The Narimanov–Moiseev modal equations for sloshing in an annular tank
A complete weakly-nonlinear modal system of the Narimanov–Moiseevtype is derived for sloshing in an upright annular tank by using the deri-vation scheme proposed by the author for a spherical tank. The modalsystem couples the two dominant generalised coordinates responsible forthe lowest natural slo...
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| author | Timokha, A. N. Timokha, A. N. |
| author_facet | Timokha, A. N. Timokha, A. N. |
| author_institution_txt_mv | [
{
"author": "A. N. Timokha",
"institution": "Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine; Centre of Excellence AMOS, Norwegian University of Science and Technology, Trondheim, Norway"
}
] |
| author_sort | Timokha, A. N. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
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| datestamp_date | 2018-01-23T12:01:13Z |
| description | A complete weakly-nonlinear modal system of the Narimanov–Moiseevtype is derived for sloshing in an upright annular tank by using the deri-vation scheme proposed by the author for a spherical tank. The modalsystem couples the two dominant generalised coordinates responsible forthe lowest natural sloshing modes and an infinite set of the second- andthird-order generalised coordinates. |
| first_indexed | 2026-08-04T01:01:28Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2015, т. 12, № 5, 241–266
УДК 532.595
The Narimanov–Moiseev modal
equations for sloshing
in an annular tank∗
A.N. Timokha
Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine;
Centre of Excellence AMOS, Norwegian University of Science and
Technology, Trondheim, Norway
A complete weakly-nonlinear modal system of the Narimanov–Moiseev
type is derived for sloshing in an upright annular tank by using the deri-
vation scheme proposed by the author for a spherical tank. The modal
system couples the two dominant generalised coordinates responsible for
the lowest natural sloshing modes and an infinite set of the second- and
third-order generalised coordinates.
Виводиться повна слабо-нелiнiйна модальна система типу Нарiманова-
Моiсєєва, яка описує коливання рiдини у вертикальному бацi кiльце-
вого перерiзу. Використовується схема виводу, яку автор запропонував
для сферичного баку. Модальна система пов’язує двi домiнантнi уза-
гальненi координати, що вiдповiдають за першi власнi форми колива-
ння рiдини, та нескiнченну множину узагальнених координат другого
та третього порядкiв малостi.
1. Introduction
The annular upright tank belongs to the historically-first reservoir shapes
for which weakly-nonlinear modal systems describing liquid sloshing due
to a resonant excitation of the lowest natural sloshing frequencies were
derived. Interested readers can find the state-of-the-art of 1990 and
2015 in [5] and [6], respectively. Derivation of these systems adopts,
∗ 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© A.N. Timokha, 2015
242 A. N. Timokha
normally, the Narimanov–Moiseev asymptotic relationships between the
sloshing-related generalised coordinates. For axisymmetric basins, the
complete weakly-nonlinear modal system of the Narimanov–Moisev type
should couple two dominant, first-order generalised coordinates governi-
ng the lowest natural sloshing modes amplification and an infinite set of
the second- and third-order sloshing-related generalised coordinates [4].
Getting the complete systems is a rather complicated analytical task. As a
result, those exist only for the upright circular cylindrical [3] and spherical
[2] tanks. The existing Narimanov–Moiseev modal systems for the upright
annular tank are almost fully represented by the five-dimensional (three
generalised coordinates of the second order) system by Lukovsky [5, 6]
and the fourteen-dimensional (six generalised coordinates of the second
and third-order, respectively) system by Takahara & Kimura [9]. Using
the analytical procedure from [2], the present paper derives the complete
infinite-dimensional weakly-nonlinear Narimanov–Moiseev modal system
for this tank shape. The hydrodynamic coefficients of this system are vali-
dated by comparing them with those by Lukovsky [5, 6].
2. Natural sloshing modes and frequencies
h
dimensional
r
1
h
nondimensional
S
S
Q
Σ
0
0e
0b
0
S0i
h
r
r1 1
1
0
D L 0e
L
L 0i
0b
L 0
nondimensional
meridional cross−section
x
y
z
r
r
1
2
Fig. 1. Dimensional and nondimensional sketches of the mean liquid domain
Q0 in an upright annular tank. The nondimensional mean liquid depth is h,
the internal radius equals to r1 and the external radius is equal to the unit.
The mean free surface is Σ0, the wetted inner and external walls are S0i and
S0e, respectively, but S0b is the bottom. The corresponding boundaries in the
meridional cross-section are denoted by the L0∗ symbols.
The nonlinear multimodal method involves the analytically–found
Narimanov–Moiseev modal equations 243
natural sloshing modes associated with eigenfunctions of the spectral
boundary problem
∇2ϕ = 0 in Q0,
∂ϕ
∂n
= 0 on S0e, S0i, S0b,
∂ϕ
∂n
= κϕ on Σ0 (1)
in notations of fig. 1. The geometric parameters are normalised by the
larger radius r̄2. The r̄2-scaled spectral boundary problem (1) has the
analytical solution [1, 6] which is obtained by using separation of spatial
variables in the cylindrical coordinate system
ϕMi(r, z, θ) = RMi(r)ZMi(z)
cosMθ
sinMθ , M = 0, . . . ; i = 1, . . . , (2)
where
RMi(r) = αMi det
∣∣∣∣
JM (kMir)
J ′
M (kMi)
YM (kMir)
Y ′
M (kMi)
∣∣∣∣ , (3a)
ZMi(z) =
cosh(kMi(z + h))
cosh(kMih)
. (3b)
Here, JM (·) and YM (·) are the Bessel functions of the first and second
kinds, respectively, the radial wave numbers kMi are computed from the
equations R′
M,i(r1) = 0, and the normalising multipliers αMi follow from
the orthogonality condition [1]
λ(Mi)(Mj) =
∫ 1
r1
rRMi(r)RMj(r) dr = δij , i, j = 1, . . . , (4)
where δij is the Kronecker delta. The wave numbers and multipliers can
be found analytically, by using the Bessel function algebra, or numerically.
The spectral parameter κMi and the natural sloshing frequencies σMi are
κMi = kMi tanh(kMih), σ2
Mi = κMi ḡ/r̄2 = κMi g, (5)
where ḡ is the gravity acceleration.
The limit case r1 = 0 (circular cylindrical tank) implies replacing (3a)
with RMi = αMiJM (kMir) but other formulas remain the same.
3. The Stokes–Joukowski potentials
The linearised Stokes–Joukowski potentials Ω0i(r, z, θ), i = 1, 2, 3 are
harmonic functions satisfying the Neumann boundary conditions [1,
Sect. 5.4.4]:
∂Ω01
∂n
= −(znr − rnz) sin θ,
∂Ω02
∂n
= (znr − rnz) cos θ,
∂Ω03
∂n
= 0 (6)
244 A. N. Timokha
z
4
(t)
η
1
(t)
η
5
(t)
η
2
(t)
0
x
y
Σ
(t)Σ
Q (t)
S(t)
η
Fig. 2. The liquid volume evolution Q(t) with the free surface Σ(t) and the
wetted tank surface S(t) are considered in the tank-fixed coordinate system
Oxyz whose coordinate plane Oxy coincides with the mean free surface Σ0
but Oz is the symmetry axis. Small-magnitude tank motions are governed by
the generalised coordinates η1(t) (surge), η4(t) (roll), η2(t) (sway), and η5(t)
(pitch). The heave and yaw motions are not considered.
on Σ0, S0i, S0e, and S0b (see, notations in fig. 1). This implies Ω01 =
−F (r, z) sin θ, Ω02 = F (r, z) cos θ, Ω03 = 0, where
F (r, z) = rz +
∞∑
n=1
cnR1n(r)
sinh(k1n(z +
h
2 ))
cosh(k1n
h
2 )
, (7a)
cn = − 2
k1n
Pn, Pn =
∫ 1
r1
r2 R1n(r) dr. (7b)
Again, cn can be computed in both analytical and numerical ways.
4. Statement of the problem
We consider sloshing in an upright annular rigid tank performing small-
magnitude sway, surge, roll, and pitch motions (heave and yaw are not
considered) which are described by the generalised coordinates ηi(t) =
O(ǫ) ≪ 1, i = 1, 2, 4, 5 responsible for the tank-fixed coordinate system
motions. The geometric notations are given in fig. 2. An inviscid contai-
ned liquid with irrotational flows is assumed. The surface patterns Σ(t)
are governed by z = ζ(r, θ, t) and the sloshing flows are defined by
Narimanov–Moiseev modal equations 245
velocity potential Φ(r, θ, z, t); the both unknowns are given in the tank-
fixed, Oxyz-equivalent cylindrical coordinate system. The unknowns ζ
and Φ should be found from the corresponding free-surface problem
or, alternatively, from a variational principle (see, [1, 6] and references
therein). Furthermore, the modal solution is used (the natural sloshing
modes (2) and their projections on Σ0 constitute a Fourier basis):
ζ(r, z, θ) =
Ia,Ir∑
M,i
RMi(r) cos(Mθ) pMi(t)+
Ia,Ir∑
m,i
Rmi(r) cos(mθ) rmi(t), (8a)
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]
+
Ia,Ir∑
M,i
RMi(r)ZMi(z) cos(Mθ)PMi(t)
+
Ia,Ir∑
m,i
Rmi(r)Zmi(z) sin(mθ)Rmi(t), Ia, Ir → ∞ (8b)
(henceforth, the capital summation letter implies the change from zero to
Ia but the lower case indices mean the change from one to either Ia or Ir)
which introduces the sloshing-related nondimenional generalised coordi-
nates O(ǫ) . pMi(t), rmi(t) and velocities O(ǫ) . Pmi(t), Rmi(t). The
o(ǫ)-quantities associated with the nonlinear Stokes–Joukowski potential
are omitted so that (8b) linearly depends on ηi(t) as in the linear sloshing
theories (see, details in [1, Ch. 7 and 5]).
Using the Bateman–Luke variational principle, the books [1,5,6] derive
the fully-nonlinear modal system with respect to the generalised coordi-
nates and velocities. For axisymmetric tanks, the modal system can be
rewritten [2] in the form
Ia,Ir∑
M,n
∂ApAb
∂pMn
ṗMn +
Ia,Ir∑
m,n
∂ApAb
∂rmn
ṙmn =
Ia,Ir∑
M,n
App(Ab)(Mn)PMn
+
Ia,Ir∑
m,n
Apr(Ab),(Mn)Rmn, (9a)
246 A. N. Timokha
Ia,Ir∑
M,n
∂Arab
∂pMn
ṗMn +
Ia,Ir∑
m,n
∂Arab
∂rmn
ṙmn =
Ia,Ir∑
M,n
Apr(Mn),(ab)PMn
+
Ia,Ir∑
m,n
Arr(ab)(mn)Rmn, A = 0, . . . ; a, b = 1, . . . ; Ia, Ir → ∞ (9b)
(the kinematic subsystem) and
Ia,Ir∑
M,n
∂ApMn
∂pAb
ṖMn +
Ia,Ir∑
m,n
∂Armn
∂pAb
Ṙmn +
1
2
Ia,Ir∑
ML,nk
∂App(Mn)(Lk)
∂pAb
PMnPLk
+
Ia,Ir∑
Ml,nk
∂Apr(Mn),(lk)
∂pAb
PMnRlk +
1
2
Ia,Ir∑
ml,nk
∂Arr(mn)(lk)
∂PAb
RmnRlk + gΛAA, pAb
+ (η̈1 − gη5 − Sbη̈5)Λ1A,Pb = 0, (10a)
Ia,Ir∑
M,n
∂ApMn
∂rab
ṖMn +
Ia,Ir∑
m,n
∂Armn
∂rab
+
1
2
Ia,Ir∑
ML,nk
∂App(Mn)(Kl)
∂rab
PMnPLk
+
Ia,Ir∑
Nl,nk
∂Apr(Mn),(lk)
∂rab
PMnRlk +
1
2
Ia,Ir∑
ml,nk
∂Arr(mn)(lk)
∂rab
RmnRlk + gΛ,aarab
+ (η̈2 + gη4 + Sbη̈4)Λ1a,Pb = 0, A = 0, . . . ; a, b = 1, . . . , (10b)
(the dynamic subsystem), Ia, Ir → ∞, where the comma between the
indices pairs, alike (Ab), (Mn), means that the pairs do not commutate;
coefficients Pb are defined in (7b),
Sb = 2 k−1
1b tanh(k1b
h
2 ), (11)
and the Λ-tensor is introduced in section 10. The modal system (9),
(10) contains the following nonlinear functions of the sloshing-related
generalised coordinates
App(Ab)(Mn)=
1∫
r1
π∫
−π
r
[
cosAθ cosMθ G(1)
(Ab)(Mn)+sinAθ sinMθ G(2)
(Ab)(Mn)
]
dθdr,
Arr(ab)(mn)=
1∫
r1
π∫
−π
r
[
sin aθ sinmθ G(1)
(ab)(mn) + cos aθ cosmθ G(2)
(ab)(mn)
]
dθdr,
Narimanov–Moiseev modal equations 247
Apr(Ab),(mn)=
1∫
r1
π∫
−π
r
[
cosAθ sinmθ G(1)
(Ab)(mn)− sinAθ cosmθ G(2)
(Ab)(mn)
]
dθdr,
ApAb =
1∫
r1
π∫
−π
r cos(Aθ)G(0)
Ab dθdr, A
r
ab =
1∫
r1
π∫
−π
r sin(aθ)G(0)
Ab dθdr, (12)
where
G(0)
Ab = RAb(r)
∫ ζ
−h
cosh(kAb(z + h))
cosh(kAbh)
dz = RAb(r) I
(0)
(Ab),
G(1)
(Ab)(Mn)= R′
Ab(r)R′
Mn(r)I
(1)
(Ab)(Mn)+RAb(r)RMn(r)kAbkMnI
(2)
(Ab)(Mn),
G(2)
(Ab)(Mn) = AM r−2 RAb(r)RMn(r) I
(1)
(Ab)(Mn) ; (13)
I
(1)
(Ab)(Mn) =
∫ ζ
−h
cosh(kAb(z + h)) cosh(kMn(z + h))
cosh(kAbh) cosh(kMnh)
dz,
I
(2)
(Ab)(Mn) =
∫ ζ
−h
sinh(kAb(z + h)) sinh(kMn(z + h))
cosh(kAbh) cosh(kMnh)
dz.
(14)
5. Adaptive third-order modal equations
The general adaptive intermodal ordering suggests that all sloshing-
related generalised coordinates and velocities in (8) are of the same order
O(ǫ1/3). The aim is to derive a weakly-nonlinear infinite-dimensional
modal system coupling the sloshing-related generalised coordinates wi-
thout the generalised velocities. Derivation consists of the following five
steps [2].
The first step suggests the Taylor expansion by ζ of I(0)(Ab), I
(1)
(Ab)(Mn),
and I
(2)
(Ab)(Mn) by (13) and (14). By definition, ζ = O(ǫ1/3). Analysis
shows that I(0)(Ab) should be expanded up to the third order but I(1)(Ab)(Mn)
and I(2)(Ab)(Mn) require expansion up to the second order, i.e.
I
(0)
(Ab) = k−1
Ab tanh(kAbh) + ζ + 1
2κAbζ
2 + 1
6k
2
Abζ
3 + . . . , (15a)
I
(1)
(Ab)(Mn) = O(1) + ζ + 1
2 (κAb + κMn)ζ
2 + . . . , (15b)
I
(2)
(Ab)(Mn) = O(1) + κAbκMnζ +
1
2 (k
2
AbκMn + k2MnκAb)ζ
2 + . . . . (15c)
248 A. N. Timokha
Inserting (15b) and (15c) into (13) gives
G(1)
(Ab)(Mn) = O(1) + (R′
AbR′
Mn +RAbRMnκAbκMn) ζ
+ 1
2 [(κAb + κMn)R′
AbR′
Mn +RAbRMn(k
2
AbκMn + k2MnκAb)]ζ
2, (16a)
G(2)
(Ab)(Mn) = O(1) + r−2AMRAbRMn ζ
+ 1
2r
−2AM(κAb + κMn)RAbRMn ζ
2. (16b)
By the second step, ApAb and Arab should be expanded up to O(ǫ) in
terms to the sloshing-related generalised coordinates. For this purpose,
(8a) is inserted into expressions of (15a) and, thereafter, substituted into
the corresponding formulas of (12). This gives
ApAb = ΛAA,pAb +
1
2
Ia,Ir∑
MN,ij
χpp(Mi)(Nj),(Ab)pMipNj
+ 1
2
Ia,Ir∑
mn,ij
χrr(mi)(nj),(Ab)rmirnj +
1
3
Ia,Ir∑
MNK,ijl
χppp(Mi)(Nj)(Kl),(Ab)pMipNjpKl
+
Ia,Ir∑
Mnk,ijl
χprr(Mi),(nj)(kl),(Ab)pMirnjrkl, (17a)
Arab=Λ,aarab+
Ia,Ir∑
Mn,ij
χpr(Mi),(nj),(ab)pMirnj+
1
3
Ia,Ir∑
mnk,ijl
χrrr(mi)(nj)(kl),(ab)rmirnjrkl
+
Ia,Ir∑
MNk,ijl
χppr(Mi)(Nj),(kl),(ab)pMipNjrkl, Ia, Ir → ∞, (17b)
where
χpp(Mi)(Nj),(Ab) = κAbΛAMN,λ(Ab)(Mi)(Nj),
χrr(Mi)(nj),(Ab) = κAbΛA,mnλ(Ab)(mi)(nj),
χppp(Mi)(Nj)(Kl),(Ab) =
1
2k
2
AbΛAMNK,λ(Ab)(Mi)(Nj)(Kl),
χprr(Mi),(nj)(kl),(Ab) =
1
2k
2
AbΛAM,nkλ(Ab)(Mi)(nj)(kl) ,
Narimanov–Moiseev modal equations 249
χpr(Mi),(nj),(ab) = κabΛM,anλ(Mi)(nj)(ab) ,
χrrr(mi)(nj)(kl),(ab) =
1
2k
2
abΛ,amnkλ(mi)(nj)(kl)(ab) ,
χppr(Mi)(Nj),(kl),(ab) =
1
2k
2
abΛMN,akλ(Mi)(Nj)(kl)(ab)
with notations from section 10. The partial derivatives of (17) by the
generalised coordinates take the form
∂ApAb
∂pDf
= ΛAD,δbf+
Ia,Ir∑
M,i
χpp(Mi)(Df),(Ab)pMi+
Ia,Ir∑
NK,jl
χppp(Df)(Nj)(Kl),(Ab)pNjpKl
+
Ia,Ir∑
nk,jl
χprr(Df),(nj)(kl),(Ab)rnjrkl, (18a)
∂ApAb
∂rdf
=
Ia,Ir∑
m,i
χrr(mi)(df),(Ab)rmi + 2
Ia,Ir∑
Mn,ij
χprr(Mi),(nj)(df),(Ab)pMirnj , (18b)
∂Arab
∂pDf
=
Ia,Ir∑
n,j
χpr(Df),(nj),(ab)rnj + 2
Ia,Ir∑
Mn,ij
χppr(Mi)(Df),(nj),(ab)pMirnj , (18c)
∂Arab
∂rdf
= Λ,adδbf +
Ia,Ir∑
M,i
χpr(Mi),(df),(ab)pMi +
Ia,Ir∑
mn,ij
χrrr(mi)(nj)(df),(ab)rmirnj
+
Ia,Ir∑
MN,ij
χppr(Mi)(Nj),(df),(ab)pMipNj, Ia, Ir → ∞. (18d)
The third step should lead to analogous expressions for
App(Ab)(Mn), A
rr
(ab)(mn) and Apr(Ab),(mn) but up to the second-order terms,
O(ǫ2/3). The O(1)-order term can be taken from the linear modal theory.
The result is
App(Ab)(Mn) = ΛAM,δbnκAb +
Ia,Ir∑
K,l
Πp,p(Kl),(Ab)(Mn)pKl
+
Ia,Ir∑
KC,ld
Πp,pp(Kl)(Cd),(Ab)(Mn)pKlpCd +
Ia,Ir∑
kc,ld
Πp,rr(kl)(cd),(Ab)(Mn)rklrcd, (19a)
250 A. N. Timokha
Arr(ab)(mn) = Λ,amδbnκab +
Ia,Ir∑
K,l
Πr,p(Kl),(ab)(mn)pKl
+
Ia,Ir∑
KC,ld
Πr,pp(Kl)(Cd),(ab)(mn)pKlpCd +
Ia,Ir∑
kc,ld
Πr,rr(kl)(cd),(ab)(mn)rklrcd, (19b)
Apr(Ab),(mn) =
Ia,Ir∑
k,l
Πr(kl),(Ab),(mn)rkl+
Ia,Ir∑
Kc,ld
Πpr(Kl),(cd),(Ab),(mn)pKlrcd, (19c)
where
Πp,p(Kl),(Ab)(Mn) = ΛAMK,G
(11)
(Ab)(Mn),(Kl) + ΛK,AMG
(12)
(Ab)(Mn),(Kl),
Πr,p(Kl),(ab)(mn) = ΛK,amG
(11)
(ab)(mn),(Kl) + ΛamK,G
(12)
(ab)(mn),(Kl),
Πr(kl),(Ab),(mn) = ΛA,mkG
(11)
(Ab)(mn),(kl) − Λm,AkG
(12)
(Ab)(mn),(kl),
Πp,pp(Kl)(Cd),(Ab)(Mn)=ΛAMKC,G
(21)
(Ab)(Mn),(Kl)(Cd)+ΛKC,AMG
(22)
(Ab)(Mn),(Kl)(Cd),
Πp,rr(kl)(cd),(Ab)(Mn) = ΛAM,kcG
(21)
(Ab)(Mn),(kl)(cd) + Λ,AMkcG
(22)
(Ab)(Mn),(kl)(cd),
Πr,pp(Kl)(Cd),(ab)(mn)=ΛKC,amG
(21)
(ab)(mn),(Kl)(Cd)+ΛKCam,G
(22)
(ab)(mn),(Kl)(Cd),
Πr,rr(kl)(cd),(ab)(mn) = Λ,amkcG
(21)
(ab)(mn),(kl)(cd) + Λam,kcG
(22)
(ab)(mn),(kl)(cd),
Πpr(Kl),(cd),(Ab),(mn)=2[ΛAK,mcG
(21)
(Ab)(mn),(Kl)(cd)−ΛKm,AcG
(22)
(Ab)(mn),(Kl)(cd)];
G
(11)
(Ab)(Mn),(Kl) = λ′(Ab)(Mn),(Kl) + κAbκMnλ(Ab)(Mn)(Kl),
G
(12)
(Ab)(Mn),(Kl) = AMλ̄(Ab)(Mn)(Kl),
G
(21)
(Ab)(Mn),(Kl)(Cd) =
1
2 [(κAb + κMn)λ
′
(Ab)(Mn),(Kl)(Cd)
+ (k2AbκMn + k2MnκAb)λ(Ab)(Mn)(Kl)(Cd)],
G
(22)
(Ab)(Mn),(Kl)(Cd) =
1
2AM(κAb + κMn)λ̄(Ab)(Mn)(Kl)(Cd).
The partial derivatives of (19) by the generalised coordinates are
∂App(Ab)(Cd)
∂pEf
= Πp,p(Ef),(Ab)(Cd) + 2
Ia,Ir∑
M,i
Πp,pp(Mi)(Ef),(Ab)(Cd)pMi, (20a)
∂App(Ab)(Cd)
∂ref
= 2
Ia,Ir∑
m,i
Πp,rr(mi)(ef),(Ab)(Cd)rmi, (20b)
Narimanov–Moiseev modal equations 251
∂Arr(ab)(cd)
∂pEf
= Πr,p(Ef),(ab)(cd) + 2
Ia,Ir∑
M,i
Πr,pp(Mi)(Ef),(ab)(cd)pMi, (20c)
∂Arr(ab)(cd)
∂ref
= 2
Ia,Ir∑
m,i
Πr,rr(mi)(ef),(ab)(cd)rmi, (20d)
∂Apr(Ab),(cd)
∂pEf
=
Ia,Ir∑
n,j
Πpr(Ef),(nj),(Ab)(cd)rnj , (20e)
∂Apr(Ab),(cd)
∂ref
= Πr(ef),(Ab),(cd) +
Ia,Ir∑
M,i
Πpr(Mi),(ef),(Ab)(cd)pMi. (20f)
By the fourth step, the kinematic equations (9) should be resolved
with respect to the generalised velocities PAb and Rab by postulating
PAb =
1
κAb
ṗAb+
Ia,Ir∑
MN,ij
V pp(Mi),(Nj),(Ab)ṗMipNj+
Ia,Ir∑
mn,ij
V rr(mi),(nj),(Ab) ṙmirnj
+
Ia,Ir∑
MNK,ijl
V ppp(Mi),(Nj),(Kl),(Ab)ṗMipNjpKl+
Ia,Ir∑
Mnk,ijl
V prr(Mi),(nj),(kl),(Ab)ṗMirnjrkl
+
Ia,Ir∑
Mnk,ijl
V rpr(nj),(Mi),(kl),(Ab) ṙnjpMirkl, (21a)
Rab =
1
κab
ṙab +
Ia,Ir∑
Mn,ij
V pr(Mi),(nj),(ab)ṗMirnj +
Ia,Ir∑
Mn,ij
V rp(nj),(Mi),(ab)ṙnjpMi
+
Ia,Ir∑
mnk,ijl
V rrr(mi),(nj),(kl),(ab) ṙmirnjrkl +
Ia,Ir∑
MNk,ijl
V rpp(kl),(Mi),(Nj),(ab) ṙklpMipNj
+
Ia,Ir∑
MNk,ijl
V ppr(Mi),(Nj),(kl),(ab)ṗMipNjrkl, (21b)
substituting (21) into the kinematic subsystem (9), and matching the
similar quantities. The procedure derives the V -coefficients as
V pp(Mi),(Nj),(Ab) =
1
ΛAAκAb
[
χpp(Nj)(Mi),(Ab) −
Πp,p(Nj),(Ab)(Mi)
κMi
]
,
252 A. N. Timokha
V rr(mi),(nj),(Ab) =
1
ΛAAκAb
[
χrr(nj)(mi),(Ab) −
Πr(nj),(Ab),(mi)
κmi
]
,
V rp(nj),(Mi),(ab) =
1
Λaaκab
[
χpr(Mi),(nj),(ab) −
Πr,p(Mi),(ab)(nj)
κnj
]
,
V pr(Mi),(nj),(ab) =
1
Λaaκab
[
χpr(Mi),(nj),(ab) −
Πr(nj),(Mi),(ab)
κMi
]
,
V ppp(Mi),(Nj),(Kl),(Ab) =
1
ΛAAκAb
[
χppp(Mi)(Nj)(Kl),(Ab) −
Πp,pp(Nj)(Kl),(Ab)(Mi)
κMi
−
Ia,Ir∑
C,d
V pp(Mi),(Nj),(Cd)Π
p,p
(Kl),(Ab)(Cd)
; V prr(Mi),(nj),(kl),(Ab) =
1
ΛAAκAb
×
[
χprr(Mi),(nj)(kl),(Ab)−
Πp,rr(nj)(kl),(Ab)(Mi)
κMi
−
Ia,Ir∑
c,d
V pr(Mi),(nj),(cd)Π
r
(kl),(Ab),(cd)
,
V rrr(mi),(nj),(kl),(ab) =
1
Λaaκab
[
χrrr(nj)(kl)(mi),(ab) −
Πr,rr(nj)(kl),(ab)(mi)
κmi
−
Ia,Ir∑
C,d
V rr(mi),(nj),(Cd)Π
r
(kl),(Cd),(ab)
; V rpp(kl),(Mi),(Nj),(ab) =
1
Λaaκab
×
χppr(Mi)(Nj),(kl),(ab) −
Πr,pp(Mi)(Nj),(ab)(kl)
κkl
−
Ia,Ir∑
c,d
V rp(kl),(Mi),(cd)Π
r,p
(Nj),(ab),(cd)
,
V ppr(Mi),(Nj),(kl),(ab) =
1
Λaaκab
[
2χppr(Mi)(Nj),(kl),(ab) −
Πpr(Nj),(kl),(Mi)(ab)
κMi
−
Ia,Ir∑
C,d
V pp(Mi),(Nj),(Cd)Π
r
(kl),(Cd),(ab) −
Ia,Ir∑
c,d
V pr(Mi),(kl),(cd)Π
r,p
(Nj),(ab)(cd)
,
V rpr(nj),(Mi),(kl),(Ab) =
1
ΛAAκAb
[
2χprr(Mi),(kl)(nj),(Ab) −
Πpr(Mi),(kl),(Ab)(nj)
κnj
−
Ia,Ir∑
C,d
V rr(nj),(kl),(Cd)Π
p,p
(Mi),(Ab)(Cd) −
Ia,Ir∑
c,d
V rp(nj),(Mi),(cd)Π
r
(kl),(Ab),(cd)
.
Narimanov–Moiseev modal equations 253
By the fifth step, expressions (18), (20) and (21) are substituted into
the dynamic equations (10). Excluding the o(ǫ)-terms gives the required
adaptive weakly-nonlinear modal equations
Ia,Ir∑
M,i
p̈Mi
δMEδif +
Ia,Ir∑
N,j
d
pp,(Ef)
(Mi),(Nj)pNj +
Ia,Ir∑
NK,jl
d
ppp,(Ef)
(Mi),(Nj),(Kl)pNjpKl
+
Ia,Ir∑
nk,jl
d
prr,(Ef)
(Mi),(nj),(kl)rnjrkl
+
Ia,Ir∑
mn,ij
r̈mirnj
drr,(Ef)(mi),(nj)+
Ia,Ir∑
K,l
d
rrp,(Ef)
(mi),(nj),(Kl)pKl
+
Ia,Ir∑
MN,ij
ṗMiṗNj
tpp,(Ef)(Mi),(Nj) +
Ia,Ir∑
K,l
t
ppp,(Ef)
(Mi),(Nj),(Kl)pKl
+ σ2
EfpEf
+
Ia,Ir∑
Mnk,ijl
t
prr,(Ef)
(Mi),(nj),(kl)ṗMiṙnjrkl+
Ia,Ir∑
mn,ij
ṙmiṙnj
trr,(Ef)(mi),(nj)+
Ia,Ir∑
K,l
t
rrp,(Ef)
(mi),(nj),(Kl)pKl
= −(η̈1 − gη5 − Sbη̈5)δ1Eκ1f Pf ; E = 0, . . . , Ia; f = 1, . . . , Ir, (22a)
Ia,Ir∑
Mn,ij
p̈Mirnj
dpr,(ef)(Mi),(nj) +
Ia,Ir∑
K,l
d
prp,(ef)
(Mi),(nj),(Kl)pKl
+
Ia,Ir∑
m,i
r̈mi
δmeδij +
Ia,Ir∑
N,j
d
rp,(ef)
(mi),(Nj)pNj +
Ia,Ir∑
NK,jl
d
rpp,(ef)
(mi),(Nj),(Kl)pNjpKl
+
Ia,Ir∑
nk,jl
d
rrr,(ef)
(mi),(nj),(kl)rnjrkl
+ σ2
ef ref
+
Ia,Ir∑
Mn,ij
ṗMiṙnj
tpr,(ef)(Mi),(nj) +
Ia,Ir∑
K,l
t
prp,(ef)
(Mi),(nj),(Kl)pKl
+
Ia,Ir∑
MNk,ijl
t
ppr,(ef)
(Mi),(Nj),(kl)ṗMiṗNjrkl +
Ia,Ir∑
mnk,ijl
t
rrr,(ef)
(mi),(nj),(kl) ṙmiṙnjrkl
= −(η̈2 + gη4 + Sbη̈4)δ1eκ1fPf ; e = 1, . . . , Ia; f = 1, . . . , Ir, (22b)
where the natural sloshing frequencies σEf are defined by (5), Pf comes
254 A. N. Timokha
from (7b), and
d
pp,(Ef)
(Mi),(Nj) =
κEf
ΛEE
[
ΛEEV
pp
(Mi),(Nj),(Ef) +
χpp(Nj)(Ef),(Mi)
κMi
]
,
d
ppp,(Ef)
(Mi),(Nj),(Kl) =
κEf
ΛEE
[
ΛEEV
ppp
(Mi),(Nj),(Kl),(Ef) +
χppp(Ef)(Nj)(Kl),(Mi)
κMi
+
Ia,Ir∑
A,b
V pp(Mi),(Nj),(Ab)χ
pp
(Kl)(Ef),(Ab)
,
d
prr,(Ef)
(Mi),(nj),(kl) =
κEf
ΛEE
[
ΛEEV
prr
(Mi),(nj),(kl),(Ef) +
χprr(Ef),(nj)(kl),(Mi)
κMi
+
Ia,Ir∑
a,b
V pr(Mi),(nj),(ab)χ
pr
(Ef),(kl),(ab)
,
d
rr,(Ef)
(mi),(nj) =
κEf
ΛEE
[
ΛEEV
rr
(mi),(nj),(Ef) +
χpr(Ef),(nj),(mi)
κmi
]
,
d
rrp,(Ef)
(mi),(nj),(Kl) =
κEf
ΛEE
[
ΛEEV
rpr
(mi),(Kl),(nj),(Ef) +
2χppr(Kl)(Ef),(nj),(mi)
κmi
+
Ia,Ir∑
a,b
V rp(mi),(Kl),(ab)χ
pr
(Ef),(nj),(ab) +
Ia,Ir∑
A,b
V rr(mi),(nj),(Ab)χ
pp
(Kl)(Ef),(Ab)
,
t
pp,(Ef)
(Mi),(Nj) =
κEf
ΛEE
[
ΛEEV
pp
(Mi),(Nj),(Ef) +
Πp,p(Ef),(Mi)(Nj)
2κMiκNj
]
,
t
ppp,(Ef)
(Mi),(Nj),(Kl) =
κEf
ΛEE
[
ΛEEV̄
ppp
(Mi),(Nj),(Kl),(Ef) +
Πp,pp(Kl)(Ef),(Mi)(Nj)
κMiκNj
+
Ia,Ir∑
A,b
V pp(Mi),(Nj),(Ab)χ
pp
(Kl)(Ef),(Ab) +
Ia,Ir∑
A,b
Πp,p(Ef),(Mi)(Ab)
κMi
V pp(Nj),(Kl),(Ab)
,
t
rr,(Ef)
(mi),(nj) =
κEf
ΛEE
[
ΛEEV
rr
(mi),(nj),(Ef) +
Πr,p(Ef),(mi)(nj)
2κmiκnj
]
,
Narimanov–Moiseev modal equations 255
t
rrp,(Ef)
(mi),(nj),(Kl) =
κEf
ΛEE
[
ΛEEV
rpr
(mi),(Kl),(nj),(Ef) +
Πr,pp(Kl)(Ef),(mi)(nj)
κmiκnj
+
Ia,Ir∑
A,b
V rr(mi),(nj),(Ab)χ
pp
(Kl)(Ef),(Ab) +
Ia,Ir∑
a,b
Πr,p(Ef),(mi)(ab)
κmi
V rp(nj),(Kl),(ab)
,
t
prr,(Ef)
(Mi),(nj),(kl) =
κEf
ΛEE
[
ΛEEV̄
prr
(Mi),(nj),(kl),(Ef) +
Πpr(Ef),(kl),(Mi)(nj)
κMiκnj
+
Ia,Ir∑
a,b
(
V̄ pr(Mi),(nj),(ab)χ
pr
(Ef),(kl),(ab) +
1
κnj
V pr(Mi),(kl),(ab)Π
r,p
(Ef),(ab)(nj)
)
+
Ia,Ir∑
A,b
Πp,p(Ef),(Mi)(Ab)
κMi
V rr(nj),(kl),(Ab)
,
d
pr,(ef)
(Mi),(nj) =
κef
Λee
[
ΛeeV
pr
(Mi),(nj),(ef) +
χrr(nj),(ef),(Mi)
κMi
]
,
d
prp,(ef)
(Mi),(nj),(Kl) =
κef
Λee
[
ΛeeV
ppr
(Mi),(Kl),(nj),(ef) +
2χprr(Kl),(nj)(ef),(Mi)
κMi
+
Ia,Ir∑
A,b
V pp(Mi),(Kl),(Ab)χ
rr
(nj)(ef),(Ab) +
Ia,Ir∑
a,b
V pr(Mi),(nj),(ab)χ
pr
(Kl),(ef),(ab)
,
d
rp,(ef)
(mi),(Nj) =
κef
Λee
[
ΛeeV
rp
(mi),(Nj),(ef) +
χpr(Nj),(ef),(mi)
κmi
]
,
d
rpp,(ef)
(mi),(Nj),(Kl) =
κef
Λee
[
ΛeeV
rpp
(mi),(Nj),(Kl),(ef) +
χppr(Nj)(Kl),(ef),(mi)
κmi
+
Ia,Ir∑
a,b
V rp(mi),(Nj),(ab)χ
pr
(Kl),(ef),(ab)
,
d
rrr,(ef)
(mi),(nj),(kl) =
κef
Λee
[
ΛeeV
rrr
(mi),(nj),(kl),(ef) +
χrrr(nj)(kl)(ef),(mi)
κmi
+
Ia,Ir∑
A,b
V rr(mi),(nj),(Ab)χ
rr
(kl)(ef),(Ab)
,
256 A. N. Timokha
t
pr,(ef)
(Mi),(nj) =
κef
Λee
[
ΛeeV̄
pr
(Mi),(nj),(ef) +
Πr(ef),(Mi),(nj)
κMiκnj
]
,
t
prp,(ef)
(Mi),(nj),(Kl) =
κef
Λee
[
ΛeeV̄
rpp
(nj),(Mi),(Kl),(ef) +
Πpr(Kl),(ef),(Mi)(nj)
κMiκnj
+
Ia,Ir∑
a,b
V̄ pr(Mi),(nj),(ab)χ
pr
(Kl),(ef),(ab) +
Ia,Ir∑
a,b
V rp(nj),(Kl),(ab)
κMi
Πr(ef),(Mi),(ab)
+
Ia,Ir∑
A,b
V pp(Mi),(Kl),(Ab)
κnj
Πr(ef),(Ab),(nj)
,
t
ppr,(ef)
(Mi),(Nj),(kl) =
κef
Λee
[
ΛeeV
ppr
(Mi),(Nj),(kl),(ef) +
Πp,rr(kl)(ef),(Mi)(Nj)
κMiκNj
+
Ia,Ir∑
A,b
V pp(Mi),(Nj),(Ab)χ
rr
(kl)(ef),(Ab) +
Ia,Ir∑
a,b
V pr(Nj),(kl),(ab)
κMi
Πr(ef),(Mi),(ab)
,
t
rrr,(ef)
(mi),(nj),(kl) =
κef
Λee
[
ΛeeV̄
rrr
(mi),(nj),(kl),(ef) +
Πr,rr(kl)(ef),(mi)(nj)
κmiκnj
+
Ia,Ir∑
A,b
V rr(mi),(nj),(Ab)χ
rr
(kl)(ef),(Ab) +
Ia,Ir∑
A,b
V rr(mi),(kl),(Ab)
κnj
Πr(ef),(Ab),(nj)
;
V̄ ppp(Mi),(Nj),(Kl),(Ab) = V ppp(Mi),(Nj),(Kl),(Ab) + V ppp(Mi),(Kl),(Nj),(Ab),
V̄ prr(Mi),(nj),(kl),(Ab)=V
prr
(Mi),(nj),(kl),(Ab)+V
prr
(Mi),(kl),(nj),(Ab)+V
rpr
(nj),(Mi),(kl),(Ab),
V̄ pr(Mi),(nj),(ab) = V pr(Mi),(nj),(ab) + V rp(nj),(Mi),(ab),
V̄ rrr(mi),(nj),(kl),(ab) = V rrr(mi),(nj),(kl),(ab) + V rrr(mi),(kl),(nj),(ab),
V̄ rpp(kl),(Mi),(Nj),(ab)=V
rpp
(kl),(Mi),(Nj),(ab)+V
rpp
(kl),(Nj),(Mi),(ab)+V
ppr
(Mi),(Nj),(kl),(ab).
The formulas suggest finite Ia and Ir but adopting the limit Ia, Ir →
∞ gives the inifinite-dimensional system (22) where computing the
hydrodynamic coefficients implies an infinite inner summation.
6. The Narimanov–Moiseev modal equations
Assuming (a) the O(ǫ)-order small-amplitude harmonic excitations with
the forcing frequency σ close to the lowest natural sloshing frequency
Narimanov–Moiseev modal equations 257
(here,
√
gκ11), (b) there are no secondary resonances and other small
nondimensional parameters, e.g., shallow liquid depth, Moiseev [7]
(i) showed that the dominant sloshing response is then of the order
O(ǫ1/3) contributed, exclusively, by the primary excited lowest modes
(here, the two generalised coordinates p11 and r11) and (ii) derived a
necessary (secular) condition of the steady-state (time-periodic) solution
to exist. Similar intermodal relations were postulated by Narimanov [8].
For the axisymmetric tanks, due to the trigonometric algebra with respect
to the angular coordinate, the Narimanov–Moiseev intermodal relations
deduce the following ordering for the generalised coordinates
p11 ∼ r11 = O(ǫ1/3), p0j ∼ p2j ∼ r2j = O(ǫ2/3),
r1(j+1) ∼ p1(j+1) ∼ p3j ∼ r3j = O(ǫ), j = 1, 2, . . . , (23)
but the other generalised coordinates rkl ∼ pkl = o(ǫ), k ≥ 4 and,
therefore, these can be neglected in the Narimanov–Moiseev asymptotic
scheme. The latter means that Ia = 3 but Ir may vary from 1 to infinity.
Using (23) and neglecting the o(ǫ)-terms, tedious but straightforward
derivations reduce (22) to the complete Narimanov–Moseev modal system
p̈11 + σ2
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
]
= −(η̈2 + gη4 + Sbη̈4)κ11P1, (24a)
r̈11 + σ2
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 + Sbη̈4)κ11P1; (24b)
p̈2k + σ2
2kp2k + d7,k(ṗ
2
11 − ṙ211) + d9,k(p̈11p11 − r̈11r11) = 0, (25a)
258 A. N. Timokha
r̈2k + σ2
2kr2k + 2d7,kṗ11ṙ11 + d9,k(p̈11r11 + r̈11p11) = 0, (25b)
p̈0k + σ2
0kp0k + d8,k(ṗ
2
11 + ṙ211) + d10,k(p̈11p11 + r̈11r11) = 0; (25c)
p̈3k + σ2
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, (26a)
r̈3k+σ
2
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; (26b)
p̈1n + σ2
1np1n + d16,n(p̈11p
2
11 + r11p11r̈11) + d17,n(p̈11r
2
11 − r11p11r̈11)
+ d18,np11(ṗ
2
11 + ṙ211) + d19,n(r11ṗ11ṙ11 − p11ṙ
2
11)
+
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 − Sbη̈5)κ1n Pn, (27a)
r̈1n + σ2
1nr1n + d16,n(r̈11r
2
11 + r11p11p̈11) + d17,n(r̈11p
2
11 − r11p11p̈11)
+ d18,nr11(ṗ
2
11 + ṙ211) + d19,n(p11ṗ11ṙ11 − r11ṗ
2
11)
+
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
]
Narimanov–Moiseev modal equations 259
= −(η̈2 + gη4 + Snη̈4)κ1nPn, n = 2, ..., Ir, (27b)
where the hydrodynamic coefficients are computed by the formulas
d1 = d
ppp,(11)
(11),(11),(11) = d
rrr,(11)
(11),(11),(11) = t
ppp,(11)
(11),(11),(11) = t
rrr,(11)
(11),(11),(11),
d2 = d
prr,(11)
(11),(11),(11) = d
rpp,(11)
(11),(11),(11) =
1
2 t
prr,(11)
(11),(11),(11) =
1
2 t
prp,(11)
(11),(11),(11),
d1 − d2 = d
rrp,(11)
(11),(11),(11) = d
prp,(11)
(11),(11),(11),
d1 − 2d2 = t
rrp,(11)
(11),(11),(11) = t
ppr,(11)
(11),(11),(11),
d
(j)
3 = d
pp,(11)
(11),(2j)= d
rr,(11)
(11),(2j)= d
pr,(11)
(11),(2j)= −drp,(11)(11),(2j)= t
pr,(11)
(11),(2j) = −tpr,(11)(2j),(11)
= t
pp,(11)
(11),(2j) + t
pp,(11)
(2j),(11) = t
rr,(11)
(11),(2j) + t
rr,(11)
(2j),(11),
d
(j)
4 = d
pp,(11)
(2j),(11) = d
rr,(11)
(2j),(11) = −dpr,(11)(2j),(11) = d
rp,(11)
(2j),(11),
d
(j)
5 = d
pp,(11)
(11),(0j) = d
rp,(11)
(11),(0j) = t
pr,(11)
(0j),(11) = t
pp,(11)
(0j),(11) + t
pp,(11)
(11),(0j),
d
(j)
6 = d
pp,(11)
(0j),(11) = d
pr,(11)
(0j),(11), d7,k = t
pp,(2k)
(11),(11) = −trr,(2k)(11),(11) =
1
2 t
pr,(2k)
(11),(11),
d8,k = t
pp,(0k)
(11),(11) = t
rr,(0k)
(11),(11), d10,k = d
pp,(0k)
(11),(11) = d
rr,(0k)
(11),(11),
d9,k = d
pp,(2k)
(11),(11) = −drr,(2k)(11),(11) = d
pr,(2k)
(11),(11) = d
rp,(2k)
(11),(11),
d11,k = d
ppp,(3k)
(11),(11),(11) = −drrr,(3k)(11),(11),(11) = − 1
2d
rrp,(3k)
(11),(11),(11),
d12,k = t
ppp,(3k)
(11),(11),(11) = −trrp,(3k)(11),(11),(11) = − 1
2 t
prr,(3k)
(11),(11),(11) = t
ppr,(3k)
(11),(11),(11)
= −trrr,(3k)(11),(11),(11) =
1
2 t
prp,(3k)
(11),(11),(11),
d
(j)
13,k = d
pp,(3k)
(11),(2j) = −drr,(3k)(11),(2j) = d
pr,(3k)
(11),(2j) = d
rp,(3k)
(11),(2j),
d
(j)
14,k = d
pp,(3k)
(2j),(11) = −drr,(3k)(2j),(11) = d
pr,(3k)
(2j),(11) = d
rp,(3k)
(2j),(11),
d
(j)
15,k = t
pp,(3k)
(11),(2j) + t
pp,(3k)
(2j),(11) = −trr,(3k)(11),(2j) − t
rr,(3k)
(2j),(11) = t
pr,(3k)
(11),(2j) = t
pr,(3k)
(2j),(11),
d16,n = d
ppp,(1n)
(11),(11),(11)= d
rrr,(1n)
(11),(11),(11), d17,n = d
prr,(1n)
(11),(11),(11)= d
rpp,(1n)
(11),(11),(11),
d18,n = t
ppp,(1n)
(11),(11),(11) = t
rrr,(1n)
(11),(11),(11), d19,n = t
prr,(1n)
(11),(11),(11) = t
prp,(1n)
(11),(11),(11),
d16,n − d17,n = d
rrp,(1n)
(11),(11),(11) = d
prp,(1n)
(11),(11),(11),
d18,n − d19,n = t
rrp,(1n)
(11),(11),(11) = t
ppr,(1n)
(11),(11),(11),
d
(j)
20,n = d
pp,(1n)
(11),(2j) = d
rr,(1n)
(11),(2j) = d
pr,(1n)
(11),(2j) = −drp,(1n)(11),(2j),
260 A. N. Timokha
d
(j)
21,n = d
pp,(1n)
(2j),(11) = d
rr,(1n)
(2j),(11) = −dpr,(1n)(2j),(11) = d
rp,(1n)
(2j),(11),
d
(j)
22,n = t
pp,(1n)
(11),(2j) + t
pp,(1n)
(2j),(11) = t
rr,(1n)
(11),(2j) + t
rr,(1n)
(2j),(11) = t
pr,(1n)
(11),(2j) = −tpr,(1n)(2j),(11),
d
(j)
23,n = d
pp,(1n)
(11),(0j) = d
rp,(1n)
(11),(0j); d
(j)
24,n = d
pp,(1n)
(0j),(11) = d
pr,(1n)
(0j),(11),
d
(j)
25,n = t
pp,(1n)
(11),(0j) + t
pp,(1n)
(0j),(11) = t
pr,(1n)
(0j),(11).
One should remember that the computational formulas for the
hydrodynamic coefficients require, generally speaking, Ia = 3 (the speci-
al case Ia = 2 can be considered as an exception [6] eliminating
the differential equations (26) and (27)). Another nonnegative integer
Ir determines the number of differential equations in (25)–(27), the
summation limit in both the formulas for the hydrodynamic coeffici-
ents and the differential equations (24), (26) and (27). The complete
Narimanov–Moiseev system implies the limit Ir → ∞.
Specific equalities between the d and t-tensors are due to the tri-
gonometric algebra by the angular coordinate which are associated with
the Λ-tensors by (31). The λ-tensors by (32) do not provide any smart
properties and, therefore, should be found numerically.
7. The quality control (validation)
The Narimanov–Moiseev modal equations for comparisons can be found
in [9] and [5, 6]. Whereas [9] does not present numerical values of the
hydrodynamic coefficients, Lukovsky [5,6] computed and tabled them for
a broad set of r1 and h. These coefficients will be used for validation of
the derived Narimanov–Moiseev modal equations (24), (25).
Lukovsky [5,6] derived the modal equation, in our notations, for Ia =
2, Ir = 1 adopting the following approximate truncated modal solution
ζ(r, z, θ) =
R01(r)
R0
p0(t) +
R11(r)
R1
[p1(t) cos θ + r1(t) sin θ]
+
R21(r)
R2
[p2(t) cos 2θ + r2(t) sin 2θ] , Ri = Ri1(1), i = 0, 1, 2 (28)
(together with an analogous five-term approximation of the velocity
potential) which implies the link between (28) and (8a) expressed by
pi(t) = Ri pi1(t). The five-dimensional modal system by Lukovsky takes
the form [6, Eqs. (4.1.15)–(4.1.19)]
Narimanov–Moiseev modal equations 261
2
h
5
4
3
1
−4
1 1.5 2 2.5 3 3.5 0
4
2
0
−2
0.5
6
h
10
9
8
7
−4
3
0 0.5 1 1.5 2 2.5 3 3.5
1
0
−1
−2
−3
2
Fig. 3. The hydrodynamic coefficient of the Narimanov–Moiseev modal system
within the framework of Lukovsky’s [5, 6] five-dimensional approximation
(Ia = 2, Ir = 1 in our computational formulas) versus the nondimensional
liquid depth h. The circular cross-section, r1 = 0. The solid lines denote our
calculations, but the circles correspond to the tabled values from [5,6] rescaled
according to (30). The integers at the graphs denote: 1 – d1, 2 – d2, 3 – d
(1)
3 , 4
– d
(1)
4 , 5 – d
(1)
5 , 6 – d
(1)
6 , 7 – d7,1, 8 – d8,1, 9 – d9,1, and 10 – d10,1.
µ1(r̈1 + σ2
1r1) + d1(r
2
1 r̈1 + r1ṙ
2
1 + r1p1p̈1 + r1ṗ
2
1) + d2(p
2
1r̈1 + 2p1ṙ1ṗ1
− r1p1p̈1 − 2r1ṗ
2
1)− d3(r2r̈1 − r2p̈1 + ṙ1ṗ2 − ṗ1ṙ2) + d4(r1p̈2 − p1r̈2)
+ d5(p0r̈1 + ṙ1ṗ0) + d6r1p̈0 = −Pη̈2(t), (29a)
µ1(p̈1 + σ2
1p1) + d1(p
2
1p̈1 + r1p1r̈1 + p1ṙ
2
1 + p1ṗ
2
1) + d2(r
2
1 p̈1 − r1p1r̈1
+ 2r1ṙ1ṗ1 − 2p1ṙ
2
1) + d3(p2p̈1 + r2r̈1 + ṙ1ṙ2 + ṗ1ṗ2)− d4(p1p̈2 + r1r̈2)
+ d5(p0p̈1 + ṗ1ṗ0) + d6p1p̈0 = −Pη̈1(t), (29b)
µ0(p̈0 + σ2
0p0) + d6(r1r̈1 + p1p̈1) + d8(ṙ
2
1 + ṗ21) = 0, (29c)
µ2(r̈2 + σ2
2r2)− d4(p1r̈1 + r1p̈1)− 2d7ṙ1ṗ1 = 0, (29d)
µ2(p̈2 + σ2
2p2) + d4(r1r̈1 − p1p̈1) + d7(ṙ
2
1 − ṗ21) = 0 (29e)
that restores our hydrodynamic coefficients (computed with Ia = 2, Ir =
1 in all the formulas) as follows
d1 =
d1R2
1
µ1
, d2 =
d2R2
1
µ1
, d
(1)
3 =
d3R2
µ1
, d
(1)
4 = −d4R2
µ1
,
262 A. N. Timokha
2
3
5
1
h
=0.21r
4
3.5
−1
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
−2
h
r
8
7
9
10
6
=0.21
−5
−4
−3
−2
−1
0
0 0.5 1 1.5 3 2 2.5
−9
−8
−7
−6
3.5
2
=0.4
h
1r
3
5
4
1
−2
2 2.5 3 3.5 1 0.5 0
8
6
4
2
0
1.5
=0.4
h
1r
6
10
9
8
7
−10
2 2.5 3 3.5 1 0.5 0
0
−2
−4
−6
−8
1.5
Fig. 4. The same as in fig. 3 but for the annular cross-section with r1 = 0.2
and 0.4.
d
(1)
5 =
d5R0
µ1
, d
(1)
6 =
d6R0
µ1
, d7,1 = −d7R2
1
R2µ2
, d8,1 =
d8R2
1
R0µ0
,
d9,1 = −d4R2
1
R2µ2
, d10,1 =
d6R2
1
R0µ0
. (30)
Using our formulas with Ia = 2, Ir = 1 and the tabled hydrodynamic
coefficients by Lukovsky [5, 6] (rescaled by (30)) gives almost identical
results; the difference is always detected being less than 0.1%. This fact
Narimanov–Moiseev modal equations 263
1
h
=0.6
4
3
2
1
r
3.5
−1
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3
−2
9
6
10
7
r =0.6
h
3.5
−8
−6
−4
−2
0
0 0.5 1 1.5 2 2.5 3
−10
3
r
1
h
=0.8
2
4
5 1
−2
2 2.5 3 3.5 1 0.5 0
8
6
4
2
0
1.5
h
r
6
10
9
8
7
1=0.8−12
1.5 2 2.5 3 3.5 0.5 0
0
−2
−4
−6
−8
−10
1
Fig. 5. The same as in fig. 3 but for the annular cross-section with r1 = 0.6
and 0.8. The tabled numbers in [6] for d5 and d8 as r1 = 0.6 are clearly wrong
demonstrating a discontinuous character on h; these are excluded from our
comparisons.
is illustrated in figs. 3–5.
264 A. N. Timokha
Table 1. Computed hydrodynamic coefficients d1 and d2 for h = 1 versus Ir.
r1 = 0 r1 = 0.4 r1 = 0.8
Ir d1 d2 d1 d2 d1 d2
1 1.758023 -1.083355 1.549476 -1.011521 2.658499 -0.7886150
2 1.761798 -1.081642 1.670994 -0.894068 2.661058 -0.7860675
3 1.762160 -1.081505 1.677407 -0.889331 2.661086 -0.7860614
4 1.762240 -1.081476 1.678871 -0.888116 2.661098 -0.7860503
5 1.762266 -1.081467 1.679219 -0.887935 2.661100 -0.7860501
... ... ... ... ... ...
14 1.762287 -1.081460 1.679490 -0.887773 2.661101 -0.7860490
15 1.762287 -1.081460 1.679491 -0.887773 2.661101 -0.7860490
16 1.762288 -1.081460 1.679492 -0.887771 2.661101 -0.7860490
... ... ... ... ... ...
20 1.762288 -1.081460 1.679493 -0.887771 2.661101 -0.7860490
25 1.762288 -1.081460 1.679493 -0.887771 2.661101 -0.7860490
8. The modal systems of larger dimensions
The comparative study from the previous section validated our derivati-
ons of the Narimanov–Moiseev modal system for the very particular,
Ia = 2 and Ir = 1. An extra analysis is needed to ensure us that the
derived expressions are valid for a larger system dimension. A difficulty
is that increasing Ip ≥ 2 yields s series of new differential equations and
changes the hydrodynamic coefficients at the cubic polynomial terms wi-
th respect to the generalised coordinates and their derivatives. These
hydrodynamic coefficients, d1, d2, d11,k, d12,k, d16,k and d18,k, are functi-
ons of the upper summation limit Ir. This means, in particular, that d1
and d2 in the Lukovsky modal system (29) cannot be used for validation
of our Narimanov–Moiseev modal systems with a larger dimension, as
Ir ≥ 2.
The coefficients d1 and d2 versus Ir are illustrated in Table 1 for the
nondimensional liquid depth h = 1. The table confirms that d1 and d2
change with Ir, but not dramatically. Lukovsky’s computations [6] with
Ir = 1 can be adopted as a rough approximation and Ir = 2 stabilises,
at least, two significant figures of these hydrodynamic coefficients.
Narimanov–Moiseev modal equations 265
9. Conclusions
The complete Narimanov-Moiseev weakly-nonlinear modal system is deri-
ved for sloshing in an upright annular tank. The derivations are vali-
dated by comparison with numerical results on the hydrodynamic coeffi-
cients by Lukovsky [6]. Further studies should focus on derivations of the
corresponding formulas for the hydrodynamic forces and moments. The
general Lukovsky formulas would facilitate that. Another problem is a
study of the modal system applicability. This should include an analysis
of the secondary resonance occurrence, an estimate of damping due to
the flow separation at the central pile as well as establishing a clear
strategy on how many higher modes (driven by Ir) are needed to approxi-
mate steady-state and transient wave patterns. The latter task implies a
quantitative comparison with experiments which can be found, e.g. in [9].
10. Notations
Axisymmetric shape yields two algebras for the natural sloshing modes,
in angular and radial directions. The angular components leads to the
Λ-tensor whose elements are
ΛM...N,i...j =
∫ π
−π
cos(Aθ) . . . cos(Mθ) · sin(iθ) . . . sin(jθ) dθ. (31)
They can be computed by recursive formulas
ΛM,i = 0, Λ,ij = πδij , ΛMN, = πδMN , M
2 +N2 6= 0, Λ00, = 2π,
ΛM...NK,i...j =
1
2 (ΛM...|N−K|,i...j + ΛM...|N+K|,i...j),
ΛM...N,i...ljk =
1
2 (ΛM...|j−k|,i...l − ΛM...|j+k|,i...l)
following from the corresponding trigonometrical relations.
The radial component introduces the λ-tensors defined by the
formulas
λ(Ab)...(Mn) =
∫ 1
r1
rRAb(r) . . .RMn(r) dr,
λ′(Ab)(Mn),(Cd)...(Ef)=
∫ 1
r1
rR′
Ab(r)R′
Mn(r) · RCd(r) . . .REf (r) dr,
λ̄(Ab)...(Mn) =
∫ 1
r1
r−1 RAb(r) . . .RMn(r) dr (32)
266 A. N. Timokha
which should, generally speaking, be computed, numerically, except for
the case (4).
[1] Faltinsen O.M., Timokha A.N. Sloshing.— Cambridge, New York:
Cambridge: Cambridge University Press, 2009.— 686 p.
[2] Faltinsen O.M., Timokha A.N. Multimodal analysis of weakly nonlinear
sloshing in a spherical tank // Journal of Fluid Mechanics.— 2013.— 719.—
P. 129–164.
[3] Lukovsky I., Ovchynnykov D., Timokha A. Asymptotic nonlinear multi-
modal method for liquid sloshing in an upright circular cylindrical tank.
Part 1: Modal equations // Nonlinear Oscillations.— 2012.— 14, 4.—
P. 512–525.
[4] Lukovsky I., Timokha A. Combining Narimanov–Moiseev’ and Lukovsky–
Miles’ schemes for nonlinear liquid sloshing // Journal of Numerical and
Applied Mathematics.— 2011.— 105, 2.— P. 69–82.
[5] Lukovsky I.A. Introduction to nonlinear dynamics of rigid bodies with the
cavities partially filled by a fluid.— Kiev: Naukova Dumka, 1990.— 296 p.
(in Russian).
[6] Lukovsky I.A. Nonlinear dynamics: Mathematical models for rigid bodies
with a liquid.— Berlin: De Gruyter, 2015.— 400 p.
[7] Moiseev N.N. On the theory of nonlinear vibrations of a liquid of fini-
te volume // Journal of Applied Mathematics and Mechanics.— 1958.—
22, 5.— P. 860–872.
[8] Narimanov G. S. Movement of a tank partly flled by a fluid: the taking
into account of non-smallness of amplitude // Prikl. Math. Mech.— 1957.—
21.— P. 513–524 (in Russian).
[9] Takahara H., Kimura K. Frequency response of sloshing in an annular
cylindrical tank subjected to pitching excitation // Journal of Sound and
Vibration.— 2012.— 331, 13.— P. 3199–3212.
|
| id | oai:trim.imath.kiev.ua:article-41 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:01:28Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a5/a19f144e887f736f83b9e3c8a659bea5.pdf |
| spelling | oai:trim.imath.kiev.ua:article-412018-01-23T12:01:13Z The Narimanov–Moiseev modal equations for sloshing
in an annular tank Модальні рівняння Наріманова-Моісєєва для хлюпання рідини в соосних баках Timokha, A. N. Timokha, A. N. A complete weakly-nonlinear modal system of the Narimanov–Moiseevtype is derived for sloshing in an upright annular tank by using the deri-vation scheme proposed by the author for a spherical tank. The modalsystem couples the two dominant generalised coordinates responsible forthe lowest natural sloshing modes and an infinite set of the second- andthird-order generalised coordinates. Виводиться повна слабо-нелiнiйна модальна система типу Нарiманова-Моiсєєва, яка описує коливання рiдини у вертикальному бацi кiльце-вого перерiзу. Використовується схема виводу, яку автор запропонувавдля сферичного баку. Модальна система пов’язує двi домiнантнi уза-гальненi координати, що вiдповiдають за першi власнi форми колива-ння рiдини, та нескiнченну множину узагальнених координат другогота третього порядкiв малостi. Інститут математики НАН України 2015-12-01 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/41 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 5 (2015): Mathematical problems of mechanics and computational mathematics; 241-266 Сборник Трудов Института математики НАН Украины; Том 12 № 5 (2015): Математичні проблеми механіки та обчислювальної математики; 241-266 Збірник Праць Інституту математики НАН України; Том 12 № 5 (2015): Математичні проблеми механіки та обчислювальної математики; 241-266 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/41/16 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Timokha, A. N. Timokha, A. N. The Narimanov–Moiseev modal equations for sloshing in an annular tank |
| title | The Narimanov–Moiseev modal equations for sloshing
in an annular tank |
| title_alt | Модальні рівняння Наріманова-Моісєєва для хлюпання рідини в соосних баках |
| title_full | The Narimanov–Moiseev modal equations for sloshing
in an annular tank |
| title_fullStr | The Narimanov–Moiseev modal equations for sloshing
in an annular tank |
| title_full_unstemmed | The Narimanov–Moiseev modal equations for sloshing
in an annular tank |
| title_short | The Narimanov–Moiseev modal equations for sloshing
in an annular tank |
| title_sort | narimanov–moiseev modal equations for sloshing
in an annular tank |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/41 |
| work_keys_str_mv | AT timokhaan thenarimanovmoiseevmodalequationsforsloshinginanannulartank AT timokhaan thenarimanovmoiseevmodalequationsforsloshinginanannulartank AT timokhaan modalʹnírívnânnânarímanovamoísêêvadlâhlûpannârídinivsoosnihbakah AT timokhaan modalʹnírívnânnânarímanovamoísêêvadlâhlûpannârídinivsoosnihbakah AT timokhaan narimanovmoiseevmodalequationsforsloshinginanannulartank AT timokhaan narimanovmoiseevmodalequationsforsloshinginanannulartank |