Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations
Combining the variational method by Lukovsky – Miles and the Narimanov – Moiseev asymptotics, a nonlinear modal system describing the resonant liquid sloshing in an upright circular cylindrical tank is derived. Sloshing occurs due to a small-amplitude periodic or almost-periodic excitation with the...
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Lukovsky, I. Ovchynnykov, D. Timokha, A. 2021-02-01T15:44:55Z 2021-02-01T15:44:55Z 2011 Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations / I. Lukovsky, D. Ovchynnykov, A. Timokha // Нелінійні коливання. — 2011. — Т. 14, № 4. — С. 482-495. — Бібліогр.: 41 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/175504 517.9 Combining the variational method by Lukovsky – Miles and the Narimanov – Moiseev asymptotics, a nonlinear modal system describing the resonant liquid sloshing in an upright circular cylindrical tank is derived. Sloshing occurs due to a small-amplitude periodic or almost-periodic excitation with the forcing frequency close to the lowest natural sloshing frequency. In contrast to the existing nonlinear modal systems based on the Narimanov – Moiseev asymptotic intermodal relationships, the derived modal equations: (i) contain all the necessary (infinitely many) generalized coordinates of the second and the third orders, (ii) include exclusively nonzero hydrodynamic coefficients for which (iii) rather simple computational formulas are found. As a consequence, the modal equations can be used in analytical studies of nonlinear sloshing phenomena that will be demonstrated in the forthcoming Part 2. Комбiнуючи варiацiйний метод Луковського – Майлса та асимптотику Нарiманова – Моiсеєва, побудовано нелiнiйну модальну систему, що описує резонанснi коливання рiдини у вертикальному круговому цилiндричному резервуарi. Коливання вiдбуваються завдяки перiодичному чи майже перiодичному збуренню з частотою, близькою до першої власної частоти. На вiдмiну вiд iснуючих нелiнiйних модальних систем, якi базуються на асимптотичних спiввiдношеннях Нарiманова – Моiсеєва, побудованi модальнi рiвняння: (i) включають всi необхiднi (нескiнченну кiлькiсть) узагальненi координати другого та третього порядку, (ii) утримують винятково ненульовi гiдродинамiчнi коефiцiєнти, для яких (iii) знайдено достатньо простi обчислювальнi формули. Як наслiдок, модальнi рiвняння можна використати в аналiтичних дослiдженнях нелiнiйних явищ, що буде продемонстровано в наступнiй частинi 2. en Інститут математики НАН України Нелінійні коливання Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations Асимптотичне нелiнiйне мультимодальне моделювання сплескiв рiдини у вертикальному круговому цилiндричному резервуарi. Ч. 1. Модельнi рiвняння Асимптотическое нелинейное мультимодальное моделирование всплесков жидкости в вертикальном круговом цилиндрической резервуаре. Ч. 1. Модельные уравнения Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations |
| spellingShingle |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations Lukovsky, I. Ovchynnykov, D. Timokha, A. |
| title_short |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations |
| title_full |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations |
| title_fullStr |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations |
| title_full_unstemmed |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations |
| title_sort |
asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. part 1: modal equations |
| author |
Lukovsky, I. Ovchynnykov, D. Timokha, A. |
| author_facet |
Lukovsky, I. Ovchynnykov, D. Timokha, A. |
| publishDate |
2011 |
| language |
English |
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Нелінійні коливання |
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Інститут математики НАН України |
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Article |
| title_alt |
Асимптотичне нелiнiйне мультимодальне моделювання сплескiв рiдини у вертикальному круговому цилiндричному резервуарi. Ч. 1. Модельнi рiвняння Асимптотическое нелинейное мультимодальное моделирование всплесков жидкости в вертикальном круговом цилиндрической резервуаре. Ч. 1. Модельные уравнения |
| description |
Combining the variational method by Lukovsky – Miles and the Narimanov – Moiseev asymptotics, a nonlinear modal system describing the resonant liquid sloshing in an upright circular cylindrical tank is derived. Sloshing occurs due to a small-amplitude periodic or almost-periodic excitation with the forcing frequency close to the lowest natural sloshing frequency. In contrast to the existing nonlinear modal systems based on the Narimanov – Moiseev asymptotic intermodal relationships, the derived modal equations: (i) contain all the necessary (infinitely many) generalized coordinates of the second and the third orders, (ii) include exclusively nonzero hydrodynamic coefficients for which (iii) rather simple computational formulas are found. As a consequence, the modal equations can be used in analytical studies of nonlinear sloshing phenomena that will be demonstrated in the forthcoming Part 2.
Комбiнуючи варiацiйний метод Луковського – Майлса та асимптотику Нарiманова – Моiсеєва, побудовано нелiнiйну модальну систему, що описує резонанснi коливання рiдини у вертикальному круговому цилiндричному резервуарi. Коливання вiдбуваються завдяки перiодичному чи майже перiодичному збуренню з частотою, близькою до першої власної частоти. На вiдмiну вiд iснуючих нелiнiйних модальних систем, якi базуються на асимптотичних спiввiдношеннях Нарiманова – Моiсеєва, побудованi модальнi рiвняння: (i) включають всi необхiднi (нескiнченну кiлькiсть) узагальненi координати другого та третього порядку, (ii) утримують винятково ненульовi гiдродинамiчнi коефiцiєнти, для яких (iii) знайдено достатньо простi обчислювальнi формули. Як наслiдок, модальнi рiвняння можна використати в аналiтичних дослiдженнях нелiнiйних явищ, що буде продемонстровано в наступнiй частинi 2.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/175504 |
| citation_txt |
Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations / I. Lukovsky, D. Ovchynnykov, A. Timokha // Нелінійні коливання. — 2011. — Т. 14, № 4. — С. 482-495. — Бібліогр.: 41 назв. — англ. |
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| fulltext |
UDC 517.9
ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID
SLOSHING IN AN UPRIGHT CIRCULAR CYLINDRICAL TANK. PART 1:
MODAL EQUATIONS
АСИМПТОТИЧНЕ НЕЛIНIЙНЕ МУЛЬТИМОДАЛЬНЕ МОДЕЛЮВАННЯ
ХЛЮПАННЯ РIДИНИ У ВЕРТИКАЛЬНОМУ КРУГОВОМУ
ЦИЛIНДРИЧНОМУ РЕЗЕРВУАРI.
Ч. 1. МОДАЛЬНI РIВНЯННЯ
I. Lukovsky, D. Ovchynnykov, A. Timokha
Inst. Math. Nat. Acad. Sci. Ukraine
Ukraine, 01601, Kyiv, Tereshchenkivska str., 3
Combining the variational method by Lukovsky – Miles and the Narimanov – Moiseev asymptotics, a
nonlinear modal system describing the resonant liquid sloshing in an upright circular cylindrical tank
is derived. Sloshing occurs due to a small-amplitude periodic or almost-periodic excitation with the for-
cing frequency close to the lowest natural sloshing frequency. In contrast to the existing nonlinear modal
systems based on the Narimanov – Moiseev asymptotic intermodal relationships, the derived modal equati-
ons: (i) contain all the necessary (infinitely many) generalized coordinates of the second and the third
orders, (ii) include exclusively nonzero hydrodynamic coefficients for which (iii) rather simple computati-
onal formulas are found. As a consequence, the modal equations can be used in analytical studies of
nonlinear sloshing phenomena that will be demonstrated in the forthcoming Part 2.
Комбiнуючи варiацiйний метод Луковського – Майлса та асимптотику Нарiманова – Моiсеєва,
побудовано нелiнiйну модальну систему, що описує резонанснi коливання рiдини у вертикаль-
ному круговому цилiндричному резервуарi. Коливання вiдбуваються завдяки перiодичному чи
майже перiодичному збуренню з частотою, близькою до першої власної частоти. На вiдмiну
вiд iснуючих нелiнiйних модальних систем, якi базуються на асимптотичних спiввiдношеннях
Нарiманова – Моiсеєва, побудованi модальнi рiвняння: (i) включають всi необхiднi (нескiнченну
кiлькiсть) узагальненi координати другого та третього порядку, (ii) утримують винятково
ненульовi гiдродинамiчнi коефiцiєнти, для яких (iii) знайдено достатньо простi обчислюваль-
нi формули. Як наслiдок, модальнi рiвняння можна використати в аналiтичних дослiдженнях
нелiнiйних явищ, що буде продемонстровано в наступнiй частинi 2.
1. Introduction. Accounting for liquid sloshing loads is of importance for designing the engi-
neering constructions carrying a liquid cargo. Safety, reliability, stability, and control analysis
of the liquid containing structures have been extensively studied in the context of aircraft and
spacecraft applications, for cargo tanks of automotive vehicles, offshore platforms, and sei-
smic analysis of the elevated water tanks. The studies require comprehensive quantitative and
qualitative examining the coupled fluid-structure dynamics, its modeling and simulation on the
real-time scale. Liquid sloshing response becomes most severe in resonance conditions when
the carrying structure oscillates with a frequency close to the lowest natural sloshing frequency.
Those resonant free-surface motions are strongly nonlinear and must be described by solving
an evolution free-boundary problem in which both instant shapes of the free surface Σ(t) and
the velocity field in the liquid domain Q(t) are the unknowns.
c© I. Lukovsky, D. Ovchynnykov, A. Timokha, 2011
482 ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 4
ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID . . . 483
Under certain circumstances, one can distinguish three different approaches to solving the
nonlinear liquid sloshing problem. The first approach is the Computational Fluid Dynamics
(CFD). A broad variety of numerical methods exists which could be divided into two sub-
classes comprising potential flow, the Navier – Stokes method, and, sometimes, their hybrids
typically based on the domain decomposition method [8, 39]. The CFD methods are universal,
accurate and efficient, especially on the short-time scale when focus is on transient waves. Their
drawback is that they are, generally speaking, computational time demanding, especially for
three-dimensional problems. Furthermore, their applicability can be rather limited when the
tasks consist in simulation and classification of the so-called steady-state wave regimes occur-
ring on the long-time scale and, therefore, requiring long-time simulations with different initial
scenarios.
The second approach is purely analytical. It is developed for studying the steady-state (peri-
odic) solution expected for prescribed small-amplitude harmonic tank excitations. The analyti-
cal approach employs asymptotic methods which have been created by great mathematicians
of the XIX century in the theory of nonlinear ocean waves [3]. An extension of these methods
to nonlinear resonant sloshing in closed basins is often referred to the pioneering paper by
Moiseev [32]. More mathematical details on constructing the steady-state asymptotic solution
by solving a series of recurrence boundary value problems and deriving the so-called secularity
condition (the necessary solvability condition) coupling the forcing frequency and the dominant
response amplitude can be found in [5, 36, 37, 11, 12, 8]. The asymptotic steady-state solution
technique changes with the mean liquid depth. For finite liquid depths, the Taylor expansion of
the nonlinear free-surface conditions with respect to the mean (unperturbed) free surface leads
to cubic algebraic secularity equations and yields the so-called third-order Moiseev asymptotics
causing the dominant response amplitude be of the orderO(ε1/3) where ε is the nondimensional
forcing amplitude. The asymptotic solution methods are generally not applicable to transient
waves, and for modeling the fluid-structure interaction. Furthermore, the asymptotic steady-
state solution is only valid in a matching forcing-frequency range and for a relatively small
forcing amplitude. Forcing frequencies away from this range and increasing the forcing ampli-
tude can lead to the so-called internal (secondary, combinatory) resonance and, thereby, cause
a failure of the Moiseev intermodal asymptotic relationships (Moiseev’s asymptotics).
The third approach is associated with nonlinear multimodal methods whose application
assumes an ideal liquid with irrotational flow and no overturning and breaking waves allowed.
In this paper, we follow this approach to derive an infinite-dimensional system of nonlinear
asymptotic-type modal equations for sloshing in an upright circular cylindrical tank by combi-
ning the variational multimodal method by Lukovsky – Miles [19, 20, 29] and the Narimanov –
Moiseev intermodal asymptotic relationships [34, 35, 32] which can follow from the second
approach or, simply, be postulated. Distinguished details for such a combined variational-and-
asymptotic version of multimodal methods, its difference from others are outlined in revi-
ews [27, 11]. Readers interested in employing other versions of multimodal methods for liquid
sloshing in an upright cylindrical tank are referred to [34, 4, 35, 20, 10] (Narimanov’s modal-type
perturbation method), [33, 38, 15] (fully-nonlinear [non-asymptotic] multimodal [Perko-type]
method), [16, 17, 18] (combining the Lagrange variational principle and perturbation method),
[20, 7, 8] (combining the Bateman – Luke variational principle and perturbation method), and
references therein.
Sloshing of an ideal liquid with irrotational flow introduces a nonlinear free boundary pro-
blem with the two unknowns that are the instant free-surface shapes and the velocity potential.
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 4
484 I. LUKOVSKY, D. OVCHYNNYKOV, A. TIMOKHA
According to the multimodal method concept for liquid sloshing in tanks with upright walls, the
instant free-surface shapes should be defined by the Fourier-type solution with unknown time-
dependent coefficients qi(t) (furthermore, generalized coordinates) in the front of fi(y, z) =
= ϕi(0, y, z), i.e.,
x = f(y, z, t) =
∑
i
qi(t)fi(y, z), (1)
where ϕi(x, y, z) are the so-called natural sloshing modes. Analogous Fourier-type solution
involving ϕi(x, y, z) is used for the velocity potential. Even though there exist different versions
of nonlinear multimodal methods, all of them are developed to transform the original problem
to an infinite-dimensional system of nonlinear ordinary differential (modal) equations coupling
the generalized coordinates qi. However, since derivation of nonlinear multimodal equations
is a difficult mathematical task, each a version proposes a proper analytical way pursuing the
modal equations of desirable structure.
Except for the Perko-type methods, the derivations require a postulation of asymptotic
relationships between the generalized coordinates qi assuming a small set of dominant generali-
zed coordinates. Neglecting the nonlinear terms in qi which have the order higher than the for-
cing input signal associated with the highest-order term, O(ε), leads to the so-called asymptotic
nonlinear modal equations. Using the asymptotic modal equations helps to avoid physically-
unrealistic higher harmonics which give negligible contribution to liquid response, but may
cause the stiffness of the differential (modal) equations as it was observed for the Perko-type
simulations [15].
The Narimanov – Moiseev asymptotics [34, 35, 32] is the most-often accepted asymptotic
relationships used for deriving the asymptotic nonlinear modal systems. They follow from Moi-
seev’s asymptotic solution (the second approach) or can be postulated as it has been done
by Narimanov in his classical works [34, 35]. Adopting the asymptotic relationships reduces
the problem to calculation of the non-zero hydrodynamic coefficients at the polynomial-type
quantities in the asymptotic modal equations. Usually, the number of the nonzero coefficients
is quite limited. Bearing in mind analytical studies based on the asymptotic nonlinear modal
equations, i.e., considering the nonlinear sloshing as an object of either applied mathematics
or theoretical mechanics, strongly requires to exclude the zeros from the modal equations as
well as to provide simple and compact formulas for the nonzero hydrodynamic coefficients.
Of course, using the asymptotic nonlinear multimodal equations as a computational tool [18,
7], i.e., considering the multimodal method as a CFD approach, does not need the analytical
extraction of the zeros.
For rectangular cross-section, the Narimanov – Moiseev asymptotic relationships lead, due
to trigonometric relations between the natural sloshing modes fi, to a nine-dimensional nonli-
near asymptotic modal system. This system is explicitly derived and analytically studied in
[6, 8]. Other cylindrical tank shapes yield, generally speaking, infinite-dimensional asymptotic
multimodal systems. The latter is also true for the case of circular cross-section. The literature
presents various analytically-given finite-dimensional asymptotic modal systems [20 – 22] but
these systems couple only a few of second and third-order generalized coordinates. To the
best authors’ knowledge, the present paper firstly derives in an analytical form the infinite-
dimensional Narimanov – Moiseev’ asymptotic modal system for the circle-based tank provi-
ding that the modal equations (i) contain all the necessary generalized coordinates of the
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 4
ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID . . . 485
Fig. 1. Sketch of an upright circular cylindrical tank and the adopted nomenc-
lature. The geometrical and physical characteristics are scaled in our
analysis by the dimensional tank radius R0 so that, e.g., h in the figure
is the ratio between the mean liquid depth and R0.
second and third order following from the Narimanov – Moiseev asymptotic intermodal relati-
onships, (ii) include exclusively nonzero hydrodynamic coefficients for which (iii) rather simple
computational formulas are found. In the forthcoming Part 2, we will present analytical studies
of the nonlinear resonant sloshing based on the derived modal system and compare the analyti-
cal results with experiments.
2. Statement of the problem. 2.1. Free boundary problem. An upright circular cylindrical
tank is considered which is partly filled with an inviscid incompressible liquid with irrotati-
onal flow. Fig. 1 introduces the basic notations. No overturning waves are assumed. The time-
dependent liquid domain Q(t) is bounded by the free surface Σ(t) and the wetted tank surface
S(t). The mean liquid depth is equal to h. Henceforth in all the mathematical expressions, we
suggest that the liquid characteristics, including h and the gravity acceleration g, are scaled by
the tank radius R0 so that, everywhere in our forthcoming analysis the theoretical radius of the
tank is nondimensional and equal to 1.
The liquid motions are considered in the tank-fixed coordinate system Oxyz whose ori-
gin is situated in the center of the mean free-surface Σ0. The Ox-axis is superposed with the
tank symmetry axis. For brevity, we concentrate on the case when the tank moves translatory
with the velocity v0(t) relative to an absolute Earth-fixed coordinate system Ox′y′z′. Small-
magnitude angular forcing terms can also be accounted for by assuming that these terms are
of the highest order in the Narimanov – Moiseev asymptotic ordering. The latter procedure is
extensively discussed in [8].
The absolute velocity potential Φ(x, y, z, t), and the free surface Σ(t) are the two unknowns
which should be found from the following nonlinear free-boundary problem
∇2Φ = 0, r ∈ Q(t), (2)
∂Φ
∂ν
= v0 · ν +
ft√
1 + |∇f |2
, r ∈ Σ, (3)
∂Φ
∂ν
= v0 · ν, r ∈ S(t), (4)
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 4
486 I. LUKOVSKY, D. OVCHYNNYKOV, A. TIMOKHA
∂Φ
∂t
+
1
2
|∇Φ|2 −∇Φ · v0 + U = 0, r ∈ Σ(t). (5)
Here ν is the outer normal vector, U = (g · r) is the gravity potential with r = (x, y, z),
g = (−g, 0, 0) is the gravity acceleration vector, and x = f(y, z, t) is the free-surface equation.
For the free-boundary problem (2), typical initial conditions (at t = 0) define the initial
liquid shape and velocity field and take the form
f(y, z, 0) = ξ0(y, z), Φ(x, y, z, 0) = Φ0(x, y, z), r ∈ Q(0). (6)
2.2. Variational formulation. In 1976, Miles [29] and Lukovsky [19] independently proposed
to use the Bateman – Luke variational principle for derivation of nonlinear modal systems. Hi-
story of the Bateman – Luke principle starts from 1908, when R. Hargneaves [13] has noted that
the pressure integral can play the role of the Lagrangian in variational formulations of diverse
hydrodynamic problems. The canonical formulation of this principle for an incompressible ideal
liquid is given by Bateman [1]. Furthermore, this formulation was generalized by Luke [28] for
ocean waves and by Lukovsky [20] for liquid sloshing in a tank performing arbitrary spatial
motions. The Bateman – Luke principle for a compressible fluid can be found in [24, 25, 26, 2].
According to Lukovsky [20], the Bateman – Luke principle for (2) – (5) can be formulated
as follows: The free-boundary problem (2) – (5) is associated with the necessary extrema of the
action
W =
t2∫
t1
Ldt, (7)
where the Lagrangian L is defined by the pressure integral
L =
∫
Q(t)
(p− po) dQ = −ρ
∫
Q(t)
[
∂Φ
∂t
+
1
2
|∇Φ|2 −∇Φ · v0 + U
]
dQ (8)
and trial functions satisfy the conditions
δΦ(x, y, z, t1) = δΦ(x, y, z, t2) = δf(y, z, t1) = δf(y, z, t2) = 0. (9)
3. Nonlinear multimodal modeling. The nonlinear multimodal modeling is based on the
Fourier-type solution (1) in which qi(t) are treated as generalized coordinates of the considered
hydromechanical system. Here fi(x, y) is a complete orthogonal system of functions satisfying
the volume conservation condition
∫
Σ0
fi(x, y) dx dy = 0. In addition, one should introduce
the Fourier-type representation of the velocity potential
Φ(x, y, z, t) = v0 · r +
∑
n
Qn(t)ϕn(x, y, z), (10)
where the complete set of harmonic functions ϕn(x, y, z) satisfies both the Laplace equation in
the whole tank domain and the zero Neumann boundary conditions on the wetted body surface.
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ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID . . . 487
Normally, ϕn and fn(x, y) = ϕn(x, y, 0) are the eigenfunctions (natural sloshing modes) of
the spectral boundary problem
∇2ϕn = 0, ~r ∈ Q0;
∂ϕn
∂ν
= κnϕn, ~r ∈ Σ0;
∂ϕn
∂ν
= 0, ~r ∈ S0, (11)
whereQ0 is the mean liquid domain and S0 is the mean wetted tank surface. The natural sloshi-
ng frequencies are defined by the eigenvalues κn via σn =
√
gκn.
The aim of the multimodal modeling is to derive a system of ordinary differential equations
(modal equations) with respect to generalized coordinates qi(t). There are different analyti-
cal schemes (multimodal methods) how to do that; these are shortly outlined in Introduction.
According to [19, 20, 29], the derivation can employ the Bateman – Luke principle instead of
the free-boundary problem (2).
3.1. Lukovsky – Miles’ variational method. Lukovsky [20] showed how to use the Bateman –
Luke principle for deriving the nonlinear modal equations coupling qi(t) and Qn(t). The result
for translatory tank excitations is the following infinite-dimensional system of nonlinear ordi-
nary differential equations:∑
i
∂An
∂qi
q̇i −
∑
k
AnkQk = 0, n = 1, 2, . . . , (12)
∑
n
∂An
∂qi
Q̇n +
1
2
∑
n,k
∂Ank
∂qi
QnQk +
3∑
j=1
(v̇Oj − gj)
∂lj
∂qi
= 0, i = 1, 2, . . . , (13)
where
∂l1
∂qi
=
∫
Σ0
f2
i dS qi,
∂l2
∂qi
=
∫
Σ0
yfi dS,
∂l3
∂qi
=
∫
Σ0
zfi dS, (14)
g = (g1, g2, g3) = (−g, 0, 0), and
An =
∫
Q(t)
ϕn dQ, Ank =
∫
Q(t)
∇ϕn · ∇ϕk dQ. (15)
The nonlinear modal equations (12), (13) are a full analogy of the original free-boundary
problem. Direct simulations by the modal equations (12), (13) imply the so-called Perko’s
numerical method (see Introduction). Lukovsky & Timokha [27] pointed out that these si-
mulations can be stiff for resonant sloshing and, therefore, a certain numerical time-integration
becomes numerically unstable. This physically-unrealistic stiffness is caused by amplification of
higher harmonics which, in the reality, are highly damped due to different dissipative mecha-
nisms. An alternative is to introduce asymptotic relationship between generalized coordinates
and, thereby, exclude (”filter”) the unrealistically high harmonics.
3.2. Narimanov – Moiseev’ asymptotic intermodal relationships for circular-base tank. For
the circular-base tank, the modal solution (1) can be rewritten in the cylindrical coordinate
system as follows:
x = f(ξ, η, t) =
∞∑
m=0
∞∑
n=1
(rm,n(t) sin(mη) + pm,n(t) cos(mη))fmn(ξ), (16)
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488 I. LUKOVSKY, D. OVCHYNNYKOV, A. TIMOKHA
where fmn =
Jm(km,nξ)
Jm(km,n)
(Jm(·) is the Bessel function) and J ′m(km,n) = 0. The zeros of the last
equation defines the eigenvalues κm,n and the natural sloshing frequencies σm,n by the formulas
κm,n = km,ntanh (km,nh) and σ2
m,n = gκm,n. (17)
The generalized coordinates qi as well as rm,n and pm,n are nondimensional (scaled by
the tank radius) and one can introduce asymptotic relations between them. When the forci-
ng frequency σ is close to the lowest natural frequency σ1,1 associated with the two generalized
coordinates p1,1(t) and r1,1(t), the Narimanov – Moiseev asymptotics [32, 20, 30, 31, 27] requires
the asymptotic relation
p1,1 ∼ r1,1 = O(ε1/3), (18)
where ε � 1 implies the nondimensional forcing magnitude.
Postulating (18) and using the trigonometric algebra with respect to the angular coordinate
η, one can establish the second- and third-order generalized coordinates
p0,n ∼ p2,n ∼ r2,n = O(ε2/3); p3,n ∼ r3,n = O(ε), n = 1, 2, . . . ,
(19)
p1,m ∼ r1,m = O(ε), m = 2, 3, . . . .
Remaining generalized coordinates are of the order o(ε) and can be neglected in our nonlinear
asymptotic multimodal analysis.
4. Nonlinear asymptotic multimodal equations. The most general analytical scheme for
combining the Lukovsky – Miles variational method and the Narimanov – Moiseev asymptotics
is described in [27]. Accounting for (18) – (19), the scheme suggests the following steps:
1. Using the Taylor expansion, one should find polynomial expressions (in terms of nondi-
mensional generalized coordinates qi) for ∂An/∂qk and Ank keeping up to the O(ε2/3)-order
and ∂Ank/∂qi keeping the O(ε1/3)-terms.
2. We should find the asymptotic solution Qi = F (qk, q̇k) from modal equations (12) by
substituting previously-found asymptotic expressions for ∂An/∂qk andAnk.This solution should
neglect the o(ε)-terms.
3. We should substitute expressions Qi = F (qk, q̇k) from the previous step into modal
equations (13) and keep up to the O(ε)-terms. This will give the desirable asymptotic modal
equations.
The scheme was fully realized only for upright cylindrical tanks of rectangular shape. By
generalizing [20], the paper [23] showed that the scheme can also be applied for a circular
cylindrical tank. It is implemented in the present paper to obtain the required analytically-given
asymptotic nonlinear modal equations.
Implementing the analytical scheme with 3N (N → ∞) generalized coordinates of the
second order and 4N generalized coordinates of the third order leads to the nonlinear asympto-
tic modal equations which include the following two differential equations for the lowest-
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ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID . . . 489
order generalized coordinates p1,1 and r1,1 :
µ1,1
[
p̈1,1 + σ2
1,1p1,1
]
+ p1,1
N∑
n=1
d
(2)
0,np̈0,n +
N∑
n=1
d
(3)
0,n (p̈1,1p0,n + ṗ1,1ṗ0,n) +
+ d1
(
p2
1,1p̈1,1 + p1,1ṗ
2
1,1 + r1,1p1,1r̈1,1 + p1,1ṙ
2
1,1
)
+
+ d2
(
r2
1,1p̈1,1 + 2r1,1ṙ1,1ṗ1,1 − r1,1p1,1r̈1,1 − 2p1,1ṙ
2
1,1
)
+
+
N∑
n=1
d
(3)
2,n (p̈1,1p2,n + r̈1,1r2,n + ṗ1,1ṗ2,n + ṙ1,1ṙ2,n) +
+
N∑
n=1
d
(2)
2,n (p1,1p̈2,n + r1,1r̈2,n) = −µ1,1κ1,1
k2
1,1 − 1
v̇01, (20a)
µ1,1
[
r̈1,1 + σ2
1,1r1,1
]
+ r1,1
N∑
n=1
d
(2)
0,np̈0,n +
N∑
n=1
d
(3)
0,n (r̈1,1p0,n + ṙ1,1ṗ0,n) +
+ d1
(
r2
1,1r̈1,1 + r1,1ṙ
2
1,1 + r1,1p1,1p̈1,1 + r1,1ṗ
2
1,1
)
+
+ d2
(
p2
1,1r̈1,1 + 2p1,1ṙ1,1ṗ1,1r1,1p1,1p̈1,1 − 2r1,1ṗ
2
1,1
)
+
+
N∑
n=1
d
(3)
2,n (p̈1,1r2,n − r̈1,1p2,n + ṗ1,1ṙ2,n − ṙ1,1ṗ2,n) +
+
N∑
n=1
d
(2)
2,n (p1,1r̈2,n − r1,1p̈2,n) = −µ1,1κ1,1
k2
1,1 − 1
v̇02. (20b)
These equations include both the lowest- and second-order generalized coordinates, but the
third-order generalized coordinates are absent here. Notations for km,n (the roots of the equati-
on J ′m(km,n) = 0), κm,n (see eq. (17)), σm,n (natural sloshing frequency) as well as the translatory
velocity components v̇01(t) and v̇02(t) were explained before. The nondimensional hydrodynamic
coefficients at the nonlinear terms are defined by the end of this section.
The differential equations for finding the second-order generalized coordinates p0,n, p2,n
and r2,n take the form
2µ0,n
[
p̈0,n + σ2
0,np0,n
]
+ d
(1)
0,n
(
ṗ2
1,1 + ṙ2
1,1
)
+ d
(2)
0,n (p̈1,1p1,1 + r̈1,1r1,1) = 0, (21a)
µ2,n
[
p̈2,n + σ2
2,np2,n
]
+ d
(1)
2,n
(
ṗ2
1,1 − ṙ2
1,1
)
+ d
(2)
2,n (p̈1,1p1,1 − r̈1,1r1,1) = 0, (21b)
µ2,n
[
r̈2,n + σ2
2,nr2,n
]
+ 2d
(1)
2,nṙ1,1ṗ1,1 + d
(2)
2,n (p̈1,1r1,1 + r̈1,1p1,1) = 0. (21c)
Here n = 1, . . . , N, i.e., there is 3N ordinary differential equations for these generalized coordi-
nates. Note that equations (21) contain p1,1 and r1,1 defined by (21) and, therefore, one can say
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490 I. LUKOVSKY, D. OVCHYNNYKOV, A. TIMOKHA
that the first and second-order generalized coordinates are nonlinearly coupled by our modal
equations. However, the third-order generalized coordinates p3,n and r3,n are not presented in
(21). Equations for these generalized coordinates take the form
µ3,n
[
r̈3,n + σ2
3,nr3,n
]
+ d3
(
r1,1ṗ
2
1,1 + 2p1,1ṗ1,1ṙ1,1 − r1,1ṙ
2
1,1
)
+
+ d4
(
p2
1,1r̈1,1 + 2r1,1p1,1p̈1,1 − r2
1,1r̈1,1
)
+
N∑
n=1
d
(1)
3,n (ṗ1,1ṙ2,n + ṙ1,1ṗ2,n) +
+
N∑
n=1
d
(2)
3,n (p1,1r̈2,n + r1,1p̈2,n) +
N∑
n=1
d
(3)
3,n (p̈1,1r2,n + r̈1,1p2,n) = 0, (22a)
µ3,n
[
p̈3,n + σ2
3,np3,n
]
+ d3
(
p1,1ṗ
2
1,1 − 2r1,1ṗ1,1ṙ1,1 − p1,1ṙ
2
1,1
)
+
+ d4
(
p2
1,1p̈1,1 − 2p1,1r1,1r̈1,1 − r2
1,1p̈1,1
)
+
N∑
n=1
d
(1)
3,n (ṗ1,1ṗ2,n − ṙ1,1ṙ2,n) +
+
N∑
n=1
d
(2)
3,n (p1,1p̈2,n − r1,1r̈2,n) +
N∑
n=1
d
(3)
3,n (p̈1,1p2,n − r̈1,1r2,n) = 0, (22b)
µ1,n
[
r̈1,n + σ2
1,nr1,n
]
+ d5
(
r̈1,1r
2
1,1 + r1,1p1,1p̈1,1
)
+
+ d6
(
r1,1ṙ
2
1,1 + r1,1ṗ
2
1,1
)
+ d7
(
r̈1,1p
2
1,1 − r1,1p1,1p̈1,1
)
+
+ d8
(
ṙ1,1ṗ1,1p1,1 − r1,1ṗ
2
1,1
)
+
N∑
n=1
d
(1)
4,n (ṗ1,1ṙ2,n − ṙ1,1ṗ2,n) +
+
N∑
n=1
d
(2)
4,n (p1,1r̈2,n − r1,1p̈2,n) +
N∑
n=1
d
(3)
4,n (p̈1,1r2,n − r̈1,1p2,n) +
+ ṙ1,1
N∑
n=1
d
(1)
5,nṗ0,n + r1,1
N∑
n=1
d
(2)
5,np̈0,n + r̈1,1
N∑
n=1
d
(3)
5,np0,n = −µ1,nκ1,n
k2
1,n − 1
v̇02,
(22c)
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ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID . . . 491
µ1,n
[
p̈1,n + σ2
1,np1,n
]
+ d5
(
p̈1,1p
2
1,1 + r1,1p1,1r̈1,1
)
+
+ d6
(
p1,1ṗ
2
1,1 + p1,1ṙ
2
1,1
)
+ d7
(
p̈1,1r
2
1,1 − r1,1p1,1r̈1,1
)
+
+ d8
(
ṙ1,1ṗ1,1p1,1 − p1,1ṙ
2
1,1
)
+
N∑
n=1
d
(1)
4,n (ṗ1,1ṗ2,n + ṙ1,1ṙ2,n) +
+
N∑
n=1
d
(2)
4,n (r1,1r̈2,n + p1,1p̈2,n) +
N∑
n=1
d
(3)
4,n (p̈1,1p2,n + r̈1,1r2,n) +
+ ṗ1,1
N∑
n=1
d
(1)
5,nṗ0,n + p1,1
N∑
n=1
d
(2)
5,np̈0,n + p̈1,1
N∑
n=1
d
(3)
5,np0,n = −µ1,nκ1,n
k2
1,n − 1
v̇01,
(22d)
where n = 1, . . . , N. Equations (22) are linear in p3,n and r3,n and contain nonlinear quantities
in terms of the first- and second-order generalized coordinates.
The most important result of the present paper is that nonzero hydrodynamic coefficients in
(20) – (22) can be effectively calculated by the following quite simple formulas:
d
(1)
0,n = d
(2)
0,n −
d
(3)
0,n
2
, d
(2)
0,n =
π
2
[
2−
k2
0,n
κ0,nκ1,1
]
j(0,n)(1,1)2 ,
d
(3)
0,n = π
[
j(0,n)(1,1)2 −
1
κ2
1,1
(
j
(1,1)2
(0,n) + i(0,n)(1,1)2
)]
, d
(1)
2,n = d
(2)
2,n −
d
(3)
2,n
2
,
d
(2)
2,n =
π
2
[
j(2,n)(1,1)2 −
1
κ2,nκ1,1
(
j
(2,n)(1,1)
(1,1) + 2i(2,n)(1,1)2
)]
,
d
(3)
2,n =
π
2
[
j(2,n)(1,1)2 −
1
κ2
1,1
(
j
(1,1)2
(2,n) − i(2,n)(1,1)2
)]
,
d1 =
π
2κ1,1
[
k4
0,1
(
j(0,1)(1,1)2
)2
4κ01κ1,1
1
j(0,1)2
+ i(1,1)4 − j
(1,1)2
(1,1)2
]
+ d2,
d2 =
π
4κ1,1
(
j
(1,1)(2,1)
(1,1) + 2i(2,1)(1,1)2
)2
κ1,1κ2,1
1
j(2,1)2
− 3i(1,1)4 − j
(1,1)2
(1,1)2
,
d3 =
π
4κ1,1
(
j
(1,1)(2,1)
(1,1) + 2i(2,1)(1,1)2
)
κ1,1κ2,1
(
2i(1,1)(2,1)(3,1) − j
(1,1)(2,1)
(3,1)
)
j(2,1)2
+
+
π
4κ1,1
[
j
(1,1)2
(1,1)(3,1) − i(1,1)3(3,1)
]
+ 2d4,
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492 I. LUKOVSKY, D. OVCHYNNYKOV, A. TIMOKHA
d4 =
π
4κ1,1
(
j
(1,1)(2,1)
(1,1) + 2i(2,1)(1,1)2
)
κ2,1κ3,1
(
6i(1,1)(2,1)(3,1) + j
(2,1)(3,1)
(1,1)
)
j(2,1)2
−
− π
4κ1,1
[
(κ1,1 + κ3,1)
2κ3,1
(
3i(1,1)3(3,1) + j
(1,1)(3,1)
(1,1)2
)]
,
d
(1)
3,n = d
(2)
3,n + d
(3)
3,n −
π
2
j(1,1)(2,n)(3,1) −
j
(1,1)(2,n)
(3,1) − 2i(1,1)(2,n)(3,1)
κ1,1κ2,n
,
d
(2)
3,n =
π
2
j(1,1)(2,n)(3,1) −
j
(2,n)(3,1)
(1,1) + 6i(1,1)(2,n)(3,1)
κ2,nκ3,1
,
d
(3)
3,n =
π
2
j(1,1)(2,n)(3,1) −
j
(1,1)(3,1)
(2,n) + 3i(1,1)(2,n)(3,1)
κ1,1κ3,1
,
d
(1)
4,n = d
(2)
4,n − d
(3)
3,n −
π
2
j(1,1)(2,n)(1,2) −
j
(1,1)(2,n)
(1,2) + 2i(1,1)(2,n)(1,2)
κ1,1κ2,n
,
d
(2)
4,n =
π
2
j(1,1)(2,n)(1,2) −
j
(2,n)(1,1)
(1,1) + 2i(1,1)(2,n)(1,2)
κ2,nκ1,2
,
d
(3)
4,n =
π
2
j(1,1)(2,n)(1,2) −
j
(1,1)(1,2)
(2,n) − i(1,1)(2,n)(1,2)
κ1,1κ1,2
,
d
(1)
5,n = d
(2)
5,n + d
(3)
5,n − π
j(0,n)(1,1)(1,2) −
j
(0,n)(1,1)
(1,2)
κ1,1κ0,n
,
d
(2)
5,n = π
j(0,n)(1,1)(1,2) −
j
(0,n)(1,2)
(1,1)
κ0,nκ1,2
,
d
(3)
5,n = π
j(0,n)(1,1)(1,2) −
j
(1,1)(1,2)
(0,n) + i(0,n)(1,1)(1,2)
κ1,1κ1,2
in which, by definition,
j
(c,d)
(a,b) =
∫
ξ
(∏
fa,b(ka,bξ)
)(∏ d
dξ
fc,d(kc,dξ)
)
dξ,
i
(c,d)
(a,b) =
∫
1
ξ
(∏
fa,b(ka,bξ)
)(∏ d
dξ
fc,d(kc,dξ)
)
dξ
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ASYMPTOTIC NONLINEAR MULTIMODAL MODELING OF LIQUID . . . 493
and there are special indexing rules for i and j exemplified by the formula
j
(1,2)(0,1)(1,2)
(0,2)(2,2)(1,1) = j
(0,1)(1,2)2
(0,2)(1,1)(2,2) =
=
1∫
0
ξ(f0,2(k0,2ξ)f1,1(k1,1ξ)f2,2(k2,2ξ))
(
d
dξ
f0,1(k0,1ξ)
(
d
dξ
f1,2(k1,2ξ)
)2
)
dξ.
Eqs. (22c) and (22d) contain coefficients d5, d6, d7, and d8 which are computed by the
formulas:
d5 = −0, 51201
h1,1
− 0, 16879
h1,2
+
0, 50224
h1,2h1,1h0,1
+
0, 17969
h1,2h2,1h1,1
,
d6 = −1, 34899
h1,1
− 0, 3376
h1,2
+
1, 00448
h1,2h1,1h0,1
+
0, 37908
h2
1,1h0,1
+
0, 35938
h1,2h2,1h1,1
+
0, 23782
h2
1,1h2,1
,
d7 = −0, 11748
h1,1
− 0, 00307
h1,2
+
0, 17969
h1,2h2,1h1,1
,
d8 = −0, 68799
h1,1
− 0, 17186
h1,2
+
0, 37908
h2
1,2h0,1
+
0, 35938
h1,2h2,1h1,1
+
0, 50224
h1,2h1,1h0,1
,
where hm,n = tanh(km,nh) depends on the nondimensional depth.
It may be important for applications that the modal equations (20) – (22) can be rewritten
in the following matrix form:
Q(~q)~̈q + C~q + ~Ψ(~q; ~̇q) = V, (23)
where ~q = (q1,1; q1,2; . . . ; q1,n; q2,1; q2,2; . . . ; q2,n; . . . ; q7,1; q7,2; . . . ; q7,n)T .
5. Conclusions. Bearing in mind analytical studies of nonlinear resonant sloshing in an
upright circular-base tank, the present paper analytically derives a system of nonlinear ordi-
nary differential equations (modal system) facilitating an approximate modeling of sloshing
phenomena. The derivation uses the Narimanov – Moiseev intermodal asymptotic relationships
which cause for this tank shape an infinite number of the generalized coordinates coupled by
the system. In contrast to the existing analytically-given modal equations, the derived system
(i) contains all the necessary generalized coordinates, (ii) includes exclusively nonzero hydrody-
namic coefficients for which (iii) rather simple computational formulas are found. A use of the
modal equations in analytical studies of the nonlinear resonant sloshing will be demonstrated
in the forthcoming Part 2.
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Received 27.04.11
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