I treat the alignment problem of screw gears as a contact-line problem rather than as two separate helical-gear inspections. The screw gears I consider here are the double-helical or herringbone type, formed by two opposite-hand helical tooth sections that meet near a central relief groove. Their apparent symmetry is not guaranteed by ordinary helical-gear tolerances. The critical feature is the relative axial position of the two opposite-hand contact-line families. When those families are extended toward the symmetry plane, they form a V-shaped vertex. If the vertices of all teeth do not lie in the same axial plane, the screw gears will exhibit axial floating, uneven flank loading, periodic impact, and noise. My method measures the contact lines themselves, fits their least-squares midlines, intersects the opposite-hand midlines, and evaluates the axial scatter of the resulting vertices.
Conventional inspection of screw gears usually treats the right-hand section and the left-hand section as independent cylindrical gears. That approach can report profile, helix, and pitch deviations for each section, but it does not directly answer whether the two sections are correctly aligned with respect to each other. For screw gears, the alignment of opposite-hand flanks is a functional parameter. It controls how the load is shared between the two hand directions at a given instant. It also controls the axial force balance that screw gears are intended to provide. Therefore, I define a new measurand: the axial coordinate of the V-shaped vertex obtained from the contact lines of opposite-hand flanks. I then derive evaluation indices from the distribution of those axial coordinates around the gear circumference.
My first-person position is simple. I do not begin with a complete coordinate measurement of both flanks and then compare them indirectly. I begin with the contact line, because the contact line is the geometric entity that actually carries the load. For screw gears, a contact line on the right-hand flank and a contact line on the left-hand flank intersect in a projected axial-circumferential plane. Their intersection is the V-shaped vertex. The axial position of that vertex is the alignment error for that tooth pair. The total spread of all vertex axial positions is the total alignment deviation of the screw gears. The mean vertex axial position is the symmetry deviation. The maximum adjacent change is the adjacent alignment difference. I use these four quantities as the core evaluation set.
The motivation is both metrological and dynamic. From a metrological viewpoint, screw gears require a parameter that belongs to the pair of opposite-hand flanks, not to either flank alone. From a dynamic viewpoint, the axial position of the V-shaped vertex affects the phase between the right-hand and left-hand meshing events. A small axial offset can make one hand section engage earlier than the other. This causes axial vibration and uneven load distribution even when each individual flank satisfies its own tolerance. Therefore, I measure the relative alignment of screw gears through contact lines, and I evaluate the resulting vertex distribution as a functional quality characteristic.
The symbols I use throughout the derivation are summarized in the following table. They are chosen so that the measurement equations remain close to the physical geometry of screw gears.
| Symbol | Meaning |
|---|---|
| \(Z\) | Number of teeth of the screw gears |
| \(m_n\) | Normal module |
| \(\alpha_n\) | Normal pressure angle |
| \(\beta\) | Helix angle at the reference cylinder |
| \(\beta_b\) | Base helix angle |
| \(r_b\) | Base radius |
| \(P\) | Spiral parameter of the involute helicoid |
| \(B\) | Single tooth width of one hand section |
| \(W\) | Width of the central relief groove |
| \(u\) | Circumferential coordinate in the unwrapped plane |
| \(z\) | Axial coordinate |
| \(m_R, m_L\) | Fitted slopes of right-hand and left-hand contact lines |
| \(b_R, b_L\) | Fitted axial intercepts of right-hand and left-hand contact lines |
| \(A_i\) | V-shaped vertex of tooth pair \(i\) |
| \(f_{A,i}\) | Single alignment deviation of tooth pair \(i\) |
| \(f_{Au}\) | Adjacent tooth alignment difference |
| \(F_A\) | Total alignment deviation |
| \(f_{As}\) | Symmetry deviation, also the mean alignment deviation |
I define the geometric center of the screw gears as the \(Z\)-axis of a Cartesian coordinate system. The base circle lies in the \(XOY\) plane. The tooth profile intersects the base circle at the positive \(X\)-axis. For a right-hand involute helicoid, I write the flank as a parametric surface. The parameter \(t\) moves along the generating straight line, while \(\theta\) describes the involute rotation. The axial coordinate contains the spiral parameter \(P\), which couples rotation and translation. The parameter \(P\) is related to the base helix angle by the expression \(P = r_b \cot \beta_b\). The right-hand flank can be expressed as follows.
$$ \mathbf{r}_R(\theta,t) = \begin{bmatrix} r_b\cos\theta – t\sin\theta \\ r_b\sin\theta + t\cos\theta \\ \frac{P}{r_b}t + P\theta \end{bmatrix} $$
For the left-hand flank, the hand direction reverses. I use the same base radius and the same spiral parameter in magnitude, but the sign of the axial coupling changes. A possible left-hand representation, including an axial offset \(h\) that represents the relative section placement, is written as follows.
$$ \mathbf{r}_L(\theta,t) = \begin{bmatrix} r_b\cos\theta + t\sin\theta \\ r_b\sin\theta – t\cos\theta \\ -\frac{P}{r_b}t – P\theta + h \end{bmatrix} $$
The generating line of the right-hand flank is obtained by differentiating the parametric surface with respect to \(t\). For fixed \(\theta\), this derivative is a constant direction vector. It is the direction of the contact line in the three-dimensional space of the screw gears.
$$ \frac{\partial \mathbf{r}_R}{\partial t} = \begin{bmatrix} -\sin\theta \\ \cos\theta \\ P/r_b \end{bmatrix} $$
Similarly, the left-hand generating line has a direction vector whose axial component has the opposite sign. When I project the contact lines onto the unwrapped cylindrical plane, I obtain two straight lines with opposite slopes. The right-hand contact line is \(z_R(u)=m_R u+b_R\). The left-hand contact line is \(z_L(u)=m_L u+b_L\). In the ideal case, the slopes are \(m_R=\tan\beta_b\) and \(m_L=-\tan\beta_b\). In a real screw gear, the slopes and intercepts both contain manufacturing errors. The fitted slopes absorb helix deviations, and the fitted intercepts carry the relative axial placement of the two hand sections.
The V-shaped vertex is the intersection of these two contact lines. I solve the two line equations simultaneously. The circumferential coordinate of the vertex is \(u_A\). The axial coordinate of the vertex is \(z_A\). The axial coordinate is the quantity I care about, because it directly represents the alignment of the opposite-hand contact lines of the screw gears.
$$ u_A = \frac{b_L-b_R}{m_R-m_L} $$
$$ z_A = \frac{m_R b_L – m_L b_R}{m_R – m_L} $$
The included angle of the V shape is determined by the slopes. In the ideal case, this angle is related to the base helix angle. I write it as follows.
$$ \psi_V = 2\arctan |m_R| \approx 2\beta_b $$
This angle matters for sensitivity. If the contact lines are nearly parallel, the intersection point is poorly conditioned. If the V angle is sufficiently large, the intersection is stable and the axial coordinate of the vertex can be evaluated with high resolution. For the screw gears in my study, the helix angle is \(30^\circ\), so the V angle is large enough for a robust least-squares intersection. I nevertheless track the slope difference \(m_R-m_L\) as a conditioning indicator.
I now explain the measurement principle in the first person. I mount the screw gears on the rotary spindle of a CNC gear measuring center. The spindle axis defines the workpiece axis. The probe is mounted on a three-axis slide. The \(Z\)-axis of the slide is parallel to the spindle axis. The \(X\) and \(Y\) axes are perpendicular to the spindle axis. I first calibrate the probe and the spindle center. Then I measure the upper and lower end faces of the screw gears. From those end-face measurements, I calculate the axial coordinate of the symmetry plane. I set that symmetry plane as the zero of the axial coordinate. This step is essential because the alignment deviation of screw gears is defined with respect to the central symmetry plane, not with respect to an arbitrary chuck face.
The following table summarizes the measurement setup and the role of each element.
| Element | Function | Reason |
|---|---|---|
| Rotary spindle | Carries the screw gears and defines the axis of rotation | The vertex distribution is evaluated around the workpiece axis |
| Three-axis probe slide | Moves the probe in \(X\), \(Y\), and \(Z\) | Allows access to both hand flanks along the contact-line direction |
| \(Z\)-axis origin | Set at the symmetry plane | Makes the axial vertex coordinate equal to the alignment deviation |
| Probe calibration | Determines probe radius and center offset | Reduces systematic error in the fitted contact lines |
| End-face measurement | Locates the upper and lower faces | Defines the central symmetry plane of the screw gears |
| Thermal stabilization | Keeps the workpiece and machine at stable temperature | Reduces drift in the axial coordinate of the vertices |
For each tooth, I measure the right-hand flank contact line and the left-hand flank contact line. I do not measure the entire flank from root to tip. Instead, I measure a controlled evaluation interval near the middle of each hand section. This interval avoids the tooth ends, the fillet region, and the region affected by the relief groove. By avoiding the ends, I reduce the influence of tooth-tip chamfers, edge contact, and form modification. By avoiding the root, I reduce the influence of fillet geometry. The evaluation interval is therefore a metrological window that represents the load-carrying contact line of the screw gears.
For a single hand section of width \(B\) and a central relief groove of width \(W\), I use the following intervals. The right-hand interval starts at \(0.5W+0.2B\) and ends at \(0.5W+0.8B\). The left-hand interval is the mirror image, starting at \(-0.5W-0.8B\) and ending at \(-0.5W-0.2B\). These expressions place the measurement window at the middle sixty percent of each hand section.
$$ \mathcal{I}_R = [0.5W+0.2B,\; 0.5W+0.8B] $$
$$ \mathcal{I}_L = [-0.5W-0.8B,\; -0.5W-0.2B] $$
For the screw gears in my experiment, the single tooth width is \(B=30\) mm and the relief groove width is \(W=20\) mm. Therefore, the right-hand measurement interval is \([16,34]\) mm, and the left-hand measurement interval is \([-34,-16]\) mm. These intervals are wide enough to define a stable least-squares line, yet narrow enough to avoid the ends of the screw gears.
The following table lists the measurement and evaluation steps I use for the full-tooth contact-line method.
| Step | Action | Output |
|---|---|---|
| 1 | Mount and align the screw gears | Workpiece axis coincides with spindle axis |
| 2 | Measure upper and lower end faces | Axial coordinate of the symmetry plane |
| 3 | Set the symmetry plane to zero | Reference for all vertex axial coordinates |
| 4 | Select the evaluation interval on each hand section | Right-hand and left-hand measurement windows |
| 5 | Measure contact-line points on every tooth | Point clouds for right-hand and left-hand flanks |
| 6 | Fit each contact line by least squares | Slopes \(m_R\), \(m_L\) and intercepts \(b_R\), \(b_L\) |
| 7 | Intersect opposite-hand midlines | Vertex \(A_i\) for each tooth pair |
| 8 | Evaluate alignment indices | \(f_{A,i}\), \(f_{Au}\), \(F_A\), \(f_{As}\) |
The least-squares fit is the numerical heart of the method. For the right-hand contact line, I collect \(N\) points \((u_j,z_j)\) inside the evaluation interval. I fit \(z=m_R u+b_R\). The slope and intercept are obtained from the normal equations. I use the same procedure for the left-hand contact line. The equations below are the standard least-squares solution that I implement in the measurement software.
$$ m_R = \frac{N\sum_{j=1}^{N} u_j z_j – \sum_{j=1}^{N} u_j \sum_{j=1}^{N} z_j}{N\sum_{j=1}^{N} u_j^2 – \left(\sum_{j=1}^{N} u_j\right)^2} $$
$$ b_R = \frac{\sum_{j=1}^{N} z_j – m_R \sum_{j=1}^{N} u_j}{N} $$
For the left-hand contact line, I replace \(R\) by \(L\). The fitted slopes and intercepts define the two midlines of the V shape. I then compute the intersection. If the slope difference \(m_R-m_L\) is small, the denominator is small, and the vertex is sensitive to small errors in the fitted lines. Therefore, I monitor the slope difference and reject or re-measure any tooth whose slope difference is below a threshold.
Once I obtain the vertex axial coordinate \(z_{A,i}\) for each tooth pair, I define the single alignment deviation as follows. The index \(i\) runs from one to \(Z\). Because the symmetry plane is the axial zero, \(f_{A,i}\) is simply \(z_{A,i}\). A positive value means that the actual vertex is axially above the design symmetry plane. A negative value means that it is axially below the design symmetry plane.
$$ f_{A,i} = z_{A,i} $$
The adjacent tooth alignment difference \(f_{Au}\) is the maximum absolute difference between consecutive teeth. This parameter reflects the rate of change of the V-vertex axial position around the circumference. A large adjacent difference can excite axial vibration and noise in screw gears.
$$ f_{Au} = \max_{i=1,\ldots,Z-1} \left| f_{A,i+1} – f_{A,i} \right| $$
The total alignment deviation \(F_A\) is the algebraic difference between the maximum and minimum single deviations. It represents the full axial spread of the V-shaped vertices. This is the most direct global measure of alignment quality for screw gears.
$$ F_A = \max_{i=1,\ldots,Z} f_{A,i} – \min_{i=1,\ldots,Z} f_{A,i} $$
The symmetry deviation \(f_{As}\) is the arithmetic mean of all single alignment deviations. It describes the average axial offset of the V-vertex distribution from the design symmetry plane. A non-zero mean indicates a systematic shift of the opposite-hand contact lines.
$$ f_{As} = \frac{1}{Z}\sum_{i=1}^{Z} f_{A,i} $$
I also use two statistical parameters for completeness. The standard deviation \(\sigma_A\) measures the scatter of the vertex distribution. The root-mean-square value \(f_{A,rms}\) measures the overall magnitude of the alignment deviations. These two parameters are useful when I compare screw gears produced by different processes.
$$ \sigma_A = \sqrt{\frac{1}{Z-1}\sum_{i=1}^{Z} \left(f_{A,i}-f_{As}\right)^2} $$
$$ f_{A,rms} = \sqrt{\frac{1}{Z}\sum_{i=1}^{Z} f_{A,i}^2} $$
The evaluation indices are summarized in the following table. I use this table as the core reporting format for screw gears.
| Index | Formula | Physical meaning |
|---|---|---|
| \(f_{A,i}\) | \(z_{A,i}\) | Axial vertex position of a single tooth pair |
| \(f_{Au}\) | \(\max |f_{A,i+1}-f_{A,i}|\) | Maximum adjacent change of vertex position |
| \(F_A\) | \(\max f_{A,i}-\min f_{A,i}\) | Total axial spread of all vertices |
| \(f_{As}\) | \(\frac{1}{Z}\sum f_{A,i}\) | Mean axial offset or symmetry deviation |
| \(\sigma_A\) | Standard deviation of \(f_{A,i}\) | Scatter of the vertex distribution |
| \(f_{A,rms}\) | RMS of \(f_{A,i}\) | Overall magnitude of alignment deviation |
I now turn to the experimental evaluation. The screw gears in my test have a normal module of \(2.5\) mm, a tooth number of \(27\), a normal pressure angle of \(20^\circ\), a helix angle of \(30^\circ\), a single tooth width of \(30\) mm, and a relief groove width of \(20\) mm. The right-hand section is mounted at the upper end, and the left-hand section is measured from the opposite side. I mark all teeth from one to twenty-seven so that I can trace anomalous vertices back to the manufacturing process.
The following table lists the principal parameters of the screw gears and the measurement windows.
| Parameter | Value |
|---|---|
| Normal module \(m_n\) | \(2.5\) mm |
| Number of teeth \(Z\) | \(27\) |
| Normal pressure angle \(\alpha_n\) | \(20^\circ\) |
| Helix angle \(\beta\) | \(30^\circ\) |
| Single tooth width \(B\) | \(30\) mm |
| Relief groove width \(W\) | \(20\) mm |
| Right-hand measurement interval | \([16,34]\) mm |
| Left-hand measurement interval | \([-34,-16]\) mm |
After I measure the contact lines and fit their least-squares midlines, I obtain the vertex axial coordinates for all tooth pairs. The single alignment deviations fall in the range from \(-0.0326\) mm to \(-0.0243\) mm. All values are negative, which means that the vertices lie slightly below the design symmetry plane. The total alignment deviation \(F_A\) is \(0.0082\) mm. This value occurs between the ninth tooth and the fifteenth tooth. The adjacent tooth alignment difference \(f_{Au}\) is \(0.0060\) mm. This value occurs between the fifteenth tooth and the sixteenth tooth. The symmetry deviation \(f_{As}\) is \(0.0286\) mm. The design requirement is satisfied by the total alignment deviation and by the symmetry deviation.
The following table summarizes the experimental results for the screw gears.
| Quantity | Value | Location or remark |
|---|---|---|
| Minimum single deviation | \(-0.0326\) mm | Within the measured tooth set |
| Maximum single deviation | \(-0.0243\) mm | Within the measured tooth set |
| Total alignment deviation \(F_A\) | \(0.0082\) mm | Between tooth 9 and tooth 15 |
| Adjacent alignment difference \(f_{Au}\) | \(0.0060\) mm | Between tooth 15 and tooth 16 |
| Symmetry deviation \(f_{As}\) | \(0.0286\) mm | Mean of all single deviations |
| Design requirement | Satisfied | Based on the measured indices |
| Anomalous tooth | Tooth 15 | Large right-hand contact-line error |
I investigated the anomalous tooth. The right-hand contact line of tooth 15 has a larger error than the neighboring teeth. This produces a local change in the V-vertex axial position. The likely cause is that this tooth was used as a trial grinding tooth. During process setup, the grinding wheel helix angle was adjusted several times. Those adjustments left a larger helix deviation on this tooth. The full-tooth contact-line method detects this effect because it evaluates the relative intersection of opposite-hand contact lines rather than the absolute deviation of a single flank. This is a practical advantage of my method for screw gears.
The dynamic meaning of the indices is important. The single alignment deviation \(f_{A,i}\) affects the phase between the right-hand and left-hand meshing events for one tooth pair. The adjacent difference \(f_{Au}\) affects short-period axial vibration. The total deviation \(F_A\) affects long-period axial vibration and periodic load sharing. The symmetry deviation \(f_{As}\) affects a constant axial bias. For screw gears, these four effects can appear even when both hand sections separately satisfy ordinary helical-gear tolerances. Therefore, I report them together as a vector of functional alignment parameters.
I can also represent the vertex distribution by a Fourier series. This representation helps me separate once-per-revolution effects from higher-order effects. In the equation below, \(\phi_i\) is the angular position of tooth \(i\), \(A_k\) and \(B_k\) are Fourier coefficients, and \(\varepsilon_i\) is the residual. The zeroth-order term is exactly the symmetry deviation \(f_{As}\).
$$ f_{A,i} = f_{As} + \sum_{k=1}^{K} \left[ A_k \cos(k\phi_i) + B_k \sin(k\phi_i) \right] + \varepsilon_i $$
For screw gears, the first harmonic is often related to eccentricity or clamping error. The tooth-frequency harmonic is related to individual tooth errors. The low-order harmonics are related to gradual changes along the circumference. I use this decomposition as a diagnostic tool after the primary indices are computed. It does not replace \(F_A\), \(f_{Au}\), \(f_{As}\), and \(f_{A,i}\), but it helps me identify the source of an alignment pattern.
I also relate the alignment deviation to axial load sharing. If the two hand sections of screw gears are perfectly aligned, the axial forces cancel in an ideal manner. If the vertices are shifted, the left-hand and right-hand contact lines do not reach their load-carrying positions at the same axial coordinate. A first-order relation between the axial force imbalance and the local alignment deviation can be written as follows. The coefficient \(K_a\) represents the axial stiffness influence of the mesh.
$$ \Delta F_{axial,i} \propto K_a \left(f_{A,i}-f_{As}\right) $$
The dynamic response of screw gears can be represented by a lumped axial model. The right-hand side contains the sum of the mesh forces from all tooth pairs. Each mesh force depends on the local alignment deviation and on the rotation angle. This shows why the axial vertex distribution is not merely a static inspection number. It enters the equation of motion directly.
$$ m\ddot{z} + c\dot{z} + kz = \sum_{i=1}^{Z} F_i\left(f_{A,i}, \Omega t\right) $$
I now discuss measurement uncertainty. The full-tooth contact-line method is robust, but it still has error sources. The most important are probe radius compensation, spindle axis alignment, workpiece clamping, thermal drift, contact-line point selection, least-squares fitting, and intersection sensitivity. The following table lists these sources and the actions I take to reduce them.
| Error source | Effect on vertex coordinate | Mitigation |
|---|---|---|
| Probe radius and offset | Shifts the fitted contact line | Calibrate the probe before measurement |
| Spindle axis alignment | Creates a sinusoidal axial error | Align the workpiece and check runout |
| Clamping deformation | Changes the symmetry plane | Use uniform clamping and measure end faces |
| Thermal drift | Creates a slow axial trend | Stabilize temperature and repeat measurements |
| Point selection | Changes the fitted slope | Use the same evaluation interval for all teeth |
| Least-squares fit | Averages random point noise | Use enough points and check residuals |
| Intersection sensitivity | Amplifies slope errors | Monitor \(m_R-m_L\) and re-measure weak teeth |
The sensitivity of the vertex to slope and intercept errors can be estimated by first-order propagation. The vertex axial coordinate is a function of four fitted parameters: \(m_R\), \(m_L\), \(b_R\), and \(b_L\). If the uncertainties of these parameters are known, the uncertainty of \(z_A\) can be approximated by the following expression.
$$ \delta z_A = \sqrt{ \left(\frac{\partial z_A}{\partial m_R}\delta m_R\right)^2 + \left(\frac{\partial z_A}{\partial m_L}\delta m_L\right)^2 + \left(\frac{\partial z_A}{\partial b_R}\delta b_R\right)^2 + \left(\frac{\partial z_A}{\partial b_L}\delta b_L\right)^2 } $$
The partial derivatives show the conditioning of the intersection. The derivative with respect to the slope difference contains the intercept difference. The derivative with respect to the intercepts contains the slope difference. Therefore, a large slope difference stabilizes the vertex. A small slope difference makes the vertex sensitive to both slope and intercept errors. In my measurement of screw gears, the helix angle is large enough that the slope difference is acceptable. I nevertheless use this propagation formula as a quality check.
I also compare my full-tooth contact-line method with the conventional independent-flank method. The following table summarizes the differences for screw gears.
| Aspect | Conventional independent-flank inspection | Full-tooth contact-line method |
|---|---|---|
| Measurand | Separate deviations of right-hand and left-hand sections | Intersection of opposite-hand contact lines |
| Alignment assessment | Indirect and incomplete | Direct vertex axial coordinate |
| Functional meaning | Limited for screw gears | Strong relation to axial vibration and load sharing |
| Measurement interval | Usually full flank | Controlled middle interval |
| Output | Profile, helix, pitch of each section | \(f_{A,i}\), \(f_{Au}\), \(F_A\), \(f_{As}\) |
| Anomaly detection | May miss opposite-hand misalignment | Sensitive to relative vertex position |
My method also has clear limitations. I currently define the V-shaped vertex from the axial projection of contact lines. This captures the axial alignment distribution, but it does not fully describe the circumferential position distribution of the vertices. For screw gears with asymmetric wear or asymmetric modification, the circumferential coordinate of the vertex may also matter. In future work, I plan to extend the vertex definition to a three-dimensional point and to evaluate both axial and circumferential deviations. I also plan to combine the measured vertex distribution with a dynamic model of screw gears so that the inspection indices can be mapped directly to noise and vibration targets.
From a manufacturing viewpoint, the indices have practical meaning. A large \(F_A\) indicates that the relative axial position of the two hand sections changes around the circumference. This can come from table rotation error, workpiece eccentricity, or a gradual change in grinding conditions. A large \(f_{Au}\) indicates a sudden change between neighboring teeth. This can come from a trial tooth, a local wheel dressing problem, or an indexing error. A non-zero \(f_{As}\) indicates a systematic offset of the entire V-vertex distribution. This can come from the axial setting of the two hand sections, from the symmetry plane calibration, or from a systematic wheel position error. The following table lists these interpretations.
| Observation | Likely cause | Process action |
|---|---|---|
| Large \(F_A\) | Gradual axial misalignment around the circumference | Check table rotation, clamping, and wheel path |
| Large \(f_{Au}\) | Local tooth anomaly or trial tooth | Inspect the anomalous tooth and re-grind if required |
| Large positive \(f_{As}\) | Systematic upward axial shift | Adjust the axial setting of the hand sections |
| Large negative \(f_{As}\) | Systematic downward axial shift | Adjust the axial setting of the hand sections |
| Large \(\sigma_A\) | Random tooth-to-tooth variation | Check indexing, thermal stability, and dressing |
| Large low-order harmonic | Eccentricity or once-per-revolution error | Check centering and spindle runout |
For the screw gears in my test, the total alignment deviation of \(0.0082\) mm satisfies the design requirement. The symmetry deviation of \(0.0286\) mm also satisfies the design requirement. The anomalous tooth 15 increases the adjacent alignment difference to \(0.0060\) mm. If the design target for adjacent difference is tighter, tooth 15 should be reworked or the grinding process should be adjusted so that trial teeth are not left in the final set. This example shows that the full-tooth contact-line method is not only a research tool but also a practical inspection method for screw gears.
I can also use the measured vertex distribution to define a tolerance zone. Instead of requiring every flank to lie inside a separate tolerance band, I can require every V-shaped vertex to lie inside an axial band of width \(T_A\) around the symmetry plane. I can additionally require the adjacent change to be below \(T_{Au}\). This is a functional tolerance scheme for screw gears. The following inequalities express such a scheme. The values \(T_A\), \(T_{Au}\), and \(T_{As}\) are chosen from the dynamic requirements of the transmission.
$$ \max_i f_{A,i} – \min_i f_{A,i} \le T_A $$
$$ \max_i \left| f_{A,i+1}-f_{A,i} \right| \le T_{Au} $$
$$ \left| f_{As} \right| \le T_{As} $$
This tolerance scheme is more directly related to the behavior of screw gears than a pair of independent helical-gear tolerance schemes. It focuses on the opposite-hand relationship that defines the screw gears. It also provides a clear acceptance criterion for the full-tooth contact-line method. I believe this is the main contribution of my approach.
From the metrology side, I emphasize that the contact-line method is efficient. It does not require a dense scan of the entire tooth surface. It requires only enough points inside a well-chosen evaluation interval to fit a stable line. Because the fit averages random point noise, the method can achieve good repeatability with a moderate number of points. The evaluation interval also reduces the influence of form modification and edge effects. For screw gears, this means that the measured vertex axial coordinate reflects the load-carrying part of the tooth rather than the non-working ends.
From the algorithmic side, I use the following sequence. First, I read the point data for each tooth and each hand. Second, I project the points onto the unwrapped axial-circumferential plane. Third, I apply the evaluation interval. Fourth, I fit the least-squares line. Fifth, I compute the intersection. Sixth, I store the vertex axial coordinate. After all teeth are processed, I compute the indices. I then optionally perform Fourier decomposition and uncertainty propagation. This sequence is simple enough for online inspection and complete enough for engineering evaluation of screw gears.
The following table lists the data items I store for each tooth. These items form the traceability record for the screw gears.
| Data item | Description |
|---|---|
| Tooth index | Tooth number around the screw gears |
| Right-hand slope | \(m_R\) from least-squares fit |
| Right-hand intercept | \(b_R\) from least-squares fit |
| Left-hand slope | \(m_L\) from least-squares fit |
| Left-hand intercept | \(b_L\) from least-squares fit |
| Vertex axial coordinate | \(z_{A,i}\) or \(f_{A,i}\) |
| Vertex circumferential coordinate | \(u_{A,i}\) |
| Fit residual | RMS residual of the contact-line fit |
| Slope difference | \(m_R-m_L\), used as a conditioning check |
I also note that the method is compatible with different tooth forms, as long as the contact line can be identified in the evaluation interval. For screw gears with profile modification, the contact line may be slightly curved. In that case, a straight-line fit is still a good local approximation inside the evaluation interval. If the curvature is large, I can use a second-order fit and compute the intersection of the local tangents at the evaluation interval center. This extension keeps the same vertex concept while adapting to stronger modification.
The following equation shows a second-order extension. The right-hand contact line is approximated by a quadratic polynomial, and the left-hand contact line is approximated by another quadratic polynomial. The vertex is obtained from the intersection of the two local tangents at the center of the evaluation interval. This reduces the influence of symmetric curvature on the vertex estimate.
$$ z_R(u) = a_{R2}u^2 + a_{R1}u + a_{R0} $$
$$ z_L(u) = a_{L2}u^2 + a_{L1}u + a_{L0} $$
$$ z_{A} = z_R(u_0) + \left.\frac{dz_R}{du}\right|_{u_0}\left(u_A-u_0\right) = z_L(u_0) + \left.\frac{dz_L}{du}\right|_{u_0}\left(u_A-u_0\right) $$
For most screw gears, the first-order method is sufficient. The second-order method is a fallback for heavily modified flanks. I keep both options in my measurement software.
In conclusion, I have presented a first-person framework for measuring and evaluating the alignment of screw gears using tooth-flank contact lines. I defined the V-shaped vertex as the intersection of opposite-hand contact-line midlines. I derived the parametric flank equations, the contact-line projection, the least-squares fitting equations, and the vertex intersection equations. I defined four primary evaluation indices: single alignment deviation, adjacent alignment difference, total alignment deviation, and symmetry deviation. I added statistical and Fourier diagnostics for deeper analysis. I validated the method on screw gears with 27 teeth, a normal module of 2.5 mm, a pressure angle of 20 degrees, and a helix angle of 30 degrees. The total alignment deviation was 0.0082 mm, and the symmetry deviation was 0.0286 mm. Both satisfied the design requirement. The method detected an anomalous trial tooth through the adjacent alignment difference. This confirms that the full-tooth contact-line method is effective, convenient, and functionally meaningful for screw gears.
My main conclusion is that screw gears should not be evaluated only as two independent helical gears. The V-shaped vertex is the natural connection between the two hand directions. Its axial distribution controls axial vibration, noise, and load sharing. By measuring contact lines and intersecting their least-squares midlines, I obtain a direct and robust estimate of that distribution. I therefore recommend the full-tooth contact-line method for design analysis, process control, and final inspection of high-precision screw gears.