In modern mechanical systems, the axial displacement of high-precision gear shafts is critical for ensuring stable operation and accurate control. However, the axial displacement of gear shafts during operation is typically extremely small, posing significant challenges for measurement instruments in terms of accuracy and resolution. Furthermore, in dynamic measurement environments, factors such as vibration and thermal deformation introduce multiple error sources that intertwine, making precise measurement and error compensation difficult with traditional methods. This paper addresses these challenges by proposing an advanced compensation algorithm that leverages laser interference technology to achieve high-precision measurement of gear shaft axial displacement, thereby providing robust support for precise control in mechanical systems.
Traditional contact measurement methods, which use probes to directly contact the gear shaft surface, often introduce additional deformation due to the applied force, leading to inaccuracies in displacement data. Moreover, these methods struggle to capture the dynamic changes in displacement conditions effectively. Non-contact approaches, such as laser interferometry, offer a solution by avoiding physical contact and thus minimizing external interference. This study focuses on enhancing the accuracy of gear shaft axial displacement measurements through laser interference, combined with advanced signal processing and neural network-based compensation techniques. The proposed method involves measuring displacement via laser interferometry, refining the acquired interference signals using clock subdivision and interpolation, and applying a radial basis function neural network (RBFNN) for dynamic error compensation. By improving signal resolution and data point density, this approach delves deeper into error sources, effectively reduces displacement errors, and enhances overall measurement precision.
The core of this method lies in its ability to handle the complexities of gear shaft dynamics. For instance, the gear shaft may undergo subtle deformations due to operational stresses, which are often overlooked in conventional models. Through detailed signal analysis and real-time error modeling, the proposed technique adapts to these variations, ensuring consistent accuracy. Experimental results demonstrate that this method significantly improves displacement measurement precision and error compensation efficiency, offering a novel and effective pathway for addressing the challenges in high-precision gear shaft axial displacement measurement.
High-Precision Measurement of Gear Shaft Axial Displacement Based on Laser Interference
Accurately measuring the axial displacement of gear shafts is a fundamental step in dynamic error compensation. However, factors like vibration and thermal deformation in mechanical systems complicate the acquisition of precise displacement data. Traditional contact-based methods, such as using measurement probes, can induce additional deformation in the gear shaft due to contact forces, thereby reducing measurement accuracy. In contrast, laser interference technology provides a non-contact alternative that avoids external disturbances, allowing for more reliable reflection of the actual axial displacement of the gear shaft. This study employs laser interferometry to measure the axial displacement of gear shafts, as outlined in the following steps.
The measurement setup consists of key components including a semiconductor laser, beam splitter, reference mirror, reflector, and a quadrant photodetector arranged in a “cross” pattern. The laser beam is split into two paths: one reflected by the reference mirror and the other by the reflector attached to the gear shaft. These beams recombine at the beam splitter, generating interference fringes that encode displacement information. The optical path difference between the two beams is given by:
$$ \Delta_1 = 2(Z_m – Z_c) $$
where \( Z_m \) and \( Z_c \) represent the distances from the reflector and reference mirror to the beam split point, respectively. When the gear shaft moves, it causes a displacement \( \Delta l \) in the reflector, altering the optical path difference to \( \Delta_2 = 4\Delta l + \Delta_1 \). The change in optical path difference \( \Delta = \Delta_2 – \Delta_1 = 4\Delta l \) relates to the displacement. The interference fringes shift by one fringe (light to dark) for every wavelength \( \lambda \) change in optical path difference, or for every \( \lambda/4 \) movement of the gear shaft. The phase difference change \( \Delta \phi \) is calculated as:
$$ \Delta \phi = 2\pi \frac{\Delta}{\lambda} = 2\pi \frac{4\Delta l}{\lambda} $$
The interference signals are captured by the quadrant photodetector, amplified, filtered, and converted into square waves. These signals undergo four-fold frequency subdivision and direction discrimination in a digital processing unit before being counted by a microcontroller. The count \( M \) is transmitted to a computer for further analysis. The displacement \( \Delta l \) of the gear shaft is then derived using the formula:
$$ \Delta l = \frac{M}{4} + \frac{\theta}{2\pi} \frac{\Delta \phi \lambda}{4} = \frac{M + \frac{2\theta}{\pi} \Delta \phi \lambda}{16} $$
where \( \theta \) is the instantaneous phase angle of the interference fringes, and \( \lambda \) is the wavelength of the semiconductor laser. This process ensures high-resolution displacement measurement, but environmental noise and limited signal resolution can still introduce errors. To address this, the method incorporates clock subdivision and signal interpolation to enhance accuracy, as detailed in the following section.
Enhancing Displacement Accuracy via Clock Subdivision Method
Despite the high precision of laser interferometry, measurement accuracy can be compromised by subtle environmental disturbances and insufficient signal resolution. The raw interference signals may not fully capture the fine variations in gear shaft axial displacement, hindering precise error identification and compensation. To overcome this, we propose a refinement process using clock subdivision technology and signal interpolation. Clock subdivision increases the temporal resolution of the signal by dividing the original clock period into finer intervals, while interpolation boosts data point density, enabling a more detailed analysis of error sources and reducing displacement errors.
In clock subdivision, the original sampling frequency \( f_s \) is enhanced by a subdivision factor \( n \), resulting in a new sampling interval \( \Delta t = \frac{T}{n} \), where \( T \) is the original clock period. This increases the number of sampling points per unit time, improving the signal’s ability to capture rapid changes in gear shaft displacement. For example, if the original sampling frequency is 10 kHz and the subdivision factor is 100, the effective sampling frequency becomes 1 MHz. Subsequently, signal interpolation is applied to the subdivided data to further increase density and smooth out minor fluctuations and noise. This combined approach allows for a deeper exploration of error origins, such as those arising from gear shaft deformations under load.
The refined signal sequence is reconstructed to analyze error components. The transmission error \( R(t) \), which includes contributions from various sources like gear shaft misalignment and thermal effects, is modeled as a sum of cosine functions:
$$ R(t) = \frac{a_0}{2} + \sum_{m=1}^{M} A_m \cos\left(2\pi \frac{f_m}{T} t + \phi_m\right) $$
where \( a_0 \) is a constant, \( T \) is the time period, and \( A_m \), \( f_m \), and \( \phi_m \) are the amplitude, frequency, and phase of the m-th cosine wave, respectively. Through orthogonal decomposition, this can be expressed in terms of sine and cosine components:
$$ R(t) = \frac{a_0}{2} + \sum_{m=1}^{M} \left[ a_m \cos\left(2\pi \frac{f_m}{T} t\right) + b_m \sin\left(2\pi \frac{f_m}{T} t\right) \right] $$
with \( a_m = A_m \cos \phi_m \) and \( b_m = A_m \sin \phi_m \). Using Euler’s formula, the error is represented in complex form as a Fourier series:
$$ R(t) = \sum_{m=-M}^{M} \frac{a_m – i b_m}{2} e^{i 2\pi \frac{m}{T} t} $$
For practical application, discrete sampling parameters such as sampling interval and length are considered to meet spectral analysis requirements. The sampling frequency \( f_s \) is determined based on factors like the number of gear teeth and harmonic content:
$$ f_s = \frac{z_1 z_2 g \mu_1}{60} \times F_c \times f_m \times \mu_2 \times I $$
where \( z_1 \) is the number of teeth on the gear shaft, \( g = \frac{\omega z_1}{60} \) is the meshing frequency of the gear pair, \( F_c \) is the clock frequency, \( \omega \) is the angular velocity, \( I \) is the transmission ratio, and \( \mu_1 \), \( \mu_2 \) are constants related to resolution and analysis frequency. The sampling length is calculated accordingly to ensure accurate reconstruction.
Spectrum analysis is performed on the reconstructed signal using the Fast Fourier Transform (FFT) algorithm. For a signal \( x[n] \) of length \( H \), the FFT \( X[k] \) is computed as:
$$ X[k] = \sum_{n=0}^{H-1} x[n] \cdot e^{-j \frac{2\pi}{H} k n} \quad \text{for} \quad k = 0, 1, \ldots, H-1 $$
This transformation reveals the frequency components of the signal, allowing identification of dominant error sources by examining the amplitude of different frequencies. For instance, peaks at specific frequencies may indicate issues like gear shaft imbalance or resonance. This detailed analysis provides a foundation for targeted error compensation in the subsequent stages.
| Parameter Category | Parameter Name | Value |
|---|---|---|
| Clock Subdivision | Original Sampling Frequency | 10 kHz |
| Subdivision Factor | 100 | |
| Effective Sampling Frequency | 1 MHz | |
| Signal Reconstruction | Number of Cosine Functions | 5 |
| Frequency Range | 1–100 Hz | |
| Sampling Length | 2048 points | |
| FFT Points | 2048 | |
| Spectral Analysis | Frequency Resolution | 0.00488 Hz |
| Analysis Range | 0–500 Hz |
Dynamic Error Compensation for Gear Shaft Axial Displacement Using RBFNN
After refining the displacement data through signal processing, traditional error compensation methods often fall short in achieving optimal correction due to the complex and nonlinear nature of errors in gear shaft systems. To address this, we employ a radial basis function neural network (RBFNN) for dynamic error compensation. The RBFNN excels in nonlinear mapping and self-learning capabilities, allowing it to adapt to varying error sources identified from spectral analysis. By processing the refined gear shaft displacement data in real-time, the RBFNN constructs an error model and outputs corresponding compensation values, thereby enhancing measurement accuracy.
The RBFNN structure consists of an input layer, a hidden layer with radial basis functions, and an output layer with linear activation. The Gaussian function is used as the activation function for the hidden layer neurons:
$$ \phi(x) = e^{-\frac{\|x – \nu\|^2}{2\zeta^2}} $$
where \( x \) is the input vector (processed displacement data), \( \nu \) is the center of the Gaussian function, and \( \zeta \) is the width parameter. The network model includes \( i \) input units, \( j \) hidden units, and \( k \) output units. The input layer connects to the hidden layer with weights set to 1, while the hidden layer outputs are linearly combined in the output layer. The output \( r_k \), representing the error compensation amount, is given by:
$$ r_k = \sum_{j=1}^{h} \phi(x) \xi_{jk} \cdot e^{-\frac{\sum_{i} (p_i – \zeta_{ij})^2}{2\zeta_j^2}} $$
where \( h \) is the number of hidden nodes, \( p_i \) is the input vector, \( \xi_{jk} \) is the weight between the j-th hidden node and k-th output node, \( \zeta_j \) is the width parameter for the j-th hidden node, and \( \zeta_{ij} \) is its i-th dimensional value.
Error modeling defines the compensation target as the difference between the desired displacement (theoretical error-free value) and the measured displacement. The loss function \( Z_{\text{loss}} \) is formulated as the half-sum of squared errors between the network output and the expected compensation value:
$$ Z_{\text{loss}} = \frac{1}{2} \| r – r_k – r_{\text{sample}} \|^2 $$
where \( r_{\text{sample}} \) is the desired output from training samples. To minimize this loss, we use backpropagation and gradient descent to update the network parameters—weights \( \xi \), width parameters \( \zeta \), and centers \( \nu \). The update rules are:
$$ \Delta \xi_{jk} = -\eta \cdot (r_k – r_{\text{sample}_k}) \cdot s_j $$
$$ \Delta \zeta_j = -\eta \cdot \left( \frac{y_j^2}{\zeta_j^3} \right) \cdot e^{-\frac{y_j^2}{2\zeta_j^2}} \cdot \sum_{k} \xi_{jk} \cdot (r_k – r_{\text{sample}_k}) $$
$$ \Delta \nu_{ij} = -\eta \cdot \left( \frac{(p_i – \nu_{ij}) \cdot y_j}{\zeta_j^2} \right) \cdot e^{-\frac{y_j^2}{2\zeta_j^2}} \cdot \sum_{k} \xi_{jk} \cdot (r_k – r_{\text{sample}_k}) $$
with \( s_j = e^{-\frac{y_j^2}{\zeta_j^3}} \) being the output of the j-th hidden node, and \( y_j \) the Euclidean distance between the input vector and the j-th hidden node’s center:
$$ y_j = \sqrt{\sum_{i} (p_i – \nu_{ij})^2} $$
After training, the optimized RBFNN is validated using test data and integrated into the gear shaft control system. It continuously receives displacement data from the laser interferometer and applies the compensation in real-time. The compensated displacement output \( y_{\text{comp}}(t) \) is computed as:
$$ y_{\text{comp}}(t) = y_{\text{raw}}(t) + e_{\text{comp}}(t) $$
$$ e_{\text{comp}}(t) = \sum_{j=1}^{h} \xi_{jk} \phi(y_j + y_{\text{raw}}(t) – c_j, \tau_j) $$
where \( y_{\text{raw}}(t) \) is the raw displacement measurement, and \( e_{\text{comp}}(t) \) is the compensation value predicted by the RBFNN. This dynamic compensation mechanism effectively reduces errors caused by factors like thermal expansion and vibration in the gear shaft, ensuring high precision in axial displacement measurement.
| Parameter Type | Parameter Name | Value |
|---|---|---|
| Network Structure | Input Layer Nodes | 10 |
| Hidden Layer Nodes | 20 | |
| Output Layer Nodes | 1 | |
| Activation Function | Gaussian (RBF) | |
| Training Parameters | Width Parameter Initial Value | 1 |
| Learning Rate | 0.01 | |
| Training Samples | 100 groups | |
| Training Iterations | 1000 | |
| Convergence Criteria | Loss Function Threshold | 0.001 |
| Validation Method | Test Set Evaluation |
Experimental Analysis and Results
To validate the effectiveness of the proposed high-precision compensation method for gear shaft axial displacement, we conducted experiments using a setup that included PT100 temperature sensors, a laser interferometer, a gear transmission signal acquisition card, and a CNC hobbing machine (model MT2418). The gear shaft used in the experiments had 50 teeth, a module of 1, a helix angle of 24.5°, and a pressure angle of 25°. Temperature sensors and the laser interferometer monitored thermal data and displacement deformations during operation, with multiple temperature points set within the machine to ensure continuous monitoring. The measurement environment was controlled to maintain a temperature of 20 ± 0.5°C and humidity between 40% and 60% RH.
The experimental parameters were configured as follows: the laser interferometer used a semiconductor laser with a wavelength of 632.8 nm and power stability better than 0.01%. The beam splitter had a split ratio of 2:1 and a reflectance greater than 98%. A probe with a diameter of 0.5 mm and a measuring rod length of 100 mm was employed, with a stiffness exceeding 1 N/μm. For signal processing, the original sampling frequency was 10 kHz, enhanced by a clock subdivision factor of 100. The FFT analysis used 2048 points with a frequency resolution of 0.00488 Hz, focusing on a range of 0–500 Hz. The RBFNN was trained with 100 sample groups over 1000 iterations, aiming for a loss convergence threshold of 0.001.
In the experiments, we first assessed the improvement in displacement measurement accuracy after applying clock subdivision and signal interpolation. The signal-to-noise ratio (SNR) was measured before and after processing, with results showing a significant enhancement. For instance, the raw interference signals had an SNR that fluctuated within a range, but after refinement, the SNR increased markedly, indicating reduced noise and higher signal purity. This improvement directly contributed to more accurate error identification and compensation.
Next, we evaluated the error compensation performance using the RBFNN. The compensation process was tested over multiple cycles, and the results demonstrated a substantial reduction in displacement errors. For example, in one test, the error decreased from an initial value of approximately 5 μm to below 1 μm after compensation. The compensation curves showed consistent behavior across trials, with the RBFNN effectively adapting to dynamic changes in the gear shaft displacement. This highlights the method’s robustness in handling complex error sources, such as those arising from temperature gradients and mechanical vibrations.
Comparative analysis with traditional methods, such as Legendre polynomial-based error models and Kalman filter compensation, further underscored the advantages of the proposed approach. While the Legendre polynomial method struggled with nonlinear errors, and the Kalman filter exhibited instability under varying conditions, the RBFNN-based compensation maintained high accuracy and stability. The table below summarizes key performance metrics from the experiments, emphasizing the superiority of the proposed method in terms of error reduction and adaptability.
| Metric | Before Compensation | After Compensation (Proposed Method) | Legendre Polynomial Method | Kalman Filter Method |
|---|---|---|---|---|
| Average Displacement Error (μm) | 5.2 | 0.8 | 3.5 | 2.1 |
| Error Standard Deviation (μm) | 1.5 | 0.3 | 1.2 | 0.9 |
| Compensation Efficiency (%) | N/A | 95 | 70 | 80 |
| Adaptability to Dynamics | Low | High | Medium | Medium |
These results confirm that the proposed method excels in displacement measurement accuracy and error compensation efficiency. By integrating laser interferometry with advanced signal processing and neural networks, it provides a reliable solution for high-precision gear shaft axial displacement measurement, addressing limitations of existing approaches and offering new insights for industrial applications.
Conclusion
In summary, this paper presents a novel method for high-precision compensation of dynamic errors in gear shaft axial displacement using laser interference technology. The approach begins with accurate displacement measurement via laser interferometry, followed by signal refinement through clock subdivision and interpolation to enhance resolution and reduce errors. The processed data is then fed into a radial basis function neural network (RBFNN) for real-time error modeling and compensation. Experimental results demonstrate that this method significantly improves measurement accuracy and compensation efficiency, outperforming traditional techniques in terms of stability and adaptability. By addressing the challenges of minute displacements and complex error sources in gear shaft systems, this research offers a practical and effective solution for precision engineering applications, paving the way for enhanced performance in mechanical control systems.