tmr7 - experimental methods in marine hydrodynamics … · fast fourier transform ... flow past a...

TMR7 - Experimental Methods in MarineHydrodynamics

Time Series Analysis

Jose P. Gallardo Canabes

Department of Marine TechnologyNorwegian University of Science and Technology

November 3, 2010

J. Gallardo C. (NTNU) Time Series Analysis November 3, 2010 1 / 26

Outline of the presentation

Motivation

Some theory

Examples

Applications in Matlab


Motivation

Time series are common in engineering applications

Visual inspection is not enough

Many tools are available for the analysis


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


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!


Filtering of data

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


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


Examples

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

2 Irregular waves

3 Example of response amplitude operator (RAO)


Example 1 - Flow past a circular cylinder

Flow past a circular cylinder at Re = 100

Unsteady, viscous and laminar 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



Flow past a curved circular cylinder at Re = 500

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


Example 2 - Irregular long-crested waves

Linear theory for the statistical descriptionWave 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 3 - 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 3 - RAO with regular waves

Load data: write regw1 = load(regw1.asc) in the commandprompt

Open tstool application: write tstool in the command prompt



Step1: Import the data regw1.asc : File → Import from workspace →Array data



Step 2: Choose the variable regw1 and specify the time vector in thecurrent variable (first column)

Step3: Press next, choose create several time series and press Finish



The time series names can be renamed, rename regw15 as AccCGand regw17 as AccCG

Plot the time series to see how they look (drag to the folder TimePlots)

Choose the data of interest: Tools → Select data

With the right click over the selected data choose Keep observations



Spectral plots can also be obtained be dragging the time series to thefolder Spectral Plots

High and low frequency components can be removed using the plotby selecting frequency intervals

Right-click over the selected frequency intervals and choose Pass



The variables are exported to the workspace with class timeseries

To get the values as arrays write in the command prompt:time=Wave.time, WaveHeight=Wave.DataandResponse=AccCG.Data

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

Spectral density values: [WaveSpectrum,fw] =

pwelch(WaveHeight,[],[],[],2*pi*Fs)and[ResponseSpectrum,fr] =

pwelch(Response,[],[],[],2*pi*Fs)

RAO: RAO=sqrt(ResponseSpectrum./WaveSpectrum)



Plot spectral density and RAO

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−4 Spectral Density plot

Frequency, [rad/s]

S

Wave height [m2s]

Response [m2/s3]

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Frequency [rad/s]

RA

O[(

m/s

2 )/(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 … · fast fourier transform ... flow past a...

Documents