Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept...
-
Upload
truongphuc -
Category
Documents
-
view
218 -
download
1
Transcript of Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept...
![Page 1: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/1.jpg)
1
Multimodal Method in Sloshing Analysis:
Analytical Mechanics Concept
by Alexander Timokha
CeSOS/AMOS, NTNU, Trondheim, NORWAY & Institute of Mathematics, National Academy of Sciences of Ukraine, UKRAINE
Trondheim, 28. May, 2013
![Page 2: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/2.jpg)
2
Multimodal Method in Sloshing Analysis:
Analytical Mechanics Concept
Etymology comes from which is the most cited paper on sloshing of the last two decades
CO
NC
EPT
![Page 3: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/3.jpg)
3
Multimodal Method in Sloshing Analysis:
Analytical Mechanics Concept
The concept originally appeared in XIX century but generalized in 2000-2013 by the author together with Prof. Odd M. Faltinsen
“Liquid Sloshing Dynamics” is treated as a conservative mechanical system with infinite degrees of freedom
so that the Lagrange formalism can adopt the generalized coordinates and velocities responsible for the global liquid modes instead of working with typically- accepted hydrodynamic characteristics (velocity field, pressure, etc.)
CO
NC
EPT
Historical aspects & Ideas: Linear & Nonlinear
![Page 4: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/4.jpg)
4
IDEAS come from aircraft and spacecraft applications: when the task consists of describing the coupled dynamics of a body with cavities filled by a liquid N.E. Joukowski (1885) paper `On the motion of a rigid body with cavities filled by a homogeneous fluid’ Joukowski’ theorem for a completely filled tank: `The “rigid body-ideal irrotational incompressible fluid” mechanical system can modelled as a rigid body with a specifically-modified inertia tensor’ • Considering the fluid as “frozen” is a wrong way (boiled and fresh
eggs!!!). • The velocity field is described by the so-called Stokes-Joukowski’
potentials.
HISTO
RY
& ID
EA
S
![Page 5: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/5.jpg)
5
by the liquid: • ideal, • incompressible, • irrotational flow Translatory velocity: Instant angular velocity:
1 2 2( ) ( ( ), ( ), ( ))Ov t t t t
4 5 6( ) ( ( ), ( ), ( ))t t t t
The liquid velocity potential
0( ) ( )( , , , ) ( , , ), ( , , )Ox y v t tz t x zr x y zy r
the Stokes-Joukowski potential
FULLY FILLED TANK
HISTO
RY
& ID
EA
S
![Page 6: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/6.jpg)
6
Having known the time-independent Stokes-Joukowski potentials makes it possible to find the fluid flow for any time instant so that • the velocity field • the pressure, • the resulting hydrodynamic force and moments,
etc. are explicit functions of the six input generalized coordinates
What about liquid sloshing (free surface)?
HISTO
RY
& ID
EA
S
![Page 7: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/7.jpg)
7
LINEAR SLOSHING, 50-60’s of XX century Liquid: • ideal • incompressible • irrotational flow
1 2 2( ) ( ( ), ( ), ( ))Ov t t t t
4 5 6( ) ( ( ), ( ), ( ))t t t t
0( , , , ) ( )
( , , )
( , , ) ( , , )
( , ,( )) 0
( ) ( ) NO
N
N
N
N
N
r x y z xx y z t R t
x y
yv t z
x
t
z t t y
sloshing modes
interpreted as generalized coordinates (infinite set!!!)
Free surface generalized velocities
HISTO
RY
& ID
EA
S
Joukowski’ solution
![Page 8: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/8.jpg)
8
As long as we know • the time-independent Stokes-Joukowski potentials
(Neumann problem in the unperturbed liquid domain) • the natural sloshing modes (the spectral boundary problem
in the unperturbed liquid domain) then the free-surface elevations the hydrodynamic forces and moments (provided by the
corresponding Lukovsky formulas) are functions of the input and the generalized coordinates where the latters are the solution of the linear oscillator problem: and the hydrodynamic coefficients are integrals over and
0( , , )x y z
( , , )N x y z
60( 3)2 1 2
41 5 2 4
( ) ( ) , 1,2,... k mm mm m
km m
k
mm
g g m
( , , )N x y z
0( , , )x y z
HISTO
RY
& ID
EA
S
![Page 9: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/9.jpg)
9
( )
free surface ( ): , , , 0
subject to volume conservation
(
0
)N
Q t
t Z x y z
dQ
t
NONLINEAR MULTIMODAL METHOD: new life in 00’s
Liquid: • ideal • incompressible • irrotational flow
1 2 2( ) ( ( ), ( ), ( ))Ov t t t t
4 5 6( ) ( ( ), ( ), ( ))t t t t
0( , , , ) ( , , ,{ ( )} ( )) ( , , )N NON
N t R tx y z t r x y z x y zv
The free surface elevations, the hydrodynamic forces and moments also remain functions of the six input and infinite set of the free-surface generalized coordinates
Generalized velocities
Generalized coordinates HISTO
RY
& ID
EA
S
![Page 10: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/10.jpg)
10
THERE ARE • APPLIED
MATHEMATICAL & • PHYSICAL
PROBLEMS TO BE SOLVED TO IMPLEMENT THE NONLINEAR MULTIMODAL METHOD
HISTO
RY
& ID
EA
S
![Page 11: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/11.jpg)
11
The nonlinear multimodal method 1. The method is of analytical nature.
What are analytical limitations? 2. The Euler-Lagrange equation for
liquid sloshing dynamics 3. Physical and mathematical arguments
for choosing the generalized coordinates
4. How does it work?
The m
ultimodal m
ethod
![Page 12: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/12.jpg)
12
A. For any instant must be defined in the time-varied liquid domain and satisfy the tank surface condition
B. Derivation of the Euler-Lagrange equations with respect to the generalized coordinates and velocities
( , , )N x y z
( )Q thand-made product for each tank shape
e.g., the Bateman-Luke variational principle
MAT
HE
MAT
ICA
L LIM
ITATIO
NS
Bateman-Luke variational principle derives the Euler-Largange equation
The following problems must be solved analytically:
Assuming A (we know analytical modes):
![Page 13: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/13.jpg)
13
(Kinematic Eq.): , changingd
;d
N NK NK K
K KK
A AA F
tN
(Dynamic Eq.): 1 2 3
1 2 changing1
... 0 2
K KLK K L
K KLN N N N N
A A l l lF NF F
where
* *( ) ( )
1 * 2 * 3 *( ) ( ) ( )
d ; d ,
d ; d ; d
N N N NK N N KQ t Q t
N N NQ t Q t Q t
A Q A Q
l x Q l y Q l z Q
EU
LE
R-L
AG
RA
NG
E E
QU
ATIO
N
Application needs a finite-dimensional form in application, so … how to use the Euler-Lagrange equation, e.g., for • Steady-state and transient response? • Wave elevations? Forces and moments? Coupling? • Realistic clean tanks? • Dissipation? Internal structures effect?
![Page 14: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/14.jpg)
14
Problems to be solved analytically: A. For any instant must be defined
in the time-varied liquid domain and satisfy the tank surface condition
B. Derivation of the Euler-Lagrange equations with respect to the generalized coordinates and velocities
C. Reduction of the infinite-dimensional Euler-Lagrange equations to a finite-dimensional form
D. Accounting for specific phenomena neglected by the physical model (damping, inner structures, wall/roof impact, perforated bulkhead, and so on)
( , , )N x y z
( )Q thand-made product for each tank shape
e.g., the Bateman-Luke variational principle
Normally, asymptotic approaches
results of the last decade
PHY
SICA
L AG
RU
ME
NT
S
![Page 15: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/15.jpg)
15
HOW does it work? Let’s consider devils examples for:
1. For 2D flows 2. For 3D rectangular tanks with upright
walls 3. For complex tank shapes H
ow does it w
ork?
What should be solved? • Choosing the leading and negligible
generalized coordinates • Modifying the modal equation due to
damping, specific phenomena and internal structures
![Page 16: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/16.jpg)
16
When forcing the lowest natural sloshing frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth
ratio (h/l=0.3368…) – secondary resonances and adaptive asymptotics
Intermediate and small depths (liquid depth/tank length < 0.2) • Multiple secondary resonance • Boussinesq ordering Internal structures (e.g., bulkheads) • Small forcing –amplitude quasilinear theory • Increasing forcing amplitude – secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances.
What does it mean in terms of leading generalized coordinates?
2D in rectangular tank
![Page 17: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/17.jpg)
17
When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth ratio
(h/l=0.3368…) – secondary resonances and adaptive asymptotics
Intermediate and small depths (liquid depth/tank length < 0.2) • Multiple secondary resonance • Boussinesq ordering Internal structures (e.g., bulkheads) • Small forcing –amplitude quasilinear theory • Increasing forcing amplitude – secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances.
What does it mean in terms of leading generalized coordinates?
2D in rectangular tank
![Page 18: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/18.jpg)
18
Example 1: The Moiseev-type multimodal theory The modal solution
Moiseev proved for the steady-state (periodic) sloshing due to excitation of the lowest frequency when there are no secondary resonances
01
1
12
( , , ) ( , ) ( , ),
( , ) ( ) ( , 0),
cosh ( )cos ( )
cosh( )
O n nn
n nn
n
y z t v r y z R y z
z y t t y
n z hn y
nh
1/3 2/31 1
4
3
2
2 2
3
3
( ), ( ),
( ); ( ), 4
/ / ( ); ( ) 1,
n n
R O R O
R O R O n
l l O O
Exam
ple 1:
![Page 19: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/19.jpg)
19
21 1 2 322
2 21 1 1 2 1 2 1 1 1 1 2 1
22 2 1 1 1
2 23 3 1 2 1 1 2 1 1 1 1
4 523 1 2 3 4 2 35
1( ) ( ) ( ) ,
( ) 0,
(
)
.
(
) ( )
d d d
d d
q
K t
q q tq Kq
Modal equations
2 , 4,.( ..,)iii i
iK Nt
24 4
( )( ) ( ) ( )
i ii
gP S
l l
tK t t t
Forcing term: Exam
ple 1:
Coefficients are analytically found as functions of the liquid depth to the tank breadth ratio…
![Page 20: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/20.jpg)
20
Tran
sien
ts:
Expe
rimen
ts
Theo
ry w
ith d
iffer
ent
initi
al sc
enar
ios
Steady-state waves with the horizontal/angular harmonic forcing, the dominant amplitude parameter:
(frequency , nondimensional amplitude ),
/ 0.3368...h l / 0.3368...h l
Exam
ple 1:
![Page 21: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/21.jpg)
21
When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth
ratio (h/l=0.3368…) – secondary resonances and adaptive asymptotics
Intermediate and small depths (liquid depth/tank length < 0.2) • Multiple secondary resonance • Boussinesq ordering Internal structures (e.g., bulkheads) • Small forcing –amplitude quasilinear theory • Increasing forcing amplitude – secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances.
What does it mean in terms of leading generalized coordinates?
Exam
ple 2:
![Page 22: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/22.jpg)
22
Exam
ple 2:
Exa
mpl
e 2:
A
dapt
ive
mod
al sy
stem
s for
cr
itica
l dep
th a
nd in
crea
sing
fo
rcin
g am
plitu
de
• Nonlinear multiple frequency effects excite higher natural frequencies • Increased importance with decreasing depth and increasing forcing
amplitude • Reason for decreasing depth importance is that when the
liquid depth goes to zero • Implies that more then one mode is dominant
n n
![Page 23: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/23.jpg)
23
Different ordering of the generalized coordinates due to secondary resonances with a finite liquid depth
Exam
ple 2:
Rectangular tank, 1x1m, nearly critical depth, h/l = 0.35
Subharmonic regimes are predicted
![Page 24: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/24.jpg)
24
Increasing the forcing amplitude and accounting for roof impact, h/l=0.4
Exam
ple 2:
0.01
0.1
impact neglected impact accounted for Flow 3D
CFD: Symbols ∆ and represent numerical results by the viscous CFD-code FLOW-3D obtained for fresh water with different internal parameters of the code: ∆ for “alpha=0.5, epsdj=0.01” and for “alpha=1.0, epsdj=0.01”. Experiments: The influence of viscosity: ■, fresh water; ○, reginol-oil; □, glycerol-water 63%, ♦, glycerol-water 85%
![Page 25: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/25.jpg)
25
When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth ratio
(h/l=0.3368…) – secondary resonances and adaptive asymptotics
Intermediate and small depths (liquid depth/tank length < 0.2) • Multiple secondary resonance • Boussinesq ordering Internal structures (e.g., bulkheads) • Small forcing –amplitude quasilinear theory • Increasing forcing amplitude – secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances.
What does it mean in terms of leading generalized coordinates?
Exam
ple 3:
![Page 26: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/26.jpg)
26
Intermediate and small depths normally cause amplification of higher modes and a series of local wave breaking & overturning
Exam
ple 3:
![Page 27: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/27.jpg)
27
As a consequence, a Boussinesq-type ordering can be proven being applicable to with
1/4( / ) ( ), 1i i
O h l O iR
Tran
sien
ts:
The measured and calculated wave elevations near the wall for horizontal forcing . The solid and dashed lines correspond to experiments and the Boussinesq-type multimodal method, respectively.
/ 0.173, 0.028h l
Exam
ple 3:
![Page 28: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/28.jpg)
28
Exam
ple 3:
Steady-state due to harmonic forcing ◊ = experiments Chester & Bones
Theory by Chester Multumodal theory
/ 0.083333h l 0.001254 0.002583
Agreement is almost ideal when no wave breaking occurs and, therefore, damping does not matter. Multi-peak response curves and damping are important when dealing with secondary resonances.
![Page 29: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/29.jpg)
29
Associated damping becomes important with Decreasing depth and increasing forcing amplitude Roof impact Internal structures Normally, viscous boundary layer effect is less
important Damping terms can be incorporated into the modal equations following the strategy in Chapter 6 of «Sloshing» book. An open problem is damping due to the local free-surface phenomena: • Overturning and impact on underlying fluid • Breaking waves in the middle of the tank
Exam
ples vs. damping
![Page 30: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/30.jpg)
30
When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth ratio
(h/l=0.3368…) – secondary resonances and adaptive asymptotics
Intermediate and small depths (liquid depth/tank length < 0.2) • Multiple secondary resonance • Boussinesq ordering Internal structures (e.g., bulkheads) • Small forcing –amplitude quasilinear theory • Increasing forcing amplitude – secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances.
What does it mean in terms of leading generalized coordinates?
Exam
ple 4:
![Page 31: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/31.jpg)
31
Exam
ple 4:
Exa
mpl
e 4:
Mid
dle
scre
en
The middle-screen causes: (a) migration of the resonances and super-multipeak response curves; (b) damping; (c) disappearance of the primary resonance with increasing the solidity ratios
![Page 32: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/32.jpg)
32
• ‘Planar’ waves – waves keeps symmetry with respect to the excitation plane: occur far from the primary resonance.
• Swirling exactly at the primary resonance • Irregular (chaotic) waves. Weak chaos? • For the rectangular shape, a diagonal-type (squares-like) waves
– waves with an angle to the excitation plane
Three-dimensional: nearly steady-state wave response
3D sloshing
![Page 33: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/33.jpg)
33
Moiseev-type modal system for square base tank
3D sloshing
2 2 2 1 51 1,0 1 1 1 2 1 2 2 1 1 1 1 3 2 1 1,0 1,0 5
1 1
2 26 1 1 1 7 1 8 1 1 9 1 1 10 1 1 11 1 1 1 12 1 1
( ) ( )
( ) 0,
ga a d a a a a d a a a a d a a P S
L L
d a b b d c d a b d c b d b a d a b b d b c
2 2 2 2 41 0,1 1 1 1 2 1 2 2 1 1 1 1 3 2 1 0,1 0,1 4
1 1
2 26 1 1 1 7 1 8 1 1 9 1 1 10 1 1 11 1 1 1 12 1 1
( ) ( )
( ) 0,
gb b d b b b b d b b b b d b b P S
L L
d b a a d c d a b d c a d a b d a b a d a c
1,0 1 2,0 2 0,1 1 0,2 2 1,1 1 3,0 3 2,1 21 1,2 12 0,3 3; ; ; ; , ; ; ;a a b b c a c c b
2 22 2,0 2 4 1 1 5 1
0;a a d a a d a
2 22 0,2 2 4 1 1 5 1
0;b b d b b d b
21 1 1 1 2 1 1 3 1 1 1,1 1
ˆ ˆ ˆ 0,c d a b d b a d a b c
2 2 2 1 53 3,0 3 1 1 2 2 1 3 2 1 4 1 1 5 1 2 3,0 3,0 5
1 1
( ) 0,g
a a a q a q a q a a q a a q a a P SL L
2 2 221 2,1 21 1 6 1 7 1 1 1 8 2 9 1 10 2 1 11 1 1 12 1 1 13 1 1 1 14 1 1 15 2 1
( ) ( ) 0,c c a q c q a b b q a q a q a b q c a q a b q a b a q a c q a b
2 2 212 1,2 12 1 6 1 7 1 1 1 8 2 9 1 10 2 1 11 1 1 12 1 1 13 1 1 1 14 1 1 15 1 2
( ) ( ) 0,c c b q c q a b a q b q b q b a q c b q b a q a b b q b c q a b
2 2 2 2 43 0,3 3 1 1 2 2 1 3 2 1 4 1 1 5 1 2 0,3 0,3 4
1 1
( ) 0.g
b b b q b q b q b b q b b q b b P SL L
Modal equations with nine degrees of freedom
![Page 34: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/34.jpg)
34
For harmonic forcing, one can find analytically approximate steady-state solutions and study their stability (also analytically!!!). This makes it possible to classify the steady-state regimes and establish the frequency ranges where the regimes exist and stable. One can distinguish: 3D
sloshing
• the order (stable steady-state), • the strong chaos (treated as no stable steady-
state for leading generalized coordinates), • the weak chaos (here, irregular for higher-
order generalized coordinates)
![Page 35: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/35.jpg)
35
Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L
3D sloshing
![Page 36: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/36.jpg)
36
3D sloshing
Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L
![Page 37: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/37.jpg)
37
3D sloshing
Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L
![Page 38: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/38.jpg)
38
3D sloshing
Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L
![Page 39: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/39.jpg)
39
3D sloshing
Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L
![Page 40: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/40.jpg)
40
3D sloshing
Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L (relatively large!!!)
Local phenomena for 3D, but the classification is Ok. Why? Why is the Moissev-type model still applicable?
![Page 41: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/41.jpg)
41
The method was developed in 2012, after the book issued
Sphe
rica
l tan
ks
3D sloshing
![Page 42: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/42.jpg)
42
11 secondary resonance by (01) a0 t /2 0 44. .9h
(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065
Classification of 3D
sloshing
![Page 43: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/43.jpg)
43
(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065
secondary resonance is far from the r e0.6 angh
Classification of 3D
sloshing
![Page 44: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/44.jpg)
44
(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065
11 secondary resonance by (22) a1 t /0 1 33. .0h
Classification of 3D
sloshing
![Page 45: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/45.jpg)
45
Swirling wave patterns taken for these input parameters from T.Hysing (1976) Det Norske Veritas, Høvik, Norway
Specifically, splashing, steep wave patterns, local breaking.
Classification of 3D
sloshing
![Page 46: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/46.jpg)
46
(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065
111.0 secondary resonance by (22) at / 1.033h
Classification of 3D
sloshing
![Page 47: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/47.jpg)
47
`planar’ splashing’: ``...drops splashed from the tank wall and showered through the ullage...'' but wave patterns remain planar
for 1 splashing (planar & swirling type) is all the frequency rangeh
Classification of 3D
sloshing
![Page 48: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/48.jpg)
48
Swirling always causes secondary resonances and local phenomena when forcing amplitude increases, but classification remains correct. Again, why?
This is explained by the concept of the weak chaos: Here, a clearly steady-state by a subset of [leading] generalized coordinates and irregular motions by other [infinite set] higher-order coordinates caused by higher resonances.
Weak chaos in sloshing problem
s
![Page 49: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/49.jpg)
49
Weak chaos in sloshing problem
s
The weak chaos concept for low-dimensional Hamiltonian (conservative) systems (see, e.g. Henning et al. (2013), Physica D, 253): ORDER WEAK CHAOS CHAOS Lyapunov exponent: <0 >0, but small >0, finite «… in domains of weak chaos trajectories (by higher-order generalized coordinates) slowly diffuse into thin chaotic layers and wander through a complicated network of higher order resonances…» For our almost-conservative mechanical system with an infinite set of generalized coordinates (g.c.), this implies: ORDER WEAK CHAOS CHAOS stable by all g.c. stable by dominant g.c., unstable by all g.c. but chaos in higher-order g.c. unless a strong damping occurs For 3D sloshing, the weak chaos is the reality well modelled by the multimodal method
![Page 50: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/50.jpg)
50 Longitudinal resonant excitation with h/L=0.5 and for a square-base tank. Higher-order (secondary) resonances.
0.00817ε =
From
the
orde
r to
wea
k ch
aos a
nd,
ther
eaft
er, s
tron
g ch
aos f
or sw
irlin
g w
ith
incr
easi
ng th
e fo
rcin
g am
plitu
de
Swirl
ing
(wea
k ch
aos)
Swirl
ing
(wea
k ch
aos)
stro
ng c
haos
stro
ng c
haos
orde
r
orde
r
orde
r
orde
r
Weak chaos in sloshing problem
s
![Page 51: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/51.jpg)
51
Open problems within the framework of the same paradigm, i.e., six degrees of freedom for the rigid tank and generalized coordinates for the free surface motions:
Open problem
s: intensive
1. Different internal structures. 2. Accounting for damping due to wave
breaking, overturning, etc. 3. Complex tank shapes. 4. Order weak chaos chaos. 5. Importance of weak chaos for coupled
motions CFD are normally unapplicable on the long-time scale!
![Page 52: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/52.jpg)
52
Input has more than six (infinite) degrees of freedom: • inflow-outflow & sloshing
(damaged ship tank, wave energy, etc.);
• elastic/hyperelastic tank walls (fish farms, membrane tanks, etc.);
• ship collapse
Open problem
s: extensive
![Page 53: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/53.jpg)
53
REFERENCES Books: 1. Faltinsen, O.M., Timokha A.N. (2009): Sloshing. Cambridge University Press. 608pp. (ISBN-13:
9780521881111) Chinese Version of the book issued in 2012: P.R.C.:National Defense Industry Press. 783pp. (ISBN-13: 978-7-118-08608-3)
2. Gavrilyuk, I.P., Lukovsky, I.A., Makarov, V.L., Timokha, A.N. (2006): Evolutional problems of the contained fluid. Kiev: Publishing House of the Institute of Mathematics of NASU. 233pp. (ISBN 966-02-3949-1)
Selected papers in peer-reviewed journals: 1. Faltinsen, O.M., Timokha, A.N. (2013): Multimodal analysis of weakly nonlinear sloshing in a spherical
tank. Journal of Fluid Mechanics, 719, 129-164 2. Faltinsen, O.M., Timokha, A.N. (2012): Analytically approximate natural sloshing modes for a spherical
tank shape. Journal of Fluid Mechanics, 703, 391-401 3. Faltinsen, O.M., Timokha, A.N. (2012): On sloshing modes in a circular tank. Journal of Fluid Mechanics,
695, 467-477 4. Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O., Timokha, A. (2012): Multimodal method for linear
liquid sloshing in a rigid tapered conical tank. Engineering Computations, 29, No 2, 198-220 5. Lukovsky, I.A., Ovchynnykov, D.V., Timokha, A.N. (2012): Asymptotic nonlinear multimodal method for
liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations. Nonlinear Oscillations, 14, No 4, 512-525
6. Lukovsky, I.A., Timokha, A.N. (2011): Combining Narimanov--Moiseev' and Lukovsky--Miles' schemes for nonlinear liquid sloshing. Journal of Numerical and Applied Mathematics, 105, No 2, 69-82
7. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Effect of central slotted screen with a high solidity ratio on the secondary resonance phenomenon for liquid sloshing in a rectangular tank. Physics of Fluids, 23, Issue 6, Art. No. 062106, 1-13
8. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Analytical modeling of liquid sloshing in a two-dimensional rectangular tank with a slat screen. Journal of Engineering Mathematics, 70, 1-2, 93-109
![Page 54: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/54.jpg)
54
9. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle: Quasilinear modal analysis and experiments. Physics of Fluids, 23, Issue 4, Art. No. 042101, 1-19
10. Faltinsen, O.M., Timokha, A.N. (2011): Natural sloshing frequencies and modes in a rectangular tank with a slat-type screen . Journal of Sound and Vibration, 330, 1490–1503
11. Barnyak, M., Gavrilyuk, I., Hermann, M., Timokha, A. (2011): Analytical velocity potentials in cells with a rigid spherical wall. ZAMM, 91, No 1, 38–45
12. Faltinsen, O.M., Timokha, A.N. (2010): A multimodal method for liquid sloshing in a two-dimensional circular tank. Journal of Fluid Mechanics, 665, 457-479
13. Hermann, M., Timokha, A. (2008): Modal modelling of the nonlinear resonant fluid sloshing in a rectangular tank II: Secondary resonance. Mathematical Models and Methods in Applied Sciences, 18, N 11, 1845-1867
14. Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O., Timokha, A. (2008): Natural sloshing frequencies in rigid truncated conical tanks. Engineering Computations, 25, Issue 6, 518-540
15. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2007): Two-dimensional resonant piston-like sloshing in a moonpool. Journal of Fluid Mechanics, 575, 359-397 [Supplementary material]
16. Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu., Timokha, A. (2007): Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 2. Nonliear resonant waves. Journal of Engineering Mathematics, 57, 57-78
17. Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu., Timokha, A. (2006): Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 1. Linear fundamental solutions. Journal of Engineering Mathematics, 54, 71-88
18. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2006): Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 3. Base ratio perturbations. Journal of Fluid Mechanics, 551, 93-116
19. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2006): Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with a finite fluid depth. Physics of Fluids, 18, Art. No. 012103, 1-14
20. Gavrilyuk, I.P., Lukovsky, I.A., Timokha, A.N. (2005): Linear and nonlinear sloshing in a circular conical tank. Fluid Dynamics Research, 37, 399-429
21. Hermann, M., Timokha, A. (2005): Modal modelling of the nonlinear resonant sloshing in a rectangular tank I: A single-dominant model. Mathematical Models and Methods in Applied Sciences, 15, N 9, 1431-1458
22. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2005): Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth. Journal of Fluids and Structures, 20, Issue 1, 81-103
![Page 55: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/55.jpg)
55
23. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2005): Resonant three-dimensional nonlinear sloshing in a square base basin. Part 2. Effect of higher modes. Journal of Fluid Mechanics, 523, 199-218
24. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2003): Resonant three-dimensional nonlinear sloshing in a square base basin. Journal of Fluid Mechanics, 487, 1-42
25. Faltinsen, O.M., Timokha, A.N. (2002): Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. Journal of Fluid Mechanics, 470, 319-357
26. Lukovsky, I.A., Timokha, A.N. (2002): Modal modeling of nonlinear sloshing in tanks with non-vertical walls. Non-conformal mapping technique. International Journal of Fluid Mechanics Research, 29, Issue 2, 216-242
27. Gavrilyuk, I., Lukovsky, I.A., Timokha, A.N. (2001): Sloshing in a circular conical tank. Hybrid Methods in Engineering, 3, Issue 4, 322-378
28. Faltinsen, O.M., Timokha, A.N. (2001): Adaptive multimodal approach to nonlinear sloshing in a rectangular tank. Journal of Fluid Mechanics, 432, 167-200
29. Lukovsky, I.A., Timokha, A.N. (2001): Asymptotic and variational methods in nonlinear problems on interaction of surface waves with acoustic field. J. Applied Mathematics and Mechanics. 65, Issue 3, 477-485
30. Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I.A., Timokha, A.N. (2000): Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. Journal of Fluid Mechanics, 407, 201-234
31. Gavrilyuk, I., Lukovsky, I.A., Timokha, A.N. (2000): A multimodal approach to nonlinear sloshing in a circular cylindrical tank. Hybrid Methods in Engineering, 2, Issue 4, 463-483
![Page 56: Analytical Mechanics Concept - CESOS - NTNU Timokha.pdf · Analytical Mechanics Concept The concept originally appeared in XIX century but generalized in 2000- 2013 by ... Joukowski’](https://reader031.fdocuments.us/reader031/viewer/2022022009/5af28fa97f8b9a95468b5b3d/html5/thumbnails/56.jpg)
56
THANK YOU FOR ATTENTION