Two-Dimensional Unsteady Planing Elastic Plate Michael Makasyeyev Institute of Hydromechanics of NAS of Ukraine, Kyiv 1 [email protected].

1

[email protected]

Outline

2

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

3

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.


4

2. Hydrodynamic problem

,

,

,

,


5

2. Hydrodynamic problem ,


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 ,,

6

3. Elasticity problem ,


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…

7

4. Solution method ,


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

8




9




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

10




11




In hydrodynamic problem:


tyxFtyxFty ,,,,,,, txFtH ,, txFtH ,,

/,,, yetHNty 0/ VitN

tPtHgN ,,/2

tPtYTDt

m ,,222

2

ttxfFtY x ,,,

20

202

22 2 V

tVi

tN

12




13






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

14



4.4. Inverse Fourier transform and obtainment of integral

equations

15



4.4. Inverse Fourier transform and obtainment of integral

equations



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

,,,

16

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


20/Vg

17



4.5. Formation of general simultaneous integral equations system

18




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 ,

19





Dynamics equations:

ttVx ,01

00

2

2

, tl

c dxtxpadt

thdm

tl

c dxtxpbxadt

tdI

02

2

,

tl t


201 ,,;,,,,

txN ,

Unknown functions:

txp ,

tl t

20



4.6. Numerical solution of integral equations system by the method of discrete vortexes.Solution of the nonlinear problem of wetted length


Dynamics equations:

ttVx ,01

00

2

2

, tl

c dxtxpadt

thdm

tl

c dxtxpbxadt

tdI

02

2

,

tl t


201 ,,;,,,,

txN ,

Unknown functions:

txp ,

tl t

21



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


201 ,,;,,,,

txN ,

22



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


201 ,,;,,,,

txN ,

MDV or other

23



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


201 ,,;,,,,

txN ,

;,...,, 21 npppX

MDV or other BXtlA

24



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


201 ,,;,,,,

txN ,

;,...,, 21 npppX

MDV or other BXtlA

max,0min

ltlBXtlABXtlA

(least-squares method+ nondifferential minimization)

25

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.


Thank you for your attention

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

Two-Dimensional Unsteady Planing Elastic Plate Michael Makasyeyev Institute of Hydromechanics of NAS of Ukraine, Kyiv 1 [email protected].

Documents

solution method

modeling of hydroelasticity

hydrodynamic problem

fundamental solution

n elasticity problem

method of discrete vortexes

hydrodynamic characteristics

planing surface