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

2. 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, Edinburgh, UK

3. 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.
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

4. , , , ,
2. Hydrodynamic problem
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

5. 2. Hydrodynamic problem Boundary conditions Initial conditions or,
2. Hydrodynamic problem
(2.1)
Boundary conditions
, (2.2)
, (2.3)
(2.4)
Initial conditions
(2.5) or
(2.6)
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

6. 3. Elasticity problem Boundary conditions Initial conditions (3.1),
3. Elasticity problem
(3.1)
Boundary conditions
pinning
. (3.2)
fixed ends
. (3.3)
free ends
(3.4)
or combinations…
Initial conditions
, (3.5)
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

7. 4. Solution method General idea Classical boundary problem,
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
4.4. 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
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

8. 4.1. Generalized functions problem,
4. Solution method
4.1. Generalized functions problem
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

9. 4.1. Generalized functions problem,
4. Solution method
4.1. Generalized functions problem
In hydrodynamic problem:
In elasticity problem:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

10. 4.2. Fourier transform and fundamental solution,
4. Solution method
4.2. Fourier transform and fundamental solution
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

11. 4.2. Fourier transform and fundamental solution,
4. Solution method
4.2. Fourier transform and fundamental solution
In hydrodynamic problem:
In elasticity problem:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

12. 4.3. Solutions for generalized functions,
4. Solution method
4.3. Solutions for generalized functions
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

13. 4.3. Solutions for generalized functions,
4. Solution method
4.3. Solutions for generalized functions
In hydrodynamic problem:
In elasticity problem:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

14. 4.4. Inverse Fourier transform and obtainment of integral equations,
4. Solution method
4.4. Inverse Fourier transform and obtainment of integral equations
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

15. 4.4. Inverse Fourier transform and obtainment of integral equations,
4. Solution method
4.4. Inverse Fourier transform and obtainment of integral equations
In hydrodynamic problem:
In elasticity problem:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

16. Steady motion
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

17. 4.5. Formation of general simultaneous integral equations system,
4. Solution method
4.5. Formation of general simultaneous integral equations system
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

18. 4.5. Formation of general simultaneous integral equations system,
4. Solution method
4.5. Formation of general simultaneous integral equations system
Compound integral equation of hydrodynamics and elasticity:
Dynamics equations:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

19. 4.5. Formation of general simultaneous integral equations system,
4. Solution method
4.5. Formation of general simultaneous integral equations system
Compound integral equation of hydrodynamics and elasticity:
Dynamics equations:
Unknown functions:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

20. Solution of the nonlinear problem of wetted length,
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:
Dynamics equations:
Unknown functions:
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

21. Solution of nonlinear problem of wetted length,
4. Solution method
4.6. Numerical solution of integral equations system by method of discrete vortexes.
Solution of nonlinear problem of wetted length
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

22. Solution of nonlinear problem of wetted length,
4. Solution method
4.6. Numerical solution of integral equations system by method of discrete vortexes (MDV).
Solution of nonlinear problem of wetted length
MDV or other
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

23. Solution of nonlinear problem of wetted length,
4. Solution method
4.6. Numerical solution of integral equations system by method of discrete vortexes (MDV).
Solution of nonlinear problem of wetted length
MDV or other
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

24. Solution of the nonlinear problem of wetted length,
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
MDV or other
(least-squares method+ nondifferential minimization)
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK

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

26. Thank you for your attention
Mathematical challenges and modeling of hydroelasticity. Jun 21–24, Edinburgh, UK