1 Introduction

The parallel shaft gearbox is one of the most vital and common component of rotating machinery. It is widely used in some rotating machinery such as automobile, power generating turbines, helicopters and more. The faults of gearbox are common cause of rotating machinery breakdown, and it can generally result in financial losses even safety accidents. Therefore, gearbox fault detection plays an important role in preventing breakdown accidents of rotating machinery and reducing economic losses (Samuel and Pines 2005). When faults occur on gearbox, fault signals with periodic impact components will be generated during gearbox operation. These impulsive components in the vibration signal appear in a fixed period and form a fixed frequency, which is the fault frequency (Golafshan and Sanliturk 2015). The vibration signal analysis is a common and effective method for gearbox fault detection (McFadden and Smith 1985). Several signal analysis techniques have been employed for gearbox fault detection, such as acoustic emission technology, infrared thermal imaging technology and oil analysis technology. Gu et al. (2011) used Discrete Wavelet Transform (DWT) and envelope analysis to analyze the collected acoustic emission signals, and achieved the purpose of gearbox fault diagnosis. Lim et al. (2014) combines infrared thermal imaging technology with Support Vector Machine (SVM) to diagnose faults of rotating machinery. Hamilton and Quail (2011) summarized the oil analysis technology used in wind turbine gearbox. However, acquisition of acoustic emission signals requires a higher sampling frequency, resulting in a huge amount of data, which limits the engineering application. Infrared thermal imaging technology and oil analysis technology are more suitable for monitoring the running state of equipment, and the accurate fault detection of rotating machinery is still in the exploratory stage.

The traditional vibration analysis methods of gearbox fault diagnosis include time domain analysis (McFadden 1991; Enayet et al. 2008; Hong and Dhupia 2014) and frequency domain analysis (Feng et al. 2016; Hong and Dhupia 2014). However, the raw vibration signal of gearbox contains non-stationary and non-linear components, which makes it difficult to detect gearbox faults by traditional time domain analysis and frequency domain analysis. Thus, some time–frequency analysis methods are used for gearbox fault detection. Wavelet transform is a commonly used time–frequency analysis method, which is widely used in gearbox fault diagnosis (Yu et al. 2017). However, the effect of wavelet transform depends on the selection of suitable wavelet basis function, which limits the application of wavelet transform (Honorio et al. 2012). Norden E Huang proposed Hilbert Huang transform for the non-linear and non-stationary signals, and the Empirical Mode Decomposition (EMD) is the key technique of the Hilbert Huang transform. EMD is a self-adaptive analysis technique which can decompose non-stationary signals into a series of intrinsic mode functions (IMFs) (Li et al. 2017). However, EMD still has many shortcomings, such as the criterion of IMF, over-decomposition, end effect and modal mixing. To overcome these inherent defects, Peng et al. (2005) combines Wavelet Packet Decomposition (WPD) with EMD to ensure that each IMF obtained by EMD is a single component. Yang (2008) uses band-pass filter to preprocess the mechanical fault signal and then decomposes the signal with EMD, which can restrain many shortcomings of EMD. Tang et al. (2012) uses the improved blind source separation method to suppress the modal mixing effect of EMD. Wu and Huang (2009) proposes the Ensemble Empirical Mode Decomposition (EEMD) for the problem of modal aliasing in EMD. EMD is used to decompose the signal in this work, and then IMFs with large kurtosis are selected to reconstruct to enhance the fault characteristics.

Therefore, the deconvolution technique is proposed to identify the fault of rotating machinery. The minimum entropy deconvolution (MED) is based on searching for a linear time-invariant filter which maximizes kurtosis of the filtered signal (Obuchowski et al. 2016). Compared with the traditional deconvolution method, MED does not need any assumptions and prior knowledge. It is an iterative method which can construct filters adaptively to find sparse spikes with simple characteristics (Cabrelli 1984). The method has been applied to gearbox faults detection successfully by Endo and Randall (2007, 2009). Meanwhile, the MED also can be used to identify the rolling bearings fault (Jiang et al. 2012; Sawalhi et al. 2007), and it proved to be an effective method. Nevertheless, some drawbacks of MED emerged during application to rotating machinery fault diagnosis. One of drawbacks is that the use of MED may produce spurious impulse whatever the signal is a fault signal or a white noise signal. The other one is that MED may not obtain optimal results due to iterative operations.

In view of these problems of MED, GL. Mcdonald proposed Maximum Correlated Kurtosis Deconvolution (MCKD) for gearbox fault detection (McDonald et al. 2012). MCKD can solve the problems of MED mentioned above. But MCKD is also an iterative process, it is usually impossible to obtain the optimal filter, and the fault cycle needs to be known beforehand. In addition, extra signal resampling pretreatment is needed for non-integer fault cycle, which increases computational complexity and limits its practical application. Thus, GL. Mcdonald also proposed a new deconvolution method called Multipoint Optimal Minimum Entropy Deconvolution (MOMEDA) for gearbox fault detection (McDonald and Zhao 2017). The method solves the iterative problem and achieves good results. However, vibration signals collected from a mechanical system usually contain a large amount of noise. For this reason, it often affects the effect of the above method.

Thus, a fault detection method based on EMD and MOMEDA is proposed in this paper. Firstly, the vibration signal is decomposed to some IMFs by EMD. It is well known that kurtosis is an impulse index for vibration signal, and the kurtosis of the fault signal is usually greater than the kurtosis of the noise signal. Therefore, some large kurtosis components are selected from these IMFs to reconstruct the fault signal. Finally, the processed signals are processed by MOMEDA to achieve the purpose of fault detection. Experimental data on a gearbox with broken tooth are used to verify the performance of the proposed method.

The remains of this paper is organized as follows: Sect. 2 gives the theoretical background of the MED technique, MCKD technique and the proposed method. In Sect. 3, the performance of the described approach on experimental gearbox faulty data is presented. Finally, the conclusion of the paper is given in Sect. 4.

2 Theoretical background

2.1 Minimum entropy deconvolution

The minimum entropy deconvolution was originally proposed by Wiggins (1978) for extraction of seismic wave reflection parameters. Then the technology gradually expanded to other fields. The basic principle of MED is to design the optimal filter rely on the property that MED can enhance the impulse characteristic. After filtered, the pulse impulse component of the filtered signal is enhanced, and the calculation terminates when the maximum kurtosis value is reached.

Suppose that the input signal is x(t), and the MED filter is f(t). Thus, the output signal by using the MED filter is:

$$y\left( t \right) = x\left( t \right)f\left( t \right)$$
(1)

If the length of MED filter is L, the Eq. 1 can also be written as follows:

$${\text{y}}(t) = \sum\limits_{l = 1}^{L} {f(l)x(t - l)}$$
(2)

The kurtosis is large for the pulse impulse in vibration signal. Therefore, the kurtosis value of the filtered signal is considered as the result function to evaluate the filtering effect. The result function is calculated in the following formula:

$$K(f(l)) = \frac{{\sum\nolimits_{t = 1}^{N} {y^{4} (t)} }}{{\left( {\sum\nolimits_{t = 1}^{N} {y^{2} (t)} } \right)^{2} }}$$
(3)

where the l equals 1, 2,……L. In order to obtain the maximum of K(f(l)), the derivative of filter coefficients f(l) for the above formula is obtained, and the derivative is equal to zero so that the solution is obtained. After deducing and simplifying, the following formula can be obtained:

$$\frac{{\sum\nolimits_{t = 1}^{N} {y^{2} (t)} }}{{\sum\nolimits_{t = 1}^{N} {y^{4} (t)} }}\sum\limits_{t = 1}^{N} {\left( {y^{3} (t)\frac{\partial y(t)}{\partial f(l)}} \right)} = \sum\limits_{t = 1}^{N} {\left( {y(t)\frac{\partial y(t)}{\partial f(l)}} \right)}$$
(4)

A, b, f are respectively equal to the following formula:

$$b = \frac{{\sum\nolimits_{t = 1}^{N} {y^{2} (t)} }}{{\sum\nolimits_{t = 1}^{N} {y^{4} (t)} }}\sum\limits_{t = 1}^{N} {(y^{3} (t)d(t - l))}$$
(5)
$$A = \sum\limits_{t = 1}^{N} {\sum\limits_{p = 1}^{L} {(d(t - p)d(t - l))} }$$
(6)
$$F = \sum\limits_{p = 1}^{L} {f(p)}$$
(7)

Then it can be obtained in matrix form:

$$b = AF$$
(8)

It also can be calculated as follows:

$$F = A^{ - 1} b$$
(9)

where b is a cross-correlation vector which is obtained by calculating the correlation between the input and output signals of the inverse filter; and A is the Toeplitz autocorrelation vector that is a weighted summation of the autocorrelations of the input signals; the F is the coefficients vector for the desired inverse filter.

The implementation steps of MED can be summarized as follows:

Step 1:

Compute the autocorrelation vector A

Step 2:

Initialize the MED filter coefficients

Step 3:

The output signal is calculated by using MED filter

Step 4:

Calculate the cross-correlation matrix b according to the Eq. 8

Step 5:

Compute the filter coefficients rely on Eq. 9

Step 6:

Calculate the termination condition of the iteration as follows:

$$\delta = \frac{{\sum\nolimits_{t = 1}^{N} {y^{(i)4} (t)} }}{{\left( {\sum\nolimits_{t = 1}^{N} {y^{(i)2} (t)} } \right)^{2} }} - \frac{{\sum\nolimits_{t = 1}^{N} {y^{(i - 1)4} (t)} }}{{\left( {\sum\nolimits_{t = 1}^{N} {y^{(i - 1)2} (t)} } \right)^{2} }}$$
(10)

where i is the number of iterations, the iteration process will terminate when the variation of the kurtosis value between the iterations is lower than the threshold value.

2.2 Maximum correlated Kurtosis deconvolution

The MCKD algorithm was putted forward to emphasize the periodic pulse components for the signal with noise. The evaluation criterion of this method is to maximize the correlation kurtosis. Suppose that the signal collected by sensor is:

$$x_{n} = h_{n} *y_{n} + e_{n}$$
(11)

where xn is the original fault signal; hn is the response of the system transmission path; yn is the failure periodic impact component; en is the noise signal.

The hn masked the fault cycle impulse of the xn, and the correlation kurtosis becomes smaller. The purpose of the MCKD algorithm is use the deconvolution to highlight the fault impact components, which is hidden in the raw signal. The key of the MCKD method is to search a finite length unit impulse response filter f. The filter can detect the original pulse signal x(n) of the collected signal yn, and the specific calculation process is shown as follows:

$$y_{n} = x_{n} *f_{l}$$
(12)

The objective function of MCKD is the maximum correlation kurtosis. Firstly, the correlation kurtosis is defined as follows:

$$CK_{M} (T) = \frac{{\sum\nolimits_{n = 1}^{N} {\left( {\prod\nolimits_{m = 0}^{M} {y_{n - mT} } } \right)^{2} } }}{{\left( {\sum\nolimits_{n = 1}^{N} {y_{n}^{2} } } \right)^{M + 1} }}$$
(13)

wherein M is the quantity of sequential pulses, which is to be deconvolution. The T is the separation cycle of these pulses. Thus, the objective function is shown as follows:

$$MCKD_{M} (T) = \max_{{f_{l} }} \frac{{\sum\nolimits_{n = 1}^{N} {\left( {\prod\nolimits_{m = 0}^{M} {y_{n - mT} } } \right)^{2} } }}{{\left( {\sum\nolimits_{n = 1}^{N} {y_{n}^{2} } } \right)^{M + 1} }}$$
(14)

To maximize the correlation kurtosis, we solve

$$\begin{aligned} \frac{{dCK_{m} (T)}}{{df_{l} }} & = 0 = 2\parallel y\parallel^{ - 2M - 2} \sum\limits_{n = 1}^{N} {\left[ {\left( {\prod\limits_{m = 0}^{M} {y_{n - mT} } } \right)^{2} \left( {\sum\limits_{m = 0}^{M} {\frac{{x_{n - mT - k + 1} }}{{y_{n - mT} }}} } \right)} \right]} \\ & \quad - \,2(M + 1)\parallel y\parallel^{ - 2M - 4} \sum\limits_{n = 1}^{N} {\left( {\left( {\prod\limits_{m = 0}^{M} {y_{n - mT} } } \right)^{2} } \right)} \sum\limits_{n = 1}^{N} {y_{n} } x_{n - k + 1} \\ \end{aligned}$$
(15)

The result of the filter is shown as follows by simplification and representation in matrix form.

$$f = \frac{{\parallel y\parallel^{2} }}{{2\parallel B\parallel^{2} }}(X_{0} X_{0}^{T} )^{ - 1} \sum\limits_{n = 0}^{M} {X_{mT} } A_{m}$$
(16)
$$A_{m} = \left[ {\begin{array}{*{20}c} {y_{1 - mT}^{ - 1} (y_{1}^{2} y_{1 - T}^{2} \cdot \cdot \cdot y_{1 - MT}^{2} )} \\ {y_{2 - mT}^{ - 1} (y_{2}^{2} y_{2 - T}^{2} \cdot \cdot \cdot y_{2 - MT}^{2} )} \\ \vdots \\ {y_{N - mT}^{ - 1} (y_{N}^{2} y_{N - T}^{2} \cdot \cdot \cdot y_{N - MT}^{2} )} \\ \end{array} } \right],\quad B = \left[ {\begin{array}{*{20}c} {y_{1} y_{1 - T} \cdot \cdot \cdot y_{1 - MT} } \\ {y_{1} y_{1 - T} \cdot \cdot \cdot y_{1 - MT} } \\ \vdots \\ {y_{N} y_{N - T} \cdot \cdot \cdot y_{N - MT} } \\ \end{array} } \right]$$

The array \(X_{0} X_{0}^{T}\) is the Toeplitz autocorrelation array of xn and the inverse \((X_{0} X_{0}^{T} )^{ - 1}\) is supposed to exist. The flow of the MCKD algorithm is shown as follows:

Step 1:

Initialization deconvolution period T, number of sequential M and the length of filter

Step 2:

Calculate the \(X_{0} X_{0}^{T}\) and \(X_{mT}\) of the input signal

Step 3:

Calculate the filtered signal y

Step 4:

Calculate the Am and B from the filtered signal

Step 5:

Update filter coefficient fl

Step 6:

If the \(\Delta CK_{m} (T)\) is less than the threshold, the iteration is ended, or 3–5 steps are repeated

Step 7:

Calculate the period T of the final filtered signal

2.3 Multipoint optimal minimum entropy deconvolution

One drawback of MED is that the coefficients of the filter have been determined when iteration is performed. To overcome this shortcoming, Cabrelli (1985) has proposed a new norm called D norm, which is mainly used to deconvolution pulse signals. The D-Norm is shown as follows:

$$D(y) = \mathop {\hbox{max} }\limits_{k = 1,2,. \ldots N} \frac{{\left| {y_{k} } \right|}}{\left\| y \right\|}$$
(17)

On this basis, Geoff L. McDonald proposed the concept of Muti D-Norm and introduced a new algorithm called Multipoint Optimal Minimum Entropy Deconvolution Adjusted (MOMEDA). The Muti D-Norm is formulated as the following:

$$MDN(y,t) = \frac{1}{\left\| t \right\|}\frac{{t^{T} y}}{\left\| y \right\|}$$
(18)

where the t is a constant vector in the deconvolution, which determines the location and weight of the goal impulse. The objective function of MOMEDA is to make the MDN maximum as following:

$$\mathop {\hbox{max} }\limits_{f} MDN(y,t) = \mathop {\hbox{max} }\limits_{f} \frac{{t^{T} y}}{\left\| y \right\|}$$
(19)

To maximize the MDN, we solve

$$\frac{d}{df}\left( {\frac{{t^{T} y}}{\left\| y \right\|}} \right) = \frac{d}{df}\left( {\frac{{t_{1} y_{1} }}{\left\| y \right\|}} \right) + \frac{d}{df}\left( {\frac{{t_{2} y_{2} }}{\left\| y \right\|}} \right) + \cdot \cdot \cdot + \frac{d}{df}\left( {\frac{{t_{N - L} y_{N - L} }}{\left\| y \right\|}} \right)$$
(20)

After simplification,

$$\frac{d}{df}\left( {\frac{{t^{T} y}}{\left\| y \right\|}} \right) = \left\| y \right\|^{ - 1} (t_{1} M_{1} + t_{2} M_{2} + \cdot \cdot \cdot + t_{N - L} M_{N - L} ) - \left\| y \right\|^{ - 3} t^{T} yX_{0} y$$
(21)
$$X_{0} = \left[ {\begin{array}{*{20}c} {x_{L} } & {x_{L + 1} } & {x_{L + 2} } & \cdots & {x_{N} } \\ {x_{L - 1} } & {x_{L} } & {x_{L + 1} } & \cdots & {x_{N - 1} } \\ {x_{L - 2} } & {x_{L - 1} } & {x_{L} } & \cdots & {x_{N - 2} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {x_{1} } & {x_{2} } & {x_{3} } & {} & {x_{N - L + 1} } \\ \end{array} } \right]$$

where the \(M_{k} = \left[ {x_{k + L - 1} ,x_{k + L - 2} \cdot \cdot \cdot x_{k} } \right]^{T}\), then the equation can be written as follows.

$$\frac{d}{df}\left( {\frac{{t^{T} y}}{\left\| y \right\|}} \right) = \left\| y \right\|^{ - 1} X_{0} t - \left\| y \right\|^{ - 3} t^{T} yX_{0} y$$
(22)

The above formula is equal to zero and can be obtained:

$$\left\| y \right\|^{ - 1} X_{0} t - \left\| y \right\|^{ - 3} t^{T} yX_{0} y = 0$$
(23)

Since \(y = X_{0}^{T} f\) and assuming \((X_{0} X_{0}^{T} )^{ - 1}\) exists, the Eq. 21 becomes:

$$\frac{{t^{T} y}}{{\left\| y \right\|^{2} }}f = (X_{0} X_{0}^{T} )^{ - 1} X_{0} t$$
(24)

Hence the filter can be calculated as following:

$$f = (X_{0} X_{0}^{T} )^{ - 1} X_{0} t$$
(25)

Where the length of vector t is \((N - L + 1)\), it is same as the output signal.

2.4 Empirical mode decomposition

EMD is an adaptive signal processing method for nonlinear and non-stationary signals. The precondition of the method is that the signal is composed of a series of intrinsic mode functions (IMFs). Moreover, each of IMFs is required to satisfy the following conditions (Huang et al. 1998):

  1. (1)

    The extreme points and zero crossing points of the entire IMF signal must be equal or at least one difference (Yang et al. 2006).

  2. (2)

    For any data point, the envelope average value of its local maximum and minimum must be zero, that is, the local envelope signal is symmetrical with respect to the time axis (Cheng et al. 2006).

EMD is based on the premise that any signal is composed of many different IMFs. The process of decomposing a given signal x (t) into a number of IMFs can be summarized as follows:

  1. (1)

    Indentify all the local extrema, the upper envelope of the signal x(t) is determined according to its local maxima. Similarly, the lower envelope of the signal can be obtained rely on the local minima. Then calculate the mean of upper and lower envelope value as the m1.

  2. (2)

    The first component h1 is the difference between the signal \(x(t)\) and \(m_{1}\), the formula is as follows:

    $$x(t) - m_{1} = h_{1}$$
    (26)

\(h_{1}\) will be the first IMF when it meets the above two requirements.

  1. (3)

    When \(h_{1}\) does not meet the above two requirements, it will be used as the raw signal to duplicate the above first two steps.

    $$h_{1} - m_{11} = h_{11}$$
    (27)

Then repeat operation k times until \(h_{1k}\) becomes an IMF shown as follows:

$$h_{1(1 - k)} - m_{1k} = h_{1k}$$
(28)

Then make \(c_{1}\) equal to \(h_{1k}\) as the first IMF of signal.

  1. (4)

    The signal \(r_{1}\) is obtained after the signal \(c_{1}\) is removed from the raw signal.

    $$r_{1} = x(t) - c_{1}$$
    (29)

The \(r_{1}\) repeats the above processes as a new raw signal and the second IMF can be obtained. A raw signal can get several IMFs by repeating the above operation, the decomposition is terminated until the remaining signal \(r_{n}\) is monotonic.

Therefore, the raw signal x(t) can be written as follows.

$$x(t) = \sum\limits_{i = 1}^{n} {c_{i} + r_{n} }$$
(30)

c1, c2,…,cn represent IMFs with different frequency bands, which are arranged from high to low. Each frequency band contains different frequency components and they varies according to the raw signal x(t). The residual signal represents the overall trend of the raw signal x(t).

3 Proposed method

The proposed method of fault detection based on EMD-MOMEDA is illustrated in this section. As known, the MOMEDA is an efficient technique for detecting faults of mechanical parts. However, the working environment of the mechanical system is usually complex. For this reason, the signals measured by accelerometers, which installed on the rotating machinery, always contains large amounts of random noise. Furthermore, the signal is also doped into the noise signal during the acquisition process. These causes affect the signal processing effect of the MOMEDA method. To overcome this problem, EMD is proposed to process the raw signal to eliminate noise interference, the decomposition and reconstruction of signals using EMD can eliminate the influence of random components. Therefore, the method of fault detection based on EMD-MOMEDA is proposed in this work.

The operation flow of the new method is shown as Fig. 1.

Fig. 1
figure 1

Diagram of the proposed method

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4 Experimental data analysis

In order to make a qualitative validation of the mentioned method, the gearbox preset fault experiments was carried out to investigate the effectiveness of the new method.

4.1 Case study 1

4.1.1 Experimental setup

The test rig is composed of a 4 kW stepless speed motor, a parallel shaft gearbox, a speed torque sensor, a magnetic powder brake and signal acquisition system. The overall structure of the test bed is presented in Fig. 2. The magnetic particle brake can provide the load for the experiment. As shown in Fig. 3, the test gearbox is equipped with a set of four gears and three shafts. The input shaft of the gearbox is driven through stepless speed motor, the power is transmitted from the input shaft to the output shaft by the intermediate shaft. The magnetic powder brake can provide load, which is connected with the output end of the gearbox through a coupling. Thus the conservation of energy is achieved. In the experiment, three different degrees of broken teeth are implanted in gear 2 of intermediate shaft. The width of broken teeth fault is 2 mm, 5 mm and 10 mm respectively as shown in Fig. 4. Experiments are carried out to test the performance of each degree broken tooth fault under different speed and load combinations. Shaft rotational speed is 800 rpm, 1000 rpm and 1200 rpm, the loads are 10 Nm, 15 Nm and 20 Nm respectively. The signal acquisition system consists of a computer, four 3065B4 piezoelectric accelerometer produced by DYTRAN company, a PXI-1031 chassis of NI company, data acquisition card and Labview software. The accelerometers were mounted on the outside of the gearbox such as the location of the digital mark shown in Fig. 3. Data is sampled at 20 kHz per channel for six seconds.

Fig. 2
figure 2

Gearbox implanted fault test-rig

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Fig. 3
figure 3

Gearbox structure and sensor position

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Fig. 4
figure 4

Broken tooth fault on gear 2 with different degrees

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As shown in the Fig. 3, the number of the gear teeth has been known. Thus, the fault frequency of broken tooth is calculated and given in Table 1.

Table 1 Basic size and style requirements
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4.1.2 Experiment result analysis

In this paper, the data of 2 mm broken tooth fault under the condition of 800 rpm and 10Nm is used to investigate the effectiveness of proposed method. First of all, the traditional Fourier spectrum and cepstrum are used to analyze the fault signal and the normal state signal. The Fourier spectrum and cepstrum are shown as Figs. 5 and 6. As shown in the Fig. 5, there is no significant difference between normal signal spectrum and fault signal spectrum, no fault frequency component is found in fault signal spectrum. It is illustrate that the Fourier spectrum is difficult to identify the fault accurately. As shown in the Fig. 6, the cepstrum of fault signal also has no cycle time corresponding to fault frequency. Therefore, it is difficult to detect the gear fault only by traditional spectrum analysis. Then the method proposed in this paper is used to detect gear faults.

Fig. 5
figure 5

Fourier spectrum of normal signal and fault signal a normal signal, b fault signal

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Fig. 6
figure 6

Cepstrum of normal signal and fault signal a normal signal, b fault signal

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Fig. 7
figure 7

EMD decomposition results

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Fig. 8
figure 8

Kurtosises of all IMFs

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Firstly, the raw signal is decomposed to several IMFs by EMD. The result of signal decomposition is presented in Fig. 7. After applying the EMD to the data, the IMFs needs to be selected and reconstructed. In these IMFs, only some IMFs that contain the fault impact signals are useful. The signal components contained in the remaining IMFs are useless for fault diagnosis, such as random noise. Because kurtosis is widely used to reflect the characteristic of the shock, the kurtosis of all the IMFs are calculated are given in Fig. 8. Select IMF with kurtosis above 3.5 to reconstruct the signal. As shown in the Fig. 8, IMF 2, IMF 4, IMF 5, IMF 6 and IMF 7 are chosen to reconstruct the signal for the fault diagnosis. The reconstructed signal is processed by MOMEDA and the result is shown in Fig. 9. In the Fig. 9, the T is sample count between two consecutive fault impulses. Thus, the period of fault impulse can be computed by T and sample rate. The fault impulse frequency can be calculated by period. After calculation, the fault impulse frequency is 7.3 Hz approximately. The fault impulse frequency is consistent with the fault characteristic frequency calculated above. Therefore, the broken tooth fault can be diagnosed accurately by using proposed method in this paper, then the effectiveness of the method is proved.

Fig. 9
figure 9

The result of 2 mm broken tooth data processed by proposed method

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To illustrate the influence of EMD in the method, the signal also be processed by only using MOMEDA and the result is presented in the Fig. 10. As shown in the figure, the fault impulse can not be found obviously. Therefore, it is necessary to carry out EMD before using MOMEDA for diagnosis.

Fig. 10
figure 10

The result of 2 mm broken tooth data processed by MOMEDA

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In order to further prove the validity of the proposed method, a comparison of MCKD, MED and proposed method is also carried out on the same signal. The performance of the MCKD is shown in the Fig. 11, there are two fault impulses can be found in the figure. However, it is difficult to argue that the fault impact has periodicity. Thus, the broken tooth fault of the gear can not be identified accurately. Figure 12 shows the process result by using MED, and the fault impulse is blurred and confused in the figure. In view of this, the broken tooth fault can not be detected.

Fig. 11
figure 11

The result of 2 mm broken tooth data processed by MCKD

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Fig. 12
figure 12

The result of 2 mm broken tooth data processed by MED

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The data of 5 mm and 10 mm broken tooth fault under the condition of 800 rpm and 10Nm are used to further prove the effectiveness of the EMD-MOMEDA. The processing results are shown in the Figs. 13 and 14 below. As can be seen from the graphs, the impact pulses can be clearly highlighted from the fault signal by used proposed method. Therefore, the validity of the method can be further proved.

Fig. 13
figure 13

The result of 5 mm broken tooth data processed by proposed method

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Fig. 14
figure 14

The result of 10 mm broken tooth data processed by proposed method

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The length of the filter is the key factor that affects the effect of EMD-MOMEDA. For the same signal, the EMD-MOMEDA is used for processing with different length filter, the result is shown in Fig. 15. As can be seen from the figure, with the increasing of filter length, the impulses of signal become stronger and clearer gradually. Thus, the conclusion is drawn that the length of the filter can enhance the effect of the proposed method.

Fig. 15
figure 15

The result of 2 mm broken tooth data using EMD-MOMEDA with different length filter

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The Fig. 16 is shown the result of using MCKD with different length filter for the same signal. As shown in the Fig. 16, increasing the length of the filter can reduce the influence of noise on impact pulse. The Fig. 17 is shown the result of using MED with different length filter for the same signal. As can be seen from the figure, the length of filter can also affect the effect of the MED, but it is not obvious.

Fig. 16
figure 16

The result of 2 mm broken tooth data using MCKD with different length filter

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Fig. 17
figure 17

The result of 2 mm broken tooth data using MED with different length filter

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4.2 Case study 2

In order to further prove the feasibility of the proposed method, another preset fault experiment of gearbox was carried out. As shown in the Fig. 18, the test rig is composed of a stepless speed motor, a parallel shaft gearbox, a speed torque sensor, two vibration sensors, a magnetic powder brake and signal acquisition system. The structure of the experimental gearbox and the gear with broken tooth are shown in the Fig. 19. Sampling rate is 20,480 Hz and sampling time is 9.6 s. The Fig. 20 displays the time domain waveform of raw signal for broken tooth fault. The rotation frequency is 20 Hz in the experiment. The fault frequency of broken tooth fault is 10.38 Hz. After calculation, the period of fault impact is 1973 sample points.

Fig. 18
figure 18

Gearbox implanted fault test-rig

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Fig. 19
figure 19

Gearbox structure and gear with broken tooth fault

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Fig. 20
figure 20

Waveform of raw signal for broken tooth fault

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After processing by the proposed method, the results are shown in Fig. 21. As shown in the Fig. 21, there are periodic shock characteristics in the result graph. The fault impulse frequency is consistent with the fault characteristic frequency calculated above. Therefore, the broken tooth fault can be diagnosed accurately by using proposed method in this paper and the effectiveness of the method is proved.

Fig. 21
figure 21

The result of broken tooth data processed by proposed method

Full size image

5 Conclusions

In this work, a novel hybrid method based on EMD and MOMEDA is proposed to fault detection for gearbox. In the proposed method, EMD is used to decompose the raw signal to get IMFS and then select the IMFs with large kurtosis value to reconstruct the signal. Then the reconstructed signal is processed by MOMEDA to realize fault detection. In order to illustrate the availability of the proposed approach, a gearbox preposition fault experiment is carried out. This combined approach is applied to experimental signal. The experimental results illustrate that the method is effective for fault detection of gearbox. The experimental results also show that the method is more competitive for fault detection of gearbox compared with traditional methods such as Fourier spectrum, cepstrum, MOMEDA, MCKD and MED. The influence of different filter length on the proposed method is also analyzed, with the increasing of filter length, the effect of the proposed method is better.