Опис діяльності ювіляра і привітання

We present a review of the research into the mathematical problems of mechanics and control theory carried out at Institute of Mathematics of the National Academy of Sciences of Ukraine for the entire period of its existence.

UDC 517.9 The Bateman – Luke-type variational formulation of the free-boundary ‘sloshing’ problem is generalized to irrotational flows and unprescribed tank motions, i.e., to the case where both the tank and liquid motions should be found simultaneously for a given set of external forces applied to fixed points of the rigid tank body. We prove that the variational equation, which corresponds to the formulated problem, implies both the dynamic (force and moment) equations of the rigid body and the free-boundary problem, which describes sloshing in terms of the Clebsch potentials.

Наведено огляд дослiджень з математичних проблем механiки та теорiї керування, проведених в Iнститутi математики НАН України вiд початку його заснування.

Resonant sloshing in an upright annular tank is studied by using a new nonlinear modal theory, which is complete within the framework of the Narimanov–Moiseev asymptotics. The applicability is justified for a fairly deep liquid (the liquid-depth-to-outer-tank-radius ratio $1.5\lesssim h=\bar{h}/\bar{r}_{2}$ ) and away from the non-dimensional inner radii $r_{1}=\bar{r}_{1}/\bar{r}_{2}=0.08546$ , 0.17618, 0.27826, 0.31323, 0.31855, 0.43444, 0.46015, 0.48434, 0.68655, 0.70118. The theory is used to describe steady-state (stable and unstable) resonant waves due to a harmonic excitation with the forcing frequency close to the lowest natural sloshing frequency. We show that the surge-sway-pitch-roll excitation is always of either longitudinal or elliptic type. Existing experimental results on the horizontally excited steady-state wave regimes in an upright circular tank ( $r_{1}=0$ ) are utilised for validation. Inserting an inner pole with the radii $r_{1}\approx 0.25$ and 0.35 ( $1.5\lesssim h$ ) causes that no stable swirling and/or irregular waves exist. The response curves for an elliptic-type excitation are examined versus the minor-axis forcing-amplitude component. Stable swirling is then expected being co- and counter-directed to the angular forcing direction. Passage to the rotary (circular) excitation keeps the co-directed swirling stable for all resonant forcing frequencies but the stable counter-directed swirling disappears.

The derivation consists of five steps which were described by Faltinsen et al. (2003) and Faltinsen & Timokha (2013) for other tank shapes. In the most general case, the third-order adaptive modal system is derived suggesting that the finite liquid depth, the forcing magnitude is equal to O( ), and all generalised coordinates and velocities in (2.8) are of the lowest order O( 1/3). Furthermore, the o( )-order quantities can be neglected. The first step suggests the Taylor expansion by ζ of I (Ab), I (1) (Ab)(Mn), and I (2) (Ab)(Mn) by (2.13) and (2.14). By definition, ζ = O( 1/3). Analysis shows that I (Ab) should be expanded up to the third order but I (Ab)(Mn) and I (2) (Ab)(Mn) require expansion up to the second order, i.e. I (Ab) = k −1 Ab tanh(kAbh) + ζ + 1 2κAbζ 2 + 6 k 2 Abζ 3 + . . . , ( 1a)

We consider the most general problem of waves on the interface of two ideal fluids regarded as an ullage gas and a liquid, respectively. Separating the fast and slow time scales, we develop the differential and variational formalism for an acoustically levitating drop and determine its time-averaged shape (vibroequilibrium state of the drop). The vibroequilibrium states of the drop may differ from the spherical shape. Stable vibroequilibria are associated with the local minima of the quasipotential energy whose analytic form is also established.

The multimodal method reduces the sloshing problem with free surface to a (modal) system of nonlinear ordinary differential equations. The method was originally proposed for nonimpulsive hydrodynamic loads. However, recently it has been successfully extended to the case of sloshing-induced slamming. In the 1950–1960’s, this method was used in the computational fluid dynamics (CFD) but later was replaced by the algorithms developed in the 1990-2000’s. At present, the method plays a dual role: first , as a unique analytic tool for the investigation of nonlinear sloshing regimes, their stability, and chaos as well as for simulations when traditional CFD fails (e.g., in the case of containers with perforated screen) and, second , as a source of construction of the modal systems, which are analogs of the Korteweg–de-Vries, Boussinesq, and other equations but for bounded volumes of liquid. We present a survey of the state-ofthe-art of the problem, describe the existing modal systems, and formulate open problems.

Based on variational method, we deduce nonlinear modal equations describing the dynamics of a levitating drop. Using these equations, we construct an asymptotic modal theory for axisymmetric drop oscillations. We consider nonlinear free oscillations of the drop with a frequency close to the lowest natural frequency. The results are compared with experimental data and numerical results obtained by other authors.

Sloshing of an ideal incompressible liquid in a rigid truncated (tapered) conical tank is considered when the tank performs small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. The multimodal method, the non-conformal mapping technique and the Moiseev type asymptotics are employed to derive a finite-dimensional system of weakly nonlinear ordinary differential (modal) equations. This modal system is a generalization of that by Gavrilyuk et al 2005 Fluid Dyn. Res. 37 399-429. Using the derived modal equations, we classify the resonant steady-state wave regimes occurring due to horizontal harmonic tank excitations. The frequency ranges are detected where the 'planar' and/or 'swirling' steady-state sloshing are stable as well as a range in which all steady-state wave regimes are not stable and irregular ( chaotic) liquid motions occur is established. The results on the frequency ranges are qualitatively supported by experiments by Matta E 2002 PhD Thesis Politecnico di Torino, Torino.(Some figures may appear in colour only in the online journal)

PurposeThe purpose of this paper is to derive linear modal equations describing the forced liquid sloshing in a rigid truncated (tapered) conical tank, as well as to show how to couple these modal equations with “global” dynamic equations of a complex mechanical system carrying this tank.Design/methodology/approachDerivation of the modal equations can be based on the Trefftz variational method developed by the authors in a previous paper. Describing the coupled dynamics utilizes Lukovsky' formulas for the resulting hydrodynamic force and moment due to liquid sloshing.FindingsThe so‐called Stokes‐Joukowski potentials can be found by using the Trefftz method from the authors' previous paper with the same polynomial‐type functional basis. Coupling the modal equations with the global dynamic equations becomes a relatively simple task facilitated by Lukovsky's formulas. Using the linear multimodal method can be an efficient alternative to traditional numerical and analytical tools employed for studying the coupled vibrations of a tower with a conical rigid tank on the tower top.Practical implicationsThe derived modal equations are equipped by tables with the computed non‐dimensional hydrodynamic coefficients. Interested readers (engineers) can incorporate the modal equations into the global dynamic equations of a whole mechanical system without new computations of these coefficients.Originality/valueThe multimodal method can be an alternative to traditional numerical tools. Using the derived modal equations simplifies analytical studies and provides efficient calculations of the coupled dynamics of a mechanical system carrying a rigid tapered conical tank with a liquid.

The present paper extends the multimodal method, which is well known for liquid sloshing problems, to the free-surface problem modeling the levitating drop dynamics. The generalized Lukovsky-Miles modal equations are derived. Based on these equations an approximate modal theory is constructed to describe weakly-nonlinear axisymmetric drop motions. Whereas the drop performs almost-periodic oscillations with the frequency close to the lowest natural frequency, the theory takes a finite-dimensional form. Periodic solutions of the corresponding finite-dimensional modal system are compared with experimental and numerical results obtained by other authors. A good agreement is shown.

Combining the Lukovsky–Miles variational method and the Narimanov–Moiseev asymptotics, we deduce a nonlinear modal system describing the resonant liquid sloshing in an upright circular cylindrical tank. The sloshing occurs due to a small-amplitude periodic or an almost-periodic excitation with forcing frequency close to the lowest natural sloshing frequency. In contrast to the existing nonlinear modal systems based on the Narimanov–Moiseev asymptotic intermodal relations, the derived modal equations (i) contain all necessary (infinitely many) generalized coordinates of the second and third orders and (ii) include exclusively nonzero hydrodynamic coefficients, for which (iii) fairly simple computational formulas are found. As a consequence, the modal equations can be used in analytical studies of nonlinear sloshing phenomena, which will be demonstrated in the forthcoming Part II.

The survey collects asymptotic nonlinear modal equations of liquid sloshing theory whose derivation combines variational (Miles-Lukovsky's) and asymptotic (Narimanov-Moiseev's) methods. A particular emphasis is placed on classification of steady-state resonant solutions obtained by analyzing these equations.

Employing a variational solution of basic boundary problems of the linear modal sloshing theory in a tapered conical tank, we derive a multimodal model describing the forced liquid motions and associated hydrodynamic loads. The multimodal model can be used in problems on the fluid-structure interaction. This fact is exemplified for Sretenski’s problem on the dynamic damper as well as for coupled eigenoscillations of a water tower with a conical elevated tank. c © Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

By employing the nonlinear multimodal technique, the paper proposes an algorithm and computer code (SLOSHER) for derivation of asymptotic [third-order] nonlinear modal systems describing the nonlinear liquid sloshing in a vertical circular cylindrical tank. Applicability of the code is demonstrated for the benchmark modal system by Lukovsky (1990). A novel nonlinear modal system which takes into account most principal third-order modes is derived. c ⃝ Gavrilyuk, Hermann, Lukovsky, Ovchynnykov, Timokha

PurposeThe main purpose of this paper is to develop two efficient and accurate numerical analytical methods for engineering computation of natural sloshing frequencies and modes i the case of truncated circular conical tanks.Design/methodology/approachThe numerical‐analytical methods are based on a Ritz Treftz variational scheme with two distinct analytical harmonic functional bases.FindingsComparative numerical analysis detects the limit of applicability of variational methods in terms of the semi‐apex angle and the ratio between radii of the mean free surface and the circular bottom. The limits are caused by different analytical properties of the employed functional bases. However, parallel use of two or more bases makes it possible to give an accurate approximation of the lower natural frequencies for relevant tanks. For V‐shaped tanks, dependencies of the lowest natural frequency versus the semi‐apex angle and the liquid depth are described.Practical implicationsThe methods provide the natural sloshing frequencies for V‐shaped tanks that are valuable for designing elevated containers in seismic areas. Approximate natural modes can be used in derivations of nonlinear modal systems, which describe a resonant coupling with structural vibrations.Originality/valueAlthough variational methods have been widely used for computing the natural sloshing frequencies, this paper presents their application for truncated conical tanks for the first time. An original point is the use of two distinct functional bases.