1. 3 8 th International Conferences on Boundary Elements and other Mesh Reduction Methods 3 8 th International Conferences on Boundary Elements and other Mesh Reduction Methods Vasyl Gnitko, Kyril Degtyarev,Vasyl Gnitko, Kyril Degtyarev, Vitaly Naumenko, Elena StrelnikovaVitaly Naumenko, Elena Strelnikova A.N. Podgorny Institute of Mechanical Engineering ProblemsA.N. Podgorny Institute of Mechanical Engineering Problems Ukrainian Academy of Sciences, UkraineUkrainian Academy of Sciences, Ukraine

2. A.N. Podgorny Institute of Mechanical Engineering Problems, National Academy of Sciences, Ukraine

3. CONTENTS Introduction and problem statementIntroduction and problem statement Mode superposition method for coupled dynamic problemsMode superposition method for coupled dynamic problems Systems of the boundary integral equations and some remarks about their numerical implementationSystems of the boundary integral equations and some remarks about their numerical implementation Some numerical resultsSome numerical results Failure of cracked shellsFailure of cracked shells ConclusionConclusion

4. A.N. Podgorny Institute of Mechanical Engineering Problems The A.N.Podgorny Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine (IPMash NAS of Ukraine) is a renown research centre in power and mechanical engineering.The A.N.Podgorny Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine (IPMash NAS of Ukraine) is a renown research centre in power and mechanical engineering. IPMash has 12 research departments with a staff of 346 specialists (133 research workers, including one Academician and three Corresponding Members of NAS of Ukraine; and 31 Doctors and 77 Candidates of Science). The Institute also has a special Design-and-Engineering Bureau, and a pilot production facility.IPMash has 12 research departments with a staff of 346 specialists (133 research workers, including one Academician and three Corresponding Members of NAS of Ukraine; and 31 Doctors and 77 Candidates of Science). The Institute also has a special Design-and-Engineering Bureau, and a pilot production facility. Key research areasKey research areas optimisation of processes in power machinery, and improvement of equipment design;optimisation of processes in power machinery, and improvement of equipment design; energy saving technologies and non-conventional power engineering facilities;energy saving technologies and non-conventional power engineering facilities; predicting the reliability, dynamic strength and life of power equipment;predicting the reliability, dynamic strength and life of power equipment; simulation and computer technologies in power machine building.simulation and computer technologies in power machine building.

5. Structures under investigation Wind power station Francis turbine’s water wheelWind power station Francis turbine’s water wheel

6. Structures under investigation Kaplan turbine water wheel Hydroturbine coverKaplan turbine water wheel Hydroturbine cover

7. Structures under investigation. STORAGE TANKS

8. FLUID-STRUCTURE INTERACTION. PROBLEM STATEMENT Equations of motionEquations of motion L, М – stiffness and mass matrixes ;L, М – stiffness and mass matrixes ; U=(u, v, w) – displacement vector;U=(u, v, w) – displacement vector; Q(t) – external load;Q(t) – external load; P(t) – liquid pressureP(t) – liquid pressure Ideal incompressible liquid Ideal incompressible liquid the flow is potentialthe flow is potential Laplace equation for velocity potentialLaplace equation for velocity potential

9. PROBLEM STATEMENT Cauchy-Lagrange integral to determine P=pnCauchy-Lagrange integral to determine P=pn g is the free fall gravity acceleration, z is the vertical coordinate of a point in the liquid, p 0 is the atmospheric pressure and  l is the fluid density. g is the free fall gravity acceleration, z is the vertical coordinate of a point in the liquid, p 0 is the atmospheric pressure and  l is the fluid density. Boundary conditions on the free surfaceBoundary conditions on the free surface

10. Dynamical problem for elastic shells of revolution interacting with a fluid with the next set of boundary conditions relative to with the next set of boundary conditions relative to  fixation conditions of the shell relative to Ufixation conditions of the shell relative to U Initial conditionsInitial conditions

11. MODE DECOMPOSITION METHOD FOR COUPLED DYNAMIC PROBLEMS Displacements are linear combination of structure natural modes without including the liquidDisplacements are linear combination of structure natural modes without including the liquid u k are the normal modes of vibrations of the empty shell. u k are the normal modes of vibrations of the empty shell. Representation for velocity potentialRepresentation for velocity potential

12. Three systems of basic functions At first we obtain the natural modes and frequencies of structure without liquid – the first system of basic functionsAt first we obtain the natural modes and frequencies of structure without liquid – the first system of basic functions Second, we represent the velocity potential as a sumSecond, we represent the velocity potential as a sum and for each component consider the corresponding boundary value problem for Laplace equation.and for each component consider the corresponding boundary value problem for Laplace equation. The potential corresponds to the problem of elastic structure vibrations with the liquid but without including the force of gravityThe potential corresponds to the problem of elastic structure vibrations with the liquid but without including the force of gravity The potential Ф 2 corresponds to the problem of rigid structure vibrations with the liquid including the force of gravityThe potential Ф 2 corresponds to the problem of rigid structure vibrations with the liquid including the force of gravity

13. FIRST SYSTEM OF BASIC FUNCTIONS. MODES OF CYLINDRICAL SHELL VIBRATIONS

14. REPRESENTATION FOR VELOCITY POTENTIAL  1 To define the function  1 (elastic shell vibrations without including the force of gravity) we formulate the following boundary value problemTo define the function  1 (elastic shell vibrations without including the force of gravity) we formulate the following boundary value problem We use the representation for velocity potential  1 (shell vibrations without gravity) in the formWe use the representation for velocity potential  1 (shell vibrations without gravity) in the form It leads to boundary value problems for potentials  1 kIt leads to boundary value problems for potentials  1 k

15. Boundary value problem for velocity potential  2 (liquid vibrations in rigid shell including the force of gravity) From boundary conditions on the free surface we haveFrom boundary conditions on the free surface we have We use the representation for velocity potentialWe use the representation for velocity potential For harmonic vibrations of liquid in rigid shellFor harmonic vibrations of liquid in rigid shell

16. Evaluation of fluid velocity potential  2 At first we define the natural values  k and corresponding natural modes  k. Then we have the following representationAt first we define the natural values  k and corresponding natural modes  k. Then we have the following representation Boundary condition on the free surface leads toBoundary condition on the free surface leads to As the natural modes of liquid vibration in rigid shell are orthogonal we haveAs the natural modes of liquid vibration in rigid shell are orthogonal we have

17. The governing system of differential equations

18. EIGENVALUE PROBLEM We use the direct BEM formulation

19. BOUNDARY INTEGRAL EQUATION METHOD. SINGLE DOMAIN APPROACH

20. INTEGRAL EQUATIONS FOR SHELL OF REVOLUTION

21. A cylindrical shell with a flat bottom partially filled with the fluid Radius R = 1 m, thickness h = 0.01 m, length L = 2 m, Young’s modulus E = 2·10 5 MPa, Poisson’s ratio ν = 0.3, material’s density  = 7800 kg/m 3, the fluid density  l = 1000 kg/m 3. The filling level of the fluid H. Boundary conditions are following : to z = 0 and r = R. z = 0 and r = R.

22. Natural frequencies of cylindrical tank, H = 0.8 m m Natural frequencies, Hz Empty tank Fluid-filled tank Presented method FEM 0 123.238.158.06 291.1045.0444.71 1 148.5220.98 20.86 2145.30 77.28 77.04 2 179.7740.04 39.90 2 117.07 109.51108.89

23. A conical shell with a flat bottom partially filled with the fluid

24. Natural frequencies of conical tank m Natural frequencies, Hz Empty tank Fluid-filled tank Presented method FEM 0 14303,21004,11009,4 24588,61549,31554,1 1 13063,9645,0649,9 24148,61107,91117,2 2 11957,9555,1 569,8 23352,51016,7 1036,1

25. Free vibrations of liquid in rigid shells. Boundary value problem. Third system of basic functions. Free vibrations

26. Boundary integral equation method in the problem of liquid vibration in the rigid shell Represent ψ as the sum of double and single layer potentialsRepresent ψ as the sum of double and single layer potentials

27. INTEGRAL REPRESENTATION а) Let - (control point is on the free surface):а) Let - (control point is on the free surface): b) Let - (control point is on the shell surface):b) Let - (control point is on the shell surface):

28. The system of integral equations NotationsNotations The natural modes (third system of basic functions) and values problemThe natural modes (third system of basic functions) and values problem

29. ARITHMETIC GEOMETRIC MEAN AGM(A,B) we have applied here the approach based on the following characteristic property of the arithmetic geometric mean AGM(a,b)we have applied here the approach based on the following characteristic property of the arithmetic geometric mean AGM(a,b) To define AGM(a,b) there exist the simple Gaussian algorithmTo define AGM(a,b) there exist the simple Gaussian algorithm

30. Comparison of analytical and numerical results n=1n=2n=3n=4n=5n=5  =0 BEM3.8157.01910.18013.33316.480 Analyti- cal 3.8157.01610.17313.32416.470  =1 BEM1.6575.332 8.54011.71114.889 Analyti- cal 1.6575.329 8.53611.70614.863

31. Numerically and analytically obtained modes on free surface

32. Sloshing modes on the vertical wall Numbers 1,2,3 correspond to the first, second and third modes of liquid sloshingNumbers 1,2,3 correspond to the first, second and third modes of liquid sloshing

33. MODES OF VIBRATIONS OF CYLINDRICAL SHELL

34. Free vibration of the liquid in rigid cylindrical tanks n=1 n=2 n=3  =0 BEM3.8007.02910.206 [*]3.7967.01510.173  =1 BEM1.5835.3348.558 [*]1.5815.3258.536  =2 BEM 2.975 6.7179.997 [*]2.9706.7049.969 Frequency parameter H/R=0.7 [*] – Analytical solution

35. Free vibrations of the liquid in conical tanks

36. MODES OF LIQUID VIBRATIONS IN CONICAL SHELL

37. n+ mn+ mn+ mn+ m  =0  =1 1212 10+103.546.931.395.15 20+203.506.781.385.05 30+303.486.741.375.02 40+403.476.711.375.01 Analytical solution 3.466.701.364.97 Free vibrations of liquid in rigid conical tank Frequency parameter Modes of free surface

38. . Frequencies of empty and fluid- filled tanks  =0  =1 nnSnS nLnL Empty elastic tank Fluid-filled tank nSnS nLnL Empty elastic tank Fluid-filled tank 116.1193214.03300 211,223.2337.9446427.2328 328.2991639.15547 439.99588410.7233 5411.4410512.0857 6512.72391,248.520721.9555 72,191.101143.86282,1139.70879.7191 83,2205.252119.6273,2,1232.443178.422 94,3,2 365.795238.6954,3277.303210.007

39. CONCLUSIONS The shell vibrations coupled with liquid sloshing under the force of gravity were considered. The free vibration analysis of the elastic cylindrical shell was carried out using the proposed techniques. We introduce the representation of the velocity potential as the sum of two potentials, one of them corresponds to problem of the fluid free vibrations in the rigid shell and another one corresponds to the problem of fluid- filled elastic shell vibrations without including the gravitational component. The spectrum of frequencies for cylindrical tank was analysed.The shell vibrations coupled with liquid sloshing under the force of gravity were considered. The free vibration analysis of the elastic cylindrical shell was carried out using the proposed techniques. We introduce the representation of the velocity potential as the sum of two potentials, one of them corresponds to problem of the fluid free vibrations in the rigid shell and another one corresponds to the problem of fluid- filled elastic shell vibrations without including the gravitational component. The spectrum of frequencies for cylindrical tank was analysed. The proposed method allows us to carry out numerical simulation for different value of gravitational acceleration g. The proposed method allows us to carry out numerical simulation for different value of gravitational acceleration g.

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