TMR7 - Experimental Methods in Marine Hydrodynamics … · Fast Fourier transform ... 2 Flow past a circular cylinder and determination of the dominant ... Experimental Methods in

TMR7 - Experimental Methods in MarineHydrodynamics

Time Series Analysis

Jose P. Gallardo Canabes

Department of Marine TechnologyNorwegian University of Science and Technology

September 7, 2011

J. Gallardo C. (NTNU) Time Series Analysis September 7, 2011 1 / 32

Outline of the presentation

1 Motivation

2 Background information

3 MATLAB for data analysis

4 Examples



1 Motivation



4 Examples



1 Motivation



4 Examples



1 Motivation



4 Examples


Table of Contents

1 Motivation



4 Examples


Motivation

Time series are common in science and engineering applications

Planetary motion

Periodic and predictable. . . to somedegree

Irregular waves

Random and difficult to predict


Motivation


Planetary motion


Irregular waves



Motivation


Planetary motion


Irregular waves



Motivation

Visual inspection is not enough

0

time

ζ(t)

Many tools are available for the analysis


Motivation


0

time

ζ(t)



Motivation


0

time

ζ(t)



Motivation


0

time

ζ(t)



Table of Contents

1 Motivation



4 Examples


Fourier Series

A periodic function can be decomposed into its harmonic componentsThe function is represented as an infinite Fourier series

f (t) = a0 +∞∑k=1

ak cos(2πkt

T) + bk sin(

2πkt

T) (1)

with the coefficients a0, ak and bk defined as

a0 =1

T

∫ T

0f (t)dt

ak =2

T

∫ T

0f (t) cos(

2πkt

T)dt

bk =2

T

∫ T

0f (t) sin(

2πkt

T)dt


Fourier Series

A periodic function can be decomposed into its harmonic componentsThe function is represented as an infinite Fourier series

f (t) = a0 +∞∑k=1

ak cos(2πkt

T) + bk sin(

2πkt

T) (1)

with the coefficients a0, ak and bk defined as

a0 =1

T

∫ T

0f (t)dt

ak =2

T

∫ T

0f (t) cos(

2πkt

T)dt

bk =2

T

∫ T

0f (t) sin(

2πkt

T)dt


Fourier Integral

When T →∞ the Fourier integral is obtained

f (t) =

∫ ∞0

[A(ω) cos(ωt) + B(ω) sin(ωt)]dω (2)

with

A(ω) =1

π

∫ ∞−∞

f (t) cos(ωt)dt

B(ω) =1

π

∫ ∞−∞

f (t) sin(ωt)dt


Fourier Transform

A more compact form of expressing the Fourier integral is

f (t) =1√2π

∫ ∞−∞

f (ω)e iωtdω (3)

With the the Fourier transform of f (t)

f (ω) =1√2π

∫ ∞−∞

f (t)e−iωtdt (4)

The Fourier transform converts a function of time into a function offrequency


Fourier Transform


f (t) =1√2π

∫ ∞−∞

f (ω)e iωtdω (3)


f (ω) =1√2π

∫ ∞−∞

f (t)e−iωtdt (4)



Fourier Transform


f (t) =1√2π

∫ ∞−∞

f (ω)e iωtdω (3)


f (ω) =1√2π

∫ ∞−∞

f (t)e−iωtdt (4)



Discrete Fourier Transform (DFT)

Typically the values of f (t) are sampled at equally spaced points

The DFT of a signal f with n components will be

fn =N−1∑k=0

fke−intk , n = 0, . . . ,N − 1 (5)

Expressed in matricial form f = Ff where F is a N × N matrix

DFT requires O(N2) operations

Fast Fourier transform (FFT) lowers this requirement toO(N) log2(N) operations!





fn =N−1∑k=0

fke−intk , n = 0, . . . ,N − 1 (5)








fn =N−1∑k=0

fke−intk , n = 0, . . . ,N − 1 (5)








fn =N−1∑k=0

fke−intk , n = 0, . . . ,N − 1 (5)








fn =N−1∑k=0

fke−intk , n = 0, . . . ,N − 1 (5)





Table of Contents

1 Motivation



4 Examples


MATLAB

Fast and efficient algorithms

Relatively easy to learn and write scripts (MATLAB editor)

Several options for visualization of data

Computations on large datasets

Specific applications (MATLAB toolbox)


MATLAB







MATLAB







MATLAB







MATLAB







Time Series Tools - MATLAB

Opened by writing tstool in the command prompt

Import data

Plot the time series data

Select data subsets for analysis (filter)

Process the data (statistics)

Plot spectrum

Further analysis (power spectra) can be performed with built-in functionsof the Signal Processing Toolbox


Table of Contents

1 Motivation



4 Examples


Examples

1 Simple mathematical functions

2 Flow past a circular cylinder and determination of the dominant(Strouhal) frequency

3 Irregular waves

4 Example of response amplitude operator (RAO)


Examples



3 Irregular waves



Examples



3 Irregular waves



Examples



3 Irregular waves



Filtering of data

High-pass filter Removes low frequency componentsLow-pass filter Removes high frequency componentsBand-pass filter Select a frequency interval for filtering


Example 1 - Simple functions

Beat frequency: interference between two frequencies f1and f2

0 1 2 3 4 5

−0.2

−0.1

0

0.1

0.2

0.3

f1+f

2

t*

Function with random noise

0 2 4 6 8 10−2

−1

0

1

2


Example 1 - Simple functions

Beat frequency: interference between two frequencies f1and f2

0 1 2 3 4 5

−0.2

−0.1

0

0.1

0.2

0.3

f1+f

2

t*

Function with random noise

0 2 4 6 8 10−2

−1

0

1

2


Example 2 - Flow past a circular cylinder

Flow past a circular cylinder at different Reynolds numbers

Unsteady, viscous and laminar or turbulent flow

Strouhal number characterizes vortex shedding frequency

U∞

Boundary layer

Shear layerWake



Flow past a curved circular cylinder at Re = 100

Unsteady, viscous laminar flow

Frequency analysis of the velocity trace in the cross-stream direction




Unsteady, viscous laminar flow

Frequency analysis of the velocity trace in the cross-stream direction




Transition to turbulence

One dominant frequency, and secondary instabilities




Transition to turbulence

One dominant frequency, and secondary instabilities



Frequency analysis of the vortex shedding

One shedding frequency, St = 0.176 at Re = 100

St = 0.225 at Re = 500





St = 0.225 at Re = 500





St = 0.225 at Re = 500


Example 3 - Irregular long-crested waves

Linear theory for the statistical description

Wave elevation composed of a large number of wave components

ζ(x , t) =N∑j=1

ζaj sin(ωj t − kjx + ε)

Relation between the discrete amplitude and the wave spectrum for ωj

1

2ζ2aj = S(ωj)∆ω

0

time

ζ(t)

Frequency

S(ω

)



Basic idea with this example is to generate time series of irregular waveswith a Pierson-Moskowitz spectrum, and generate the spectrum again withthe MATLAB function pwelch

Generate the data from the script IrregWave.m (requires functionPMspectrum.m)

Calculate the sampling frequency from the time series: writeFs=1./dt in the command prompt

Use the pwelch function to get the values of spectral density andfrequency: [P,F] = pwelch(z,[],[],[],2*pi*Fs)

Plot the estimated spectrum: plot(F,P)



Values of the parameters are T1 = 10.13 s and H1/3 = 2.52 m

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5Plot of the Pierson−Moskowitz Spectrum

ω [rad/s]

S(ω

) [m

2 /s]

T1 = 10.13 [s] and H1/3 = 2.52 [m]

0 50 100 150 200 250 300−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time [s]

ζ(t)

[m]

Irregular wave

0 50 100 150 200 250 300−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time [s]

ζ(t)

[m]

Irregular wave

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5Estimated spectrum using the simulated time series

Frequency [rad/s]

S(ω

) [m

2 s]

How does the spectral estimation depends on length of the time series?Sampling frequency?


Example 4 - Response Amplitude Operator (RAO)

Given a sea state defined by the spectrum Sζ(ω), find the responsespectrum Sη(ω)

The response amplitude for a certain frequency can be found by thetransfer function Hη(ω)

Conside the state at the frequency ωj

η0j = Hη(ωj)ζaj

1

2η2

0j︸︷︷︸=Sη(ωj )∆ω

= H2η (ωj)

1

2ζ2aj︸︷︷︸

=Sζ(ωj )∆ω

Sη(ω) = H2η (ω)Sζ(ω)


Example 4 - RAO with irregular waves

Run the MATLAB script GetData.m to retrieve and plot data fromthe file test3004.mat

Open tstool application: write tstool in the commandprompt



Step1: Import the data wave, waveCarriage and heave → Import fromworkspace → Array data


Example 3 - RAO with regular waves

Step 2: Choose the variable wave and specify the time vector inanother variable (press Select Variable tab and choose time)Step3: Press next, rename the time series to FilteredWave and pressFinish



Plot the time series to see how they look (drag to the folder TimePlots); the data is already pre-processedSpectral plots can be obtained be dragging the time series to thefolder Spectral PlotsHigh and low frequency components can be removed using the plotby selecting frequency intervalsRight-click over the selected frequency intervals and choose Pass



Repeat the steps above for waveCarriage and heave (remember torename to FilteredWaveCarriage and FilteredHeave)

The variables are exported to the workspace with class timeseries

In the script RAOirreg.m the data is retrieved aszeta=FilteredWave.Data , zeta c=FilteredWaveCarriage.Data

and eta 3=FilteredHeave.Data

Sampling frequency: Fs=1/(time(2)-time(1))

Spectral density values: [WaveSpectrum,fw] =

pwelch(zeta,[],[],[],2*pi*Fs),[EncounteredWaveSpectrum,fe] =

pwelch(zeta c,[],[],[],2*pi*Fs)and [ResponseSpectrum,fr]

= pwelch(eta 3,[],[],[],2*pi*Fs)

RAO:RAO=sqrt(ResponseSpectrum./EncounteredWaveSpectrum)



Plot spectral density and RAO

0 1 2 3 4 50

1

2

3

4

5

6Spectral Density plot

Frequency, [rad/s]

S(ω

)

Wave [m2s]

Encounter Wave [m2s]

Heave Response [m2s]

0 1 2 3 4 50

0.5

1

1.5

Frequency [rad/s]

RA

O [m

/m]


References

1 O. M. Faltinsen. Sea Loads on Ships and Offshore Structures,Cambridge Ocean Technology Series, 1990.

2 C. M. Larsen & W. Lian. TMR 4180 Marine Dynamics, Departmentof Marine Technology NTNU, 2009.

3 D. E. Newland. An Introduction to Random Vibrations, Spectral andWavelet Analysis, Longman Scientific & Technical, 1993.


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TMR7 - Experimental Methods in Marine Hydrodynamics … · Fast Fourier transform ... 2 Flow past a circular cylinder and determination of the dominant ... Experimental Methods in