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Signals are converted from space domain or time to the frequency domain generally through the Fourier transform i.e. Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT). The signal information is transformed to a magnitude and phase component of each frequency using the Fourier transform.

The signal used frequency as the x-axis variable is to describe phase and frequency spectrum as well as amplitude spectrum. The signal’s frequency is studied by spectrum analysis where in

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spectrums of structures such as amplitude and phase is the important features. In research and development in addition to engineering technology, spectrum analyzer is a very important analysis of the test apparatus. The analysis involves signal’s phase spectrum and amplitude spectrum, power spectrum, FFT transform, filtering etc.

The representation of a signal in terms of its individual frequency components is known as the frequency domain representation of the acquired signal. The frequency domain representation gives more information about the signals and the system by which the signal has been generated. The Frequency domain signal processing and analysis of a vibration signal is performed by following steps which is shown in Fig.3.3.

Figure 3.3: Block diagram for frequency domain signal processing

3 . 3 . 1 Discrete Fourier Transform

The discrete-time Fourier transform (DTFT) gives an alternative depictions for discrete-time (DT) sequence. By using DTFT, a DT sequence x[k] is transford into a function X(Ω) in that DTFT frequency domain.The independent variable Ω(=wT) is confined to the range –π ≤ Ω <π. It is also contineuous in nature.The use of digital computers and specialized hardwares are increased in different applications like signal processing, monitoring,control of different machines.Hence now a days the signal transformation has focused which is appropriate for digital computations. As the signal is continuous in nature, direct realization of the DTFT is not appropriate for such a digital devices. The discrete Fourier transform (DFT) is introduced for this reason. DFT can be implimented efficiently on digital computers. The DTFT

Frequency domain signal processing

DFT/FFT

Windowing

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of the signal x (k) is a continuous function of “w”. It is discretized for storing on the digital computer. The DTFT 𝑋_2𝜋(w) of signal x (k) is written as the following equation 3.2

𝑋2𝜋(𝑤)=∑∞𝑘=−∞𝑥(𝑘)𝑒−𝑗𝑤𝑘 (3.2)

For k=all integers

Using the discrete Fourier transform (DFT), a finite list of samples of a function which is equally-spaced is converted into a finite combination of complex sinusoidal signals. That organized by their frequencies which have same sample values. It can convert the sampled function of its original domain (time or position) to the frequency domain.The DFT is the most important discrete transform (DT) which is used to perform Fourier analysis in various practical applications. In digital signal processing, the function is a signal or any quantity that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, vibration monitoring of a machine sampled over a finite time intervals (often defined by a window function).The DFT is also used to efficiently solve partial differential equation, convolution or multiplying large integers.

As DFT is dealing with a finite amount of data, DFT is implemented in computers by various numerical algorithms and in the dedicated hardware. These implementations are also employed efficiently by using the fast Fourier transform (FFT) algorithms. So that, the terms "DFT" and "FFT" are used interchangeably. The terminology is further known by the synonym finite Fourier transform of the DFT.

The series of N complex numbers x0, 𝑥1 ..., xN−1 is transformed into an N-periodic series of

complex numbers .It is performed using the DFT formula which is given in the following equation 3.3.

𝑋𝑘=∑𝑁−1𝑛=0𝑥𝑛𝑒−𝑗2𝜋𝑛𝑘/𝑁 (3.3)

From the DFT of a signal, the time domain signal is rebuild by using the inverse Fourier transfer (IDFT) which is given in the following equation 3.4.

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3.3.1.1 Windowing

In practical application only a finite number of s a m p l e s of the signal a r e obtained. When the DFT or FFT is used to extract the frequency content of a signal, it is intrinsically assumed that the data is a single period of a periodically repeating waveform. Leakage exists because of the finite time record of the input signal. An appropriate window is usually selected so as to reduce the spectral leakage. That is by applying a smoothing window function to the data before it is discrete-time Fourier transformed can greatly minimize spectral leakage.

Types of windowing: Many different types of windows are available in the LabVIEW analysis library. Some of t h e s e windows are Rectangular, Hanning, Hamming, Blackman-Harris, Triangle, Flat Top and Exponential. The above windows are explained briefly in the following sub section.

1. Rectangular window

The rectangular window (sometimes known as the boxcar or Dirichlet window) is the simplest window, equivalent to replacing all but N values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off .The rectangular windowing function is given by the following equation i.e.3.5.

W (n) =1 (3.5) Other windows are designed to moderate these sudden changes because discontinuities have undesirable effects on the discrete-time Fourier transform (DTFT) and/or the algorithms that produce samples of the DTFT. The rectangular window is the 1st order B-spline window as well as the 0th power cosine window

2. Hanning window

The Hann window named after Julius von Hann and also known as the Hanning (for being similar in name and form to the Hamming window), von Hann and the raised cosine window is defined by the following equations 3.6 and 3.7

W (n) =0.5(1-cos (2πn/N-1)) (3.6) The unlagged version is given by

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3. Hamming window

The window with these particular coefficients was proposed by Richard W. Hamming. This window is optimized to reduce the maximum side lobe for providing it a height of about one-fifth that of the Hann window. The hamming window function is expressed by the following equation 3.8.

W(n)=α-β(2πn/N-1) (3.8) Where, α=0.54

β =1-α=0.46

The unlagged version is given by

𝑊0(n)=W(n+N-1/2) (3.9)

=0.54+o.46cos (2πn/N-1) 4. Blackman window

Blackman windows are defined as:

W (n) =𝑎₀-𝑎₁cos (2πn/N-1)+𝑎₂ cos (4πn/N-1) (3.10) Where, 𝑎₀ =1-α/2; 𝑎₁ = ½; 𝑎₂=α/2

By common convention, the unqualified term Blackman window refers to α = 0.16, as this most closely approximates the "exact Blackman", with 𝑎₀= 0.42659, a1 = 0.49656, and a2 = 0.076849.

These exact values place zeros at the third and fourth side lobes. 5. Blackman-Harris window

The Blackman-Harris window function is given by the following equation

W (n)=𝑎₀-𝑎₁cos(2πn/N-1)+𝑎₂cos(4πn/N-1)+𝑎₃cos(6πn/N-1) (3.11) Where, 𝑎₀=0.35875; 𝑎₂=0.48829; 𝑎₃=0.14128; 𝑎₄=0.01168

6. Triangle window

The triangular window is defined as

W (n)=1-|𝑛 − (𝑁 − 1)/2/𝑁 − 1| (3.12) The triangular window is the 2nd order B-spline window.

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3.3.1.2 Digital filters

Filtering is the process by which the frequency content of a signal is altered. It is one of the most commonly used signal processing techniques. In modem digital signal processing the analog filters are replaced to digital filters by using various signal processing tools. It has applied in many applications that involve flexibility and programmability. Digital filters have t h e following advantages over their analog equivalent:

 They are programmable

 They are stable and predictable

 They do not drift with temperature or humidity and do not require precision components  They have a superior performance-to-cost ratio

Digital filters in LabVIEW have features to control parameter such as filter order, cutoff frequencies, amount of ripple, and stop band attenuation. LabVIEW offers wide range of Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) filters such as Butterworth, Chebyshev , Chebyshev II or Inverse Chebyshev , Bessel, and Elliptic. The IIR and FIR filters are described briefly.

I. Infinite Impulse Response

Infinite impulse response (IIR) is a property of digital signal processing systems. Systems having this property are identified as IIR systems. IIR system has an impulse response function which is non-zero through an infinite length of period. The IIR filters are different to finite impulse response (FIR) filters. FIR filters have fixed duration of impulse responses. The easiest analog IIR filter is an RC filter which is made up of a single resistor(R) supplied to a node then it is shared with a only one capacitor(C). These filters have an exponential impulse response distinguished by an RC time constant. Because the exponential function is asymptotic to a limit, and thus never settles at a fixed value, the response is considered infinite [26].

In digital IIR filters, the output feedback of the filter is immediately traceable in the equations which are defining the output. It is very obvious that unlike FIR filters, it is mandatory to carefully judge the "time zero" case in which the outputs of the filter have not clearly explained. Design of digital IIR filters is usually dependent on their analog equivalents because there are ample of works, resources, and straightforward design techniques regarding analog feedback

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filter design. In general, when a digital IIR filter is be implemented, an analog filter(e.g. Chebyshev filter, butterworth filter, Elliptic filter) is first implemented and then it is converted to digital filter by applying discretization techniques such as Impulse invariance or bilinear transform. The Chebyshev filter, Butterworth filter, and the Bessel filter are all included in IIR filters. These filters are explained with a precise manner in the following sections.

A. Chebyshev filters

Chebyshev filters are analog or digital filters having a steeper roll-off and more pass band ripple(type I) or stop band ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are consequent from Chebyshev polynomials.Because of the passband ripple inherent in Chebyshev filters; the ones that have a smoother response in the pass band but a more irregular retort in the stopband are favored for some applications.

These are the most frequent Chebyshev filters. The gain (or amplitude) response as a function of angular frequency ω of the nth-order low-pass filter is given by following equation 3.13.

𝐺𝑛(ω)=|𝐻𝑛(𝑗𝜔)|=1/√1 +∈2 𝑇𝑛2(𝜔/𝜔0) (3.13)

Where, ∈=ripple factor. 𝜔0=cut-off frequency

𝑇𝑛( )= Chebyshev polynomial of the nth order.

B

.

The Butterworth filter is a form of signal processing filter considered to have as flat a frequency response as possible in the passband. It is also referred to as a maximally flat magnitude filter. An ideal electrical filter should have uniform sensitivity for the wanted frequencies and also completely reject the redundant frequencies. Such an ideal filter cannot be achieved but it is shown in succession closer approximations were obtained with increasing numbers of filter elements of the right values. At the time, filters generated considerable ripple in the passband, and the choice of component values was highly interactive. Butterworth showed that a low pass

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filter could be designed whose cutoff frequency was normalized to 1 radian per second and whose frequency response (gain) is given the following equation3.14.

Gain=G(ω)=�1/1 + 𝜔2𝑛 (3.14) Where, ω=Angular frequency

N=Number poles in the filter C. Bessel filter

A Bessel filter is a category of linear filter with a maximally flat group delay (maximally linear phase response). An audio crossover system is used based on the Bessel filters. The entire passband of Analog Bessel filters are characterized by almost constant group delay across, the wave shape of filtered signals in the pass band is preserved.The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. The Bessel filter has better shaping factor, flatter phase delay, and flatter group delay than a Gaussian of the same order, though the Gaussian has lower time delay.

Bessel low-pass filter is characterized by its transfer function which shown in the following equation 3.15.

H(s)=𝜃_𝑛(0)/𝜃_𝑛(s/𝜔₀) (3.15) Where 𝜃_𝑛(s)=Reverse Bassel polynomial

𝜔0=Frequency chosen to give the desired cut-off frequency

II. Finite Impulse Response

In a signal processing system, as finite impulse response (FIR) filter settles to zero in finite duration of time, the impulse response of the filter is of finite duration. This is opposite to infinite impulse response (IIR) filters because it has internal feedback and may persists to respond indefinitely.FIR filters can be analog or digital and discrete-time or continuous-time in nature. The term digital filter is introduced because these filters are operated on discrete-time signals. The term finite impulse response is introduced as the filter output is computed with a weighted, finite sum of present, past, and anticipative values of the filter input, i.e.

Y (n) =∑𝑀_𝑘=−𝑀2 𝑏_𝑘

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Where both 𝑀₁ and 𝑀₂ are finite.

The simplest FIR filters which is a 3 term moving average filter which is given in the following equation.

Y (n) =(x [n+1] +x[n] +x [n-1]) (3.17)

A FIR filter is based on a feed-forward difference equation which is presented by the equation (3.17).The meaning of feed-forward is that there is no feedback of past or future outputs form the present output. It is related to just input terms. Block diagram for a FIR filter is presented in Fig.3.4.

x (n) y(n)

X (z) Y (z)

Figure 3.4: Block diagram for a FIR filter

In LabVIEW implementation of the vibration signal monitoring system, The user is allowed to select various design of filters. These filters are: Butterworth, Chebychev, Inverse Chebychev, all Elliptic and Bessel. These are available in the LabVIEW software in the block format. For each selected filter, the user can set the cut off frequency to perform lowpass, highpass, bandpass, and bandstop filtering.

3.4 Methodology

Signal analysis usually means analyzing signals in time and frequency domain, it is the basis of signal transmission and processing. For m ach i n e fault diagnosis, time-domain analysis provides a limited amount of information that can only roughly determine whether the machinery and equipment faults and fault extent, but it cannot determine fault type and fault location. Therefore, it is also necessary to analyze signal in frequency domain, such as power spectrum analysis. The time domain and frequency domain signal processing, monitoring and analysis is performed by software design of vibration testing and analysis system, which is described by the flow chart followed by an algorithm. Flow chart for vibration signal monitoring system is presented in Fig.3.5.

ℎ𝑏(n)

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Figure 3.5: Flow chart for vibration signal analysis

The algorithm for vibration signal monitoring and analysis is discussed in following steps:- (i). First a machine measuring point is selected in terms of acceleration, velocity and

displacement in normal running condition of machine.

(ii). Comparison of above three parameters with machine measuring point is done after vibration is created in the machine.

(iii). High sampling frequency or high acquisition rate is set.

(iv). The vibration signal processing and analysis is done in time domain and frequency domain for finding the fault of vibration signal.

(v). Time domain analysis is done by first and second integration of acceleration of vibration signal.

(vi). The spectrum analysis is performed by fast Fourier transform (FFT), windowing, digital filter.

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(viii). Fault regions are obtained and result is displayed. Hence the control action can be taken.

By following the above steps the vibration monitoring can be implemented by LabVIEW simulation system which is discussed in the next section.

3.4.1 Description of the flow chart

The flow chart which is shown in Fig.3.5 is described briefly by which the vibration monitoring system is easy to perform. In the normal condition of the machine, motor measuring point is selected. In the running condition of the machine, vibration signal is created. When vibration occurs in machine three parameters are changed. These are displacement, velocity and acceleration of machine. Accelerometer transducer is directly proportional to the velocity, displacement, acceleration and frequency of signal. By above factors, the machinery condition can be accurately diagnosed. Then the control action can be taken on the machine. The techniques for controlling of vibration signal are discussed in the next chapter.

Accelerometer transducer converts the physical signal that is vibration signal into analog signal. The analog signal will go to data acquisition card. Here the sampling rate or acquisition rate has to set. Data acquisition card must have high acquisition rate. Data acquisition card must have high acquisition rate, because if the vibration of the machine increases, consequently velocity of the machine also increases. The acquisition rate must also be increased to prevent aliasing effect of the wave form and to calculate the component frequency of the displacement. So data acquisition card must be taken, which have high acquisition rate. Data acquisition card gives the digital signal. It converts analog signal to digital signal. The digital signal from the data acquisition card is in time domain form.

Firstly, interfering signals among vibration signal is filtered out by band-pass filter in testing motor; and then to get speed signal and displacement signal through first integral and quadratic integral, and it is available to calculate speed RMS and peak-peak of displacement; finally, frequency information can be obtained by means of power spectral analysis. The all the above methods are implemented using the LabVIEW software which discussed in the simulated result and discussion chapter.