A Composite Method of Marine Shafting’s Fault Diagnosis by Ship Hull Vibrations Based on EEMD

The fault diagnosis is always a key issue in the security ﬁeld of marine propulsion system. There are obvious problems like the unsteady working of sensors, distortion of original data, and ambivalent feature information from marine shafting’s vibration or motion. It is therefore critical to develop a more eﬀective method to identify the fault information so that the safety of marine propulsion system can be pre-estimated. Hence, a composite method which is based on the ensemble empirical mode decomposition (EEMD) and coupled with the autocorrelation method (AM), the fast Fourier transform (FFT), is mixed and applied to identify the fault information of marine shafting during its operating by hull vibration. The contrastive analysis of the three methods and fault feature study are then conducted to assess the eﬀectiveness of the proposed method thoroughly and validated by the author previously. The research indicates that the composite method is available to fault diagnosis of marine shafting by hull vibration which coupled the shafting vibration with fault feature.


Introduction
As indispensable "links" of ship propulsion torque and thrust transmissions, the ship propulsion shafting system is the key part of a ship. Marine propulsion shafting mainly consists of propeller, intermediate shaft, intermediate bearing, thrust shaft, thrust bearing, stern tube, stern tube bearing and other devices as shown in details in Figure 1. Ship propulsion shafting are affected by some external loads or periodic excitations, such as propeller excitation [1], main engine excitation, wave loads, engine room environmental vibration loads, and so on. During ship navigating, it perhaps happens that the bearings serious wear phenomena derived from the oscillations and whirl of the oil film. e shafting vibration coupled fault information can be transferred constantly to ship hull by the oil film of bearings. e effective extracting fault information is the key work that has drawn much attention of many researchers all around the world. In 2010, Jayaswal P., Verma S. N. et al. [2] investigated the fault diagnosis methods for vibration signals by combining wavelet transform with neural networks and fuzzy logic. In 2016, Khang H. V. et al. [3] used windowed Fourier transform to clearly identify fault characteristic frequencies in time spectrums. Especially, Huang N. E. Reference [4], Zhaohua W. [5], et al. proposed an Empirical Mode Decomposition (EMD) method and ensemble empirical mode decomposition (EEMD), which are now widely used in fault diagnosis. en the methods were combined and applied to identify the preset bearing failure modes of benches [6], to extract fault features from vibration signals [7], to reduce noise [8] and analyze fault diagnosis of rotating machinery [9] or rolling bearing [10][11][12][13][14][15][16]. In 2011, Zhou T. T. [17] proposed a method of partial ensemble empirical mode decomposition (PEEMD) for fault diagnosing of marine shafting for the first time. en Chen Y [18] and He Y [19] used the envelope analysis method with spectrum kurtosis (SK) to identify marine shafting vibrations. And the above methods certainly are available in fault diagnosing, but the signal is usually limited to vibration sources such as the bearings or shaft, not the hull. erefore a Composite method is proposed and applied to identify the marine shafting's fault from the hull vibration, and some simulations and real ship tests are carried out in this paper.

Empirical Mode
Decomposition. EMD methods can be applied to decompose the signal adaptively to the series of intrinsic mode functions (IMF) of the frequency components which distributes from high frequency to low frequency. Each IMF component contains different characteristic in time domain which can represent the real physical information of the signals. e two IMF conditions need to meet during decomposing process [20]: the number of extreme value is equal to zero value in the original signal, and the up-envelope curve and down-envelope curve is symmetrical to time axis in whole time domain. e decomposing process is shown as follows: Step 1.
e cubic spline curve is adopted to draw the upenvelope curve and down-envelope curve of vibration signal, then the maximum and minimum of them are extracted to calculate the mean m 1 . e first IMF component is restructured as follows.
Here, h 1 is the first IMF component, x(t) is the original vibration signal, m 1 is the mean value of s1 and s2 which are respectively the up-envelope curve and down-envelope curve of the vibration signal drawn by connecting the local maximum point and local minimum point with a cubic spline curve.
Step 2. If h 1 does not meet the IMF conditions, smoothing process will be done and h 1 will be used as the original data as following. h 12 is the result of repeating the second smoothing process.
Here, m 11 is the mean of up-envelope curve and downenvelope curve of h 1 . e smoothing step will be done continually until the result meet the IMF conditions. So the h 1k can be got finally, which becomes the first IMF component and is denoted by c 1 .
where k is the times of repeating smoothing process.
Step 3. When c 1 is separated from the original signal, the first residual function r 1 is left as equation (4).
en the similar smoothing process will be used to get the next residual function until the final residual function is a monotonic function as (4).
Lastly, the signal can be expressed as following function by EMD method.

Ensemble Empirical Mode
Decomposition. EMD methods can decompose original signals into a series of intrinsic mode functions (IMF) and a residual component, and which has the adaptability, orthogonality and completeness. Otherwise EMD method still has some theoretical problems like endpoint effecting, mean curve construction, and mode mixing. Hence, EEMD method was composite to curb modal aliasing phenomena. A flow chart of the EEMD method is illustrated in Figure 2, and the details are as follows: (i) Step 1: add the random Gaussian white no ise n i (t) to the original vibration signal x(t), so the noiseadded signal x i (t) can be obtained as follows: where the subscript i is the serial number, M is the ensemble number. (ii) Step 2: according to equation (7), the matrix of noise-added signal can be expressed as [x 1 (t), x 2 (t), . . . , x M (t)] (i � 1, 2, . . . , M),. en the EMD is adopted to decompose the noise-added signal x i (t)(i � 1, 2, . . . , M) and the IMF components are obtained which can be described as . So the matrix M with j time of Gaussian white noise can be expressed as equation (8).
Step 3: the ensemble means of the corresponding IMF components then is calculated by the equation (9), so the final result of that signal can be obtained more effectively.

Autocorrelation Method.
e autocorrelation functions describe the dependence of the same sample functions of random vibrations between different instantaneous amplitudes. e expressions of the autocorrelation functions of discrete random vibration signals can be derived as follows: where R xx (t) represents the autocorrelation function, x(i) is the sample function, k is the serial number which belong to the natural number. And the autocorrelation function is described in (10).
Here, τ is value of the time domain, Δt denotes the interval time of sampling. For the autocorrelation functions, the maximum value is R xx (0) when τ � 0 and the minimum approaches to zero when τ ⟶ ∞. So the value of autocorrelation function is limited in the range of zero and R xx (0).
Autocorrelation functions are one of the important parameters of stochastic vibration signal analysis. ey also reflect the degrees of smoothness and steepness of a waveform. erefore, autocorrelation functions are often used to detect periodic vibration components from a random vibration signal contains in the practical engineering project. e autocorrelation functions of periodic components will maintain the original periodic without attenuation and it can be applied to qualitatively analyze the fault features of marine shafting from hull vibration which is also periodic.

Method Developing.
In fact, the fault signals of marine shafting system are characterized with periodicity and can easily be overwhelmed by strong background noise, so it is difficult to identify fault information accurately in measured signals.
e composite method combines the ensemble empirical mode decomposition (EEMD) innovatively, the autocorrelation method (AM), and the fast Fourier transform (FFT), which is abbreviated as EEAF. e EEAF has displayed good adaptability, orthogonality, and completeness in our research. In its analysis process, the measured signals are first decomposed by the EEMD method and the original signals with some strong background noise are decomposed to a series of IMF components.
at also improves effectively the ratio of signal-to-noise for the periodic components. en, autocorrelation analysis is done for obtaining the autocorrelation function of each IMF component, which is applied to determine the periodicity of IMF components. Finally, the frequencies and amplitudes of the periodic signals can be effectively extracted from the IMF components with periodic signals by filtering, excluding and spectrum analysis. e process of EEAF is shown in Figure 3 and is divided into the following steps.
(i) Step 1: the decomposing of the measured original signal by EEMD is performed, which is presented in Chapter 2.2 previously. And the IMF components are obtained.

Start Input Signal x(t)
Initializing the average number of times M Adding white noise n i (t) Figure 2: Decomposing process of EEMD.

Shock and Vibration 3 (ii)
Step 2: to construct the autocorrelation function R i (τ) of the IMF components as equation (12).
where T is the period time of IMF component. For discrete signals, (11) can be transformed into the following: where N represents the length of the related data, n denotes the number of delays, i is the time serial number; τ indicates the delay timed. e R i (nΔt) can be used to eliminate the random disturbance noise with aperiodic features, and identify the IMF components with periodic features.
(iii) Step 3: the IMF components with periodicity are extracted by FFT analysis. e periodic component characteristic quantities in the IMF components such as frequencies and amplitudes, can be obtained.

Numerical Verification
e followed case is done to verify the EEAF method. e designated signal is composed of the periodic components and random environmental noise, which can be expressed as the following equation.

x(t)
Adding white noise n m (t) where s 0 is the random Gaussian white noise, s 1 , s 2 , s 3 , s 4 and s 5 are five different periodic functions. e random Gaussian white noise has an amplitude standard deviation of 0.2, and the ensemble number is set as M � 100. So the waveform of the designated signal is shown in Figure 4. rough the EEMD method, the modes and residual components are extracted and shown in Figure 5. e designated signal has been decomposed to nine IMF components and one residual component. en select the IMF component with periodic feature to avoid confusion, the autocorrelation analysis of nine IMF components are conducted. e results are shown in Figure 6. In Figure 6, the autocorrelation functions of IMF1, IMF2, IMF3, IMF4, and IMF5 are periodic obviously, and they have retained the original periodicity without attenuation. Subsequently, they are retained and used for the further analysis. However, the autocorrelation function waveforms of the remaining components have obvious periodicity, but disordered. e effect of Gaussian white noise plays a main factor of the result, and it should be eliminated.
Finally, the five IMF components with periodicity, IMF1, IMF2, IMF3, IMF4, and IMF5, are transformed by the fast Fourier transform. e feature quantities of IMF component such as frequency and amplitude are extracted. e frequency domain of the five IMF components is presented in Figure 7, in which the preset frequencies (100 Hz, 50 Hz, 30 Hz, 15 Hz and 5 Hz) are successfully recognized. Furthermore, the amplitudes corresponding to the five frequencies are obtained, those are 0.62162, 0.986, 0.7856, 0.5932, and 0.5829 respectively. To compare the data of Table 1, the extracted frequencies are exactly the same as the preset frequencies, but there are little changes in amplitude observed with the values for both within a 3% error range because of the slight confusion. In summary, the EEAF method is available to extract the periodic component with fault feature through the case study.

Test and Discussion
In actual ship, the complexity of the hull vibrations are much greater than that of the analog signals. e test is done in a 64000DWT bulk carrier, whose total length is 199.90 m, the length between perpendicular lines is 194.5 m, the molded width is 32.26 m, and the molded depth is 18.50 m. In addition, its design draft, deadweight and design speed are 11.30 m, 63,800 ton, 15.60 knot respectively. During sea trial of this ship, the temperature of the stern bearings rises rapidly to 87°C, and a high temperature alarm is raised (Note: the alarm value is set at 60°C). So the abnormal wear of the stern bearings occurs.  Table 2. e data show that there are slight differences in the frequencies of the vibration data collected from the two measuring points. However, the deviations are less than 5%. Furthermore the data of Table 2 are used to draw the curves of frequency and amplitude in Figure 9. In Figure 9(a), the frequencies of hull vertical vibrations are distributed mainly in 5th order line and 2nd order trend line. e amplitudes of hull vertical vibrations are shown in Figure 9(b), and the amplitudes of the 5th order vibration signals present a trend of increasing, decreasing and then increasing again. e phenomena happens in some ships with misalignments of marine propulsion shafting.
at is to say, the amplitude of hull vertical vibration in 5th order increases with the increase of the shaft speed. erefore, it indicates that the propulsion shafting of this ship is misaligned.

Longitudinal Vibration.
e data from the sensors of No. 3 and No. 4 are the signals of hull longitudinal vibration. Similar to the aforementioned processes, an EEAF method is used to analyze the test data and the main parameters of the extracted information from periodic longitudinal vibration are obtained, as detailed in Table 3.
In Figure 10(a), the frequencies are mainly distributed in the 5th order trend line. As shown in Figure 10(b), the amplitudes increase with the increasing of the shaft speed until the tail bearing failure occurs. e corresponding peak speed of the 5th order curve is observed to be extremely close to the resonance speed of this marine shaft torsional vibration and longitudinal vibration.

Transverse Vibration.
e transverse vibrations of hull come from the sensors of No. 7 and No. 8 in Figure 8.
rough the EEAF method, the transverse vibrations of the hull are extracted and the main data filled in Table 4. Figure 11 shows the characteristics of the periodic components in the hull longitudinal vibration under the different operating conditions which are presented in Table 4. As shown in Figure 11        these frequencies in the 5th harmonics increase firstly and then decrease with the increase of the shaft speed. When the tail bearings are in fault, the amplitudes are observed to increase once again.

Conclusions
In this study, a novel method of extracting periodic fault information of marine propulsion shafting by the hull vibrations is proposed, which is referred to as the EEAF method. rough the numerical verification, test and discussion, the EEAF method is available to extract the useful frequencies and amplitudes connected with marine shafting's fault. In addition, this method has alleviated the modal aliasing problems successfully, and can decompose and extract the periodic feature quantities which characterize the fault features of marine shafting from the measured signals more accurately. e fault features extracted from the hull vibrations are similar with the phenomena of marine shafting misalignment. So the fault information obtained by EEAF method from hull vibrations can be used to pre-estimate the operating condition of marine shafting qualitatively. However, there should be more experiments and tests to develop the effectiveness and quantitative of the fault diagnosis of the EEAF method.

Data Availability
e test data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.