tmr7 - experimental methods in marine hydrodynamics … · ... time series analysis september 7,...

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)

J. Gallardo C. (NTNU) Time Series Analysis September 7, 2011 12 / 32

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

2 Background information

3 MATLAB for data analysis

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)


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 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.


tmr7 - experimental methods in marine hydrodynamics … · ... time series analysis september 7,...

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