Introduction

A vibration exciter is a device attached to vibration machinery to generate excitation force, which has been widely used in the industries of architecture, mining, aerospace industry, petroleum, etc.1,2. The vibration exciter works in harsh working environments such as high temperatures and dust for a long time, and its rolling bearings are prone to failure due to high amplitude alternating impact loads. If bearing faults cannot be detected in a timely manner, they may cause damage to the exciter equipment, reduce work efficiency and product quality, and result in economic losses or even casualties3,4. The method based on vibration signal analysis is considered one of the most effective methods for bearing fault detection5,6. The environmental noise during the operation of the vibration exciter is severe, and the vibration level generated by the eccentric block is relatively high Therefore, the weak vibration generated by bearing damage is often submerged in strong background noise and various vibration interferences7. In addition, the fault signal of a rolling bearing is an amplitude modulation-frequency modulation (AM-FM) signal, and the fundamental frequency (envelope) of the modulation signal is the fault feature of a rolling bearing8,9. Therefore, seeking demodulation techniques that can extract the weak fault information from the vibration signals with strong noise interference is the key to fault diagnosis of exciter rolling bearings.

In the past few decades, the envelope analysis technique based on the band-pass filtering of bearing vibration signal has often been applied to the demodulation of the modulated signal10,11. However, correctly selecting the bandwidth of the filter has limited the development of this technology12. Recently, a parameter-free demodulation method, the Teager–Kaiser Energy Operator (TKEO), which can enhance fault impact energy, has been widely applied in bearing fault diagnosis13,14,15. The TKEO has a strong ability to track amplitude and frequency, so the subtle changes caused by external noise fluctuations can be reflected in the energy operator, ultimately affecting the accuracy in estimating instantaneous frequencies and the fault detection results16,17. Therefore, it is often used in combination with various noise reduction methods. Azergui et al.18 proposed a signal analysis method based on wavelet packet transform (WPT) and TKEO to extract fault features of the rolling bearing from vibration signals. Gu et al.19 proposed an adaptive variational mode decomposition and Teager energy operator (AVMD-TEO) method for early fault diagnosis of rolling bearings. Shi et al.16 reconstructed and denoised the original vibration signal by establishing a series of bandpass filters, and then used the TKEO energy spectrum of the denoised signal for bearing fault diagnosis. Bendjama et al.20 proposed a joint time–frequency analysis method based on Morlet wavelet filtering (MWF), eigentime scale decomposition (ITD), and TKEO to extract pulse characteristics of bearings in order to evaluate the running state of bearings. Despite all this, the joint of TKEO and de-noising pre-processing method not only faces the problem of selecting tuning parameters but also increases computational costs and reduces algorithm efficiency. For this reason, three enhanced energy operators, symmetric higher-order analytical energy operator (SHAEO), multi-resolution symmetric difference energy operator (MSDAEO), and symmetric higher-order frequency weighted energy operator (SHFWEO) with strong anti-interference performance against noise have been proposed successively21,22,23. Without pre-filtering, the three enhanced energy operators can extract bearing fault features submerged in heavy background noise, greatly reducing the running time of the program and improving its efficiency. Thus, this paper applies these three enhanced energy operators to the fault diagnosis of exciter bearings. In addition, comparative studies are also conducted on these three enhanced energy operators based on the vibration screen exciter bearing fault diagnosis test bench.

The paper hereafter is organized as follows: symmetric higher-order analytical energy operator (SHAEO), multi-resolution symmetric difference energy operator (MSDAEO), and symmetric higher-order frequency weighted energy operator (SHFWEO) are introduced in "Symmetric higher-order analytical energy operator (SHAEO)", "Multi-resolution symmetric difference energy operator (MSDAEO)", "Symmetric higher-order frequency weighted energy operator (SHFWEO)" sections respectively. The theoretical comparison and analysis of the anti-noise and anti-interference performance of the three energy operators were conducted in "Anti-noise performance evaluation" and "Anti-interference performance evaluation" sections, respectively. "Vibration exciter bearing fault diagnosis experiment" section evaluated and compared the performance of SHAEO, MSDAEO, and SHFWEO through fault diagnosis experiments on the exciter bearings of the vibrating screen. "Conclusions" section concludes the paper.

Symmetric higher-order analytical energy operator (SHAEO)

Given the following continuous signals:

$$x\left( t \right) = A\left( t \right)\cos \psi \left( t \right),$$
(1)

its analytic form can be defined as follows:

$$X\left( t \right) = x\left( t \right) + i\tilde{x}\left( t \right) = \left| {X\left( t \right)} \right|\left[ {\cos \psi \left( t \right) + j\sin \psi \left( t \right)} \right] = A\left( t \right)e^{j\psi (t)} ,$$
(2)

where, \(\tilde{x}\left( t \right) = H\left[ {x\left( t \right)} \right],\) H[·] represents the Hilbert transformation (HT).

The instantaneous envelope and phase of x(t) can be denoted as (3) and (4), respectively:

$$A\left( t \right) = \pm \left| {X\left( t \right)} \right| = \pm \sqrt {x^{2} \left( t \right) + \tilde{x}^{2} \left( t \right)} ,$$
(3)
$$\psi \left( t \right) = \arctan \frac{{\tilde{x}\left( t \right)}}{x\left( t \right)}.$$
(4)

The relationship between the instantaneous phase and the instantaneous frequency (IF) of a signal is defined as follows:

$$\omega \left( t \right) = \dot{\psi }\left( t \right)$$
(5)

Therefore, according to Eq. (5), it can be obtained that:

$$\omega \left( t \right) = \dot{\psi }\left( t \right) = \frac{{x\left( t \right)\dot{\tilde{x}}\left( t \right) - \dot{x}\left( t \right)\tilde{x}\left( t \right)}}{{A^{2} \left( t \right)}},$$
(6)
$$x\left( t \right)\dot{\tilde{x}}\left( t \right) - \dot{x}\left( t \right)\tilde{x}\left( t \right) = A^{2} \left( t \right)\omega \left( t \right).$$
(7)

Equation (7) contains the information of amplitude modulation (AM) and frequency modulation (FM). Therefore, a new energy operator, the analytic energy operator (AEO), is established, which is defined as follows:

$$\Theta \left[ {x\left( t \right)} \right] = x\left( t \right)\dot{\tilde{x}}\left( t \right) - \dot{x}\left( t \right)\tilde{x}\left( t \right).$$
(8)

Its discrete form is

$$\Theta \left[ {x\left( n \right)} \right] = x\left( n \right)\dot{\tilde{x}}\left( n \right) - \dot{x}\left( n \right)\tilde{x}\left( n \right).$$
(9)

Using the cross energy between x(t) and its higher-order derivative, a higher-order energy transformation called higher-order AEO is developed based on Eq. (9), which is defined as follows:

$$\Theta_{k} \left[ {x^{\left( m \right)} \left( t \right)} \right] = x\left( t \right)H\left[ {x^{\left( m \right)} \left( t \right)} \right] - x^{\left( m \right)} \left( t \right)H\left[ {x\left( t \right)} \right],$$
(10)

where, \(x^{m} \left( t \right) \equiv \left\{ {\begin{array}{*{20}l} {d^{m} x\left( t \right)/dt^{m} ,\quad m \ge 1} \hfill \\ {x\left( t \right),\quad {\text{ }}m = 0} \hfill \\ {\int_{{ - \infty }}^{t} {x^{{\left( {m + 1} \right)}} \left( \tau \right)d\tau ,\quad {\text{ }}m \le - 1} } \hfill \\ \end{array} } \right.\) represents a signal and its m-th derivative (or integral).

Substituting x(n) for x(t) and using the two-sample forward difference to estimate the derivative, the discrete form of TEO is obtained as follows:

$$\Psi \left[ {x\left( n \right)} \right] = x^{2} \left( n \right) - x\left( {n + 1} \right)x\left( {n - 1} \right).$$
(11)

Three-sample central symmetric difference is expressed as follows:

$$\dot{x}\left( n \right) = \frac{{x\left( {n + 1} \right) - x\left( {n - 1} \right)}}{2}.$$
(12)

According to the recurrence formula, its higher-order version can be expressed as follows:

$$x^{\left( m \right)} \left( n \right) = \frac{{x^{{\left( {m - 1} \right)}} \left( {n + 1} \right) - x^{{\left( {m - 1} \right)}} \left( {n - 1} \right)}}{{2^{m} }}.$$
(13)

Substituting Eq. (13) into (10) gives

$$\Theta_{m} \left[ {x^{\left( m \right)} \left( n \right)} \right] = x\left( n \right)H\left[ {\frac{{x^{{\left( {m - 1} \right)}} \left( {n + 1} \right) - x^{{\left( {m - 1} \right)}} \left( {n - 1} \right)}}{{2^{m} }}} \right] - \frac{{x^{{\left( {m - 1} \right)}} \left( {n + 1} \right) - x^{{\left( {m - 1} \right)}} \left( {n - 1} \right)}}{{2^{m} }}H\left[ {x\left( n \right)} \right]$$
(14)

Literature24 points out that the higher the order, the better the suppression effect of the energy operator on interference. However, the continuous differential operation leads to the introduction of high-frequency noise during demodulation, and it is necessary to determine the appropriate order. Here, the optimal difference order is determined by studying the noise reduction effect of the energy operators with different orders. In the following test, we only explore SHAEO from first-order to fifth-order. SHAEO demodulation with different orders is performed for 10 noisy signals (with signal-to-noise ratios ranging from − 10 to − 1 dB). The estimation results are plotted in Fig. 1.

Fig. 1
figure 1

The SNRs before and after SHAEO processing with different orders.

Full size image

Obviously, SHAEO at the five different orders can all improve the SNR. Among them, the third order has the best denoising effect. Consequently, the order m of the higher-order derivative in this paper is set to 3 and the third-order SHAEO can be expressed as follows:

$$\begin{aligned} \Theta_{3} \left[ {x^{\left( 3 \right)} \left( n \right)} \right] = &\, x\left( n \right)H\left[ {\frac{{x^{\left( 2 \right)} \left( {n + 1} \right) - x^{\left( 2 \right)} \left( {n - 1} \right)}}{8}} \right] - \frac{{x^{\left( 2 \right)} \left( {n + 1} \right) - x^{\left( 2 \right)} \left( {n - 1} \right)}}{8}H\left[ {x\left( n \right)} \right] \\ { = } & \frac{1}{32}\left\{ \begin{gathered} x\left( n \right)H\left[ {x(n + 3) - 3x(n + 1) + 3x(n - 1) - x(n - 3)} \right] - \hfill \\ \left[ {x(n + 3) - 3x(n + 1) + 3x(n - 1) - x(n - 3)} \right]H\left[ {x\left( n \right)} \right] \hfill \\ \end{gathered} \right\}. \\ \end{aligned}$$
(15)

For simplicity, Eq. (15) can be expressed as follows:

$$\Theta_{3} \left[ {x^{\left( 3 \right)} \left( n \right)} \right]{ = }\frac{1}{32}\left\{ \begin{aligned} & {\text{Re}} (n){\text{Im}} (n + 3) - {\text{Re}} (n + 3){\text{Im}} (n) + 3{\text{Re}} (n + 1){\text{Im}} (n) - 3{\text{Re}} (n){\text{Im}} (n + 1) \\ & + 3{\text{Re}} (n){\text{Im}} (n - 1) - 3{\text{Re}} (n - 1){\text{Im}} (n) + {\text{Re}} (n - 3){\text{Im}} (n) - {\text{Re}} (n){\text{Im}} (n - 3) \\ \end{aligned} \right\}$$
(16)

Multi-resolution symmetric difference energy operator (MSDAEO)

Due to the fact that symmetric difference sequences can replace forward difference sequences to improve the accuracy of instantaneous frequency estimation, and nonadjacent sampling points can reduce calculation error by chance and measurement errors caused by noise interference25,26 , a three-point symmetric difference with a signal interval of k is used to represent the first-order difference, i.e.,

$$\dot{x}\left( n \right) = \frac{{x\left( {n + k} \right) - x\left( {n - k} \right)}}{2}.$$
(17)

Substituting Eq. (17) into Eq. (9) yields a new energy transformation called the Multi-resolution symmetric difference energy operator (MSDAEO), which is represented as follows:

$$\begin{aligned} \Phi \left[ {x(n)} \right] = &\, \frac{{\left[ {x\left( {n + k} \right) - x\left( {n - k} \right)} \right] \cdot h\left[ {x\left( n \right)} \right]}}{2} - \frac{{h\left[ {x\left( {n + k} \right) - x\left( {n - k} \right)} \right] \cdot x\left( n \right)}}{2} \\ { = } &\, \frac{1}{2}\left\{ {{\text{Re}} (n + k){\text{Im}} (n) - {\text{Re}} (n - k){\text{Im}} (n) - {\text{Re}} (n){\text{Im}} (n + k) + {\text{Re}} (n){\text{Im}} (n - k)} \right\}, \\ \end{aligned}$$
(18)

where \(h\left[ n \right] = H\left[ {x\left( n \right)} \right]\) denotes the discrete Hilbert transform, k is the optimal resolution parameter determined by the maximum signal-to-noise ratio improvement (SNRI) proposed in22.

Symmetric higher-order frequency weighted energy operator (SHFWEO)

Instantaneous energy of the continuous signal \(x(t) = A\cos (\omega t + \varphi )\) based on signal processing is defined as follows:

$$S\left[ {x\left( t \right)} \right] = \left| {x\left( t \right)} \right|^{2} = \left| {x\left( t \right) + jH\left[ {x\left( t \right)} \right]} \right|^{2} = A^{2} \left( t \right).$$
(19)

Equation (19) only contains amplitude information. However, the instantaneous energies of unit amplitude signals with different frequencies may vary27. In addition, it is difficult to track impact signals with small transient amplitudes. Take the derivation function as a filter, and the Frequency-weighted Energy Operator (FW-EO)27 is defined as follows:

$$\Gamma \left[ {x\left( t \right)} \right]{ = }\left| {\dot{x}\left( t \right) + jH\left[ {\dot{x}\left( t \right)} \right]} \right|^{2} { = }\dot{x}^{2} \left( t \right){ + }H\left[ {\dot{x}\left( t \right)} \right]^{2} .$$
(20)

Using the cross energy between x(t) and its higher-order derivative, the high-order FW-EO is proposed based on Eq. (20), which is defined as follows:

$$\Gamma_{k} \left[ {x\left( t \right)} \right] \equiv \left[ {x,x^{m} } \right] = \left| {x^{(m)} \left( t \right) + jH\left[ {x^{m} \left( t \right)} \right]} \right| = \left[ {x^{(m)} \left( t \right)} \right]^{2} + H\left[ {x^{\left( m \right)} \left( t \right)} \right]^{2} ,m = 0, \pm 1, \pm 2,...$$
(21)

Its discrete form is

$$\Gamma_{m} \left[ {x\left( n \right)} \right] = \left| {x^{(m)} \left( n \right) + jH\left[ {x^{m} \left( n \right)} \right]} \right| = \left[ {x^{(m)} \left( n \right)} \right]^{2} + H\left[ {x^{\left( m \right)} \left( n \right)} \right]^{2} ,m = 0, \pm 1, \pm 2,...$$
(22)

Substituting Eq. (13) into (22) yields the symmetric high-order frequency weighted energy operator (SHFWEO), i.e.,

$$\Gamma_{k} \left[ {x\left( n \right)} \right] = \left[ {x^{\left( m \right)} \left( n \right)} \right]^{2} + H\left[ {x^{\left( m \right)} \left( n \right)} \right]^{2} = \left[ {\frac{{x^{{\left( {m - 1} \right)}} \left( {n + 1} \right) - x^{{\left( {m - 1} \right)}} \left( {n - 1} \right)}}{{2^{m} }}} \right]^{2} + H\left[ {\frac{{x^{{\left( {m - 1} \right)}} \left( {n + 1} \right) - x^{{\left( {m - 1} \right)}} \left( {n - 1} \right)}}{{2^{m} }}} \right]^{2}$$
(23)

Let the discrete signal be \(x\left( n \right) = A\cos \left( {\Omega n + \alpha } \right),\) then the m-th difference of x(n) is denoted as follows:

$$x^{\left( m \right)} \left( n \right) = A\sin^{m} \Omega \cos \left( {\Omega n + \frac{m\pi }{2}} \right)$$
(24)

The Hilbert transform of x(m)(n) is

$$H\left[ {x^{\left( m \right)} \left( n \right)} \right] = A\sin^{m} \Omega \sin \left( {\Omega n + \frac{m\pi }{2}} \right)$$
(25)

Substitute Eqs. (24) and (25) into Eq. (23), while \(0 < \Omega \le \frac{\pi }{2},\) i.e., \(\sin \Omega \approx \Omega ,\) yields:

$$\Gamma_{k} \left[ {x\left( n \right)} \right] = \left[ {A\sin^{m} \Omega \cos \left( {\Omega n + \frac{k\pi }{2}} \right)} \right]^{2} + \left[ {A\sin^{m} \Omega \sin \left( {\Omega n + \frac{m\pi }{2}} \right)} \right]^{2} = A^{2} \sin^{2m} \Omega \approx A^{2} \Omega^{2m} .$$
(26)

Equation (26) contains both amplitude modulation (AM) and frequency modulation (FM) information, so the symmetric higher-order frequency-weighted energy operator (SHFWEO) has demodulation characteristics.

Anti-noise performance evaluation

The anti-noise performance of energy operators affects demodulation accuracy. Therefore, studying the effectiveness of each energy operator under different background noise levels is of great significance. Thus, we evaluate the noise robustness by comparing the difference in SNR between the input and output signals. The vibration signal of a faulty bearing can be expressed as:

$$x(t) = \sum\limits_{i} {A_{m} } e^{{ - 2\alpha \pi f_{n} (t - iT)}} \sin \left( {2\pi f_{n} \sqrt {1 - \alpha^{2} } (t - iT)} \right),$$
(27)

where \(A_{m} { = }3\) represents the amplitude of the fault signal, \(\alpha { = 0}{\text{.1}}\) is the damping factor, \(f_{n} { = }2000{\text{Hz}}\) is the resonance frequency, T = 0.01 s is the fault period.

In the test, the SNR of the input signal ranges from − 10 to − 1 dB. Then, ten noisy signals were subjected to SHAEO, MSDAEO, SHFWEO, and TKEO processing to obtain the signal-to-noise of the input and output signals, as shown in Fig. 2. Obviously, all four energy operators can improve the SNR of the signal. The three enhanced energy operators have better noise resistance performance than TKEO, and MSDAEO provides the best results.

Fig. 2
figure 2

The SNRs of input and output signals.

Full size image

Anti-interference performance evaluation

The anti-interference performance of three energy operators, SHAEO, MSDAEO, and SHFWEO, is evaluated using the bearing fault model in reference28, which is denoted as follows:

$$s\left( t \right) = x(t) + I(t) = \sum\limits_{m = 1}^{M} {Ae^{ - \beta t} \cos \left( {\omega_{r} t + \phi } \right)} + \sum\limits_{l = 1}^{L} {B_{l} \cos \omega_{l} t} ,$$
(28)

where \(x(t) = \sum\limits_{m = 1}^{M} {Ae^{ - \beta t} \cos \left( {\omega_{r} t + \phi } \right)}\) denotes the bearing fault signal, A is the amplitude, \(\beta\) is the damping coefficient, \(\omega_{r}\) represents resonance frequency, \(I(t){ = }\sum\limits_{l = 1}^{L} {B_{l} \cos \omega_{l} t}\) is harmonic interference signal, \(B_{l}\) and \(\omega_{l}\) represent the amplitude and frequency of the l-th harmonic interference, respectively. The analytic form of s(t) is:

$$s\left( t \right) = \left[ {\sum\limits_{m = 1}^{M} {Ae^{ - \beta t} \cos \left( {\omega_{r} t + \phi } \right)} + \sum\limits_{l = 1}^{L} {B_{l} \cos \omega_{l} t} } \right]{ + }i\left[ {\sum\limits_{m = 1}^{M} {Ae^{ - \beta t} \sin \left( {\omega_{r} t + \phi } \right)} + \sum\limits_{l = 1}^{L} {B_{l} \sin \omega_{l} t} } \right]$$
(29)

Substituting the real and imaginary parts of Eq. (29) into Eq. (16) yields:

$$\Theta_{3} \left[ {s^{\left( 3 \right)} \left( t \right)} \right]{ = }\frac{1}{32}\left\{ {\sum\limits_{m = 1}^{M} {\left( {A^{2} \omega_{r}^{2} - 3A^{2} \beta^{2} \omega_{r} } \right)} e^{ - 2\beta t} + c(t)e^{ - \beta t} + \sum\limits_{l = 1}^{L} {B_{l}^{2} } \omega_{l}^{3} } \right\},$$
(30)

where,

$$c\left( t \right) = \sum\limits_{l = 1}^{L} {\left[ {AB_{l} \omega_{l}^{3} + B_{l} \left( {3A\omega_{r} \beta^{2} - A^{2} \omega_{r}^{2} } \right)} \right]} {\text{cos}}\left[ {\omega_{r} t - \omega_{l} t + \varphi } \right] - \sum\limits_{l = 1}^{L} {\left( {AB_{l} \beta^{3} - 3A\beta \omega_{r}^{2} } \right)\sin \left( {\omega_{r} t - \omega_{l} t - \varphi } \right)}$$
(31)

According to Eq. (30), the third-order SHAEO consists of three parts: the demodulation part of the bearing fault signal \(\sum\limits_{m = 1}^{M} {\left( {A^{2} \omega_{r}^{2} - 3A^{2} \beta^{2} \omega_{r} } \right)} e^{ - 2\beta t} ,\) the high-frequency part \(c(t)e^{ - \beta t} ,\) and the harmonic interference \(\sum\limits_{l = 1}^{L} {B_{l}^{2} } \omega_{l}^{3} .\) Then, the SIR of the bearing fault signal after third-order SHAEO demodulation can be expressed as:

$$SIR\left[ {\Theta_{3} \left( {s\left( t \right)} \right)} \right] = \frac{{\frac{1}{{T_{p} }}\int\limits_{0}^{{T_{p} }} {\left[ {\left( {A^{2} \omega_{r}^{2} - 3A^{2} \beta^{2} \omega_{r} } \right)^{2} e^{ - 4\beta t} } \right]} dt}}{{\frac{1}{{T_{l} }}\int\limits_{0}^{T} {\left[ {\sum\limits_{l = 1}^{L} {B_{l}^{2} } \omega_{l}^{3} } \right]}^{2} dt}} \approx \frac{{A^{4} \omega_{r}^{6} }}{{4T_{p} \beta \sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{6} } }}$$
(32)

Substitute the real and imaginary parts of Eq. (31) into (18) yields:

$$\Phi \left[ {s(t)} \right] = \sum\limits_{m = 1}^{M} {A^{2} \omega_{r} } e^{ - 2\beta t} + c(t)e^{ - \beta t} + \sum\limits_{l = 1}^{L} {B_{l}^{2} } \omega_{l}$$
(33)
$$\lambda \left( t \right) = \sum\limits_{l = 1}^{L} {AB_{l} \omega_{r} \omega_{l} \cos \left( {\omega_{r} t - \omega_{l} t + \phi } \right)} + \sum\limits_{l = 1}^{L} {AB_{l} \beta \omega_{l} \sin \left( {\omega_{r} t - \omega_{l} t + \phi } \right)}$$
(34)
$$SIR\left[ {\Phi \left( {s\left( t \right)} \right)} \right] = \frac{{\frac{1}{{T_{p} }}\int\limits_{0}^{{T_{p} }} {\left( {A^{2} \omega_{r} e^{ - 2\beta t} } \right)}^{2} dt}}{{\frac{1}{{T_{l} }}\int\limits_{0}^{T} {\left[ {\sum\limits_{l = 1}^{L} {B_{l}^{2} \omega_{l} } } \right]}^{2} dt}} \approx \frac{{A^{4} \omega_{r}^{2} }}{{4\beta T_{p} \sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{2} } }}$$
(35)

Substituting Eq. (26) into third-order SHFWEO yields:

$$\Gamma_{3} \left( t \right) = \left[ {s^{\left( 3 \right)} \left( t \right)} \right]^{2} + H\left[ {s^{\left( 3 \right)} \left( t \right)} \right]^{2} = A^{2} e^{ - 2\beta t} \left( {\beta^{2} + \omega_{r}^{2} } \right)^{3} + \lambda \left( t \right)e^{ - \beta } + \sum\limits_{l = 1}^{L} {B_{l}^{2} \omega_{l}^{6} }$$
(36)

where,

$$\lambda \left( t \right) = \sum\limits_{l = 1}^{L} {2AB_{l} \omega_{l}^{3} \omega_{r}^{3} \sin \left( {\omega_{r} t + \phi } \right)\sin \left( {\omega_{l} t} \right)} - \sum\limits_{l = 1}^{L} {6AB_{l} \beta^{2} \omega_{l}^{3} \omega_{r} \cos \left( {\omega_{r} t - \omega_{l} t + \phi } \right)} - \sum\limits_{l = 1}^{L} {6AB_{l} \omega_{l}^{3} \beta \omega_{r}^{2} \sin \left( {\omega_{l} t - \omega_{r} t - \phi } \right)}$$
(37)

According to Eq. (28), the third-order SHFWEO also consists of three parts: the demodulation part of the bearing fault signal \(A^{2} e^{ - 2\beta t} \left( {\beta^{2} + \omega_{r}^{2} } \right)^{3}\), the high-frequency part \(\lambda \left( t \right)e^{ - \beta }\) and the harmonic interference \(\sum\limits_{l = 1}^{L} {B_{l}^{2} \omega_{l}^{6} }\). Then, the SIR of the bearing fault signal after third-order SHFWEO demodulation can be expressed as:

$$SIR\left[ {\Gamma_{3} \left( {x\left( t \right)} \right)} \right] = \frac{{\frac{1}{{T_{p} }}\int\limits_{0}^{{T_{p} }} {\left[ {A^{2} e^{ - 2\beta t} \left( {\beta^{2} + \omega_{r}^{2} } \right)^{3} } \right]}^{2} dt}}{{\frac{1}{{T_{l} }}\int\limits_{0}^{T} {\left[ {\sum\limits_{l = 1}^{L} {B_{l}^{2} \omega_{l}^{6} } } \right]}^{2} dt}} \approx \frac{{A^{4} \left( {\beta^{2} + \omega_{r}^{2} } \right)^{6} }}{{4\beta T_{p} \sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{12} } }}$$
(38)

For ease of analysis, let \(B_{1} = B_{2} = \, \cdots = B_{M} = B\). Generally, \(\omega_{2} = 2\omega_{1} , \, \omega_{3} = 3\omega_{1} ,...,\omega_{l} = l\omega_{1} ,\) \(\omega_{l} \ll \beta ,\) \(\omega_{l} \ll \omega_{r} ,\) then, combining Eqs. 32 and 38, it can be obtained that

$$\xi { = }\frac{{SIR\left[ {\Gamma_{3} \left( {s\left( t \right)} \right)} \right]}}{{SIR\left[ {\Theta_{3} \left( {s\left( t \right)} \right)} \right]}} = \frac{{\left( {{{\left( {\beta^{2} + \omega_{r}^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\beta^{2} + \omega_{r}^{2} } \right)} {\omega_{r} }}} \right. \kern-0pt} {\omega_{r} }}} \right)^{6} }}{{{{\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{12} } } \mathord{\left/ {\vphantom {{\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{12} } } {\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{6} } }}} \right. \kern-0pt} {\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{6} } }}}}{ = }\frac{{\left( {{{\left( {\beta^{2} + \omega_{r}^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\beta^{2} + \omega_{r}^{2} } \right)} {\omega_{r} }}} \right. \kern-0pt} {\omega_{r} }}} \right)^{6} }}{{{{\sum\limits_{l = 1}^{L} {\omega_{l}^{12} } } \mathord{\left/ {\vphantom {{\sum\limits_{l = 1}^{L} {\omega_{l}^{12} } } {\sum\limits_{l = 1}^{L} {\omega_{l}^{6} } }}} \right. \kern-0pt} {\sum\limits_{l = 1}^{L} {\omega_{l}^{6} } }}}} > 1,$$
(39)
$$\xi { = }\frac{{SIR\left[ {\Theta_{3} \left( {s\left( t \right)} \right)} \right]}}{{SIR\left[ {\Phi_{3} \left( {s\left( t \right)} \right)} \right]}} = \frac{{\omega_{r}^{4} }}{{{{\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{6} } } \mathord{\left/ {\vphantom {{\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{6} } } {\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{2} } }}} \right. \kern-0pt} {\sum\limits_{l = 1}^{L} {B_{l}^{4} \omega_{l}^{2} } }}}}{ = }\frac{{\omega_{r}^{4} }}{{{{\sum\limits_{l = 1}^{L} {\omega_{l}^{6} } } \mathord{\left/ {\vphantom {{\sum\limits_{l = 1}^{L} {\omega_{l}^{6} } } {\sum\limits_{l = 1}^{L} {\omega_{l}^{2} } }}} \right. \kern-0pt} {\sum\limits_{l = 1}^{L} {\omega_{l}^{2} } }}}} > 1.$$
(40)

Therefore, \(SIR\left[ {\Phi_{3} \left( {s\left( t \right)} \right)} \right] < SIR\left[ {\Theta_{3} \left( {s\left( t \right)} \right)} \right] < SIR\left[ {\Gamma_{3} \left( {s\left( t \right)} \right)} \right].\) It can be concluded that SHFWEO has the best anti-interference performance, followed by SHAEO, and MSDAEO has the worst.

Harmonic interference signals with frequencies of 67 Hz, 115 Hz, and 133 Hz are added to the bearing fault vibration signal represented by Eq. (33). The faulty bearing signal and the composite signal with harmonic interference are shown in Fig. 3.

Fig. 3
figure 3

(a) Faulty bearing signal, (b) composite signal.

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TKEO, SHAEO, MSDAEO, and SHFWEO are applied on the composite signal, the results are shown in Fig. 4. It can be seen that SHFWEO has the best effect on suppressing the interference frequency components, followed by SHAEO and MSDAEO. The results are consistent with the theoretical analysis.

Fig. 4
figure 4

Energy spectra of the composite signal: (a) TEO spectrum, (b) SHAEO spectrum, (c) MSDAEO spectrum, (d) SHFWEO spectrum.

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Vibration exciter bearing fault diagnosis experiment

The vibration screen exciter bearing fault diagnosis experiments are conducted to evaluate the performance of SHAEO, MSDAEO, and SHFWEO. The vibration screen device and the measurement point ___location of the faulty bearing are shown in Fig. 5. The model of the vibrating screen is SDM00 which is a dual axis and dual motor multifunctional vibrating screen. Its vibration exciter is composed of two transmission shafts, eccentric blocks, bearings, and bearing seats. Two synchronous motors drive the exciter to rotate in opposite directions at equal speeds, generating centrifugal inertia forces on two eccentric mass blocks. Both motor models are Y90S-6 and the rotational frequency is fr = 16.7 Hz. The faulty bearing with specifications listed in Table 1 is shown in Fig. 6. According to the theoretical fault frequency formula of the bearing, the characteristic frequency of the bearing outer race fault is calculated as follows:

$$f_{o} = \frac{{f_{r} }}{2}Z\left( {1 - \frac{d}{D}\cos \theta } \right){ = }104.2\; \, ({\text{Hz}})$$
(41)
Fig. 5
figure 5

(a) Vibrating screen experimental device and (b) measuring point ___location.

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Table 1 Bearing specifications.
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Fig. 6
figure 6

Outer race fault bearing of vibration exciter.

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In this experiment, two sizes of eccentric blocks are prepared, as shown in Fig. 7. The two different vibration levels, high and low, are realized by replacing the two kinds of eccentric blocks. Firstly, the fault bearing and small eccentric block are installed on the exciter to work for a period of time. A DYTRAN 3274A1 accelerometer is installed on the small platform of the exciter bearing seat. A high-performance multi-channel INV3018C data acquisition instrument is used to collect the fault vibration signals of the exciter rolling bearing. The sampling frequency is 20 kHz, and each channel collects 65,534 points. Then, the signal of the fault bearing is collected for analysis. The collected signal and its frequency spectrum are shown in Fig. 8. The spindle rotation frequency (16.7 Hz) and the 5th harmonic of the bearing outer race fault characteristic frequency (520.7 Hz) can be identified in Fig. 8b.

Fig. 7
figure 7

(a) Large eccentric block, (b) small eccentric block.

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Fig. 8
figure 8

The original signal under the low vibration level: (a) time-___domain waveform, (b) frequency spectrum.

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Three energy operators, SHAEO, MSDAEO and SHFWEO, are used to process the signal. The resolution k of MSDAEO is determined according to the following SNRI index:

$$SNRI = 10\log \left[ {\frac{{\frac{1}{N}\sum\nolimits_{n = 1}^{N} {\left| {\left. {v\left( n \right)} \right|^{2} } \right.} }}{{\frac{1}{N}\sum\nolimits_{n = 1}^{N} {\left| {\left. {s\left( n \right) - v\left( n \right)} \right|^{2} } \right.} }}} \right]$$
(42)

Calculate the SNRI of MSDAEO at different resolutions and list the results in Table 2. Select the resolution k = 2 when the SNRI is at its maximum. The demodulation results of the three energy operators on the original signal are shown in Fig. 9. It can be seen that under the low vibration level, the processing results of the three energy operators are similar, and they can effectively identify the outer race fault characteristic frequency of bearing and its 2–5 harmonics. The difference is that due to the fusion of higher-order differential sequences in SHAEO and SHFWEO, the amplitude in the spectrogram is higher, especially in SHAEO, which has the highest amplitude among the three.

Table 2 SNRI of MSDAEO at different resolutions.
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Fig. 9
figure 9

Energy spectra in the case of the low vibration level: (a) SHAEO spectrum, (b) MSDAEO spectrum, (c) SHFWEO spectrum.

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Gaussian noise with SNR =  − 8 dB was added to the collected fault signal at low vibration levels to evaluate the noise resistance performance of the three methods. The noise signal and its spectrum are shown in Fig. 10. It can be seen that after adding noise, the fault impact characteristics of the time-___domain signal are submerged, and the noise in the spectrum is enhanced.

Fig. 10
figure 10

Signal after noise addition: (a) time ___domain diagram, (b) spectrogram.

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Three methods, namely SHAEO, MSDAEO, and SHFWEO, were used to process the noisy signal. The resolution k of MSDAEO was set to 2 according to the maximum SNRI criterion. The demodulated results by the three energy operators are shown in Fig. 11.

Fig. 11
figure 11

Energy spectra of the signal after noise addition: (a) SHAEO spectrum, (b) MSDAEO spectrum, (c) SHFWEO spectrum.

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Compared with Fig. 9, the background noise in the SHAEO spectrum of the noisy signal drowned out the bearing outer race fault characteristic frequency (fo) and its 5th harmonic (5fo). fo in the SHFWEO spectrum is submerged, and only its 2–5 harmonics can be seen. Although the MSDAEO spectrum has increased noise compared to before noise addition, it can still recognize the outer fault characteristic frequency and its 2–5 harmonics, which has the best anti-noise performance among the three energy operators.

The vibration interference and edge frequency energy generated by centrifugal load vary at different vibration levels. Then, we disassemble the vibration exciter and replace the four small eccentric blocks with four large ones to operate the vibrating screen at high vibration levels to study the differences between the three energy operators under different vibration levels. The time-___domain waveform and frequency spectrum of the collected fault signal are shown in Fig. 12. Compared with Fig. 8a, it can be observed that the sinusoidal waveform is more prominent under high vibration levels, which is because as the vibration level of the vibrating screen increases, the imbalance phenomenon of the transmission shaft caused by the presence of eccentric blocks becomes more and more outstanding. The bearing rotation frequency (16.7 Hz) and the 5th harmonic (520.7 Hz) of the outer race fault characteristic frequency can be identified in the spectrum diagram.

Fig. 12
figure 12

The original signal under the high vibration level: (a) time-___domain waveform, (b) frequency spectrum.

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Three energy operators, SHAEO, MSDAEO, and SHFWEO, are also used to demodulate the fault signals under the high vibration level. The resolution of MSDAEO is set to 3 according to the maximum SNRI criterion. The processing results of the three methods are shown in Fig. 13. Compared with the processing results of low vibration levels (Fig. 9), the noise in the SHAEO spectrum is significantly enhanced, and the edge frequencies are more prominent, which is because the high-order differential sequence fused in SHAEO increases the amplitude of the fault characteristic frequency and that of the edge frequency. The MSDAEO spectrum has the minimum noise but severe edge frequency and interference, which proves that MSDAEO can effectively improve the SNR of signals but has limited ability to enhance the SIR. SHFWEO has the minimum interference and side frequency among the three, which verifies its excellent anti-interference ability. As the vibration level of the vibrating screen increases, the fault characteristic frequency in the SHFWEO spectrum shows the most significant increase. That is because SHFWEO was developed based on the concept of energy and is more sensitive to the signal’s energy. The larger the signal energy, the stronger the extraction ability of SHFWEO.

Fig. 13
figure 13

The energy spectra in the case of the high vibration level: (a) SHAEO spectrum, (b) MSDAEO spectrum, (c) SHFWEO spectrum.

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In summary, among the three energy operators, MSDAEO has the most outstanding ability to improve the SNR of signals but the weakest effect to improve signal SIR, which is because MSDAEO contains a symmetric difference sequence improved by non-adjacent sampling points to improve SNR of the signal but has no high-order differential sequences to improve SIR of the signal and effectively suppress the influence of vibration interference. Once the energy of vibration interference is too large, its performance will decrease. Therefore, it is not recommended to use MSDAEO under high vibration interference energy conditions, but SHWEO and SHAEO are recommended. This operator can be used in conjunction with denoising algorithms such as modal decomposition that has excellent anti-interference performance, thereby expanding its applicability. SHFWEO performs the best in improving the SIR of signals but is sensitive to signal energy, which is because SHFWEO is developed from the concept of energy, and simultaneously uses high-order differential sequences to improve SIR and symmetric differential sequences to improve SNR. Therefore, it has good impact feature extraction performance in strong noise and multiple interference situations. Its performance largely depends on the magnitude of signal energy, and if the signal energy is too small, its application will be limited to a certain extent. Therefore, SHFWEO is not suitable for use in combination with methods such as modal decomposition and wavelet decomposition, which can decompose a signal into multiple sub signals or filter out a large amount of signal information, thereby reducing the energy of the signal. When extracting weak fault features, SHFWEO can be combined with denoising algorithms that enhance bearing fault pulses. SHAEO contains a high-order differentiation sequence to improve SIR of the signal and symmetric differential sequences to enhance SNR of the signal, thereby suppressing noise and significantly increasing the amplitude of fault characteristic frequencies. But this method will also increase the amplitude of other frequencies, such as the amplitude of the edge frequency, which will cause obvious edge frequency interference and affect judgment. This operator was not developed based on the concept of energy, so its anti-interference performance is not as good as SHFWEO. The adjacent three-point symmetric difference sequences fused by this operator have lower noise resistance than MSDAEO with a symmetric difference sequence improved by non-adjacent sampling points. But it is suitable for a wide range of occasions, even in situations with strong background noise and large vibration interference. SHAEO can be used in conjunction with most denoising pre-processing algorithms, including denoising methods that reduce signal energy.

Conclusions

The theoretical derivation of three energy operators based on technology fusion, namely symmetric higher-order analytical energy operator (SHAEO), multi-resolution symmetric difference energy operator (MSDAEO), and symmetric higher-order frequency weighted energy operator (SHFWEO), is given. A comparative study was conducted on the three energy operators through theoretical analysis and fault diagnosis experiments of exciter bearing under three working conditions: low vibration level, low vibration level with noise, and high vibration level. The research results indicate that MSDAEO has the most outstanding ability to improve the SNR of signals but the weakest effect to improve signal SIR. SHFWEO performs the best in improving the SIR of signals but is sensitive to signal energy. SHAEO can increase the amplitude and improve the SNR and SIR of the signal. Finally, the pre-processing methods that can be used jointly with the three energy operators in different application scenarios are given.