Identifying Inner Race Faults in Deep Groove Ball Bearing Using Nonlinear Mode Decomposition and Hilbert Transform

Singh, Swapna; Yelve, Nitesh P.

doi:10.1115/1.4065767

Graphical Abstract Figure

Abstract

This study focuses on the analysis of vibration-based signatures obtained from deep groove ball bearings with faults on the inner race. Various time−frequency-based methods are commonly used to diagnose faults in bearings. However, due to the non-self-adaptive nature of these methods and the nonlinear and nonstationary signals produced by the faults, mode decomposition techniques are seen as promising methods. This article presents a novel approach based on Nonlinear Mode Decomposition (NMD), which decomposes the complex signal into nonlinear modes. The data are taken from an online database of deep groove ball bearing with inner race faults of different sizes. These data are then subjected to NMD to extract nonlinear modes. Statistical parameters are applied to select a subset of significant nonlinear modes from the complete set. Finally, the Fast Fourier Transform is applied to the Hilbert Transform (HT) of the selected modes to see fault frequency and its higher harmonics resulting from nonlinearity. Additionally, the instantaneous frequency and instantaneous phase, two key parameters acquired from the HT, are also plotted for normal and faulty bearings, and the results are discussed in the article. The proposed method offers a valuable approach for accurately detecting and diagnosing deep groove ball-bearing faults.

1 Introduction

Bearing is an important part of a machinery and any fault in it can lead to a sudden failure, impacting the entire machinery. Early detection of faults allows for timely replacements, preventing costly breakdowns, and ensuring bearings operate within their estimated lifespan. As any machine operates, dynamic forces create vibrations, forming a unique “vibration signature.” With time, components wear out, altering the applied forces and, subsequently, the signature. Thus, the vibration signature carries crucial information about the machine's condition [1]. Bearings can experience different types of faults, such as inner race faults, outer race faults, faults in rolling elements, and faults in the cage. These faults generate vibrations, leading to changes in the vibration signature. Various signal processing techniques, including time domain and frequency domain analysis, are employed [2] to detect these changes. In previous studies [3], Fourier analysis, a frequency domain technique, is predominantly utilized to identify faults in different machine components, including bearings. This is because each component of the rotating machinery typically produces a distinct frequency.

Faults in the bearing produce nonlinear contact phenomena [4] because when the load is applied, the mass center of the inner race shifts, leading to the contact deformation between the rolling elements and races. The nonlinear springs connecting these elements to the raceways adhere to the principles of the Hertzian contact deformation theory. When a ball or a rolling element passes through the defect area, a collision impact occurs, and it leads to nonlinearity in the signal (see Fig. 1). The collision impact system of the rolling element bearing happens in three stages. First, the roller comes in contact with the beginning sidewall of the fault, which generates a vibration of an inner race, rolling element, and outer race. Second, it comes in contact with the end sidewall of the fault, for which it travels a distance that is equal to the width of the fault. Here, the contact is between the rolling elements and the outer race, and this leads to a rigid collision. Finally, the rolling element moves away from the end sidewall of the fault, hence the collision impact system vanishes [5]. Thus, the faults in the inner race, outer race, rolling element, and cage generate distinct frequencies. The vibrations in the rolling element bearings are characterized by nonlinear and nonstationary behavior, whereas Fourier Transform is effective for stationary and linear signals [6].

Fig. 1

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Bearing with inner race fault

The nonlinearity of the signals is proven by many researchers by preparing a nonlinear dynamic model of a faulty bearing similar to the one shown in Fig. 2, where the bearing is treated as a spring−mass model, and the contact between the bearing and raceway is considered as a nonlinear spring [7–9]. In addition to the nonlinear modeling of a defective bearing, Ahmadi et al. [10] improved the model by considering the finite size of the rolling elements as point masses. They compared previous models with the new ones and observed that the present work gives better results for low and high frequencies. Zheng et al. [5] also modeled the rolling bearing fault as a nonlinear system. However, this analysis is based on the collision impact. They utilized the Hertzian contact theory and Lagrange method to derive the equations. They obtained velocity and acceleration responses of the collision impact system using the Newmark-β method and obtained responses from the simulation, experiment, and the empirical formulae for outer race faults, inner race faults, and both outer and inner race faults at 60 rpm, 200 rpm, and 400 rpm, respectively. For all three cases, the fast Fourier transform (FFT) shows the rotation frequency, respective fault frequency, and the higher harmonics of fault frequency as a result of nonlinearity.

Fig. 2

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Vibration model of a bearing

Bearing signals are usually composed of multi-components, and in many cases, they are nonstationary and nonlinear, i.e., each constituent component exhibits time variability in terms of amplitude, phase, and frequency. Each component can be considered as an amplitude modulation and frequency modulation (AM-FM) oscillatory mode. Therefore, an arbitrarily complicated multi-component signal can be modeled as a superposition of several AM-FM components. Studying nonstationary signals requires accessing instantaneous parameters like amplitude and frequency per component [11]. However, instantaneous frequency applies solely to single-frequency elements, therefore complex signals must be broken into mono-components with distinct instantaneous frequencies [12]. Proper conditions must be used to ensure the separation of a signal into the monocomponents. Using these conditions, an adaptive mode decomposition algorithm can be developed, allowing intricate signals to be broken into mono-components, enabling estimation of their amplitude, phase, and frequency [13].

Various mode decomposition techniques have been utilized in the literature. Feng et al. [6] and Smith [6,14] introduced Empirical Mode Decomposition (EMD) for an adaptive breakdown of complex multi-component signals into individual mono-components, effectively separating components based on conditions like extrema and zero crossings. Loutridis [15] demonstrated EMD's efficacy in detecting gear root cracks and Peng et al. [3] demonstrated the same in diagnosing rolling bearings and rotors. However, EMD exhibits drawbacks such as widely varying single Intrinsic Mode Functions (IMFs) and mode mixing, which Lei et al. [16] addressed using Enhanced Ensemble Empirical Mode Decomposition (EEMD) with correlation-based IMF selection. In contrast, utilizing data smoothing, the Local Mean Decomposition (LMD), proposed by Smith [17], retains frequency and amplitude variations more effectively than EMD. Wang and Liang [18] illustrated the advantages of LMD and found it to be superior to EMD for early fault detection. Variational Mode Decomposition (VMD), introduced by Dragomiretskiy and Zosso [19], offers robust multi-component signal decomposition into AM-FM components, effectively suppressing noise and enhancing fault features. Salunkhe et al. [20] and Salunkhe and Desavale [21] also developed a novel method for fault detection in rolling element bearing using Variational Mode Decomposition and Independent component analysis. Recently, Iatsenko [22], Wang et al. [23], and Xiao et al. [22–24] presented the Nonlinear Mode Decomposition (NMD) principle, emphasizing its potential in separating oscillations and dynamics characterization, although its application in machinery fault diagnosis remains relatively unexplored, despite its promising advantages.

Considering the drawbacks associated with various mode decomposition techniques, including their lack of adaptability, sensitivity to noise, and generation of nonmeaningful modes, it is observed that NMD overcomes these limitations and proves to be suitable for intricate signal analysis. Therefore, the present study uses NMD to extract meaningful modes from a complex signal obtained from a faulty rotating machinery, specifically, a deep groove ball bearing with an inner race fault. This investigation also aims at identifying statistical parameters capable of effectively characterizing faulty components, thus serving as indicators for selecting nonlinear modes associated with the fault data. Subsequently, the Hilbert Transform (HT) [25] is applied to the nonlinear modes featuring fault-related characteristics, followed by the execution of the FFT on the resulting output. The remaining part of the paper has four distinct sections. Section 2 explains the research methodology proposed by the authors. Section 3 comprehensively details all the steps of the NMD, offering explanations and accompanying figures demonstrating the outcomes derived from applying each algorithmic step to a simulated signal. Section 4 explains the results obtained after applying NMD to the actual signal data acquired from an online database for various fault sizes. This section also discusses the results related to statistical parameters, HT, FFT, and instantaneous parameters. Finally, the concluding remarks of the present study are given in Sec. 5.

2 Methodology

The present study employs NMD to extract meaningful modes from a complex signal obtained from a deep groove ball bearing with an inner race fault. This section outlines the methodology of the work adopted in this paper. It involves obtaining the nonlinear modes by applying NMD on both simulated signal and real data. Once these modes are obtained, the question arises about identifying the mode containing a fault. The presence of a fault tends to increase the effect of collision and nonlinearity in the signal, which in the case of bearing fault can be observed as the fault frequency and its higher harmonics, respectively [5]. Since NMD is applied to break down the signal into modes; with decreasing order of nonlinearity, it can be hypothesized that the first nonlinear mode would contain the highest nonlinearity and likely to contain the fault frequency too. However, to validate this assumption, two key attributes of fault signals, impulsiveness, and amplitude, can be quantified using statistical parameters [19]. This investigation also aims at pinpointing the statistical parameters that could effectively portray the traits of faulty components, serving as indicators for selecting nonlinear modes with the fault data. Additionally, two more parameters, instantaneous frequency and instantaneous phase, are derived from the HT [26]. The flowchart in Fig. 3 shows the outline of the work presented in this paper). Each block of the flowchart is explained in detail in the subsections.

Fig. 3

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Flowchart showing the methodology

2.1 Nonlinear Mode Decomposition.

Any complex signal constitutes a blend of amplitude and phase dynamics arising from the various components within the signal. Real-world signals typically comprise multiple elements, further compounded by noise. Fourier transform does not work here because it does not represent time-dependent oscillations as single entities, and it decomposes the signal into multiple tones which does not make any physical sense. The following way is to extend Fourier analysis when there are amplitude or frequency-modulated signals wherein the Fourier transform is computed for relatively short sections of the signal. Therefore, to extract the individual components of the signal, a refined approach is needed to differentiate the different components within a single time series. Time−frequency representation (TFR) techniques emerge as an ideal fit for this purpose. These techniques are divided into two categories: linear and quadratic. The linear category involves two types of TFRs: The Window Fourier Transform (WFT) and the Wavelet transform (WT). The quadratic category includes methods such as Wigner-Ville and Choi-Williams. Among these, the linear TFRs provide a straightforward and effective means of extracting and reconstructing individual signal components [27].

NMD is an adaptive mode decomposition technique that breaks the signal into nonlinear modes. The method comprises two main steps: Initially, the signal is decomposed into the constituent harmonics (modes/oscillations) using TFRs. The next step is to extract all the possible components from the decomposed modes or oscillations derived from the preceding step. In order to extract the component, the first step is to identify the component using the ridge curve extraction method [6,22], followed by its reconstruction using the ridge method or direct method. Finally, true harmonics are identified, followed by the noise test which determines the presence of noise within the residual signal. In the present study, the algorithm [28] is applied first to simulated signals to illustrate the process (Sec. 3). Following to this, it is implemented on actual data (Sec. 4) to detect faults.

2.2 Online Database.

Since the simulated signals used to explain the NMD algorithm do not contain fault frequencies, real signals from an online database are used as a case study. The Bearing Data Center [29] is a platform that offers access to test data regarding both normal and defective ball bearings. The data are taken from this website for different fault conditions and is analyzed in Sec. 4. Once the data are obtained, NMD is applied to give nonlinear modes. The data containing the inner race fault on a 6205-2RS JEM SKF deep groove ball bearing with specifications are given in Table 1. The corresponding fault frequencies are given in Table 2. The first column in Table 2 lists different types of fault frequencies. The second column explains the formula for each fault frequency wherein f_s = speed of the shaft, N_b = number of balls, B_d = ball diameter, P_d = pitch diameter, and $ϕ = contact angle$ ⁠. The third column gives the value of the respective fault frequency, which is supposed to be multiplied by the shaft speed in revolutions per second [29].

Table 1

Bearing specifications (all dimensions are in mm) [29]

Inside diameter	Outside diameter	Thickness	Ball diameter	Pitch diameter
25.00122	51.99888	15.00124	7.94004	39.0398

Table 2

Fault frequency formulae and calculated values [1]

Defect frequencies	Formulae	Value × RPM (Hz)
Ball Pass Frequency of Inner Race (BPFI)	$f_{BPFI} = f_{s} \frac{N_{b}}{2} (1 + \frac{B_{d}}{P_{d}} \cos ϕ)$	5.4152
Ball Pass Frequency of Outer Race (BPFO)	$f_{BPFO} = f_{s} \frac{N_{b}}{2} (1 - \frac{B_{d}}{P_{d}} \cos ϕ)$	3.5848
Ball Spin Frequency (BSF)	$f_{BSF} = f_{s} \frac{P_{d}}{2 B_{d}} (1 - \frac{B_{d}^{2}}{P_{d}^{2}} co s^{2} ϕ)$	4.7135
Fundamental Train Frequency (FTF)	$f_{FTF} = \frac{f_{s}}{2} (1 - \frac{B_{d}}{P_{d}} \cos ϕ)$	0.39828

Defect frequencies	Formulae	Value × RPM (Hz)
Ball Pass Frequency of Inner Race (BPFI)	$f_{BPFI} = f_{s} \frac{N_{b}}{2} (1 + \frac{B_{d}}{P_{d}} \cos ϕ)$	5.4152
Ball Pass Frequency of Outer Race (BPFO)	$f_{BPFO} = f_{s} \frac{N_{b}}{2} (1 - \frac{B_{d}}{P_{d}} \cos ϕ)$	3.5848
Ball Spin Frequency (BSF)	$f_{BSF} = f_{s} \frac{P_{d}}{2 B_{d}} (1 - \frac{B_{d}^{2}}{P_{d}^{2}} co s^{2} ϕ)$	4.7135
Fundamental Train Frequency (FTF)	$f_{FTF} = \frac{f_{s}}{2} (1 - \frac{B_{d}}{P_{d}} \cos ϕ)$	0.39828

2.3 Statistical Parameters.

Furthermore, statistical parameters are used to quantify the distinctive attributes of fault signals such as impulsiveness and amplitude [30]. These parameters are computed for all the nonlinear modes obtained from NMD. The mode exhibiting the highest value of the statistical parameter should show the fault frequency and its higher harmonics, as conversely, the maximum amplitude of fault frequency and its higher harmonics would bring maximum change in the statistical parameters. Therefore, the mode with the highest value of the statistical parameter is determined, and its HT is calculated. Finally, the FFT of this HT is plotted to show the frequency content.

After referring to the literature [27], statistical parameters such as root-mean-square (RMS), kurtosis × RMS (KR), variance, standard deviation, root, RMS × variance (RV) and RMS × standard deviation (RS) are utilized in the present work.

RMS is the most common vibration indicator used as a health indicator for machines. RMS is a measurement of the amplitude and it represents the intensity of the signal which is given as

R (t)_{RMS} = \sqrt{\frac{1}{υ} \sum_{n = 1}^{υ} λ_{n}^{2}}

(1)

where

v

denotes the total number of data points in the signal,

λ_{n}

represents the amplitude of the nth data point, and

R (t)_{RMS}

represents the signal's intensity. The RMS value can be expressed in units of acceleration, velocity, or displacement, depending on the signal's specific unit of measurement. In the case of a fault-free bearing, the RMS value of vibration tends to be lower. However, this value rises proportionally with the fault size.

Variance

(Δ^{2})

is defined as the deviation of the vibration signal energy from the mean value and it is given as

Δ^{2} = \frac{1}{υ - 1} \sum_{n = 1}^{υ} {(λ_{n} - μ)}^{2}

(2)

where

μ

is mean of the signal. Standard deviation can be defined as the square root of the variance and it is given as

Δ = \sqrt{\frac{1}{υ - 1} \sum_{n = 1}^{υ} {(λ_{n} - μ)}^{2}}

(3)

KR combines kurtosis, which measures impulsiveness, with RMS, which assesses amplitude. Combinations such as RV and RS are also utilized in the present work.

2.4 Hilbert Transform and Fast Fourier Transform.

The Hilbert transform (HT) (see Eq. ) [31] is carried out for the nonlinear modes with the higher values of the statistical parameters [32]. Subsequently, FFT is carried out, which distinctly reveals the peaks corresponding to the fault frequency and its higher harmonics. Results for two different cases of fault sizes are discussed in the Sec. 4. Also, the instantaneous parameters such as instantaneous frequency

w_{i} (t)

and instantaneous phase

φ_{i} (t)

of nonlinear modes (see Eqs. and ) derived from HT are explored in the same section for normal and faulty conditions of the bearing.

H [N_{i} (t)] = \tilde{N_{i}} (t) = \frac{1}{π} \int_{- \infty}^{\infty} \frac{N_{i} (τ)}{t - τ} d τ

(4)

where

\tilde{N_{i}} (t)

is the HT of the ith nonlinear mode

N_{i} (t)

⁠, and integral implies its Cauchy principal value.

\tilde{N_{i}} (t)

is the convolution integral of

N_{i} (t)

with

1 / π t

⁠.

\tilde{N_{i}} (t) = N_{i} {(t)}^{*} \frac{1}{π t}

(5)

An analytical signal

A_{i} (t)

can be given as

A_{i} (t) = N_{i} (t) + j H [N_{i} (t)] = α_{i} (t) e^{j φ_{i} (t)}

(6)

where

α_{i} (t) = \sqrt{N_{i}^{2} (t) + H^{2} [N_{i} (t)]}

⁠,

φ_{i} (t) = \arctan \frac{H [N_{i} (t)]}{N_{i} (t)}, and w_{i} (t) = \frac{d φ_{i} (t)}{d t}

(7)

where

α_{i} (t)

is the instantaneous amplitude,

φ_{i} (t)

is the instantaneous phase, and

w_{i} (t)

is the instantaneous frequency derived from HT.

3 Nonlinear Mode Decomposition of a Simulated Signal

The primary objective of NMD is to break down a given signal into a set of nonlinear modes. In order to achieve this, four steps must be considered:

Accurately extract the fundamental harmonic of a nonlinear mode from the signal's TFR.
Identify potential candidates for all of its possible harmonics based on their properties.
Distinguish the true harmonics corresponding to the same a nonlinear mode.
Subtract the resulting nonlinear mode (obtained by summing together all the true harmonics) from the signal and repeat the procedure on the remaining portion until a predetermined stopping criterion is satisfied.

The effectiveness of the NMD algorithm is showcased here using a simulated signal. This signal incorporates both amplitude and frequency modulation. An appropriate TFR, precisely the continuous wavelet transform (CWT), is applied to the generated signal to decompose it into possible harmonics. Further steps involve extracting the ridge curves, conducting signal reconstruction, and deriving nonlinear modes. Following is the comprehensive breakdown of the steps employed for NMD.

3.1 Amplitude Modulation/Frequency Modulation.

An amplitude and frequency modulated signal are created using Eq. to demonstrate the effectiveness of NMD. The carrier frequency is set at 500 Hz, while the frequency modulated component

(f_{FM})

has a modulating frequency of 50 Hz, and the amplitude modulated component

(f_{AM})

exhibits a modulating frequency of 50 Hz as well. The signal is sampled at 2000 Hz. This signal shows (see Fig. 4(a)) clear amplitude and frequency modulation, with its instantaneous amplitude and frequency fluctuating over time. Figure 4(b) shows the FFT of the generated signal.

x (t) = [1 + 0.6 \cos (2 π f_{AM} t + π / 3)] \times \cos [2 π f_{carrier} t + \cos (2 π f_{FM} t)]

(8)

Fig. 4

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Simulated signal: (a) signal in the time domain and (b) FFT of signal

3.2 Extraction and Reconstruction of the Main Component

3.2.1 Continuous Wavelet Transform.

During the process of conducting NMD, once data are acquired from either a machinery or a simulated signal, CWT is used to extract and reconstruct the components present in the signal [33]. A signal x(t) can be decomposed into consecutive wavelets using CWT,

W_{x} (ω, t)

as shown in Eq. . Figures 5 and 6 illustrate the signal decomposition into wavelets. Fig. 5 shows decomposed wavelets over time for the simulated signal, while Fig. 6 presents a scalogram for a single wavelet [3].

\begin{aligned} W_{x} (ω, t) & = \int_{- \infty}^{\infty} x^{+} (u) ϕ^{*} [\frac{ω (u - t)}{ω_{ϕ}}] \frac{ω d u}{ω_{ϕ}} \\ = \frac{1}{2 π} \int_{0}^{\infty} e^{i η t} \hat{x} (η) {\hat{ϕ}}^{*} (\frac{ω_{ϕ} η}{ω}) d η \end{aligned}

(9)

where

\hat{x} (η), x^{+} (t), ⟨ x (t) ⟩

and

std [x (t)]

are Fourier transform, positive frequency part, time average, and standard deviation of signal, respectively, and are given as

\hat{x} (η) = \int_{- \infty}^{\infty} x (t) e^{- i η t} d t \Leftrightarrow x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \hat{x} (η) e^{i η t} d η

(10)

\begin{aligned} x^{+} (t) & = \int_{0^{+}}^{\infty} \hat{x} (η) e^{i η t} d η, ⟨ x (t) ⟩ = \frac{1}{T} \int x (t) d t, and \\ std [x (t)] & = \sqrt{⟨ {[x (t)]}^{2} ⟩ - {[⟨ x (t) ⟩]}^{2}} \end{aligned}

(11)

where T denotes the time duration of the signal

x (t)

⁠.

Fig. 5

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Decomposed wavelets of a simulated signal

Fig. 6

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Wavelet transform of a simulated signal (single wavelet)

3.2.2 Ridge Curve Extraction.

Once the signal is decomposed into its components such as corresponding instantaneous amplitudes, phases, and frequencies using CWT, some components of interest are extracted. However, it is necessary to recognize the component of interest. One should locate the specific time−frequency area within the existing TFR where this component is prominently situated. Once this region is determined, it becomes feasible to check the attributes of the corresponding component by applying a designated reconstruction approach [28]. In a signal containing multiple AM/FM components along with noise, the wavelet transform's resolution properties for time modulation and frequency separation among components are important. Under appropriate circumstances, each AM/FM component displays a distinct peak in the wavelet. A succession of such peaks linked to a particular AM/FM component forms a ridge curve as shown in Fig. 7 with its associated frequency designated as $ω_{p}$ [23,24]. Now, as the curve is extracted, the amplitude, phase, and frequency of the corresponding component can be reconstructed.

Fig. 7

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Extracted ridge curve of a simulated signal (middle sinusoidal curve: ridge curve and top and bottom curves: supports)

3.2.3 Main Harmonic Reconstruction.

After the ridgeline curve is extracted, the main harmonic or component is reconstructed as shown in Fig. 8. The attributes such as amplitude, phase, and frequency of the component can be directly derived from the TFR at specific ridge points. The component extracted here generally represents a harmonic of some nonlinear modes. It is assumed that it corresponds to a fundamental, i.e., the first harmonic. For the sub-harmonic, its ridge line closest to the peak sequence in the direction of amplitude increases. After finding the sub-harmonic ridge line, the ridge method can be used to reconstruct the sub-harmonics.

Fig. 8

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Reconstructed signal (simulated)

3.2.4 Identifying True Harmonics.

The process outlined in the previous section generally results in a harmonic candidate, which may not necessarily be a true harmonic. Therefore, even if the nonlinear harmonic does not inherently possess a specific harmonic, a signal will still be generated, comprising noise or components closely matching the frequency of the presumed harmonic. Thus, once the nth harmonic candidate is extracted, the subsequent step is to ascertain its validity. A method of surrogate data is employed here to address this challenge that evaluates the harmonic against the null hypothesis of independence between the first harmonic (fundamental harmonic) (see Eqs. (12)–(14)) and the extracted harmonic candidate. Initially, a metric is chosen to gauge the extent of interdependence between the behaviors of two harmonics. Subsequently, this metric is computed for the original harmonic and a set of surrogates, which are intentionally generated time series in line with the null hypothesis being examined. If the original metric value falls outside the distribution of its surrogate counterparts, it signifies genuine interdependence, confirming the harmonic as true (see Eqs. (15)–(18)) otherwise, it is dismissed as false.

For fundamental harmonic,

\frac{A^{n} (t)}{A^{1} (t)} = a_{n} = constant

(12)

ψ^{n} (t) - n ψ^{1} (t) = constant

(13)

ϑ^{n} (t) = n ϑ^{1} (t)

(14)

For true harmonic,

δ_{A, ψ, ϑ}^{(n)} \in [0, 1]

(15)

where A,

ψ

⁠, and

ϑ

represent the amplitude, phase, and frequency of the harmonics, respectively,

δ_{A}^{(n)} = \exp {- \frac{\sqrt{{[A^{(n)} (t) A^{(1)} (t) - A^{(1)} (t) A^{(n)} (t)]}^{2}}}{A^{(1)} (t) A^{(n)} (t)}}

(16)

δ_{ψ}^{(n)} = a | \exp {i [ψ^{(n)} (t) - n ψ^{(1)} (t)]} |

(17)

δ_{ϑ}^{(n)} = \exp {- \frac{\sqrt{{[ϑ^{(n)} (t) - n ϑ^{(1)} (t)]}^{2}}}{ϑ^{(n)} (t)}}

(18)

Ideally, for true harmonics, one should have $δ_{A, ψ, ϑ}^{(n)} = 1$ ⁠, but noise and interference with the other components introduce errors, making the consistencies smaller than unity.

3.2.5 Noise Detection.

After gathering the true harmonics, they are summed to create the nonlinear mode component, which is subsequently subtracted from the original signal until only noise remains in the residual signal. The surrogate test approach, coupled with Fourier transformation, is employed here to determine the presence of noise within the residual signal.

Figure 9 shows the decomposition of the generated signal (see Fig. 4) into nonlinear modes. It shows the plot of all 52 nonlinear modes amplitudes over time, whereas Figs. 10(a)–10(j) shows them one-by-one over time. The NMD explained in this section, using a simulated signal, is further employed on actual faulty bearing signals taken from an online database [29] and the results thereof are discussed in the following section.

Fig. 9

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Fifty-two nonlinear modes for a simulated signal

Fig. 10

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Nonlinear modes obtained by applying NMD on simulated signal: (a) nonlinear modes 1–5, (b) nonlinear modes 6–10, (c) nonlinear modes 11–15, (d) nonlinear modes 16–20, (e) nonlinear modes 21–25, (f) nonlinear modes 26–30, (g) nonlinear modes 31–35, (h) nonlinear modes 36–40, (i) nonlinear modes 41–45, and (j) nonlinear modes 46–52

4 Results and Discussion

Two specific cases of different fault sizes, viz., 0.007 in. and 0.021 in. are selected from the online database, and the entire process, as detailed in Sec. 2, is carried out. The results thereof are presented in this section. For Case 1, all the steps explained in Sec. 2 are discussed, whereas for Case 2, only the results are given. Additionally, the instantaneous parameters obtained from the HT are presented for these two cases.

4.1 Case 1.

In this case, a dataset with a sampling frequency of 48,000 Hz for a fault size of 0.007 in., speed of 1772 rpm, and 1 HP load on the motor is considered.

Step 1: Calculating Nonlinear Modes Using NMD

The time domain signal derived from these data is presented in Fig. 11(a), while Fig. 11(b) displays the FFT of complete data. Since the data are nonlinear and nonstationary, the FFT does not show the fault frequencies. Therefore, the data are subjected to NMD, and the whole signal is decomposed into nonlinear modes, which are shown both together and separately with respect to time in Figs. 12 and 13, respectively. Figure 13 displays five consecutive modes in each subfigure.

Step 2: Utilizing Statistical Parameters

Fig. 11

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Dataset taken from the online database [29]: (a) time domain signal and (b) FFT of the whole signal

Fig. 12

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Fifty-two nonlinear modes obtained by applying NMD on real data taken from an online database for fault size 0.007 in. (Case 1)

Fig. 13

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NMD analysis results for fault size 0.007 in. (Case 1): (a) nonlinear modes 1–5, (b) nonlinear modes 6–10, (c) nonlinear modes 11–15, (d) nonlinear modes 16–20, (e) nonlinear modes 21–25, (f) nonlinear modes 26–30, (g) nonlinear modes 31–35, (h) nonlinear modes 36–40, (i) nonlinear modes Ms 41–45, and (j) nonlinear modes 46–52

The statistical parameters are estimated for all the decomposed nonlinear modes. These modes are then sorted in the descending order of the values of the statistical parameters. The first four sorted nonlinear modes are shown in Figs. 14–19 for all the statistical parameters. It is observed that the fault frequency (BPFI) and its higher harmonics are seen with the maximum amplitudes in the first nonlinear mode, i.e., the nonlinear mode with the highest statistical parameter value. As the statistical parameter values for the nonlinear modes reduce, these amplitudes also decrease.

Fig. 14

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Nonlinear modes and their FFTs for the first four4 highest values of RMS: (a) highest RMS, (b) second highest RMS, (c) third highest RMS, and (d) fourth highest RMS

Fig. 15

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Nonlinear modes and their FFTs for the first four highest KR values: (a) highest KR, (b) second highest KR, (c) third highest KR, and (d) fourth highest KR

Fig. 16

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Nonlinear modes and their FFT for first four values of variance: (a) highest variance, (b) second highest variance, (c) third highest variance, and (d) fourth highest variance

Fig. 17

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Nonlinear modes and their FFT for first four values of SD: (a) highest SD, (b) second highest SD, (c) third highest SD, and (d) fourth highest SD

Fig. 18

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Nonlinear modes and their FFT for first four values of RMS × variance: (a) highest RMS × variance, (b) second highest RMS × variance, (c) third highest RMS × variance, and (d) fourth highest RMS × variance

Fig. 19

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Nonlinear modes and their FFT for the first four values of RMS × SD: (a) highest RMS × SD, (b) second highest RMS × SD, (c) third highest RMS × SD, and (d) fourth highest RMS × SD

Similar to the case above, one more case with a fault size of 0.021 in. is examined. This case is discussed for all the above-stated statistical parameters. The reason for considering one more case is to ensure that the observations made from the above-stated case are correct and consistent.

4.2 Case 2.

In this case, data with a sampling frequency of 48,000 Hz, fault size of 0.021 in., speed of 1772 rpm, and motor with 1 HP load are considered. Here, nonlinear modes corresponding to the highest values of all six6 statistical parameters (RMS, KR, variance, SD, RV, and RS) are shown (see Fig. 20) as a similar trend of the reduction in amplitudes of fault frequency (BPFI) and its higher harmonics is observed for the sorted nonlinear modes.

Fig. 20

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Nonlinear modes and their FFTs for the highest values of statistical parameters for fault size 0.021 in.: (a) RMS, (b) KR, (c) Variance, (d) SD, (e) RV, and (f) RS

From these results (Figs. 14–20), it is verified that the nonlinear mode with the highest values of the statistical parameters vividly shows fault frequency (BPFI) and its higher harmonics as conversely, their highest amplitudes bring maximum change in the statistical parameters. Another important fact observed here is that the first decomposed nonlinear mode has the highest values for all the statistical parameters (see Fig. 21). Thus, from Fig. 21, it can be understood that the collision effect and nonlinearity corresponding to the fault are dominant in the first nonlinear mode and they increase with the increase in the fault size, as a clear increase in the values of the statistical parameters can be seen for the first nonlinear mode in Figs. 21 and 22 with the increase in the fault size.

Fig. 21

View large Download slide

Statistical parameters for first 15 nonlinear modes for different fault sizes: (a) RMS, (b) KR, (c) variance, (d) SD, (e) RV, and (f) RS

Fig. 22

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Highest values of statistical parameters for different fault sizes

4.3 Exploring the Significance of Instantaneous Parameters for Fault Diagnosis.

The significance of the instantaneous parameters, such as instantaneous frequency and instantaneous phase obtained from the HT for fault diagnosis, is discussed in this section.

4.3.1 Instantaneous Frequency.

Figures 23(a)–23(f) show the instantaneous frequency plots obtained after implementing HT on the first decomposed nonlinear mode for all six statistical parameters under normal and faulty conditions. Instantaneous frequency plays a vital role in describing the physical significance of the mono-components or modes obtained after applying the NMD algorithm. This parameter helps in determining the variation of frequency with respect to the time. This parameter also helps reconstruct the signal using TFR (CWT in the present work) [33]. Additionally, when comparing this parameter between normal and faulty signals, it is observed that in the case of faulty signals, the instantaneous frequency exhibits a downward shift (see Fig. 23).

Fig. 23

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Comparison of instantaneous frequency for normal and faulty conditions for different fault sizes and different statistical parameters: (a) RMS, (b) KR, (c) variance, (d) SD, (e) RV, and (f) RS

4.3.2 Instantaneous Phase.

Figures 24(a)–24(f) show the instantaneous phase of the nonlinear mode with the highest statistical parameter, i.e., the first nonlinear mode, obtained after applying the HT on it. The instantaneous phase is obtained in both normal as well as faulty conditions. It can be observed from Figs. 24(a)–24(f) that for all the statistical parameters, there is a shift in the instantaneous phase for faulty conditions compared to the normal condition. Between fault sizes, a significant difference is also observed for the instantaneous phase. Thus, instantaneous frequency and instantaneous phase can serve as a fault indicator, if their plots are known for the healthy condition.

Fig. 24

View large Download slide

Comparison of instantaneous phase for normal and faulty conditions for different fault sizes and different statistical parameters: (a) RMS, (b) KR, (c) variance, (d) SD, (e) RV, and (f) RS

5 Conclusions

Based on the implementation of NMD, utilizing statistical parameters, and exploring instantaneous properties for different bearing fault data, it can be concluded that nonlinear modes obtained from the NMD give promising results for fault diagnosis. The collision effect and nonlinearity corresponding to the fault are found to be dominant in the first nonlinear mode as all the statistical parameters showed their highest values for the first nonlinear mode. This is also evident from the fact that the fault frequency and its higher harmonics are seen for the first nonlinear mode. The collision effect and nonlinearity in the signal are also found to increase with the fault size, as the amplitude of the fault frequency and its higher harmonics increase with the increase in the size of the fault. However, to make it a general conclusion this result needs to be checked with the other types of faults too. The instantaneous frequency and instantaneous phase obtained from the HT may serve as good fault indicators if their values corresponding to the healthy conditions are known. Thus, the present research shows novel techniques that use NMD, statistical parameters, and instantaneous properties to diagnose inner-race faults on the bearing.

Acknowledgment

The work described in this paper is financially supported by the IRCC Seed Grant from the Indian Institute of Technology Bombay, Mumbai, India (Project No. RD/0521-IRCCSH0-015).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

Abbreviations

AM-FM =: Amplitude Modulation and Frequency Modulation
BPFI =: Ball Pass Frequency of Inner race
EEMD =: Enhanced Ensemble Empirical Mode Decomposition
EMD =: Empirical Mode Decomposition
FFT =: Fast Fourier Transform
HT =: Hilbert Transform
IMFs =: Intrinsic Mode Functions
NMD =: Nonlinear Mode Decomposition
RV =: RMS × Variance
TFR =: Time−frequency representation
VMD =: Variational Mode Decomposition
WFT =: Window Fourier Transform
WT =: Wavelet Transform

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Identifying Inner Race Faults in Deep Groove Ball Bearing Using Nonlinear Mode Decomposition and Hilbert Transform

Abstract

1 Introduction

2 Methodology

2.1 Nonlinear Mode Decomposition.

2.2 Online Database.

2.3 Statistical Parameters.

2.4 Hilbert Transform and Fast Fourier Transform.

3 Nonlinear Mode Decomposition of a Simulated Signal

3.1 Amplitude Modulation/Frequency Modulation.

3.2 Extraction and Reconstruction of the Main Component

3.2.1 Continuous Wavelet Transform.

3.2.2 Ridge Curve Extraction.

3.2.3 Main Harmonic Reconstruction.

3.2.4 Identifying True Harmonics.

3.2.5 Noise Detection.

4 Results and Discussion

4.1 Case 1.

4.2 Case 2.

4.3 Exploring the Significance of Instantaneous Parameters for Fault Diagnosis.

4.3.1 Instantaneous Frequency.

4.3.2 Instantaneous Phase.

5 Conclusions

Acknowledgment

Conflict of Interest

Data Availability Statement

Nomenclature

Abbreviations

References

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Identifying Inner Race Faults in Deep Groove Ball Bearing Using Nonlinear Mode Decomposition and Hilbert Transform Free

Abstract

1 Introduction

2 Methodology

2.1 Nonlinear Mode Decomposition.

2.2 Online Database.

2.3 Statistical Parameters.

2.4 Hilbert Transform and Fast Fourier Transform.

3 Nonlinear Mode Decomposition of a Simulated Signal

3.1 Amplitude Modulation/Frequency Modulation.

3.2 Extraction and Reconstruction of the Main Component

3.2.1 Continuous Wavelet Transform.

3.2.2 Ridge Curve Extraction.

3.2.3 Main Harmonic Reconstruction.

3.2.4 Identifying True Harmonics.

3.2.5 Noise Detection.

4 Results and Discussion

4.1 Case 1.

4.2 Case 2.

4.3 Exploring the Significance of Instantaneous Parameters for Fault Diagnosis.

4.3.1 Instantaneous Frequency.

4.3.2 Instantaneous Phase.

5 Conclusions

Acknowledgment

Conflict of Interest

Data Availability Statement

Nomenclature

Abbreviations

References

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Cited By

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Identifying Inner Race Faults in Deep Groove Ball Bearing Using Nonlinear Mode Decomposition and Hilbert Transform