Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing
A modal method is developed for the problem of the oscillation of a liquid in a square-section tank, which performs periodic horizontal and angular motions of small amplitude. The analysis shows that the dominant wave component is exclusively determined by the first harmonic of periodic excitation....
Збережено в:
| Дата: | 2017 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2017
|
| Онлайн доступ: | https://trim.imath.kiev.ua/index.php/trim/article/view/60 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Завантажити файл: |  |
Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552587497046016 |
|---|---|
| author | Timokha, A. N. Тимоха, А. Н. Тимоха, О. М. |
| author_facet | Timokha, A. N. Тимоха, А. Н. Тимоха, О. М. |
| author_institution_txt_mv | [
{
"author": "A. N. Timokha",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Timokha, A. N. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-13T11:57:10Z |
| description | A modal method is developed for the problem of the oscillation of a liquid in a square-section tank, which performs periodic horizontal and angular motions of small amplitude. The analysis shows that the dominant wave component is exclusively determined by the first harmonic of periodic excitation. Equivalent harmonic motions are reciprocal or elliptic types. The steady-state resonant wave regimes for this type of perturbation are studied. |
| first_indexed | 2026-08-04T01:01:48Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2016, т. 13, № 3, 266–280
УДК 532.595
Resonant sloshing in a square-base
tank due to an angular-and-horizontal
periodic forcing*
A.N. Timokha1,2
1 Institute of Mathematics of NAS of Ukraine, Kyiev;
2 Centre of Excellence “AMOS”, Norwegian University of Science and
Technology, Trondheim, Norway; atimokha@gmail.com
Розробляється модальний метод для задачi про коливання рiдини у
резервуарi квадратного перерiзу, який виконує перiодичнi горизон-
тальнi та кутовi рухи малої амплiтуди. Аналiз показує, що домiнан-
тна хвильова компонента виключно визначається першою гармонi-
кою перiодичного збурення. Еквiвалентнi гармонiчнi рухи є зворотно-
поступального чи елiптичного типу. Вивчено усталенi резонанснi хви-
льовi режими для таких типiв збурення.
Разрабатывается модальный метод для задачи про колебания жид-
кости в резервуаре квадратного сечения, который совершает пери-
одические горизонтальные и угловые движения малой амплитуды.
Анализ показывает, что доминантная волновая компонента исклю-
чительно определяется первой гармоникой периодического возбужде-
ния. Эквивалентные гармонические движения являются возвратно-
поступательными или эллиптического типов. Изучено установившиеся
резонансные волновые режимы для такого типа возмущений.
*The work was partly supported by the Grant № 0112U001015. The author also
acknowledges the financial support of the Centre of Autonomous Marine Operations
and Systems (AMOS) whose main sponsor is the Norwegian Research Council (Project
number 223254–AMOS).
c○ Timokha A.N., 2016
Resonant sloshing in a square-base tank due to an angular-and- . . . 267
𝐈𝐧𝐭𝐫𝐨𝐝𝐮𝐜𝐭𝐢𝐨𝐧
The paper [1] originated theoretical studies on resonant sloshing in a
square-base tank performing either longitudinal (along parallel walls) or
diagonal harmonic excitations with the forcing frequency close to the
lowest natural sloshing frequency. A weakly-nonlinear multimodal the-
ory was developed. The results on the steady-state wave regimes were
validated by experiments. The forthcoming parts [2] and [3] focused on
amplification of the higher natural sloshing modes and the base ratio per-
turbation. The studies [1–3] were followed up by many researchers who
adopted numerical methods [9–11] and their own versions of the mul-
timodal theory [6–8, 12]. New model tests were also done in [7, 10, 11].
The main focus was on investigating the planar, nearly-diagonal (squares-
like), swirling and irregular resonant steady-state sloshing. As in [2, 4],
the papers [6, 8, 9, 12] also investigated the energy transfer from lower to
higher natural sloshing modes. A novelty was an experimental and nu-
merical analysis of the steady-state resonant sloshing for an oblique (nei-
ther longitudinal nor diagonal) horizontal harmonic forcing [6, 7, 10,11].
The present paper suggests an arbitrary periodic (not necessarily har-
monic!) combined surge-sway-roll-pitch periodic tank motion with a small
amplitude and generalises [1] to identify stable and unstable steady-state
resonant sloshing regimes. This implicitly implies, that (i) the forcing fre-
quency 𝜎 is close to the lowest natural sloshing frequency 𝜎1, (ii) the two
lowest degenerated (Stokes) natural sloshing modes give the dominant
asymptotic contribution, and (iii) the secondary resonance phenomena
can be neglected and, therefore, the Narimanov-Moiseev asymptotic the-
ory is applicable.
𝟏 𝐒𝐭𝐚𝐭𝐞𝐦𝐞𝐧𝐭
A rigid square base tank is partially filled by a perfect incompressible
liquid with the mean depth . Irrotational liquid flows are assumed. The
tank moves with a small amplitude (relative to the base size) by surge,
sway, roll, and pitch; the heave and yaw are zeros. The liquid sloshing is
considered in the non-inertial coordinate system 𝑂𝑥𝑦𝑧 which is fixed with
the rigid tank so that 𝑂𝑥𝑦-plane coincides with the mean free surface Σ0
and 𝑂𝑧 passes through the centre of Σ0. Figure 1 introduces basic nota-
tions including the translatory 𝒗𝑂(𝑡) and instant angular 𝝎(𝑡) velocities
of the tank. The free surface Σ(𝑡) : 𝑧 = 𝑓(𝑥, 𝑦, 𝑡) and the absolute velocity
268 Timokha A. N.
6
η 4 )
(η 5 )
Q(t)
Σ(t)
z
O
0Σ
surge
roll
x
sway
pitch
y
( )
(
1 )
η
η2
( 3= = 0η η )
(
Figure 1. Sketch of a square-base tank which moves periodically by
surge, sway, roll, and pitch so that the tank translatory velocity is 𝒗𝑂(𝑡) =
(𝑣𝑂1(𝑡), 𝑣𝑂2(𝑡), 0) = (𝜂̇1(𝑡), 𝜂̇2(𝑡), 0) and the instant angular velocity is 𝝎(𝑡) =
(𝜔1(𝑡), 𝜔2(𝑡), 0) = (𝜓̇1(𝑡), 𝜓̇2(𝑡), 0) = (𝜂̇4, 𝜂̇5, 0). The 𝑂𝑥𝑦𝑧 system is rigidly
fixed with the tank so that the mean free surface Σ0 belongs to 𝑂𝑥𝑦 and the
origin is in the centre of the rectangle Σ0.
potential Φ(𝑥, 𝑦, 𝑧, 𝑡) must be simultaneously found from the correspond-
ing free-surface problem or its variational analogy [5]. The task consists
of finding an approximate analytical steady-state solution of the problem,
𝑓(𝑥, 𝑦, 𝑡+2𝜋/𝜎) = 𝑓(𝑥, 𝑦, 𝑡) and Φ(𝑥, 𝑦, 𝑧, 𝑡+2𝜋/𝜎) = Φ(𝑥, 𝑦, 𝑧, 𝑡), where
𝜎 is the circular frequency of the periodic tank motions.
The Narimanov–Moiseev nonlinear multimodal theory of the resonant
sloshing is used. This assumes that 𝜎 is close to the natural sloshing fre-
quency 𝜎1 of the two standing Stokes cross-wave modes and these modes
are amplified with the lowest asymptotic order 𝑂(𝜖1/3) where 𝜖 ≪ 1 (all
geometric parameters are scaled by the breadth=width=𝐿1 so that the
tank cross-section becomes a unit square, := /𝐿1 and 𝑔 := 𝑔/𝐿1,
where 𝑔 is the gravity acceleration) is associated with the nondimen-
sional forcing amplitude. As explained in Chapters 8 and 9 of [5], the
Narimanov–Moiseev theory may fail due to the secondary resonant phe-
nomena whose occurrence in the square base tank is expected at critical
and small liquid depths as well as when the forcing amplitude increases
causing the surface-wave breaking and fragmentations [4]. The modal
theory starts with the Fourier (modal) representation
𝑓(𝑥, 𝑦, 𝑡) =
∑︁
𝑖,𝑗≥0,𝑖+𝑗 ̸=0
𝛽𝑖,𝑗(𝑡)𝑓
(1)
𝑖 (𝑥)𝑓
(2)
𝑗 (𝑦), (1)
Resonant sloshing in a square-base tank due to an angular-and- . . . 269
where 𝑓
(1)
𝑖 (𝑥) 𝑓
(2)
𝑗 (𝑦) are the natural sloshing modes and
𝑓
(1)
𝑖 (𝑥) = cos(𝜋𝑖(𝑥+ 1/2)), 𝑓
(2)
𝑖 (𝑦) = cos(𝜋𝑖(𝑦 + 1/2)), 𝑖 ≥ 0 (2)
are the Stokes modes. Instead of working with the original fully-nonlinear
problem, we adopt the weakly-nonlinear approximate (modal) system of
ordinary differential equations coupling 𝛽𝑖,𝑗(𝑡) [1]:
𝑎̈1 + 𝜎2
1,0𝑎1 + 𝑑1(𝑎̈1𝑎2 + 𝑎̇1𝑎̇2) + 𝑑2(𝑎̈1𝑎
2
1 + 𝑎̇21𝑎1) + 𝑑3𝑎̈2𝑎1
+ 𝑑6𝑎̈1𝑏
2
1 + 𝑏̈1(𝑑7𝑐1 + 𝑑8𝑎1𝑏1) + 𝑑9𝑐1𝑏1 + 𝑑10𝑏̇
2
1𝑎1 + 𝑑11𝑎̇1𝑏̇1𝑏1 + 𝑑12𝑏̇1𝑐̇1
= −𝑃1,0(𝜂1 − 𝑆1,0𝜂5 − 𝑔𝜂5) = 𝐾𝑥(𝑡), (3a)
𝑏̈1 + 𝜎2
0,1𝑏1 + 𝑑1(𝑏̈1𝑏2 + 𝑏̇1𝑏̇2) + 𝑑2(𝑏̈1𝑏
2
1 + 𝑏̇21𝑏1) + 𝑑3𝑏̈2𝑏1 + 𝑑6𝑏̈1𝑎
2
1
+ 𝑎̈1(𝑑7𝑐1 + 𝑑8𝑎1𝑏1) + 𝑑9𝑐1𝑎1 + 𝑑10𝑎̇
2
1𝑏1 + 𝑑11𝑎̇1𝑏̇1𝑎1 + 𝑑12𝑎̇1𝑐̇1
= −𝑃0,1(𝜂2 + 𝑆0,1𝜂4 + 𝑔𝜂4) = 𝐾𝑦(𝑡), (3b)
𝑎̈2 + 𝜎2
2,0𝑎2 + 𝑑4𝑎̈1𝑎1 + 𝑑5𝑎̇
2
1 = 0; 𝑏̈2 + 𝜎2
0,2𝑏2 + 𝑑4𝑏̈1𝑏1 + 𝑑5𝑏̇
2
1 = 0, (3c)
𝑐1 + 𝑑1𝑎̈1𝑏1 + 𝑑2𝑏̈1𝑎1 + 𝑑3𝑎̇1𝑏̇1 + 𝜎2
1,1𝑐1 = 0, (3d)
𝑎̈3 + 𝜎2
3,0𝑎3 + 𝑎̈1(𝑞1𝑎2 + 𝑞2𝑎
2
1) + 𝑞3𝑎̈2𝑎1 + 𝑞4𝑎̇
2
1𝑎1 + 𝑞5𝑎̇1𝑎̇2
= −𝑃3,0(𝜂1 − 𝑆3,0𝜂5 − 𝑔𝜂5), (4a)
𝑐21+𝜎
2
2,1𝑐21+ 𝑎̈1(𝑞6𝑐1+ 𝑞7𝑎1𝑏1)+ 𝑏̈1(𝑞8𝑎2+ 𝑞9𝑎
2
1)+ 𝑞10𝑎̈2𝑏1+ 𝑞11𝑐1𝑎1+
+ 𝑞12𝑎̇
2
1𝑏1 + 𝑞13𝑎̇1𝑏̇1𝑎1 + 𝑞14𝑎̇1𝑐̇1 + 𝑞15𝑎̇2𝑏̇1 = 0, (4b)
𝑐12 + 𝜎2
1,2𝑐12 + 𝑏̈1(𝑞6𝑐1 + 𝑞7𝑎1𝑏1) + 𝑎̈1(𝑞8𝑏2 + 𝑞9𝑏
2
1) + 𝑞10𝑏̈2𝑎1 + 𝑞11𝑐1𝑏1+
+ 𝑞12𝑏̇
2
1𝑎1 + 𝑞13𝑎̇1𝑏̇1𝑏1 + 𝑞14𝑏̇1𝑐̇1 + 𝑞15𝑎̇1𝑏̇2 = 0, (4c)
𝑏̈3 + 𝜎2
0,3𝑏3 + 𝑏̈1(𝑞1𝑏2 + 𝑞2𝑏
2
1) + 𝑞3𝑏̈2𝑏1 + 𝑞4𝑏̇
2
1𝑏1 + 𝑞5𝑏̇1𝑏̇2
= −𝑃0,3(𝜂2 + 𝑆0,3𝜂4 + 𝑔𝜂4), (4d)
270 Timokha A. N.
where 𝛽1,0 = 𝑎1, 𝛽2,0 = 𝑎2, 𝛽0,1 = 𝑏1, 𝛽0,2 = 𝑏2, 𝛽1,1 = 𝑐1, 𝛽3,0 = 𝑎3, 𝛽2,1 =
𝑐21, 𝛽1,2 = 𝑐12, 𝛽0,3 = 𝑏3,
𝑃𝑖,0 = 𝑃0,𝑖 =
2
𝜋𝑖
tanh(𝜋𝑖)[(−1)𝑖−1], 𝑆𝑖,0 = 𝑆0,𝑖 =
2
𝜋𝑖
tanh(𝜋𝑖/2) (5)
and 𝜎1 = 𝜎0,1 = 𝜎1,0, 𝜎
2
𝑖,𝑗 = 𝑔𝜋
√︀
𝑖2 + 𝑗2𝜆𝑖,𝑗 tanh(𝜋
√︀
𝑖2 + 𝑗2). The ex-
plicit expressions for the hydrodynamic coefficient and the corresponding
tables are given in [1] and [5]. Following [5], we re-denote, the nondimen-
sional generalised coordinates 𝜂𝑖(𝑡) = 𝑂(𝜖) ≪ 1 determining the periodic
surge, sway, roll and pitch tank motions.
The modal system (3)–(4) is equivalent to the original free-surface
problem within the framework of the Narimanov-Moiseev asymptotic ap-
proximation. It makes it possible to analyse steady-state regimes, their
stability as well as transient waves. Chapter 9 by [5] outlines other mod-
ified weakly-nonlinear modal theories which account for the secondary
resonance.
𝟐 𝐀𝐬𝐲𝐦𝐩𝐭𝐨𝐭𝐢𝐜 𝐬𝐭𝐞𝐚𝐝𝐲-𝐬𝐭𝐚𝐭𝐞 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧𝐬 𝐨𝐟 (𝟑)–(𝟒)
Following [1], we introduce the lowest-order approximation of the steady-
state solution
𝑎1 = 𝐴 cos𝜎𝑡+𝐴 sin𝜎𝑡+ 𝑜(𝜖1/3); 𝑏1 = 𝐵̄ cos𝜎𝑡+𝐵 sin𝜎𝑡+ 𝑜(𝜖1/3) (6)
responsible for the first two Stokes cross-waves, by 𝑓
(1)
1 (𝑥) and 𝑓
(2)
1 (𝑦).
We substitute (6) into (3c), (3d) and (3a), (3b) and (4) to get the second-
and third-order terms of the steady-state solution, respectively. Gather-
ing the first Fourier harmonic components in (3) yields a solvability con-
dition appearing as the following system of nonlinear algebraic equations⎧⎪⎪⎪⎨⎪⎪⎪⎩
1○: 𝐴
[︀
Λ +𝑚1(𝐴
2 +𝐴2) +𝑚2𝐵̄
2 +𝑚3𝐵
2
]︀
+(𝑚2−𝑚3)𝐴𝐵̄𝐵 = 𝜖𝑥,
2○: 𝐵
[︀
Λ +𝑚1(𝐵
2 + 𝐵̄2) +𝑚2𝐴
2 +𝑚3𝐴
2
]︀
+(𝑚2−𝑚3)𝐴𝐴𝐵̄ = 𝜖𝑦,
3○: 𝐴
[︀
Λ +𝑚1(𝐴
2 +𝐴2) +𝑚2𝐵
2 +𝑚3𝐵̄
2
]︀
+(𝑚2−𝑚3)𝐴𝐵̄𝐵 = 𝜖𝑥,
4○: 𝐵̄
[︀
Λ +𝑚1(𝐵
2 + 𝐵̄2) +𝑚2𝐴
2 +𝑚3𝐴
2
]︀
+(𝑚2−𝑚3)𝐴𝐴𝐵 = 𝜖𝑦
(7)
with respect to the lowest-order wave amplitudes 𝐴,𝐴,𝐵, 𝐵̄ = 𝑂(𝜖1/3),
where Λ = 𝜎̄2
1−1 = 𝜎2
1/𝜎
2−1. The 𝑂(𝜖)-order nondimensional amplitude
parameters 𝜖𝑥, 𝜖𝑥, 𝜖𝑦 and 𝜖𝑦 are the first Fourier harmonic components
Resonant sloshing in a square-base tank due to an angular-and- . . . 271
in the right-hand sides of (3a) and (3b)
𝜖𝑥 =
2
𝑇𝜎2
∫︁ 𝑇
0
cos𝜎𝑡𝐾𝑥(𝑡) d𝑡; 𝜖𝑥 =
2
𝑇𝜎2
∫︁ 𝑇
0
sin𝜎𝑡𝐾𝑥(𝑡) d𝑡,
𝜖𝑦 =
2
𝑇𝜎2
∫︁ 𝑇
0
cos𝜎𝑡𝐾𝑦(𝑡) d𝑡; 𝜖𝑦 =
2
𝑇𝜎2
∫︁ 𝑇
0
sin𝜎𝑡𝐾𝑦(𝑡) d𝑡.
(8)
At least one from 𝜖𝑥, 𝜖𝑥, 𝜖𝑦 and 𝜖𝑦 should not be zero. Henceforth,
1. The 𝑂𝑥-axis direction is chosen to get
√︁
𝜖2𝑦 + 𝜖2𝑦 ≤
√︀
𝜖2𝑥 + 𝜖2𝑥 ̸= 0.
2. An appropriate time-phase shift 𝑡 := 𝑡+ 𝜓0 is used to achieve
𝜖𝑥 = 0 and 0 ≤
√︁
𝜖2𝑦 + 𝜖2𝑦 = 𝜖𝑦 ≤ 𝜖𝑥 ̸= 0. (9)
3. The derivation line
0 = (𝐴 1○−𝐴 3○)− (𝐵̄ 2○−𝐵 4○) ≡ −𝜖𝑥𝐴+ (𝜖𝑦𝐵̄ − 𝜖𝑦𝐵) (10)
deduces the solvability condition
𝐴 = 𝛿𝐵̄−𝛿𝐵,where 𝛿 = 𝜖𝑦/𝜖𝑥, 𝛿 = 𝜖𝑦/𝜖𝑥; 0 ≤
√︀
𝛿2 + 𝛿2 ≤ 1. (11)
4. The Moiseev asymptotic condition
Λ = 𝜎̄2
1 − 1 = 𝜎2
1/𝜎
2 − 1 = 𝑂(𝜖2/3) (12)
is adopted providing all quantities in (7) are of the equal asymptotic
order 𝑂(𝜖).
5. The nondimensional coefficients 𝑚𝑖 = 𝑚𝑖() are independent of 𝜎,
their values were computed in [1] to show that 𝑚1 > 𝑚2,𝑚3 > 𝑚2
as well as 𝑚3 > 𝑚1,𝑚3 > 0,𝑚1 < 0,𝑚2 < 0 for > 0.3368.... This
and other critical depths leading to zeros for the appearing linear
combinations of 𝑚1 are avoided in the analysis.
Even though a uniform periodic tank motion is assumed, the dom-
inant wave contribution (6) is uniquely determined by the first Fourier
harmonics of 𝜂𝑖(𝑡), the higher Fourier harmonics only influence the 𝑂(𝜖)
272 Timokha A. N.
y
x
yε
εx
yε yε
x
y y
ε
~
εx
(b) (c)(a)
x x
Figure 2. Three schematic trajectories of the equivalent horizontal harmonic
tank motions that classify the periodic tank excitations by their first Fourier
harmonics (8) with (9). The case (a) implies the reciprocating excitation type
(longitudinal, diagonal and oblique excitations are particular cases) occurring
for 𝜖𝑦 = 0. The elliptic excitation type in the case (b) (𝜖𝑦 = 0) suggests the
major-axis of the ellipse belongs to 𝑂𝑥. The oblique elliptic excitation type in
the panel (c) corresponds to 𝜖𝑦𝜖𝑦 ̸= 0.
asymptotic terms so that they do not affect the stability of the con-
structed stead-state solutions. For any periodic tank motion, one can
introduce an equivalent harmonic horizontal tank excitation
−𝑃1𝜂
*
1(𝑡) = 𝜖𝑥 cos𝜎𝑡; −𝑃1𝜂
*
2(𝑡) = 𝜖𝑦 cos𝜎𝑡+𝜖𝑦 sin𝜎𝑡; 𝜂
*
4 = 𝜂*5 = 0, (13)
(𝑃1 = 𝑃1,0 = 𝑃0,1) leading to the same equations (7) and, therefore, the
periodic solution within to the 𝑂(𝜖) terms. According to (13), the tank
moves along the quadratic curve
(𝜀2𝑦+𝜀
2
𝑦)𝑥
2+𝜀2𝑥 𝑦
2−2𝜀𝑥𝜀𝑦 𝑥𝑦 = 𝜀2𝑥𝜀
2
𝑦 (𝜀𝑥 = 𝜖𝑥/𝑃1, 𝜀𝑦 = 𝜖𝑦/𝑃1, 𝜀𝑦 = 𝜖𝑦/𝑃1)
(14)
in the horizontal plane. The curve is either an straight line (𝜖𝑦 = 0) or
an ellipse (𝜖𝑦 ̸= 0).
We classify the periodic resonant tank excitations by the trajectories
(13). The reciprocating excitation type with 𝜖𝑦 = 0 implies the tank
oscillates along an interval in figure 2 (a). The particular cases are lon-
gitudinal (𝜖𝑦 = 𝜖𝑦 = 0), diagonal (𝜖𝑦 = 0, |𝜖𝑦| = 𝜖𝑥 ̸= 0), and oblique
(𝜖𝑦 = 0, 0 < |𝜖𝑦| < 𝜖𝑥) excitations. The elliptic tank excitation type with
𝜖𝑦 ̸= 0 is shown in figure 2 (b,c). When 𝜖𝑦 = 0, the elliptic trajectory
possesses the axisymmetric shape in figure 2 (b).
The asymptotic steady-state solutions and their stability [1] are dete-
rmined by 𝐴,𝐵, 𝐵̄ (𝐴 = 𝛿𝐵̄ − 𝛿𝐵) which should be analytically found
from (7) as functions of Λ (𝜎/𝜎1). The result is the response curves in
Resonant sloshing in a square-base tank due to an angular-and- . . . 273
the four-dimensional space (𝜎/𝜎1, 𝐴,𝐵, 𝐵̄). The main twofold task con-
sists of describing the response curves and, based on (6), classifying the
corresponding steady-state wave regimes, which are best characterised by
the lowest-order asymptotic wave component
𝑧 = 𝑆(𝑥, 𝑦;𝐴, 𝐵̄) cos𝜎𝑡+ 𝑆(𝑥, 𝑦; 𝛿𝐵̄ − 𝛿𝐵,𝐵) sin𝜎𝑡+ 𝑜(𝜖1/3), (15)
where
𝑆(𝑥, 𝑦; 𝑎, 𝑏) = (𝑎𝑓
(1)
1 (𝑥) + 𝑏𝑓
(2)
1 (𝑦)) (16)
is the combined Stokes mode.
According to (15), there are two types of the wave patterns. When
(𝐴, 𝐵̄) and (𝛿𝐵̄ − 𝛿𝐵,𝐵) are parallel vectors (one of them can be zero),
(15) implies a standing resonant wave by a combined Stokes mode. Par-
ticular cases are the so-called planar, diagonal and nearly-diagonal (squares-
like) steady-state wave regimes specified in [1] for longitudinal and diag-
onal excitations. Whereas (𝐴, 𝐵̄) and (𝛿𝐵̄− 𝛿𝐵,𝐵) are not parallel, (15)
defines a swirling wave, where an almost flat crest travels around each of
the four sides with an almost flat trough on the opposite side.
𝟑 𝐓𝐡𝐞 𝐫𝐞𝐜𝐢𝐩𝐫𝐨𝐜𝐚𝐭𝐢𝐧𝐠 𝐞𝐱𝐜𝐢𝐭𝐚𝐭𝐢𝐨𝐧 𝐭𝐲𝐩𝐞
This excitation type is illustrated by trajectories in figure 2 (a) and,
according to (11), it needs
𝛿 = 0, 𝐴 = −𝛿𝐵, −1 < 𝛿 ≤ 1 (17)
in (7), 𝛿 = tan𝛼, where 𝛼 is the angle between the excitation direction
and 𝑂𝑥. The lowest-order approximation (15) takes then the form
𝑧 = 𝑆(𝑥, 𝑦;𝐴, 𝐵̄) cos𝜎𝑡+𝐵𝑆(𝑥, 𝑦;−𝛿, 1) sin𝜎𝑡+ 𝑜(𝜖1/3) (18)
where the first combined Stokes mode depends on 𝐴 and 𝐵̄.
𝐒𝐭𝐚𝐧𝐝𝐢𝐧𝐠 𝐫𝐞𝐬𝐨𝐧𝐚𝐧𝐭 𝐰𝐚𝐯𝐞 𝐛𝐲 𝑆(𝑥, 𝑦;𝐴, 𝐵̄). Substituting (17) into 2○
and 3○ of (7) transforms these equations to the form 𝐵 [...] = 0. This
implies that 𝐵 ≡ 0 is a particular solution, which determines the standing
wave by the combined Stokes mode 𝑆(𝑥, 𝑦;𝐴, 𝐵̄) where 𝐴 and 𝐵̄ should
be found from 1○ and 4○ of (7). Because 1○ takes the form 𝐴 [...] = 𝜖𝑥 ̸= 0,
𝐴 ̸= 0 and the latter two equations 1○ and 4○ can be rewritten in the
form{︃
𝐵̄
[︀
(𝑚1 −𝑚2)𝐵̄
2 + (𝜖𝑥/𝐴− (𝑚1 −𝑚2)𝐴
2)
]︀
− 𝛿𝜖𝑥 = 0,
Λ = 𝜖𝑥/𝐴−𝑚1𝐴
2 −𝑚2𝐵̄
2, 𝑚1 ̸= 𝑚2.
(19)
274 Timokha A. N.
The system defines the response curves in the (𝜎/𝜎1, 𝐴, 𝐵̄) space. The
curves can be parametrised by 𝐴. The procedure suggests taking 𝐴 ̸= 0,
solving the depressed cubic with respect to 𝐵̄ (which has from one to three
real roots) and, after getting these roots, computing Λ = 𝜎2
1/𝜎
2 − 1 =
Λ(𝐴, 𝐵̄(𝐴)).
Longitudinal excitation type. When 𝛿 = 0, the depressed cubic in
(19) has the zero root 𝐵̄ = 0 and may have the two real roots ±|𝐵̄|
coming from 𝐵̄2 = 𝜖𝑥/𝐴/(𝑚2 − 𝑚1) + 𝐴2 > 0. The first root implies,
according to (18), the so-called planar standing wave by the first Stokes
mode, 𝑧 = 𝐴𝑓
(1)
1 (𝑥) cos𝜎𝑡, but the other two roots imply the so-called
nearly-diagonal (squares-like) steady-state wave regimes, which are, in
fact, the two standing resonant waves in terms of the combined Stokes
modes 𝑆(𝑥, 𝑦;𝐴, |𝐵̄|) and 𝑆(𝑥, 𝑦;𝐴,−|𝐵̄|)).
Diagonal excitation type. When 𝛿 = 1, the depressed cubic in (19) has
the real root 𝐵̄ = 𝐴 which corresponds to the so-called diagonal steady-
state wave (the standing wave by the combined Stokes mode 𝑆(𝑥, 𝑦; 1, 1)).
It may also have two real roots coming from the quadratic equation (𝑚1−
𝑚2)(𝐵̄
2+𝐴𝐵̄)+𝜖𝑥/𝐴 = 0. These two roots determine the aforementioned
nearly-diagonal (squares-like) steady-state wave regimes.
There are no obvious analytical solutions of the depressed cubic for
the oblique excitation type with 0 < 𝛿 < 1 and, therefore, it should be
solved numerically. The number of real roots depends on the discriminant
Δ1(𝐴)=−(𝑚1 −𝑚2)
[︂
4
(︁𝜖𝑥
𝐴
− (𝑚1 −𝑚2)𝐴
2
)︁3
+ 27(𝑚1 −𝑚2)𝛿
2𝜖2𝑥
]︂
. (20)
When Δ1 > 0, the depressed cubic has three different real roots, the case
Δ1 = 0 implies two real roots one of which has the double multiplicity,
and, finally, the negative discriminant causes only one real root.
𝐒𝐰𝐢𝐫𝐥𝐢𝐧𝐠. When 𝐵 ̸= 0, one can divide (10) by 𝐵 and, provided by
(17), express 4○ through 1○, 2○ and 3○. These three equations can, after
tedious derivations, be rewritten in the form
𝛿2𝐵̄3 + 𝛿𝐴(2− 𝛿2)𝐵̄2 +𝐴2(1− 2𝛿2)𝐵̄ + 𝛿
[︀
𝜖1(1− 𝛿2)−𝐴3
]︀
= 0,
𝜖1 =
𝜖𝑥(𝑚2 −𝑚1)
(𝑚2 −𝑚3)(𝑚1 −𝑚3)
, (21a)
𝐵2 =
𝐴
[︀
(𝑚1 −𝑚3)𝐴
2 + (𝑚2 −𝑚1)𝐵̄
2 + 𝛿(𝑚2 −𝑚3)𝐴𝐵̄
]︀
− 𝜖𝑥
𝛿(𝑚2 −𝑚3)𝐵̄ + (𝑚1 −𝑚3 + 𝛿2(𝑚2 −𝑚1))𝐴
> 0,
(21b)
Resonant sloshing in a square-base tank due to an angular-and- . . . 275
Λ = 𝜖𝑥/𝐴+𝛿(𝑚2−𝑚3)𝐵̄𝐵
2/𝐴−𝑚1𝐴
2−𝑚2𝐵̄
2−(𝑚3+𝛿
2𝑚1)𝐵
2. (21c)
The expressions assume that 𝐴 ̸= 0 along a response curve (the fact can
be proved) and the denominator 𝛿(𝑚2−𝑚3)𝐵̄+(𝑚1−𝑚3+𝛿
2(𝑚2−𝑚1))𝐴
is not zero as well.
The formulas (a), (b) and (c) in (21) consequently determine an an-
alytical solution and the response curves in the four-dimensional space
(𝜎/𝜎1, 𝐴, 𝐵̄, |𝐵|) parametrically defined by 𝐴: by taking a real 𝐴 ̸= 0, we
solve the cubic equation (21a) with respect to 𝐵̄ (analytically, by using
Cardano’s formulas, or numerically), compute ±|𝐵| (if exist) by (21b)
and Λ (the forcing frequency ratio 𝜎/𝜎1) by (21c). The cubic equation
equation (21a) may have from three to one real number that depends on
the discriminant
Δ2(𝐴) = −𝜖1𝛿4(𝛿2 − 1)[27𝜖1𝛿
2(𝛿2 − 1) + 4𝐴3(𝛿2 + 1)3], (22)
which is identical to zero for longitudinal (𝛿 = 0) and diagonal (|𝛿| = 1)
excitations. The analytical solution (21) transforms to a standing wave
when the vectors (𝐴, 𝐵̄) and (−𝛿, 1) are parallel (𝐴 = −𝛿𝐵̄). Substituting
𝐴 = −𝛿𝐵̄ into (21a) leads to 𝛿𝜖1(1−𝛿2) = 0 which, again, is only possible
for the two limit cases 𝛿 = 0 and |𝛿| = 1.
Longitudinal excitation type. When 𝛿 = 0, (17) ⇒ 𝐴 = 0 and (21a)
⇒ 𝐵̄ = 0. Remaining 𝐴,𝐵 and Λ are governed by (21b) and (21c) which
are equivalent to the relationships in [1]. This solution is the swirling
steady-state wave regime by the Stokes modes 𝑓
(1)
1 (𝑥) and 𝑓
(2)
1 (𝑦) which
may occur in the two opposite directions due to 𝐵 = ±|𝐵| in (18).
Diagonal excitation type. When 𝛿 = 1, the cubic equation (21a) has,
according to Δ2 ≡ 0, two real roots. The root of the single multiplic-
ity is 𝐵̄ = 𝐴 but the root 𝐵̄ = −𝐴 has the double multiplicity. The
first root determines swirling by the two perpendicular combined Stokes
modes 𝑆(𝑥, 𝑦; 1, 1) and 𝑆(𝑥, 𝑦;−1, 1) which can also occur into two differ-
ent directions since 𝐵 is defined within to the sign. The root 𝐵̄ = −𝐴 is
mathematically impossible as leading to 𝐵2 · 0 = 𝐴 · 0− 𝜖𝑥 ̸= 0 in (21b).
𝟒 𝐓𝐡𝐞 𝐚𝐱𝐢𝐬𝐲𝐦𝐦𝐞𝐭𝐫𝐢𝐜 𝐞𝐥𝐥𝐢𝐩𝐭𝐢𝐜 𝐞𝐱𝐜𝐢𝐭𝐚𝐭𝐢𝐨𝐧 𝐭𝐲𝐩𝐞
This excitation type is associated with the elliptic trajectory of the equiv-
alent horizontal tank motions in figure 2 (a). Mathematically, this implies
𝜖𝑦 = 𝛿𝜖𝑥 = 0 in (7) so that (11) leads to
𝐴 = 𝛿𝐵̄, 0 < 𝛿 ≤ 1. (23)
276 Timokha A. N.
Without less of generality, the counterclockwise direction along the el-
liptic orbit is chosen. The limit case 𝛿 = 1 corresponds to the rotary
(circular) excitation type. The lowest-order asymptotic approximation
(15) gives
𝑧 = 𝑆(𝑥, 𝑦;𝐴, 𝐵̄) cos𝜎𝑡+ 𝑆(𝑥, 𝑦; 𝛿𝐵̄, 𝐵) sin𝜎𝑡+ 𝑜(𝜖1/3) (24)
in terms of the two combined Stokes modes.
𝐒𝐰𝐢𝐫𝐥𝐢𝐧𝐠 𝐛𝐲 𝐭𝐡𝐞 𝐭𝐰𝐨 𝐒𝐭𝐨𝐤𝐞𝐬 𝐦𝐨𝐝𝐞𝐬. Substituting (23) into (7) trans-
forms 3○ and 4○ to the form 𝐵̄[...] = 0 that means that 𝐴 = 𝐵̄ = 0 is
a particular solution of the secular system. The two non-zero amplitude
parameters 𝐴 and 𝐵 can be found from 1○ and 2○ rewritten in the form{︃
𝐵
[︀
(𝑚1 −𝑚3)𝐵
2 + (𝜖𝑥/𝐴− (𝑚1 −𝑚3)𝐴
2)
]︀
− 𝛿𝜖𝑥 = 0, 𝐴 ̸= 0,
Λ = 𝜖𝑥/𝐴−𝑚1𝐴
2 −𝑚3𝐵
2,
(25)
which gives an analytical solution and the corresponding response curves
in the space (𝜎/𝜎1, 𝐴,𝐵) parametrically defined as functions of 𝐴. The
procedure suggests solving the depressed cubic which may have from one
to three real roots depending on the discriminant
Δ3(𝐴)=−(𝑚1 −𝑚3)
[︂
4
(︁𝜖𝑥
𝐴
− (𝑚1 −𝑚3)𝐴
2
)︁3
+ 27(𝑚1 −𝑚3)𝛿
2𝜖2𝑥
]︂
. (26)
Because 𝐴 = 𝐵̄ = 0, (24) defines the steady-state swirling by the Stokes
modes 𝑓
(1)
1 (𝑥) and 𝑓
(2)
1 (𝑦). The signs of 𝐴 and 𝐵 are determined by (25)
and, therefore, the swirling direction is defined as well.
Passage to the longitudinal excitation type. When 𝛿 → 0, the elliptic
orbit in figure 2 (a) flattens and this excitation type transforms to the
longitudinal excitations along the 𝑂𝑥 axis. In this limit, the depressed
cubic has the real root 𝐵 = 0 which implies the planar steady-state
wave regime. The two other real roots ±|𝐵| are computed by 𝐵2 =
𝜖𝑥/𝐴/(𝑚3−𝑚1)+𝐴
2 > 0; these define two swirling waves whose direction
depends on the transients stage.
The rotary excitation with 𝛿 = 1. The depressed cubic in (25) has then
the real root 𝐵 = 𝐴 which corresponds to the co-called rotary (swirling)
wave. The two other real roots come from the quadratic equation (𝑚1 −
𝑚3)(𝐵
2 +𝐴𝐵) + 𝜖𝑥/𝐴 = 0 with respect to 𝐵.
𝐒𝐰𝐢𝐫𝐥𝐢𝐧𝐠 𝐛𝐲 𝐭𝐡𝐞 𝐭𝐰𝐨 𝐜𝐨𝐦𝐛𝐢𝐧𝐞𝐝 𝐒𝐭𝐨𝐤𝐞𝐬 𝐦𝐨𝐝𝐞𝐬. When 𝐵̄ ̸= 0, divid-
ing (10) by 𝐵̄ makes it possible to express 2○ via 1○, 3○ and 4○. These
Resonant sloshing in a square-base tank due to an angular-and- . . . 277
three equations can be rewritten in the form
𝛿2𝐵3 + 𝛿𝐴(2− 𝛿2)𝐵2 +𝐴2(1− 2𝛿2)𝐵 + 𝛿
[︀
𝜖2(1− 𝛿2)−𝐴3
]︀
= 0,
𝜖2 =
𝜖𝑥(𝑚3 −𝑚1)
(𝑚2 −𝑚3)(𝑚2 −𝑚1)
, (27a)
𝐵̄2 =
𝐴
[︀
(𝑚2 −𝑚1)𝐴
2 + (𝑚1 −𝑚3)𝐵
2 + 𝛿(𝑚2 −𝑚3)𝐴𝐵
]︀
+ 𝜖𝑥
𝛿(𝑚2 −𝑚3)𝐵 + (𝑚2 −𝑚1 + 𝛿2(𝑚1 −𝑚3))𝐴
> 0,
(27b)
Λ = 𝜖𝑥/𝐴−𝛿(𝑚2−𝑚3)𝐵̄
2𝐵/𝐴−𝑚1𝐴
2−𝑚3𝐵
2−(𝑚2+𝛿
2𝑚1)𝐵̄
2, (27c)
which defines the analytical solution and the response curves in the space
(𝜎/𝜎1, 𝐴,𝐵, |𝐵̄|) (parametrised by 𝐴 ̸= 0). The equation (27) has from
one to three real roots depending on the discriminant
Δ4(𝐴) = −𝜖2𝛿4(𝛿2 − 1)[27𝜖2𝛿
2(𝛿2 − 1) + 4𝐴3(𝛿2 + 1)3]. (28)
The solution (27) implies the swirling steady-state wave regime by the
two combined Stokes modes (24) which becomes a standing resonant
wave by a combined Stokes wave when 𝛿𝐵̄ = 𝐴𝐵 (the two vectors (𝐴, 𝐵̄)
and (𝛿𝐵̄, 𝐵) are parallel). Requiring this condition in (27b) deduces the
algebraic algebraic equation
−𝐴3+𝐴𝐵[𝛿𝐵+(1− 𝛿2)𝐴2]− 𝛿𝜖𝑥
𝑚2 −𝑚1
= 0, 𝐴𝐵 > 0, 0 < 𝛿 ≤ 1, (29)
which constitutes, together with (27a), an algebraic system to find (𝐴,𝐵)
for which (24) implies the standing wave.
Passage to the longitudinal reciprocating excitation. When 𝛿 → 0,
𝐴 = 0 from (23) and 𝐵 = 0 from (27a). The two equations, (27c)
and (27b), describe the two standing squares-like resonant waves, 𝑧 =
[𝐴𝑓
(1)
1 (𝑥)± |𝐵̄|𝑓 (2)1 (𝑦)] cos𝜎𝑡+ 𝑜(𝜖1/3), which were described in [1].
The rotary excitation type implies 𝛿 = 1 in (27). The two real roots of
(27a) are 𝐵 = 𝐴 and 𝐵 = −𝐴 (of the double multiplicity, Δ4 ≡ 0 as 𝛿 =
1). The second root contradicts to (27b) as causing 𝐵̄2 ·0 = 𝐴 ·0+𝜖𝑥 ̸= 0,
but 𝐵 = 𝐴 ̸= 0 in (27b) and (27c) leads to the steady-state swirling by
the combined Stokes modes (24), where
Λ = −2(𝑚1 +𝑚2 −𝑚3)𝐴
2 + 𝜖𝑥(1−𝑚2 − 2𝑚2 +𝑚3)/𝐴, 𝐴 ̸= 0 (30)
provided by 𝐵 = 𝐴, 𝐵̄2 = 𝐴2 + 𝜖𝑥/𝐴 > 0, (𝑚1 + 𝑚2 − 𝑚3) ̸= 0 and
(1−𝑚1 − 2𝑚2 +𝑚3) ̸= 0.
278 Timokha A. N.
𝟓 𝐓𝐡𝐞 𝐨𝐛𝐥𝐢𝐪𝐮𝐞 𝐞𝐥𝐥𝐢𝐩𝐭𝐢𝐜 𝐞𝐱𝐜𝐢𝐭𝐚𝐭𝐢𝐨𝐧 𝐭𝐲𝐩𝐞
When all forcing amplitudes in (7) are not zeros, 𝜖𝑥 𝜖𝑦 𝜖𝑦 ̸= 0, the first
Fourier harmonic component of the periodic tank excitations determines
an oblique elliptic orbit in figure 2 (c) in terms of the equivalent horizontal
tank motions. The equation (11) contains both non-zero coefficients,
𝐴 = 𝛿𝐵̄ − 𝛿𝐵, 𝛿 𝛿 ̸= 0. (31)
The amplitude parameter 𝐴 ̸= 0 since assuming the zero transforms (7)
to the contradiction 𝐴𝐵̄𝐵 = 0, 𝐴[...] = 𝜖𝑥 ̸= 0, 𝐵̄[...] = 𝜖𝑦 ̸= 0, 𝐵[...] =
𝜖𝑦 ̸= 0. This means that dividing (10) by 𝐴 ̸= 0 makes 1○ derivable from
other three secular equations as (31) is satisfied.
Moreover, whereas 𝛿𝛿 ̸= 0, 𝐴𝐵̄𝐵 ̸≡ 0 along a response curve. A
tedious derivation reduces finding the semi-analytical solution to getting
real roots of
9∑︁
𝑖=0
𝑐
(0)
𝑖 𝐵𝑖𝐵̄9−𝑖+𝜖3
6∑︁
𝑖=0
𝑐
(1)
𝑖 𝐵𝑖𝐵̄6−𝑖+𝜖23
3∑︁
𝑖=0
𝑐
(2)
𝑖 𝐵𝑖𝐵̄3−𝑖 = 0; 𝜖3 =
𝜖𝑥
𝑚2 −𝑚3
,
(32)
where coefficients 𝑐
(0)
𝑖 , 𝑐
(1)
𝑖 and 𝑐
(2)
𝑖 are the polynomials by 𝛿, 𝛿 and the
quadratic functions by 𝑚1,𝑚2 and 𝑚3. The equality (32) can be con-
sidered as, for instance, an algebraic equation with respect of 𝐵 when
𝐵̄ ̸= 0 is a real parameter. At least one real root must exist. After
getting all real roots of (32), (31) computes 𝐴 but the following formulas
consequently compute 𝐴 and 𝜎/𝜎1,
𝐴 = −𝐴
[︀
𝜖3(𝛿𝐵𝑚3 − 𝛿𝐵̄𝑚2 +𝑚1𝐴)
+𝐵𝐵̄((𝑚1 −𝑚2)𝐵
2 + (𝑚1 −𝑚3)𝐵̄
2 + (𝑚2 +𝑚3)𝐴
2)
]︀
/
[︀
𝐴2((𝑚2 −𝑚1)𝐵̄
2 − (𝑚3 −𝑚1)𝐵
2] +𝐵2𝐵̄2(𝑚2 −𝑚3)
]︀
, (33a)
Λ = −𝑚1(𝐴
2 +𝐴2)−𝑚2𝐵
2 −𝑚3𝐵̄
2 − (𝑚2 −𝑚3)𝐴𝐵𝐵̄/𝐴. (33b)
As a consequence, changing 𝐵̄ ̸= 0, (32), (31) and (33) determine from
one to nine response curves in the space (𝜎/𝜎1, 𝐴, 𝐵̄, 𝐵).
This semi-analytical solution defines the steady-state resonance wave
patterns (15). They imply a swirling wave in the most general case but it
can also imply a standing wave when the vector (𝐴, 𝐵̄) and (𝛿𝐵̄− 𝛿𝐵,𝐵)
are parallel, namely, when the condition
𝐴𝐵 = 𝐵̄(𝛿𝐵̄ − 𝛿𝐵) (34)
is satisfied.
Resonant sloshing in a square-base tank due to an angular-and- . . . 279
𝟔 𝐂𝐨𝐧𝐜𝐥𝐮𝐬𝐢𝐨𝐧𝐬
Using the Narimanov-Moiseev approximate modal theory for the resonant
sloshing in a square-base tank, we study the steady-state wave regimes
occurring due to a periodic small-magnitude sway-surge-roll-pitch motion
of the tank. The only first Fourier harmonics of the periodic forcing mat-
ters for classifying the resonant steady-state surface waves. This makes
it possible to introduce an equivalent horizontal harmonic tank forcing,
which causes the same steady-state resonant waves in terms of the first
and second asymptotic components. This equivalent forcing can be iden-
tified as of either reciprocating or elliptic type.
The analytical solutions for the reciprocating and elliptic excitation
types are constructed. Existence of standing and swirling wave regimes
in certain frequency ranges is confirmed. For the elliptic excitation type,
only swirling-type waves are theoretically possible.
[1] Faltinsen O.M., Rognebakke O.F., Timokha A.N. Resonant three-
dimensional nonlinear sloshing in a square base basin // Journal of Fluid
Mechanics.— 2003.— 487. — P. 1–42.
[2] Faltinsen O.M., Rognebakke O. F., Timokha A.N. Resonant three-
dimensional nonlinear sloshing in a square base basin. Part 2. Effect of
higher modes // Journal of Fluid Mechanics.— 2005.— 523. — P. 199–
218.
[3] Faltinsen O.M., Rognebakke O. F., Timokha A.N. Resonant three-
dimensional nonlinear sloshing in a square base basin. Part 3. Base ratio
perturbations // Journal of Fluid Mechanics.— 2006.— 551. — P. 93–
116.
[4] Faltinsen O.M., Rognebakke O. F., Timokha A.N. Transient and steady-
state amplitudes of resonant three-dimensional sloshing in a square base
tank with a finite fluid depth // Physics of Fluids.— 2006.— 18,
Art. No. 012103.— P. 1–14.
[5] Faltinsen O.M., Timokha A.N. Sloshing.— Cambridge University Press,
2009.— 686 p.
[6] Ikeda T., Harata Y., Osasa T. Internal resonance of nonlinear sloshing
in rectangular liquid tanks subjected to obliquely horizontal excitation //
Journal of Sound and Vibration.— 2016.— 361. — P. 210–225.
[7] Ikeda T., Ibrahim R. A., Harata Y., Kuriyama T. Nonlinear liquid sloshing
in a square tank subjected to obliquely horizontal excitation // Journal of
Fluid Mechanics.— 2012.— 700. — P. 304–328.
280 Timokha A.N.
[8] Pilipchuk V.N. Nonlinear interactions and energy exchange between liquid
sloshing modes // Physica D.— 2013.— 263. — P. 21–40.
[9] Wu Chih-Hua, Chen Bang-Fuh Sloshing waves and resonance modes of
fluid in a 3D tank by a time-independent finite difference method // Ocean
Engineering.— 2009.— 36.— P. 500–510.
[10] Wu Chih-Hua, Chen Bang-Fuh, Hung Tin-Kan Hydrodynamic forces in-
duced by transient sloshing in a 3d rectangular tank due to oblique hor-
izontal excitation // Computers and Mathematics with Applications.—
2013.— 65. — P. 1163–1186.
[11] Wu Chih-Hua, Faltinsen O.M., Chen Bang-Fuh Analysis on shift of na-
ture modes of liquid sloshing in a 3d tank subjected to oblique horizontal
ground motions with damping devices // Advances in Mechanical Engi-
neering.— 2013.— Article ID 627124.— P. 1-24.
[12] Zhang Hong-Shang, Wu Peng-Fei, Liu Wen-Bai The analysis of second-
order sloshing resonance in a 3-D tank // Journal of Hydrodynamics.—
2014.— 26, no. 2.— P. 309-315.
1. Луковський І.О., Гаврилюк І.О., Василик В.Б., Ситник Д.О.
2. Біленко В. І., Божонок К. В., Дзядик С. Ю., Стеля О. Б.
Інтегро–апроксимаційний алгоритм
Вступ
Постановка задачі
Алгоритм
Похибка алгоритму
Застосування a–методу для алгебраїчно–нелінійних рівнянь гіперболічного типу
Задача Дирихле для алгебраїчно–нелінійних рівнянь еліптичного типу на прямокутнику
Наближений розв'язок початкової задачі для алгебраїчно–нелінійних рівнянь параболічного типу на прямокутнику
Сплайн–алгоритм
Монотонна схема для рівняння конвекції–дифузії
Висновки
3. Василик В.Б., Макаров В.Л., Ситник Д.О.
Вступ
Регуляризація та явне зображення розв'язку
Вибір контуру інтегрування
Чисельний метод
4. Веселовська Г.М.
5. Грушковская В.В.
Введение
Построение модельной системы
Условия устойчивости
Оценка скорости убывания решений
Пример: оценка скорости затухания колебаний маятниковой системы с частичной диссипацией
Выводы
6. Дзюбенко Г.А.
Вступ
Допоміжні факти
Доведення Теореми ??
7. Діденко Ю.Ф., Денисенко В.І.
8. Елишевич М.А.
Постановка задачи
Полученный результат
Пример
9. Константинов А.В., Лимарченко О.С., Кинебас К.В., Паранькина О.Ю.
Введение
Объект исследования и математическая модель
Результаты вычислительных экспериментов
Выводы
10. Мазко О.Г., Кусій С.М.
Вступ
Допоміжні твердження
Лінійні системи з керованими і спостережуваними виходами
Статичний регулятор по вимірюваному виходу
Динамічний регулятор
Алгоритм побудови динамічного регулятора
Приклад. Гасіння коливань лінійного осцилятора.
Висновок
11. Працьовитий М. В., Маслова Ю. П.
Вступ
Функція Радемахера і ряди Уолша
Узагальнення функцій Радемахера
Узагальнення функцій Уолша
12. Працьовитий М.В., Чуйков А.С.
Вступ
Оператори лівостороннього та правостороннього зсуву елементів ланцюгового дробу
Інші функції, пов'язані з оператором T(x)
13. Новицький В.В., Зінчук М.О., Коломійчук О.П., Тетерятник О.В.
Вступ
Оптимальне керування лінійними неперервними майже консервативними системами
Оптимальне керування лінійними дискретними майже консервативними системами
14. Осауленко Р. Ю.
Вступ
Перетворення, які зберігають хвости Qs–зображення чисел
Група перетворень, які зберігають частоти цифр Qs–зображення числа
Приклад функції, яка зберігає частоти, але не зберігає хвости зображення Qs-ірраціональних чисел
15. Слинько В.І., Кравчук С.В.
Постановка задачі.
Основний результат.
Умови стійкості
16. Солодун А. В.
Постановка задачи
Численные результаты
17. Ситник Д.О.
Вступ
Sinc–апроксимація
Sinc-апроксимація функції за її значеннями поза інтерполяційною сіткою
18. Сосницький С.П.
Вступ
Про рівняння збуреного руху в околі стаціонарних лагранжевих трикутників
Теорема про орбітальну нестійкість лагранжевих стаціонарних рухів у задачі трьох тіл
Висновок
19. Сосницький С.П.
Вступ
Про достатні умови відсутності осцилюючих симетричних рухів
20. Чернецька Л.О.
21. Timokha A.N.
Statement
Asymptotic steady-state solutions of (??)–(??)
The reciprocating excitation type
The axisymmetric elliptic excitation type
The oblique elliptic excitation type
Conclusions
22. Shlepakov L.N.
Main relationships for a non-inflated system
Construction of enlarged systems
Defining the task mathematical programming
Case of multiple channels with same probability characteristics in the same system of channels.
23. Shidlich A.L.
Approximative characteristics
Main results
Order estimates for some functionals and their applications
Proof of Theorems ?? and ??.
24. Луковський І.О., Стороженко В.О.
25. Луковський І.О., Пустовойтов М.О.
|
| id | oai:trim.imath.kiev.ua:article-60 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:01:48Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/eb/9c322378650162d1ef6d5e3da92716eb.pdf |
| spelling | oai:trim.imath.kiev.ua:article-602018-02-13T11:57:10Z Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing Резонансные плескания жидкости в баке квадратного сечения при угловых и горизонтальных возбуждениях Резонансні хлюпання рідини в баці квадратного перерізу при кутових та горизонтальних збурень Timokha, A. N. Тимоха, А. Н. Тимоха, О. М. A modal method is developed for the problem of the oscillation of a liquid in a square-section tank, which performs periodic horizontal and angular motions of small amplitude. The analysis shows that the dominant wave component is exclusively determined by the first harmonic of periodic excitation. Equivalent harmonic motions are reciprocal or elliptic types. The steady-state resonant wave regimes for this type of perturbation are studied. Разрабатывается модальный метод для задачи про колебания жидкости в резервуаре квадратного сечения, который совершает периодические горизонтальные и угловые движения малой амплитуды. Анализ показывает, что доминантная волновая компонента исключительно определяется первой гармоникой периодического возбуждения. Эквивалентные гармонические движения являются возвратно-поступательными или эллиптического типов. Изучено установившиеся резонансные волновые режимы для такого типа возмущений. Розробляється модальний метод для задачi про коливання рiдини у резервуарi квадратного перерiзу, який виконує перiодичнi горизонтальнi та кутовi рухи малої амплiтуди. Аналiз показує, що домiнантна хвильова компонента виключно визначається першою гармонiкою перiодичного збурення. Еквiвалентнi гармонiчнi рухи є зворотно-поступального чи елiптичного типу. Вивчено усталенi резонанснi хвильовi режими для таких типiв збурення. Інститут математики НАН України 2017-12-22 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/60 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 13 No. 3 (2016): Mathematical problems of mechanics and computational mathematics; 266-280 Сборник Трудов Института математики НАН Украины; Том 13 № 3 (2016): Математичні проблеми механіки та обчислювальної математики; 266-280 Збірник Праць Інституту математики НАН України; Том 13 № 3 (2016): Математичні проблеми механіки та обчислювальної математики; 266-280 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/60/55 Авторське право (c) 2016 Праці Інституту математики НАН України |
| spellingShingle | Timokha, A. N. Тимоха, А. Н. Тимоха, О. М. Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| title | Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| title_alt | Резонансные плескания жидкости в баке квадратного сечения при угловых и горизонтальных возбуждениях Резонансні хлюпання рідини в баці квадратного перерізу при кутових та горизонтальних збурень |
| title_full | Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| title_fullStr | Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| title_full_unstemmed | Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| title_short | Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| title_sort | resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/60 |
| work_keys_str_mv | AT timokhaan resonantsloshinginasquarebasetankduetoanangularandhorizontalperiodicforcing AT timohaan resonantsloshinginasquarebasetankduetoanangularandhorizontalperiodicforcing AT timohaom resonantsloshinginasquarebasetankduetoanangularandhorizontalperiodicforcing AT timokhaan rezonansnyepleskaniâžidkostivbakekvadratnogosečeniâpriuglovyhigorizontalʹnyhvozbuždeniâh AT timohaan rezonansnyepleskaniâžidkostivbakekvadratnogosečeniâpriuglovyhigorizontalʹnyhvozbuždeniâh AT timohaom rezonansnyepleskaniâžidkostivbakekvadratnogosečeniâpriuglovyhigorizontalʹnyhvozbuždeniâh AT timokhaan rezonansníhlûpannârídinivbacíkvadratnogopererízuprikutovihtagorizontalʹnihzburenʹ AT timohaan rezonansníhlûpannârídinivbacíkvadratnogopererízuprikutovihtagorizontalʹnihzburenʹ AT timohaom rezonansníhlûpannârídinivbacíkvadratnogopererízuprikutovihtagorizontalʹnihzburenʹ |