Dynamic modeling of the motions of variable-shape wave energy converters

2025-08-18 0 0 2.59MB 29 页 10玖币
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Dynamic modeling of the motions of variable-shape wave energy
converters
Mohamed A. Shabaraa,∗,Ossama Abdelkhalika
aDepartment of Aerospace Engineering, Iowa State University, Ames, IA, 50011, USA
ARTICLE INFO
Keywords:
Wave Energy Converter
Point Absorber
Flexible Material WEC
Variable Shape Wave Energy Con-
verter
Variable Geometry Wave Energy Con-
verter
Spherical Shell
Lagrangian Mechanics
Rayleigh-Ritz Approximation
ABSTRACT
In the recently introduced Variable-Shape heaving wave energy converters, the buoy changes its
shape actively in response to changing incident waves. In this study, a Lagrangian approach for
the dynamic modeling of a spherical Variable-Shape Wave Energy Converter is described. The
classical bending theory is used to write the stress-strain equations for the flexible body using
Love’s approximation. The elastic spherical shell is assumed to have an axisymmetric vibrational
behavior. The Rayleigh-Ritz discretization method is adopted to find an approximate solution for
the vibration model of the spherical shell. A novel equation of motion is presented that serves as
a substitute for Cummins equation for flexible buoys. Also, novel hydrodynamic coefficients that
account for the buoy mode shapes are proposed. The developed dynamic model is coupled with
the open-source boundary element method software NEMOH. Two-way and one-way Fluid-
Structure Interaction simulations are performed using MATLAB to study the effect of using a
flexible shape buoy in the wave energy converter on its trajectory and power production. Finally,
the variable shape buoy was able to harvest more energy for all the tested wave conditions.
Nomenclature
= Damping Coefficient (Ns/m)
= Young Modulus of Elasticity (MPa)
= Sphere thickness (m)
= Volume (m3)
= Time (sec)
= Sphere Radius (m)
= Material Destiny (kg/m3)
= Poisson’s Ratio
= Rayleigh-Ritz Coefficient
= Frequency (rad/sec)
𝟏= Identity matrix = [𝟏1𝟏2𝟏3]
Subscripts and Superscripts
hydro = hydrodynamic
= buoyancy
hst = hydrostatic
 = Power Take-off Unit
= Water
ref = reference
rad = radiation
†= Pseudo Inverse
This document is the results of a research project funded by The National Science Foundation.
∗Corresponding Author
mshabara@iastate.edu (M.A. Shabara); ossama@iastate.edu (O. Abdelkhalik)
https://www.aere.iastate.edu/ossama/ (O. Abdelkhalik)
ORCID(s): 0000-0002-8942-1412 (M.A. Shabara); 0000-0003-4850-6353 (O. Abdelkhalik)
Shabara & Abdelkhalik: Preprint submitted to Elsevier Page 1 of 29
arXiv:2210.17297v3 [math.DS] 12 Dec 2022
2
1. Introduction
Oceans are colossal reservoirs of energy of particularly high density energy [1]. The total theoretical ocean energy
potential is estimated to be 29.5PWh/yr [2], which is more than the US electric power needs in 2020. Despite its
significant potential, ocean energy is still a very small portion of the overall renewable energy production [3].
One widely used concept for harvesting wave power is the heaving Wave Energy Converter (WEC). In its simplest
form, this point absorber device may consist of a floating buoy connected to a vertical hydraulic cylinder (spar) attached
at the bottom of the seabed. As the buoy moves due to the wave and control forces, the hydraulic cylinders drive
hydraulic motors, and the motors drive a generator [4]. The forces on the floating buoy are the excitation, radiation,
and hydrostatic forces [5]. The excitation force is due to the wave field and the buoy’s geometry. The motion of the
buoy itself creates waves which in turn create the radiation forces. The hydrostatic force accounts for the buoyancy
force and weight of the buoy. In most of the current wave harvesting devices, the WEC has a Fixed-Shape Buoy (FSB).
The equation of motion for a heave-only 1-DoF FSB WEC is [6]:
 () =
excitation force

∫∞
−∞
()(−, )+
radiation force

− () − ∫
−∞
()(−)−(1)
where is the buoy mass, is the control force, is the time, is the heave displacement of the buoy from the sea
surface, and is the hydrostatic force that reflects the spring-like effect of the fluid. The is the wave surface elevation
at buoy centroid, is the excitation force, and is the impulse response function defining the excitation force in
heave. The radiation force is , where is a frequency-dependent added mass, and is the impulse response function
defining the radiation force in heave.
For a FSB, the convolution integral part of the radiation force in Eq. (1) can be approximated using a state space
model of states, = [1,⋯,  ], which outputs the radiation force [7]:
=+and =,(2)
where the constant radiation matrices and are obtained by approximating the impulse response function in the
Laplace domain, as detailed in several references such as [8].
Two important aspects impact the energy converted from oceans: the control force and the buoy’s shape. For an
FSB WEC, linear dynamic models are widely used in control design e.g. [4,9–17]. Usually, the control is designed
to maximize the mechanical power of the WEC; many references adopt different approaches to achieve optimality.
Often, the resulting control forces have a spring-like component in addition to the resisting force [18,19]. Hence, the
Power Take-Off (PTO) unit needs to have a bidirectional power flow capability, which is typically complex and more
expensive. In analyzing the shapes of FSB WECs, the use of any non-cylindrical shape requires the use of non-linear
hydro models [20,21]. This is the reason that most studies assume cylindrical shape of the FSB WEC.
From an economic perspective, the cost of having the complex bidirectional power flow PTO to maximize harvested
energy is high. Moreover, the structure of a FSB WEC needs to be designed to withstand very high loads at peak times
despite operating at a much less load most of the time. This impacts the structural design and increases the cost. To
mitigate this peak load, geometry controlled OSWEC was recently proposed in references [22–25], where controllable
surfaces, along with a wave-to-wave control, are used to maximize power capture, increase capacity factor, and reduce
design loads. The latter controlled-geometry OSWEC changes shape only when the wave climate changes, and hence
it can be considered similar to the case of an FSB WEC when it is not in the transition from one geometry to another.
A geometry control of the overtopping WEC is proposed in reference [26]. The slope angle and crest freeboard of
the device is made adaptive to the sea conditions by geometry control. Reference [27] proposed a variable flap angle
pitching device. The resonance characteristics of the WEC can be altered by controlling the angle of the flap. Later,
reference [28] proposed a floating airbag WEC that has a longer resonance period without implementing phase control.
The concept of a variable-shape buoy (VSB) WEC was recently introduced to reduce the complexity of the
PTO. A VSB WEC changes its shape continuously. The wave/WEC interaction produced by the VSB WEC can be
leveraged to produce more power without adding complexity to the PTO unit. Specifically, Zou et al. [29] proposed the
Variable-Shape point absorber; their original design comprises a pressurized gas chamber attached to a set of multiple
controllable moving panels. This VSB WEC is controlled by a simple linear damping PTO unit [29]. A low-fidelity
3
dynamic model is derived in [29] to demonstrate the superiority of the VSB WEC compared to the FSB WEC. The
average power harvested using the VSB WEC in [29] is about 18% more compared to the FSB WEC.
In another study, references [30,31] present three-dimensional two-way Fluid-Structure Interaction (FSI) high
fidelity simulations, using the ANSYS software package, to simulate a spherical VSB WEC. The WEC in [30,31] has
a hyper-elastic hollow shell of radius 2m. The internal volume contains trapped gas that helps in creating a restoring
moment. The device is simulated in a numerical wave tank (NWT) with dimensions 80 × 60 × 60 m3, and damping
regions at the sides and at the outlet of the NWT. The free surface height was at 40 m. Their simulation captured the
highly non-linear behavior of the VSB WEC and showed an enhancement in the heave displacement and velocity for
the VSB WEC compared to a similar-size FSB WEC. Reference [32] presents a study that uses a similar approach but
applies a passive control force. The PTO force is dependent on the WEC heave velocity (   = − ), where is
a constant damping coefficient. The results showed an increase in the heave displacement and velocity for the VSB
WEC over the similar-size FSB WEC. The results also show an increase in the harvested energy of about 8%.
As can be seen from the above discussion, studies on VSB WECs currently use either high fidelity numerical
software tools or rough approximate low fidelity tools [33] for simulations. The high fidelity tools are computationally
expensive, and the above low fidelity simulations cannot capture important features in this FSI phenomenon.
In this work, Lagrangian mechanics and Rayleigh-Ritz approximation are used to derive novel equation of motions
for variable shape wave energy converters (Eqs.(106),(126) and (128)). Novel expressions for the generalized added
mass, damping, hydrostatic, and excitation forces and coefficients are also derived analytically for two-way and one-way
FSI schemes (sections 4and 6). The paper is organized as follows: In section 2, the kinematics of the flexible buoy
are presented, and then the kinetic and potential energies for the special case of spherical buoys are presented [34],
noting that the use of spherical buoys is only to demonstrate the utility of the model, and the derived aforementioned
generalized hydrodynamic/hydrostatic coefficients and forces can be used with any flexible buoy geometry. In section
3, the equations of motion for flexible shell buoys are derived using Lagrangian mechanics for the free unconstrained
vibration. In section 4, a generalized form for the hydrodynamic forces and coefficients are proposed, for regular and
irregular waves. Section 5presents the most general form of the proposed equation of motion for VSB WECs and
discusses the two-way FSI scheme for the proposed model. To reduce the computational time associated with the
two-way FSI schemes, Reynolds averaging is applied to obtain novel hydrodynamic coefficients for the one-way FSI
in section 6. The model validation is discussed in section 7. Finally, the numerical simulation results for the one-way
and two-way FSI for regular and irregular waves are presented in section 8.
2. Kinetic and Potential Energies of Spherical Shell Buoys
The derivation of the equation of motion in the current study uses Lagrangian mechanics; hence, the calculation
of the kinetic and potential energies is required. This section starts with the layout of the used reference frames, then
a description of the domain discretization technique used in the current study. The system’s kinematics are derived
in subsection 2.2; the result is then used to calculate the kinetic and potential energies in subsections 2.3 and 2.4,
respectively.
Consider a flexible buoy for which the non-deformed shape is spherical. As shown in Fig. (1), the inertial frame is
denoted as
𝒂and can be described as:
𝒂=
𝒂1,
𝒂2,
𝒂3(3)
The body-fixed frame
𝒔is attached to the buoy’s center of mass. Any point on the buoy’s surface can be specified using
the two coordinates and , as illustrated in Fig. (1). Consider an infinitesimal mass at the surface of the buoy; we
introduce the reference frame
𝒆which is attached to that infinitesimal mass on the buoy’s surface before deformation,
and its third axis
𝒆3is aligned with the radius of the non-deformed buoy shape.
Hence, the reference frame
𝒆is obtained by rotating
𝒔by an angle around the
𝒔3then by an angle around the
second intermediate frame as follows.
(, ) = 2()3()(4)
where ()represents a fundamental transformation matrix of a single rotation of angle about the coordinate ,
= 2,3. The reference frame
𝒄is centered at the infinitesimal mass on the surface such that
𝒄3is normal to the surface.
The angle is the angle between
𝒔3and
𝒄3. In the analysis presented in this paper, it is assumed that the deformations
4
Figure 1: Deformed (Solid Black Line) and non-deformed Buoy (Dashed Blue Line)
are axisymmetric about the
𝒂3axis; hence the axis
𝒄2is always perpendicular to the page. The axes
𝒂2,
𝒔2, and
𝒆2
are also perpendicular to the page. If the shape is not deformed from its original spherical shape, then the frames
𝒆
and
𝒄coincide. The frames
𝒆and
𝒄become different, in general, when the shape is deformed. For the FSB WEC the
reference frames
𝒆and
𝒄coincide; this applies to the VSB WEC at the initial time before deformation.
References [30,32,35] carried out high-fidelity FSI simulations using ANSYS for spherical VSB WECs, and they
found that the steady-state response for the VSB WECs is close to being axisymmetric; thus in this work the VSB
WEC response is assumed to be axisymmetric to simplify the analysis.
The coordinate transformation matrix from the
𝒂frame to the
𝒔frame is computed in this paper using the 3-2-1
Euler angle sequence as (, , ) = 1()2()3().
Since the changes in the buoy shape are assumed axisymmetric, we can express the deformation vector (displace-
ment) as a function of only the angle and the time . This deformation vector can be expressed in the
𝒆frame as:
𝒓(, ) = (, ) 0 (, )(5)
where the second component (normal to the page plane) is set to zero because of the axisymmetry of the deformation,
(, )is the displacement component in the
𝒆1direction, and (, )is the displacement component in the
𝒆3direction.
In this paper, each of these displacement components is assumed a series of separable functions; that is, each term in
their series can be expressed as a product of two functions, one of them depends only on and the other depends only
on . Moreover, the Rayleigh-Ritz approximation is used to obtain an approximate solution for the displacement vector
as discussed in the following section.
2.1. Rayleigh-Ritz Approximation
The approximation method used in this work is the Rayleigh-Ritz method. Each component of the displacement
vector
𝒓 is assumed to have the following form [34,36,37]:
(, ) =
=1
Ψ
()() = [Ψ
1… Ψ
]

𝚿
1()
⋮
()
𝜼()
=𝚿
()𝜼()(6)
5
(, ) =
=1
Ψ
()() = [Ψ
1… Ψ
]

𝚿
1()
⋮
()
𝜼()
=𝚿
()𝜼()(7)
where the functions Ψ
and Ψ
are trial (admissible) functions of and the functions are functions of time ,∀
= 1,⋯, . Therefore, the displacement vector can be expressed in the
𝒆frame as follows:
𝒓(, ) =
𝚿
()
𝟎
𝚿
()
𝚽
𝜼() = 𝚽()𝜼()(8)
For a spherical shape buoy, the Legendre functions of the first kind [34,38] can serve as shape functions for the
Ritz-Rayleigh method to satisfy the essential geometrical (Dirichlet) boundary conditions [36,39–42] as follows:
Ψ
() = (cos())
, and Ψ
() = (1 + (1 + ))Ω2
1−Ω2
(cos()) (9)
where the coefficients of the equations above form an eigenvector for the Legendre differential equation, i.e. the constant
"A" can take any real value. Ω2
is a dimensionless frequency parameter expressed as [34,37,40]:
Ω2
=1
2(1 − 2)(
±
2− 4
)(10)
where is the Poisson’s ratio, and
=(+ 1) − 2, ∈ℤ+(11)
= 1 + 2+1
12 [(+ 1)2−2](12)
= 3(1 + ) + +1
2
2
(+ 3)(+1+)(13)
From [37,39,40] the natural frequencies in radians per second for spherical shells are calculated using Eq. (14)
2
=
2Ω2
(14)
where is Young’s Modulus, is the non-deformed radius of the shell, and is the density of the shell material.
When = 0, the vibration mode corresponds to the breathing mode (volumetric or pulsating modes) which is a pure
radial vibration mode [40,42,43]. For  > 0, the ±sign in Eq. (10) yields the modes corresponding to the membrane
vibration modes and bending vibration modes. The bending vibration modes are obtained when using the negative
sign; these modes are sensitive to the ∕ratio. On the other hand, the membrane modes are insensitive to the change
in the ∕ratio. Due to the extensional motion of the buoy, only the membrane vibration modes are used in the current
work. To obtain the approximated equations of motion using the Rayleigh-Ritz method, the approximated displacement
vector needs to be substituted in the kinetic and strain energy equations as follows.
2.2. Kinematics of a Flexible Spherical Buoy - Free Vibration
This subsection is concerned with calculating the
𝒓 vector as it is crucial for the calculation of the kinetic
energy of the spherical shell due to the translation and rotational motions as well as the deformation of the sphere
摘要：

Dynamicmodelingofthemotionsofvariable-shapewaveenergyconverters?MohamedA.Shabaraa,<,OssamaAbdelkhalikaaDepartmentofAerospaceEngineering,IowaStateUniversity,Ames,IA,50011,USAARTICLEINFOKeywords:WaveEnergyConverterPointAbsorberFlexibleMaterialWECVariableShapeWaveEnergyCon-verterVariableGeometryWaveEne...

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