Abstract
This paper proposes an acoustic signal-aided detection method for evaluating the transmission accuracy of spiral bevel gears based on Particle Swarm Optimization (PSO) and Maximum Correlated Kurtosis Deconvolution (MCKD). The method combines a gear rolling inspection machine with a developed acoustic signal acquisition system to conduct real-time acoustic signal acquisition experiments to detect the quality of tooth surface contact marks on spiral bevel gears. By analyzing the effects of different axial deviations on gear mesh acoustic signals, it is found that an increase in axial deviation leads to a rise in the amplitude of mesh frequency and its harmonics, significantly affecting gear transmission accuracy. Practical examples verify the feasibility of the proposed method, providing an effective auxiliary means for assessing the meshing performance of spiral bevel gears.
Keywords: Spiral bevel gear, acoustic signal detection, transmission accuracy, contact mark, signal processing
1. Introduction
Spiral bevel gears, due to their high overlap ratio, smooth meshing, and high load-bearing capacity, are widely used in complex machinery equipment [1-2]. However, during operation, the transmission accuracy and stability of the entire system can be affected by factors such as installation errors. In engineering practice, engineers often assess the meshing performance of spiral bevel gears by observing contact marks, which is highly subjective and relies heavily on their experience, making it difficult to accurately reflect the operational status of the gears [3]. Therefore, exploring new methods for detecting the transmission accuracy of spiral bevel gears is crucial for improving the performance of gear transmission systems.
2. Literature Review
In recent years, scholars worldwide have conducted in-depth research on tooth surface contact marks of spiral bevel gears. Men et al. [4] used a camera to capture images of contact marks and explore the relationship between the mark area and gear installation distance. Su et al. [5] studied the influence of installation errors on tooth surface contact marks. Simon et al. [6] employed Tooth Contact Analysis (TCA) technology to investigate the meshing performance of spiral bevel gears. Wang et al. [7] analyzed contact marks with installation errors. While these studies focus on qualitative analysis of contact marks, there is still a lack of comprehensive methods for assessing transmission accuracy.
Acoustic measurement methods, as non-contact and real-time detection techniques, have gained increasing attention. Hu et al. [8] studied the resonance frequency and dynamic stress variations of turbine engine gears using acoustic waveguide and resistance strain gauge technology. Guo et al. [9] proposed a non-contact measurement method based on acoustic detection for the traveling wave resonance of aeroengine bevel gears. Luan et al. [10] developed a derived noise measurement method and system dynamic calibration method, providing a new approach for precise detection of aeroengine bevel gears.
These acoustic detection methods capture and analyze the sound signals generated during gear transmission, enabling precise detection of gear performance. The sound signals arise from the complex vibrations caused by gear interactions and collisions, further transforming into acoustic waves with superimposed frequencies. To recover the original impact pulses from complex sound signals, McDonald et al. [11] introduced MCKD, which effectively separates periodic pulse components from noise. Li et al. [15] improved MCKD performance by optimizing its parameters using PSO.
However, the application of acoustic signals for assessing transmission accuracy in spiral bevel gears remains limited. This paper aims to address this gap by proposing an acoustic signal-aided detection method based on PSO-MCKD, enhancing the accuracy and efficiency of spiral bevel gear transmission accuracy detection.
3. Contact Marks of Spiral Bevel Gears
Contact marks, left on the tooth surfaces during spiral bevel gear meshing, are crucial indicators of gear transmission performance. They reflect the meshing quality, shape, size, and position, significantly influencing the smooth operation and lifespan of gears. Axial deviations, inevitable during gear installation, can lead to unreasonable tooth surface contact marks, causing increased vibrations, noise, and compromised transmission system stability. Axial deviations can be classified into large wheel deviations (ΔG) and small wheel deviations (ΔT).
In engineering practice, gear meshing performance is often evaluated using empirical or iterative adjustment methods, which are time-consuming and ineffective. The close correlation between tooth surface contact marks and acoustic characteristics motivates the introduction of acoustic measurement methods to assess contact mark rationality and enhance the accuracy of spiral bevel gear meshing performance evaluation.
4. Acoustic Signal Processing Method Based on PSO-MCKD
PSO-MCKD aims to extract periodic impact signals from strong noise environments. MCKD effectively highlights signal features, enabling sound signal detection even at low signal-to-noise ratios. By combining PSO and maximum correlated kurtosis indicators, PSO-MCKD excels in adaptive parameter selection, feature extraction from noisy environments, global optimization, and robustness. This section introduces the basic principles of MCKD and PSO and outlines the acoustic signal processing flow.
4.1 Basic Principles
4.1.1 Maximum Correlated Kurtosis Deconvolution (MCKD)
MCKD essentially treats the acquired signal as the input and employs maximum correlated kurtosis to find an optimal finite impulse response filter. The mathematical model is given by:
y(u)=h(u)∗x(u)+e(u)
where y(u) is the input signal, h(u) is the transmission response, x(u) is the periodic pulse, and e(u) is noise. The maximum correlated kurtosis KMC(T) is defined as:
K_{MC}(T) = \max_f \left[ \frac{\sum_{r=1}^{R} \left( \prod{1/(M+1)}
where T is the deconvolution period and M is the maximum shift. The optimal filter f is:
f=∥x∥4∥x2∥2∥B2∥(Y0Y0T)−1∑m=0MY−mTAm
where matrices B, Ym, and Am are defined in the original MCKD literature [11].
4.1.2 Particle Swarm Optimization (PSO)
PSO searches for the optimal solution by leveraging the cooperation and information sharing among particles. Each particle has velocity (v) and position (s) and independently explores the optimal solution space. The velocity and position updates are:
vk+1jd=wvkjd+c1η1(pkjd−skjd)+c2η2(gkkd−skjd)
sk+1jd=skjd+vk+1jd
where w is the inertia weight, c1 and c2 are learning factors, and η1 and η2 are random numbers between 0 and 1. To balance exploration and exploitation, the inertia weight and learning factors are dynamically adjusted.
4.2 Acoustic Signal Processing Flow
The PSO-optimized MCKD process. Initially, PSO algorithm parameters are set, and the particle swarm is initialized. The fitness function values are calculated to determine local and global optima. During iterations, particle positions and velocities are updated until the termination condition is met, outputting the optimal parameters [L, M].
The spiral bevel gear acoustic signal detection flow. The acquired sound signal is processed using PSO-optimized MCKD to find optimal parameters L and M. Then, MCKD processes the signal, and the resulting components are separated to obtain the Hilbert envelope spectrum for analysis.
5. Experimental Setup and Signal Analysis
5.1 Experimental Setup
A real-time acoustic signal acquisition test rig for spiral bevel gears was designed, comprising a numerically controlled Y9550 bevel gear rolling inspection machine and an acoustic signal acquisition system. The acquisition system includes a high-performance COINV INV9206 single-directional sound pressure sensor and an Advantech PCI-1742U data acquisition card. The test rig is configured to capture and analyze sound signals from gear meshing under varying axial deviations.
5.2 Signal Analysis
To verify the reliability of the proposed method, six sets of spiral bevel gear pairs with varying axial deviations were tested. Table 1 lists the basic gear parameters, and Table 2 details the test gear types and axial deviations. The rotating frequency fr and meshing frequency fm were calculated using Equations (9) and (10), respectively.
fr=60n
fm=Z1fr
where n is the spindle speed and Z1 is the number of teeth on the pinion.
The acquired sound signals were analyzed in the frequency domain to identify changes in mesh frequency and harmonic amplitudes with varying axial deviations.
5.2.1 Forward Axial Deviation Analysis
The frequency domain analysis of sound signals with no deviation (ΔT = 0 mm), small deviation (ΔT = +0.05 mm), and large deviation (ΔT = +0.1 mm). As the axial deviation increases, the amplitudes of mesh frequency and its harmonics rise significantly (Table 3).
5.2.2 Reverse Axial Deviation Analysis
Similarly, The frequency domain analysis for negative axial deviations. Again, as the deviation increases, the mesh frequency and harmonic amplitudes rise (Table 4).
6. Verification with Contact Mark Analysis
To further validate the acoustic signal-aided detection method, contact marks on high-precision spiral bevel gear., were analyzed. The contact marks under varying axial deviations. As deviations increase, contact mark shapes and sizes change, confirming the acoustic signal analysis results.
7. Conclusion
This paper proposes an acoustic signal-aided detection method for assessing the transmission accuracy of spiral bevel gears based on PSO-MCKD. Experiments with six sets of gears under varying axial deviations verified the method’s feasibility. Key findings include:
- Design of Acoustic Signal Detection Experiments: Accurate detection of spiral bevel gear transmission accuracy and operating status was achieved through sound signal acquisition, processing, and analysis. This significantly improved detection accuracy and efficiency.
- PSO-MCKD Signal Processing: The proposed method effectively optimized MCKD algorithm precision and noise reduction performance. By enhancing the kurtosis value of original signals, it facilitated the extraction of weak transient impulses, achieving superior results in spiral bevel gear acoustic signal processing.
- Axial Deviation Effects: Experimental results revealed that increasing axial deviation leads to rises in mesh frequency and harmonic amplitudes, significantly impacting gear transmission accuracy. This finding supports acoustic signal-aided detection for evaluating contact mark-based transmission accuracy.
The proposed method enhances the accuracy and efficiency of spiral bevel gear transmission accuracy detection, overcoming limitations of traditional visual inspection methods. It provides a valuable tool for precision assessment in industrial applications.