vibrationdata 1 unit 5 the fourier transform. vibrationdata 2 courtesy of professor alan m. nathan,...

VibrationdataVibrationdata

1

Unit 5

The Fourier TransformThe Fourier Transform


2

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


3

Unit 5

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

1. Mathematician Approach

2. Engineering Approach

Even within each approach there are many equations.


4

Continuum of Signals

Random 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


5

Frequency Example

-5

5

0

0 0.5 1.0 1.5 2.0

TIME (SEC)

AC

CE

L (

G)

TIME HISTORY EXAMPLE


6

Resolving 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


7

Equation of Previous Signal

t22π2sin1.2t16π2sin1.5t10π2sin1.0ty

At the risk of short-circuiting the process, the equation of the signal in the previous figure 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 Need 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


8

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


9

Textbook 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


10

Textbook 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)


11

Magnitude 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


12

Real Time History

Practical 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.


13

Fourier 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

f = 0

Dirac Delta function


14

Characteristics of the Continuous Fourier Transform

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

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


15

Characteristics 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 ………


16

Discrete 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


17

Discrete 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


18

Discrete 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 / 1 =fΔ

fΔk(k) frequency


19

Nyquist 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


20

Half 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.


21

Previous 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.


22

Spectrum 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


23

Full 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


24

Full Amplitude Fourier Transform

Note that k = 0 is a special case

The Fourier transform at this frequency is already at full-amplitude

Consider a sine wave with an amplitude of 1 G and a frequency of 1 Hz

This signal would simply have a full-amplitude Fourier magnitude of 1 G at 1 Hz, as shown in the next figure


25

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


26

Full 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


27

Full 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


28

Unit 5 Exercise 1

A time history has a duration of 20 seconds.

What is the frequency resolution of the Fourier transform?


29

Unit 5 Exercise 2

a) Generate sine time history using:

Amplitude = 7, Frequency = 2 Hz, Duration 100 sec, Sample rate = 100

b) Save sine function as ASCII text file.

c) Then take the Fourier transform of sine function:

vibrationdata > Fourier transform mean removal=yes, window=rectangular, min freq = 0 Hz, max freq = 10 Hz


30

Unit 5 Exercise 3

Calculate Fourier transforms and identify dominant frequencies for these time histories from previous webinar units:

tuning_fork.txt - time (sec) & unscaled sound pressuredrop.txt - time (sec) & accel(G)channel.txt - time (sec) & accel(G)

Use: vibrationdata > Fourier transform mean removal=yes, window=rectangular, Min Freq=0 Hz

Data Max Freq (Hz)

tuning_fork 1000

drop 100

channel 1000

vibrationdata 1 unit 5 the fourier transform. vibrationdata 2 courtesy of professor alan m. nathan,...

Documents