Research on Instantaneous Angular Speed Signal Separation Method for Planetary Gear Fault Diagnosis

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1. Introduction

Planetary gearboxes are characterized by their small size, high transmission ratio, and high efficiency, and are widely used in transportation, mining and metallurgy, wind power generation, and other fields. In practical operation, as a key component for transmitting torque, they are subjected to complex dynamic overload forces. The gears are prone to failure, and once a failure occurs, it can lead to serious consequences, even causing injuries or fatalities. Therefore, it is of great significance to conduct research on planetary gear systems.

Damage perception of transmission components is mainly based on the analysis of linear vibration signals. Various signal processing methods have been applied to extract gear fault characteristics, including time-frequency analysis [1] , wavelet transform [2] , empirical mode decomposition [3] , signal sparse decomposition [4] , and others. However, compared to other transmission components, conducting damage perception of planetary gear trains based on vibration signal analysis is more challenging. Due to the unique motion characteristics of planetary gear trains, even in a healthy state, theoretically, planetary gear trains will produce fault characteristic frequency components. For example, the fault characteristic frequency of the ring gear is equal to the frequency at which the planetary gear passes through the ring gear, naturally also equal to the frequency at which the planetary gear train passes through the sensor; similarly, the fault characteristic frequency of the sun gear is equal to the frequency at which the planetary gear passes through the sun gear; and regardless of whether there is damage to the planetary gear train, the frequency components of the planetary gear passing through the sun gear or ring gear always exist.

Considering that tooth damage leads to a decrease in meshing stiffness and transmission smoothness, which is the fundamental physical basis for perceiving gear damage, compared to traditional linear vibration signals, the response of instantaneous speed to changes in transmission smoothness is more direct and sensitive. Moreover, the torsional vibration signal obtained based on speed measurement is not affected by transmission paths, is not affected by multi-interface energy dissipation and multi-path signal superposition attenuation, and is less affected by high-frequency vibration components. In addition, the speed signal is the physical quantity that the state perception system needs to measure. Using the speed signal instead of the vibration signal not only is expected to obtain more effective approaches for state monitoring and fault prediction but also can reduce the hardware and testing costs of perception systems.

In recent years, a large number of scholars at home and abroad have conducted extensive research on encoder signals. Bourdon et al. [5] proposed a filter defined in the angular frequency domain, through which the detection of the filter can quantify the magnitude of angular velocity changes related to faults in the outer race of bearings. Liu et al. [6] derived a method for solving the time-varying meshing stiffness when tooth root crack faults occur based on the energy method, and established the corresponding dynamic model to extract the torsional vibration contained in the encoder response signal under crack damage conditions. Feng et al. [7] proposed extracting gear fault features from the ratio spectrum of the reciprocal of the resampled time intervals at equal angles, overcoming the difficulties of fault diagnosis under variable speeds. It is noted in the research that the encoder signal at low speeds not only contains the interested fault components but also contains same-order components caused by factors such as manufacturing or installation errors. These two components are usually mixed together, making it difficult to extract fault characteristics. Guo et al. [8] proposed a feature extraction and detection method based on the relative value of fault characteristic indicators to address the difficulty of extracting fault characteristic orders in encoder signals due to sun gear tooth root crack faults at low speeds.

The analysis methods mentioned above for encoder signals mostly achieve final fault diagnosis based on fault characteristic frequencies. However, due to the unique motion characteristics of planetary gear trains, their fault characteristic signals are often overwhelmed by other components, resulting in unclear fault characteristics and difficulty in achieving good results.

The time-domain synchronous averaging analysis method is an effective method for analyzing time-domain features of signals. By superimposing and averaging signals of interest over a certain period, it can reduce the influence of background noise and other components, and enhance the representation of vibration characteristics related to the target gear in the signal. McFadden [9] obtained meshing vibration signals of individual planetary gears through windowed time-domain synchronous averaging analysis, while Smidt [10] installed sensors on the planetary frame and monitored the gearbox’s condition through time-domain synchronous averaging analysis. Yip [11] preprocessed vibration signals using the time-domain synchronous averaging method and then extracted feature indicators from the preprocessed results to achieve gearbox fault diagnosis. Abboud [12] proposed a generalized time-domain synchronous averaging algorithm that can be applied under variable speed and load conditions.

This paper addresses the challenging issues faced by planetary gear systems, such as abundant sources of vibration, complex signal coupling and modulation mechanisms, significant influences from transmission paths, and difficulties in damage information separation. It proposes a method for fault diagnosis of planetary gears through the separation of instantaneous angular speed signals. This method can diagnose faults in planetary gear systems and assess the severity of the faults, providing strong support for the life prediction and operation and maintenance of planetary gear systems. Firstly, encoder pulse signals are obtained through a built-in encoder. Based on this, the IAS signals are calculated using the Hilbert transform. Then, the time-domain synchronous averaging technology is applied to obtain the time-domain synchronous average signal of the IAS of the planetary gear, thus achieving the fault diagnosis of the planetary gear train. Experimental results validate the effectiveness of the calculated IAS signals, demonstrating that the time-domain synchronous averaging technology can highlight impact characteristics, effectively separate and extract fault impacts, greatly reduce the testing cost of experiments, and provide an effective tool for the fault diagnosis of planetary gear trains.

2. Method for Acquiring Instantaneous Angular Velocity

In the operation of a planetary gearbox, angular displacement represents the change in speed and depends on two components: the average speed and the fluctuation component.

$θ = (\bar{ω} \pm ω) t$ (1)

Optical encoders are commonly used to obtain angular displacement. Due to the variation in rotor speed, the pulse sequence generated by the encoder also changes. Therefore, the angular velocity can be calculated by the time interval, thus allowing the calculation of angular displacement.

Due to the presence of a varying component in angular velocity, the measured pulse sequence is a phase-modulated signal. Assuming that this modulated signal is bandpass-filtered, only the content around the fundamental frequency or its harmonics is retained. Therefore, the bandpass signal can be represented as:

$s (t) = A_{c} \cos (ω_{c} t + ϕ_{c}) + \sum_{i} β_{i} \sin (ω_{i} t + ϕ_{i})$ (2)

Where $\sum_{i} β_{i} \sin (ω_{i} t + ϕ_{i})$ represents the modulating signal with a period, and the subscript c represents the carrier signal.

Neglecting the initial phase $ϕ_{c}$ and representing the phase variation of the summation term in equation (2) as $ϕ (t)$ , the modulating signal can be simplified as:

$s (t) = A_{c} \cos [w_{c} t + ϕ (t)]$ (3)

The relationship between constant speed and carrier frequency is:

$\bar{ω} = \frac{ω_{c}}{n}$ (4)

Where n is the number of pulses per revolution (PPR), defined by the encoder model. Similarly, the variation in angular velocity can also be expressed as:

$ω (t) = \frac{d ϕ (t)}{d t}$ (5)

The carrier frequency of the signal can be calculated through Fourier transform. However, demodulation techniques must be used to obtain changes in angle and thus obtain the IAS signal.

In signal processing, a signal that lacks negative frequency components is referred to as an analytic signal. The analytic representation is the complex form of the actual measured modulated signal. Therefore, the analytical form, denoted as s(t), is represented as the real part, with its Hilbert transform represented as the imaginary part.

$s_{a} (t) = s (t) + j H [s (t)] = A_{c} e^{j [ω c t + ϕ (t)]}$ (6)

The Hilbert transform of s(t) is:

$H [s (t)] = A_{c} \sin [ω_{c} t + ϕ (t)]$ (7)

To obtain the signal of angle variation, the carrier frequency component in equation (6) is first eliminated by multiplying with the complex signal $e^{- j w_{c} t}$ .

$e^{j ϕ (t)} = e^{j [ω c t + ϕ (t)]} e^{- j ω c t}$ (8)

IAS signal is

$ϕ (t) = \tan^{- 1} \frac{Imag [e^{j ϕ (t)}]}{Real [e^{j ϕ (t)}]}$ (9)

Finally, by taking the time derivative of, the signal of velocity variation or IAS can be obtained.

3. Time-Domain Synchronous Averaging Technique

The time-domain synchronous averaging technique is a common signal processing method, which mainly includes three steps: data segmentation, resampling, and averaging processing. In the application to gears, the purpose is to enhance the representation of vibration characteristics related to the target gear in the signal, while reducing the influence of background noise and other components, and extracting specific periodic signals of interest. The specific working principle is shown in Figure 1.

In the time-domain synchronous averaging technique, the vibration signal x(t) is first segmented based on the speed pulse signal to obtain the vibration signal for each revolution cycle. Due to variations in operating conditions, environmental factors, etc., the number of data points in each revolution cycle of the vibration signal may vary. Therefore, each segment of data needs to be resampled before averaging to obtain the time-domain synchronous average signal. This process can be represented as:

$x_{T S A} (t) = \frac{1}{N} \sum_{i = 1}^{N - 1} x (t + i L T_{s})$ (10)

Where x_TSA(t) is the time-domain synchronous average signal, N is the number of averages, L is the number of sampling points in each revolution cycle, T_s is the sampling interval, and the period of the vibration signal x(t) is T₀ = LT_s.

If the signal x(t) is composed of a periodic signal y(t) with period T₀ and a white noise signal n(t), then the signal x(t) can be represented as:

$x (t) = y (t) + n (t)$ (11)

Figure 1. Time domain synchronous averaging schematic.

Dividing the signal x(t) into N segments according to the period T₀ of the signal y(t) and taking the average of the corresponding points in each segment, we obtain:

$x_{T S A} (t) = y (t) + \frac{1}{\sqrt{N}} n (t)$ (12)

The averaging process exploits the uncorrelated nature of white noise. At this point, the output noise is from the original signal x(t), and the signal-to-noise ratio is improved.

4. Method for Separating Instantaneous Angular Velocity Signals for Planetary Gear Fault Diagnosis

The flowchart of the method for separating instantaneous angular speed signals for planetary gear fault diagnosis is shown in Figure 2, which includes two steps: obtaining instantaneous angular velocity and obtaining time-domain synchronous average signals.

4.1. Obtaining Instantaneous Angular Speed Signals

1) Perform Fourier transform on the real signal s(t) to transfer the signal to the frequency domain. Calculate the amplitude spectrum based on the FFT result, and

Figure 2. Methodology flow chart.

find the position corresponding to the carrier frequency in the amplitude spectrum, which is equal to the pulse count multiplied by the rotor speed.

2) Set the FFT results within the negative frequency range to zero. This achieves the real analytic representation required by equation (7). Maintain the FFT results around the carrier frequency, setting all other results in the positive frequency range to zero. Shift the filtering to the right side of the carrier frequency to the beginning of the positive frequency range, and shift the filtering on the left side of the carrier frequency to the negative frequency range of the filtering window. This result is the angle variation signal obtained by equation (9), which removes the influence of the carrier frequency.

3) By taking the time derivative of the angular displacement, the signal of velocity variation or IAS can be obtained.

4.2. Obtaining Time-Domain Synchronous Average Signals

Based on the encoder pulse signal, the IAS signal is segmented to obtain a signal within one revolution cycle. Each segment of data is resampled, and then the number of sampling points for each gear meshing is determined based on the number of teeth on the ring gear. This process is repeated for one revolution of the planetary gear, and finally, averaging is performed to obtain the time-domain synchronous average signal.

5. Experimental Signal Verification

To validate the effectiveness of the method, a test bench for simulating planetary gear faults was constructed as shown in Figure 3. The test bench consists of a variable frequency motor, a torque and speed sensor, a planetary gearbox, an encoder, a radial loader, a loading device, and a coupling. The test system used consists of vibration acceleration sensors, photoelectric encoders, acquisition cards, and data conditioning modules. The sensitivity of the vibration acceleration sensor is 10.02 mV/ms⁻² and is installed at the top and horizontal ends of the planetary gearbox. The photoelectric encoder used generates 1024 speed pulses per revolution. The parameters of the planetary gearbox are shown in Table 1. The structure of the planetary gear system and a broken tooth fault on a planetary gear are shown in Figure 4.

During the experiment, the speed was set to 600 rpm, with a sampling frequency of 51.2 kHz and a sampling time of 180 s, for both the healthy and faulty states of the planetary gear. The local encoder pulse signals for the healthy and faulty states of the planetary gear are shown in Figure 5 and Figure 6. When a gear fault exists, amplitude modulation occurs due to load fluctuations, and frequency modulation occurs due to speed fluctuations, leading to changes in the fault characteristic frequencies. The characteristic frequencies of the planetary gearbox were calculated based on the actual measured speed signals and are shown in Table 2 and Table 3.

Figure 3. Planetary gear train failure simulation test bench.

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