Two-Dimensional Unsteady
Planing Elastic Plate
Michael Makasyeyev
Institute of Hydromechanics of NAS of Ukraine, Kyiv
M_Makasyeyev@ukr.net
1
Outline
1. Introduction
1.1. Motivations
2. Hydrodynamic problem
3. Elasticity problem
4. Solution method
5. Numerical results
6. Conclusion
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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1. Introduction
1.1. 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.
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2. Hydrodynamic problem
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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 x, t    x, t 
2. Hydrodynamic problem
 x, y, t   0
y0
(2.1)
Boundary conditions
N x,0, t    px, t   g x, t 
  x   ,
 y x,0, t   N x, t 
  x   ,
x ,  y  0
y  
x  l t 
x  l t 
(2.2)
px, t   0,
(2.3)
 x, t    x, t 
(2.4)
x, t   ht   t x  f x, t 
x  0,
x  l t ,
0  x  l t 
ht   0, t 
N   / t  V0 / x
Initial conditions
 x, y,0  0 x, y 
or
 x,0  0 x
t x, y,0  1x, t 
t x,0  1 x 
(2.5)
(2.6)
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3. Elasticity problem
4 f
2 f
2 f
D 4  T 2  m 2  px, t   p x, t 
x
x
t
(3.1)
Boundary conditions
pinning
2 f
2 f
f 0, t   f l max , t   2 0, t   2 l max , t  .0
x
x
fixed ends
.f
f
f 0, t   f l max , t   0, t   l max , t   0
x
x
free ends
2 f
2 f
3 f
3 f
0, t   2 lmax , t   3 0, t   3 lmax , t   0
x 2
x
x
x
or combinations…
(3.2)
(3.3)
(3.4)
Initial conditions
f, x,0  f 0 x
f
x,0  f1 x
t
(3.5)
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
General idea
Classical boundary problem
4.1. Generalized functions
problem
4.2. Fourier transform and fundamental
solution
4.3. Solutions for generalized
functions
4.5. Formation of general
simultaneous integral equations
system
4.4. Inverse Fourier transform and
obtainment of integral equations
4.6. Numerical solution of integral equations system by the method of discrete vortexes.
Solution of the nonlinear wetted length problem
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.1. Generalized functions
problem
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.1. Generalized functions
problem
In hydrodynamic problem:
  x,0, t   y   y x,0, t   y
In elasticity problem:
4 f
2 f
2 f
D 4  T 2  m 2 
x
x
t
3


  p x, t   px, t 0,lmax x    ck h0k t  3k  x   hlk t  3k  x  lmax 
k 0
1, x  0, l max 
0, x  0, l max 
 0,lmax  x   
 k  f
0, t 
h0k t  
x k
 k  f
lmax , t 
hlk t  
x k
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.2. Fourier transform and fundamental
solution
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.2. Fourier transform and fundamental
solution
In hydrodynamic problem:
, y, t   F  x, y, t   , F  x, y, t 
, y, t   NH , t e
N
2
 y

H  , t   F  x, t 
H  , t   F  x, t 
N   / t  iV0
/
2

2
N  2  2iV0  2V0
t
t
/   g H  , t    P , t 
2
In elasticity problem:
 2
2
2
m 2   D  T
 t

Y , t   P , t 

Y  , t   Fx  f x, t  , t 
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.3. Solutions for generalized
functions
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.3. Solutions for generalized
functions
In hydrodynamic problem:
t
D
H  , t     P , eiV0  t  D ,  t d   H 0    , t   H1  D , t eiV0t
t


0


D , t   sin g  t / g 
H 0    F 0 x  
H1    F 1 x  
In elasticity problem:
1 t
sin   t   
sin   t


Y  , t  
P

,

d








Y

cos


t

Y

0
1
m 0
  
  
     12   2
1  D / m
 2  T / m
Y0    F  f 0 x
Y1    F  f1 x 
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.4. Inverse Fourier transform and
obtainment of integral equations
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.4. Inverse Fourier transform and
obtainment of integral equations
In hydrodynamic problem:
 x, t  
l t  t
  ps, K0 x  s  V0   t ,  t d ds  0 x  V0t, t 
0 0
K 0 x, t   Fx
1
 

t

sin g  t   

2
2x
 g

S x    sin
0
In elasticity problem:
t lmax
g 1
sgn x S   f 01   C   f 02  
2 x 3 / 2
x
x


2
s 2 ds
C x    cos
0
2
s 2 ds
3
 ps, N x  s, t   dsd  k0h0k t N k x, t   hlk t N k x  lmax , t   f 0 x, t   f1 x, t 
0 0
f 0 x, t   F 1Y0  cos t 
3k 
t
1  sin   t 



N x, t  
F
x
,
t


N
x
,
t

k
 x 3k N x, t   d
sin   t 
2     
1 
 x, t   
0
f1 x, t   F Y1  
   

Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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Steady motion
H  , t   
i
2 g
P 
t


  e

i V0 sgn    g



e

i V0 sgn 
 g
 d

 
i
P 
lim H  , t   reg

P   V0 sgn    g   V0 sgn    g
2
t 
 V0  g 2 g


 V0 sgn    g 
2 g
V0
2
    
  g /V0
2
lim H  , t   H 0  , t 
t 
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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
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4. Solution method
4.5. Formation of general simultaneous
integral equations system
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.5. Formation of general simultaneous
integral equations system
Compound integral equation of hydrodynamics and elasticity:
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
Dynamics equations:
l t 
d 2 hc t 
m
 a  px, t dx   0
dt 2
0
l t 
d 2 t 
Ic
 a  x  b  px, t dx
2
dt
0
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.5. Formation of general simultaneous
integral equations system
Compound integral equation of hydrodynamics and elasticity:
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
Dynamics equations:
l t 
d 2 hc t 
m
 a  px, t dx   0
dt 2
0
l t 
d 2 t 
Ic
 a  x  b  px, t dx
2
dt
0
Unknown functions:
px, t 
l t 
t 
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4. Solution method
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:
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
Dynamics equations:
l t 
d 2 hc t 
m
 a  px, t dx   0
dt 2
0
l t 
d 2 t 
Ic
 a  x  b  px, t dx
2
dt
0
Unknown functions:
px, t 
l t 
t 
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.6. Numerical solution of integral equations system by method of discrete vortexes.
Solution of nonlinear problem of wetted length
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
l t 
d 2 hc t 
m
 a  px, t dx   0
dt 2
0
l t 
d 2 t 
Ic
 a  x  b  px, t dx
2
dt
0
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.6. Numerical solution of integral equations system by method of discrete vortexes (MDV).
Solution of nonlinear problem of wetted length
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
l t 
m
Ic
d 2 hc t 
 a  px, t dx   0
dt 2
0
MDV or other
l t 
d 2 t 


a
 x  b px, t dx
dt 2
0
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.6. Numerical solution of integral equations system by method of discrete vortexes (MDV).
Solution of nonlinear problem of wetted length
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
l t 
m
Ic
d 2 hc t 
 a  px, t dx   0
dt 2
0
l t 
d 2 t 


a
 x  b px, t dx
dt 2
0
MDV or other
Al t   X  B
X   p1, p2 ,..., pn ;  
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, 2010. Edinburgh, UK
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4. Solution method
4.6. Numerical solution of integral equations system by the method of discrete vortexes (MDV).
Solution of the nonlinear problem of wetted length
l t  t
  ps, K1 x  s  V0   t ,  t   K 2 x  s, x, s;  t , t , d ds 
  1 x  V0t , t   N x, t 
0 0
l t 
m
Ic
d 2 hc t 
 a  px, t dx   0
dt 2
0
Al t   X  B
MDV or other
l t 
d 2 t 


a
 x  b px, t dx
dt 2
0
X   p1, p2 ,..., pn ;  
 Al t  X  B  Al t  X  B 
min
l t 0,lmax 
(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.
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Thank you for your attention
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