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Preliminary
ANALYSIS OF FORCE RESPONSE DATA FROM TESTS ON
A MODEL OF A TRUSS STRUCTURE SUBJECTED TO PLUNGING BREAKING WAVES
By
Alf Tørum
Norwegian University of Science and Technology Department of Civil and Transport Engineering
NTNU-IBAT/MB TN January 2013
24 May 2012 Version 21 January 2013.
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TABLE OF CONTENT 1. GENERAL ................................................................................................................................. 3 2. TESTS ON MODEL TRUSS STRUCTURE .......................................................................... 7 3. METHOD OF ANALYZING WAVE SLAM MING FORCES .......................................... 16 3.1. General on force identification in structural dynamics ................................................. 16 3.2. “Single-degree-of-freedom” vs. “Multiple- degrees-of-freedom” ................................. 16 3.3. General on the applied analysis method ........................................................................... 20 3.4. Frequency response function (FRF) ................................................................................. 21 3.5. FRF applied to the wave slamming forces....................................................................... 27 3.6. Possible reasons for the high wave force response when the wave broke a head of the structure ......................................................................................................... 39 4. CONSIDERATIONS ON THE ACCURACY OF THE APPLIED ANALYSIS METHOD .................................................................................................................................... 42 5. WAVE SLAMMING FORCE CALCULATIONS ................................................................. 40 6. PROBABILITY OF OCCURRENCE OF PLUNGING BREAKING WAVES .................. 52 7. DISCUSSION OF RESULTS ................................................................................................... 55 ACKNOWLEDGEMENT ............................................................................................................ 60 REFERENCES .............................................................................................................................. 60
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1. GENERAL Wave forces from non-breaking waves on a slender vertical pile are commonly calculated according to the Morison equation,
dztuDCdzuDuCdFdFdF MwDwMD ∂∂
+=+=4
5.02πρρ (1)
where ρw is the mass density of water, CD is the drag coefficient, CM is the inertia coefficient, D is the pile diameter, u is the water particle velocity and t is time. If the waves break against the pile, Figure 3, a slamming force may occur on part of the pile, ληb. The total force is then
sMD FFFF ++= (2) The slamming force is commonly written as:
bbsws DCCF ληρ 25.0= (3) where Cs is a slamming force factor, Cb is the breaking wave celerity (the water particle velocity is set equal to the wave celerity at breaking), λ is the curling factor which indicates how much of the wave crest is active in the slamming force, Figure 1. The nature of the slamming force is indicated in Figure 2. The slamming force has a short duration, τp, but high intensity. The duration of the slamming force is somewhere in the range τp = 0.20 D/Cb – 0.5D/Cb. The force - time history is given differently by different researchers. Figure 3 shows the most used force – time histories. The most recent force – time history is the one marked “own model” by Wienke and Oumeraci (2005). In Figure 3, t = time, V = water velocity, R = cylinder radius, f1 = line force per unit length, ρ = mass density of water
Figure 1. Breaking wave leading to possible slamming force.
Figure 4. The nature of the slamming force.
Cs was set to Cs = π by Goda et al. (1966). They obtained λ-values of approximately λ = 0.4. Different values of Cs have later been obtained by different researchers, but Cs = π has frequently been used. One of the latest investigations on wave slamming forces on cylinders in a large model scale test set-up has been carried out by Wienke and Oumeraci (2005). They set, on theoretical grounds, Cs = 2π and obtained values of λ as shown in Figure 4 for different
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inclination of the pile. Note that since Cs = 2π, the results of Wienke and Oumeraci (2005) give approximately twice the slamming force obtained by Goda et al. (1966).
Figure 3. Different time histories of the line force. T=time, R=cylinder radius, V=cylinder velocity. Wienke and Oumeraci (2005).
All previous tests, except those by Wienke and Oumeraci (2005), have been carried out at a fairly small scale with cylinder diameters typically 5 – 10 cm in diameter. Wienke and Oumeraci (2005), carried out tests in a large wave flume with a cylinder with diameter 0.70 m, water depths approximately 4 m and with wave heights up to 2.8 m. They used “artificial breaking waves” in the sense that they programmed the wave generator to generate plunging waves in “deep” water at or very close to the cylinder in “deep” water.
Figure 6. Curling factor vs. pile inclination. Wienke and Oumeraci (2005). Currently new investigations are being carried out by Professor Oumeraci, Leichweiss Institut für Wasserbau, University of Braunschweig, Germany, and one of his PhD students, on slamming forces on a single pile in depths were waves break (plunging) due to depth limitations. (Personal communication between Hocine Oumeraci and Alf Tørum) Ros (2011), Arntsen et al. (2011) carried out tests on wave slamming forces on a single pile. Figure 5 shows the test pile, where local force responses were measured at different elevations. Figure 6 shows a time series of the responses from one of the tests. Although the waves are so-called regular waves there is considerable variation of the response from wave to wave. The reason for this is not exactly known, but it may be due to small variations in the front slope from wave to wave and seems to be inherent scatter in such tests.
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Figure 5. Instrumented cylinder (dimensions in cm). The striped zones represent the force transducers (rings). The gap between each transducer measures 0.01 cm. Wave sketch is not in scale and is
included for illustrative purpose only.
Figure 6. Measured response at the third transducer from above, Figure 5, for regular waves.
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Ros (2011), Arntsen et al. (2011) obtained force intensities along the pile as shown in Figure 7. Similar triangular force intensities was also obtained by Sawaragi and Nochino (1984) and Tanimoto et al. (1986), which is in contradiction to the assumption of a uniform force intensity as assumed by Goda et al. (1966) and by Wienke and Oumeraci (2005).
Figure 7. Slamming force intensity Fs along the pile. T = 2.2 s and H = 28 cm. Z = 0 is at the still water line. tp: maximum peak force intensity instant.
Table 1 show comparison between the results of the slamming forces from plunging breaking waves obtained on the 6 cm diameter single pile used by Ros (2011) and the results of different researchers. Table 1 show that the results of Wienke and Oumeraci (2005) give the highest slamming force. The Wienke and Oumeraci (2005) method was, as mentioned, based on large scale tests in the Large Wave Flume on a pile with diameter 0.70 m and with wave heights in the flume of approximately 2.5 m. The reason why Wienke and Oumeraci (2005) obtained higher forces is not clear, but it could be due to scale effects related to the smaller scales used by the other researchers?
Table 1. Comparison on the test set-up used by Ros (2011).
Calculated total forces based on different studies
Study Cs-value λmaxVertical force
distribution
Total slamming force, N
Wienke and Oumeraci (2005) 2π 0.46 Uniform 88
Goda (1966) π 0.40 Uniform 39
Sawaragi and Nochino (1986)
π 0.90 Triangular 44
Tanimoto et al. (1986)
π 0.66 Triangular 32
Ros (2011) 4.3 0.67 Triangular 36.
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2. TESTS ON MODEL TRUSS STRUCTURE Miriam Aashamar has, as part of her Master thesis, Aashamar (2012), carried out tests on wave forces on a model truss structure as shown in Figures 8, 9 and 10. We have in this note looked briefly into more details of the responses that were measured. Figure 8 shows the test set-up. The model scale was 1:50 in relation to water depth, e.g. the model water depth was 33.3 cm at the middle of the structure, corresponding to 16.65 m prototype values. The water depth in the deeper horizontal part of the flume is 89.5 cm. The width of the wave flume is 1.00 m. The bottom slope in front of the model structure is 1:10.
Figure 8. Model test set-up of Aashamar (2012). The wave forces/responses were measured by two force transducer at the top of the structure and two force transducers at the bottom, Figure 11. The waves were measured at two locations: 1. Some distance ahead of the model structure in 90 cm water depth in the wave flume, and 2. Half way between the wave flume wall and one of the front legs of the structure, Figure 9. It has to be born in mind that the wave measurements at the location of wave breaking may be somewhat uncertain because of air entrained in the water due to the wave breaking.
Figure 9. 1:50 scale model of truss support structure for wind turbines.
Wave gauge
Slamming forces
Slamming forces
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The wave slamming forces have a short duration and what is usually recorded are the force responses since there will be some dynamic effects due to the stiffness of the force measuring system and the model support structures. The challenge of the data analysis is to resolve the wave impact force from the measured responses. For a one-degree-of-freedom system the response is depending on the form of the impulse and the ratio between the duration of the impact impulse and the natural period of oscillations of the system. Figure 12 shows ratio between the response and the force for different impulse forms. The duration of the impact is set differently by different researchers, but is in the range
1 (0.25 0.5)b
DtC
= − (4)
where D = pile diameter, Cb is the celerity of the breaking wave. In our case (the 1:50 scale model) Cb = 2.1 m/s. Aune (2011). The vertical legs have a diameter of D = 0.016 m. This gives t1 in the range t1= 0.0019s- 0.0038. The natural period of oscillations of the model structure is obtained from pluck tests. Figure 26 shows results of pluck tests by plucking of the structure with an impulse hammer (see later). The maximum response is for the sum of all the four force transducers. The pluck tests reveal that the natural period of oscillation is approximately T = 0.02 s. This gives t1/T in the range t1/T = 0.095 – 0.019. If we assume a triangular pulse the response will be somewhere in the range 0.4 – 0.8 of the force according to Figure 12. We will come back to this issue under Chapter 3.
Figure 10. Model truss structure. Dimensions in mm. The structure has the same apperance from all four sides.
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Figure 11. Model structure with the force transducers.
Figure 12. Ratio between response and force for a one-degree-of-freedom system as a function of the ratio of the impulse duration and natural period of oscillation. Wienke and Oumeraci (2005) carried out tests on wave slamming forces from plunging breaking waves on a 0.70 m diameter single pile in the Large Wave Channel in Hannover, Figure 13.
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Figure 13. Test set-up, Wienke and Oumeraci (2005).
Figure 14. The six steps of the wave force analysis procedure by Wienke and Oumeraci (2005). Figure 14 shows the steps in their wave force analysis procedure. 1. This is the recorded force response signal. 2. This is the measured Morison force from a wave slightly below wave breaking height. 3. Shows the measured response – the Morison force from 2, e.g. the dynamic part of the response.. 4. From 3. to 4. De-convolution (Duhamel integral) has been applied and the wave slamming force has been obtained,. In 5 the total wave force has been obtained as a sum 2+4. Finally this force has been applied to the pile and the response result, 6, is compared to the measured response.
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We should note that Wienke and Oumeraci (2005) considered their structural system as a “single-degree-of freedom” system. Ros (2011), Arntsen et al. (2011) applied basically the same approach as Wienke and Oumeraci (2005), but obtained as mentioned smaller slamming forces than Wienke and Oumeraci (2005). It should be noted that the ratio of the impulse duration and the natural period of oscillations for the Ros tests were approximately 5 – 8 the Duhamel integral results are not so sensitive to the exact ratio, Figure 12, triangular impulse. The force response data from the tests on the model structure, Figure 9, has a different appearance than the forces response data on a single cylinder. Figure 15 shows the force time trace for a non-breaking wave. Total force means that the force responses of all four force transducers have been added together. The sampling frequency was 19200 Hz (19200 samples per second). There is some noise in the force time trace, but no sign of any wave slamming force. Except for the noise, he recorded forces are considered to be the “Morison forces”. The two peaks in the force time trace, at times approximately 2.67 s and 2.87 s, occurs when the wave hits first the front section and second when the wave hit the rear section. The wave height at the structure is in this case approximately 22 cm, while the wave height in “deep” water is approximately 17 cm. The legend M054e150-1 signifies the following: M = Miriam (Master student), 054 = frequency f = 0.54 s-1, e150 = eccentricity index for the wave generator, higher index value means a higher wave height, 2 means the second test run with the shown settings of the wave generator. The numbers in the figure title, 40600 – 85000 indicate which data points have been used for the analysisi
Figure 15. Total force response for a non-breaking wave. Wave height in “deep” water Ho ≈ 0.17 m, wave at structure H ≈ 0.22 m Wave period Tm = 1.85 s. Prototype values, scale 1:50: H0 ≈ 8.5 m, H ≈
11.0 m and wave period T = 13.1. Figures 16 and 17 show an examples of the total force and the wave height for one particular run when the wave break at the structure,. The response is a mixture of Morison forces and wave slamming forces. The high frequency part of the time trace is supposed to be the response due to the wave slamming forces. Figure 18 shows the highest response recorded during the test series of Aashamar (2012). We will come back to further discussions on this case.
2 2.5 3 3.5 4 4.5-10
-5
0
5
10
15
20
Time, s
Tota
l res
pons
e fo
rce,
N. W
ave
heig
ht, c
m
Mf054e150-1 40600 - 85000. Total response force. Waves
Force responseWave at structureWave in "deep" water
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Total force response for a non-breaking wave. Wave height in “deep” water Ho ≈ 0.17 m, wave at structure H ≈ 0.22 m Wave period Tm = 1.85 s. Prototype values, scale 1:50: H0 ≈ 8.5 m, H ≈ 11.0 m
and wave period Tp = 13.1.
Figure 16. Total force response for a breaking wave. Wave height in “deep” water, Ho ≈ 0.20 m, breaking wave at structure, Hb ≈ 0.23 m. Wave period T = 1.85 s. Prototype values, scale 1:50: Ho ≈ 10 m, Hb ≈11.5 m, wave period T = 13.1 s.
Figure 17. Total force response for a breaking wave. Waves height in “deep” water, Ho ≈ 0.22 m, breaking wave at structure, Hb ≈ 0.23 m. Wave period T = 1.85 s. Prototype values, scale 1:50: Ho =≈11.0 m, Hb ≈ 11.5 m, wave period T = 13.1 s.
2 2.5 3 3.5 4 4.5-10
-5
0
5
10
15
20
25
30
Time, s
Tota
l res
pons
e fo
rce,
N. W
ave
heig
ht, c
m
Mf054e170-1 40600 - 85000. Total response force. Waves
Force responseWave at structureWave in "deep" water
2 2.5 3 3.5 4 4.5-15
-10
-5
0
5
10
15
20
25
30
Time, s
Tota
l res
pons
e fo
rce,
N. W
ave
heig
ht, c
m
Mf054e180-1 40600 - 85000. Total response force. Waves
Force responseWave at structureWave in "deep" water
13
Figure 18.Time expansion for the highest response during all test series.. The shown force response time traces does not exhibit such clear slamming force response as found for single piles, e.g. Wienke and Oumeraci (2005), Ros (2011), Arntsen et al. (2011), who applied the Duhamel integral approach. Hence we will look for other methods for analysis. Figure 19 show the time traces for total force response, and wave heights for a specific run, while Figure 20 shows a time expanded portion of the force response and the wave height at the structure for the same trace. In this case the waves break and the wave measurement become somewhat uncertain because of air entrainment.
3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5-10
0
10
20
30
40
50
60
70
Time, s
Forc
e, N
. Wav
e he
ight
, cm
MF054e265-1. 72000 - 86000 Total force. Wave height
Total forceWave
14
Figure 19. Total force response for the time series Mf054e200-2 and waves at the structure and in “deep” water.
Figure 20. Force response data for tests with regular waves with frequency 0.54 Hz or wave period T = 1.85 s. (prototype Tp = 13.1 s). The wave height was approximately 0.22 m. Air entrapped in the water during wave breaking influences to some extent the wave measurements.
0 1 2 3 4 5 6 7 8-15
-10
-5
0
5
10
15
20
25
30
35
Time, s
Tota
l res
pons
e fo
rce,
N. W
ave
heig
ht, c
mMf054e200-2 1 - 150000. Total response force. Waves
Force esponseWave at structureWave in "deep" water
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-5
0
5
10
15
20
25
30
35
Time, s
Forc
e, N
. Wav
e he
ight
, cm
MF054e200. 130000 - 150000 Total force. Wave height
Total forceWave
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It is certainly not so easy to analyze the data from the truss structure as it is from the tests on a single cylinder, e.g. Wienke and Oumeraci (2005). When testing a single cylinder the highest response occur at the first response, e.g. Figure 14, and thereafter there is a damped oscillation of the structure. In our case there are also indications of smaller slamming response forces in the beginning of the force response time series before the maximum slamming response force occurs, Figures 15 - 20. The reason for this is not precisely known, but there might be slamming forces against the several bracings, as also indicated in Figure 9. Our main interest are the wave slamming forces. Wienke and Oumeraci (2005) carried out tests with almost breaking waves and considered the forces then measured as the Morison forces, Figure 14, “Measured quasistatic force”. They then deducted this Morison force from the measured response force and arrived at the slamming force response from the plunging breaking waves. We used another approach to arrive at the wave slamming response, namely by low pass filtering the measured force response, Figure 21. We have first filtered the response force, deducted this filtered force from the total force response. The thus obtained high frequency response was then again filtered and deducted from the previous obtained high frequency response. It is this “double filtered” response force that are shown in Figure 21 and again in Figure 22. We have therefore adopted another approach than Wienke and Oumeraci (2005) to arrive at the wave slamming force from the wave slamming response force.
Figure 21. Total force response, filtered force response, high frequency part of the response and wave height at the structure.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-10
-5
0
5
10
15
20
25
30
35
Time, s
Forc
e, N
. Wav
e he
ight
, cm
MF054e200-2. 24000 - 40000. Total force, filtered total, filteredfiltered total, Wave height
TotalFiltred(Total-Filtered)-FlteredfilteredWave
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Figure 22. Filteredfiltered force response and wave at the structure. 3. METHODS OF ANALYSING WAVE SLAMMING FORCES 3.1. General on force identification in structural dynamics We are facing the problem of force identification in structural dynamics, i.e. we measure a short duration and high intensity response. The force measuring system is not “ideal”. For such short duration loads it is almost impossible to build a measuring system stiff enough not to be influenced by the dynamics of the transducers, the structure itself and the support structure. Thus force identification in structural dynamics has become an important issue, e.g. Ewins (2000), Lourens (2012). We will use a method based on the linear spectrum and a linear transfer function, see later. But we will first make some considerations on the degree of freedom of our structure.
3.2.” Single-degree-of freedom” vs. “Multiple-degree-of freedom” system? Our model structure represents a multiple degree of freedom structure. Even if the structure is infenitely stiff, the model structure represents a two-degree-of freedom structure as shown in the following. From Figure 23 it is seen that the response of the top two transducers is much larger than on the two bottom force transducers. This indicates that the structure may experience a translation x1 and a rotation θ as indicated in Figure 24.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-10
-5
0
5
10
15
20
Time, s
Hig
h fre
quen
cy fo
rce,
N. W
ave,
cm
MF054e200-2. 24000 - 40000. Total - filteredfiltered total. Wave
ForceWave
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Figure 23. Total top and bottom response forces.
Figure 24. Translation and rotation of the structure. Positive direction of θ is in the anticlockwise direction. In the sketch θ is thus negative. We assume that the mass of the structure is m, including the added mass, evenly distributed along the height l of the structure. We assume further that the resultant wave slamming force F(t) attacks at a height l1 above bottom level. This impact force gives response forces R1(t) at the top force transducer
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-5
0
5
10
15
20
25
Time, s
Forc
e, N
. Wav
e, c
m
MF054e200-2. 100000 - 140000. Total top force. Total bottom force. Wave.
WaveBottom forceTop force
x1
θ
l1
l
P(t) R1(t)
l/2
C
R2(t)
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and R2(t) at the bottom force transducers. The center of gravity C is assumed to be located at l/2 from the sea bottom.
Figure 25.Translational and rotational motion of a body in the xy-plane. C = center of gravity. Timoshenko and Young (1951) With reference to Figure 25 the principle of angular momentum in plane motion leads to the following general expression for the angular momentum, Timoshenko and Young (1951):
( )ang c c c c cMO m x y y x I θ= − +
(5) where xc = horizontal position of the center of gravity, yc = vertical center of gravity, Ic = moment of inertia around the center of gravity. In our case yc = l/2, y = 0, xc = x1+ 0.5∙l∙θ. Note that in Eq. (5) θ is positive in the anticlockwise direction.
/2 22
0
212
l
cm mlI y dyl
= =∫ (6)
This gives in our case:
1
2 2 2
1 1
( )2
( )2 4 12 2 3
ang c c c c clMO my x I my x I
l l ml ml mlm x x
θ θ θ
θ θ θ
= − − = − + −
= − − − = − −
(7)
or
2' ''
1 1 1 2 1 2
2' ''
1 1 1 1 2 1 1
( )( )( ) ( ) ( )
2 3 2 2 2
( )( ) ( ) 22 3 2 2 2
angMO ml ml l l lx P t l c xl c l R t R tt
ml ml l l lx P t l c x l c l k x l kx
θ θ
θ θ θ
∂= − − = − − + + + −
∂
= − − = − − + + + + −
(8)
when 1 1( ) ( )R t k x lθ= + and 2 1( ) 2R t kx= where k= spring constant (one spring at the upper end and
two springs at the lower end with the same spring constants), 1 'c l x is a ”rectilinear” damping force
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with a “fictious” moment arm l’, and 2 ''c l θ is a “rotational” damping force with a “fictious” moment arm l’’. This again gives
2 2' ''
1 1 1 1 2 1( )( )2 2 3 2 2
ml l ml l lx c l x k x c l k P t lθ θ θ+ − + + + = −
(9)
This is a rather complicated expression to further arrive at the force P(t). One may have to consider a multi-degree of freedom system. We have not considered this further at this stage. However, looking at Figure 24 it is seen that the responses of the bottom force transducers are small at the time points of maximum response of the top force transducer. As a first approximation we may then set x1 = 0 and obtain
2
3angmlMO θ≈ − (10)
Using now the principle of angular motion and equating the rate of change of MOang to the moment of all external forces with respect to the bottom pivoting point, we may write:
22
1 1 1( )3ang
mlMO Pl R l c Pl kl ct
θ θ θ θ∂= − = − + + = − + +
∂ (11)
or
22
13ml c kl Plθ θ θ+ + = (12)
Since we now can setxl
θ = , where x is the deflection at the top transducer, we obtain:
13ml cx x klx Pl
l+ + = (13)
where k = spring constant for the top force transducer and c is a damping coefficient. The un-damped natural period of oscillation for this system is:
2
2 2
/ 32 2 23
on
I ml mTkl kl k
π π π= = = (14)
The moment arm l1 is unknown. But the results of Ros (2011) show that the wave slamming force has its highest intensity at an elevation 0.175 m above still water level when the water depth is 0.33 m. The approximate arm length for the 1:50 scale model, for which the water depth is 0.30 m, is then set to:
10.175 0.1750.30 0.30 0.460.33 0.33
l d d m= + = + = (15)
20
We have however, for simplisity and to test if the analysis procedure gives reasonable results, made the structure into a single degree of freedom structure by adding all the four force transducers into a total response force. This is as far as we understand the same approch and assuption as Wienke and Oumearci (2005) used 3.3. General on the applied analysis method. We will follow a procedure as described by Määtänen,(1979). Prof. Määttänen used this method to resolve ice forces from measured response forces on structures subjected to moving ice. But the method should also be applicable for wave slamming loads. The method is different from the method used by Wienke and Oumeraci (2006) when analyzing wave slamming forces on single piles. The wave slamming response of a truss structure is, as we have seen, more complex than on a single pile. AS far as we know this analysis method has not been used for wave slamming forces on truss structures before. The measured response force f(t) can be expanded into Fourier integral – extension of Fourier series expansion - and in case of forced vibration will be:
1( ) ( ) ( )2
i tFf t H S e dωω ω ω
π
∞
∞
= ∫ (16)
where H (ω) is the frequency response function (FRF) and SF (ω) is the linear spectrum (not power spectrum) of the forcing function. The transfer function in our case is obtained by pluck tests on the structure with an impulse hammer, Figure 26. Normally pluck tests are carried out by plucking the structure at several locations. However, in this case we know from previous tests, Ros (2011), Arntsen et al. (2011), the approximate location of the wave slamming resultant load. Plucking was therefore carried out at and in the vicinity of the location of the resultant force, or approximately 17 cm above still water level (SWL), Figure 23. The impulse hammer measures the impulse force by a force transducer in the tip of the hammer. From the measured impulse force and from the simultaneously measured response forces in the four force transducers, the frequency response function H(ω) is obtained. The Fourier transform for Eq. (5) gives
( ) ( ) ( ) ( )i tF fH S f t e dt Sωω ω ω
∞−
∞
= =∫ (17)
which is the linear spectrum of the measured force f(t). From this SF(ω) can be solved as:
( )
( )( )
fF
SS
Hω
ωω
= (18)
and the Inverse Fourier Transform( IFFT) gives the requested real wave slamming force:
( )1( )2 ( )
f i tSF t e d
Hωω
ωπ ω
∞
∞
= ∫ (19)
Although the above equations look complicated, they can be easily solved by programs in e.g. Matlab. We have used the Matlab environment.
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3.4. Frequency response function (FRF). The frequency response function (FRF) H(ω) is a calibration factor. In our case it has been obtained by using an impulse hammer as shown in Figure 26.
Figure 26. Impulse hammer. Plucking should be made at the location of the force attack. We do not know precisely where wave forces are located. But from Ros (2011), Arntsen et al. (2011), we know, as m,entioned, that the resultant of the wave slamming force is located in the areas marked 1 -6 in Figure 27. We therefore plucked the structure at 7 locations: At each front section corner columns: Approximately 12, 17 and 22 cm above still water level. In addition in the middle of the top plate as indicated in Figure 27. For each plucking point several plucks were made. Points 2 and 5 are located at the approximate level of the maximum wave slamming force, Figure 7, Ros (2011), Arntsen et al. (2011). All pluck tests were carried out when the water level was at the still water level durring the wave test runs. Added mass and still water damping is thus incorporated in the test results.
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Figure 27. Approximate location of the impuls hammer hit points. Points 1,2 and 3 are on the vertical leg closest to the viewer, Figure 9, while points 4, 5 and 6 are on the rear vertical leg. Figure 28 shows the impuls hammer force and the total response of the structure for one particular pluck test. The total response is, as mentioned, the sum of all the four force transducers. Figure 29 shows an expanded time window of the impuls hammer force and the total response of the structure. It is seen from Figures 28 and 29 that there is a high frequency component in the beginning of the time series with a frequency of approximately 300 Hz. We do not know precisely where this high frequency component is coming from. We have in Figure 30 filtered the response signal, but we have not used the filtered signal in our further analysis. The filtered signal reveals a natural period of oscillations of approximately Tn = 0.0225 s or a natural frequency of oscillations of approximately fn = 45 Hz. Figure 29 reveals also that there is a natural period of oscillations, which has a high damping, of approximately 0.0035 s or a frequency of approximately 285 Hz. The FRF is now:
, ( )( )
( )Total hammer
Hammer
SH
Sω
ωω
= (20)
where
, ,( ) ( ) i tTotal hammer Total hammerS f t e dtωω
+∞−
−∞
= ∫ (21)
and
1, 4 2, 5 3, 6
≈ 17 cm
7
Wave direction
23
( ) ( ) i tHammer HammerS f t e dtωω
+∞−
−∞
= ∫ (22)
We have tested the frequency response funcion by applying it to the Totalhammer response, using Eq. (21), where Sf(ω) = STotal,hammer(ω). The result is shown in Figure31. The agreement with the “original” hammer force, Figure 29, is good.
Figure 28. The impuls hammer force and the total response of the structure.
Figure 29. The impuls hammer force and the total response of the structure – expanded time view.
6 6.1 6.2 6.3 6.4 6.5 6.6-50
0
50
100
150
200
Time, s
Tota
l sup
port
forc
e, H
amm
er fo
rce,
N
Mhammer2-1. !16500 - 125000
TotalHammer
6.065 6.07 6.075 6.08 6.085 6.09 6.095 6.1-50
0
50
100
150
200
Time, s
Tota
l sup
port
forc
e, H
amm
er fo
rce,
N
Mhammer2-1. 116500 - 116800
TotalHammer
24
Figure 30. The total response force and the filtered total response force.
Figure 31. The FRF concept used on the “Total” response force.. Figure 32 shows the power spectra of the hammer force and the total response force.
6.04 6.06 6.08 6.1 6.12 6.14 6.16 6.18-60
-40
-20
0
20
40
60
80
100
120
Time, s
Forc
e, N
.
Hammer response and fitered hammer response force. Mhammer2-1. 116500 - 125000.
Total forceFiltered total force
6.065 6.07 6.075 6.08 6.085 6.09 6.095 6.1-50
0
50
100
150
200
Time, s
Ham
mer
forc
e, N
Mhammer2-1. 116500 - 125000
25
Figure 32. Power spectra of hammer force and total force. We will also test the method in another way. Eq. (17) give the linear spectrum of the force response Sf(ω) when H(ω) and SF(ω) are known. We have obtained H(ω) from the hammer load. For a given impulse load, e.g. a half- sinusoidal load, SF (ω) can be calculated. The response force is obtained from the following Inverse Fourier Transform (IFFT):
1( ) ( ) ( )2
i tFF t H S e dωω ω ω
π
∞
∞
= ∫ (23)
Our “system” is damped. The damping of our system is obtained from the time series of the damped oscillations, Figures 28 . The damping factor is obtained through the logarithmic decrement
(24)
Where xi is the amplitude of the i-th oscillation and xi+n is the amplitude of the i+n-th oscillation. The relation between the logarithmic decrement δ and the damping factor ζ is
2
2 21πςδ πςς
= ≈−
, for small damping factors. (25)
or 1
2ς δ
π≈ (26)
0 50 100 150 200 2500
1
2
3
4
5
6
7
8x 104
Frequency, Hz
Rel
ativ
e va
lues
Mhammer2-1. Time points 116000 - 125000. Power spectra
Hammer forceTotal force
1 ln i
i n
xn x
δ+
=
26
We obtain from Figure 26 (the low pass filtered signal) x1 ≈ 76 N and x2 ≈ 36 N, giving logarithmic decrement δ = 0.74 and damping factor ζ = 0.12. The only conditions with damping we have an analytical solution at hand for, is for the response for a step excitation. The response force is then:
2
21 sin( 1
1
nt
R o neF F t
ζω
ζ ως
− = − − +Φ
− (27)
Where Fo = step force, ωn = 2πfn = natural angular frequency of oscillations.
21tan ζζ−
Φ =
Figure 33 shows the response as calculated from Eq. (23) and the analytical solution. Our “filtered” calculations based on Eq. (23) underestimate the response with approximately 10 %. This is not an unreasonable estimate taking into account our many “steps”.
Figure 33. Response for damping factor ζ = 0.12 for a step load of 10 N. Calculations based on Eq. (23) and analytical response.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
2
4
6
8
10
12
14
16
18
20
Time, s
Forc
e, N
Response force, step load, analytical ksi = 0.12. Hammer2-1, 116500 - 125000
LoadResponse based on hammerFiltered response based on hammerAnalytical response
27
3.5. FRF applied to the wave slamming response forces. We have applied the outlined procedure to obtain the wave slamming force for specific response force time series, MF054e200_2 and MF054e265_1, wave frequency 0.54 Hz, wave period T = 1.85 s, as obtained by Miriam Aashamar for her Master thesis, Ashamar (2012). We have further analyzed the time series MF048e280_1 of Miriam Aashamar’s test data, wave frequency 0.48 Hz, wave period T = 2.08 s. It should be noted that the analysis is a simplified analysis based on, as mentioned, the assumption of a single degree of freedom system subjected to a “Total” force. We have, as mentioned, filtered away the low frequency part of the response force as shown in Figures 21 and 22. The low frequency part is assumed to be the Morison type forces or the quasi static wave force. The IFFT of S(ω)/H(ω), Figure 34, for the test series MF054e200_2 has a high frequency component. This high frequency component has been filtered away and we obtain a force time series as shown in Figure 35. This filtering is further discussed in Chapter 4. The time series of Figure 35 resembles the time series shown in Figures 22 and 23, except that the forces are smaller. This indicates that the applied forces are smaller than the response forces. The total force will be the high frequency forces plus the Morison forces. Comparing Figures 35 and 22 it is seen that the ratio between the maximum response and the maximum force is approximately 9.5 N/6,5 N = 1.46. If we assume that the slamming force impulse is triangular in shape, Figure 12 indicates that the ratio between the duration of the impulse and the natural frequency of oscillation is approximately 0.8 or duration time t1 ≈ 0.019 s when T = 0.0225 s, Chapter 3.3. This again means that the duration time is significantly larger than indicated in Chapter 2, t1 ≈ 0.003. This latter duration time is based on wave slamming on a single cylinder. A possible reason for the apparent longer duration time for the truss structure may be that the wave hits at slightly different time points at different parts of the structure. The apparent highest response for all the tests was measured for the test series Mf054e265_1, wave frequency f = 0.54 Hz, wave period T = 1.85s. Figure 36 shows the whole time series of the total response force for this series, while Figure 37 shows a time expansion around t = 4 s. Figures 38 and 39 show response forces and waves, and waves respectively. This response is especially interesting since in this case the wave broke ahead of the structure and surged against the structure illustrated by the small wave crest height at the structure for the largest response, Figures 37 - 39. The deep water wave height for test series Mf054e265-1, Figure 39, is approximately Hdeep = 35 cm, while for test series MF054e200, Figures 19 and 20, the deep water wave height is approximately Hdeep = 25 cm. This indicates that wave slamming forces may occur for a range of deep water wave heights. It is mentioned again that since the waves break the wave measurements at the structure is somewhat uncertain, Figure 37, due to air content.. As the response force is higher for tests MF054e265, when the waves break ahead of the structure, than for test MF054e200, when the waves break more directly on the structure, the surging wave for MF054e265 may give relatively higher forces on the bottom force transducers than for the tests MF054e200. Figures 41 and 42 show that this is the case. We have selected the part of the time series of Figure 36 where the response is highest, or around t = 4 s. Figure 43 show an expanded part of the time series of Figure 38. It is noted that the wave crest height is lower in this case than for the tests series Mf054e200_1, Figure 22. We have treated the time series of Figure 37 as we treated the time series of Figure 22. Figure 43 shows the total force response, filtered force response, high frequency part of the response and wave height, while Figure 44 shows the filteredfiltered force response and wave separately.
28
Finally Figure 45 shows the low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) for the response force, Figure 44, using Mhammer2.1 plucking test results. We have subsequently analyzed data for Mf054e265 for the remaining pluck points, Figures 47 – 50. It appears that the pluck points on one side of the structure, Points 1, 2 1nd 3 gives almost the same maximum force, 25 N. The pluck points 4, 5 and 6 give also the same maximum, force, approximately 20 N. Pluck point 7 give a maximum force of approximately 25 N. Pluck point 7 is though located at a position where we know that the slamming force will not occur. If we compare Figure 45 and Figure 44 it is seen that the ratio between the maximum response and the maximum force is approximately 42N/32.5 N≈ 1.30. If we assume that the slamming force impulse is triangular in shape, Figure 12 indicates that the ratio between the duration of the impulse and the natural frequency of oscillation is approximately 0.7 or duration time t1 ≈ 0.015 s when T = 0.0225 s, Chapter 3.3. This again means that the duration time is significantly larger than indicated in Chapter 2, t1 ≈ 0.003. If we assume that the slamming force impulse is triangular in shape, Figure 12, we are beyond the response ratio for this assumption. Part of the test series Mf048e280 (f=0.48 Hz or T = 2.08 s) was filmed. The film shows clearly that the wave broke ahead of the structure. We have analyzed the wave that gave one of the highest responses for this test series. Figure 52 shows the total response force and the wave measured at the structure, while Figure 53 shows the “total response – filteredfiltered” response. Figure 54 shows the obtained wave slamming force. It is interesting to note again that the wave slamming force is approximately half of the wave slamming force response.
Figure 34. Inverse Fast Fourier Transform of S(ω)/H(ω) for the response force, Figure 22.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-8
-6
-4
-2
0
2
4
6
8
10
12
Time, s
Forc
e, N
MF054e200-2. 24000 - 40000. iFFT S(w)/H(w)
29
Figure 35. Lowpass filtered Inverse Fast Fourier Transform of Figure 34.
Figure 36. The time series were the highest total response force occurred for all tests.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-6
-4
-2
0
2
4
6
8
Time, s
Forc
e, N
MF054e200-2. 24000 - 40000. Filtered IFFT S(w)/H(w)
0 5 10 15 20 25-20
-10
0
10
20
30
40
50
60
70
Time, s
Tota
l res
pons
e fo
rce,
N
Mf054e265-1 0 - 420112. Total response force
30
Figure 37. Time expansion for the highest response, Figure 36.
Figure 38. Total force response for the time series Mf054e265-1 and waves at the structure and in “deep” water.
3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5-10
0
10
20
30
40
50
60
70
Time, s
Forc
e, N
. Wav
e he
ight
, cm
MF054e265-1. 72000 - 86000 Total force. Wave height
Total forceWave
0 2 4 6 8 10 12-20
-10
0
10
20
30
40
50
60
70
Time, s
Tota
l res
pons
e fo
rce,
N. W
ave
heig
ht, c
m
Mf054e265-1 1 - 200000. Total response force. Waves
Force esponseWave at structureWave in "deep" water
31
Figure 39.Time series Mf054e265-1. Waves at the structure and in “deep” water.
Figure 40. Time series Mf054e200-2. Waves at the structure and in “deep” water.
0 2 4 6 8 10 12-20
-15
-10
-5
0
5
10
15
20
25
Time, s
Wav
e he
ight
, cm
Mf054e265-1 1 - 200000. Waves
Wave at structureWave in "deep" water
0 1 2 3 4 5 6 7 8-15
-10
-5
0
5
10
15
20
Time, s
Wav
e he
ight
, cm
Mf054e200-2 1 - 150000. Waves
Wave at structureWave in "deep" water
32
Figure 41. Time series Mf054e200-2. Total response at the top and bottom force transducers.
42. Time series Mf054e265-1. Total response at the top and bottom force transducers.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-5
0
5
10
15
20
25
Time, s
Rsp
onse
forc
e, N
Mf054e200-2 24000 - 40000. Response forces
Total response topTotal response bottom
3.7 3.75 3.8 3.85 3.9 3.95-5
0
5
10
15
20
25
30
35
40
45
Time, s
Rsp
onse
forc
e, N
Mf054e265-1 72000 - 75000. Response forces
Total response topTotal response bottom
33
Figure 43. Time series Mf054e265-1. Total force response, filtered force response, high frequency part of the response and wave height.
Figure 44. Time series Mf054e265-1. Total response - filteredfiltered force response, and wave at structure.
3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5-20
-10
0
10
20
30
40
50
60
70
Time, s
Forc
e, N
. Wav
e he
ight
, cm
MF054e265-1. 72000 - 86000. Total force, filtered total, filteredfiltered total, Wave height
TotalFiltred(Total-Filtered)-FlteredfilteredWave
3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5-20
-10
0
10
20
30
40
50
Time, s
Hig
h fre
quen
cy fo
rce,
N. W
ave,
cm
MF054e265-1. 72000 - 86000. Total - filteredfiltered total. Wave
ForceWave
34
Figure 45. Time series Mf054e265-1. Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force of Figure 44, using hammer plucking Mhammer2-1.
Figure 46 Time series Mf054e265-1 .Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force, Figure 44, using hammer plucking Mhammer5-1.
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-15
-10
-5
0
5
10
15
20
25
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer2-1, Filtered IFFT S(w)/H(w)
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-10
-5
0
5
10
15
20
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer5-1, Filtered IFFT S(w)/H(w)
35
Figure 47. Time series Mf054e265-1 .Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force, Figure 44, using hammer plucking Mhammer1-1.
Figure 48. Time series Mf054e265-1 .Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force, Figure 44, using hammer plucking Mhammer3-1.
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-20
-15
-10
-5
0
5
10
15
20
25
30
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer1-1, Filtered IFFT S(w)/H(w)
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-15
-10
-5
0
5
10
15
20
25
30
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer3-1, Filtered IFFT S(w)/H(w)
36
. Figure 49. Time series Mf054e265-1 .Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force, Figure 44, using hammer plucking Mhammer4-1.
Figure 50. Time series Mf054e265-1 .Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω)
of the response force, Figure 44, using hammer plucking Mhammer6-1.
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-10
-5
0
5
10
15
20
25
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer4-1, Filtered IFFT S(w)/H(w)
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-10
-5
0
5
10
15
20
25
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer6-1, Filtered IFFT S(w)/H(w)
37
Figure 51 Time series Mf054e265-1 .Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force, Figure 44, using hammer plucking Mhammer7-1.
Figure 52. Time series Mf048e280-1. Total force response and wave height at the structure .
3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2-15
-10
-5
0
5
10
15
20
25
Time, s
Forc
e, N
Mf054e265-1. 72000-86000. Mhammer7-2, Filtered IFFT S(w)/H(w)
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3-10
0
10
20
30
40
50
60
Time, s
Forc
e, N
. Wav
e he
ight
, cm
MF048e280-1. 42000-56000. Total force. Wave height
Total forceWave
38
Figure 53. Time series Mf048e280-1. Total response - filteredfiltered force response, and wave at structure.
Figure 54.Time series Mf048e289-1. Low pass filtered Inverse Fast Fourier Transform of S(ω)/H(ω) of the response force, Figure 53,using hammer plucking Mhammer5-1. MF048e280.
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3-15
-10
-5
0
5
10
15
20
25
Time, s
Hig
h fre
quen
cy fo
rce,
N. W
ave,
cm
MF048e280-1. 42000 - 56000. Total - filteredfiltered total. Wave
ForceWave
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-8
-6
-4
-2
0
2
4
6
8
10
12
Time, s
Forc
e, N
MF048e280-1. 42000 - 56000. Mhammer5-1 Filtered IFFT S(w)/H(w)
39
3.6. Possible reason(s) for the high forces when the waves broke ahead of the truss structure. It was mentioned that waves for the time series Mf054e265-1 also broke ahead of the truss structure. Figure 55 shows a snap-shot of the breaking wave for this test series. It is seen that the wave break ahead of the structure. The high forces are then not induced by the wave breaking as an almost vertical wall against the structure. Endresen and Tørum (1992) measured extremely high vertical forces on an elevated pipeline through the surf zone. The pipeline was orientated normal and almost normal to the coastline. No water wave kinematics were measured, but it was speculated that the high vertical forces were due to large eddies with high water particle accelerations, causing high inertia forces when the wave broke as plunging breaking..
Figure 55. Snapshot, Mf054265-1. Ting and Kirby (1994) measured wave kinematics and turbulence for surging and plunging breaking waves on a bottom slope of 1:35. They run their tests for 102 consecutive waves and analyzed the mean water particle velocity and the turbulent velocities by the phase averaging method. Figures 56 and 57 show some of their results for the plunging waves. In this case the wave height and the water depth at the breaking point were Hb = 0.128 m and hb = 0.156 m respectively. The wave period was 5.0 s. The measurements were taken at location approximately 2.6 m behind the wave breaking point (after the wave had broken) at a still water depth of h = 0.072m.
We may consider '2u = urms as the root-mean-square value of the turbulence velocity. The maximum turbulence velocity is then approximately umax = 2√2 urms. Using the results of Figure 57 we find that u’max is approximately u’max = 0.66 m/s. Figures 58 and 59 show raw data from one of the tests by Ting and Kirby (1994). These data are taken at a location approximately 1 m behind the wave breaking point and 5 cm below the still water line. The maximum turbulence velocity u’max of Figure 59 is inferred as approximately u’max = 0.7 m/s. The raw data apparently confirm the above findings of the maximum u’max = 0.66 m/s. The maximum mean water particle velocity, û, is obtained from Figure 56 as approximately 0.28 m/s. The turbulence may thus have increased the drag force considerably. Since the “period” of the turbulent variation is short, the water particle accelerations may become large and cause large inertia forces. This is currently speculation why the forces of test run Mf054e265 – 1 becomes so large for waves breaking ahead of the structure, but this speculation should be pursued further.
40
Figure 56. Results of Ting and Kirby (1994).
Figure 57. Results of Ting and Kirby (1994).
41
Figure 58. Horizontal particle velocities. From raw data kindly sent by professor Kirby, University of Delaware, USA. Wave period T = 5 s., still water depth d = 12.6 cm, measurements 5 cm below the still water line.
Figure 59. . Horizontal particle velocities. From raw data kindly sent by professor Kirby, University of Delaware, USA. Expanded from the time series of Figure 58.
0 20 40 60 80 100 120 140 160 180 200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, s
Vel
ocity
, m/s
Kirby, IV-level -0.05 m,n40500c0
139 140 141 142 143 144 145 146 147-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, s
Vel
ocity
, m/s
Kirby, IV-level -0.05 m,n40500c0
42
4. CONSIDERATIONS ON THE ACCURACY OF THE APPLIED ANALYSIS METHOD The applied analysis method for obtaining the wave slamming forces is one of more. The question is then how good is the method? We do not have a firm answer to this question. But we will make some considerations on this issue. Figure 60 shows the power spectrum of the total high frequency response force for test MF054e200-1, Figures 21 and 22. It is seen that there are many frequencies present with a high peak at approximately 45 Hz, the natural period of oscillation. The high peak value at approximately 10 Hz and the smaller peaks in-between reflects probably the not-so-clean time signal, Figure 21.
Figure 60. MF054e200-2. Power spectrum of total response force.
Figure 61 shows the linear transfer function squared and Figure 62 show the same on semi log scale. We have on Figure 63 plotted the power spectrum of S(ω)/H(ω). The peaks of the spectrum occur at different frequencies than for the response force, Figure 60. This is apparently due to the values of the transfer functions at different frequencies. There is some power at approximately 200 Hz in the power spectrum, Figure 60. On the other hand the transfer function, Figure 61, has a low value around this frequency. Hence the value of IFFT becomes very high for the frequency slightly below 200 Hz. One may thus argue that the high value of the power spectrum for frequencies approximately 200 Hz, and also some of the other frequencies, may be due to the low values of the transfer function and may thus not be physically correct. It thus seems justified to low-pass filter the IFFT of S(ω)/H(ω), Figure 64. Figure 65 shows the low pass filtered time series of Figure 64. We may then take the power spectrum of the IFFT series of Figure 64 (which is the same as shown in Figure 63) and multiply it with H(ω), SS(ω)*H(ω). The resulting spectrum, which should be the response spectrum, is shown in Figure 66.. The agreement Between Figure 66 and Figure 60 is reasonably good, as one would expect.
0 50 100 150 200 2500
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Frequency, Hz
Rel
ativ
e va
lues
MF054e200-2. Timepoints 24000 -40000. Power spectrum
43
We have applied the same procedure on the test MF054e200-2, using hammer pluck Mhammer7-2. The results are shown in Figures 68 – 73. The approximate same results are obtained as for pluck Mhammer2-1, Figure 65 vs. Figure 73. One may thus believe that the test procedure and the analysis method give reasonable results.
Figure 61. Hammer2-1. Transfer function squared
0 50 100 150 200 2500
2
4
6
8
10
12
14
Frequency, Hz.
Tran
sfer
func
tion.
(-)
Transfer function. Hammer2-1, 116500 - 125000
44
Figure 62. Hammer2-1. Linear transfer function squared. Semi log scale.
Figure 63. M054e200-2. Power spectrum of IFFT: SS(ω)= (S(ω)/H(ω)), which should be the power
spectrum of the applied force.
0 50 100 150 200 25010-3
10-2
10-1
100
101
102
Frequency, Hz
Tran
sfer
func
tion
Mhammer2-1. 116500 - 121000. Transfer function
0 50 100 150 200 2500
1000
2000
3000
4000
5000
6000
Frequency, Hz
Rel
ativ
e va
lues
MF054e200-2. Timepoints 24000 - 30000. Power spectrum of IFFT(S(w)/H(W)
45
Figure 64. IFFT of S(ω)/H(ω) of Figure 63, which should be the applied force.
Figure 65. MF054e200-2. Filtered force of Figure 64.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-10
-5
0
5
10
15
Time, s
Forc
e, N
MF054e200-2. 24000 - 40000. Mhammer2-1. IFFT S(w)/H(w)
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-6
-4
-2
0
2
4
6
8
Time, s
Forc
e, N
MF054e200-2. 24000-40000. Hammer2-1. Filtered IFFT SF=Sf(w)/H(w)
46
Figure 66. MF054e200-2. Power spectrum of the force spectrum, Figure 64, multiplied with the transfer function H(ω): SS(w).*H(w). Compare with Figure 60.
Figure 67. Hammer at point 7 and total response.
0 50 100 150 200 2500
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Frequency, Hz
Rel
ativ
e va
lues
MF054e200-2. Timepoints 24000 - 40000. Power spectrum of IFFT(SS(w).*H(W)) = S(w)
4.54 4.56 4.58 4.6 4.62 4.64 4.66 4.68 4.7-50
0
50
100
150
200
250
300
350
400
Time, s
Forc
e,N
Mhammer7-2. 87000 - 95000
TotalHammer
47
Figure 68 Mhammer2-2. Spectra for hammer force and total response for hammer in Point 7.
Figure 69. Mhammer7-2. Transfer function squared.
0 50 100 150 200 2500
1
2
3
4
5
6
7
8
9x 104
Frequency, Hz
Rel
ativ
e va
lues
Mhammer7-2. Time points 87000 - 95000. Power spectra
Hammer forceTotal force
0 50 100 150 200 25010-3
10-2
10-1
100
101
102
Frequency, Hz
Tran
sfer
func
tion
Mhammer7-2. 87000 - 95000. Transfer function squared
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Figure 70. MF054e200-2.Mhammer7-2.Power spectrum of applied force.
Figure 71. MF054e200-2. Hammer7-2.Power spectrum of SS(ω)*H(ω).
0 50 100 150 200 2500
500
1000
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2500
3000
3500
4000
4500
5000
Frequency, Hz
Rel
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e va
lues
MF054e200-2. Timepoints 24000 - 30000. Mhammer7-2 Power spectrum of IFFT(S(w)/H(W)) =SS(w)
0 50 100 150 200 2500
500
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1500
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2500
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3500
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Frequency, Hz
Rel
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MF054e200-2. Timepoints 24000 - 40000. Mhammer7-2. Power spectrum of IFFT(SS(w).*H(W)) = S(w
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Figure 72. MF054e200-2, Mhammer7-2. Unfiltered
Figure 73. MF054e200-2.Mhammer7-2. Filtered force of Figure 72. Compare with Figure 65.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-25
-20
-15
-10
-5
0
5
10
15
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25
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Forc
e, N
MF054e200-2. 24000 - 40000. Mhammer7-2. IFFT S(w)/H(w)
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1-6
-4
-2
0
2
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8
Time, s
Forc
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MF054e200-2. 24000-40000. Hammer7-1. Filtered IFFT SF=Sf(w)/H(w)
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5. WAVE SLAMMING FORCE CALCULATIONS NEK IEC 61400-3 (2009) gives some guidance to calculate wave slamming forces. It is made a distinction between slap and slam forces. According to their definition: “Wave slap and slam forces occur when a member is suddenly immersed in water. Wave slam occurs when an approximate horizontal member is engulfed by a rising water surface as a wave passes. The highest slamming forces occur for members at mean water level and the slam force is close to the vertical. Wave slap is associated with breaking waves and affects members of any inclination but in the plane perpendicular to the wave direction. The highest forces occur above the mean water level. In both instances, the force is applied impulsively and dynamic response of the structure is therefore of importance. The actual hydrodynamics of slamming are highly complex. The member geometry, the precise shape of the water surface and the presence of entrained air at the instant of the slam occurring have a significant effect on the slam force”. NEK IEC 61400-3 (2009) refers to the method developed by Wienke (2001), which is also referred to in Wienke and Oumeraci (2005), and recommends Cs = 2π and λ = 0.5. We have made some calculations of the forces on our model structure, following the recommendations given in NEK IEC 61400-3 (2009). We have assumed the crest elevation to be ηcrest = 0.17 m as seen on Figure 20. Then ληcrest = 0.085 m. The wave slamming force is calculated according to the following procedure.
2 22 0.5 0.5s w s vert b w s br br bF C D C C D l Cρ λη ρ= ⋅ + (12) Where ρw = specific density of water = 1000 kg/m3, Dvert = diameter of the vertical columns = 0.016 m, Dbr = diameter of the bracings = 0.012 m, Cb = celerity of the wave at breaking = 2.1 m/s, lbr = length of bracings within the hit area ≈ 0.38 m. The calculated force is thus:
2 22 0.5 1000 2 0.016 2.1 0.5 0.085 0.5 1000 2 0.016 0.38 2.1 18 84.2 102.2sF Nπ π= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ = + = The calculated wave slamming force is thus much larger than the measured maximum force, 102.2 N vs. approximately 20 - 25N, Figures 47 and 49. It should be noted that the bracings contrubutes most to the calculated slamming forces com pared to the vertical legs, 84.2 N vs. 18 N. Admittedly there has been no investigations on wave slamming forces on cylinders aligned similar to the bracings in our truss structure with respect to wave directions. Other possible reasons for the apparent discrepancy is also discussed in Chaper 6. It should also be noted that Hildebrandt and Schlurmann (2012) obtained smaller slamming forces than Wienke and Oumeraci (2005), but not all details of Hildebrandt and Schlurmann (2012) is available yet (Hildebrandt PhD thesis)..
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Figure 75. Platform model with the wave slamming hit area. Wave direction into the paper plane. Measures in cm.
ληcrest = 8.5 cm
ηcrest = 17 cm
Wave slamming force hit area
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6. PROBABILITY OF OCCURANCE OF PLUNGING WAVES. Waves break as spilling breaker, plunging breaker, collapsing breakers and surging breaker. Fig. 76 indicates different breaking forms for regular waves on a uniformly sloping bottom. The breaking form is to a large extent governed by the Iribarren number or the surf similarity parameter, ξ0, defined as:
00
2
tan2 HgT
αξπ
= , where H0 = deep water wave height and T = wave period.
If we consider the wave period for design purposes in the North Sea, say T = 14 s, generally speaking spilling breakers occurs for gently sloping bottoms, while plunging breakers occurs more easily on steeper sloping bottoms. In our tests we have used a rather steep sloping bottom, 1:10, in order to be sure that we obtained plunging breaking waves. It is known that for a wind turbine farm on the Thornton bank outside the Belgian coast the bottom slope on the outer edge is very steep and plunging breaking waves were specified for design considerations.
Fig. 76. Types of wave breaking on impermeable slopes and related ξ0-values. Not much work has been done on the probability of occurrence of plunging breaking waves and more has to be done. However, Reedijk et al. (2009) looked at the probability of plunging breaking dependent on the surf similarity parameter and a breaker parameter B defined as:
1/4 1/4 5/4,0 0 ,0 0( / )( / )s sB H h h L H L h−= =
Where Hs,0 is the significant wave height in deep water, h is the water depth, L0 is the
deep water wave length, 2
0 2gTLπ
= , where g is the acceleration due to gravity and L0 is
the deep water wave length. Aashamar (2012) carried out also some tests with irregular waves. Some results are shown in Figure ? The given waves are those measured in “deep” water or at h = 0.90 m.
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It is seen that the maximum measured responses are comparable with the responses measured with regular waves.
Figure 76 Total force response for a test with irregular waves and the wave height analysis (lower part). Aashamar (2012). Figure 78 shows the results of Reedijk et al (2009). The result obtained by Aashamar for the conditions of Figure ? is also plotted in the figure. It is seen that the probability of having plunging breaking is large, which is also inferred from Figure ?
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Figure 79. Probability of occurrence of plunging breaking waves. The lower bound and upper bound signifies the border line between the surging breaking and the plunging breaking, and the plunging breaking and the surging breaking respectively. The percentages give the percentage of plunging breaking waves for different parameters ξ0 and B. Reedijk et al. (2009). The cross is the results of the test series shown in Figure 76, Aashamar (2012).
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7. DISCUSSION OF THE RESULTS As Figure 25 shows there is a significant variation of the response forces from wave to wave, similarly as Ros (2011) also found, Figure 8, for the local force meters on a single pile. It is interesting to note that the largest force response occurred when the wave broke some distance ahead of the structure. This was also found by Sawaragi and Nochino (1984) for a single vertical cylinder. After breaking the wave in our case surged against the structure and imposed forces with a slamming character in both the top and bottom transducers, Figure 32, while for a case when the wave plunged directly on the structure there were indications of less slamming on the bottom transducers, Figure 31. May be we have to reconsider the concept of Figure 1 for total forces/responses? Previously the duration time of the slamming force has been derived partly from the idealized conditions when a horizontal falling cylinder is hitting still water. There are indications from our analysis that the duration time in our case is larger than obtained from this ideal approach. One possible reason for this may be because the wave may hit different parts of the truss structure at slightly different time points. Most of the previous tests on breaking wave slamming forces has been on vertical cylinders where the ratio cylinder diameter/wave height, D/H, has been larger than in our case. Apelt and Piorewicz (1987) found that the maximum slamming force occurred when D/H was approximately D/H = 2.0. Their investigation covered D/H as low as approximately 0.5. For D/H = 0.5 the wave slamming force was approximately 40 – 50% of the maximum slamming force, depending to some extent on the wave steepness. In the Wienke and Oumeraci case, Wienke and Oumeraci (2005) the minimum D/H value was approximately 0.7 m/2.0 m = 0.35, while in our case the minimum D/H is approximately 1.6cm/24 cm = 0.066. Extrapolations from Apelt and Pierowics indicate that it is expected to get significantly lower wave slamming forces on our truss structure members than on the mono-pile tested by Wienke and Oumeraci (2005). Other factors that may influence the results are scale effects. There is significant air entrained in the water during the breaking process and which may influence the results differently in the model and in reality. To our knowledge the applied analysis method has not been used for wave slamming forces on truss structures before. Although this has been a simplified analysis, the used method is promising. Further analysis should be made by using the other hammer plucking tests and look at the individual response forces for each force transducer. May be one has consider the system as a “Multi degree system”. The calculations we have carried out on the wave slamming forces on the truss structure we have tested shows that the calculated forces according to the NEK IEC 61400-3 (2009) standard are much larger than we have estimated from our analysis. As we have discussed there may be different reasons for that: 1. Lack of simultaneous wave actions on different parts of the front section of the structure? 2. Small diameters in relation to wave height? 3. Scale effects? 4. Air entrainment? 5. Analysis methods? 6. We may not have the most unfavorable form of the wave when it hits the structure? (more vertical, jfr. Wienke and oUmeraci (2005) 7. Others? Slamming forces is supposed to occur on the vertical legs as well as on the bracings of a truss structure. It is thus a challenging task to resolve the slamming forces on the individual members of the
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truss structure. Large scale tests have been planned Large Wave Chanel in Hamburg, (scale 1:8) of the same structure as we have tested in scale 1:50. During these tests it is planned to measure wave slamming forces locally on vertical leg and on some bracings in the expected breaking wave hit area, in addition to the total wave forces on the structure. Figures 80 – 82 indicate the force measurement instrumentation on this structure. For a Master thesis in the spring of 2013 we have planned to carry out some tests to explore the simultaneous action on different parts of the truss structure: 1. Measuring simultaneously the forces on two vertical cylinders placed parallel to the wave crest with a spacing between them corresponding to the distance between the two front vertical legs. 2. Measuring the forces on a section corresponding to the front section of the truss structure. 3. Measuring forces on a section corresponding to a side section of the truss structure.
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Figure 80. Wave direction into the paper plane. Local force measurements and forces on bracings (shadowed). . ηmax is approximate maximum expected crest height. Ru,max is approximate maximum expected runup on pile. The original plan to measure the bracing forces on the front section is shown in Figure A4, but based on our experience with the 1:50 scale model (see front page) we recommend this bracing force measuement sheme in the front section
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Figure 81. Local force measurements on bracings (shadowed). . ηmax is approximate maximum expected crest height. Ru,max is approximate maximum expected runup on pile. The original plan to measure the bracing forces on the front section is shown in Figure A4, but based on our experience with the 1:50 scale model (see front page) we recommend this bracing force measuement sheme in the front section.
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Figure 82. Force gtransducers at top and bottom to measure total force. Bracing forces to be measured on the shadowed bracings. ηmax = approximate maximum expected crest height.
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ACKNOWLEDGEMENT The writer acknowledges the many helpful discussions with professor Mauri Määttänen, who made him aware of the applied analysis method. The writer also appreciates the discussions with and help from PhD student Torodd Nord. The writer acknowledges the co-operation with Master student Miriam Aashamar, who he supervised during her Master thesis work. All the data that have been used in this note was taken by her. She has also analyzed her data by a Matlab computer program developed by the writer. The writer has elaborated somewhat further on the analysis of Miriam Aashamar’s data. REFERENCES Aashamar, M. (2012): Wave slamming forces on truss support structures for wind turbines. Master thesis, Norwegian University of Science and Technology (NTNU), Department of Civil and Transport Engineering, June 2012. Apelt, C.J. and Piorewicz, J. (1987): Laboratory studies of breaking wave forces acting on vertical cylinders in shallow water. Coastal Engineering, 11, pp 263 – 282, 1987. Arntsen, Ø. Ros, X. and Tørum, A. (2011): Impact forces on vertical pile from plunging breaking waves. Proc.
International Conference on “Coastal Structures, Japan, 2011. Endresen, H.K and Tørum, A. (1992): Wave force son a pipeline through the surf zone. Coastal Engineering, 18
(1992) 267 – 281, Elsevier. Ewins, D.J. (2000): Modal testing; theory, practice and application. Second edition. Research Studies Press LTD, Baldock, Hertfordshire, England. Goda, Y., Haranaka, S., and Kitahata, M., (1966): Study of impulsive breaking wave forces on piles. Report of
Port and Harbor Research Institute, Japan, Vol. 5. No. 6, pp. 1–30 (in Japanese). Concept also in English language in Watanabe, A. and Horikawa, K. (1974): Breaking wave forces on large diameter cell. Proc.14th Intern. Conf. on Coastal Eng. pp 1741–1760.
Hildebrandt, A. and Schlurmann (2012): Breaking wave kinematics, local pressures, and forces on a tripod structure. Abstract to the 33rd Intenational Conference on Coastal Engineering, Santander, Spain, July 2012. Lourens, E.-M (2012): Force identification in structural dynamics. PhD thesis, Katholieke Universiteit Leuven, Faculty of Engineering, Kastelpark Arenberg 40, B-3001 Leuven, Belgium. NEK IEC 61400-3 (2009): Norwegian electrotechnical publication. Wind turbines, Part 3: Design requirements
for offshore wind turbines. Norsk elektronisk komite/Norsk nasjonalkomite for International Electrotechnical Commision, IEC.
Määttänen, M. (1979): Laboratory tests for dynamic ice-structure interaction. Proceedings “Port and Ocean Engineering under Arctic Conditions” (POAC), Norwegian Institute of technology, 1979. Reedijk, J.S., Muttray, M. and Bergmann, H. (2009): Risk awareness – key to sustainable design approach for breakwater armouring. Proceedings of the conference “Coasts, marine Structures and bfrakwaters. Edinburgh, UK, September 2009. Ros,X. (2011): Impact forces from plunging breaking waves on a vertical pile. Master thesis, Technical
University of Catalonia, Barcelona, Spain, carried out at the Department of Civil and Transport Engineering, NTNU, Trondheim, Norway, under the Erasmus Socrates student exchange program. In preparation (December 2010).
Sawaragi, T. and Nochino, M. (1984): Impact forces of nearly breaking waves on a vertical circular cylinder. Coastal Engineering in Japan, Vol. 27, pp 249–263.
Tanimoto, K., Takahashi, S., Kaneko, T. and Shiota, K., (1986): Impulsive breaking wave forces on an inclined pile exerted by random waves. Proc. 20th Intern. Conf. on Coastal Eng. pp 2288–2302.
Ting, F.C.K. and Kirby, J.T. (1994): Observation of undertow and trubulence in a laboratory surf zone. Coastal Engineering, 24, 1994 pp. 51 – 80.
Wienke, J. (2001): Druckschlagbelastung auf slanke zylindrische Bauwerke durch brechende Wellen – theoretishe und grossmasstäbliche Laborumuntrsuchungen – PhD thesis, TU Brauinschweig,, Germany.
Wienke, J. and Oumeraci, H., (2005): Breaking wave impact force on vertical and inclined slender pile-theoretical and large-scale model investigations. Coastal Engineering 52 (2005) pp 435–462.
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