Shock and Vibration

Volume 2015 (2015), Article ID 735853, 13 pages

http://dx.doi.org/10.1155/2015/735853

## Application of Self-Adaptive Wavelet Ridge Demodulation Method Based on LCD to Incipient Fault Diagnosis

^{1}College of Mechanical Engineering, Hunan University of Arts and Science, Changde 415003, China^{2}China College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China

Received 19 November 2014; Revised 20 March 2015; Accepted 5 April 2015

Academic Editor: Lei Zuo

Copyright © 2015 Songrong Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

When a local defect occurs in gearbox, the vibration signals present as the form of multicomponent amplitude modulation and frequency modulation (AM-FM). Demodulation analysis is an effective way for this kind of signal. A self-adaptive wavelet ridge demodulation method based on LCD is proposed in this paper. Firstly, multicomponent AM-FM signal is decomposed into series of intrinsic scale components (ISCs) and the special intrinsic scale component is selected in order to decrease the lower frequency background noise. Secondly, the genetic algorithm is employed to optimize wavelet parameters according to the inherent characteristics of signal; thirdly, self-adaptive wavelet ridge demodulation wavelet for the selected ISC component is performed to get instantaneous amplitude (IA) or instantaneous frequency (IF). Lastly, the characteristics frequency can be obtained to identify the working state or failure information from its spectrum. By two simulation signals, the proposed method was compared with various existing demodulation methods; the simulation results show that it has higher accuracy and higher noise tolerant performance than others. Furthermore, the proposed method was applied to incipient fault diagnosis for gearbox and the results show that it is simple and effective.

#### 1. Introduction

Fault diagnosis technique is of great significance to guarantee the normal operation of mechanical and electrical equipment. When a localized defect occurs in gearbox, the vibration signals present as the form of multicomponent amplitude modulation and frequency modulation (AM-FM) [1], expressed as a frequency family on the spectrum, which contains the center frequency and its harmonic frequency. For this kind of signals, some demodulation techniques have been used to find the fault feature information. Hilbert demodulation method is widely used in machinery fault diagnosis [2, 3], but there exists window effect and end effect of Hilbert transforms inevitably, resulting in greater demodulation error. The energy separation algorithm (ESA) appears much popular in recent years for the application to machinery fault diagnosis [4–7], because it is suitable to extract the local dynamic characteristics of nonstationary signal. However, ESA requires that the processed signal should be narrow-band monocomponent [4, 5]. In addition, ESA is sensitive to noise [8]. Compared with the above time domain demodulation methods, the wavelet ridge demodulation technique is time-frequency domain demodulation method, which uses continuous wavelet transform (CWT) to obtain instantaneous amplitude (IA) information and instantaneous frequency (IF) information [8, 9]. In general, the analytic Morlet wavelet is used as the basic wavelet due to its similarity to the fault associated impacts [10–13]. But, the analytic Morlet wavelet parameters, which are center frequency and shape factor, would exert a great impact on the results of wavelet ridge demodulation. In order to select the proper parameters, some techniques have been employed [10–12]. Unfortunately, there is no mature theory to tell us how to choose them. In addition, there are few methods, which can select both center frequency and shape factor of Morlet wavelet to obtain the optimal time-scale resolution. Here, genetic algorithm (GA), which not only has better ability to search the optimal solution but also has fast convergence, is introduced to obtain the two optimal parameters according to the analyzed signal local characteristics, and Morlet wavelet with optimal parameters using GA is called self-adaptive wavelet. Therefore we will utilize self-adaptive wavelet ridge demodulation approach to obtain better demodulation results in this paper.

On the other hand, to greatly eliminate the background noise and improve the demodulation accuracy, multicomponent AM-FM signals should be decomposed into monocomponent AM-FM signals before using self-adaptive wavelet ridge demodulation approach. Empirical mode decomposition (EMD) method [3, 4, 14, 15] or local mean decomposition (LMD) method [16–18] is widely employed to decompose multicomponent AM-FM signal into monocomponent AM-FM signals in general. However, EMD method still has theoretical limitations, such as frequency confusion, overshooting, undershooting, end effect, and the emergence of negative frequency components of nonphysical meaning. Compared to the EMD method, LMD method avoids these problems to some extent, but its computing speed is much slower than EMD. Local characteristic-scale decomposition (LCD) is a new data-driving signal analysis method. Based on the inherent characteristics of the signal itself, the LCD method can decompose a complex multicomponent AM-FM into several intrinsic scale components (ISC). Simultaneously, each ISC component is a monocomponent AM-FM signal which has obvious physical meaning. Our team’s research works show that compared with the LMD and EMD method, LCD not only avoids the shortcomings of EMD and LMD, but also owns much faster computing speed [19–21]. Therefore, LCD method is used to decompose the multicomponent gearbox fault vibration signal to a number of ISCs at first. Subsequently, one or several interesting ISCs are selected as analyzed component. After that, noise would be greatly removed to clearly demodulate fault-associated features component from the selected ISCs.

In summary, targeting the demodulation solution of the multicomponent AM-FM vibration signal with low signal-noise ratio produced by gearbox failures, we present a self-adaptive wavelet ridge demodulation method based on LCD for fault diagnosis. The rest of the paper is organized as follows. In Section 2, the wavelet ridge demodulation principle based on LCD is introduced. The process to get a self-adaptive wavelet based on GA is described. We describe the proposed method and the simulation study is provided in Section 4. The proposed method is applied for incipient fault diagnosis of gearbox in Section 5. Finally, we offer conclusions in Section 6.

#### 2. Wavelet Ridge Demodulation Principle Based on LCD

A real signal of monocomponent can be expressed as . When the instantaneous frequency of the signal is much larger than the amplitude modulation frequency, expressed as , this real signal monocomponent is called a progressive single frequency signal. Then its analytical signal can be written as

The expression of the instantaneous frequency is

After selecting a mother wavelet whose analytical wavelet is expressed as , can be transformed by continuous wavelet transformation as follows:where is the scale parameter, is the translation parameter:

As to any scale parameter and translation parameter , suppose only has a first-order stagnation , and then the first-order stagnation satisfies and ; that is,

Wavelet ridge is defined as a collection of all the points which meet on a phase plane. The expression of the collection is , where is a Hardy real space. A point on the wavelet ridge line is called wavelet ridge point. Obviously, according to formula (5), there is

Here can be seen that instantaneous frequency can be extracted from the wavelet ridge points.

Also, the wavelet coefficients of signal about can be expressed aswhere is the Fourier transform of . And the wavelet coefficients modulus is initially defined as

As to wavelet ridge point, it can be seen that from formula (6). So the wavelet coefficients modulus of wavelet ridge point is further expressed as

So, it can be seen that signal instantaneous frequency can be gained after the wavelet ridge is extracted, which is expressed aswhere is the center frequency of ; that is, .

At the same time, the signal instantaneous amplitude can be expressed as

As presented above, it is clear that the demodulation analysis of monocomponent AM-FM signal based on wavelet ridge is feasible. However, most vibration signals produced by gearbox failures are generally multicomponent AM-FM signals. They should be decomposed into monocomponent AM-FM signals by appropriate time-frequency signal processing method before demodulation. In this paper, LCD method is employed to accomplish the signal decomposition.

The LCD method has the assumptions that a complex signal consists of a number of ISCs (Intrinsic Scale Component, ISC) and any two ISCs are independent of each other. In the entire data segment, ISC must meet the following two conditions.(I)The maximal value is positive, the minimum value is negative, and the data set are monotonic between any two adjacent extreme points.(II)Let all the extreme points be written as , ; the line determined by any two adjacent extreme points and can be expressed as

Remembering the value of at the as , the relation between and should meet the following:when , .

Based on this definition, a complex multicomponent AM-FM signal can be decomposed into the sum of a finite number of ISCs and a residual signal. Each ISC is a monocomponent AM-FM signal whose instantaneous frequency has specific physical meaning. That is,where is the th ISC component and is the residual signal.

#### 3. Self-Adaptive Wavelet

##### 3.1. Morlet Wavelet Frequency Resolution

When there is local failure for gear, the fault gear teeth will stimulate system to produce a convergent impact response and the vibration signal collected by the acceleration sensor shows the obvious multicomponent modulation characteristic. Therefore, as mentioned above, we can adopt wavelet ridge demodulation based on LCD to extract the fault feature. In order to match this kind of signal, analytic Morlet wavelet with impact feature is chosen, which is defined as

The Fourier transform of is represented aswhere is the shape factor and is the center frequency, whose numerical values determine the speed of the waveform vibration damping, respectively; from formula (16), Morlet wavelet quality factor is . So, the best frequency resolution can be gained by adjusting and , which can result in a good time-scale accumulation.

##### 3.2. The Procedure of Obtaining Self-Adaptive Wavelet

Sparse degree of wavelet coefficients can characterize the degree of similarity between the basic wavelet function and signal. The energy entropy of wavelet can indicate this sparse degree, which shows accumulation performance of wavelet coefficients. As to each specific scale , the wavelet energy entropy is defined aswhere is the probability of energy distribution (), is the wavelet energy, is the wavelet coefficient, and is the total wavelet energy within the time scale plane.

Accordingly, the wavelet energy entropy is taken as the objective function during selecting the optimal wavelet parameters. To optimize wavelet parameters, genetic algorithm (GA) and particle swarm optimization (PSO) [22, 23] are two widely utilized approaches. In this paper, GA is employed to optimize either envelope factor or center frequency with wavelet energy entropy as the fitness function. That wavelet with optimal parameters is called self-adaptive wavelet. The procedure to get self-adaptive wavelet is described as follows.

*Step 1. *Set search prime range and population size of parameters, and , and randomly generate initial population. In this paper, the population size is set to 100 and the parameters are, respectively, encoded as 10-bit binary string chromosomes by binary coding method.

*Step 2. *Make wavelet decomposition of the signal and calculate the fitness value of each individual according to formula (17). Then, sort the fitness values by size.

*Step 3. *Based on individual fitness value in the search space, individuals are screened and evolved by a series of genetic manipulation, selection, reproduction, crossover, mutation, and so forth to constantly update and select populations.

*Step 4. *
In this step, it is determined whether iteration satisfies the termination condition or not. If satisfied, the optimal solution is finished. If not satisfied, go to Step 2 until the optimal solution is got. The optimization procedure of wavelet parameters via GA is shown in Figure 1.