# Variational methods of solving dynamic problems for fluid-containing bodies

- 44 Downloads

## Abstract

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.

## Keywords

body with cavities body-fluid system variational approach free surface nonlinear sloshing natural frequencies## Preview

Unable to display preview. Download preview PDF.

## REFERENCES

- 1.M. Ya. Barnyak and I. A. Lukovskii, “Determining the natural frequencies and modes of a perfect fluid in a vessel of high relative depth,” Prikl. Mekh., 10, No. 5, 109–115 (1974).Google Scholar
- 2.M. Ya. Barnyak and I. A. Lukovskii, “The method of orthogonal projections in the problem on natural sloshing of a fluid in a vessel,” Dynamics and Stability of Multidimensional Systems [in Russian], Inst. Mat. AN USSR, Kiev (1974), pp. 77–85.Google Scholar
- 3.M. Ya. Barnyak and I. A. Lukovskii, “A modification of the variational method for solving the problem on natural sloshing of a fluid in a vessel,” Prikl. Mekh., 13, No. 7, 83–89 (1977).Google Scholar
- 4.M. Ya. Barnyak, I. A. Lukovskii, and G. A. Shvets, “Numerical implementation of the variational method for solving the problem on natural sloshing of a fluid in a vessel,” Dynamics and Stability of Controllable Systems [in Russian], Inst. Mat. AN USSR, Kiev (1977), pp. 62–73.Google Scholar
- 5.M. V. Bekker and I. A. Druzhinin, “Natural sloshing of a fluid in a spherical cavity. Anumerical-analytical alternative of the variational method,” Izv. RAN, Zh. Vych. Mat. Tekhn. Fiz., 40, No. 4, 633–637 (2000).MathSciNetGoogle Scholar
- 6.I. B. Bogoryad, I. A. Druzhinin, G. Z. Druzhinina, and É. E. Libin, Introduction to Dynamics of Vessels with Fluid [in Russian], Izd. Tomskogo Univ., Tomsk (1977).Google Scholar
- 7.A. N. Guz, V. D. Kubenko, and A. É. Babaev, “Dynamics of shell systems interacting with a liquid,” Int. Appl. Mech., 38, No. 3, 260–301 (2002).Google Scholar
- 8.L. V. Dokuchaev and I. A. Lukovskii, “A method for determining the hydrodynamic characteristics of a moving vessel with partitions,” Izv. AN SRSR, Mekh. Zhidk. Gaza, No. 4, 205–213 (1968).Google Scholar
- 9.N. E. Zhukovskii, Motion of a Body with Cavities Filled with a Homogeneous Dropping Fluid [in Russian], Vol. 2 of the 5-volume Collected Works [in Russian], Gosnauchtekhizdat, Moscow-Leningrad (1931).Google Scholar
- 10.K. S. Kolesnikov, Liquid-Fuel Rocket as a Control Object [in Russian], Mashinostroenie, Moscow (1969).Google Scholar
- 11.A. N. Komarenko, I. A. Lukovskii, and S. F. Feshchenko, “Eigenvalue problem with a parameter in the boundary conditions,” Ukr. Mat. Zh., 17, No. 6, 22–30 (1965).MATHGoogle Scholar
- 12.A. N. Komarenko and I. A. Lukovskii, “Stability of nonlinear sloshing of a fluid in a vessel moving harmonically,” Prikl. Mekh., 10, No. 10, 97–102 (1974).Google Scholar
- 13.V. D. Kubenko and V. V. Dzyuba, “Axisymmetric interaction problem for a sphere pulsating inside an elastic cylindrical shell filled with and immersed into a liquid,” Int. Appl. Mech., 38, No. 10, 1210–1219 (2002).Google Scholar
- 14.V. D. Kubenko, P. S. Koval’chuk, L. G. Boyarshina et al., Nonlinear Dynamics of Axisymmetric Bodies with Fluid [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
- 15.P. S. Koval’chuk and L. A. Kruk, “Chaotic modes of forced nonlinear vibrations of fluid-filled cylindrical shells,” Int. Appl. Mech., 39, No. 12, 1452–1457 (2003).Google Scholar
- 16.H. L. Lamb, Hydrodynamics, 6th ed., Cambridge Univ. Press (1932).Google Scholar
- 17.O. S. Limarchenko, “Variational formulation of the problem on the motion of a tank with a fluid,” Dokl. AN USSR, Ser. A, No. 10, 904–908 (1978).Google Scholar
- 18.O. S. Limarchenko, “Analyzing the efficiency of discrete models in solving problems for a tank with a fluid under impulsive excitation,” Mat. Fiz. Nonlin. Mekh., 4, 44–48 (1985).Google Scholar
- 19.O. S. Limarchenko, “A direct method for solving the problem on the spatial motion of a body-fluid system,” Prikl. Mekh., 19, No. 8, 77–84 (1983).Google Scholar
- 20.I. O. Lukovskii, “Low-amplitude wave motions of a homogeneous incompressible fluid in vessels having the form of bodies of revolution,” Dop. AN URSR, No. 8, 1013–1017 (1961).Google Scholar
- 21.I. O. Lukovskii, “Equations of disturbed motion of a body with a cavity that has the form of a body of revolution and is partially filled with a fluid,” Dop. AN URSR, No. 6, 749–753 (1962).Google Scholar
- 22.I. A. Lukovskii, “The coefficients of the equations of disturbed motion of a body with axisymmetric cavities partially filled with a fluid,” Proc. Conf. Young Scientists of the Institute of Mathematics AS USSR [in Russian], Kiev (1963), pp. 106–110.Google Scholar
- 23.I. A. Lukovskii, “An approximate method for determining the hydrodynamic coefficients of the equations of disturbed motion of a body with cavities partially filled with a fluid,” Hydroaeromechanics [in Russian], Issue 1, Izd. KhGU, Kharkov (1965), pp. 62–72.Google Scholar
- 24.I. A. Lukovskii, “Applying the eigenfunction expansion method to solve boundary-value problems in the theory of disturbed motion of a fluid-containing body,” Proc. 1st Republ. Math. Conf. Young Researchers [in Russian], Issue 1, Inst. Mat. AN USSR, Kiev (1965), pp. 462–469.Google Scholar
- 25.I. A. Lukovskii, “Calculating the characteristics of motion of a fluid in a cavity in the form of an ellipsoid of revolution,” Prikl. Mekh., 1, No. 7, 101–106 (1965).Google Scholar
- 26.I. A. Lukovskii, “Determining the hydrodynamic characteristics of disturbed motion of a body with cavities separated by radial partitions and partially filled with a fluid,” Hydroaeromechanics, Issue 1, Izd. KhGU, Kharkov (1965),pp. 53–61.Google Scholar
- 27.I. A. Lukovskii, “Studying the motion of a body with a nonlinearly sloshing fluid,” Prikl. Mekh., 3, No. 6, 119–127 (1967).Google Scholar
- 28.I. A. Lukovskii, “Solving problems for a sloshing fluid in vessels of complex geometry,” Mat. Fiz., 3, 274–283 (1967).Google Scholar
- 29.I. O. Lukovskii, “Solving the nonlinear problem on the natural sloshing of the fluid in vessels of arbitrary geometry,” Dop. AN URSR, Ser. A, No. 3, 207–210 (1969).Google Scholar
- 30.I. A. Lukovskii, “Studying the nonlinear sloshing of the fluid in a vessel in the form of a body of revolution,” Mat. Fiz., 9, 57–72 (1971).MathSciNetGoogle Scholar
- 31.I. A. Lukovskii, Studying the Nonlinear Sloshing of Fluid in Moving Conic Vessels [in Russian], Mat. Fiz., 10, 70–79 (1971).Google Scholar
- 32.I. A. Lukovskii, Nonlinear Sloshing of Fluid in Vessels of Complex Geometry [in Russian], Naukova Dumka, Kiev (1975).Google Scholar
- 33.I. A. Lukovskii, “Methods of approximate solution of nonlinear dynamic boundary-value problems for a finite fluid with free boundary,” Mathematization of Knowledge and Scientific and Technological Advance [in Russian], Naukova Dumka, Kiev (1975), pp. 129–148.Google Scholar
- 34.I. A. Lukovskii, “The variational method in nonlinear dynamic problems for a finite fluid with free surface,” Vibration of Structures with Fluid [in Russian], TsNTI “Volna,” Moscow (1976), pp. 260–265.Google Scholar
- 35.I. A. Lukovskii and A. S. Korneyeva, “Spatial nonlinear motions of the fluid in a right circular cylinder,” Dynamics and Stability of Controllable Systems [in Russian], Inst. Mat. AN USSR, Kiev (1977), pp. 93–103.Google Scholar
- 36.I. A. Lukovskii, Variational Method in Nonlinear Problems of the Theory of Motion of a Body with a Cavity Partially Filled with a Fluid [in Russian], Preprint 78.22, Inst. Mat. AN USSR, Kiev (1978).Google Scholar
- 37.I. A. Lukovskii and A. M. Pil’kevich, “Studying the nonlinear sloshing of fluid in coaxial circular cylinders by the variational method,” Boundary-Value Problems of Mathematical Physics [in Russian], Inst. Mat. AN USSR, Kiev (1978), pp. 79–91.Google Scholar
- 38.I. A. Lukovskii, “Variational formulation of nonlinear dynamic boundary-value problems for a finite fluid executing prescribed three-dimensional motion,” Prikl. Mekh., 16, No. 2, 102–108 (1980).Google Scholar
- 39.I. A. Lukovskii, “Applying Ostrogradskii’s variational principle to solve nonlinear dynamic problems for a body with cavities containing a fluid,” Dynamics and Stability of Mechanical Systems [in Russian], Inst. Mat. AN USSR, Kiev (1980), pp. 3–15.Google Scholar
- 40.I. A. Lukovskii, “Determining interaction forces in the nonlinear dynamic problem for a body with a cavity containing a fluid,” Nonlinear Boundary-Value Problems [in Russian], Inst. Mat. AN USSR, Kiev (1980), pp. 181–190.Google Scholar
- 41.I. A. Lukovskii, “Approximate method for solving nonlinear dynamic problems for a fluid in a moving vessel,” Prikl. Mekh., 17, No. 2, 89–96 (1981).Google Scholar
- 42.I. A. Lukovskii, “Determining the Zhukovskii potentials for the nonlinear wave motions of the fluid in a vessel executing angular displacements,” Proc. 4th Seminar on Dynamics of Rigid and Elastic Bodies Interacting with Fluid [in Russian], Izd. Tomsk. Gos. Univ., Tomsk (1981), pp. 128–139.Google Scholar
- 43.I. A. Lukovskii, “Determining the natural frequencies and modes of the fluid in a vessel using Bateman’s variational principle,” Analytical Methods for Dynamic Study of Complex Systems [in Russian], Inst. Mat. AN USSR, Kiev (1982), pp. 3–11.Google Scholar
- 44.I. A. Lukovskii, M. Ya. Barnyak, and A. N. Komarenko, Approximate Methods for Solving Dynamic Problems for a Finite Fluid [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
- 45.I. A. Lukovskii and A. M. Pil’kevich, Nonlinear Equations of Spatial Motions of a Body with a Cylindrical Cavity Containing a Fluid [in Russian], Preprint 84.18, Inst. Mat. AN USSR, Kiev (1984).Google Scholar
- 46.I. A. Lukovskii and A. M. Pil’kevich, Tables of Hydrodynamic Coefficients for the Nonlinear Equations of Spatial Motions of a Body with a Cylindrical Cavity Containing a Fluid [in Russian], Preprint 84.39, Inst. Mat. AN USSR, Kiev (1984).Google Scholar
- 47.I. A. Lukovskii and A. M. Pil’kevich, “Determining the apparent moments of inertia of a fluid by the variational and perturbation methods,” Proc. 5th Seminar on Dynamics of Rigid and Elastic Bodies Interacting with a Fluid [in Russian], Izd. Tomsk. Gos. Univ., Tomsk (1984), pp. 102–112.Google Scholar
- 48.I. A. Lukovskii and A. N. Bilyk, “Forced nonlinear sloshing of a fluid in moving axisymmetric conic vessels,” Numerical-Analytical Methods for Dynamic and Stability Studies of Multidimensional Systems [in Russian], Inst. Mat. AN USSR, Kiev (1985), pp. 12–26.Google Scholar
- 49.I. A. Lukovskii and A. M. Pil’kevich, “Sloshing of a fluid in a rectangular tank,” Direct Methods in Dynamic and Stability Problems for Multidimensional Systems [in Russian], Inst. Mat. AN USSR (1986), pp. 13–18.Google Scholar
- 50.I. A. Lukovskii, “Applying the variational principle to derive the nonlinear equations of disturbed motion of a body-fluid system,” Spacecraft Dynamics and Space Exploration [in Russian], Mashinostroenie, Moscow (1986), pp. 182–194.Google Scholar
- 51.I. A. Lukovskii and G. A. Shvets, “A rapidly converging variational algorithm for the problem on natural sloshing of a fluid in a vessel,” Spacecraft Dynamics and Space Exploration [in Russian], Mashinostroenie, Moscow (1986), pp. 137–142.Google Scholar
- 52.I. A. Lukovskii, V. A. Trotsenko, and V. I. Usyukin, Interaction of Thin-Walled Elastic Elements with a Fluid in Moving Containers [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
- 53.I. A. Lukovskii and A. M. Pil’kevich, “Sloshing of a fluid in a cylindrical container with convex bottom,” Applied Problems of Dynamics and Stability of Multidimensional Systems [in Russian], Inst. Mat. AN USSR, Kiev (1987),pp. 15–21.Google Scholar
- 54.I. A. Lukovskii, Introduction to Nonlinear Dynamics of a Body with Cavities Containing a Fluid [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
- 55.I. A. Lukovskii, “Constructing finite-dimensional models in dynamics of a finite fluid based on variational principles,” Modeling Dynamic Interaction in Body-Fluid Systems [in Russian], Inst. Mat. AN USSR, Kiev (1990), pp. 5–15.Google Scholar
- 56.I. O. Lukovskii, “Variational principle in the nonlinear theory of motion of floating bodies with containers partially filled with a fluid,” Dop. NAN Ukrainy, No. 8, 63–66 (1993).Google Scholar
- 57.I. A. Lukovskii, O. S. Limarchenko, and A. N. Timokha, “Nonlinear models in applied dynamic problems for bodies containing a fluid with free surface,” Prikl. Mekh., 28, No. 11, 75–83 (1992).MathSciNetGoogle Scholar
- 58.I. O. Lukovskii, Boundary-Value Problems in the Hydrodynamic Theory of Ships with Compartments Containing Liquid Cargo [in Russian], Preprint 94.16,Inst. Mat. NAN Ukrainy, Moscow (1994).Google Scholar
- 59.I. O. Lukovskii and M. Ya. Barnyak, “A modified variational method for solving the problem on natural sloshing of a fluid in an oblique cylinder,” Dop. NAN Ukrainy, No. 5, 62–66 (1997).Google Scholar
- 60.I. A. Lukovskii, “Determining the forces of interaction in dynamic problems for a floating body containing a fluid,” Prikl. Mekh., 33, No. 9, 74–81 (1997).MathSciNetGoogle Scholar
- 61.I. O. Lukovskii, “The theory of nonlinear sloshing of a finite, weakly viscous, incompressible fluid,” Dop. NAN Ukrainy, No. 10, 80–84 (1997).Google Scholar
- 62.I. A. Lukovskii, “Nonlinear equations of wave motions of a fluid in moving tanks in the form of a circular cone,” Nonlinear Boundary-Value Problems of Mathematical Physics and Their Application [in Russian], Inst. Mat. NAN Ukrainy, Kiev (1998), pp. 139–143.Google Scholar
- 63.I. A. Lukovskii and G. F. Zolotenko, “Numerical simulation of the sloshing of a fluid in a moving closed rectangular vessel,” GidroMekh., 72, 72–87 (1998).Google Scholar
- 64.I. O. Lukovskii, “Solving spectral problems in the linear theory of sloshing of a fluid in finite tanks,” Dop. NAN Ukrainy, No. 5, 53–58 (2002).Google Scholar
- 65.I. O. Lukovskii and D. V. Ovchinnikov, “Nonlinear mathematical model of the fifth order in the problem on the sloshing of a fluid in a cylindrical vessel,” Dynamic and Stability Problems for Multidimensional Systems [in Russian], Inst. Mat. NAN Ukrainy, Kiev (2003), pp. 119–160.Google Scholar
- 66.I. O. Lukovskii and O. V. Solodun, “Nonlinear model of motion of a fluid in cylindrical vessels with compartments,” Dop. NAN Ukrainy, No. 5, 50–55 (2001).Google Scholar
- 67.I. O. Lukovskii and O. V. Solodun, “Studying the forced nonlinear sloshing of a fluid in circular cylindrical vessels using a seven-mode model of the third order,” Dynamic and Stability Problems for Multidimensional Systems [in Russian], Inst. Mat. NAN Ukrainy, Kiev (2003), pp. 161–179.Google Scholar
- 68.I. A. Lukovskii and A. N. Timokha, “Nonlinear theory of sloshing of the fluid in moving containers: classical and nonclassical problems,” Problems of Analytical Mechanics and Its Applications [in Russian], Inst. Mat. NAN Ukrainy, Kiev (1999), pp. 169–201.Google Scholar
- 69.I. A. Lukovskii, G. F. Zolotenko, and A. M. Pil’kevich, “Comparative analysis of two variational models in the nonlinear theory of relative motion of fluid,” Prikl. GidroMekh., 5, No. 4, 12–43 (2003).MathSciNetGoogle Scholar
- 70.A. I. Lur’e, Analytical Mechanics [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
- 71.G. N. Mikishev and B. I. Rabinovich, Dynamics of a Body with Cavities Partially Filled with a Fluid [in Russian], Mashinostroenie, Moscow (1968).Google Scholar
- 72.G. N. Mikishev, Experimental Methods in Spacecraft Dynamics [in Russian], Mashinostroenie, Moscow (1978).Google Scholar
- 73.N. N. Moiseev, “Theory of nonlinear sloshing of a finite fluid,” Prikl. Mat. Mekh., 22, 612–621 (1958).MathSciNetGoogle Scholar
- 74.N. N. Moiseev and V. V. Rumyantsev, Dynamics of Bodies with Cavities Containing a Fluid [in Russian], Nauka, Moscow (1965).Google Scholar
- 75.N. N. Moiseev and A. A. Petrov, Numerical Methods for Calculating the Natural Frequencies of a Finite Fluid[inRussian], VTs AN SSSR, Moscow (1966).Google Scholar
- 76.G. S. Narimanov, “Motion of a vessel partially filled with a fluid executing low-amplitude motions,” Prikl. Mat. Mekh., 21, No. 4, 513–524 (1957).MathSciNetGoogle Scholar
- 77.G. S. Narimanov, L. V. Dokuchaev, and I. A. Lukovskii, Nonlinear Dynamics of Fluid-Carrying Aircraft [in Russian], Mashinostroenie, Moscow (1977).Google Scholar
- 78.M. V. Ostrogradskii, Complete Works [in Russian], Vol. 1, Izd. AN USSR, Kiev (1959).Google Scholar
- 79.B. I. Rabinovich, Introduction to Launch Vehicle Dynamics [in Russian], Mashinostroenie, Moscow (1975).Google Scholar
- 80.O. V. Solodun, “Studying forced sloshing in circular cylindrical vessels with a diametral partition,” Nonlinear Oscillations, 5, No. 1, 90–106 (2002).MATHMathSciNetGoogle Scholar
- 81.J. J. Stocker, Water Waves, Interscience, New York (1957).Google Scholar
- 82.E. M. Stazhkov, “A rapidly converging method based on the optimal choice of the origin of coordinates in the liquid sloshing problem,” Proc. 3rd Seminar on Dynamics of Elastic and Rigid Bodies Interacting with Fluid [in Russian], Izd. Tomskogo Univ., Tomsk (1978), pp. 125–132.Google Scholar
- 83.V. A. Trotsenko, “Solving dynamic boundary-value problems for a fluid in horizontal cylindrical containers with partitions,” Nonlinear Oscillations, 6, No. 3, 401–427 (2003).MATHMathSciNetGoogle Scholar
- 84.S. F. Feshchenko, I. A. Lukovskii, B. I. Rabinovich, and L. V. Dokuchaev, Method for Determining the Apparent Masses of a Fluid in Moving Containers [in Russian], Naukova Dumka, Kiev (1969).Google Scholar
- 85.H. N. Abramson, The Dynamic Behavior of Liquids in Moving Containers, NASA SP-106, Washington (1966).Google Scholar
- 86.H. N. Abramson, W. H. Chu, and D. D. Kana, “Some studies of nonlinear lateral sloshing in rigid containers,” J. Appl. Mech., 33, No. 4, 66–74 (1966).Google Scholar
- 87.H. Bateman, Partial Differential Equations in Physics, Dover, New York (1944).Google Scholar
- 88.H. F. Bauer, “Nonlinear mechanical model for the description of propellant sloshing,” AJAA J., 4, No. 9, 1662–1668 (1966).Google Scholar
- 89.F. T. Dodge, D. D. Kana, and H. N. Abramson, “Liquid surface oscillation in longitudinally excited rigid cylindrical containers,” AIAA J., 3, No. 4, 685–695 (1965).CrossRefMATHGoogle Scholar
- 90.O. M. Faltinsen, O. F. Rognebakke, I. A. Lukovsky, and A. N. Timokha, “Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth,” J. Fluid Mech., 407, 201–234 (2000).MATHADSMathSciNetGoogle Scholar
- 91.O. M. Faltinsen, O. F. Rognebakke, and A. N. Timokha, “Resonant three-dimensional nonlinear sloshing in a square base basin,” J. Fluid Mech., 487, 1–47 (2003).MATHADSMathSciNetGoogle Scholar
- 92.I. P. Gavrilyuk, I. A. Lukovsky, and A. N. Timokha, “A multimodal approach to nonlinear sloshing in a circular cylindrical tank,” Hybrid Methods in Engineering, 2, No. 4, 463–483 (2000).Google Scholar
- 93.V. D. Kubenko, P. S. Koval’chuk, and L. A. Kruk, “On multimode nonlinear vibrations of filled cylindrical shells,” Int. Appl. Mech., 39, No. 1, 85–92 (2003).Google Scholar
- 94.P. S. Koval’chuk, N. P. Podchasov, and V. V. Kholopova, “Analysis of nonlinear bulging of liquid-filled cylindrical shells under local dynamic loading,” Int. Appl. Mech., 39, No. 3, 312–317 (2003).Google Scholar
- 95.P. S. Koval’chuk and V. G. Filin, “Circumferential traveling waves in filled cylindrical shells,” Int. Appl. Mech., 39, No. 2, 192–196 (2003).Google Scholar
- 96.J. C. Luce, “A variational principle for a fluid with a free surface,” J. Fluid Mech., 27, 395–397 (1976).ADSGoogle Scholar
- 97.P. McIver, “Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth,” J. Fluid Mech., 201, 243–257 (1989).MATHADSMathSciNetGoogle Scholar
- 98.J. W. Miles, “Nonlinear surface waves in closed basins,” J. Fluid Mech., 75, Pt. 3, 419–448 (1976).MATHADSMathSciNetGoogle Scholar
- 99.J. W. Miles, “Internally resonant surface waves in a circular cylinder,” J. Fluid Mech., 149, 1–14 (1984).MATHADSMathSciNetGoogle Scholar