two-dimensional unsteady planing elastic plate michael makasyeyev institute of hydromechanics of nas...
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Outline
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1. Introduction1.1. Motivations
2. Hydrodynamic problem 3. Elasticity problem4. Solution method5. Numerical results6. Conclusion
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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1. Introduction1.1. Motivations
A planing surface is being experienced high forces from the water and it might result in the deformations. As consequences, hydrodynamic characteristics might change and even cause the damage of the hull. At the unsteady motion, for example, on the wave surface, the forces can increase manifold and can have dynamical character. It increases the chance of negative effects. The laws of change of pressure distribution, trim angle, wetted length and draft at the planing of elasticity deformable has not been studied. I am interested in useing approaches and methods of wing theory, in particular the method of singular integral equations.
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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2. Hydrodynamic problem
,
,
,
,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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2. Hydrodynamic problem ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
0,, tyx 0y (2.1)
txN ,0, txgtxp ,, x , (2.2)
txNtxy ,,0, x , (2.3)
0, yx y (2.4)
xVtN // 0
Boundary conditions
(2.5)
(2.6)
yxyx ,0,, 0 txyxt ,0,, 1
xx 00, xxt 10,
Initial conditions
or
tlx
tlx
,,0,0, tlxxtxp
txfxtthtx ,,
tlx0 txtx ,,
tth ,0
txtx ,,
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3. Elasticity problem ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
Boundary conditions
Initial conditions
(3.1) txptxpt
fm
x
fT
x
fD ,,
2
2
2
2
4
4
0,,0,,0 max2
2
2
2
max
tlx
ft
x
ftlftf . (3.2)
0,,0,,0 maxmax
tlx
ft
x
ftlftf
. (3.3)
0,,0,,0 max3
3
3
3
max2
2
2
2
tlx
ft
x
ftl
x
ft
x
f(3.4)
xfxf 00, xfxt
f10,
,
(3.5)
pinning
fixed ends
free ends
or combinations…
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
General idea
Classical boundary problem
4.2. Fourier transform and fundamental solution
4.3. Solutions for generalized functions
4.4. Inverse Fourier transform and obtainment of integral equations 4.5. Formation of general
simultaneous integral equations system
4.6. Numerical solution of integral equations system by the method of discrete vortexes.Solution of the nonlinear wetted length problem
4.1. Generalized functions problem
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.1. Generalized functions problem
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.1. Generalized functions problem
ytxytx y ,0,,0, In hydrodynamic problem:
In elasticity problem:
2
2
2
2
4
4
t
fm
x
fT
x
fD
3
0max
330,0 max
,,k
klk
kkkl lxthxthcxtxptxp
tx
fth
k
k
k ,00
tlx
fth
k
k
lk ,max
max
max,0 ,0,0
,0,1max lx
lxxl
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.2. Fourier transform and fundamental solution
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.2. Fourier transform and fundamental solution
In hydrodynamic problem:
In elasticity problem:
tyxFtyxFty ,,,,,,, txFtH ,, txFtH ,,
/,,, yetHNty 0/ VitN
tPtHgN ,,/2
tPtYTDt
m ,,222
2
ttxfFtY x ,,,
20
202
22 2 V
tVi
tN
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.3. Solutions for generalized functions
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.3. Solutions for generalized functions
In hydrodynamic problem:
In elasticity problem:
t
tVi dtDePtH0
,,, 0
gtgtD /sin, xFH 00 xFH 11
tVietDHtt
DH 0,, 10
t
dt
Pm
tY0
sin,
1,
t
YtYsin
cos 10
22
1 mD /1 mT /2 xfFY 00 xfFY 11
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.4. Inverse Fourier transform and obtainment of integral
equations
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.4. Inverse Fourier transform and obtainment of integral
equations
In hydrodynamic problem:
In elasticity problem:
tx,
tl t
dsdttVsxKsp0 0
00 ,, ttVx ,00
21
02
sin,x
ttg
gFtxK x
txt
FtxN ,sin
2
1,
3
0max0 ,,
kklkkk tlxNthtxNth txftxf ,, 10
t
k
k
k dtxNx
txN0
3
3
,, tYFtxf cos, 0
10
t
YFtxfsin
, 11
1
02012/3
sgn1
2fCfSx
x
g
x
dssxS0
2
2sin
x
dssxC0
2
2cos
t l
ddstsxNsptx0 0
max
,,,
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Steady motion
tH ,
tgVi
ePg
i
sgn0
2
degVi
sgn0
gV
PregtH
t 20
,lim
gVPg
i sgn
20 gV sgn0
20
0
2sgn
V
ggV
tHtHt
,,lim 0
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
20/Vg
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.5. Formation of general simultaneous integral equations system
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.5. Formation of general simultaneous integral equations system
Compound integral equation of hydrodynamics and elasticity:
Dynamics equations:
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.5. Formation of general simultaneous integral equations system
Compound integral equation of hydrodynamics and elasticity:
Dynamics equations:
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
Unknown functions:
txp ,
tl t
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.6. Numerical solution of integral equations system by the method of discrete vortexes.Solution of the nonlinear problem of wetted length
Compound integral equation of hydrodynamics and elasticity:
Dynamics equations:
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
Unknown functions:
txp ,
tl t
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.6. Numerical solution of integral equations system by method of discrete vortexes.Solution of nonlinear problem of wetted length
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.6. Numerical solution of integral equations system by method of discrete vortexes (MDV).Solution of nonlinear problem of wetted length
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
MDV or other
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.6. Numerical solution of integral equations system by method of discrete vortexes (MDV).Solution of nonlinear problem of wetted length
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
;,...,, 21 npppX
MDV or other BXtlA
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4. Solution method ,
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
4.6. Numerical solution of integral equations system by the method of discrete vortexes (MDV).Solution of the nonlinear problem of wetted length
ttVx ,01
00
2
2
, tl
c dxtxpadt
thdm
tl
c dxtxpbxadt
tdI
02
2
,
tl t
dsdttsxsxKttVsxKsp0 0
201 ,,;,,,,
txN ,
;,...,, 21 npppX
MDV or other BXtlA
max,0min
ltlBXtlABXtlA
(least-squares method+ nondifferential minimization)
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6. Conclusions
,
,
,
,
,
The unsteady 2D-theory of hydroelasticity planing plate is created. Basically it is the 2D-linearized theory of unsteady motion of small displacement body with elastic bottom on the free surface. At the V0=0 we have the theory of floating body. The cases of steady motion and harmonic motion have been obtained at the time t trending to infinity. Difficulties: 1) The obtaining of inverse generalized Fourier transformation for some functions. 2) Some problems in numerical procedures of wetted length definition as time-depended function.
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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Thank you for your attention
26Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK