Two-Dimensional Unsteady Planing Elastic Plate Michael Makasyeyev Institute of Hydromechanics of NAS...

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M_Makasyeyev@ukr.net

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

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,

,

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 ,

22

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

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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

Thank you for your attention

26Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK