In modern mechanical systems, gear transmissions play a pivotal role due to their efficiency and reliability. However, errors in manufacturing and assembly, such as gear shaft axis parallelism deviations, significantly impact transmission accuracy, dynamic behavior, vibration, and noise. Traditional measurement methods for gear shaft parallelism errors often rely on contact techniques requiring high-precision reference planes or complex image processing algorithms, which can introduce limitations in accuracy and applicability. This paper proposes a non-contact optical measurement approach based on the Digital Image Correlation Method (DICM) to dynamically assess gear shaft axis parallelism errors. By leveraging DICM’s full-field displacement measurement capabilities and rigid-body kinematics principles, the spatial orientation of gear shafts is determined, enabling precise calculation of parallelism errors and transmission deviations. The method is validated through experimental studies on a single-stage reduction gear pair, demonstrating its effectiveness in real-time monitoring and error analysis.
The Digital Image Correlation Method is a non-contact, full-field optical technique that tracks surface deformations by analyzing digital images. It involves selecting a subset of pixels from a reference image and correlating it with deformed images to compute displacement vectors. For gear shaft measurements, DICM captures the motion of target points on gear shaft end faces, facilitating the derivation of rotational and translational matrices. The fundamental equation for DICM displacement calculation is given by:
$$ u_x = x’ – x, \quad v_y = y’ – y $$
where \( u_x \) and \( v_y \) represent the displacement components in the x and y directions, respectively, and \( (x, y) \) and \( (x’, y’) \) denote the coordinates of points in the reference and deformed images. To ensure accurate tracking, the gear shaft surfaces are coated with speckle patterns, which provide random灰度 variations essential for correlation analysis. The correlation function, typically based on normalized cross-correlation or sum of squared differences, is optimized to achieve sub-pixel accuracy, with DICM systems offering displacement resolutions as fine as 0.01 pixels in-plane and 0.03 pixels out-of-plane.
The gear shaft axis parallelism error is defined in terms of two orthogonal planes: the reference plane H and the vertical plane V. The reference plane H contains the reference gear shaft axis and passes through a point defined by the intersection of the other axis and the mid-width plane of the gear. The vertical plane V is perpendicular to H and passes through the same point, parallel to the reference axis. The parallelism errors \( \Delta\gamma_{xz} \) and \( \Delta\gamma_{yz} \) are projections of the spatial angle between gear shaft axes onto planes H and V, respectively. If the direction vector of the reference gear shaft axis is \( \mathbf{n}_1 = (n_{1x}, n_{1y}, n_{1z}) \) and that of the other gear shaft axis is \( \mathbf{n}_2 = (n_{2x}, n_{2y}, n_{2z}) \), the errors are computed as:
$$ \Delta\gamma_{xz} = \arccos\left( \frac{n_{1x}}{\sqrt{n_{1x}^2 + n_{1z}^2}} \right) – \arccos\left( \frac{n_{2x}}{\sqrt{n_{2x}^2 + n_{2z}^2}} \right) $$
$$ \Delta\gamma_{yz} = \arccos\left( \frac{n_{1y}}{\sqrt{n_{1y}^2 + n_{1z}^2}} \right) – \arccos\left( \frac{n_{2y}}{\sqrt{n_{2y}^2 + n_{2z}^2}} \right) $$
Thus, determining the gear shaft axis direction vectors \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) is crucial for parallelism error measurement.
To obtain the gear shaft axis direction vectors, DICM is applied to measure coordinate changes of target points on gear shaft end faces. Based on rigid-body kinematics, the motion of a gear shaft from position 1 to position 2 can be described by a rotation matrix \( \mathbf{R} \) and a translation vector \( \mathbf{v} \). For a set of non-collinear points \( \mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n \) in the reference image and their corresponding points \( \mathbf{p}_1, \mathbf{p}_2, \dots, \mathbf{p}_n \) in the deformed image, the transformation is given by:
$$ \mathbf{p}_i = \mathbf{R} \mathbf{a}_i + \mathbf{v}, \quad i = 1, 2, \dots, n $$
The rotation matrix \( \mathbf{R} \) is orthogonal, satisfying \( \mathbf{R}^T \mathbf{R} = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix. The average vectors \( \bar{\mathbf{a}} \) and \( \bar{\mathbf{p}} \) are computed as:
$$ \bar{\mathbf{a}} = \frac{1}{n} \sum_{i=1}^n \mathbf{a}_i, \quad \bar{\mathbf{p}} = \frac{1}{n} \sum_{i=1}^n \mathbf{p}_i $$
A matrix \( \mathbf{M} \) is defined as:
$$ \mathbf{M} = \frac{1}{n} \sum_{i=1}^n (\mathbf{p}_i \mathbf{a}_i^T) – \bar{\mathbf{p}} \bar{\mathbf{a}}^T $$
The rotation matrix \( \mathbf{R} \) is then derived from the eigenvalues and eigenvectors of \( \mathbf{M}^T \mathbf{M} \). Let \( D_{11} \) and \( D_{22} \) be the eigenvalues, and \( \mathbf{V} \) be the matrix of eigenvectors. Then, \( \mathbf{R} \) is expressed as:
$$ \mathbf{R} = \begin{bmatrix} \frac{1}{D_{11}} \mathbf{m}_1 & \frac{1}{D_{22}} \mathbf{m}_2 & \frac{1}{D_{11} D_{22}} (\mathbf{m}_1 \times \mathbf{m}_2) \mathbf{V}^T \end{bmatrix} $$
where \( \mathbf{m}_1 \) and \( \mathbf{m}_2 \) are columns of \( \mathbf{M} \mathbf{V} \). The translation vector \( \mathbf{v} \) is calculated as:
$$ \mathbf{v} = \bar{\mathbf{p}} – \mathbf{R} \bar{\mathbf{a}} $$
Alternatively, the gear shaft motion can be described as a rotation about an axis and a translation along it. Let \( \mathbf{n} \) be the unit direction vector of the gear shaft axis, and \( \phi \) be the rotation angle. The rotation matrix can be decomposed as:
$$ \mathbf{R} = \mathbf{I} \cos\phi + \mathbf{n} \mathbf{n}^T (1 – \cos\phi) + [\mathbf{n}]_\times \sin\phi $$
where \( [\mathbf{n}]_\times \) is the skew-symmetric matrix of \( \mathbf{n} \). The rotation angle \( \phi \) and axis vector \( \mathbf{n} \) are obtained from:
$$ \sin\phi = \frac{1}{2} \sqrt{ (\mathbf{R}_{32} – \mathbf{R}_{23})^2 + (\mathbf{R}_{13} – \mathbf{R}_{31})^2 + (\mathbf{R}_{21} – \mathbf{R}_{12})^2 } $$
and
$$ \mathbf{n} \sin\phi = \frac{1}{2} \begin{bmatrix} \mathbf{R}_{32} – \mathbf{R}_{23} \\ \mathbf{R}_{13} – \mathbf{R}_{31} \\ \mathbf{R}_{21} – \mathbf{R}_{12} \end{bmatrix} $$
By applying this to both driving and driven gear shafts, their axis direction vectors \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) are determined, allowing parallelism error computation.
The experimental setup for measuring gear shaft parallelism errors utilizes a Vic-3D high-speed DICM system. This system includes two high-speed cameras with Nikon 60 mm f/2.8D wide-angle lenses, LED area lights, and a calibration target with a 20 mm dot spacing for a field of view of 200–300 mm. The cameras have a resolution of 1280 × 1024 pixels, and the software Vic-Snap is used for displacement and strain analysis. The displacement measurement accuracy is 0.01 pixels in-plane and 0.03 pixels out-of-plane, corresponding to approximately 0.01 mm for a field of view of 300 mm × 240 mm. To enhance measurement precision, rigid plates are attached to the gear shaft end faces, and speckle patterns with diameters of 1 mm are applied, ensuring sufficient pixel coverage for correlation.
The experiment involves a single-stage reduction gear pair with a driving gear shaft having 24 teeth and a driven gear shaft with 40 teeth. The gear shafts are rotated at a constant speed of 10 rpm under no-load conditions to ensure steady motion. Images are acquired at a sampling frequency of 125 Hz for 35 seconds, limited by the camera’s onboard memory. Prior to acquisition, the system is calibrated using the calibration target to determine intrinsic and extrinsic parameters. In Vic-Snap, analysis regions are selected on the gear shaft end faces, with a subset size of 21 pixels and a step size of 7 pixels. Displacement data from multiple target points are exported for post-processing.
The following table summarizes the key parameters of the experimental setup:
| Parameter | Value |
|---|---|
| Gear Shaft Teeth (Driving/Driven) | 24/40 |
| Rotation Speed | 10 rpm |
| Camera Resolution | 1280 × 1024 pixels |
| Sampling Frequency | 125 Hz |
| Subset Size | 21 pixels |
| Step Size | 7 pixels |
| Field of View | 300 mm × 240 mm |
From the displacement data, the rotation matrices and translation vectors for both gear shafts are computed. The axis direction vectors \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) are derived, and the parallelism errors \( \Delta\gamma_{xz} \) and \( \Delta\gamma_{yz} \) are calculated. Additionally, the rotation angles \( \phi_1 \) and \( \phi_2 \) of the driving and driven gear shafts are obtained, enabling the determination of transmission error (TE), defined as the difference between actual and theoretical displacement:
$$ TE = \phi_2 – \frac{z_1}{z_2} \phi_1 $$
where \( z_1 \) and \( z_2 \) are the tooth numbers of the driving and driven gear shafts, respectively.
The results indicate that the gear shafts rotate at nearly constant speeds, with mean driving gear shaft speed of 10.17 rpm and driven gear shaft speed of 6.10 rpm, consistent with the gear ratio. The parallelism errors exhibit periodic behavior, corresponding to the meshing cycle of gear pairs. For the driving gear shaft, the axis direction vector \( \mathbf{n}_1 \) projects onto the H and V planes with angles relative to the Z-axis, as shown in the following table:
| Plane | Angle with Z-axis (degrees) |
|---|---|
| H (XZ) | θ_{1H} |
| V (YZ) | θ_{1V} |
Similarly, for the driven gear shaft, the axis direction vector \( \mathbf{n}_2 \) projects as:
| Plane | Angle with Z-axis (degrees) |
|---|---|
| H (XZ) | θ_{2H} |
| V (YZ) | θ_{2V} |
The parallelism error \( \Delta\gamma_{xz} \) on the H plane has a peak-to-peak value of 0.64°, with positive and negative peaks of 0.31° and -0.33°, respectively, and a mean of 0.01°. On the V plane, \( \Delta\gamma_{yz} \) shows a peak-to-peak value of 0.64°, with peaks of 0.41° and -0.22°, and a mean of 0.01°. The periodicity aligns with the gear meshing cycle, which for this gear pair corresponds to 1080° rotation of the driving gear shaft or 1800° of the driven gear shaft.
The transmission error curve, derived from the rotation angles, also displays periodic fluctuations, with amplitude variations reflecting the influence of gear shaft misalignments. The following equation summarizes the relationship between gear shaft motion and transmission error:
$$ TE(t) = \phi_2(t) – \frac{24}{40} \phi_1(t) $$
where \( t \) denotes time. The results validate the DICM-based approach for dynamic measurement of gear shaft parallelism errors and transmission behavior.
In conclusion, the Digital Image Correlation Method provides an effective non-contact solution for measuring gear shaft axis parallelism errors. By integrating DICM with rigid-body kinematics, the spatial orientation of gear shafts is accurately determined, enabling computation of parallelism errors and transmission deviations. The experimental results demonstrate the method’s capability to capture dynamic changes in gear shaft alignment, with high precision and full-field measurement advantages. This approach facilitates real-time monitoring of gear systems under operational conditions, contributing to improved design and maintenance practices. Future work could explore applications in complex gear assemblies and the integration of DICM with other sensing technologies for comprehensive error analysis.
The proposed method underscores the importance of accurate gear shaft alignment in mechanical transmissions. By repeatedly analyzing gear shaft motions through DICM, engineers can identify misalignments early, reducing wear and enhancing system longevity. The use of mathematical formulations, such as the rotation matrix and axis direction vectors, ensures robust error quantification, while the optical nature of DICM eliminates the need for physical contact, minimizing interference with gear operation. Overall, this study highlights the potential of DICM as a versatile tool for gear shaft error measurement in various industrial applications.