We consider the problem of nonlinear oscillations of an ideal incompressible liquid in a tank of a bodyof-revolution shape. It is shown that the ordinary way of application of perturbation techniques results in the violation of solvability conditions of the problem. To avoid this contradiction we introduce some additional conditions and revise previously used approaches. We construct a discrete nonlinear model of the investigated problem on the basis of the Hamilton-Ostrogradskii variational formulation of the mechanical problem, preliminarily satisfying the kinematic boundary conditions and solvability conditions of the problem. Numerical examples testify to the efficiency of the constructed model.

Mechanical and mathematical aspects of nonlinear properties for dynamical behavior of liquid with a free surface in tanks are described. Basic principles for dynamical simulation of the mentioned systems in the range of manifestation of nonlinear properties are proposed. Criteria for construction of lowdimensional models are developed. Some properties of such models for certain examples demonstrate the potential of the suggested approach.

Investigation of dynamics problems of a fluid with a free surface is fraught with significant difficulties caused both by the need to satisfy the boundary condition on a previously unknown boundary, and the substantial nonlinearity of the problem which appears principally in the boundary conditions and the fluid interaction with the elastic or solid walls of the tank.Special complexities are observed in quantitative investigations of problems of this kind, especially problems of the dynamics of joint motion of bodies with a fluid. Despite the urgency of such problems, as is noted in [7,9], say, there is just an insignificant number of papers on this problem.In particular, it is shown [i, 4-6, i0] that the application of variational and numerical methods permits reduction of the investigation of such problems to the investigation of a system of ordinary differential equations and to problems of linear algebra, for whose solution powerful computational methods have been developed that predetermined the success of applying numerical and variational methods to the investigation of dynamics problems of a fluid with a free surface.i. A variational method of investigating nonlinear dynamics problems of the combined motion of a tank and a fluid partially filling it, based on the application of the Hamilton--Ostrogradskii principle to the system of tank-fluid with free surface, is elucidated in [4,5]. The direct method of L. V. Kantorovich, generalized to the particular case of nonlinear problems of mathematical physics with a free boundary, is used for the solution of the variational problem.Its distinguishing feature is the construction of a series expansion of the velocity potential ~ and the perturbations of the liquid free surface ~ that satisfy the kinematic boundary conditions on the tank walls identically (because of the selection of the coordinate functions), and on the free surface to the accuracy of fourth-order quantities in ~ (this is achieved by substituting the expansions for ~ and ~ in the kinematic boundary condition on the free surface with the subsequent determination of the interrelation of the coefficients of their series expansions).Let us note that the method to be applied to determine the dependences of the coefficients of the series expansions of ~ and ~ is analogous to the known method [ii] ; however, it is constructed on the basis of the kinematic boundary condition on the liquid free surface written in differential form, is used for the case of a moving tank, and eliminates the need for approximate inversion of the matrix with variable elements, Realization of the proposed modification of the direct method results in a nonlinear system of ordinary differential equations that allows for a simple numerical solution for arbitrary methods of dynamical excitation.In contrast to the method of Narimanov [8,9], the need to solve sequences of linear boundary-value problems of mathematical physics is eliminated here.The system equations of motion obtained on the basis of the proposed method have the fol...

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Variation of resonant properties of the system liquid-structure caused by changes of distribution of normal frequencies is under consideration. It was shown that different types of mobility of carrying body (translational or rotational motion with different types of constraints) causes growth of normal frequencies, which correspond to antisymmetric oscillations of a liquid free surface, while the rest of frequencies do not change. General arrangement of normal frequencies, which corresponds to the case of immovable reservoir, is considerably violated. In this case some new types of internal resonances in liquidstructure systems are manifested. Two basic problems with redistribution of sequence of normal frequencies were investigated, namely, parametric resonance of movable in translational direction cylindrical reservoir in the Faraday generalized problem and forced motion of liquid in cylindrical reservoir on pendulum with different lengths of suspension. Some general regularities of development of dynamical processes in these systems are discussed.