# Could mass eccentricity explain the formation of orbits in wind turbines?

Aljoscha Sander
Energy and Sustainability Research Institute Groningen
University of Groningen
Groningen, the Netherlands
Institute for Integrated Product Development
University of Bremen
Bremen, Germany
<EMAIL_ADDRESS>
&Andreas Haselsteiner
University of Bremen
Institute for Integrated Product Development
ForWind - Center for Wind Energy Research
Bremen, Germany
<EMAIL_ADDRESS>
&Bas Holman
<EMAIL_ADDRESS>

###### Abstract

The kinematics of offshore wind turbines are of great importance when
installing the turbines, as the motions of the components during craning
operations are a limiting factor. Most critical is the installation of the
blades: the blade’s bolts need to be inserted into the rotor flange, an
operation that requires great precision. Both the blade and the turbine
undergo environmental loading, leading to relative motions between the blade
root and the hub during installation. Results from an offshore wind farm
installation measurement campaign showed, that the partially installed
turbines show intricate patterns of motion (orbits) in the horizontal plane.
The mechanism behind the formation of these orbits remains elusive so far.

In this paper, we present a novel torsional coupling mechanism linking motions
in the fore-aft and side-side direction. It can explain the formation of
orbits that change direction.

_Keywords_ Offshore Wind $\cdot$ Offshore Wind Turbine Installation $\cdot$
Structural Dynamics

##  1 Introduction

Offshore wind is growing continuously and is on its way to become one of the
pillars of green energy. Recent cost reductions in offshore wind (29% between
2010 and 2019) have mostly been due to increasing turbine size (IRENA, 2020).
Increasing turbine size, however, leads to increasing difficulties during
installation: larger components are more challenging to handle and require
more space on an installation vessel. Today, most turbines are installed
component by component, including the blades, a process generally referred to
as single blade installation (Jiang, 2021). Blades are installed utilizing a
specialized tool, referred to as an installation yoke, that securely fastens
the blade during craning operations. Once the blade reaches hub height, a
guiding pin is used to land the blade’s bolts in the corresponding holes in
the rotor flange. During this operation, relative motions between the blade
root and the rotor hub greatly hamper the secure landing of the blade in the
rotor’s flange. As these relative motions are driven by environmental loading
on both the turbine and the blade, they are difficult to understand and even
more so to predict.

In a measurement campaign conducted during the installation of the offshore
wind farm “Trianel Windpark Borkum II” off the coast of Germany in the German
Bight, the kinematics of blades and partially installed turbines were
investigated (Sander et al., 2020a, b; Sander, 2020). These measurements
revealed, that the partially installed turbines, consisting of a monopile
foundation, transition piece, the tower and the nacelle, were displaying
intricate patterns of motions — orbits — during single blade installation. A
two-minute orbit captured during the installation of the wind park Trianel
Windpark Borkum II is shown in Figure 1. In these orbits, amplitude and
direction of motion change swiftly. It stands to reason that these orbits
influence the installation of blades and may even be responsible for an
increase in installation difficulty.

Figure 1: Formation of orbits as observed during the installation of the
offshore wind park “Trianel Windpark Borkum II.” Note: to enhance visibility,
the orbits are not to scale.

Several mechanisms may in theory explain the formation of orbits. In this
paper, we present how mechanical coupling between the fore-aft and side-side
first motion leads to the formation of orbits in a cantilevered beam with an
eccentric mass present at the free end.

We focus on one specific state of an offshore wind turbine during
installation: the hammerhead configuration. In this configuration, the
foundation, transition piece, tower, and nacelle have already been installed,
and the next installation step would be the installation of the first blade.
In this configuration, the turbine is top-heavy and, due to the nacelle’s and
rotor’s weight, also nose-heavy. As for the turbines installed at Trianel
Windpark Borkum II, the centre of gravity of the nacelle is approximately 0.3
m shifted towards the rotor with respect to the tower axis. Figure 2 depicts
the hammerhead configuration as well as the simplified mechanical system we
derive to investigate the behavior of partially installed turbines during
installation.

Figure 2: Hammerhead configuration (left) and simplified mechanical model of
the hammerhead configuration (right) as a cantilevered beam with an eccentric
top mass $m_{e}$. Wind and wave forces induce a loading force and moment
$F,M$. The cantilevered beam has a mass $m$, a moment of inertia $J$, an
elasticity $K$, and a structural damping $\delta$. The eccentric mass at the
beam’s free end also has a moment of inertia $J_{e}$

A partially installed offshore wind turbine in the hammerhead configuration
resembles a cantilevered beam with an eccentric mass at the free end. There
are two main directions when viewing the cantilevered beam from above: Fore-
aft (North-South) and side-side (East-West). Consider that the eccentric mass
is located north of the tower center on the fore-aft axis. If the beam
undergoes lateral vibrations in the side-side direction the inertia of the
eccentric mass leads to torsion of the cantilevered beam.

This twisting of the beam along its main axis has two effects:

  * a)

The eccentric mass starts to oscillate around the tower center, effectively
turning into a torsional oscillator

  * b)

Due to the circular motion of the eccentric mass around the tower center,
transfer of momentum from the initial direction of motion — side-side — to the
second direction — fore-aft — takes place.

The torsional oscillation of the eccentric mass in illustrated in Figure 3.

Figure 3: Simplified kinematics of a cantilevered beam with an eccentric mass
at the free end undergoing harmonic oscillations in the side-side direction.
When viewed from above, the inertia of the eccentric mass induced rotation
around the axis of the beam, leading to transfer of momentum.

In this paper, we set out to investigate this effect of torsional coupling
with a table-top experiment, finite element simulations, and a reduced order
model approach.

The paper is subdivided into four sections: First, results from a table-top
experiment (section 2), capable of producing orbits that change direction and
shape over time, are shown. A section on finite element simulations (section
3) of a comparable system follows. Finally, a discrete system of two bodies,
described with differential equations (section 4) is presented. The final
section (section 5) provides a discussion and suggests future research into
the topic.

##  2 Table-Top Experiment: Cantilevered Beam with an Eccentric Mass

###  2.1 Experimental Setup

As a first proof of concept, we built a table-top experiment that allows us to
demonstrate the effect of an eccentric mass on the vibration of a cantilevered
beam. As a beam, we chose a wooden stick (massive beech, 0.006 m diameter)
with a free length of 0.7 m. At the free end of the beam, we placed a
3D-printed lever that allowed us to add M8 bolts (0.028 m length, 0.0129 kg
each) to control the amount of eccentric mass. Lever length was 0.082 m
weighing a total of 0.026 kg. The lever was oriented, such that the eccentric
mass was positioned north of the beam’s axis in the fore-aft plane. The side-
side direction is thus defined as perpendicular to the fore-aft direction.

A second beam made of fibre glass (diameter 0.008 m) was placed next to the
cantilevered beam. Figure 4 shows the experimental setup. The second beam was
used to displace the cantilevered beam from its resting position by pulling
the cantilevered beam towards the second beam using a thin cotton thread. The
initial displacement of the top of the beam was 0.135 m and three different
masses (no bolts, two bolts, four bolts) were tested in the experiment. To
release the cantilevered beam from its initial, displaced position, the thread
was set on fire using a lighter. For each mass, the experiment was repeated
three times. A MEMS-based inertial measurement unit (MPU 9255, InvenSense TDK,
Tokyo, Japan) placed on top of the cantilevered beam measured linear
acceleration and angular velocity with a sampling rate of 100 Hz. Measurements
were recorded using a Raspberry Pi 400 and analyzed using a jupyter notebook
with python3.

Figure 4: Experimental setup of the table-top experiment. (A) indicates the
eccentric mass, added to the cantilevered beam via a 3D-printed lever.

We applied the following post-processing steps to the recorded data:

First, obtained accelerations in fore-aft and side-side direction were
resampled to 50 Hz using linear interpolation. As the data is indexed by the
time of the recording, the time series of each experiment had to be normalized
to enable comparison of the different time series. The first peak in the fore-
aft acceleration signal in each recorded experiment served as a normalization
time stamp; each signal was cropped, such that it starts 0.5 s prior to the
first peak and lasts for 120 s.

Following time normalization, the data was filtered by applying a Butterworth
high-pass filter with a cut-off frequency of 0.5 Hz, order 11 and padding of
25 s to remove any transient effects. The filter was applied forwards and
backwards in time to avoid introducing an artificial phase delay. Afterwards,
the accelerations and the angular velocities were integrated using a second
order trapeze type integration, yielding the velocity vector of the beam’s tip
and the torsion angle of the beam. To obtain the displacement vector, the
velocity vector was filtered again with the same Butterworth filter,
integrated again and finally both torsion angle and displacement vector were
filtered one last time with the same Butterworth filter.

To compare experimental measurements with finite element simulations and the
3DOF model (both without damping), the observed displacement from the
experimental results was normalized to remove the influence of damping from
the observed orbits. To this end, the observed decay in displacement amplitude
over time is estimated using an exponential function of the following form:

$A=A_{0}e^{\gamma t}+a$ (1)

where $A_{0}$ is the initial amplitude, $\gamma$ is the damping coefficient,
and $a$ is a constant offset to account for the finite time series. To obtain
orbits free of damping, the displacement vector is scaled by the exponential
decay function.

###  2.2 Experimental Results

Figure 5, Figure 6, and Figure 7 each show the calculated position as a
function of time, the angular velocity along the beam main axis and the
absolute acceleration in the fore-aft-side-side plane as well as the
corresponding orbits.

Figure 5: Position, angular velocity, and absolute acceleration (left) and
Lissajous figures (orbits; right) for the low mass scenario (right).

Figure 6: Accelerations, angular velocity and absolute acceleration (left) and
Lissajous figures (orbits; right) for the medium mass scenario.

Figure 7: Accelerations, angular velocity and absolute acceleration (left) and
Lissajous figures (orbits; right) for the high mass scenario.

The area enclosing all observed orbits seems to be a function of the eccentric
mass. In the case of the medium mass scenario, the area is almost quadratic,
whereas for the low and high mass scenarios the area is more distorted. For
each mass scenario, three runs of the experiment are shown, and the observed
orbits are in close resemblance between consecutive runs. With increasing
mass, decay is slower. Furthermore, for the medium and high mass scenarios,
the angular velocity around the beam axis increases after the initial release
and torsion seems to last longer than in the low mass scenario. In all three
mass scenarios, torsion of the eccentric mass around the beam’s main axis can
be observed. It seems that, as long as torsion takes place, changes in
direction - and thus orbits - are present.

##  3 Finite Element Simulations of a Cantilevered Beam with an Eccentric Mass

###  3.1 Simulation Setup

A numerical model based on the simplified system shown in Figure 2 was created
using MATLAB, presented in Figure 8. The cantilevered beam was modeled as a
hollow cylinder made of homogeneous linear elastic material (structural steel
s235) and with a constant cross-section (diameter: 1.5 m, wall thickness: 15
mm) along its height (30 m), supported by linear uncoupled springs (Equation
8). The nacelle was modeled as a concentrated mass ($\mathbf{m_{t}}$) with
inertia and an additional eccentric mass in both the horizontal and the
vertical plane, closely resembling the table-top experiment.

Figure 8: FE model.

The equations of motion of the undamped system are

$\mathbf{M\ddot{\mathbf{u}}}+\mathbf{K}\mathbf{u}=\mathbf{f},$ (2)

where $\mathbf{u}\in\mathbb{R}^{6}$ describes the concentrated mass’
translational ($x,y,z$) and rotational ($\theta_{1},\theta_{2},\theta_{3}$)
position as a column vector.

The equations were obtained by superposition of the equations of motions of
the concentrated mass ($\mathbf{M}_{top}$) at the free end of the cantilevered
beam, the equations for the support springs and the equations for the
elasticity of the cantilevered beam itself
($\mathbf{K}=\mathbf{K_{mt}}+\mathbf{K_{found}}+\mathbf{K}_{beam}$). The
cantilevered beam was modeled based on linear one-dimensional finite elements
with 12 degrees of freedom each. Euler Bernoulli beam theory was used to deal
with bending. Using the Euler Lagrange equation with a rotation matrix based
on Tait–Bryan angles and using Taylor series for linearization, the mass
matrix (Equation 3), stiffness matrix (Equation 5), and gravitational force
(Equation 7) of the eccentric concentrated top mass were found. For the case
study, the numerical values of the top mass matrix are shown in Equation 4,
the numerical values for the stiffness matrix are shown in Equation 6, and the
numerical values for the foundation spring matrix $\mathbf{K_{found}}$ are
listed in Equation 8:

$\mathbf{M}_{top}=\begin{bmatrix}m_{t}&0&0&0&m_{t}\,z&-m_{t}\,y\\\
0&m_{t}&0&-m_{t}\,z&0&m_{t}\,x\\\ 0&0&m_{t}&m_{t}\,y&-m_{t}\,x&0\\\
0&-m_{t}\,z&m_{t}\,y&I_{c}+\left({z}^{2}+{y}^{2}\right)\,m_{t}&-m_{t}\,x\,y&-m_{t}\,x\,z\\\
m_{t}\,z&0&-m_{t}\,x&-m_{t}\,x\,y&I_{c}+\left({z}^{2}+{x}^{2}\right)\,m_{t}&-m_{t}\,y\,z\\\
-m_{t}\,y&m_{t}\,x&0&-m_{t}\,x\,z&-m_{t}\,y\,z&I_{c}+\left({y}^{2}+{x}^{2}\right)\,m_{t}\end{bmatrix},$
(3)

$\mathbf{M}_{top}=10^{4}\begin{bmatrix}1.7&0&0&0&0.2&0\\\
0&1.7&0&-0.2&0&1.4\\\ 0&0&1.7&0&-1.4&0\\\ 0&-0.2&0&4.2&0&-0.6\\\
0.2&0&-1.4&0&8.2&0\\\ 0&1.4&0&-0.6&0&8.0\end{bmatrix},$ (4)

$\mathbf{K}_{m_{t}}=\textnormal{diag}\begin{bmatrix}0&0&0&-g\,m_{t}\,z&-g\,m_{t}\,z&0\end{bmatrix},$
(5)

$\mathbf{K}_{m_{t}}=10^{4}\textnormal{diag}\begin{bmatrix}0&0&0&-1.9&-1.9&0\end{bmatrix},$
(6)

$\mathbf{f}_{g,m_{t}}=\begin{bmatrix}0&0&-g(\,m_{t}+m_{t})&-g\,m_{t}\,y&g\,m_{t}\,x_{e}&0\end{bmatrix}^{T},$
(7)

$\mathbf{K}_{found}=10^{11}\,\textnormal{diag}\begin{bmatrix}2.44&2.44&1.48&0.41&0.41&0.31\end{bmatrix},$
(8)

where diag[] denotes the diagonal elements of $6\times 6$ matrices where non-
diagonal elements are 0.

As loads, gravitational force and an initial deformation of 0.71 m due to a
horizontal force applied at the center of the top of the cantilevered beam
were considered. To account for numerical diffusivity imposed by linearization
of the numerical model, small initial deformations to induce instabilities
were applied.

###  3.2 Simulation Results

The response to gravitational force and an initial deformation of 0.71 m due
to a horizontal force applied at the top of the cantilevered beam with an
angle of 66° with respect to the eccentricity of the top mass is presented in
Figure 9 and Figure 10. The pattern found in the simulations (Figure 9) appear
to be quite similar to the pattern found in the experiments, as presented in
Figure 7. As in the experiment, torsion around the beam’s main axis, albeit
small in value, coincided with the presence of orbits. If, however, the
initial displacement was in line with the eccentric mass or the eccentric mass
was removed completely, no orbits appeared.

Figure 9: Formation of orbits as observed in the finite element model. The
titles of the subfigures denote the time interval of the highlighted orbits.
Figure 10: Selected interval of fore-aft (x), side-side (y) and torsion angle
$\Theta_{z}$ time series from the finite element model.

##  4 A Vibration Model with three Degrees of Freedom

A common approach to study vibrations is to reduce the real system into a
simplified model that consists of discrete components. In this approach, one
aims to describe the real system as simple as possible to generate the main
dynamics of the real system. Many vibration phenomena can be sufficiently
described with systems with only one or two degrees of freedom. Here, we
present a model with three degrees of freedom that seems to capture important
dynamics of the real system.

###  4.1 Model Derivation

We model the real system in two dimensions (Figure 11). The tower’s bending is
modeled as a linear spring, which is connected to a fixed point that
represents the tower’s foundation, and to a first body that represents the
tower’s top section. The tower’s torsion is modeled with a torsional spring
that connects a second body with a fixed angle in the inertial reference
frame. This second body is pinned to the first body such that it translates
with the first body, but rotates freely. Consequently, the system’s three
degrees of freedom are: the first body’s position in $x$ and $y$ direction and
the second body’s orientation $\theta$.

By applying the physical laws of conservation of linear and angular momentum,
we derived the equations of motions for the system. The system’s equations of
motion read:

$(m_{1}+m_{2})\ddot{x}-\cos(\theta)m_{2}d\ddot{\theta}+\sin(\theta)m_{2}d\dot{\theta}^{2}+k_{1}x=0,$
(9)
$(m_{1}+m_{2})\ddot{y}-\sin(\theta)m_{2}d\ddot{\theta}-\cos(\theta)m_{2}d\dot{\theta}^{2}+k_{1}y=0,$
(10)
$(I_{zz}+m_{2}d^{2})\ddot{\theta}-\cos(\theta)m_{2}d\ddot{x}-\sin(\theta)m_{2}d\ddot{y}+k_{2}\theta=0,$
(11)

where $m_{1}$ and $m_{2}$ are the masses of the two bodies, $I_{zz}$ is the
second body’s moment of inertia, $d$ is the distance between them, $k_{1}$ is
the stiffness of the spring that connects the first body with the origin,
$k_{2}$ is the torsional stiffness of the spring between the first and second
body, $x$ and $y$ denote the position of the first body’s center and $\theta$
denotes the orientation of the second body (Figure 11). Dots are used to
denote time derivatives.

The equations can be arranged to have only acceleration on the left hand side:

$\ddot{x}=\frac{1}{m_{1}+m_{2}}\left(\cos(\theta)m_{2}d\ddot{\theta}-\sin(\theta)m_{2}d\dot{\theta}^{2}-k_{1}x\right),$
(12)
$\ddot{y}=\frac{1}{m_{1}+m_{2}}\left(\sin(\theta)m_{2}d\ddot{\theta}+\cos(\theta)m_{2}d\dot{\theta}^{2}-k_{1}y\right),$
(13)
$\ddot{\theta}=\frac{1}{I_{zz}+m_{2}d^{2}}\left(\cos(\theta)m_{2}d\ddot{x}+\sin(\theta)m_{2}d\ddot{y}-k_{2}\theta\right).$
(14)

Several terms couple translation ($x$ and $y$) with rotation ($\theta$): for
example, the acceleration in the $x$ direction, $\ddot{x}$, is equal to terms
that contain $\dot{\theta}$ and $\ddot{\theta}$ (Equation 12). Such coupling
terms are necessary to appear in the equation as otherwise orbits would not
change direction (for example, a mass that is initially displaced would
indefinitely move in a straight line or a mass that is in an initial orbit
would stay in it indefinitely).

The translational acceleration ($\ddot{x}$ and $\ddot{y}$ in Equations 12 and
12) is affected by the second mass’ centrifugal and Euler forces. Centrifugal
force acts in the $-\hat{r}$ direction (a coordinate system is shown in Figure
11) and reads,

$m_{2}d\dot{\theta}^{2},$ (15)

and Euler force acts in the $\hat{\theta}$ direction and reads

$m_{2}d\ddot{\theta}.$ (16)

The equation for $\ddot{\theta}$ (Equation 14) was derived by applying
conservation of angular momentum around the center of mass $G$. The force that
acts at the location where the second body is pinned to the first body depends
on the first body’s acceleration in $x$ and $y$ direction such that there are
also coupling terms with $\ddot{x}$ and $\ddot{y}$ in the equation for
$\ddot{\theta}$.

As expected, the equations also show that if $d=0$ the coupling terms
disappear:

$(m_{1}+m_{2})\ddot{x}=-k_{1}x,$ (17) $(m_{1}+m_{2})\ddot{y}=-k_{1}y,$ (18)
$I_{zz}\ddot{\theta}=-k_{2}\theta.$ (19)

###  4.2 Model Results

Figure 11: A vibration model with three degrees of freedom (DOF).

With parameter values similar as in the table top experiment, the vibration
model leads to similar motions: orbits with changing major axis direction
appear (Figure 12). With parameters values that we consider similar to a
typical modern offshore wind turbine in the hammerhead configuration (tower
and nacelle installed, but not the blades yet), orbit direction changes do not
occur. If the torsional stiffness is very high such that torsional motions are
very low, the orbit is stable (Figure 13; we consider this scenario typical
for a modern wind turbine). In a unfavorable wind turbine configuration where
torsional stiffness is lower and the nacelle’s center of mass is farther away
from the foundation’s center, orbits that change appear (Figure 14). Similarly
as in the finite element model, if the initial displacement is exactly in line
with the eccentric mass, no orbits appear.

Quantity | Table top experiment | Offshore turbine | Offshore turbine | Unit
---|---|---|---|---
|  | (Typical configuration) | (Unfavorable configuration) |
$m_{1}$ | 0.0093 | 300$\times$103 | 300$\times$103 | kg
$m_{2}$ | 0.044 | 400$\times$103 | 400$\times$103 | kg
$k_{1}$ | 4.5 | 3$\times$106 | 3$\times$106 | N m-1
$k_{2}$ | 0.001 | 4$\times$109 | 4$\times$108 | Nm deg-1
$d$ | 0.038 | 0.3 | 1.0 | m
$I_{zz}$ | 5$\times$10-5 | 4$\times$107 | 8$\times$107 | kg m2
Initial condition |  |  |  |
$x(t=0)$ | 0.1 | 1 | 1 | m
$y(t=0)$ | 0.1 | 1 | 1 | m
$\dot{x}(t=0)$ | 0 | 1 | 1 | m s-1
$\dot{y}(t=0)$ | 0 | 0 | 0 | m s-1
Table 1: Values used in the 3DOF vibration model that roughly correspond to
the table top experiment and the offshore measurements. The resulting
kinematics are presented in Figures 12, 13, and 14 Figure 12: Kinematics of
the 3DOF model in the table top configuration (Table 1). Figure 13: Kinematics
of the 3DOF model in a typical wind turbine configuration (Table 1). Figure
14: Kinematics of the 3DOF model in a unfavorable wind turbine configuration
(Table 1).

##  5 Discussion

###  5.1 Table-Top Experiment with a Cantilevered Beam

Even though the table-top experiment was conducted with very limited resources
and meant as a demonstrator experiment only, the formation of orbits with
changing directions was clearly observable. More so, the reproducibility of
the orbits was high, which suggests that the observed dynamics are stable and
can be explained by the properties of the structure instead of external forces
such as airflow acting on the structure. In future experiments, the influence
of initial and boundary conditions of the experiment should be investigated as
it stands to reason, that even without an eccentric mass present at the free
end of the beam, orbits will appear in an experiment as simple as ours, given
the difficulty of aligning the beam with Earth’s gravitational vector. If the
beam is tilted with respect to gravity, a certain mass eccentricity will
always be induced. Furthermore, in a future study, before designing a new
experiment, the governing parameters such as the Scrouton number should be
derived from offshore wind turbines by applying similitude first and then
incorporated into the design.

###  5.2 Finite Element Simulations of a Cantilevered Beam

With the linear finite element model, similar orbits as observed in the table-
top experiment have been found. Differences may be because the experiments
were performed for initial deformations exceeding linear bending behavior of
the cantilevered beam, which is not accounted for by the finite element model.
However, the similarity between the numerical and experimental results
suggest, that the non-linear effects play a minor role in the formation of
orbits. In future studies, a more realistic model of a partially installed
turbine including non-linear bending could be investigated as well as more
realistic, stochastic loading to determine the full mechanics of the partially
installed turbine.

###  5.3 Three Degrees of Freedom Model

The three degrees of freedom (3DOF) model presented in this paper leads to
orbits comparable to the table-top experiment if the parameters of the
differential equations are chosen, such that they are close to the parameters
of the table-top experiment. However, for parameters resembling a typical
offshore wind turbine, orbits were stable and hardly changed direction. This
needs to be investigated in the future. Furthermore, damping could be added as
additional coupling terms between translation and torsion are to be expected
if damping is included. The configuration of the 3DOF model was created as a
direct abstraction of the real system. Alternatively, one could derive a
simple vibration model by starting from a mathematical description of a
cantilevered beam formulation in three dimensions. Then, one would simplify
and neglect terms until a model with few degrees of freedom would remain. Such
an approach could be used to decrease model complexity in a step-wise process
and to investigate individual model assumptions.

###  5.4 Implications for the Kinematics of Partially Installed Wind Turbines

So far, a coupling mechanism between translational vibrations sufficient to
explain the observed orbits in offshore wind turbines undergoing installation
is missing. The torsional coupling presented here seems to be a valid
candidate; however, given the nature of the governing differential equations,
it is to be expected that the torsional coupling itself is a highly complex
and possibly chaotic effect. More studies, with parameters closer to real
turbines, need to be investigated rigorously. It seems, nevertheless, likely,
that some degree of the torsional coupling described here is present in
offshore wind turbines as their nacelles are notoriously nose heavy to counter
rotor thrust when operational.

Combining torsional coupling and the stochastic loads inflicted upon the
structure by wind and waves could be a sufficient explanation for the
formation of orbits and their erratic kinematics during single blade
installation.

## Data availability

All data of the table-top experiment and the corresponding code is available
on github: https://github.com/k323r/2021_preprint_eccentric-mass. Similarly,
the repository contains code that was used to simulate the dynamics of the
3DOF model.

## References

  * IRENA (2020) IRENA .  Renewable Power Generation Costs in 2019. Abu Dhabi, 2020. 144\.
  * Jiang (2021) Jiang Zhiyu.  Installation of Offshore Wind Turbines: A Technical Review  // Renewable and Sustainable Energy Reviews. IV 2021. 139\. 110576.
  * Sander (2020) Sander Aljoscha.  Oscillations of Offshore Wind Turbines Undergoing Installation I: Raw Measurements. X 2020.
  * Sander et al. (2020a) Sander Aljoscha, Haselsteiner Andreas F., Barat Kader, Janssen Michael, Oelker Stephan, Ohlendorf Jan-Hendrik, Thoben Klaus-Dieter.  Relative Motion During Single Blade Installation: Measurements From the North Sea  // ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering. XII 2020a.
  * Sander et al. (2020b) Sander Aljoscha, Meinhardt Christian, Thoben Klaus-Dieter.  Monitoring of Offshore Wind Turbines under Wave and Wind Loading during Installation  // XI International Conference on Structural Dynamics. Athens, Greece, 2020b. 2189–2205.


# The Joy of Neural Painting

Ernesto Diaz-Aviles
Libre AI
<EMAIL_ADDRESS>
Claudia Orellana-Rodriguez
Libre AI
<EMAIL_ADDRESS>
Beth Jochim
Libre AI
<EMAIL_ADDRESS>
We provide the code of our Neural Painters implementaion and notebooks to
reproduce our experiments at: https://github.com/libreai/neural-painters-x

###### Abstract

_Neural Painters_ is a class of models that follows a GAN framework to
generate brushstrokes, which are then composed to create paintings. GANs are
great generative models for AI Art but they are known to be notoriously
difficult to train. To overcome GAN’s limitations and to speed up the Neural
Painter training, we applied Transfer Learning to the process reducing it from
days to only hours, while achieving the same level of visual aesthetics in the
final paintings generated. We report our approach and results in this work.

##  1 Introduction

Artists do not learn how to paint pixel by pixel. However, most of the current
generative AI Art methods are still centered to teach machines how to ’paint’
at the pixel-level in order to achieve or mimic some painting style, e.g.,
GANs-based approaches and style transfer. This might be effective, but not
very intuitive, specially when explaining this process to artists, who are
more familiar with colors and brushstrokes.

Part of our goal is to make more accessible the advances of AI to groups of
artists who do not necessarily have a deep technical background. We want to
explore how the creative process is enriched by the interaction between
creative people and creative machines. To this end, we need to teach a machine
how to paint as a human would do it: using brushstrokes and combining colors
on a canvas (Figure 1).

Figure 1: Digital paintings generated using our approach and enlarged with
super-resolution [1]. A combination of brushstrokes help visualize ImageNet
classes and can also “paint” arbitrary images minimizing a perceptual loss.

##  2 Learning Neural Painters Faster

Neural Painters [5] are a class of models that can be seen as a fully
differentiable simulation of a particular non-differentiable painting program,
in other words, the machine “paints” by successively generating brushstrokes
(i.e., actions that define a brushstrokes) and applying them on a canvas, as
an artist would do.

These actions characterize the brushstrokes and consist of 12-dimensional
vectors defining the following variables: (i) Start and end pressure; (ii)
Brush size; (iii) Color (RGB) of the brushstroke; (iv) Brush coordinates
defining the brushstroke’s shape.

The goal of the Neural Painter is to translate these vectors of actions into
brushstrokes on a canvas.

One of the Neural Painters architectures is based on GANs. GANs are generative
models widely use for AI Art, but they are known to be notoriously difficult
to train, specially due to requiring a large amount of data, and therefore,
needing large computational power on GPUs. They also require a lot of time to
train and are sensitive to small hyperparameter variations. Reproducing the
results of the original Neural Painters paper [5] we obtained similar results
with our implementation, in terms of brushstrokes quality, but it took us two
days for training to get descent aesthetics using a single GPU on a
Colaboratory notebook [6].

To overcome these known GANs limitations, we leverage _Transfer Learning_ [8,
3] principles that allow us significantly reducing the Neural Painter training
time.

Transfer learning is a very useful technique in Machine Learning, e.g., the
ImageNet models trained as classifiers, are largely used as powerful image
feature extractors, in NLP, word embeddings, learned unsupervised or with
minimal supervision, have been very useful as representations of words in more
complex language models.

The fundamental idea, is to learn a model or feature representation on a task,
and then transfer that knowledge to another related task, without the need to
start from scratch, and only do some fine-tuning to adapt the model or
representation parameters on that task.

More precisely, since GANs main components are the Generator and Critic the
idea is to pre-train them independently, that is in a non-adversarial manner,
and do transfer learning by hooking them together after pre-training and
proceed with the adversarial training, i.e., GAN mode. This process has shown
to produce remarkable results [1] and is the one we follow here. The main
steps are as described as follows:

  1. 1.

Pre-train the Generator with a non-adversarial loss, e.g., using a feature
loss (also known as perceptual loss) [4]

  2. 2.

Freeze the pre-trained Generator weights

  3. 3.

Pre-train the Critic as a Binary Classifier (i.e., non-adversarially) using
the pre-trained Generator (in evaluation mode with frozen model weights) to
generate fake brushstrokes. That is, the Critic should learn to discriminate
between real images and the generated ones. This step uses a standard binary
classification loss, i.e., Binary Cross Entropy, not a GAN loss

  4. 4.

Transfer learning for adversarial training (GAN mode): continue the Generator
and Critic training in a GAN setting (Faster)

#### From Brushstrokes to Paintings.

Once the Generator training process is completed, we have a machine that is
able to translate vectors of actions to brushstrokes, in order to generate a
painting we use Intrinsic Style Transfer [5], similar in spirit to Neural
Style Transfer [2] but which does not require a style image. Intuitively, the
features of the content input image and the one produced by the Neural Painter
should be similar. To implement the process, we freeze the Generator model
weights and learn a set of action vectors that when input to the Generator
will produce brushstrokes, that once combined, will create a painting given an
input content image. This process can be also used to visualize ImageNet
classes. The image features are extracted using a VGG16 [7] network as a
feature extractor.

##  3 Conclusion

In this work we have combined the benefits of Neural Painters and Transfer
Learning to reduce the model training time, while retaining the quality
aesthetics for the brushstrokes and paintings generated. Our contribution
allows practitioner artists to experiment with more models that require hours
and not days to train. We are currently working on an approach that does not
require a GAN framework, i.e., using only non-adversarial perceptual losses,
to further reduce training time and computational resources for artwork
generation.

#### Acknowledgments.

We would like to thank Reiichiro Nakano for helping us clarifying doubts
during the implementation of our Neural Painters and for his supportive and
encouraging comments and feedback.

## References

  * [1] Jason Antic, Jeremy Howard, and Uri Manor.  Decrappification, deoldification, and super resolution.  https://www.fast.ai/2019/05/03/decrappify/, 2019.
  * [2] Leon A. Gatys, Alexander S. Ecker, and Matthias Bethge.  A neural algorithm of artistic style, 2015.
  * [3] Ian Goodfellow, Yoshua Bengio, and Aaron Courville.  Deep Learning.  MIT Press, 2016.  http://www.deeplearningbook.org.
  * [4] Justin Johnson, Alexandre Alahi, and Li Fei-Fei.  Perceptual losses for real-time style transfer and super-resolution, 2016\.
  * [5] Reiichiro Nakano.  Neural painters: A learned differentiable constraint for generating brushstroke paintings.  CoRR, abs/1904.08410, 2019.
  * [6] Google Research.  Collaboratory.  https://colab.research.google.com/, 2020.
  * [7] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition, 2015\.
  * [8] S. Thrun and L. Pratt.  Learning to Learn.  Springer US, 2012.

## Supplementary Material

##  Appendix A Learning Neural Painters Faster. Details.

###  A.1 Pre-train the Generator with a Non-Adversarial Loss

The training set consists of labeled examples where the input corresponds to
an action vector and the corresponding brushstroke image to the target. The
input action vectors go through the Generator, which consists of a fully-
connected layer (to increase the input dimensions) and of a Deep Convolutional
Neural Network connected to it. The output of the Generator is an image of a
brushstroke. The loss computed between the images is the feature loss or
perceptual loss [4]. The process is depicted in Figure 2

Figure 2: Pre-train the Generator using a (non-adversarial) feature loss.

###  A.2 Freeze the pre-trained Generator

After pre-training the Generator using the non-adversarial loss, the
brushstrokes look like the ones depicted in Figure 3. A set of brushstrokes
images is generated that will help us pre-train the Critic in the next step.

Figure 3: Sample Brushstrokes from the Generator Pre-trained with a Non-
Adversarial Loss.

###  A.3 Pre-train the Critic as a Binary Classifier

We train the Critic as binary classifier (Figure 4), that is, the Critic is
pre-trained on the task of recognizing true vs generated brushstrokes images
(Step (2)). We use is the Binary Cross Entropy as binary loss for this step.

Figure 4: Pre-train the Critic as a Binary Classifier.

###  A.4 Transfer Learning for Adversarial Training (GAN mode)

Finally, we continue the Generator and Critic training in a GAN setting as
shown in Figure 5. This final step is much faster that training the Generator
and Critic from scratch as a GAN.

Figure 5: Transfer Learning: Continue the Generator and Critic training in a
GAN setting. Faster.

One can observe from Figure 3 that the pre-trained Generator is doing a decent
job learning brushstrokes. However, there are still certain imperfections when
compared to the true strokes in the dataset.

Figure 6: Sample Brushstrokes from the Generator after Adversarial Training
(GAN mode).

Figure 6 shows the output of the Generator after completing a single epoch of
GAN training, i.e., after transferring the knowledge acquired in the pre-
training phase. We can observe how the brushstrokes are more refined and,
although slightly different to the true brushstrokes, they have interesting
textures, which makes them very appealing for brushstrokes paintings.

###  A.5 Painting Generation using Brushstrokes

Once the Generator training process is completed, we have a machine that is
able to translate vectors of actions to brushstrokes, but how do we teach the
machine to paint like a Bob Ross’ apprentice?

To achieve this, we use Intrinsic Style Transfer [5], similar in spirit to
Neural Style Transfer [2] but which does not require a style image.
Intuitively, the features of the content input image and the one produced by
the Neural Painter should be similar.

To implement the process we freeze the Generator model weights and learn a set
of action vectors that when input to the Generator will produce brushstrokes,
that once combined, will create a painting given an input content image.The
image features are extracted using a VGG16 [7] network as a feature extractor,
denoted as CNN in Figure 7, which depicts the whole process.

Note that the optimization process is targeted to learn the tensor of actions,
while the remaining model weights are not changed, that is, the ones of the
Neural Painter and CNN models.

Figure 7: Painting with Neural Painters using Intrinsic Style Transfer. Figure
8: Progressive steps of Neural Painting generating brushstrokes to optimize a
the perceptual loss of the image at the left. Figure 9: Neural Painter
painting Pandas. Figure 10: Neural Painter painting Hippos. Figure 11: Neural
Painter painting Jaguars. Figure 12: Neural Painter painting Ticks. Figure 13:
Neural Painter painting Loggerhead Turtles. Figure 14: Neural Painter painting
Leatherback Turtles. Figure 15: Neural Painter painting Mud Turtles. Figure
16: Neural Painter painting Loggerhead Owls.