The effect of marine growth and damage severity on the modal parameters of offshore wind turbine supporting structures: an experimental study

2.1. State space

The system identification utilized in obtaining the modal parameters of the laboratory based experimental structure is based on the stochastic subspace identification (SSI).

In SSI, a data matrix:

is used, which comprises of an $m$-variant time series sampled $n_t$ times at a sampling rate of $f_s$. At any time instant $k$, the system output vector is ${\mathbf{y}}_k\in R^m$.

The randomly excited discrete state space system of order $n$ can be written as:

where $\mathbf{x}$ is the state vector, ${\mathbf{A}}_D\in R^{n\times n}$ is the discrete system matrix and ${\mathbf{C}}_D\in R^{m\times n}$ is the discrete output matrix. $\mathbf{w}$ and $\mathbf{v}$ both $\in R^m$ are zero mean white Gaussian noise terms.

The calculation of Eigenvalues ${\lambda }_i$ and Eigenvectors ${\varphi \text{'}}_i$ is performed on a system matrix formulated in continuous time. This conversion is done by:

Using the system matrix in continuous time ${\mathbf{A}}_c$, the eigenvectors and eigenvalues can be calculated by solving the Eigenvalue problem:

Eigenvalues ${\lambda }_i$ can be further used to obtain natural frequencies $f_i$ and damping ratios ${\zeta }_i$ as follows:

2.2. The stochastic subspace identification

The Stochastic Subspace Identification (SSI) is a method used in estimating a stochastic state space system such as in Eq. (2). Introduced by Van Overschee and de Moor [16, 17] based on the Eigensystem Realization Algorithm (ERA) by Juang [18], it is considered a powerful tool in system identification [19].

2.3. The data driven stochastic subspace identification

The data driven SSI uses blocks of output signals stacked into a matrix, in which future row spaces are projected into the past row spaces [16, 17]. This is done via a block Hankel matrix $Y_h\in R^{2.m.i\times n_t-2i}$ that is built by stacking 2i data blocks with $n_t-2i$ samples from $\mathbf{Y}$ as follows:

where ${\mathbf{Y}}_{hp}$ is the referred to as the past, and ${\mathbf{Y}}_{hf}$ is considered the future. The orthogonal projection of row spaces from ${\mathbf{Y}}_{hf}$ onto ${\mathbf{Y}}_{hp}$ is defined as:

where ${\left({\mathbf{Y}}_{hp}{\mathbf{Y}}_{hp}\right)}^{-1}$ is the Moore-Penrose pseude-inverse [17].

The projection can be further decomposed to estimate the state space matrices as in [6]. A detailed derivation of the procedure can be found in [16].

2.4. Extraction of modal parameters

A problem arising when using SSI-models is to find an optimal model order, which is done via iterative estimations. Such estimations are not universally valid and would require continuous adjustment for each new experimental setup.

To overcome this problem, so-called stable paths are used. In such procedure, a large number of model orders are calculated, resulting in a number of spurious solutions. Utilizing stabilization diagrams, the obtained modal parameters are plotted versus the model order. Only those results that appear to be unchanging over several model orders are considered physical solutions and represent the modal properties of the analyzed system. There is a multitude of approaches used in identifying such stable paths as shown including Reynders [20], Allemang [21] and Magalhaes [22]. However, according to [6], those approaches rely on clustering data points which is rather expensive.

The approach adopted in this research is further detailed in [6]. In this approach, stable paths are defined to be those in which the variation of frequency, damping and Modal Assurance Criteria (MAC) are below predetermined values.

3.1. Structure, test facility and hardware

A scaled experimental model of an offshore wind turbine monopile supporting structure, shown in Fig. 1 is used in this study.

Fig. 1Test structure and dimensions

The structure consists of a 3.5 m long UPVC pipe, having a diameter of 0.21 m and a wall thickness of 5 mm. The pipe is fixed to the ground via a 1 m×1 m steel plate of 0.05 m thickness, and is fitted with a 53 kg mass on top.

The structure has a scale of 1:30 and was derived from the 5 MW benchmark turbine defined by the National Renewable Energy Laboratory (NREL). The scaling was done utilizing the Froude-Cauchy similarity criteria. The Froude criteria is usually used in hydraulic engineering when the dominant forces arise from gravitational waves. The additional consideration of the Cauchy condition is required to be able to correctly simulate the rigidity of the structure and thus the dynamic behavior of the system.

The structure is tested in a 3D wave basin shown in Fig. 2. It is 40 m long and 24 m wide, with a maximum water depth of 1 m capable of producing multi-directional sea states with a maximum wave height of around 0.47 m. The 3D wave machine (snake wave maker with 72 individual wave paddles) allows the generation of almost all required states.

Fig. 2Wave basin

The structure is fitted with 6 tri-axial accelerometers at various positions shown in Fig. 3. Furthermore, a laser-type displacement sensor is fitted at the top to measure the lateral displacement of the top mass. All measurements are conducted at a sampling rate of 500 Hz.

The application of marine growth on the structure is done via a synthetic sheet, shown in Fig. 4 of varying thickness. The specifications of the synthetic sheet are compliant with those stated in [8].

Fig. 3Sensor locations on the test structure

Fig. 4Synthetic sheet used as marine growth replacement

Damage was introduced via 3 symmetrically placed saw cuts at a height of 0.9 m, as shown in Fig. 5.

Fig. 5Symmetrical damage application on the test structure

3.2. Test cases

In total, 7 test setups were investigated. Three undamaged cases with increasing marine growth and 4 damaged cases with maximum marine growth.

The marine growth cases can be detailed as follows: a growth free case, an intermediate growth case, simulated by 1 layer of the synthetic and finally, a severe growth case simulated by 3 layers.

Furthermore, 5 cases of damage severity were investigated. A damage free case, in which no saw cuts were introduced, and damaged cases having a saw cut depth of 2 mm, 7 mm, 12.5 mm and 25 mm.

All damaged cases were coupled with the most severe marine growth case. This allows to test a structure having both marine growth and damage, providing means to distinguish between changes caused by marine growth and those caused by damage. Furthermore, damage usually happens at a later stage of the lifespan of a structure, in which it is safe to assume that marine organisms have already grown on the structure.

Waves are generated according to the JONSWAP spectrum [23]. To generate the waves via the JONSWAP spectrum, multiple inputs have to be provided, especially average wave height and wave period. Due to mechanical limitations of the wave generator, larger waves cannot be generated at higher frequencies due to mechanical limitations of the wave generator.

For each test setup, a minimum of 500 waves are generated to achieve a good statistical distribution. Furthermore, each test setup is carried out 5 times.

Before wave tests were performed, a free vibration tests was done. This involves pulling a cable connected to the top mass to a predetermined displacement (20 mm and 30 mm), then releasing the cable allowing the structure to vibrate freely. Such tests can estimate the first eigenfrequency of the structure, which is used to calibrate the period of the generated waves.

Wave tests of varying wave heights and periods are shown in Table 1. The periods were chosen to be multipliers of the first eigenfrequency of the test structure.

A summary of the considered test setups is shown in Table 2.

4.1. Free vibration tests

The results with regard to the first frequency are shown in Fig. 8. Two significant observations can be made with regard to the effect of marine growth and damage severity on the first frequency.

On one hand, the thickness and extent of modeled marine growth has no observable effect on the first frequency. This can be due to the shape of the first mode, where the largest displacements are situated at the tower top.

On the other hand, a clear effect of damage on the first frequency is observed. There is a downward trend in which the frequency does decrease when the damage severity is increased.

Fig. 8First frequency obtained via SSI utilizing stable paths (free vibration tests)

For the second frequency, the results are shown in Fig. 9. It is interesting to note that marine growth has a notable impact on the second frequency of the test structure. This can be attributed due to the fact that the second mode has a larger displacement at the position of the marine growth, which typically occurs near the water line where oxygen and sunlight levels allow such growth.

Furthermore, there is no notable change between an undamaged and damaged test structure with regard to the second frequency. There is, however, a jump between the 2 mm and 7 mm damage scenarios, resulting from water flowing into the test structure, as the wall thickness is 5 mm and is already breached when a 7 mm damage scenario is introduced.

Fig. 9Second frequency obtained via SSI utilizing stable paths (free vibration tests)

As for the third mode, shown in Fig. 10, similar observations may be drawn: marine growth has an impact on the frequency whilst damage has not. Furthermore, the same jump between the 2 mm and 7 mm damage scenarios exists.

Fig. 10Third frequency obtained via SSI utilizing stable paths (free vibration tests)

4.2. Wave tests

In these tests the structure is subjected to approximately 500 waves generated via the JONSWAP spectrum.

In comparison with the results of the tests under free vibration circumstances, the results of tests utilizing generated waves have a higher deviation. This is also attributed due to the interaction between generated waves and the test structure.

Nevertheless, the findings that were reported in the free vibration tests are still valid in the case of wave tests.

For the first frequency, shown in Fig. 11, damage had a more significant effect than marine growth.

Fig. 11First frequency obtained via SSI utilizing stable paths (wave tests)

For the second frequency, shown in Fig. 12 as well as the third frequency shown in Fig. 13, marine growth had a higher effect than damage.

The smaller effect of marine growth, which is applied near the water line where oxygen and sunlight levels are just right, on the first frequency can be due to the shape of the first mode, where the largest displacements are situated at the tower top. The shape of the second and third modes are also responsible for the higher effect of marine growth on those modes.

Fig. 12Second frequency obtained via SSI utilizing stable paths (wave tests)

Fig. 13Third frequency obtained via SSI utilizing stable paths (wave tests)

The step change between 2 mm and 7 mm damage cases still exists as water pours inside the test structure when saw cuts exceed 5 mm.

4.3. Damping

The identified damping for the first three modes is shown in Table 3. It is noteworthy that damage has a smaller impact on damping than marine growth. Although there exists studies that do not agree with this outcome, the damping ratio was found to be insensitive to damage. This is due to the artificial nature of damage itself, as it is introduced via saw cuts rather than cracks. Natural cracks to create new energy dissipation paths via friction that increase the damping of the structure. On the contrary, damaging a structure via saw cuts does not generate the same friction effect. Thus, it is expected for saw cuts not to affect the damping characteristics of the test structure.