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VibrationdataVibrationdataUnit 6a

The Fourier TransformThe Fourier Transform

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VibrationdataVibrationdata

Courtesy of Professor Alan M. Nathan,University of Illinois at Urbana-Champaign

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VibrationdataVibrationdataUnit 6a

Many, many different equations for Fourier Transforms . . . . . . . . . .

1. Mathematician Approach

2. Engineering Approach

Even within each approach there are many equations.

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VibrationdataVibrationdataContinuum of Signals

Stationary vibration signals can be placed along a continuum in terms of the their qualitative characteristics.

A pure sine oscillation is at one end of the continuum. A form of broadband random vibration called white noise is at the other end.

Reasonable examples of each extreme occur in the physical world. Most signals, however, are somewhere in the middle of the continuum.

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VibrationdataVibrationdataFrequency Example

-5

5

0

0 0.5 1.0 1.5 2.0

TIME (SEC)

AC

CE

L (

G)

TIME HISTORY EXAMPLE

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VibrationdataVibrationdataResolving Spectral Components

The time history appears to be the sum of several sine functions.

What are the frequencies and amplitudes of the components?

Resolving this question is the goal of this Unit.

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VibrationdataVibrationdataEquation of Previous Signal

At the risk of short-circuiting the process, the equation of the signal in Figure 1 is

The signal thus consists of three components with frequencies of 10, 16, and 22 Hz, respectively.

The respective amplitudes are 1.0, 1.5, and 1.2 G.

In addition, each component could have had a phase angle. In this example, the phase angle was zero for each component.

Thus, we seek some sort of "spectral function" to display the frequency and amplitude data. Ideally, the spectral function would have the form shown in the next figure.

t22π2sin1.2t16π2sin1.5t10π2sin1.0ty

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VibrationdataVibrationdata

0.5

1.0

1.5

2.0

00 5 10 15 20 25

FREQUENCY (Hz)

AC

CE

L (

G)

"SPECTRAL FUNCTION" OF TIME HISTORY EXAMPLE

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VibrationdataVibrationdataTextbook Fourier Transform

The Fourier transform X(f) for a continuous time series x(t) is defined as

where

- < f <

The Fourier transform is continuous over an infinite frequency range.

Note that X(f) has dimensions of [amplitude-time].

- dttf2πj-expx(t)=X(f)

1j

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VibrationdataVibrationdataTextbook Inverse Fourier Transform

The inverse transform is

Thus, a time history can be calculated from a Fourier transform and vice versa.

- dftf2πj+expX(f)=x(t)

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VibrationdataVibrationdataMagnitude and Phase

Note that X(f) is a complex function.

It may be represented in terms of real and imaginary components, or in terms of magnitude and phase.

The conversion to magnitude and phase is made as follows for a complex variable V.

bjaV

2b2aVMagnitude

)arctan(b/aVPhase

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VibrationdataVibrationdataReal Time History

Practial time histories are real.

Note that the inverse Fourier transform calculates the original time history in a complex form.

The inverse Fourier transform will be entirely real if the original time history was real, however.

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VibrationdataVibrationdataFourier Transform of a Sine Function

The transform of a sine function is purely imaginary. The real component,

which is zero, is not plotted.

tf2 πsinAx(t) ˆ

Im a g in a r y X ( f )

f̂f̂

ffδ

2A ˆ

ffδ

2A ˆ

f

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VibrationdataVibrationdataCharacteristics of the Continuous Fourier Transform

1. The real Fourier transform is symmetric about the f = 0 line.

2. The imaginary Fourier transform is antisymmetric about the f = 0 line.

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VibrationdataVibrationdataCharacteristics of the ContinuousFourier Transform

This Unit so far has presented Fourier transforms from a mathematics point of view.

A more practical approach is needed for engineers ………

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VibrationdataVibrationdataDiscrete Fourier Transform

An accelerometer returns an analog signal.

The analog signal could be displayed in a continuous form on a traditional oscilloscope.

Current practice, however, is to digitize the signal, which allows for post-processing on a digital computer.

Thus, the Fourier transform equation must be modified to accommodate digital data.

This is essentially the dividing line between mathematicians and

engineers in regard Fourier transformation methodology.

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VibrationdataVibrationdataDiscrete Fourier Transform Equations

The Fourier transform is

The corresponding inverse transform is

1N...,1,0,kfor,1N

0nnk

N2πjexpnx

N1

kF

1N...,1,0,nfor,1N

0knk

N2πjexpkFnx

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VibrationdataVibrationdataDiscrete Fourier Transform Relations

Note that the frequency increment f is equal to the time domain period T as follows

The frequency is obtained from the index parameter k as follows

where

k is the frequency domain index

t is the time step between adjacent points

T / 1 =fΔ

fΔk(k) frequency

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VibrationdataVibrationdataNyquist Frequency

The Nyquist frequency is equal to one-half of the sampling rate.

Shannon’s sampling theorem states that a sampled time signal must not contain components at frequencies above the Nyquist frequency.

Otherwise an aliasing error occurs.

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VibrationdataVibrationdataHalf AmplitudeDiscrete

-1.0

-0.5

0

0.5

1.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

N = 16384SR = 32 samples/sect = ( 1 / 32 ) secT = 512 secf = ( 1 / 512 ) Hz

Dashed line is line of symmetryat Nyquist frequency

FREQUENCY (Hz)

AC

CE

LE

RA

TIO

N (

G)

HALF-AMPLITUDE DISCRETE FOURIER TRANSFORM y(t) = 1 sin [ 2 ( 1 Hz) t ] G

This is the imaginary component of the transform. The real component is zero.

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VibrationdataVibrationdataPrevious Fourier Transform Plot

Note that the sine wave has a frequency of 1 Hz.

The total number of cycles is 512, with a resulting period of 512 seconds.

Again, the Fourier transform of a sine wave is imaginary and antisymmetric. The real component, which is zero, is not plotted.

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VibrationdataVibrationdataSpectrum Analyzer Approach

Spectrum analyzer devices typically represent the Fourier transform in terms of magnitude and phase rather than real and imaginary components.

Furthermore, spectrum analyzers typically only show one-half the total frequency band due to the symmetry relationship.

The spectrum analyzer amplitude may either represent the half-amplitude or the full-amplitude of the spectral components.

Care must be taken to understand the particular convention of the spectrum analyzer.

Note that the half-amplitude convention has been represented in the equations thus far.

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VibrationdataVibrationdataFull Amplitude Fourier Transform

The full-amplitude Fourier transform magnitude would be calculated as

12N,...,1kfor

1N

0nnk

N2πjexpnx

N1magnitude2

0k for 1N

0nnx

N1magnitude

kG

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VibrationdataVibrationdataFull Amplitude Fourier Transform

Note that k = 0 is a special case. The Fourier transform at this frequency is already at full-amplitude.

For example, a sine wave with an amplitude of 1 G and a frequency of 1 Hz would simply have a full-amplitude Fourier magnitude of 1 G at 1 Hz, as shown in the next figure.

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VibrationdataVibrationdata

0

0.5

1.0

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

N = 16384t = ( 1 / 32 ) secT = 512 secf = ( 1 / 512 ) Hz

FREQUENCY (Hz)

AC

CE

LE

RA

TIO

N (

G)

FULL-AMPLITUDE, ONE-SIDED DISCRETE FOURIER TRANSFORM OF y(t) = 1 sin [ 2 (1 Hz) t ] G

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VibrationdataVibrationdataFull Amplitude Fourier TransformTime History Example

-5

5

0

0 0.5 1.0 1.5 2.0

TIME (SEC)

AC

CE

L (

G)

TIME HISTORY EXAMPLE

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VibrationdataVibrationdataFull Amplitude One-Sided

0

0.5

1.0

1.5

2.0

5 10 15 20 25

N = 400t = 0.0025 secT = 2 secf = 0.5 Hz

FREQUENCY (Hz)

AC

CE

LE

RA

TIO

N (

G)

FULL-AMPLITUDE, ONE-SIDED DISCRETE FOURIER TRANSFORM OF TIME HISTORY IN FIGURE 1

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VibrationdataVibrationdataUnit 6a Exercise 1

Convert the following complex number into magnitude and phase:

x = 5 + j 9

Use program EngMath.exe

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VibrationdataVibrationdataUnit 6a Exercise 2

A time history has a duration of 20 seconds.

What is the frequency resolution of the Fourier transform?

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VibrationdataVibrationdataUnit 6a Exercise 3

Recall file white.out from Unit 5.

Take the Fourier transform using program fourier.exe.

Plot both the FT_half.out and Ft_full.out files.

How does the amplitude vary with frequency?

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VibrationdataVibrationdataUnit 6a Exercise 4

Plot the half-sine time history hs.txt.

Then take the Fourier transform.

How does the amplitude vary with frequency?

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VibrationdataVibrationdataUnit 6a Exercise 5

1. Generate white noise time history using generate.exe

Std dev = 7

0 to 20 sec

Sample rate = 500

10000 samples

Output filename = seven.dat

2. Take Fourier transform.

Call ft_full.out into EasyPlot

(continued on next page)

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VibrationdataVibrationdataUnit 6a Exercise 5 (continued)

Call ft_full.out into Excel

1. Multiply each spectral magnitude by 0.707 to convert from peak to RMS.

2. Square each spectral RMS value to convert to mean square.3. Sum the mean square values.4. Take the square root of the sum.5. The result is the standard deviation from the frequency domain.

6. Take the standard deviation of seven.dat in the time domain.

The standard deviation from the time domain should be the same as that from the frequency domain