Data Analysis
Experimental Methods in Marine Hydrodynamics
Lecture in week 36
Objectives of this lecture:
• Give you an overview of the most important methods of
data analysis in use in experimental marine
hydrodynamics
• Give some examples of how to do data analysis using
Matlab
Covers Chapter 10 in the lecture notes
1
Contents
• Typical types of tests:
–
–
–
–
•
•
•
•
1.
2.
3.
4.
Static tests
Decay tests
Regular wave tests
Irregular wave tests
Filtering
Spectral Analysis
Statistical Analysis
Special analyses:
– Slamming
– Ventilation and air injection to waterjets
– Sea sickness incidence
2
Static tests
• Tests expected to give a constant measured value
– Example: Ship resistance, propulsion and open water tests
• Only the mean value is of interest
• Take care to avoid transient effects at start-up
• Notice that even for tests of stationary phenomena like ship
resistance in calm water, there will be oscillations in the
signal
– To create a reliable average at least ten oscillations should be
included in the time window
140
2
RTm
1.98
Speed
130
Model Resistance R
110
1.92
100
1.9
1.88
90
1.86
80
1.84
70
1.82
60
3
1.8
10
Transient part
15
20
Stationary part
25
Tim e [s e conds ]
30
35
40
Carriage speed [m/s]
1.94
Tm
[N]
1.96
120
Decay Tests
•
•
•
Model is oscillated and then
released, and response is
measured
Provides information about
natural period and linear and
quadratic damping terms
Very useful for lightly damped
systems
– Well suited: Natural period and
damping in roll
– Difficult, but possible: pitch
– Close to impossible: heave
4
Logaritmic decrement:
  ln(
xi
)
xi1
Analysis of
decay tests
The damping ratio:
p
p


pcr 2 M 0
For low damping ratios
(<0.2):
  2
The logarithmic decrement:
  ln(
5
xi
)
xi 1
Equivalent damping:
2C
pEQ  2 M 0 
0
Alternative analysis of decay tests
- fitting of equivalent theoretical system
7
6
5
http://www.ivt.ntnu.no/imt/courses/tmr7/resources/decay.pdf
4
3
Response
2
1
0
-1
-2
-3
-4
-5
6
-6
40
50
60
Time [s]
70
80
Filtering
• Low-pass filtering with hardware filter in experiment
– To remove Nyquist-phenomena (down-folding)
– Filter-frequency: lower than half the sampling frequency
• Additional filtering may be applied in analysis.
Purposes:
– Remove noise:
• High-frequent noise: Low-pass filter
• Noise at particular frequencies: Notch-filter, Low-pass
– Remove low-frequent and constant values
• High-pass or band-pass
7
Amplitude
Filters
Ideal characteristic
Real characteristic
Low pass filter
High pass filter
Band pass filter
Frequency
8
"Notch" filter
Real-time filters influences the phase of
the signal!
•
An actual filter alters both the
magnitude and the phase of the
signal, but not its frequency
 Filtered signals to be compared
with respect to phase should be
filtered with the same filter
(type and setting)
From ”Measurement and Data Analysis for Engineering and Science”
by P.F. Dunn
9
Understanding filtering
Asymmetric filtering (used in real-time)  phase lag
2.5
2
Averaging window
1.5
1
0.5
0
-0.5
0
10
20
30
40
50
60
70
80
90
100
-1
-1.5
-2
-2.5
Symmetric filtering (can only be used after the test)  no phase lag
Averaging window
2.5
2
1.5
1
0.5
0
-0.5
0
10
20
30
40
50
60
70
-1
-1.5
-2
-2.5
12
Now!
80
90
100
Data acquisition without filtering
•
It is OK to do data acquisition without filtering as long as there is
virtually no signal above half the sampling frequency
– so the noise is not folded down into the frequency range of interest
•
Requires high sampling frequency
– (>100 Hz, depending on noise sources)
•
Requires knowledge of noise in unfiltered signal
– Spectral analysis, use of oscilloscope
•
Unfiltered data acquisition eliminates the filter as error source, and
eliminates the problem of phase shift due to filtering
– Drawbacks:
• Must have good control of high-frequency noise
• High sampling frequency means large data files
13
Data acquisition with filtering
• The phase shift is only a problem if channels with filtering
is compared to channels without filtering, or with different
filtering
• Phase-shift problems frequently occur when digital and
analog sensors are combined
(since digital sensors can’t be filtered using the conventional analog hardware
filters)
• Phase-shift problems might also occur when one is
sampling different sensors with different frequency
(since filter frequency is usually selected dependant on sampling frequency)
14
Filtering as part of the analysis
• Filters are applied to the digital signals
– No special hardware is required
– One can play around with different filter settings to obtain the
optimum result
• Symmetric1 filters might be applied
– No phase shift problems
• Note that even if symmetric filters are used so phase shift
is avoided, there might still be attenuation of the signal at
other frequencies than the filter frequency
1What
15
we call symmetric filters are called ‘Anti-Causal Zero-phase Filter’ in Matlab
Filtering in Matlab
From Matlab help – signal processing toolbox:
• Many options!
• Signal Processing
Toolbox contains a
large number of
relevant time-series
analysis tools,
including filters
• The tstool time
series analysis tool
also offers filtering
• Use Matlab help!
16
Aims of analysis of regular wave tests
• Response amplitude
• Response amplitude operator (divide by wave amplitude)
• Phase angle (between wave at reference location and
response)
• Response frequencies
– In non-linear systems you often get response at other frequencies
than the excitation (wave) frequency
• Take care to leave out transient response at the start of the
time series!
17
Analysis methods of regular wave tests
•
”Visual” analysis of time series
–
–
–
–
•
Needs nice, noise-free signals (can be obtained by filtering)
Work-intensive
Not very accurate, especially phase
Gives good understanding and overview of the results
Fourier series analysis
– Gives amplitude and phase of the response at the specified (or measured)
wave frequency (and higher harmonics of the wave frequency)
•
Least-squares fit to measured signal of a sum of harmonic components
with different phase and frequency
– Novel technique
– Suitable for time-series of relatively short durations (few oscillations)
•
Spectral analysis
– Gives information about frequency distribution of response
– Not suitable method to get response amplitudes
•
18
Estimate the amplitude from the standard deviation: amp  2  st.dev
Example of accuracy of estimating
amplitude from st.dev. in regular waves
3.5
3
2.5
2
1.5
1
0.5
0
8002
19
8004
8006
8000
8002
sqrt(2)*stdev(Waveheight 1)
sqrt(2)*stdev(Acc1_FP)
RAO Acc1_FP sqrt(2)*stdev
8005
8001
8002
8003
Fourier analysis Waveheight 1
Fourier analysis Acc1_FP1
RAO Acc1_FP Fourier
8006
20
21
Regular Wave 8002, H=1.3 m, T=6 s (spec)
22
Fourier Analysis
• A periodic signal can be fully described by an infinite sum
of harmonic components:
 2 t 
 4 t 

f (t )  a0  a1 cos 
a
cos
Fourier series:
 2

 
T
T




 2 t 
 4 t 
b1 sin 

b
sin
 2

 
 T 
 T 
 2 kt 
 2 kt 

f (t )  a0   ak cos 
b
sin
 k


T


 T 
k 1

In compact notation:
T
1
a0   f (t )dt
T 0
2
 2 kt 
ak   f (t ) cos 
 dt
T 0
 T 
T
With coefficients
defined as:
23
2
 2 kt 
bk   f (t ) sin 
 dt
T 0
T


T
Fourier analysis of regular waves
• The primary period T of the signal is determined:
– Automatically from the frequency of maximum energy of the
signal
– Manually (for instance specified as input wave period)
– Automatically from the frequency of maximum energy of a
reference signal (for instance a wave probe)
 2 kt 
 2 kt 

f (t )  a0   ak cos 
b
sin
 k


 T 
 T 
k 1

Output from MARINTEK in-house time-series analysis system:
• Mean value, amplitude and phase of the primary period
and higher harmonics are determined by Fourier analysis
24
About Fourier series models in Matlab
• To fit a Fourier Series to a time series:
1.
2.
3.
4.
25
Create a new time series containing just the relevant window
Open the Curve Fitting Toolbox (>> cftool)
Choose Fourier from the model type list.
Use Fit Options to control the fit.
Choosing window
Full time series of vertical acceleration
Create new arrays for time and
acceleration with just the needed
data:
4
Range of stable response
3
2
>> Time_tr=Time(9302:12890);
>> Acc1_FP_tr=Acc1_FP(9302:12890);
X: 1.289e+04
Y: 2.389
1
0
-1
3
-2
2
-3
-4
1
0
2000
4000
6000
8000
10000 12000 14000 16000
18000
0
-1
-2
26
-3
0
500
1000
1500
2000
2500
3000
3500
4000
Invoke the curve fitting tool >>cftool
27
Fourier Integral
• When T in the Fourier Series the Fourier Integral is
obtained

In compact notation:
f (t )   A( ) cos t   B( ) sin t d 
0
A( ) 
With coefficients
defined as:
B( ) 
1

1



f (t ) cos t  dt



f (t ) sin t  dt

Note that T means that the signal no longer needs to be periodic!
28
The Fourier series is applicable only to periodic signals, while the
general Fourier Integral (Fourier Analysis) is applicable also to
non-periodic signals
Fourier integral in compact, complex form
A more compact form
of the Fourier integral:
With the Fourier
transform of f(t):
29
1
f (t ) 
2
ˆf ( )  1
2


fˆ ( )eit d 




f (t )e  it dt
Discrete Fourier Transform (DFT)
• Typically, the values of f(t) are sampled at equally spaced
points (constant sampling interval)
• The DFT of a signal f with n components will be:
1
fˆn 
N
N 1
fe
k 0
 i(2 n k N )
k
• DFT requires O(N2) operations in the computer
30
Fast Fourier Transform
FFT
• FFT is a computer algorithm for calculation of the complex
Fourier integrals (in discrete form)
• Conventional Discrete Fourier Transform (DFT) can also
be implemented in a computer
– DFT requires n2 multiplications
– FFT requires n·log(n) multiplications
– FFT is more accurate (due to fewer multiplications)
• FFT is a core function of all digital data analysis
31
Matlab: use functions Y=fft(x) and y=ifft(X) to compute the
Fourier and inverse Fourier transforms using the FFT algorithm.
See fft and/or Fourier transform in Matlab help for (a lot)
more info
Irregular wave tests
• Direct representation of the full scale sea condition
• Typically wanted results:
– Response spectra
– Response spectrum parameters:
• Spectral moments
• Standard deviation
• Peak period
– Response amplitude operator (RAO)
– Statistical results:
• Max and min values,
• Information about statistical distribution
• Extreme value statistics (extrapolation using the statistical
distribution)
• Weibull plots etc.
32
Properties of stochastic processes
• Stationary: - Statistical properties constant with time
• Homogeneous: - Statistical properties constant in space
• Ergodic: - Time can be replaced by space as primary
variable without changing the statistical properties
• The wave environment is commonly assumed to be a
stationary, ergodic process
• This assumption greatly simplifies the analysis, and is a
necessity for all established analysis methods
•
33
It is not exactly true in the towing tank – wave amplitude tends to
decrease with distance from the wave maker, due to viscous damping
The stochastic wave process in the
laboratory
•
•
The wave maker produces a stationary wave output
Viscous damping means that the wave amplitude will decrease with
distance from the wave maker
– Might be of some significance, but only for long towing tanks and small
waves
•
Dissipation increases with distance from wave maker
– means that part of the energy at one frequency will dissipate to nearby
frequencies
– Regular waves are no longer completely regular
•
Wave reflections (from models and tank walls) introduces nonstationary and non-homogeneous effects
– Important to have a minimum of wave reflections
34
Autocorrelation function

Limit  1 T
Rxx ( ) 
  x(t ) x(t   )dt 
T   T 0

 is the standard deviation and m is the average value
35
Figures from Newland (1984)
Cross-correlation function

Limit  1 T
Rxy ( ) 
  x(t ) y (t   )dt 
T   T 0

Cross-correlation function for
two sine waves with y(t) lagging
x(t) by an angle 
36
Figures from Newland (1984)
Spectral Analysis
• A stationary random process is not periodic
– We cannot use Fourier analysis directly to obtain information
about frequency distribution
• The frequency distribution of a stationary random process
is determined by Fourier analysis of the autocorrelation
function:

S xx ( )   R xx ( )e (  i ) d

where
37

Limit  1 T
Rxx ( ) 
  x(t ) x(t   )dt 
T   T 0

Cross Spectral Analysis
• The cross spectrum is found by Fourier analysis of the
cross-correlation function:
S xy ( ) 


Rxy ( )e(  i ) d

• The linear complex transfer function is the ratio between
spectrum and cross-spectrum:
S xy ( )
H ( ) 
S xx ( )
38
The Response Amplitude Operator
RAO:
H ( ) 
2
39
S yy ( )
S xx ( )
Meaning of spectral moments
• The n’th moments of the spectrum is defined as:

mn    n S ( )d
0
• Standard deviation of response:   m0
• Significant value of response:
• Average period of response:
• Average zero crossing period :
40
x1
3
 4 m0
m0
T1 
m1
m0
T2 
m2
Output from
MARINTEK
batch-oriented
Time Series
analysis system
(timsas)
41
42
Statistical distributions
• The probability distribution function, P(x), is the
probability that a general value of the process x(t) is less
than or equal to the value of x
P ( x )  P ( x (t )  x)
dP ( x)
p( x) 
dx
• The probability density function:
• The probability that a<x(t)<b is given by the probability
density function such that:
b
P ( a  x(t )  b)   p ( x) dx
43
a
Probability distributions used in the
study of wave generated responses
• The distribution of the process itself, e.g. the distribution
of the wave elevation x(t) and the measured response y(t)
 Gaussian distribution
• The distribution of amplitudes; e.g. distribution of the
wave amplitudes xA and measured response amplitudes, yA
in the tests.
 Rayleigh distribution
44
Rayleigh distribution of amplitudes
• Follows from the assumption that the elevation itself is
Gaussian
 1  x   2 
X
• The cumulative distribution: P( x)  1  exp   
 
 2   X  
•
Here is the mean or expected value of x(t) defined as:

 X  E  x    xp( x)dx

•
 is the variance of x(t), defined as:
2
 X2  E  x   X    E  x 2    X 2

45

Rayleigh distribution
•
For a measured time series with
N samples the mean value and
the variance are calculated as:
1
X 
N
N
x
i 1
i
N
1
( xi   X )
 X2 

N  1 i 1
46
Non-linear response
• The response y(t) follows a Rayleigh distribution only if it
is a linear function of the wave elevation x(t)
• To describe non-linear response it is common to use the
more general Weibull distribution:
 1  x A   X G 
P ( x A )  1  exp   
 
 
 G  
– G=2 gives the Rayleigh distribution
– G=1 gives the Exponential distribution
47
Weibull plots
P(xA)-axis plotted as
ln ln1  P(x A )
Horisontal axis: response amplitude yA
48
Horisontal axis: response amplitude yA
Significant values
• Significant maxima:
– the mean of the highest one-third of the crest-to-zero values of xA,
• Significant minima:
– the mean of the highest one-third of the trough-to-zero values of xA,
• Significant double amplitude:
– mean of the highest one-third of the maximum to minimum values of xA
49
Maximum/Minimum Values
• Maximum Value:
– Measured maximum value in the record
• Minimum Value:
– Measured minimum value in the record
• Largest double amplitude:
– Largest measured crest to trough value in the record
50
Skewness
• The Skewness is defined as the third central moment of the
m
process:
 1  E ( x   X )3   33

X
• The skewness provide an indication of the degree of
asymmetry in the probability distribution about the mean
value.
• For a Gaussian distribution the skewness will therefore be
zero
• Skewness is nonzero for a Rayleigh distribution.
51
Kurtosis
• The Kurtosis is defined as:
 2  E ( x   X )4   3 
m4
X
4
3
• The Kurtosis is zero for a Gaussian process
52
Example of statistical analysis results
53
Examples of special analyses:
• Slamming
• Ventilation and air injection to waterjets
• Sea sickness incidence
54
Slamming analysis
• Definition of slamming threshold value(s)
– Typically 50 kPa, but depends heavily on context
• Counting (automatically) the number of slams above
different threshold levels
• Detailed analysis of the time series of each slam reveals
properties of the slam, the transducer and the model
dynamic response
55
56
Air suction to waterjets
Propulsion test, Seastate 5 (Hs=2.4 m)
Torque, Starboard jet [Nm]
12
10
Limit for air suction
8
6
4
2
0
57
5.115
10.230
15.345
Time [seconds]
20.460
Analysis of air suction to waterjets
• Number of occurrences, given in groups as
percentage of nominal torque value.
58
GUIDE
RAIL
CABIN
NBDL
ROLL
AXIS
TILT
TABLE
+
PITCH
AXIS
MOVING
Sea sickness incidence
A-FRAME
HEAVE
GUIDE
RAIL
• Estimation of sea sickness incidence is based on:
– Measurement of motions and accelerations of the model/ship
– Measurement of motion sickness incidence MSI (percentage of
people vomiting) to vertical accelerations of different frequency,
amplitude and duration
• Empirical relations of motion sickness incidence (MSI) as
function of frequency, RMS amplitude, and duration
available in ISO standard ISO 2631 1-4
59
60
Summary
• How to analyze decay tests
• The difference between analog real-time and digital
filtering in analysis
– the phase-shift of analog filtering can be avoided when filtering in
the analysis
• Short summary of the essentials of Fourier analysis
– Fourier series to analyze periodic signals (regular waves)
– Discretized Fourier integral to analyze an irregular wave response
• Information about specialized analyses:
– Slamming
– Air suction to waterjets
– Sea sickness
61
Some tips on data analysis in Matlab
• Import time series into Matlab
– Mat-files can be opened directly
– Binary files from Catman can be imported using catman_read.m
– It is also possible to read Excel-files and various ASCII file
formats – see Matlab help
• tstool is a graphical user interface for working with
time series
– The same methods can be applied from the command line
– Code can be generated by tstool to automate the process
– Suitable for generating time series plots, filtering, statistics,
plotting of spectra etc.
• Curvefitting tools, including Fourier Series fitting, is
collected in the cftool GUI
• Use Matlab help
62