4.1 Measurement coordinate system
The evaluation coordinate system described in Section 3 is implemented on a CNC gear measuring machine. The gear is mounted on the machine with its axis coinciding with the spindle rotation axis. The machine has four axes: Z (axial), R (radial), T (tangential), and θ (rotation). The probe is installed on a slide that can move in the R, T, and Z directions. The measuring coordinate system is aligned with the machine axes.
4.2 Axial measurement reference
Before measuring contact lines, it is essential to establish the axial measurement reference, i.e., the position of the central symmetry plane. The procedure is as follows. First, the probe is manually moved to touch the upper face of the gear. The Z-coordinate of the upper face is recorded as Up_Z. Then the probe moves down and touches the lower face; the Z-coordinate is recorded as Down_Z. The central plane position is then computed as:
$$Mid\_Z = \frac{Up\_Z + Down\_Z}{2} \tag{10}$$
The current Z-axis position is then set to -Dis_Z, where Dis_Z is the distance between the current probe position and the central plane. In this way, the Z-origin is placed on the symmetry plane, and all subsequent axial coordinates are measured relative to this plane.
4.3 Contact line measurement procedure
For each tooth, two contact lines on the same flank are measured: the line on the left-hand half and the line on the right-hand half. The probe moves along the contact line in the T-Z plane while the gear remains stationary (or rotates as needed). The measurement starts at one end of the evaluation range and ends at the other. For the upper half (right-hand gear in our setup), the starting Z-coordinate is W/2 + B/5 and the ending Z-coordinate is W/2 + 4B/5, where W is the gap width. For the lower half, the coordinates are symmetric with respect to the central plane. The probe path is designed so that the measured line is exactly the contact line on the corresponding flank.
In each measurement, the probe touches the flank and traverses along the contact line. The machine’s grating sensors record the T and Z coordinates at a specified sampling interval. After completing one tooth, the gear is rotated by one pitch, and the process is repeated for all teeth. The collected data are then fitted using the least-squares method described in Section 3.4.
5. Measurement Software
5.1 Software structure
To realize the measurement automatically, we developed a software package in C++ for the CNC gear measuring machine. The software consists of three main modules:
- Parameter input module: The user enters the gear basic parameters (module, number of teeth, pressure angle, helix angle, face width, gap width, etc.) and selects the measurement configuration (which flank, starting point, end point, sampling density).
- Measurement module: This module controls the probe and the rotary table movements using the library functions provided by the measuring machine. It also performs the axial reference establishment.
- Data processing module: This module reads the sampled data, performs least-squares fitting, computes the symmetry error for each tooth, and plots the error curves.
5.2 Library functions
The measurement software uses several low-level library functions:
StartLampOnandEndLampOn: control the status lamps to indicate the start and end of measurement.IsStartHitandIsEndHit: detect whether the start or end button has been pressed.motordrv: generates the interpolation motion of the four axes according to the specified parameters.is_motor_stop: checks whether the motor has stopped.TouchProbe: performs a probing movement along a given axis with a specified speed.sample11: reads the grating counter values of the axes at the sampling instant.counterwriteandcounterste: set the coordinate registers for the axes.
5.3 Motion control program
The motion control program follows the flowchart given in the original paper. After the measurement is started, the program reads the parameters and sets the motors to automatic mode. The probe moves to the starting position. Then it approaches the workpiece until contact is detected. If the probe is over-traveled, an alarm is raised. After the contact line is traversed, the sampling data are recorded. The program checks whether the number of sampled points has reached the preset density. If not, the measurement is repeated with a smaller sampling interval. After the measurement is complete, the program enters the data processing stage and updates the report.
5.4 Data processing program
The data processing routine first extracts the sampled coordinates (T, Z). For each tooth, the points are fitted to two straight lines. The intersection point of these lines is computed, and its Z-coordinate is the symmetry error fAi. The results are stored and displayed as a bar chart or a line chart. The adjacent differences and total error are also calculated.
6. Experiments
6.1 Workpieces and setup
Two herringbone gears were tested. The first gear (small module) has the parameters listed in Section 3.6. The second gear (large module) has the same tooth number and pressure angle, but the normal module is 3.9 mm, the face width of one half is 47 mm, and the gap width is 20 mm. The gears were mounted on a C40 CNC gear measuring machine. The evaluation range for the small-module gear was from 16 mm to 34 mm (face width 30 mm, gap 20 mm). For the large-module gear, the evaluation range was from 19 mm to 48 mm (face width 47 mm, gap 20 mm).
6.2 Measurement results for the small-module gear
The contact-line symmetry errors of all 27 teeth were measured. The results are summarized in Table 6.1(a).
| Evaluation index | Value / mm |
|---|---|
| Range of fAi | -0.0201 ~ -0.0002 |
| Adjacent difference fAu | 0.0101 |
| Total error FA | 0.0199 |
| Mean error fAs | -0.0113 |
| Measurement time / min | 105 |
It can be seen that the symmetry errors are all negative except for tooth #15, which has a value of -0.0002 mm. The negative sign indicates that the intersection points are systematically shifted below the central plane. The total error of about 0.02 mm is acceptable for this class of gears.
6.3 Measurement results for the large-module gear
Similarly, the large-module gear was measured. The results are given in Table 6.1(b).
| Evaluation index | Value / mm |
|---|---|
| Range of fAi | -0.0322 ~ -0.0245 |
| Adjacent difference fAu | 0.0056 |
| Total error FA | 0.0077 |
| Mean error fAs | -0.0281 |
| Measurement time / min | 105 |
The large-module gear exhibits a more consistent behavior, with all single-tooth errors lying between about -0.03 mm and -0.02 mm. The difference between maximum and minimum is only 0.0077 mm, indicating a better uniformity of symmetry across the teeth.
6.4 Comparison with the cross-section method
The same gears were also measured using the fast cross-section method implemented on the same measuring machine. The comparison of the two methods for the small-module gear is shown in Table 6.2(a).
| Error item | Contact-line method | Cross-section method | Absolute difference |
|---|---|---|---|
| Range of fAi / mm | -0.0201 ~ -0.0002 | -0.0309 ~ -0.0020 | 0.0108 ~ 0.018 |
| fAu / mm | 0.0101 | 0.0215 | 0.0114 |
| FA / mm | 0.0199 | 0.0228 | 0.0029 |
| fAs / mm | -0.0113 | -0.0243 | 0.0130 |
| Time / min | 105 | 10 | 95 |
For the large-module gear, the comparison is presented in Table 6.2(b).
| Error item | Contact-line method | Cross-section method | Absolute difference |
|---|---|---|---|
| Range of fAi / mm | -0.0322 ~ -0.0235 | -0.0369 ~ -0.0271 | 0.0036 ~ 0.047 |
| fAu / mm | 0.0056 | 0.0101 | 0.0045 |
| FA / mm | 0.0087 | 0.0098 | 0.0011 |
| fAs / mm | -0.0281 | -0.0301 | 0.0020 |
| Time / min | 105 | 10 | 95 |
From the tables, it is clear that the contact-line method yields smaller error values than the cross-section method. For the small-module gear, the differences in the single-tooth values are around 10 μm, while the total error difference is about 3 μm. For the large-module gear, the differences are smaller, around 2–5 μm. The cross-section method is faster (10 minutes) but provides less stable results because it samples only a limited number of discrete points on the flank. The contact-line method, although ten times slower, offers higher accuracy and better repeatability. Therefore, for high-precision evaluation, the contact-line method is recommended; for rapid quality control, the cross-section method can be used.
6.5 Repeatability verification
To verify the measurement repeatability, the gears were measured five times under the same conditions. The results for the small-module gear are summarized in Table 6.3.
| No. | fAi / mm | fAu / mm | FA / mm | fAs / mm |
|---|---|---|---|---|
| 1 | -0.0201 ~ -0.0002 | 0.0101 | 0.0199 | -0.0113 |
| 2 | -0.0179 ~ -0.0012 | 0.0114 | 0.0167 | -0.0121 |
| 3 | -0.0200 ~ -0.0006 | 0.0106 | 0.0194 | -0.0115 |
| 4 | -0.0188 ~ -0.0005 | 0.0111 | 0.0183 | -0.0119 |
| 5 | -0.0197 ~ -0.0009 | 0.0095 | 0.0188 | -0.0120 |
| Mean | — | 0.0105 | 0.0186 | -0.0118 |
| Std. dev. | — | 0.0007 | 0.0012 | 0.0003 |
For the large-module gear, the repeated results are given in Table 6.4.
| No. | fAi / mm | fAu / mm | FA / mm | fAs / mm |
|---|---|---|---|---|
| 1 | -0.0322 ~ -0.0235 | 0.0101 | 0.0199 | -0.0113 |
| 2 | -0.0330 ~ -0.0237 | 0.0114 | 0.0197 | -0.0115 |
| 3 | -0.0334 ~ -0.0225 | 0.0106 | 0.0201 | -0.0121 |
| 4 | -0.0324 ~ -0.0234 | 0.0111 | 0.0194 | -0.0114 |
| 5 | -0.0315 ~ -0.0223 | 0.0095 | 0.0192 | -0.0115 |
| Mean | — | 0.0105 | 0.0197 | -0.0116 |
| Std. dev. | — | 0.0007 | 0.0003 | 0.0003 |
The standard deviations of the evaluation indices are very small (0.001 mm or less), confirming that the contact-line method is highly repeatable. The small-module gear shows a slightly larger variation in FA (0.0012 mm), but this is still acceptable.
7. Measurement Uncertainty Analysis
According to the Guide to the Expression of Uncertainty in Measurement, the standard uncertainty is evaluated using the standard deviation of repeated measurements. Since each gear is normally evaluated based on a single measurement, we take the standard deviation of the measurement results as the standard uncertainty for the corresponding error index. The expanded uncertainty is obtained by multiplying the standard uncertainty by a coverage factor k. We here use k = 2, which gives a confidence level of approximately 95%.
7.1 Small-module gear
Using the five repeated measurements, we calculate the standard uncertainty and expanded uncertainty for fAu, FA, and fAs. The results are listed in Table 7.1.
| Error index | Standard uncertainty / mm | Expanded uncertainty (k=2) / mm |
|---|---|---|
| fAu | 0.0007 | 0.0014 |
| FA | 0.0012 | 0.0024 |
| fAs | 0.0003 | 0.0006 |
In addition, we analyzed the uncertainty for each individual tooth. The mean values, standard deviations, standard uncertainty, and expanded uncertainty for each tooth are shown in the original paper (Appendix 2). For example, tooth #15 exhibits a distinct behavior with a very small error and low uncertainty. This tooth was identified as a test-ground tooth during manufacture, which explains its different symmetry condition.
7.2 Large-module gear
The corresponding uncertainty values for the large-module gear are given in Table 7.2.
| Error index | Standard uncertainty / mm | Expanded uncertainty (k=2) / mm |
|---|---|---|
| fAu | 0.0007 | 0.0014 |
| FA | 0.0003 | 0.0006 |
| fAs | 0.0003 | 0.0006 |
The expanded uncertainty of the total error for the large-module gear is only 0.0006 mm, which is very small. This demonstrates that the contact-line measurement method provides not only high accuracy but also excellent measurement reproducibility.
8. Conclusion
In this paper, a performance-oriented method for measuring and evaluating the symmetry of herringbone gears based on tooth flank contact lines has been proposed and experimentally verified. The main conclusions are as follows:
- The contact line of a herringbone gear reflects the meshing performance. The symmetry error of a tooth is defined as the axial distance between the intersection point of the left-hand and right-hand contact lines and the central symmetry plane. This definition is physically meaningful and directly related to the axial vibration and load distribution of the gear pair.
- A mathematical model of the contact lines was established in a suitable coordinate system. The least-squares fitting method was used to reconstruct the actual contact lines from discrete measurement points. The theoretical and numerical examples show that the proposed evaluation algorithm yields results consistent with those of the cross-section method (difference less than 0.0003 mm).
- The measurement principle was implemented on a CNC gear measuring machine. The axial reference is established by measuring the two end faces of the gear. A dedicated software package was developed to control the probe motion, acquire data, and compute the symmetry errors.
- Two herringbone gears with different modules were measured. The contact-line method gave smaller and more stable symmetry errors compared with the fast cross-section method. The differences between the two methods are in the range of 2–13 μm, depending on the gear size and the error index. The measurement time of the contact-line method is about 105 minutes, while the cross-section method takes only 10 minutes. Therefore, the contact-line method is suitable for high-precision evaluation, whereas the cross-section method is convenient for rapid inspection.
- The repeatability of the contact-line method is excellent. The standard uncertainty of the total error is less than 0.0012 mm for the small-module gear and less than 0.0003 mm for the large-module gear. The expanded uncertainty (k=2) is at most 0.0024 mm, which confirms the reliability of the proposed method.
The research provides a new and effective tool for the measurement and evaluation of herringbone gear symmetry. In future work, we plan to optimize the measuring speed and integrate the algorithm into a more user-friendly industrial software interface. Moreover, the influence of the contact-line symmetry error on the dynamic response of herringbone gear systems will be further investigated experimentally.