Damped resonant steady-state sloshing in an upright circular tank
The nonlinear Narimanov–Moiseev-type modal system with linear damping terms is employed to study the damped steady-state resonant sloshing in an upright circular tank. Estimating the damping coefficients (ratios) by using Miles’ formula shows that the damping may matter for laboratory tanks. An asympt...
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Інститут математики НАН України
2017
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Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552854832545792 |
|---|---|
| author | Raynovskyy, I. A. Timokha, A. N. Райновский, И. А. Тимоха, А. Н. Райновський, І. А. Тімоха, О. М. |
| author_facet | Raynovskyy, I. A. Timokha, A. N. Райновский, И. А. Тимоха, А. Н. Райновський, І. А. Тімоха, О. М. |
| author_institution_txt_mv | [
{
"author": "I. A. Raynovskyy",
"institution": "Institute of Mathematics"
},
{
"author": "A. N. Timokha",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Raynovskyy, I. A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-13T11:53:50Z |
| description | The nonlinear Narimanov–Moiseev-type modal system with linear damping terms is employed to study the damped steady-state resonant sloshing in an upright circular tank. Estimating the damping coefficients (ratios) by using Miles’ formula shows that the damping may matter for laboratory tanks. An asymptotic steady-state solution of the modal system is derived for a prescribed cyclic tank motion with four (sway/surge/pitch/roll) degrees of freedom; the forcing frequency is close to the lowest natural sloshing frequency. The steady-state response curves by the two lowestorder natural sloshing mode amplitudes are examined versus the semi-axes ratio of the artificial elliptic orbit. |
| first_indexed | 2026-08-04T01:06:03Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2017, т. 14, №2, 180–204
УДК 532.595
Damped resonant steady-state
sloshing in an upright circular tank∗
I. A. Raynovskyy 1, A.N. Timokha 1,2
1 Institute of Mathematics of NAS of Ukraine, Kyiv;
ihor.raynovskyy@gmail.com
2 Centre of Excellence AMOS, Norwegian University of Science and
Technology, Trondheim, Norway; atimokha@gmail.com.
The nonlinear Narimanov–Moiseev-type modal system with linear damp-
ing terms is employed to study the damped steady-state resonant sloshing
in an upright circular tank. Estimating the damping coefficients (ratios) by
using Miles’ formula shows that the damping may matter for laboratory
tanks. An asymptotic steady-state solution of the modal system is de-
rived for a prescribed cyclic tank motion with four (sway/surge/pitch/roll)
degrees of freedom; the forcing frequency is close to the lowest natural
sloshing frequency. The steady-state response curves by the two lowest-
order natural sloshing mode amplitudes are examined versus the semi-axes
ratio of the artificial elliptic orbit.
Використовуючи нелiнiйну модальну систему Нарiманова–Мойсеєва iз
лiнiйними коефiцiєнтами демпфування, вивчаються усталенi демпфо-
ванi резонанснi хлюпання рiдини у цилiндричному баку. Оцiнка коефi-
цiєнтiв демпфування за допомогою формули Майлза показує, що дем-
пфування може мати значення для лабораторних посудин. Знайдено
асимптотичний усталений розв’язок модальної системи для заданого
руху цилiндричної посудини iз чотирма (sway/surge/pitch/roll) ступе-
нями вiльностi; частота збурення є близькою до найнижчої власної ча-
стоти коливань рiдини. Розглянуто залежнiсть амплiтудно-частотних
характеристик, що пов’язуються iз амплiтудами двох нижчих власних
форм коливання рiдини вiд спiввiдношення пiвосей елiптичної орбiти.
∗The work is partly supported by the Grant № 0117U004077. The second author
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© Raynovskyy I.A., Timokha A. N., 2017
Damped resonant steady-state sloshing in an upright circular tank 181
1 Introduction
Steady-state resonant sloshing in an upright circular cylindrical tank due
to a harmonic longitudinal excitation with the forcing frequency close to
the lowest natural frequency has been studied by many authors, theo-
retically and experimentally, starting from the 60’s. Most recent reviews
can be found in [10, 12, 18]. In the theoretical studies, the linear liquid
damping caused by boundary layer and bulk viscosity effects (see deriva-
tions of the associated logarithmic decrements in [5, 14]) was normally
neglected. The latter was supported by experiments with industrial con-
tainers whose geometric dimensions count in metres but [2,7,9,16,17,19]
showed that the damping may matter for certain small laboratory tanks
incl. bioreactors. Reasons are an increasing viscosity of the bioliquids
and a growing effect of the dynamic contact angle.
Bioreactors normally perform orbital periodic motions with the forc-
ing frequency close to the lowest natural sloshing frequency. Those tank
excitations were considered in [3,15] within the framework of an inviscid
potential flow model. The undamped steady-state sloshing regimes were
classified for an elliptic horizontal forcing by using an infinite-dimensional
Narimanov–Moiseev-type nonlinear modal system, which couples the gen-
eralised coordinates of the natural sloshing modes. The present paper
includes the linear damping effect into the steady-state analysis of [3].
In § 2, we introduce the Narimanov–Moiseev-typemodal system equip-
ped with linear damping terms whose (damping) coefficients are associ-
ated with boundary layer and bulk viscosity effects onto the correspond-
ing natural sloshing modes. A theoretical estimate of the damping coef-
ficients is given in § 3 following Miles’ analysis [14]. Numerical analysis
with the tap water shows that these coefficients are negligibly small for
industrial containers but, indeed, the linear damping may matter for
smaller tanks.
An analytical asymptotic periodic solution of the Narimanov-Moiseev–
type modal system is constructed in § 4. The solution describes reso-
nant steady-state wave regimes when tank performs an orbital motion
with four degrees (sway/surge/roll/pitch) of freedom. These regimes are
asymptotically equivalent to those occurring due to an elliptic horizontal
tank excitation as has been postulated in [3]. The present study focuses
on the damped steady-state wave regimes and their stability versus the
ellipse semi-axes ratio 0 ≤ δ ≤ 1 whose limit values 0 and 1 correspond
to longitudinal and rotary prescribed tank motions, respectively. Clas-
182 Raynovskyy I. A., Timokha A. N.
sification of these regimes requires analysing a secular system of four
nonlinear algebraic equations coupling the lowest-order amplitudes a, b̄
and ā, b, which correspond to sin and cos components of the two low-
est natural sloshing modes. The system has an analytical solution for
the undamped case [3, 15] but it does not in the studied case. A reason
is appearance of the damping-caused phase-lags. The secular system is
then re-written in terms of the integral amplitudes A =
√
A2 + ā2 and
B =
√
b̄2 + b2. In § 5, the steady-state results are interpreted in terms
of response curves drawn in the (σ/σ1, A,B) space where σ is the forcing
frequency and σ1 is the lowest natural sloshing frequency.
2 Statement of the problem
An incompressible inviscid liquid with an irrotational flow is considered
partly filling an upright circular rigid tank of the radius r0. The tank per-
forms a small-magnitude prescribed periodic sway/surge/roll/pitch mo-
tion, which is described by the r0-scaled generalised coordinates η1(t),
η2(t) and η4(t), η5(t), respectively, as shown in figure 1. The yaw type
tank motions cannot excite sloshing within the framework of the invis-
cid flow model but the heave (vertical) oscillations are not considered
in the present paper (see a review on the parametrically-excited slosh-
ing in [6]). The problem is studied in the nondimensional statement,
which is based on the characteristic size r0 and time 1/σ, where σ is
the forcing frequency. A small parameter 0 < ǫ ≪ 1 is introduced. It
is associated with the nondimensional periodic forcing magnitude, i.e.
ηi(t) = O(ǫ), i = 1, 2, 4, 5.
Figure 1 illustrates the time-dependent liquid domain Q(t) with the
free surface Σ(t) (governed by the single-valued function z = ζ(r, θ, t))
and the wetted tank surface S(t); Φ(r, θ, z, t) is the velocity potential.
The unknowns, ζ and Φ, are defined in the tank-fixed coordinate system.
They can be found from either the corresponding free-surface problem or
its equivalent variational formulation.
Using the Fourier-type representations of ζ and Φ and the aforemen-
tioned variational formulation makes it possible to derive [3] an approx-
imate system of ordinary differential equations (nonlinear modal equa-
tions) with respect to the time-dependent coefficients in these represen-
tations. The coefficients are interpreted as generalised coordinates and
velocities, respectively. The Fourier basis consists of the natural sloshing
Damped resonant steady-state sloshing in an upright circular tank 183
z
4
(t)
η
1
(t)
η
5
(t)
η
2
(t)
0
x
y
Σ
(t)Σ
Q (t)
S(t)
η
Figure 1. The liquid domain Q(t) is confined by the free surface Σ(t) and the
wetted tank surface S(t). Sloshing is considered in the tank-fixed coordinate
system Oxyz whose coordinate plane Oxy coincides with the mean (hydro-
static) free surface Σ0; Oz is the symmetry axis. Small-magnitude periodic
tank excitations are governed by generalised coordinates η1(t) (surge), η4(t)
(roll), η2(t) (sway), and η5(t) (pitch).
(eigen) modes ϕMi following from the spectral boundary problem
∇2ϕ = 0 in Q0,
∂ϕ
∂n
= 0 on S0,
∂ϕ
∂n
= κϕ on Σ0,
∫
Σ0
ϕdS = 0, (1)
where Q0 is the mean (hydrostatic) liquid domain confined by the mean
free surface Σ0 and the wetted tank surface S0 (figure 1). The problem
(1) has the analytical solution [4, 10]:
ϕMi(r, z, θ) = RMi(r)ZMi(z)
cos
sin (Mθ), M = 0, . . . ; i = 1, . . . , (2a)
RMi(r) = αMiJM (kMir), ZMi(z) =
cosh(kMi(z + h))
cosh(kMih)
, (2b)
where JM (·) is the Bessel functions of the first kind, the radial wave
numbers kMi are determined by the transcendental equation R′
Mi(1) = 0
(equivalent to J ′
Mi(kMi) = 0) and the normalising multipliers αMi follow
from the orthogonality condition
λ(Mi)(Mj) =
∫ 1
r1
rRMi(r)RMj(r) dr = δij , i, j = 1, . . . , (3)
where δij is the Kronecker delta. The nondimensional eigenvalues κMi
and the dimensional natural sloshing frequencies σMi are computed by
184 Raynovskyy I. A., Timokha A. N.
the formulas
κMi = kMi tanh(kMih) (J ′
Mi(kMi) = 0) and σ2
Mi = κMi g/r0, (4)
where g = 9.81 [m/s2] is the dimensional gravity acceleration.
Because we consider small-amplitude angular tank motions, the modal
representations require to know the linearised Stokes-Joukowski poten-
tials Ω0i(r, z, θ), i = 1, 2, 3 (see definition and analytical details in [3,12]).
The potentials are harmonic functions satisfying the Neumann boundary
conditions
∂Ω01
∂n
= −(znr − rnz) sin θ,
∂Ω02
∂n
= (znr − rnz) cos θ,
∂Ω03
∂n
= 0 (5)
on Σ0 and S0, where nr and nz are the outer normal components in the
r- and z- directions, respectively. The Stokes-Joukowski potentials have
the analytical form Ω01 = −F (r, z) sin θ, Ω02 = F (r, z) cos θ, Ω03 = 0,
where
F (r, z) = rz −
∞∑
n=1
2Pn
k1n
R1n(r)
sinh(k1n(z +
1
2h))
cosh(12k1nh)
; Pn =
1∫
r1
r2 R1n(r) dr.
(6)
Based on (2a) and (6), the Fourier representation takes the form [3]
ζ(r, θ, t) =
Iθ,Ir∑
M,i
RMi(r) cos(Mθ) pMi(t) +
Iθ,Ir∑
m,i
Rmi(r) sin(mθ) rmi(t), (7a)
Φ(r, θ, z, t) = η̇1(t) r cos θ+η̇2(t) r sin θ+F (r, z)[−η̇4(t) sin θ+η̇5(t) cos θ]
+
Iθ ,Ir∑
M,i
RMi(r)ZMi(z) cos(Mθ)PMi(t)
+
Iθ,Ir∑
m,i
Rmi(r)Zmi(z) sin(mθ)Rmi(t), (7b)
Iθ, Ir → ∞, where pMi(t) and rmi(t) are the free-surface generalised
coordinates but PMi(t) and Rmi(t) are the generalised velocities. Fur-
thermore, all capital summation letters imply changing from zero to Iθ
but the lower case indices mean changing from one to either Iθ or Ir.
Damped resonant steady-state sloshing in an upright circular tank 185
In [3], using the Bateman–Luke variational formalism, the modal rep-
resentation (7), the Narimanov-Moiseev asymptotic relations
p11 ∼ r11 = O(ǫ1/3), p0j ∼ p2j ∼ r2j = O(ǫ2/3),
r1(j+1) ∼ p1(j+1) ∼ p3j ∼ r3j = O(ǫ), j = 1, 2, . . . , Ir; Ir → ∞ (8)
and the Moiseev resonant detuning (measuring how close is the forcing
frequency to the lowest natural sloshing frequency)
σ̄2
11 − 1 = O(ǫ2/3), σ̄Mi = σMi/σ (9)
(see an extensive discussion on what (8) and (9) mean for axisymmetric
tanks in [11]), the following weakly-nonlinear system of ordinary differ-
ential (modal) equations with respect to the generalised coordinates were
derived
p̈11 + 2ξ11σ̄11ṗ11 + σ̄2
11p11 + d1p11
(
p̈11p11 + r̈11r11 + ṗ211 + ṙ211
)
+ d2 [r11(p̈11r11 − r̈11p11) + 2ṙ11(ṗ11r11 − ṙ11p11)]
+
Ir∑
j=1
[
d
(j)
3 (p̈11p2j + r̈11r2j + ṗ11ṗ2j + ṙ11ṙ2j) + d
(j)
4 (p̈2jp11 + r̈2jr11)
+d
(j)
5 (p̈11p0j + ṗ11ṗ0j) + d
(j)
6 p̈0jp11
]
=−(η̈1 − gη5 − S1η̈5)κ11P1, (10a)
r̈11 + 2ξ11σ̄11ṙ11 + σ̄2
11r11 + d1r11
(
p̈11p11 + r̈11r11 + ṗ211 + ṙ211
)
+ d2 [p11(r̈11p11 − p̈11r11) + 2ṗ11(ṙ11p11 − ṗ11r11)]
+
Ir∑
j=1
[
d
(j)
3 (p̈11r2j − r̈11p2j + ṗ11ṙ2j − ṗ2j ṙ11) + d
(j)
4 (r̈2jp11 − p̈2jr11)
+d
(j)
5 (r̈11p0j + ṙ11ṗ0j) + d
(j)
6 p̈0jr11
]
= −(η̈2 + gη4 + S1η̈4)κ11P1; (10b)
p̈2k + 2ξ2kσ̄2k ṗ2k + σ̄2
2kp2k + d7,k(ṗ
2
11 − ṙ211) + d9,k(p̈11p11 − r̈11r11) = 0,
(11a)
r̈2k+ 2ξ2kσ̄2k ṙ2k +σ̄2
2kr2k+2d7,kṗ11ṙ11+d9,k(p̈11r11+r̈11p11) = 0, (11b)
p̈0k + 2ξ0kσ̄0kṗ0k + σ̄2
0kp0k+ d8,k(ṗ
2
11 + ṙ211)+ d10,k(p̈11p11 + r̈11r11) = 0;
(11c)
186 Raynovskyy I. A., Timokha A. N.
p̈3k + 2ξ3kσ̄3k ṗ3k + σ̄2
3kp3k + d11,k
[
p̈11(p
2
11 − r211)− 2p11r11r̈11
]
+ d12,k
[
p11(ṗ
2
11 − ṙ211)− 2r11ṗ11ṙ11
]
+
Ir∑
j=1
[
d
(j)
13,k(p̈11p2j − r̈11r2j)
+ d
(j)
14,k(p̈2jp11 − r̈2jr11) +d
(j)
15,k(ṗ2j ṗ11 − ṙ2j ṙ11)
]
= 0, (12a)
r̈3k + 2ξ3kσ̄3k ṙ3k + σ̄2
3kr3k + d11,k
[
r̈11(p
2
11 − r211) + 2p11r11p̈11
]
+ d12,k
[
r11(ṗ
2
11 − ṙ211) +2p11ṗ11ṙ11] +
Ir∑
j=1
[
d
(j)
13,k(p̈11r2j + r̈11p2j)
+d
(j)
14,k(p̈2jr11 + r̈2jp11) +d
(j)
15,k(ṗ2j ṙ11 + ṙ2j ṗ11)
]
=0, k = 1, ..., Ir; (12b)
p̈1n + 2ξ1nσ̄1nṗ1n + σ̄2
1np1n + d16,n(p̈11p
2
11 + r11p11r̈11)
+d17,n(p̈11r
2
11−r11p11r̈11)+d18,np11(ṗ211+ ṙ211)+d19,n(r11ṗ11ṙ11−p11ṙ211)
+
Ir∑
j=1
[
d
(j)
20,n(p̈11p2j + r̈11r2j) + d
(j)
21,n(p11p̈2j + r11r̈2j)
+d
(j)
22,n(ṗ11ṗ2j + ṙ11ṙ2j) + d
(j)
23,np̈11p0j + d
(j)
24,np11p̈0j + d
(j)
25,nṗ11ṗ0j
]
= −(η̈1 − gη5 − Snη̈5)κ1n Pn, (13a)
r̈1n + 2ξ1nσ̄1nṙ1n + σ̄2
1nr1n + d16,n(r̈11r
2
11 + r11p11p̈11)
+d17,n(r̈11p
2
11−r11p11p̈11)+d18,nr11(ṗ211+ ṙ211)+d19,n(p11ṗ11ṙ11−r11ṗ211)
+
Ir∑
j=1
[
d
(j)
20,n(p̈11r2j− r̈11p2j)+d
(j)
21,n(p11r̈2j−r11p̈2j)+d
(j)
22,n(ṗ11ṙ2j− ṙ11ṗ2j)
+d
(j)
23,nr̈11p0j + d
(j)
24,nr11p̈0j + d
(j)
25,nṙ11ṗ0j
]
= −(η̈2 + gη4 + Snη̈4)κ1nPn, n = 2, ..., Ir. (13b)
The hydrodynamic coefficients are functions of the nondimensional liquid
depth h/r0. The system is equipped with the linear damping (framed)
Damped resonant steady-state sloshing in an upright circular tank 187
terms, in which the damping coefficients are associated with the logarith-
mic decrements of the corresponding natural sloshing modes. By defini-
tion, the modal system (10)–(13) includes all up to the O(ǫ)-order terms
as Ir → ∞; rkl ∼ pkl = o(ǫ), k ≥ 4 are neglected. The system needs
either initial or periodicity condition. The latter leads to the resonant
steady-state wave regimes (solutions). The system (10)–(13) without
damping terms (undamped sloshing) was extensively validated in [3] by
comparisons with experiments for longitudinal resonant excitations.
3 Linear damping coefficients
When using (10)–(13) implicitly assumes the following conditions [3]:
• The nondimensional generalised coordinates ηi(t), i = 1, 2, 4, 5, are
the given 2π-periodic functions,
ηi(t)=η
(0)
ia +
∞∑
k=1
[
η
(k)
ia cos(kt)+µ
(k)
ia sin(kt)
]
, η
(k)
ia ∼ µ
(k)
ia = O(ǫ), (14)
where the lowest-order harmonic component is not zero, i.e.
∑
i=1,2,4,5
|η(1)ia |+ |µ(1)
ia | 6= 0. (15)
• The Moiseev detuning condition (9) is satisfied.
• There are no resonance amplifications of higher-order generalised
coordinates pmj, rmj , m j 6= 1,
m− σ̄1k ≥ O(1), σ̄mi = σmi/σ, m, k ≥ 2;
σ̄2
0i − 4 ∼ σ̄2
2i − 4 ∼ σ̄2
3i − 9 ∼ σ̄2
1(i+1) − 9 ≥ O(1), i ≥ 1.
(16)
The second raw of (16) means that there are no secondary reso-
nances [3].
• Because (10)–(13) neglects the o(ǫ)-order terms, the linear damping
terms matter, if and only if,
ξ11 = O(ǫ2/3), ξ2i ∼ ξ0i = O(ǫ1/3), ξ3i ∼ ξ1n = O(1), (17)
i ≥ 1, n ≥ 2 in the corresponding differential equations.
188 Raynovskyy I. A., Timokha A. N.
The damping ratios ξMi were theoretically and experimentally esti-
mated by many authors starting from the 50’s [4,14]. For low-viscous liq-
uids, the theoretical estimates can be asymptotically expressed in terms
of the Galilei number [1], Ga (regarded as a ratio between gravity and
viscous forces),
δ = Ga−1/4 =
√
ν/(g1/2r
3/2
0 ) ≪ 1, (18)
where ν is the kinematic viscosity. The lowest-order asymptotic contribu-
tion, ξsurfMi = O(δ), is associated with the laminar boundary layer effect
on the wetted tank surface; ξsurfMi can be rather accurately approximated
by using the Keulegan analytical technique [8]. The second-order asymp-
totic contribution, ξbulkMi = O(δ2), is due to the bulk viscosity. According
to [13, 14], a good agreement with experiments requires to account for
both the contributions, i.e.
ξMi = ξsurfMi + ξbulkMi . (19)
Miles [14] have found an analytical expression for (19). Using an
alternative analytical scheme, we re-derived Miles’ approximation as
ξsurfMi = δ
µ
(1)
Mi +
1
2R2
Mi(1)(µ
(2)
Mi + µ
(3)
Mi)
2
√
2κ
5/4
Mi µ
(0)
Mi
, (20a)
ξbulkMi = δ2
[
2k2Mi
κ
1/2
Mi
− R2
Mi(1)µ
(2)
Mi
2κ
3/2
Mi µ
(0)
Mi
]
, (20b)
where
µ
(0)
Mi =
1∫
0
rR2
Mi(r)dr, µ
(1)
Mi =
1∫
0
rR′2
Mi(r)dr +M2
1∫
0
R2
Mi(r)
r
dr,
µ
(2)
Mi =M2
(
tanh(kMih)
kMi
+
h
cosh2(kMih)
)
,
µ
(3)
Mi = k2Mi
(
tanh(kMih)
kMi
− h
cosh2(kMih)
)
.
(21)
The Galileo number Ga= gr30/ν
2 is a function of the kinematic vis-
cosity, the container radius r0 and g. Along with the Bond number
Bo= ρgr20/Ts (ρ is the liquid density and the Ts is the surface tension),
Damped resonant steady-state sloshing in an upright circular tank 189
r
02
543
21
n
ξ
0
0
0.7 0.8 0.9 1 0.5 0.4 0.3 0.2 0.1
0.025
0.02
0.015
0.01
0.005
0.6
543
21
i
ξ1
0r
2
0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.025
0.02
0.015
0.01
0.005
1
2
43
5
21
k
ξ
0r
2
0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.025
0.02
0.015
0.01
0.005
1
2
5
43
1
2
n
ξ
0r
3
0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.025
0.02
0.015
0.01
0.005
1
Figure 2. The theoretical damping rates 2ξMi by (19)–(21) for M = 0, 1, 2, 3.
The tap water with ν = 10−6 [m2/s] and g = 9.81 [m/s2]. The second index is
used to mark the curves.
the Galilieo number is well-known in the microgravity hydromechanics [1].
Bo and Ga are regarded as ratios between gravitational (mass forces)
and surface/viscous forces, respectively. When 100.Bo, the surface ten-
sion effect can be neglected. For the tap liquid at the Earth conditions
with ρ = 103 [kg/m3] and Ts = 0.073 [N/m], this inequality leads to
0.05 [m]. r0, which is required in our study because the modal system
(10)–(13) does not account for the surface tension.
Figure 2 demonstrates the damping rates versus r0 for the tap wa-
ter. In view of the asymptotic relations (17), one can expect that the
linear damping is negligible small for the industrial containers, whose the
190 Raynovskyy I. A., Timokha A. N.
nondimensional forcing amplitude ≈ 0.001 looks practically impossible.
For small-size laboratory containers (e.g., bioreactors), 2ξ11 may how-
ever satisfy (17). This is especially due to a higher viscosity of bioliquids
as well as the dynamic contact angle effect [7, 8] and the surface/liquid
contamination [5], which increase the amount dissipation.
4 Steady-state resonant solution
Following [3], we construct an asymptotic periodic solution of (10)–(13)
in terms of ǫ1/3 ≪ 1, which is associated with the primary excited gen-
eralised coordinates p11(t) and r11(t). The right-hand sides of (10) take
the form
σ2P1κ11
∞∑
k=1
[
(kη
(k)
1a − (kS1 − g/σ2)η
(k)
5a ) cos(kt)
+(kµ
(k)
1a − (kS1 − g/σ2)µ
(k)
5a ) sin(kt)
]
,
σ2P1κ11
∞∑
k=1
[
(kη
(k)
2a + (kS1 − g/σ2)η
(k)
4a ) cos(kt)
+(kµ
(k)
2a + (kS1 − g/σ2)µ
(k)
4a ) sin(kt)
]
.
(22)
Because of (9), neglecting the higher-order terms, o(ǫ), allows for re-
placing g/σ2 → g/σ2
11 and, therefore, amplitudes of the first Fourier
harmonics are
ǫx = P1κ11(η
(1)
1a − [S1 − g/σ2
11]η
(1)
5a ), ǭx = P1κ11(µ
(1)
1a − [S1 − g/σ2
11]µ
(1)
5a ),
ǭy = P1κ11(η
(1)
2a + [S1 − g/σ2
11]η
(1)
4a ), ǫy = P1κ11(µ
(1)
2a + [S1 − g/σ2
11]µ
(1)
4a ).
(23)
Here, ǫx and ǭx appear in the front of cos t and sin t and imply the forcing
components in the Ox direction, but ǭy and ǫy correspond to the cos t
and sin t forcing harmonic along the Oy axis.
Because the first harmonic is not zero (see, (15)), one can assume that
ǫ2x + ǭ2x 6= 0 and, introducing a phase-lag for the input time,
ǫx > 0, ǭx = 0. (24)
This means that the resonant forcing is the same as if the tank performs
Damped resonant steady-state sloshing in an upright circular tank 191
the artificial horizontal harmonic motions
(κ11P1)η1(t) = ǫx cos t; (κ11P1)η2(t) = ǭy cos t+ǫy sin t; η4(t) = η5(t) = 0
(25)
along the elliptic trajectory
ǫ2y + ǭ2y
ǫ2x
x2 + y2 − 2
ǭy
ǫx
xy = ǫ2y. (26)
Without lost of generality, one can assume that the elliptic forcing occurs
counterclockwise and, by rotating the Oxy plane around the Oz axis leads
to a canonic form of (26) so that
ǫx > 0, ǫy ≥ 0, ǭy = ǭx = 0. (27)
Henceforth, we assume that (27) is satisfied in the appropriate coordinate
system with the corresponding phase-lag for the input forcing signal.
To find an asymptotic steady-state solution of the modal system for
the elliptic type excitations by (27), we follow the Bubnov–Galerking
procedure in [3] by posing the lowest-order components of the primary
excited modes as
p11(t) = a cos t+ ā sin t+O(ǫ), r11(t) = b̄ cos t+ b sin t+O(ǫ), (28)
where a, ā, b̄, and b are of O(ǫ1/3). The corresponding lowest-order free-
surface elevations by (28) are a superposition of the two out-of-phase
angular modes
ζ(r, θ, t)=R11(r)
[
(a cos θ + b̄ sin θ) cos t+ (ā cos θ + b sin θ) sin t
]
+o(ǫ2/3).
(29)
This implies a swirling wave unless (a cos θ+ b̄ sin θ) and (ā cos θ+ b sin θ)
define congruent patterns, which happens if and only if
a b = ā b̄. (30)
The condition (30) means that (29) determines a standing wave.
The second- and third-order generalised coordinates can be found from
(11) and (12), (13), respectively. They are
p0k(t) = s0k(a
2 + ā2 + b2 + b̄2)
192 Raynovskyy I. A., Timokha A. N.
+ s1k
[
(a2 − ā2 − b2 + b̄2) cos 2t+ 2(aā+ bb̄) sin 2t
]
+ o(ǫ), (31a)
p2k(t) = c0k(a
2 + ā2 − b2 − b̄2)
+ c1k
[
(a2 − ā2 + b2 − b̄2) cos 2t+ 2(aā− bb̄) sin 2t
]
+ o(ǫ), (31b)
r2k(t)=2c0k(ab̄+bā)+2c1k
[
(ab̄− bā) cos 2t+(ab+ āb̄) sin 2t
]
+o(ǫ), (31c)
where
s0k = 1
2
(
d10,k − d8,k
σ̄2
0k
)
, s1k =
d10,k + d8,k
2(σ̄2
0k − 4)
,
c0k = 1
2
(
d9,k − d7,k
σ̄2
2k
)
, c1k =
d9,k + d7,k
2(σ̄2
2k − 4)
.
(32)
Substituting (28) and (31) into (10) and gathering the first harmonic
terms, cos t and sin t, lead to the solvability (secular) equations
1© :a
[
(σ̄2
11 − 1) +m1(a
2 + ā2 + b̄2) +m3b
2
]
+ ā[(m1 −m3)b̄b+ ξ] = ǫx,
2© : ā
[
(σ̄2
11 − 1) +m1(a
2 + ā2 + b2) +m3b̄
2
]
+ a[(m1 −m3)b̄b− ξ] = 0,
3© : b
[
(σ̄2
11 − 1) +m1(b
2 + b̄2 + ā2) +m3a
2
]
+ b̄[(m1 −m3)āa− ξ] = ǫy,
4© : b̄
[
(σ̄2
11 − 1) +m1(b
2 + b̄2 + a2) +m3ā
2
]
+ b[(m1 −m3)āa+ ξ] = 0
(33)
with respect to a, ā, b̄ and b; here, ξ = 2ξ11, where coefficients m1 and
m3 are computed by the formulas
m1=− 1
2d1+
Ir∑
j=1
[
c1j
(
1
2d
(j)
3 − 2d
(j)
4
)
+s1j
(
1
2d
(j)
5 − 2d
(j)
6
)
−s0jd(j)5 −c0jd(j)3
]
,
(34a)
m3 = 1
2d1 − 2d2
+
Ir∑
j=1
[
c1j
(
3
2d
(j)
3 −6d
(j)
4
)
+s1j
(
− 1
2d
(j)
5 +2d
(j)
6
)
−s0jd(j)5 +c0jd
(j)
3
]
. (34b)
Coefficients m1 and m3 are functions of h/r0 and the forcing frequency
σ̄11. Utilising (9) shows that the latter dependence can be neglected by
substituting σ = σ11 into the corresponding expressions. Dependence on
σ remains only in the (σ̄2
11 − 1)-quantity of (33).
Damped resonant steady-state sloshing in an upright circular tank 193
After finding a, ā, b̄ and b from (33), we can easily compute the second-
and third-order components of the asymptotic solution. The second-
order quantities are associated with (31), namely, are fully determined
by a, ā, b̄ and b, but the third-order components are affected by the higher
harmonics in (22) as well as the lowest harmonics in the right-hand side
of (13).
The linear Lyapunov method and the multitiming technique can be
employed to study stability of the constructed asymptotic steady-state
solution. This suggests introducing the slowly varying time τ = 1
2ǫ
2/3t
and expressing the perturbed solutions as
a1 = (a+ α(τ)) cos t+ (ā+ ᾱ(τ)) sin t+ o(ǫ1/3),
b1 = (b̄+ β̄(τ)) cos t+ (b + β(τ)) sin t+ o(ǫ1/3),
(35)
where a, ā, b and b̄ come from (33). Inserting (35) into the modal equa-
tions, gathering terms of the lowest asymptotic quantities order and keep-
ing linear terms in α, ᾱ, β and β̄ lead to the following linear system of
ordinary differential equations
s′ + ξs+ S s = 0, (36)
where s = (α, ᾱ, β, β̄)T , the prime is the differentiation by τ , and the
matrix S has the following elements
s11 = −2m1aā− (m1 −m3)bb̄; s13 = −2m1āb− (m1 −m3)ab̄,
s12=−(σ̄2
11−1)−m1(a
2+3ā2+b2)−m3b̄
2; s14=−2m3āb̄− (m1−m3)ab,
s21=(σ̄2
11−1)+m1(3a
2 + ā2 + b̄2) +m3b
2; s22=2m1aā+ (m1 −m3)bb̄,
s23 = 2m3ab+ (m1 −m3)āb̄; s24 = 2m1ab̄+ (m1 −m3)āb,
s31 = 2m1ab̄+ (m1 −m3)bā; s32 = 2m3āb̄+ (m1 −m3)ab,
s33=2m1bb̄+(m1−m3)aā; s34 = (σ̄2
11 − 1) +m1(b
2 + 3b̄2 + a2) +m3ā
2,
s41 = −2m3ab− (m1 −m3)āb̄; s42 = −2m1āb− (m1 −m3)ab̄,
s43=−(σ̄2
11− 1)−m1(3b
2+ b̄2+ ā2)−m3a
2; s44=−2m1bb̄−(m1−m3)aā.
The fundamental solution s = exp(λτ)a of (36) follows from the spec-
tral matrix problem [(λ+ ξ)E+S]a = 0, where λ are the unknown eigen-
values and a are the corresponding eigenvectors. Computations give the
following characteristic biquadratic equation
(λ+ ξ)4 + s1(λ + ξ)2 + s0 = 0, (37)
194 Raynovskyy I. A., Timokha A. N.
where s0 is the determinant of S and s1 is a complicated function of the
elements of S. The eigenvalues λ can be expressed as −ξ±√
x1,2, where
x1,2 = 1
2 (−s1 ±
√
s21 − 4s0) are two solutions of the quadratic equation
x2 + s1x + s0 = 0. The fixed-point solution (associated with a, ā, b and
b̄) is asymptotically stable (α, ᾱ, β and β̄ exponentially decay with τ) if
and only if the real component of λ is strongly negative.
In the limit case ξ → 0, the stability condition (ℜ[λ] < 0) takes the
following form
s21 − 4s0 ≥ 0 & s0 ≥ 0 & s1 ≥ 0. (38)
For O(ǫ2/3) = ξ > 0, the stability condition can be written as the
alternative
either s21 − 4s0 ≥ 0 & − s1 +
√
s21 − 4s0 ≤ 0 (⇔ s0 ≥ 0 & s1 ≥ 0) ,
or s21 − 4s0 ≥ 0 & − s1 +
√
s21 − 4s0 > 0&
√
1
2
(
−s1 +
√
s21 − 4s0
)
< ξ,
or s21 − 4s0 < 0 &
√
2
√
s0 − s1 < ξ. (39)
The procedure of finding an analytical solution of the secular system
(33) with ξ = 0 (damping is neglected) is in some detail described in [3].
The procedure cannot be generalised to the studied damped sloshing case
with ξ = O(ǫ2/3). A reason is that the damping causes the two phase-
lags, ψ and ϕ, for the two lowest (perpendicular, along the Ox and Oy
directions) modes and, as a consequence, normally, all four amplitude
parameters a, ā and b, b̄, are not zero, in the contrast to [3], where the
authors proved that ā = b̄ = 0 for longitudinal and elliptic forcing types.
A physically-relevant form of (33) should therefore couple the ‘integral’
lowest-order amplitudes A,B and the phase-lags ψ, ϕ:
A =
√
a2 + ā2 and B =
√
b̄2 + b2 > 0, (40a)
a = A cosψ, ā = A sinψ, b̄ = B cosϕ, b = B sinϕ. (40b)
Inserting (40) into expressions ā 1©− a 2©, b̄ 3©− b 4©, a 1©+ ā 2© and
b 3©+ b̄ 4© of (33) derives the following alternative secular equations
{
1 :A[Λ +m1A
2 + (m3 −F)B2] = ǫx cosψ, 3 :A[DB2 + ξ] = ǫx sinψ,
2 :B[Λ +m1B
2 + (m3 −F)A2] = ǫy sinϕ, 4 :B[DA2 − ξ] = ǫy cosϕ,
(41a)
Damped resonant steady-state sloshing in an upright circular tank 195
F = (m3 −m1) cos
2(α) = (m3 −m1)/(1 + C2),
D = (m3 −m1) sin(α) cos(α) = (m3 −m1)C/(1 + C2),
(41b)
where
Λ = σ̄2
11 − 1, α = ϕ− ψ, C = tanα, 0 ≤ ǫy ≤ ǫx 6= 0,
(F(α) and D(α) are the π-periodic functions of the phase-lags difference
α). The secular systems (33) and (41) are mathematically equivalent, i.e.,
getting known A,B, ψ, ϕ from (41) computes a, ā, b, b̄ and vice versa.
In terms of definitions (40) and (41), the standing wave condition (30)
is equivalent to
sinα = 0 ⇔ C = 0 (42)
so that the non-zero C implies swirling.
5 Response curves in the (σ/σ11, A, B) space
Longitudinal forcing. Undamped resonant steady-state sloshing due
to longitudinal excitations (ǫy = 0, ξ = 0) was analysed in [3] to prove
that ā = b̄ = 0 is fulfilled for any admissible input parameters and there
exist two physically-different solutions of (33) corresponding to the so-
called planar standing wave (b = 0 and (30) is satisfied) and swirling
(a b 6= 0 in (30)). In terms of the secular equations (40) and (41) with
ξ = 0, these two steady-state solutions imply B = 0, sinψ = 0, C = 0
and AB 6= 0, sinψ = cosϕ = 0 (C = ±∞), respectively. In addition,
one should remember that swirling consists of two identical angular pro-
gressive waves occurring in counter- and clockwise directions, these two
waves correspond to C = +∞ and −∞, respectively.
For the non-zero damping ξ 6= 0 and ǫy = 0 (damped steady-state
sloshing due to longitudinal excitations), the secular system (41) has the
same two physically-different solutions implying planar sloshing with the
zero transverse amplitude B = 0 but A,ψ are computed by 1
2 + 3
2 =
= A2 [(Λ +m1A
2)2 + ξ2] = ǫ2x; 0 < A ≤ ǫx
ξ
; ψ = arccos
A(A+m1A
2)
ǫx
,
(43)
and swirling with B 6= 0, which can be computed by rewriting (41) in
196 Raynovskyy I. A., Timokha A. N.
the form
A
[
Λ +m1A
2 +
m1 +m3 C
2
1 + C2
B2
]
= ǫx cosψ;
A
[
(m3 −m1)C
1 + C2
B2 + ξ
]
= ǫx sinψ;
B2 = − 1
m1
[
Λ +
m1 +m3 C
2
1 + C2
A2
]
> 0; A2 =
ξ (1 + C2)
(m3 −m1)C
> 0.
(44)
Consequently substituting expressions for A2 and B2 of (44) into the
square sum of the first row equations, derives the cubic equation with
respect to C:
Pl(C) = q3C
3 + q2C
2 + q1C + q0 = 0, (45)
where
q3 = ξ3 (m1 +m3)
2 > 0, q2 = 2ξ2Λ (m2
3 −m2
1),
q1 = ξ
[
4 ξ2m2
1 + Λ2 (m1 −m3)
2
]
, q0 = ǫ2xm
2
1 (m1 −m3).
B
A
/
11
0.4
0.20
0.1
0.2
0.85 0.9
0.3
0.95
0.4
1 01.05 1.1 1.15
E
H
1E
2
(a)
B
A
/
11
0.4
0.20
0.05
0.1
0.85
0.15
0.9
0.2
0.95 0
0.25
1 1.05
0.3
1.1
E E
H
H
P
2
21
1
(b)
Figure 3. Response curves in the (σ/σ11, A,B)-space for the longitudinal
harmonic forcing in the Oxz-plane, h/r0 = 1.5, the nondimensional forcing
amplitude η1a = 0.01 (η2a = 0). The branches are computed by using (43) for
planar (B = 0) and (44) is used for swirling (B > 0) wave regimes. The bold
lines mark stable solutions. The undamped sloshing (ξ = 0) is presented in (a)
and the damped one (ξ = 0.02) is shown in (b). There are no stable steady-
state sloshing between E1 and E2 where irregular (chaotic) waves are expected.
Curves on the (σ/σ11, A) plane correspond to the planar wave regime. The
damping causes the response curves do not go to infinity. An extra bifurcation
point P exists where swirling emerges from the planar wave branching.
Damped resonant steady-state sloshing in an upright circular tank 197
Illustrative response curves for undamped (a) and damped (b) steady-
state sloshing are shown in figure 3. The computations were made with
h/r0 = 1.5, the forcing amplitude is η1a = 0.01, and the damping co-
efficients ξ = 2ξ11 = 0.02 (in (b)). The response curves for undamped
sloshing in (a) were discussed in [3, 4, 10]. For this case, we see the
branches belonging to the plane (σ/σ11, A) responsible for planar wave
regime. The stable planar sloshing is located to the left of E1 and to the
right of E2. The planar sloshing waves become unstable in a neighbour-
hood of the primary resonance σ/σ11 = 1 where stable swirling (to the
right ofH) and irregular waves (there are no stable steady-state sloshing)
between E1 and H are predicted. The non-zero damping removes infinite
points as shown in (b). However, stability ranges of planar and swirling
waves are weakly affected by ξ = 0.02 so that positions of E1, E2 and H1
(replaces H) determining these ranges are almost the same. A novelty is
two points H2 and P , which can be treated as bifurcation points where
swirling emerges from the planar steady-state sloshing. The steady-state
swirling branching constitutes an arc, which is pinned at these two points.
As mentioned in [3], the undamped steady-state sloshing is charac-
terised by piece-wise values of the phase lags ψ and ϕ: sinψ = 0 for
the planar wave regime and sinψ = cosϕ = 0 for swirling. The damping
makes the phase lags by amplitude (frequency) dependent functions keep-
ing C = tan(ϕ−ψ) ≥ 0. For the planar wave regime by (42), ϕ = ψ± π,
but swirling causes finite and positive C > 0, C 6= +∞. The latter in-
equality follows from the last expression of (44), in which m3 > m1. The
positive numbers C = tanα > 0 are the roots of (45). The phase lag
ψ comes from the first two equations of (42) for any given point on the
(σ/σ11, A,B) curves. For each ψ of the swirling wave regime, there are
two different phase lags ϕ1 = ψ+α and ϕ2 = ψ+α±π. Physically, these
two phase lags ϕ1,2 for each point on the arc P,H1, H2, E2 mean that
two physically-identical swirling waves (clockwise and counterclockwise)
are possible.
Elliptic forcing (ǫy = δǫx, 0 < δ < 1). The undamped sloshing with
ξ = 0 in [3] was characterised by ā = b̄ = 0. Indeed, using 3 and 4 as
well as taking b̄ 1© − a 4© and ā 3© − b 4© make it possible to derive the
linear algebraic system
ā− δb̄ = ξ(A2 +B2)/ǫx, δā− b̄ = ξ(m1 −m3)(AB/ǫx) sinα, (46)
with respect to ā and b̄ whose determinant is equal to δ2 − 1. Obviously,
(46) has only trivial solution ā = b̄ = 0 as ξ = 0 and 0 ≤ δ < 1.
198 Raynovskyy I. A., Timokha A. N.
As a consequence, the undamped elliptically-forced sloshing turns 3
and 4 into identities, but the phase lags ψ and ϕ satisfy the condition
sinψ = cosϕ = 0 and, therefore, cosα = 0 and sinα = ±1. The ampli-
tudes A = |a| and B = |b| and, therefore, the remaining two equations
1 and 2 read as
A2[Λ +m1A
2 +m3B
2] = ǫ2x, B2[Λ +m1B
2 +m3A
2] = δ2ǫ2x. (47)
This system with respect to A2 and B2 can be analytically solved as
described in [3].
The present paper focuses on the damped sloshing with ξ 6= 0 in (41a).
Obviously, the system (46) has then no trivial solution. The phase lags
ϕ and ψ are rather complicated functions of the input parameters. Both
the amplitudes A, B and the phase lags ϕ, ψ should be found from the
nonlinear system (41a).
Let us exclude ϕ and ψ and reduce (41a) to a system of three equations
with respect A, B and C. For this purpose, we insert ϕ = ψ+α into the
right-hand sides of 2 and 4 and substitute ǫx cosψ and ǫ sinψ taken
from 1 and 3 . The result is the following linear system of homogeneous
equations
(δA)[cosα(D(C)B2 + ξ) + sinα(Λ +m1A
2 + (m3 −F(C))B2)]
−B[Λ +m1B
2 + (m3 −F(C))A2] = 0,
(δA)[cosα(Λ +m1A
2 + (m3 −F(C))B2)− sinα(D(C)B2 + ξ)]
−B[D(C)A2 − ξ] = 0,
with respect to δA and B. The system must have a nontrivial solution.
This leads to the zero-determinant condition
ξ(A2 −B2)D(C) −F(C)[ξ2 + (Λ +m1(A
2 +B2))2]/(m3 −m1)
+A2B2D2(C) = 0, (48)
which couples A2, B2 and C. Another two equations with respect to A2,
B2 and C come from 1
2
+ 3
2
and 2
2
+ 4
2
and take the form
{
A2[(Λ +m1A
2 + (m3 −F)B2)2 + (DB2 + ξ)2] = ǫ2x,
B2[(Λ +m1B
2 + (m3 −F)A2)2 + (DA2 − ξ)2] = δ2ǫ2x.
(49)
The system (48), (49) is a base for getting the response curves in the
(σ/σ11, A,B) space. One can prove that C 6= 0 since C = 0 leads to (48)
Damped resonant steady-state sloshing in an upright circular tank 199
⇒ ξ2+(Λ+m1(A
2+B2))2 = 0 and (49) ⇒ A2[ξ2+(Λ+m1(A
2+B2))2] =
ǫ2x 6= 0, simultaneously. Physically, C 6= 0 means that there are no
standing wave regimes for the elliptic forcing. All steady-state sloshing
regimes are swirling. Our numerical experiments show that C > 0 as
(m3 −m1) > 0.
The authors do not know how to get an analytical solution of (48),
(49). A numerical scheme is used. We define F and D as functions of
0 < β < 1
F(β) = (m3 −m1)β, D(β) = (m3 −m1) sgn(C)
√
β(1− β). (50)
When C > 0, a simple analysis shows that
0 < A <
ǫx
ξ
, 0 < B2 ≤ min
[
1
D(β)
(ǫx
A
− ξ
)
,
δ2ǫ2x
(D(β)A2 − ξ)2
]
, (51)
which determines the fixed interval for A but the interval for B2 is de-
termined by A and β. The first equation of (49) computes the two real
Λ1,2 for any given 0 < β < 1 and A,B2 satisfying (51) as follows
Λ1,2 = −m1A
2 − (m3 −F(β))B2 ±
√
ǫ2x
A2
− (D(β)B2 + ξ)2. (52)
Furthermore, to solve (48), (49) for any fixed A belonging to the corre-
sponding interval of (51)
1) we introduce a mesh 0 < β1 < β2 < ... < βk < ... < βK < 1;
2) for any fixed βk ∈ {βn}, we solve the two equations (follow from
the second equation of (49))
[Λj +m1B
2 + (m3 −F(β))A2]2 + [D(β)A2 − ξ]2 =
δ2ǫ2x
B2
, j = 1, 2,
(53)
(associated with + and − in expression (52)) with respect to B2
on the interval by (51); the result is a set of positive roots B2
k,j,i =
B2
i (A, βk, j), j = 1, 2, for each A and βk;
3) each root Bi(A, βk, j) is subsequently substituted into (48):
ξ(A2−B2
i,k,j)D(βk)−F(βk)[ξ
2+(Λj+m1(A
2+B2
i,k,j))
2]/(m3−m1)
200 Raynovskyy I. A., Timokha A. N.
+A2B2
i,k,jD2(βk) = 0 (54)
to detect the mesh interval (βk, βk+1), where the left-hand side of
(54) changes the sign;
4) an iterative procedure is used to compute β ∈ (βk, βk+1) and the
corresponding Bi(A, β, j).
The algorithm computes the solution for any fixed A. Varying A in the
interval by (51) outputs response curves in the (σ/σ11, A,B) space, which
are presented in figures 4–7.
B
A
/
11
0
0.05
0.1
0.15
0.2
0.25
0.3
0.4
0.2
0 1.11.0510.950.90.85
E
E
R
H
1
H
1
1
2
2
R2
A
B
/
11
0
0.05
0.1
0.15
0.2
0.25
0.3
0.3
0.2
0.1
1.11.050 10.950.90.85
H
H
E
E
R
R
1
1
2
1
2
2
Figure 4. Response curves in the (σ/σ11, A,B)-space for the steady-state reso-
nant sloshing due to an elliptic counterclockwise forcing with η1a = 0.01, η2a =
δη1a; δ = 0.05 in the left panel and 0.2 in the right one; ξ = 0.02. All re-
sponse curves correspond to swirling but some subbranches are close to the
(σ/σ11, A)-plane that means that sloshing is of an almost standing (planar)
wave type. The branch containing E1,H1, H2, E2 implies swirling, which co-
directed with the elliptic forcing but the loop-like branch with R1 and R2 marks
the counter-directed swirling. The bold lines imply stability.
Figure 4 shows that, by introducing a non-zero positive δ implying an
elliptic counterclockwise forcing with a small semi-axis along Oy splits
the arc PH1H2E2 in figure 3 (b), whose points determine two identical
co- and counterclockwise swirling waves, into two different branches. The
first branch contains points E1, H1H2, E2; it exists far from the primary
resonance zone, where the co-directed stable swirling wave is close to a
standing planar wave. The corresponding subbranches are to the left
of E1 and to the right of E2. Another stability subbranch is the piece
H1H2. The second loop-like branch with R1 and R2 implies swirling,
which is counter-directed to the forcing. This swirling is stable on R1R2.
Damped resonant steady-state sloshing in an upright circular tank 201
A
B
/
11
0
0.05
0.1
0.15
0.2
0.25
0.3
0.3
0.2
0.1
1.11.050 10.950.90.85
E
R
H
H
H
E
1
2
2R1
H3
4
1
2
/
11
B
A
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.3 1.20.2 1.10.1 10.90
E
1E
2
H
2
H4
H3 3R
4R
R
R1 H
2
1
Figure 5. The same as in figure 4 but for δ = 0.3 (the left panel) and 0.45
(the right one). An extra range of the co-directed (with respect to the forc-
ing) swirling H3H4 appears with increasing δ but the stable counter-directed
swirling (points R1R2, R3R4) vanishes with increasing δ.
B
A
/
11
0
0.05
0.1
0.15
0.2
0.25
0.3
0.3
0.35
0.2
0.1
1.21.151.11.0510 0.950.90.85
E
EH
H
H
H
1
2
1
2
4
3
0
0.05
0.1
0.15
0.2
0.25
0.4
0.3
0.35
0.4
0.3
0.2
0.1 1.21.151.11.0510 0.950.90.85
E
1
E2
A
B
/
11
Figure 6. The same as in figure 4 but for δ = 0.5 (the left panel) and 0.8
(the right one). In contrast to the undamped case in [3], there are no counter-
directed swirling at all.
There is the frequency range between E1 and H1 where the theory does
not predict stable steady-state sloshing and irregular (chaotic) waves are
expected.
Increasing the semi-axes ratio δ decreases the loop-like R branch re-
sponsible for the counter-directed swirling. Figure 5 demonstrates this
fact for δ = 0.3 and 0.45. Decreasing the R-branch means that the linear
damping makes the counter-directed swirling impossible when passaging
from longitudinal to rotary forcing types. In the contrast, the theoretical
undamped analysis in [3] shows that the counter-directed propagating
wave exists and may be stable in a frequency range for any 0 < δ < 1.
202 Raynovskyy I. A., Timokha A. N.
/
11
A
B
0.4
0.3
0.20
0.85
0.1
0.9 0.10.95
0.2
1 1.05 1.1
0.3
01.15 1.2
0.4
E2
1E
Figure 7. The same as in figure 6 but for an almost rotary forcing with
δ = 0.95 (the left panel) and rotary excitations with δ = 1.0 (the right panel).
The non-zero ξ leads to vanishing the R branch as δ increases. When
ξ = 0.02, this happens for δ slightly lower then 0.5. As a consequence,
we do not see this branch in figure 6, where δ = 0.5 and 0.8.
In figures 5 and 6, we also see extra islands of stability H3H4 and
R3R4. After vanishing the R branch, the island H3H4 grows up to con-
nect other stable subbranches.
Rotary forcing (ǫx = ǫy). The undamped analysis [3] showed that the
rotary (orbital circular) forcing causes a non-uniqueness of the steady-
state solution, i.e. the secular system becomes degenerated. The paper
established wave regimes consisting of a superposition of swirling waves
in both directions and a planar standing wave. However, only co-directed
swirling has been stable according to [3].
To study the damped sloshing by using the secular system (48), (49)
with δ = 1 and ξ 6= 0, we recall that C 6= 0 but the limit C → +∞ is
possible. This limit implies the co-directed rotary wave. It transforms
(48) to A2 = B2 and the two equations (49) become equivalent
A = B > 0; A2(Λ + (m1 +m3)A
2)2 + ξ2) = ǫ2x. (55)
For the rotary co-directed sloshing, D = F = 0, that makes it possible to
restore ψ and ϕ− ψ = π/2. Numerical experiments show that (48), (49)
with ξ 6= 0 has no solution except the analytical solution (55).
Figure 7 illustrates the passage δ → 1 by drawing the response curves
with δ = 0.95 and 1. This hard-spring response is experimentally con-
firmed in [17].
Damped resonant steady-state sloshing in an upright circular tank 203
6 Conclusions
The multimodal theory of the steady-state sloshing in an upright circular
tank from [3] is generalised by adding the linear damping terms. We show
that the linear damping matter for relatively small containers, which
could be laboratory tanks, e.g., bioreactors. Because any periodic orbital
tank forcing with four degrees of freedom (sway/surge/pitch/roll) are,
within to higher order contributions, equivalent to an artificial elliptic
horizontal tank excitation, the study concentrates on the steady-state
wave regimes occurring due to these elliptic excitations with different
semi-axes ratios δ.
For the longitudinal forcing, the linear damping leads to extra bifur-
cation points on the response curves where swirling emerges from the pla-
nar wave regime. Each point on the swirling-related branch implies two
physically identical but counter-directed progressive angular waves. Non-
zero semi-axes ratio splits the branch into two non-connected parts, one
of which corresponds to swirling, which is co-directed with the forcing,
but another part implies a counter-directed swirling. For smaller ratios,
both parts exist and there are frequency ranges where the correspond-
ing steady-state waves are stable. Increasing the semi-axes ratio makes
the second branching part (counter-directed swirling) smaller so that it
finally vanishes at a certain δ. This is opposite to the undamped case [3],
when the counter-directed swirling was co-existing for any 0 < δ ≤ 1.
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1. Голуб А.П., Лисенко Л.О.
2. Кіфоренко Б.М., Ткаченко Я.В., Васильєв І.Ю.
Постановка задачі
Оптимальні керування
Результати розрахунків
3. Коломійчук О.П., Новицький В.В.
Вступ
Ідентифікація системи з кососиметричною невиродженою матрицею коефіцієнтів методом квазілінеаризації
4. Кононов Ю.Н., Джуха Ю.А.
Введение
Постановка задачи
Метод решения
Собственные частоты совместных колебаний упругой мембраны и жидкости
Устойчивость осесимметричных колебаний упругой мембраны при перегрузке
5. Константінов О.В., Новицький В.В.
Математична модель механічної системи ``резервуар – рідина з вільною поверхнею''
Побудова програмного керування та керування зі зворотним зв'язком
Результати чисельного моделювання
6. Лимарченко О.С., Нефьодов О.О.
Вступ
Математична модель системи
Результати чисельного моделювання
Висновки
7. Лимарченко В.О., Лимарченко О.С., Сапон М.М.
Вступ
Математична модель системи
Аналіз числових прикладів
Висновки
8. Луковський І.О.
Крайова задача теорії просторового руху резервуара, цілком заповненого ідеальною нестисливою рідиною
Варіаційний принцип у задачі про просторовий рух пружного резервуара, цілком заповненого рідиною
Визначення сил взаємодії між пружними стінками резервуара і рідиною
9. Мазко О.Г., Кусій С.М.
Вступ
Допоміжні твердження
Динамічний регулятор по вимірюваному виходу
Зважений рівень гасіння обмежених збурень.
Динамічний регулятор зі збуреннями
Дискретні системи з керованими і спостережуваними виходами
Приклад. Двомасова механічна система
Висновок
10. Працьовитий М.В., Свинчук О.В.
Вступ
Основний об'єкт
Розподіл значень функції f(x) при заданому розподілі випадкового аргументу
11. Сатур О.Р.
Існування граничних координат
Нерухомі точки динамічної системи конфлікту з притягальною взаємодією
12. Солодун А.В.
Постановка задачи
Нелинейные модальные системы
Модальные представления и
Кинематические и динамические уравнения
Нелинейная форма модальных уравнений
Бесконечномерная система нелинейных асимптотических модальных уравнений третьего порядка
Асимптотика Моисеева-Нариманова
Общие бесконечномерные нелинейные асимптотические модальные уравнения
13. Сосницький C.П.
Рівняння руху для обмеженої задачі трьох тіл
Про деякі важливі рівності в еліптичній обмеженій задачі трьох тіл
Про рух малої частки по координаті
14. Троценко Ю.В.
Постановка задачи
Вариационная формулировка задачи
Построение решений
Некоторые результаты расчетов
15. Тугай Г.В.
Попередні відомості
Побудова матриці Якобі, асоційованої з сингулярно збуреним оператором
Висновки
16. Raynovskyy I.A., Timokha A.N.
Introduction
Statement of the problem
Linear damping coefficients
Steady-state resonant solution
Response curves in the (/11,A,B) space
Conclusions
17. Timokha A.N.
Statement
Almost steady-state asymptotic solution
Conclusion
|
| id | oai:trim.imath.kiev.ua:article-345 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:03Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/32/4e3421070e9de3f47a9c069bce13f232.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3452018-02-13T11:53:50Z Damped resonant steady-state sloshing in an upright circular tank Затухающие резонансные стационарные колебания в вертикальном круговом резервуаре Затухаючі резонансні стаціонарні коливання у вертикальному круговому резервуарі Raynovskyy, I. A. Timokha, A. N. Райновский, И. А. Тимоха, А. Н. Райновський, І. А. Тімоха, О. М. The nonlinear Narimanov–Moiseev-type modal system with linear damping terms is employed to study the damped steady-state resonant sloshing in an upright circular tank. Estimating the damping coefficients (ratios) by using Miles’ formula shows that the damping may matter for laboratory tanks. An asymptotic steady-state solution of the modal system is derived for a prescribed cyclic tank motion with four (sway/surge/pitch/roll) degrees of freedom; the forcing frequency is close to the lowest natural sloshing frequency. The steady-state response curves by the two lowestorder natural sloshing mode amplitudes are examined versus the semi-axes ratio of the artificial elliptic orbit. Используя нелинейных модальную систему Нариманова-Моисеева с линейными коэффициентами демпфирования, изучаются установившиеся демпфированные&nbsp; резонансные колебания жидкости в цилиндрическом баке. Оценка коэффициентов демпфирования с помощью формулы Майлза показывает, что демпфирования может иметь значение для лабораторных сосудов. Найдено асимптотический устоявшийся решение модальной системы для заданного движения цилиндрической сосуды с четырьмя (sway / surge / pitch / roll) степенями вольности; частота возмущения близка к самой низкой собственной частоты колебаний жидкости. Рассмотрены зависимость амплитудно-частотных характеристик, связанных с амплитудой двух низших собственных форм колебаний жидкости от соотношения пiвосей эллиптических орбиты. Використовуючи нелiнiйну модальну систему Нарiманова–Мойсеєва iз лiнiйними коефiцiєнтами демпфування, вивчаються усталенi демпфованi резонанснi хлюпання рiдини у цилiндричному баку. Оцiнка коефiцiєнтiв демпфування за допомогою формули Майлза показує, що демпфування може мати значення для лабораторних посудин. Знайдено асимптотичний усталений розв’язок модальної системи для заданого руху цилiндричної посудини iз чотирма (sway/surge/pitch/roll) ступенями вiльностi; частота збурення є близькою до найнижчої власної частоти коливань рiдини. Розглянуто залежнiсть амплiтудно-частотних характеристик, що пов’язуються iз амплiтудами двох нижчих власних форм коливання рiдини вiд спiввiдношення пiвосей елiптичної орбiти. Інститут математики НАН України 2017-10-31 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/345 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 14 No. 2 (2017): Mathematic problems of mechanics and computational mathematics; 180-204 Сборник Трудов Института математики НАН Украины; Том 14 № 2 (2017): Математические проблемы механики и вычислительной математики; 180-204 Збірник Праць Інституту математики НАН України; Том 14 № 2 (2017): Математичні проблеми механіки та обчислювальної математики; 180-204 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/345/356 Авторське право (c) 2017 Праці Інституту математики НАН України |
| spellingShingle | Raynovskyy, I. A. Timokha, A. N. Райновский, И. А. Тимоха, А. Н. Райновський, І. А. Тімоха, О. М. Damped resonant steady-state sloshing in an upright circular tank |
| title | Damped resonant steady-state sloshing in an upright circular tank |
| title_alt | Затухающие резонансные стационарные колебания в вертикальном круговом резервуаре Затухаючі резонансні стаціонарні коливання у вертикальному круговому резервуарі |
| title_full | Damped resonant steady-state sloshing in an upright circular tank |
| title_fullStr | Damped resonant steady-state sloshing in an upright circular tank |
| title_full_unstemmed | Damped resonant steady-state sloshing in an upright circular tank |
| title_short | Damped resonant steady-state sloshing in an upright circular tank |
| title_sort | damped resonant steady-state sloshing in an upright circular tank |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/345 |
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