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\begin_body

\begin_layout Title
VIVyD – A code for simulating the generalized VIV model
\end_layout

\begin_layout Author
V.
 Denoël
\end_layout

\begin_layout Abstract
- it is a generic model.
 It can be degenerated into various known generalized VIV models.
 The code is provided open so that many other users can use it
\end_layout

\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents

\end_inset


\end_layout

\begin_layout Section
Introduction
\end_layout

\begin_layout Standard
This report presents a numerical implementation for the solution of a system of differential equations that model 2-D Vortex-Induced Vibrations (VIV) with added stochastic forcing in the wake equation.
 This model extends the recent model proposed by Rigo et al.
 [REF],
 and might be useful to simulate a direct response of the model in view of identifying its parameters from experimental data.
\end_layout

\begin_layout Standard
The system features:
\end_layout

\begin_layout Itemize
a generalized cubic expression for the nonlinearity in the wake equation,
 responsible for the limit cycle behavior
\end_layout

\begin_layout Itemize
a coupling between structural displacement and fluid forces,
 through the position of the wake
\end_layout

\begin_layout Itemize
a coupling between fluid forces and structural displacement,
 through position,
 velocity and acceleration of the cylinder
\end_layout

\begin_layout Itemize
additional stochastic loading terms in the wake equation representing the near-wall turbulence in the wake of the cylinder
\end_layout

\begin_layout Standard
In the direct integration approach,
 the governing equations are integrated using the 
\family typewriter
odeint
\family default
 function from 
\family typewriter
SciPy
\family default
.
 
\end_layout

\begin_layout Standard
In the forward identification problem,
 XXX [to de developed]
\end_layout

\begin_layout Section
Mathematical Description of the Model
\end_layout

\begin_layout Subsection
Governing Equations
\end_layout

\begin_layout Standard
The system of equations solved is based on the following set of dimensionless differential equations:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\ddot{y} & +2\xi_{s}\dot{y}+y=M_{0}q+\eta_{1},\label{eq:goveq_1}\\
\ddot{q} & +\Omega\left(aq^{2}+b\dot{q}^{2}-\varepsilon\right)\dot{q}+\Omega^{2}q=A_{2}\ddot{y}+A_{1}\dot{y}+A_{0}y+\eta_{2},\\
\dot{\eta}_{1} & =-\nu_{1}\eta_{1}+\sigma_{1}\,w_{1}(t),\\
\dot{\eta}_{2} & =-\nu_{2}\eta_{2}+\sigma_{2}\,w_{2}(t),\label{eq:goveq_4}
\end{align}

\end_inset

where:
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $y\left(\tau\right)$
\end_inset

 and 
\begin_inset Formula $\dot{y}\left(\tau\right)$
\end_inset

 represent the displacement and velocity of the structure,
 respectively,
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $q\left(\tau\right)$
\end_inset

 and 
\begin_inset Formula $\dot{q}\left(\tau\right)$
\end_inset

 represent the displacement and velocity of the fluid-induced motion,
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\eta_{1}\left(\tau\right)$
\end_inset

 and 
\begin_inset Formula $\eta_{2}\left(\tau\right)$
\end_inset

 represent the additive stochastic forcing,
 obtained by coloring white noise processes 
\begin_inset Formula $w_{1}\left(\tau\right)$
\end_inset

 and 
\begin_inset Formula $w_{2}\left(\tau\right)$
\end_inset

,
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\xi_{s}$
\end_inset

,
 and 
\begin_inset Formula $\Omega$
\end_inset

 are system parameters related to the structural damping and the scaled upstream wind velocity
\end_layout

\begin_layout Itemize
\begin_inset Formula $\varepsilon$
\end_inset

,
 
\begin_inset Formula $a$
\end_inset

,
 and 
\begin_inset Formula $b$
\end_inset

 are used to parameterize the nonlinear restoring force in the wake equation
\end_layout

\begin_layout Itemize
\begin_inset Formula $\nu_{1,2}$
\end_inset

 and 
\begin_inset Formula $\sigma_{1,2}$
\end_inset

 represent the characteristic frequency and intensity of the noise used in both the structural and wake equations,
\end_layout

\begin_layout Itemize
\begin_inset Formula $A_{0}$
\end_inset

,
 
\begin_inset Formula $A_{1}$
\end_inset

,
 
\begin_inset Formula $A_{2}$
\end_inset

 and 
\begin_inset Formula $M_{0}$
\end_inset

 are system parameters related to the fluid-structure interaction
\end_layout

\begin_layout Standard
The terms 
\begin_inset Formula $w_{1}(\tau)$
\end_inset

 and 
\begin_inset Formula $w_{2}\left(\tau\right)$
\end_inset

 represent two independent unit-intensity white noises,
 used to create the Ornstein-Uhlenbeck processes 
\begin_inset Formula $\eta_{1}\left(\tau\right)$
\end_inset

 and 
\begin_inset Formula $\eta_{2}\left(\tau\right)$
\end_inset

.
\end_layout

\begin_layout Subsection
Scaling and non-dimensionalization
\end_layout

\begin_layout Standard
Equations (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:goveq_1"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:goveq_4"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

) are written in a dimensionless version,
 as a function of the dimensionless time
\begin_inset Formula 
\[
\tau=\omega_{0}t
\]

\end_inset

where 
\begin_inset Formula $t$
\end_inset

 is the physical time and 
\begin_inset Formula $\omega_{0}$
\end_inset

 is the natural circular frequency of the structural system.
 Parameter 
\begin_inset Formula $\Omega$
\end_inset

 is used to represent the mistuning between the structure and the fluid system.
 It is related to the upstream wind velocity.
\end_layout

\begin_layout Standard
The structural and fluid variables 
\begin_inset Formula $y\left(\tau\right)$
\end_inset

 and 
\begin_inset Formula $q\left(\tau\right)$
\end_inset

 represent the dimensionless motion and inclination of the wake lamina.
 Usually the structural displacement is expressed as a function of the crossflow diameter 
\begin_inset Formula $D$
\end_inset

,
\begin_inset Formula 
\[
y\left(\tau\right)=\frac{\text{y}\left(t\left(\tau\right)\right)}{D}
\]

\end_inset

where 
\begin_inset Formula $\text{y}(t)$
\end_inset

 is the transverse displacement.
 The fluid coordinate 
\begin_inset Formula $q\left(\tau\right)$
\end_inset

 may have a different meaning depending on which model is considered,
 but typically it is a scaled version of the inclination of the lamina,
 whenever interpreted as such,
 or alternatively the fluctuating lift coefficient.
\end_layout

\begin_layout Subsection
Relation to existing wake-oscillator models
\end_layout

\begin_layout Standard
In this section,
 we review some existing models of the literature and indicate how they can be expressed in the form of the general dimensionless version given above.
 Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Symbols"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 summarizes a few symbols which are common to all models.
\end_layout

\begin_layout Standard
\begin_inset Float table
placement document
alignment document
wide false
sideways false
status open

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="2">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\rho$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 air density (usually 1.25 kg/m
\begin_inset Formula $^{3}$
\end_inset

)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\text{St}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 Strouhal number
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f_{0}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 natural frequency (in Hz)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\omega_{0}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 natural circular frequency (in rad/s)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $D$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 crossflow diameter (in m)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $V$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 upstream wind velocity (in m/s)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f_{s}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
:
 shedding frequency (in Hz)
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
Symbols
\begin_inset CommandInset label
LatexCommand label
name "tab:Symbols"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Hartlen-Currie (1970)
\end_layout

\begin_layout Standard
Hartlen and Currie's model of VIV is usually recognized as a major step in the derivation of models able to reproduce the synchronization features of the phenomenon 
\begin_inset CommandInset citation
LatexCommand cite
key "hartlen1970lift"
literal "false"

\end_inset

.
 Their model reads
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\ddot{x}_{r}+2\xi_{s}\dot{x}_{r}+x_{r} & =a\Omega^{2}c_{L}\label{eq:hc_1}\\
\ddot{c}_{L}+\left(\frac{\gamma}{\Omega}\dot{c}_{L}^{2}-\alpha\Omega\right)\dot{c}_{L}+\Omega^{2}c_{L} & =b\dot{x}_{r}\label{eq:hc_2}
\end{align}

\end_inset

where 
\begin_inset Formula $x_{r}=x/D$
\end_inset

 is the transverse displacement expressed in diameters,
 and 
\begin_inset Formula 
\[
a=\frac{\rho D^{2}}{8\pi^{2}\text{St}^{2}\left(M/L\right)}\quad;\quad\Omega=\frac{f_{s}}{f_{0}}=\text{St}\frac{V}{f_{0}D}
\]

\end_inset

where 
\begin_inset Formula $M/L$
\end_inset

 is the mass per unit length of the body and other symbols are reported in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Symbols"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

.
 Parameters 
\begin_inset Formula $\alpha$
\end_inset

,
 
\begin_inset Formula $\gamma$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

 are tuned on experimental data.
 They notice that the limit cycle amplitude,
 for small 
\begin_inset Formula $\alpha\Omega$
\end_inset

,
 is
\begin_inset Formula 
\[
C_{L_{0}}=\left(\frac{4\alpha}{3\gamma}\right)^{1/2}
\]

\end_inset

which defines the ratio 
\begin_inset Formula $\alpha/\gamma$
\end_inset

 when 
\begin_inset Formula $C_{L_{0}}$
\end_inset

 is determined experimentally.
 To compare meaningful information with experimental data,
 they use the following values for their parameters
\begin_inset Formula 
\[
\text{St}=0.2,\quad a=0.00218,\quad\alpha=0.02,\quad\gamma=0.667,\quad b=0.4.
\]

\end_inset

The chosen values for 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\gamma$
\end_inset

 results in 
\begin_inset Formula $C_{L0}=0.2$
\end_inset

,
 which,
 together with 
\begin_inset Formula $\text{St}=0.2$
\end_inset

,
 are assumed to represent the subcritical range of Reynolds number.
\end_layout

\begin_layout Standard
It is observed that equations (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:hc_1"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

)-(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:hc_2"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

) can be recast in the format of (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:goveq_1"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

)-(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:goveq_4"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

).
 By comparing the coefficients in the different terms,
 the equivalence summary in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Summary"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 (p.
 
\begin_inset CommandInset ref
LatexCommand pageref
reference "tab:Summary"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

) can be obtained.
\end_layout

\begin_layout Subsubsection
Tamura (1979,
 1980,
 1983)
\end_layout

\begin_layout Standard
Tamura's wake oscillator 
\begin_inset CommandInset citation
LatexCommand cite
key "tamura1980wake"
literal "false"

\end_inset

 is described by the following equations
\begin_inset Formula 
\begin{align}
Y''+\left(2\eta+n(f+C_{D})\frac{\Omega}{S^{\star}}\right)Y'+Y & =-\frac{fn\Omega^{2}}{S^{\star2}}\alpha\label{eq:tam_1}\\
\alpha''-2\zeta\Omega\left(1-\frac{4f^{2}}{C_{L0}^{2}}\alpha^{2}\right)\alpha'+\Omega^{2}\alpha & =-m^{\star}Y''-\Omega S^{\star}Y'\label{eq:tam_2}\\
C_{L} & =-f\left(\alpha+\frac{S^{\star}}{\Omega}\dot{Y}\right)\nonumber 
\end{align}

\end_inset

where 
\begin_inset Formula $Y=y/D$
\end_inset

,
 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $C_{L}$
\end_inset

 are the three dimensionless variables in the model.
 The last equation is a memoryless transformation of the angular position of the wake 
\begin_inset Formula $\alpha$
\end_inset

 and the transverse oscillating frequency 
\begin_inset Formula $\dot{Y}$
\end_inset

 which can be used to determine a physical representation for the lift coefficient.
\end_layout

\begin_layout Standard
This model was analytically derived from a Birkhoff oscillator,
 assuming a wake lamina is oscillating behind the cylindrical body.
 It is based on the earlier works of Funakawa who derived a similar model but with a linear equation for the wake,
 which is therefore unable to model all features related to the limit cycle,
 in particular synchronization and phase entrainment.
 All parameters of Tamura's model have a physical meaning since they are derived from a model:
 
\begin_inset Formula $S^{\star}=2\pi\text{St}$
\end_inset

,
 
\begin_inset Formula $f$
\end_inset

 chosen as 
\begin_inset Formula $f=1.16$
\end_inset

 is related to a geometric quantity used to express the lift on a fixed cylinder,
 as a function of the inclination of the lamina,
 
\begin_inset Formula $C_{D}=1.2$
\end_inset

 is the drag coefficient,
 
\begin_inset Formula $n=\rho D^{2}/2\left(M_{c}/s\right)$
\end_inset

 is the mass ratio (
\begin_inset Formula $M_{c}/s$
\end_inset

 is the mass per unit length of the cylinder),
 
\begin_inset Formula $\zeta=0.038$
\end_inset

 represents the negative damping associated with the effect of the discharged vortex,
 and 
\begin_inset Formula $m^{\star}=0.625$
\end_inset

 is another geometric parameter representing the position of the centroid of the lamina.
 Finally 
\begin_inset Formula $C_{L_{0}}$
\end_inset

 is the amplitude of the lift coefficient for a fixed cylinder.
 In the original paper it is tentatively proposed to use 
\begin_inset Formula $C_{L_{0}}=0.4$
\end_inset

 since several authors had observed RMS values of 
\begin_inset Formula $C_{L_{0}}$
\end_inset

 in the range 
\begin_inset Formula $\left[0.2;0;6\right]$
\end_inset

.
\end_layout

\begin_layout Standard
In his paper 
\begin_inset CommandInset citation
LatexCommand cite
key "tamura1980wake"
literal "false"

\end_inset

 illustrates the results of his model by comparison with various experiments.
 The mass ratio 
\begin_inset Formula $n$
\end_inset

 varies in the different simulations to match experimental tests:
\end_layout

\begin_layout Itemize
Feng (19xx):
 
\begin_inset Formula $n=0.00257$
\end_inset

,
 i.e.
 
\begin_inset Formula $M_{0}=-\frac{fn\Omega^{2}}{(2\pi\text{St})^{2}}=-0.0019\Omega^{2}$
\end_inset


\end_layout

\begin_layout Itemize
Ferguson (19xx):
 
\begin_inset Formula $n=0.00330$
\end_inset

,
 i.e.
 
\begin_inset Formula $M_{0}=-\frac{fn\Omega^{2}}{(2\pi\text{St})^{2}}=-0.00242\Omega^{2}$
\end_inset


\end_layout

\begin_layout Itemize
Yamaguchi (19xx):
 
\begin_inset Formula $n=0.00178$
\end_inset

,
 i.e.
 
\begin_inset Formula $M_{0}=-\frac{fn\Omega^{2}}{(2\pi\text{St})^{2}}=-0.00131\Omega^{2}$
\end_inset


\end_layout

\begin_layout Standard
It is observed that equations (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:tam_1"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

)-(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:tam_2"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

) can be recast in the format of (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:goveq_1"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

)-(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:goveq_4"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

).
 By comparing the coefficients in the different terms,
 the equivalence summary in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Summary"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 (p.
 
\begin_inset CommandInset ref
LatexCommand pageref
reference "tab:Summary"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

) can be obtained.
\end_layout

\begin_layout Subsubsection
Fachinetti,
 de Langre,
 Biolley
\end_layout

\begin_layout Subsubsection
Rigo,
 Andrianne,
 Denoël (2022)
\end_layout

\begin_layout Standard
The Rigo et al.
 model is a wake model only.
 It is already written in a dimensionless form,
 very similar to the generic format discussed and implemented here:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\ddot{q}-\left(\alpha q^{2}+\beta q\dot{q}+\gamma\dot{q}^{2}+\delta\right)\dot{q}+q=\eta.
\]

\end_inset

Comparison with the general form shows that 
\begin_inset Formula $\Omega=1$
\end_inset

,
 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\ddot{q}+\Omega\left(aq^{2}+b\dot{q}^{2}-\varepsilon\right)\dot{q}+\Omega^{2}q=A_{2}\ddot{y}+A_{1}\dot{y}+A_{0}y+\eta_{2}.
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Denoël
\end_layout

\begin_layout Subsubsection
Krenk
\end_layout

\begin_layout Subsubsection
Ogink-Metrikine
\end_layout

\begin_layout Subsubsection
Summary
\end_layout

\begin_layout Standard
\begin_inset Float table
placement document
alignment document
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="14" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
HC
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
FDLB
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
RAD
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
KX
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
OM
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y\left(\tau\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x_{r}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $q\left(\tau\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $c_{L}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\alpha$
\end_inset


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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $q\left(\tau\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\xi_{s}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\xi_{s}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\xi_{s}+\frac{\Omega n(f+C_{D})}{4\pi\text{St}}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\Omega$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\Omega$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\Omega$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\varepsilon$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\alpha$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\begin_inset Formula $\zeta$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\delta$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $a$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $8\Omega\zeta\frac{f^{2}}{C_{L_{0}}^{2}}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-\alpha$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $b$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{\gamma}{\Omega^{2}}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-\gamma$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $A_{0}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $-$
\end_inset


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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-m^{\star}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $a\Omega^{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-\frac{fn\Omega^{2}}{(2\pi\text{St})^{2}}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-$
\end_inset


\end_layout

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Summary of the recasting of existing VIV models in the general dimensionless format (
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).
 Legend:
 Hartlen-Currie (HC),
 Tamura (T),
 Fachinetti et al.
 (FDLB),
 Rigo et al.
 (RAD),
 Denoël (D),
 Krenk-XXX (KX),
 Ogink-Metrikine (OM).
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Numerical values proposed by some authors to model VIV in the subcritical Reynolds regime.
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name "tab:Summary-1"

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\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Relation to existing spectral models
\end_layout

\begin_layout Section
Numerical Method for the Direct Integration
\end_layout

\begin_layout Standard
The system of equations is solved numerically using the 
\family typewriter
odeint
\family default
 function from the 
\family typewriter
SciPy
\family default
 library,
 which applies a variable-coefficient adaptive step-size integration method.
 The time series 
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\end_inset

 spans a duration of 2500 units,
 discretized into 5000 steps.
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To simulate realistic VIV dynamics under stochastic forcing,
 we generate a white noise process 
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\end_inset

,
 which drives the stochastic component of the system through the equation for 
\begin_inset Formula $\dot{n}_{0}$
\end_inset

.
 This noise is interpolated to match the time steps of the solution.
\end_layout

\begin_layout Section
Statistical Analysis
\end_layout

\begin_layout Standard
After solving the system of differential equations,
 we analyze the solution by computing:
 
\end_layout

\begin_layout Itemize
The mean and standard deviation of the displacement 
\begin_inset Formula $y_{0}$
\end_inset

,
 
\end_layout

\begin_layout Itemize
The envelope of the solution using a simple maxima-based approach,
 
\end_layout

\begin_layout Itemize
Histograms of the solution and its envelope to study the distribution of the response.
 
\end_layout

\begin_layout Section
Validations
\end_layout

\begin_layout Subsection
Tamura 1983 - OK
\end_layout

\begin_layout Standard
Tam81 wrong ??
\end_layout

\begin_layout Standard
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placement document
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	filename figures/validation_TamuraFeng1979.pdf
	lyxscale 20
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\end_inset


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\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout

\end_layout

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\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
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\begin_inset Graphics
	filename figures/validation_Tamura1983.pdf
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\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Hartlen-Currie 1970
\end_layout

\begin_layout Standard
\begin_inset Float figure
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	filename figures/validation_HartlenCurrie.pdf
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\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout

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\end_inset


\end_layout

\begin_layout Section
Results
\end_layout

\begin_layout Standard
The resulting time series is analyzed for different values of the system parameters 
\begin_inset Formula $\xi_{s}$
\end_inset

 and 
\begin_inset Formula $\eta$
\end_inset

.
 The output includes both the statistical properties of the structural displacement and the fluid-induced motion,
 as well as the statistical properties of the envelope of these motions.
 The results are saved for further post-processing.
\end_layout

\begin_layout Section
Numerical Method for the Inverse Identification Problem
\end_layout

\begin_layout Standard
to be developed
\end_layout

\begin_layout Section
Conclusion
\end_layout

\begin_layout Standard
This numerical approach provides a way to simulate and statistically analyze Vortex-Induced Vibrations in a structure under stochastic forcing.
 The solution includes both the time evolution of the system and statistical properties,
 which are crucial for understanding the long-term behavior of VIV in stochastic environments.
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "/Users/vincentDenoel/Dropbox/001_ULg/001_Recherches/001_Vent/_VIV/references/VIV-bibliography"
options "plain"

\end_inset


\end_layout

\end_body
\end_document
