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Identifying Wave Loads During Random Seas Using Structural Response
Michael Vigsø*, Rune Brincker** and Christos T. Georgakis*
*Department of Engineering, Aarhus University, Denmark.
**Department of Civil Engineering, Technical University of Denmark, Kgs. Lyngby, Denmark
ABSTRACT
Structural monitoring is increasingly becoming everyday business in
the offshore industry. The monitoring may target the strain estimation
or focus on tracking the changes in the dynamic properties of the
structure in order to predict damages at remote / or possibly subsea
locations. This paper will show that by monitoring the structural
response, it is also possible to indirectly estimate the wave loading
acting on the system. This information can be used to increase
confidence in the load probability models for the structural design or
aid the health monitoring procedure. During ambient vibration, the
principles of operational modal analysis (OMA) are applied to harvest
the dynamic properties of the structure. Successively, a dynamic
model is formulated and used to calculate the loading from a random
sea state using the response of the structure. A laboratory experiment
is conducted in a wave flume at LASIF, Marseille, France, where a
scaled offshore model is equipped with accelerometers to monitor the
structural response during a random sea. The study shows that it is
possible to use the structure as a dynamic load cell and monitor the
loads occurring in actual conditions. Both the short time variations
and the load spectra can be computed successfully using the structural
response.
KEYWORDS: Wave Loading; Indirect Measurements; Opera-
tional Modal Analysis; Offshore Structures
INTRODUCTION
In the field of offshore structures, an increase is seen in the subject of
monitoring. Recently, TOTAL announced that as for the redevelop-
ment of the Tyra field, the platform Tyra East will be equipped with no
less than 100000 sensors (Beck, 2018). Most of these will, of course,
target the production processes, but the monitoring scope will also in-
clude the structural performance. The aim of structural monitoring
may be plentiful, for instance with regards to operational limitations
such as heading, static deformation or vibration level. The vibration
pattern can be used for health diagnostics, and since offshore structures
are prone to fatigue damages, monitoring their well-being is essential
for ensuring safety and reliability.
Due to the nature of offshore structures, it is common to monitor the
surrounding sea state, but more interesting is the wave loading itself
rather than the surface elevation. The wave loading depends on the ge-
ometry and surface properties of the structure, but also on its dynamics.
Therefore, it is challenging to obtain the load measurements directly
and hence we need to rely on indirect methods. By monitoring the
response of a structure, it is, however, possible to identify the loading
during operational conditions with the actual boundary conditions.
Indirect (wave) load identification has been used in ship design, but
it is becoming more common in the field of fixed offshore structures
as well. Within industry practice, wind turbines or oil and gas instal-
lations are exploited for this. For example the "digital twin" concept
introduced by Ramboll Oil and Gas as part of their monitoring pro-
gramme (Tygesen et al., 2018a), (Tygesen et al., 2018b), (Perisic et al.,
2014a) and (Perisic and Tygesen, 2014b). Within academia, early stud-
ies (Jensen et al., 1992) suggest the feasibility of wave load estimates
using the structural response on a simulated case study. Later studies
(Noppe et al., 2016; Fallais et al., 2016; Petersen et al., 2019; Maes
et al., 2018; Vigsø et al., 2018a) demonstrate how wave loads can be
estimated using existing load identification techniques. The studies in-
clude both numerical and experimental work and aim to quantify loads
from a variety of wave types, including breaking waves.
In this paper, we will add to the study of indirect methods for wave
load identification by monitoring a scaled model in a wave flume. The
aim of this study will be to estimate the load variations during random
seas.
The mathematical notation used is denoting matrices by a double un-
derline and vectors by a single underline.
THEORY
For the load identification process, we shall rely on the time domain
technique using the modally reduced model. The method is adopted
from (Lourens and Fallais, 2019) with reference to (Lourens et al.,
2012). The principle is that a modal analysis is performed and modal
parameters are obtained for the structure at in-place conditions. The
modal parameters are then fed into a recursive modal description to-
gether with response measurements from the structure, and by using
the means of a linear Kalman filter, an estimate of the wave loading is
obtained in real-time.
Published in:
Procedings of the International Ocean and Polar Engineering Conference, ISOPE
Publishing date: 2019
The performance of the method is sensitive to assumptions regarding
the spatial distribution of the load and hence a wave gauge is used to
fix the loading to the wetted area of the structure during the random
sea.
The algorithm is recapped here in a condensed form suited for the need
of this study. We start out by defining the state modal coordinates:
x(t) = "Φ0
0Φ#ζ(t)(1)
Here ζ(t)is the state vector in modal coordinates while x(t)is the state
vector in physical coordinates. Φcontains the mode shapes, expanded,
mass normalized and arranged in columns. We arrange the natural
frequencies ωiand the modal damping ξiin diagonal matrices such
that: 1
Ω=
ω10 0
0...0
0 0 ωn
,and Γ=
2ξ1ω10 0
0...0
0 0 2ξnωn
(2)
For the classic continuous state space in modal coordinates we have:
˙
ζ(t) = "0I
−Ω2−Γ#
| {z }
Ac
ζ(t) + "0
ΦTSp#
| {z }
Bc
p(t)(3)
Here, we assume that the hydrodynamic added mass and damping are
embedded in the modal parameters and thus do not appear directly in
the state equation. Another important aspect of this formulation is the
separation of the load into the spatial distribution Spand the scaling
p(t). Note that the loading may be composed of multiple sources and
thus the spatial distribution will be a matrix with load sources arranged
in columns, and the scaling will be a corresponding vector. For this
study, however, we assume that only one load source is present, i.e.
the load caused by the waves. The subscript cindicates that the state
matrices are on continuous form.
Since the spatial distribution is not constant during the wave loading,
but instead will relate to the water runup on the pile, we will adjust the
spatial distribution accordingly using the wave gauge measurements.
The position of the wave gauge may be seen in Fig. 2. Consequently,
the modal load is adjusted as:
q(t) = ΦTSpp(t)→q(t) = ΦTSp(η(t)) p(t)(4)
Here, η(t)is the water surface elevation near the structure. The prin-
ciple is sketched in Fig. 1. The distribution of the load is derived as
an inertia dominated load from a linear wave, given a peak frequency
and significant wave height matching the wave spectrum. The distri-
bution is then (with similarity to Wheeler stretching (Wheeler, 1969))
stretched such that it follows the surface elevation as shown in Fig. 1.
Augmented state
Now, since we wish to obtain an estimate on the input rather than the
response, we expand the state vector to include the load (referred to as
augmented state vector). That is:
ζa(t) = "ζ(t)
p(t)#(5)
1This formulation is based on proportionally damped systems and thus might be vio-
lated by the hydrodynamic contribution.
Sp (t)
a)
h (t)Sp (t + Dt)
b)
h (t + Dt)
Fig. 1. Principle of time varying load distribution. The spatial
distribution Sp(t)is stretched by the information from the
wave gauge η(t).
In Eq. (5) p(t)shall be interpreted as the load scalar function. The
subscript aindicates augmented state.
We assume that the augmented state equation in discrete form can be
written for time step k+1 as:
ζa(k+1) = Aaζa(k) + w(k)
γ(k)(6)
Here, w(k)and γ(k)are the noise processes which account for dis-
crepancies in the relation (more about these later). The discrete state
transition matrix Aais written as:
Aa="A B
0I#(7)
where
A=eAcdt ,and B=A−IAc
−1Bc(8)
Now we have defined the augmented state equation in discrete form
and hence we can implement the Kalman filter for a real-time input
estimate.
Kalman filter
We assume that the load scalar p(k)remains constant during the time
steps and that variations are caused only by a stochastic process also
known as a random walk. Hence:
p(k+1) = p(k) + γ(k)(9)
Here, γis assumed to be Gaussian.
Since the resolution of the mode shapes is greater than the number
of sensors, we must define the selection matrix. In general, we wish
to link the measurement z(k)to the system state ζ(k). Since only
accelerations are measured, the state observation equation reduces to
(in discrete form):
z(k) = h−SaΦ Ω2−SaΦ Γ SaΦ ΦTSp(η(k))iζa(k) + v(k)
(10)
The selection matrix Sais a matrix that picks the degrees of freedom
(DOFs) which coincide with the sensors. For example, sensor 1 coin-
cides with DOF 8 and sensor 2 with DOF 7. This yields:
Sa=
00000001. . .
00000010. . .
.
.
....
nsensors×nDOF
(11)
An important ingredient for the Kalman filter algorithm is the error
covariance matrices (noise models). In lack of information, we often
assume that the noise processes are Gaussian and mutually uncorre-
lated. Hence for the augmented error covariance matrix (with single
loading) we get:
Ew(k)
γ(k)wT(k)γ(k)="Q0
0S#δkl (12)
Here, δkl is the Kronecker delta to ensure that the covariance matrix
remains diagonal. We will assume a constant value in the system error
covariance matrix Q.
In Eq. (9) we have defined the load as a Gaussian random walk. In
order to capture any sharp variation in the load, the associated error
covariance Smust be increased. Several power of magnitude is needed
compared to the values in Qand R.
The measurement noise is also assumed to provide an error covariance
matrix with constant values on the diagonal. That is:
Ev(k)vT(k)=Rδkl (13)
The remaining part of the algorithm follows the general linear Kalman
filter, but with an update in state matrices for each time step given the
wave gauge measurements. (Kalman, 1960)
EXPERIMENT
An experiment campaign was conducted at the wave flume facility at
LASIF, Marseille University in France, 2018. The laboratory frame-
work covers a bottom-fixed mono-pile structure subjected to wave
loading. Although the model does not descend from any physical
(full scale) counterpart, it is roughly scalable as 1/100 given the con-
ditions in the North Sea. The model with key dimensions is shown in
Fig. 2. The pile is made from hollow sections leaving the inside dry
at all times. The sections are bolted together through internal flanges.
The model is made from plexiglass and equipped with 16 uniaxial ac-
celerometers (Brüel & Kjær type 4508-B 100mV/g). Eight of these are
facing the incoming waves (red), and eight are facing the transverse
direction (blue) as indicated in Fig. 2 b). A wave gauge is installed
to monitor the surface elevation near the structure. It is positioned at
the front of the pile with a transverse offset of 200 mm from the pile
centre. The model is attached to an ATI load cell which captures the
global loading on the system including dynamic amplification. This is
used for verification purposes only. The dynamic characteristics may
be seen in Table 1.
A unidirectional random sea is released towards the structure from the
front, and the corresponding vibrations are captured by the sensors
along with the surface elevation of the passing waves.
Wave Spectrum
The waves are created as a series of random waves generated from a
JONSWAP (Joint North Sea Wave Project) wave spectrum. A sample
of the wave train is shown in Fig. 3 along with the spectrum. The
structure is tested for 20 minutes at a sampling rate of 1024 Hz. The
spectrum is made using the Welch averaging technique with segments
of 32 s and a 50 % overlap (Welch, 1967).
Load cell
Max water level
Accelerometer
200
400
150
3
5
200
typ
900
385
a) b) c)
50
Wave gauge
AFT
FWD
STB
PORT
1
2
6
8
10
14
16
12
3
5
7
9
4
15
11
15
13
13
4
Fig. 2. Experimental setup. a) shows the side view cross section including the overall dimensions of the model, b) indicates the sensor position and
direction, c) shows a picture from the experiment. Dimensions are in mm. The figure is taken from (Vigsø et al., 2018b).
0 0.5 1 1.5 2 2.5 3
Frequency [Hz]
0
1
2
3
4
5
6
Wave Spectrum S(f) [m2/Hz]
Peak Period Tp =1.5 s
Significant wave height Hs = 6.4 cm
520 522 524 526 528 530
Time [s]
-4
-2
0
2
4
Surface elevation [cm]
a) b)
Fig. 3. Random sea state: a) shows the wave spectrum of the random waves measured from a wave gauge in close proximity to the pile. b) shows a
sample of the unfiltered wave gauge time signal. The significant wave height is calculated as the mean of the 1/3 highest waves using zero
down-crossing separation.
246810
Frequency [Hz]
-110
-100
-90
-80
-70
-60
-50
Singular values of spectral matrix
[dB rel. to 1 g2/Hz]
520 522 524 526 528 530
Time [s]
-2
-1
0
1
2
3
Acceleration of sensor 9 [g]
10-3
a) b)
Fig. 4. Structural accelerations: a) shows the acceleration spectrum in terms of the singular values (Brincker and Ventura, 2015). Only the three
most significant singular values are shown in descending order. b) shows a sample of the corresponding (unfiltered) accelerations for
sensor 9 during the random sea. Sensor 9 is positioned above the water level and is facing the incoming waves.
Acceleration Spectrum
As a consequence of the random waves, the structure will initiate some
vibrations. The vibrations are shown in Fig. 4 in both the frequency
domain and the time domain. The frequency domain representation is
made using the Welch averaging technique with segments of 32 s and a
50 % overlap. The spectra from the 16 accelerometers are decomposed
into singular values and the three most significant are shown.
From Table 1 and Fig. 3 a) we note that the natural frequency of the
structure is well above the main frequency content of the waves and
thus expect the response to be mainly quasi-static. When we examine
the frequency content of the accelerations shown in Fig. 4 a), we see
how the wave peak frequency fpis scattered between 1.0fpand 2.5fp.
The second thing that appears is how the natural frequency is repre-
sented in the response at approximately 8.4 Hz. The remaining modes
between 35 and 50 Hz are barely visible in the signal.
From the acceleration spectrum in Fig. 4 a) it is noted that cross vibra-
tions are present during the experiment as the second singular value
near 8.6 Hz corresponds to the first bending mode in the transverse
direction. See also Table 1.
Modal Analysis
The acceleration spectrum generated from the one-directional random
waves yields a poor basis for operational modal analysis (OMA). In-
stead, the OMA is based on additional loading, either by including
wind or brush strokes by the testing team. The modal analysis is omit-
ted from this paper and the reader may refer to (Vigsø et al., 2018b) for
further details. The first five modes are summarized in Table 1. The
Table 1. Estimated modal parameters. See (Vigsø et al., 2018b) for
more details.
Natural frequency Damping ratio Comment on mode shape
fn[Hz] ξn[%]
8.37 3.7 First bending mode
8.66 3.5 First bending mode, trans.
37.6 4.9 First torsional mode
40.3 3.7 Second bending mode
47.0 4.4 Second bending transverse
246810
Frequency [Hz]
-100
-80
-60
-40
-20
0
20
Load Spectrum
[dB rel. to 1 (Nm)
2/Hz]
520 522 524 526 528 530
Time [s]
-10
-5
0
5
10
Overturning moment [Nm]
Estimated
Measured
a) b)
Fig. 5. Global loading: a) shows the load spectrum (moment) during the random sea. Red is the estimated load while black is measured by the load
cell. b) shows a sample of the corresponding time signal. The load cell data is low-pass filtered at 50 Hz.
experimental mode shapes are scaled and expanded using a surrogate
FE model such that the full field deformation of the structure can be
estimated. The effect of the hydrodynamic added mass is included as a
non-structural mass that only affects the horizontal inertia and not the
torsional. The added mass is included in the updated FE model and is
mainly used to scale the mode shapes.
Wave Loading
The global loading on the structure is measured by a load cell at the
base. The load cell measures all six degrees of freedom, but we shall
confine ourselves to the overturning moment in the direction of the
waves. Once again, the spectrum is computed by means of Welch
averaging with segments of 32 s and a 50 % overlap, and shown in
Fig. 5 a). The data from the load cell is contaminated by harmonic
noise at 50 Hz. Hence the data shown in Fig. 5 has been low-pass
filtered to compensate for this. When comparing the time signal in
Fig. 5 b) and Fig. 3 b), we see that the structure experiences the max-
imum loading before the apparent wave reaches its maximum crest
height.
When examining the spectrum in Fig. 5 a), it is seen that the resonant
frequency of the structure is repeated in the reaction forces. This is
due to the dynamic interaction between the loading and the system.
RESULTS
During random sea, the structural accelerations and wave gauge mea-
surements are recorded and fed into the algorithm. The algorithm then
provides a real-time estimate on the wave load (red). The estimated
wave load is compared to the measurements from the load cell at the
base (black). The comparison is shown in Fig. 5 in terms of the over-
turning moment. We see that the overall trend is captured through the
indirect method, although some discrepancies are seen:
When examining the load spectra in Fig. 5 a), we see that the estimated
spectrum includes some low frequency content that is not seen by the
load cell. We expect that this is a result from the performance of the
accelerometers at low frequency (<0.5 Hz).
Next, we observe how the energy is less near the resonant frequency
of 8.4 Hz. We recall the objective of the algorithm which is to identify
the input, i.e. the wave loading, while it should omit any dynamic
amplification from the structure. Since the peak at 8.4 Hz represents a
dynamic amplification from the first mode, it also represents a flaw in
the identification algorithm as it should not have been represented.
In Fig. 5 b), a comparison is made between the estimated input (wave
load) and the measured reaction force in the time domain. Again, since
the measured force includes the dynamics of the structure and the es-
timates do not, it is not completely comparable. However, since the
primary part of the response is of a quasi-static nature, the difference
should be limited and we will not pursue to adjust for this.
When using a real-time algorithm as presented in this paper, it is com-
mon to encounter a phase shift in the estimate compared to the mea-
sured. Note that the result shown in Fig. 5 b) is adjusted for this by an
offset of 0.09 s.
DISCUSSION
When applying this type of Kalman filter approach for load identifi-
cation, it is a challenge when only accelerations are available as the
state estimates are likely to drift. Filtering and detrending may im-
prove this issue. As an alternative, one can convert an accelerometer
into a pseudo displacement sensor through integration in the frequency
domain, however, this makes the real-time implementation more chal-
lenging.
The performance of the Kalman filter is dependent on the tuning of
the noise models, i.e. the error covariance matrices in Eq. (12) and
Eq. (13). The parameters chosen for this study are as follows:
Qii
∼100,S∼109and Rii ∼10 (14)
We admit that access to the measured load is very convenient when
tuning these parameters for the Kalman filter.
In Eq. (4), we have assumed that the spatial distribution is constant
in time apart from the stretching. This is a crude simplification for
a random sea as waves with shorter wavelengths will concentrate the
load near the surface and vise versa. The spatial distribution could also
have been refined in such way that it adds a contribution from a drag
dominated scenario and an inertia dominated scenario. This has not
been pursued in this paper.
If wave breaking or slamming occur near the structure, the frequency
content of the accelerations is expected to broaden and hence yield
modal activity from higher modes. In order to estimate these loads,
more modes are likely needed and possibly another local load model
for the impact area. This will be the focus for future study.
CONCLUSION
Through this study, we have experienced the challenges associated
with both indirect and direct measurements of wave loading when
dealing with both quasi-static and dynamic response. Although chal-
lenges remain, we have demonstrated the feasibility of indirect esti-
mation of wave loads on a structure during random seas. The results
are based on experimental system identification (OMA) and driven by
acceleration measurements. Through simultaneous monitoring of the
surrounding sea, the wave loading was constrained to the wetted area
of the structure.
We have examined a scenario where the natural frequency of the struc-
ture is well above the wave peak frequency. This yields a high portion
of quasi-static response. Note that for actual offshore installations, the
frequency gap between the peak wave and the structure may be smaller
and thus the dynamic response may be larger, i.e. the Valdemar plat-
form in the North Sea, (Skafte et al., 2014).
ACKNOWLEDGEMENTS
The authors acknowledge the funding received from the Centre for Oil
and Gas – DTU/Danish Hydrocarbon Research and Technology Centre
(DHRTC).
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