Feature Recognition on Friction Induced Vibration of Water-Lubricated Bearing under Low Speed and Heavy Load

Abstract

Under low speed and heavy load operating conditions, the marine water-lubricated bearing (WLB) is often in a mixed lubrication state. This leads to abnormal friction-induced vibrations and noise in underwater vehicles. An empirical mode decomposition (EMD) based method for identifying friction-induced vibration characteristics of WLB was proposed, and friction-induced vibration experiments of WLB were carried out. The results showed that the specific pressure and lubricant temperature significantly affected friction-induced vibrations while the cooling water flow hardly had an effect. Studying the monotonicity of the eigenvalues of IMF energy, time-domain standard deviation, bias angle, and center of gravity frequency can visualize the trend of the bearing friction-induced vibration signals. Among them, the critical point of friction-induced vibrations could be analyzed by the inflection point of IMF energy, standard deviation, bias angle, and the sudden increase of the center of gravity frequency. The study results were valuable for revealing the friction-induced vibration mechanism of WLB and providing an important reference for design optimization.

1. Introduction

Applying water-lubricated bearing (WLB) in the defense industry has important strategic and practical significance [1,2]. In recent battles, especially in the application of underwater vehicles, WLBs have received wide attention and recognition from researchers because of their excellent vibration damping and impact resistance [3,4]. These characteristics play an essential role in improving the stealthiness of underwater vehicles. It has been shown that when an underwater vehicle is in service, the effective distance of the sonar monitoring enemy will be reduced by half for every 6 dB reduction in the noise intensity of the sound source. Under low speed, heavy load, start-stop, and other operating conditions, WLBs may cause friction-induced vibrations and generate noise due to poor lubrication. This friction-induced vibrations noise is one of the primary sources of naval radiation noise [5,6,7].

However, due to the non-linearity of the bearing rubber materials and the specificity of the working conditions, the friction-induced vibrations and noise mechanism of WLBs are still in need to be completely revealed [8]. The research on friction noise of WLBs could be traced back to the 1980s in the United States and some other countries, when their underwater vehicles’ technology reached the ‘silent’ level. Until now, the research on the generation and control mechanism of friction and vibration mechanisms of WLBs is a hot research topic in dynamics and tribology [9]. The effective identification of abnormal friction-induced vibration characteristics of water-lubricated bearings mainly depends on manual sensors to determine whether the bearings produce abnormal noise, so as to determine whether the bearings meet the standard. On the one hand, if the background noise is strong, it is difficult to be observed. On the other hand, WLBs will be heard by the ear with the advancement of material technology. High-frequency vibration can sometimes be suppressed, but other frequency bands are difficult to hear, especially high-frequency vibrations are difficult to detect. Therefore, the method to accurately identify the friction-induced vibration characteristics of WLBs under non-acoustic high frequency is needed, which is of great significance for studying the vibration mechanism of WLBs and the design of vibration reduction. Through the analysis of the research status of WLB friction-induced vibrations, there are mainly the following problems.

Currently, the research on the identification of friction-induced vibrations of WLBs mostly stays at the level of qualitative research. For example, it is difficult to accurately obtain the critical point of abnormal friction-induced vibration and its influencing factors by detecting abnormal noise through noise analyzers or by analyzing the vibration amplitude of bearings using existing time and frequency domain methods. The main reason is that the abnormal friction-induced vibrations are relatively weak and easy to hide in the noise generated by the complex test environment. Extracting the friction-induced vibrations characteristics of WLBs from the complex vibration signals is the key to solving these contradictions.

Section 2 reviews the mechanistic aspects of frictional vibration in WLBs and how to identify frictional vibrations. Section 3 presents the proposed method for identifying frictional vibrations. Section 4 describes the test rig and the experimental procedure. Section 5 presents the results obtained to demonstrate the feasibility of the proposed method using the results obtained. Section 6 presents the outlook for the application of the method and the shortcomings of this paper. Finally, the authors draw their conclusion in Section 7.

2. Related Works

2.1. Mechanistic Analysis of Friction-Induced Vibrations

The mechanism of frictional vibration is highly complex because it is an interdisciplinary study containing vibration, material science, tribology, and motion stability. The current studies on this subject can be broadly classified into three categories: analytical methods [10], experimental studies [11], and finite element methods [12,13].

In the study of analytical methods, simple single-degree-of-freedom system models were applied to analyze dynamic problems of systems, such as viscous slip mechanisms [14,15], friction coefficients related to the negative slope of velocity mechanisms [16], or problems belonging to modal coupling theory [17]. Although the equation of motion (EOM) could resolve these system problems efficiently, these solutions could not fully describe the virtual vibration behavior of WLBs due to the deformation of various materials and the inherent properties of friction. In experimental studies, Tuononen [18] calculated the relative sliding velocities of the contact surfaces between the glass plate and the rubber sample through digital image correlation. Then, he analyzed the rubber stick-slip mechanism by combining friction. In contrast, Yan [19] applied a high-speed camera to measure and track the front side of a water-lubricated rubber stern bearing a test block. In the study applying finite element methods, researchers proposed theories, such as relaxation vibration mechanism, friction force-velocity curve negating slope mechanisms, and mode coupling mechanisms in the research on friction noise in WLBs [20,21,22]. In these methods, the complex modal analysis has extensive use because it has been verified as an easier way to solve the inherent frequency of the bearing system. Nagy [23] studied the braking noise through transient analysis and considered the non-linear effect of the disc-block friction, and the results showed a good reliability.

2.2. Frictional Vibration Identification Method

Although many experts have conducted many numerical simulations, bench tests, and studies on the factors influencing the friction-induced vibrations of WLBs in naval propulsion shaft systems, the frictional vibration mechanism has not been fully clarified. This phenomenon arises from the lack of a real and reliable theoretical basis for the study of the friction-induced vibration mechanism and numerical simulation because the current conventional test methods cannot monitor and conduct qualitative and quantitative analyses for the vibration of the tail shaft system under friction excitation.

Sun [24] argued that the friction-induced vibrations could reflect the changes in the frictional wear state of the frictional vice, so he proposed that the change law of the vibration signal eigenvalue K could reflect the changes in the frictional vibration. Meanwhile, many other experts and scholars also put forward different methods to analyze frictional vibration. For example, Li applied a harmonic wavelet packet transformation to effectively extract features of the weak frictional vibration [25]. Otherwise, Chen considered time-frequency analysis as a suitable method for friction-induced vibration analysis [26]. Time-frequency analysis applies to the analysis of smooth or non-smooth signals. It has been applied to many fields, including the physical condition evaluation of mechanical systems [27], tribology [28,29], and structural vibration analysis [30].

Although academia achieved some breakthroughs in various aspects of friction and vibration, the noise generated by friction-induced vibrations cannot be fully recognized because of the test methods and technical means [31,32,33]. At present, most vibration signal analysis techniques cannot entirely reflect the characteristics of friction-induced vibrations. The qualitative and quantitative analyses of the friction-induced vibration signals of the water-lubricated bearing still need further study.

Therefore, a characteristic evaluation method based on the empirical modal decomposition (EMD) was proposed. The bearing vibration signal was recorded through sensors, and EMD decomposed the friction-induced vibration characteristic signal. Following the extraction of the eigenvalues of the feature signal, the critical characteristics of friction-induced vibrations were identified by judging the variation of each characteristic value. Finally, these data were applied to study the influence of friction-induced vibrations.

3. Feature Recognition Method on Friction-Induced Vibrations of WLBs

3.1. Identifying Method

The friction-induced vibrations are reflected in the envelope waveform, which can be used to identify the features of friction-induced vibrations. There are three mainstream analysis methods of bearing vibration signals, including the short-time Fourier analysis, wavelet analysis, and the Hilbert–Huang transform [34]. The short-time Fourier analysis has the disadvantage of not being able to take into account the need for frequency and time resolution, and the wavelet analysis has the disadvantage of lacking adaptivity compared to the Hilbert–Huang transform. The Hilbert–Huang transform is a signal processing method for analyzing non-linear and non-stationary signals. It consists of two main steps: EMD and the Hilbert transform. EMD is a data-driven method that decomposes non-stationary signals into a set of intrinsic mode functions (IMFs) and a residual term. Each IMF represents a local oscillatory mode with different scales and frequency features. By applying the Hilbert transform to each IMF, its instantaneous frequency and amplitude information can be obtained. The Hilbert transform is a method to convert real signals into complex signals by computing the analytic signal of the signal. EMD is commonly used as a pre-processing stage for the Hilbert–Huang Transform. The vibration signal caused by friction is categorized as a high-frequency vibration signal. EMD is essentially the smoothing of a signal, which results in a step-by-step decomposition of fluctuations or trends at different scales in the signal and generates a series of data sequences with different characteristic scales. Each data sequence is an IMF. Therefore, the signal of friction-induced vibrations can be accurately revealed. Moreover, the EMD is a direct decomposition rather than a filtering processing method, which is suitable for extracting and analyzing friction-induced vibration signals.

The EMD method allowed for the separation of IMF characteristic components from the friction-induced vibration signal, enabling the evaluation of the level of bearing friction vibrations. The IMF eigenvalues of the friction-induced vibrations are displayed in

Table 1. Eigenvalues of the friction-induced vibrations in WLBs.

Noise Spectrum	Eigenvalue
IMF Energy	IMF Energy Eimf
	IMF Energy ratio $P_i$
Time domain signals	Standard deviation Pt
	Bias angle $K_3$
	Form factor $S_f$
	Margin factor $C_e$
Frequency domain signals	Center of gravity frequency $F_c$
	Frequency standard deviation $R_f$

. For a more detailed understanding of the specific eigenvalues, please refer to [35].

The steps for identifying the eigenvalues of frictional vibrations are depicted in the left portion of Figure 1. The methodology for extracting the eigenvalues in both the time and frequency domains is thoroughly described in the right section of the same figure. These eigenvalues play a critical role in assessing the frictional vibrations.

The vibration signal of each start and stop test was recorded through the acquisition system. Once the vibration envelope generated by the friction-induced vibrations was detected in the test bearing, the vibration data underwent the EMD [36]. As shown in Figure 2, the time-domain signal of the test bearing is subjected to empirical mode decomposition. For the friction-induced vibration signal of this test bearing, too few IMF components cannot separate the friction-induced vibration characteristic signal. However, too many IMFs will dilute the decomposed friction-induced vibration characteristic signal, resulting in the great increase in computational work. The research result shows that the decomposition of the five IMF components is optimal. The time-domain signal of the first IMF best matches the envelope image in the original signal. The highest amplitude of its envelope image is regarded as the original signal, and the frequency-domain features also correspond to the high-frequency vibrations above.

In conclusion, the first IMF component is the characteristic component of the friction-induced vibrations, and the other components are the vibration signals of other specific frequency bands.

According to the characteristic components of friction-induced vibrations, the eigenvalues in Table 1 were extracted. In Section 4, different working conditions changed under the same test factors, then the change curve of each eigenvalue with the working parameters was drawn. Based on the change in each eigenvalue and sensitivity analysis, the eigenvalues were selected to establish the friction-induced vibration critical feature identification method. Finally, the friction-induced vibration critical features could be determined through the characteristics of the screened eigenvalue curve.

3.2. Test Scheme

The friction-induced vibrations of the water-lubricated bearing often occurs in the low-speed phase of heavy loads, such as start and stop conditions. A test scheme simulated the ship’s propulsion system, indicating that the rotational speed was reduced from high speed to 0 r/min. Given that the loading system exerts pressure on the bearing in the Y direction, it is imperative to evaluate the Y direction vibration of the test bearing. An acceleration sensor was arranged in the Y direction of the test bearing, and the Y direction sensor was also arranged in other important parts of the test stand. The specific installation details will be comprehensively discussed in Section 4.1. The acquisition system is shown in Figure 3.

During start and stop, the vibration envelope phenomenon occurred in the Y direction of the bearing (Figure 4a). Under the same operating conditions, no vibration envelope was found in the start and stop test at the 0.2 MPa specific pressure, but the vibration envelope appeared at the 1.0 MPa specific pressure (Figure 4b). The initial suspicion was caused by friction-induced vibrations.

The following steps can confirm that the vibration envelope phenomenon is caused by the bearing friction vibrations:

(1)

Check the friction moment diagram. According to Figure 5a, the starting point of the vibration envelope and the sudden increase of the friction moment correspond to each other. The sudden increase of the friction torque point indicates the bearing has entered the mixed lubrication stage from the hydrodynamic lubrication stage. In the meantime, the bearing liner and the rotating shaft form a pair of friction pairs;

(2)

View the waterfall graph. Figure 5b shows that the characteristic frequency of the envelope waveform appearing in this test bearing during the start and stop test was around 6 kHz and 8 kHz, and the appearance time was in the low-speed phase of the shutdown experiment. The vibration test conducted by Jin [37] on the SSB-100 marine water-lubricated tail shaft test showed that the characteristic frequency of all components of the test stand were within 3 kHz. Studies showed that the test stand components could not generate high-frequency vibrations;

(3)

Compare the vibration of each measurement point. Kim [38,39] stated that the friction-induced vibrations caused by WLBs is mainly self-excited. Researchers compared the Y direction time-frequency diagrams of other measurement points (the frequency is about 6 kHz and 8 kHz) and the same low-speed phase. In Figure 5b, the vertical direction vibration of various components are indicated as follows: T-Y represents the vibration of the test bearing, M-Y represents the vibration of the motor, B-Y represents the vibration of the test stand base, R1-Y represents the vibration of the left-side rolling bearing, R2-Y represents the vibration of the right-side rolling bearing, and H-Y represents the vibration of the loading device located beneath the test bearing. The comparison results of the vibration waveform and amplitude show that the vibration amplitude of the test bearing in this region is much higher than other measurement points (Figure 5b). So the friction-induced vibrations generated by the test bearing causes the envelope phenomenon on the time-domain diagram.

4. Test Rig and Experimental Procedure

4.1. Test Rig

The critical characteristics identification test for the water-lubricated bearing’s friction-induced vibrations was conducted on the SSB-100 marine water-lubricated tail shaft test bench at the Wuhan University of Technology. The test bench mainly consisted of the test, acquisition, loading, and lubrication systems. According to Figure 6 and Figure 7, the test bearing was mounted in the bearing housing, the test shaft was mounted through the test bearing, and the test shaft was mounted on the no. 1 support bearing and no. 2 support bearing. Next, the hydraulic cylinder was mounted under the bearing housing, and the load was directly applied to the test bearing through the hydraulic loading system. The loading system varied the loading force by adjusting the oil pressure. Then, the two ends of the bearing housing of the test bearing were the water outlet and water inlet, and the flow valve of the lubrication system was applied to control the cooling water flow of the water-lubricated bearing. Meanwhile, the standard skeleton was sealed at both ends of the bearing seat to ensure that the seal friction and friction noise were as low as possible with less leakage. Equipment parameters are shown in

Table 2. Basic parameters of each item of equipment.

Parts	Parameters	Value
Motor	Maximum speed	980 r/min
	Rated power	30 kW
	Rated torque	292.3 N·m
	Max. load	20 kN
Torque meter	Power supply	DC24 V
	Torque range	±500 N·m
	Rotational speed range	0~2000 r/min
	Output signal	0~10 V
	Model	LH~S08~50 kN
	Output sensitivity	2.0 ± 10% mV/V
Temperature sensor	type	PT100
	Measure range	50–300 °C
	accuracy	0.1 °C
Water pump	Power	800 W
	Power supply	AC380 V
	Rated flow rate	2 m3/h
	Rated head	15 m

. Finally, the acceleration sensor was arranged in the no. 1 rolling bearing seat, test bearing seat, no. 2 rolling bearing seat, motor, loading device and the base of the Y direction (vertical test bearing direction). A total of six sensors were applied to obtain the vibration data of each measuring point.

As shown in Figure 8, the data acquisition system for this experiment consists of an acceleration sensor, a B&K acquisition card, and a remote monitoring device. The sensor was manufactured by B&K with a sensitivity of 9.815 mVg⁻¹ and a frequency range of 1 to 10 kHz. During the acquisition, the sampling frequency was 32,768 Hz, the sampling time was 70 s, and the number of sampling points was 2,293,760.

4.2. Test Bearing

As shown in Figure 8, the bearing sleeve of the test bearing applied was stainless steel with a bearing sleeve thickness n of 20 mm. The lining is a polymer composite material with Young’s modulus of 1720 MPa, a density of 1.421 g/cm³, a Poisson ratio of 0.4, and a lining thickness m of 20 mm. The test rotating shaft uses ZQSn10-2 bushings with a journal outer diameter of $\varnothing {150}_{-0.04}^0$ mm, length of 150 mm, roundness and cylindricity of IT7 grade, and coaxiality of IT7 grade.

4.3. Experimental Procedure

Following soaking the test bearing for 48 h, the running-in test was carried out on the test bench until the bearing and the shaft were in a good lubrication state (the torque was relatively stable). In the test, the speed change of each start and stop test was constant to the control variables, which was decreased from 450 r/min to 0 r/min at a constant speed, and the processing time was 70 s. During deceleration, the specific pressure, flow rate, and temperature remained unchanged. The acquisition system recorded the vibration data of each measuring point during the whole shutdown process.

Following 20 h of the running-in test, the specific pressure (0.2/0.3/0.4/0.5/0.6/0.7/0.8/0.9/1.0 MPa), flow rate (2/4/6/8/10/12 L/min), and temperature (20/30/40/50/60 °C) were changed before each test. The specific test process is shown in Figure 9. In the pressure test, the flow rate is maintained at 5 L/min while the temperature is kept at 23 °C. In the flow test, the pressure is set to 0.8 MPa and the temperature is fixed at 30 °C. In the temperature test, the pressure is held at 0.85 MPa and the flow rate is maintained at 5 L/min. These conditions are selected to simulate the real-life conditions of a ship in operation, as well as to facilitate the observation of the frictional vibration phenomenon under heavy load and low flow rate conditions. Additionally, each test focuses on only one variable, allowing for the examination of the impact of individual variables on the results of frictional vibration identification.

The bearing’s specific pressure is calculated as follows.

$$P=\frac{W_0-G}{dl}$$

(1)

In this formula, P is the specific pressure, d is the average diameter of the bearing (150 mm), l is the length of the bearing, the pressure transducer shows the data as W₀, and the test bearing and bearing seat are weighed as G.

5. Result

5.1. Analysis of Friction Vibration Characteristics of WLBs

In summary, the friction vibration is characterized by the envelope, the change of working condition factors is to affect the lubrication state, once the lubrication state deteriorates to a certain degree the friction vibration will appear, that is, the envelope phenomenon, so the identification of the envelope can assess the degree of friction vibration.

As shown in Figure 10, when the flow rate is 5 L/min, and the cooling water is 23 °C, For each experimental condition, the bearing uniformly decelerates from an initial velocity of 450 r/min to 0 r/min over a period of 70 s. The time-frequency diagram of the test bearing in the Y direction shows significantly enhanced peak clusters at 6 kHz and 8 kHz with the increase of specific pressure. Then, after 0.5 MPa, the amplitude of vibration and the range of peak groups on the time-frequency diagram shows a substantial increase.

As shown in Figure 11, the specific pressure of the test bearing was at 0.8 MPa, with a lubricant temperature of 30 °C. As the flow rate increased from 2 L/min to 12 L/min, changes in the flow rate of the test bearing did not affect the amplitude changes in the time-frequency diagram.

In Figure 12, the specific pressure of the test was 0.85 MPa, and the cooling water flow rate was 5 L/min. During the cooling, water rose from 20 °C to 60 °C, a slight increase happened in the amplitude of friction-induced vibrations on the time-frequency diagram in the Y direction of the test bearing. The results showed that the effect of temperature on friction-induced vibrations was similar to the specific pressure test, but was not significant.

As shown in Figure 13a, the amplitude of the friction-induced vibrations of the test bearing at the specific pressure of 0.2 to 0.4 MPa in the time-domain diagram is small. However, the friction-induced vibration amplitude of the test bearing gradually increases at the specific pressure of 0.5 to 1.0 MPa. The increase in the cooling water flow in Figure 13b does not affect the vibration waveform in the time-domain diagram. The rising temperature in Figure 13c boosts the vibration amplitude of the overall waveform in the time domain. The overall trend in the time-domain diagram is consistent with the waveform pattern shown in the time-frequency diagram in Figure 8, Figure 9 and Figure 10. Based on this, the proportion of friction-induced vibration signals in the time-domain diagram is very obvious, which proves that the influence of friction-induced vibrations on the whole shaft system vibration cannot be ignored.

5.2. The Effect of the Working Condition Factor on IMF Energy

Based on the analysis in the previous section, this section uses the EMD method to analyze the vibration signal, extract the IMF energy eigenvalues, as well as the energy ratio, and study the influence law of the working condition factors on it.

Figure 14 shows the effects of specific pressure, flow rate, and temperature on the IMF energy and IMF energy ratio. IMF energy reflects the total energy of each component of the vibration signal. The IMF energy ratio represents the energy of each component of the vibration signal as a percentage of the total energy of the original signal.

As shown in Figure 14a, the IMF energy and IMF energy ratio tend to be stable under the specific pressure from 0.2 to 0.4 MPa. However, it shows a monotonic increasing trend when the specific is above 0.5 MPa. The IMF energy increased from 1530 m/s² at 0.5 MPa to 7378 m/s² at 1 MPa, and the IMF energy ratio increased from 10% at 0.5 MPa to 42% at 1.0 MPa.

As shown in Figure 14b, the IMF Energy and IMF energy ratio do not change much with the increasing flow rate. The IMF energy is maintained from 1800 m/s² to 2050 m/s², while the IMF energy ratio is maintained from 15% to 17%. Meanwhile, the flow rate does not affect the IMF energy and IMF energy ratio. This reflects that the flow rate has little effect on the eigenvalues.

According to Figure 14c, as the temperature increases, the IMF energy to IMF energy ratio increases gradually in general. The IMF energy increases from 5481 m/s² at 20 °C to 18,111 m/s² at 60 °C. In the meantime, the percentage of IMF energy increases from 32% at 20 °C to 62% at 60 °C. Both the IMF energy and the IMF energy ratio increase.

In general, It can be seen that the trend of IMF energy and IMF energy ratio is the same as the vibration in the time-domain and frequency-domain plots from Figure 10, Figure 11, Figure 12 and Figure 13. It shows that the IMF energy and IMF energy ratio can directly reflect the flow for the test bearing when friction-induced vibrations cause energy change. Specific pressure and temperature have a significant effect on the IMF energy characteristic value, while the flow rate has a small effect.

Some eigenvalues were selected for study with their increased ratios to explore the differences of different eigenvalues in identifying the critical characteristics of bearing friction-induced vibrations.

Based on the analysis of the influence of the working condition factors on the friction vibration eigenvalues, the following IMF energy eigenvalues are selected to explore their sensitivity to friction vibration. The specific pressure and temperature have a significant effect on friction-induced vibrations, while the flow rate has no obvious effect. Therefore, the effect of the flow rate is not discussed below.

According to the analysis of frictional vibration characteristics in the time domain and frequency domain, it is known that frictional vibration occurs at 0.5 MPa for specific pressure. Next, the sensitivity analysis of the eigenvalues will be performed around the critical frictional vibration condition.

The growth ratio $a_i$ is the percentage of each eigenvalue $n_i$ minus the previous eigenvalue $n_{i-1,}$ divided by the previous eigenvalue $n_i$, which is applied to analyze the sensitivity of each eigenvalue to friction-induced vibrations under different test conditions.

$$a_i=\frac{n_i-n_{i-1}}{n_i}\times 100\%$$

(2)

As shown in Figure 15a, It is known that at the specific pressure of 0.5 MPa, there is a sudden change in the growth ratio of the eigenvalue, so it is considered that the specific pressure of 0.5 MPa is the critical specific pressure for friction vibration. At this time, the IMF energy value was between 1300 and 1700 m/s², maintains a growth ratio of 4% to 23%. Following the 0.5 MPa test, the IMF energy continues to increase with the improving specific pressure, with a growth ratio of 20–55%. Compared with the 0.2–0.4 MPa test, the growth ratio increases significantly. As a result, IMF energy is susceptible to friction-induced vibration signals in the specific pressure test. In the temperature range of 20–60 °C, the IMF energy growth ratio has an upward trend between 10–60% with the temperature increase. The IMF energy continues to rise from 5k m/s² to 18k m/s². The IMF energy increases little at a low temperature but rapidly at a high temperature, indicating that the IMF energy is highly sensitive to the friction-induced vibration signals in the temperature test.

5.3. The Effect of the Working Condition Factors on Time Domain Eigenvalues

In this paper, the time domain eigenvalues of the vibration signal will be extracted for analysis, The standard deviation reflects the degree of dispersion of the data. The bias angle reflects the asymmetry of the vibration signal. If there is friction or collision in one direction, resulting in asymmetry of the vibration waveform, the bias angle index will increase. The form factor indicates the difference and distortion between the actual waveform and the standard sine wave and is often used for the diagnosis of faults in the low-frequency domain. The margin factor identifies the wear condition of the bearing.

Figure 16 shows the effects of specific pressure on the 4-time domain eigenvalue. As shown in Figure 16a, the standard deviation and bias angle in the time domain shows a monotonous increase under the influence of the specific pressure from 0.5 to 1.0 MPa. Simultaneously, the form factor and margin factors show a monotonous decrease as the specific pressure rises. The standard deviation, bias angle, and margin factors all mutate at 0.5 MPa. The standard deviation decreases from 0.027 m/s² at 0.4 MPa to 0.025 m/s² at 0.5 MPa. The bias angle decreases from 0.1 × 10⁻³ at 0.4 MPa to 0.08 × 10⁻³ at 0.5 MPa. The margin factor decreases from 200.41 at 0.4 MPa to 143.93 at 0.5 MPa. Combined with Figure 15, the frictional vibration will occur after the specific pressure of 0.5 MPa, so the time domain standard deviation and bias angle can be used as an important reference to determine the frictional vibration of the bearing.

As shown in Figure 17a,b, the bias angle, standard deviation, margin factor, and form factor in the time-domain signal remain constant as the flow rate increases. The standard deviation is between 0.25 m/s² and 0.3 m/s², The bias angle is maintained between 0.7 × 10⁻³ and 0.9 × 10⁻³. The form factor is maintained at about 1.5 and the margin factor is maintained between 120 and 80. The results show that the flow rate has little effect on the time domain eigenvalues.

Figure 18a,b shows that the time-domain standard deviation and bias angle in the time-domain signal gradually increases with the rising temperature. At the same time, the form factor and margin factor also show a monotonous decrease. The temperature rises from 20 °C to 60 °C. The standard deviation increased from 0.048 m/s² to 1.08 m/s². The bias angle increases from 0.049 × 10⁻³ to 0.089 × 10⁻³, The form factor decreases from 0.32 to 1.08, The margin factor decreases from 1.55 to 1.24 and the margin factor decreases from 80.90 to 20.95. The results show that the time domain eigenvalues have a high sensitivity to changes in temperature.

The increase in standard deviation and bias angle reflects the gradual increase in the steepness of the peak group caused by the friction-induced vibrations. The decreasing margin factor represents a decrease in the stability of its vibration waveform. In contrast, a decrease in the form factor represents a steep waveform, which also verifies the judgment of the time-frequency and time-domain plots.

In summary, as expected, specific pressure and temperature have significant effects on most of the time domain eigenvalues, while the flow rate have little effect. Among the four time domain eigenvalues, the form factor is less sensitive to the frictional vibration and can hardly reflect the timing of the frictional vibration of the bearing. The standard deviation and bias angle are more sensitive to frictional vibration. Compared with IMF energy, the sensitivity of time domain eigenvalues to frictional vibration is relatively weak.

Based on these results, the monotonic changes in standard deviation, bias angle, margin factor, and form factor can all reflect the performance changes of the test bearing after the occurrence of frictional vibration. Furthermore, it is possible to determine whether there is a critical point of bearing friction-induced vibrations by observing the sudden changes in standard deviation, bias angle, and margin factor in the time domain.

As shown in Figure 19, rare frictional vibrations occur at a specific pressure from 0.2 to 0.4 MPa, when the standard deviation growth ratio in the time domain is between 6% and 11%, with the bias angle growth ratio from 40% to 50%. Under the specific pressure from 0.5 to 1.0 MPa, the growth ratio of the standard deviation in the time domain is between 10% and 25%, and the growth ratio of the bias angle is between 10% and 76%. Once the specific pressure exceeds 0.5 MPa, the growth ratio of the standard deviation and bias angle in the time domain tend to increase continuously. The impact of temperature on the growth ratio of standard deviation in the time domain ranges from 6% to 27%. Meanwhile, the effect on the growth ratio of the bias angle also ranges from 10% to 65%. The time-domain standard deviation and bias angle keep increasing with the rising temperature. Moreover, the time-domain standard deviation and bias angle increase more rapidly at high temperatures than at low temperatures. Therefore, the time-domain standard deviation and bias angle of the time-domain eigenvalues can reflect the degree of frictional vibration.

5.4. The Effect of Working Condition Factors on the Frequency Domain Eigenvalues

This section studies the influence of working condition factors on frequency domain eigenvalues. The frequency standard deviation is used to describe the dispersion of the power spectrum energy distribution. The center of gravity frequency can describe the frequency of the signal components with larger components in the spectrum, reflecting the distribution of the signal power spectrum.

Figure 20 shows the influence of specific pressure, flow rate, and temperature on the frequency standard deviation and center of gravity frequency. As shown in Figure 20a, as the specific pressure increases, the frequency standard deviation shows a decreasing trend and the center of gravity frequency shows an increasing trend. The frequency standard deviation drops from 4 kHz at 0.2 MPa to 3 kHz and the center of gravity frequency is maintained between 6 and 7 kHz under the specific pressure from 0.2 to 0.4 MPa. Furthermore, it is within 8 kHz to 10 kHz under the range from 0.5 to 1.0 MPa. According to these findings, the specific pressure of 0.5 MPa is the sudden change point of friction-induced vibrations generation in this test.

As shown in Figure 20b, During the flow rate increase from 2 L/min to 12 L/min, the frequency standard deviation and the center of gravity frequency vary very little. The frequency standard deviation fluctuates between 3.8 kHz and 4.2 kHz. The center of gravity frequency remains stable between 7 kHz and 8 kHz.

As shown in Figure 20c, as the temperature rises from 20 °C to 60 °C, the standard deviation of the frequency gradually decreases and the center of gravity frequency gradually increases. The frequency standard deviation decreases from 3.5 kHz to 2.9 kHz. The center of gravity frequency increases from 9 kHz to 10 kHz.

In summary, the frequency standard deviation and center of gravity frequency are susceptible to increases in specific pressure and temperature, and the flow rate has little effect on them. The results show that the center of gravity frequency are more sensitive to the friction vibration caused by the increase of specific pressure than the frequency standard deviation.

As shown in Figure 21, almost no friction-induced vibrations appear at a specific pressure from 0.2 to 0.4 MPa, when the center of gravity frequency is between 6k and 7 kHz, and the growth ratio remains between 1% and 6%. It is noteworthy that when the specific pressure reaches 0.5 MPa, the center of gravity frequency changes abruptly to 8 kHz, and the growth ratio reaches 18.54%, verifying that the specific pressure of 0.5 MPa is a sudden change point. When the specific pressure is from 0.5 to 0.7 MPa, the center of gravity frequency continues to rise, and the growth ratio is between 0% and 4%. Under the specific pressure of 0.8 MPa, the center of gravity frequency rises to 9 kHz, with a growth ratio of 11.52%. When the specific pressure is from 0.8 to 1.0 MPa, the growth ratio is from 0% to 5%. From 20 °C to 60 °C, the center of gravity frequency increases continuously from 9 kHz to 10 kHz, and the growth ratio remains from 0.5% to 7%. The center of gravity frequency has a rapid growth at low temperatures, and the growth ratio of the center of gravity frequency is slow with increasing temperature. This finding indicates that the effect of temperature on the center of gravity frequency is limited. Therefore, the center of gravity frequency in the frequency-domain eigenvalues can sensitively reflect the degree of friction-induced vibrations.

6. Discussion and Prospect

A new idea for analyzing bearing friction-induced vibrations is proposed. The critical feature identification method of bearing friction-induced vibrations based on empirical modal decomposition (EMD) and can find the critical point of frictional vibration by combining the IMF energy, standard deviation, inflection point of bias angle, and the sudden increase of the center of gravity frequency. Meanwhile, the monotonic variation of these eigenvalues can visualize the bearing frictional vibration signal variation. The following potential applications exist for this study:

• Evaluation of frictional vibration performance based on each eigenvalue parameter;
• Frictional vibration feature recognition based on machine learning of each eigenvalue parameter;
• The evaluation of frictional vibration performance is conducted using cutting-edge signal processing techniques such as EEMD and VMD;
• The identification of frictional vibration characteristics for bearings during start-up tests is carried out.

However, the critical feature identification method of bearing friction-induced vibrations has some limitations, including the following two points:

• Researchers did not consider the effect of the initial speed on the friction-induced vibrations when the stop test factor was changing;
• At present, the method is only applicable to one test bearing, which is a single test method to obtain friction-induced vibrations.

7. Conclusions

This paper proposes an empirical mode decomposition (EMD) based method to identify the frictional vibration characteristics of WLBs, which can identify the critical characteristics of bearing frictional vibration. Researchers conducted a corresponding experimental study to analyze the effects of specific pressure, flow rate, and temperature on friction-induced vibrations. The following conclusions were drawn:

• In this paper, the data acquisition system successfully monitored the envelope signal in the start and stop tests. At the same time, the envelope signal was confirmed to be the friction-induced vibration signal through the judicial process. The results show that the frictional vibration can be judged by the envelope phenomenon on the time domain;
• The eigenvalues extracted by identifying the critical features of friction-induced vibrations show that both the specific pressure and temperature have a significant effect on the friction-induced vibrations of the test bearing, while the flow rate has little impact on the frictional vibration of the bearing. IMF energy is the most direct reflection of the friction-induced vibration signal changes in the eigenvalue, IMF energy tends to be stable under the specific pressure from 0.2 to 0.4 MPa. However, it shows a monotonic increasing trend when the specific is above 0.5 MPa. The IMF energy increased from 1530 m/s² at 0.5 MPa to 7378 m/s² at 1 MPa;
• This paper proposes an EMD-based method for characterizing the frictional vibration of WLBs, extracts the eigenvalues, and investigates the key parameters for evaluating critical frictional vibration. The IMF energy, standard deviation, bias angle, and center of gravity frequency all showed abrupt changes at 0.5 MPa in the specific pressure test. The critical point of frictional vibration can be found by combining the IMF energy, standard deviation, inflection point of bias angle, and sudden increase of center of gravity frequency.