Experimental Ship Hydrodynamics Martinus Putra-Xu Cheng.pdf

EXPERIMENTAL SHIP HYDRODYNAMICS

-SIGNAL PROCESSING-

XU CHENG

MARTINUS PUTRA WIDJAJA

FLICIEN BONNEFOY

EMSHIP

EUROPEAN MASTERS COURSE

IN INTEGRATED ADVANCED SHIP DESIGN

COLE CENTRALE DE NANTES

2015
http://www.researchgate.net/profile/Felicien_Bonnefoy

EXPERIMENTAL SHIP HYDRODYNAMICS 2015

2 EMSHIP 5th COHORT- COLE CENTRALE DE NANTES

List of Contents INTRODUCTION ................................................................................................................ 3

REGULAR WAVES ............................................................................................................. 3

Time Domain Analysis ...................................................................................................... 3

Preliminary Frequency Analysis ........................................................................................ 4

Frequency Analysis of the steady regime ........................................................................... 5

IRREGULAR WAVES ......................................................................................................... 8

List of Figure & Table

FIGURE 1. REGULAR WAVE FREE SURFACE ELEVATION .......................................................... 3

FIGURE 2. FAST FOURIER TRANSFORM OF REGULAR WAVE SIGNAL ........................................ 4

FIGURE 3. THE ENLARGED RESULTS FROM FFT OF REGULAR WAVE SIGNAL ........................... 5

FIGURE 4. THE COMPARISON BETWEEN "FFT.M" AND "FOURIER.M" ......................................... 6

FIGURE 5. THE COMPARISON OF TIME SELECTION ON "FOURIER.M" ......................................... 6

FIGURE 6. IRREGULAR FREE SURFACE WAVE ELEVATION ....................................................... 8

FIGURE 7. IRREGULAR WAVE SIGNAL PROCESSING BY "FOURIER.M" ....................................... 9

FIGURE 8. THE COMPARISON OF SPECTRAL DENSITY WITH DIFFERENT SEGMENT OF PERIOD .. 10

TABLE 1. THE OBVIOUS COMPARISON OF SPECTRAL DENSITY ............................................... 10

TABLE 2. THE STATISTICAL RESULTS OF DIFFERENCE TSEGMENT ......................................... 11



INTRODUCTION

This report analyses about the signal processing in terms of wave signal used on the wave

basin of ECN/LHEEA. The main objective of this study is to understand how to analyze the

regular and irregular waves by applying the Fourier transform in MATLAB. A difference of

input parameter is applied in order to understand how it will effected the behavior of the

results.

REGULAR WAVES

The three different parts are introduced in order to understand how to analyze the signal.

These three parts are based on a binary file named regular.mat which contain 3 variables :

1. The sampling frequency (f_samp) = 60 Hz

2. The time which stored in vector, (time in seconds)

3. The wave elevation which measured by resistive probe, (wave in meters)

Time Domain Analysis

Figure 1. Regular Wave Free Surface Elevation



After loading "regular.mat", we will obtain 2 vectors which are time, wave and the sampling

frequency. Then we can type time(end) and length(wave) in order to obtain the signal

duration and number of data points which is 51.083 s and 3066 data points. By observing the

data inside the time vector, a constant step is observed as dt and this value is 0.0167 s.

Therefore, the relation between dt and the sampling frequency (60 Hz) is

There are two regions observed inside the wave signal by plotting the time and wave as in

figure 1. The low and developing region is called transient region and the harmonic part

commonly known as the steady region. There is a low noise existing at both regions, this

might happen due to electrical noise, etc. From around 6 s, the wave-maker starts to generate

the wave and at around 13th s the steady wave occurs until the end. The wave amplitude is

around 0.32 m and the wave period is around 2 second. Since the noise detected in the

transient region is only between -0.005 to 0.005 m, the influence of the noise is relatively

small compared to the amplitude of the steady wave signal. So this noise will not have a big

effect to the analysis.

Preliminary Frequency Analysis

Figure 2. Fast Fourier Transform of Regular Wave Signal

The result in figure 2 has a unique shape which creating a symmetric value about the x-

plane=30 Hz which means both two sides have the same information. So, the frequency can



be divided by using the Shannon or Nyquist criterion ( f < f_samp/2) in order to simplify the

computation. Then a Fourier transform could be done as depicted in figure 3.

Figure 3. The Enlarged Results from FFT of Regular Wave Signal

After zooming in at the main peak, we are able to see one obvious peak where the value is

160 mm at 0.5 Hz. Also a little peak is observed at twice the first's peak frequency which the

value is around 20mm. The frequency which has the highest peak is the frequency of the

wave (1/2second= 0.5 Hz) and the highest amplitude of the FFT results is half of the wave

amplitude (approximate 0.16 m).

Frequency Analysis of the steady regime

The access to gain the information about the frequency for each amplitude of the initial wave

signal could be obtained by zooming in into the graph. This should be done in order to have a

better result by using a Fourier series code that had been developed by LHEEA. We need to

choose a signal from the steady region and also from peak to peak or trough to trough. In this

case, t_begin=13 second and t_end=48.8667 second which chosen from peak to peak.

The corresponding signal selection based on this command

selection1=wave(time>t_begin & time



Figure 4. The Comparison between "fft.m" and "Fourier.m"

There is a significant difference between the approaches of FFT function and Fourier

transform from LHEEA on the peak signal as could be observed in figure 4. In the first case,

it has two peak and unsteady signal around the hill of the peak, while by using the Fourier.m

from ECN/LHEEA, better information could be observed due to a steady signal which allow

the main and secondary peak to be seen clearer.

Figure 5. The Comparison of Time Selection on "Fourier.m"



From the above Figure 5, we can see two different color of results, the red one shows the

result with integer period, from peak to peak, while the blue one presents the result with the

integer plus half more of the period. As can be seen, there is only one higher peak in the red

one, which means the highest amplitude of the steady part of the wave at 0.5 Hz. In the other

hand, two points occur at the peak of the blue result, the reason is that the time selection starts

from one peak to another trough of the wave which will create two different amplitude.

Finally, the time selection based on the integer value (peak to peak, trough to trough) should

be used in order to know the maximum wave amplitude and at which frequency it will occur.

This is also due to the fact that from the practical point of view, the biggest amplitude will be

used for the further analysis.

Bonus Question (just present what we understand from studying this example)

For twice of the peak frequency, , water depth for ECN Wave Basin is 5m. Then,

we can calculate the wavenumber k, which is equal to 4.02

The formula of 2nd

Stokes solution for elevation is:

here, . for , .

So, the above equation can be simplified as:

So the final amplitude for 2nd order item is calculated as :

In the other hand, the amplitude of the second order Stokes Solution has the following

relation:

After running the Fourier Function, a new parameter, will be created. In this part,

From the above Figure 5, it is observed that the amplitude of twice the peak frequency is

around 0.0163m.



IRREGULAR WAVES

After loading the file irreg.mat, we know that this time the signal is quite different from the

previous regular wave as can be observed from Figure 6 below. The wave elevation is quite

unstable and there is a large fluctuation between the range of around -0.05m to 0.05m. The

period of the wave is a little less than 2 seconds, around 1.8s.

Figure 6. Irregular Free Surface Wave Elevation

We get the Figure 6 via Fourier transform on the whole signal. It is not too easy to understand

this spectrum and the points on the spectrum fluctuate too strong along frequency axis. The

reason is that the original wave signal is an irregular wave in time domain and different

frequencies of the signal have different amplitudes.



Figure 7. Irregular Wave Signal Processing by "Fourier.m"

In order to analyze the irregular waves, a Fourier transform which had been developed by

ECN could be used as in regular waves. But, as can be seen in figure 7, the result is still

unclear and too fluctuates. Therefore another built in function in Matlab called pwelch.m

should be used in order to estimate the power spectral density which could help to gain the

useful information from the above result. Before using this function, the different lengths of

time segment must be set. In this case, a set of different time segment had been chosen as

follow, (Seg1=7s, Seg2=17s, Seg3=23S and Seg4=37s) to investigate the effect on the results.

For instance, the syntax of first segment, 7 seconds, can be expressed as following:

[S1,freq1] = pwelch (signal, [], [], seg1*f_samp, f_samp);



Figure 8. The Comparison of Spectral Density with Different Segment of Period

The final Figure 8 can be obtained after superimposing the above four spectra. As we can see,

there are four different situations of the spectral density distribution and both the longest and

shortest time segments are not good because of the reasons shown in Table 1:

Table 1. The Obvious Comparison of Spectral Density

T_segment Frequency Interval Numbers of Points Problem

7 s short large large less few information

37s long small small much large fluctuation

Based on the figure 8, a qualitative conclusion could be done in terms of the proper selection

of the time segment. Thus the reasonable lengths of the time segments in this situation are the

middle between two values (17-23 seconds). But the 23 seconds of the time segment is the

best one from all the four time lengths, which is marked as blue line in the figure 8.

After developing such kind of graph, a further analysis could be done in order to obtain the

information about the wave such as peak period (Tp), significant wave height (Hs), mean

period (Tm), up-crossing period (Tz), and the analytical (Tp). This could be done as follow,

For each segment length, evaluate following statistical values



Tp directly from graph

For each segment length, we calculate the based on the following formula:

We can employ the MatLab expression:

Trapz (freq, S.* freq.^n), with n=-2, 0, 1, 2

In this part, because we know that f (1) = 0, so 1/f will be infinite, so for every frequency in

this part will start from the second value to the end. Then, we get the values of , , ,

for each segment. Finally, the different parameters of period are obtained based on these

spectrum moments. The final results are listed in the following Table 2

Table 2. The Statistical Results of Difference Tsegment

T_seg Hs Tm Tz Tp Tp / Tz Relative

Error

7 s 0.1033m 1.7392s 1.6435s 2.1247s 1.293 8.89%

17s 0.0952m 1.6828s 1.5818s 2.1223s 1.342 4.91%

23s 0.099 m 1.6946s 1.5972s 2.1129s 1.323 6.42%

37s 0.096 m 1.7089s 1.6033s 2.1694s 1.353 4.06%

The relation between Tp and Tz had been calculated as shown in the above table. The relative

error between the relationship of Tp/Tz in Bretschneider spectrum (1.408) with the calculated

results also shown in the table 2. The lowest relative error is occur for time segment 17 and

37 second, therefore the proper time segment should be located around this range. But for 23

second of time segment has an increase of relative error which means another time segment

should be analyzed except 23 second which has the good correlation with the Bretschneider



spectrum relationship. The 37 seconds of time segment has the lowest relative error, thus this

one has the best correlation to obtain proper information of the wave. Nevertheless, a

narrower time segment analysis should be done in order to get more accurate value since

there is still possibility between 17-37 seconds of time segment.

Bonus Question is already explained in the previous paragraph which is about the accuracy

of the different time segment to obtain the good information from the irregular wave.

CONCLUSION

The parameters required in the Fourier analysis for irregular wave are:

Time segment in second, the length of this parameter should be neither too long nor

too short

The irregular wave data

The f_samp.

The parameters required for the Fourier analysis in regular wave are:

The sampling frequency

The time window will be depend on the selection of the t_begin and t_end.

The Regular Wave Data

Through this Lab work, we learn a useful tool to link the relation between the input or

measured regular & irregular wave data and the reaction of the ship modal. The operator

between these is called RAO (Response Amplitude Operator)

Here, is the spectral density and is the amplitude of input

is the spectral density and is the amplitude of input

Once we know the RAO of the system studied, we will be able to study its response to a

given input or measured wave, or vice versa.

Experimental Ship Hydrodynamics Martinus Putra-Xu Cheng.pdf

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