Exact Analytic Solutions of the Nonlinear Long-Wave Equations in the Case of Axisymmetric Fluid Vibrations in a Parabolic Rotating Vessel

A class of exact analytic solutions of the system of nonlinear long-wave equations is found. This class corresponds to the axisymmetric vibrations of an ideal incompressible homogeneous fluid in a rotating vessel in the shape of a paraboloid of revolution. The radial velocity of these motions is a linear function, and the azimuthal velocity and free surface displacements are polynomials in the radial coordinate with time-dependent coefficients. The nonlinear vibration frequency is equal to the frequency of the lowest mode of linear axisymmetric standing waves in the parabolic vessel.     

Transition process in a fluid layer in a rotating paraboloid in response to a sudden change in its angular velocity

The process of establishment of the rigid-body rotation state in a liquid layer in a rotating paraboloid upon a sudden increase in its angular velocity is studied theoretically and experimentally. The theoretical study is performed within the framework of linear shallow-water theory with account for the bottom Ekman friction. Analytical solutions describing the transition process are obtained and the dependence of the establishment time on the liquid depth and the radius of curvature of the paraboloid is investigated. It is shown that the effect of free surface deformation may lead to a significant increase in the establishment time. Good agreement with the results of special laboratory experiments is found.     

Nonlinear oscillations of a charged drop levitated in gravity and electrostatic fields

Analytically, on the basis of asymptotic methods, the problem of the nonlinear oscillations of a charged ideal incompressible electroconductive fluid drop levitated at rest in gravity and homogeneous electrostatic fields is solved in the quadratic approximation in two small parameters: the initial drop shape deformation amplitude and the stationary eccentricity of the equilibrium drop shape in the electrostatic field. The calculations are performed in fractional powers of the nonlinear oscillation amplitude. The nonlinear corrections to the oscillation frequencies are always negative and already present in the second-order approximation due to the stationary deformation of the drop in the external fields rather than nonlinear interaction between the modes. In the case considered, in contrast to the nonlinear oscillations of a free charged drop, the expression for the generator of the nonlinearly oscillating drop shape contains terms proportional to the oscillation amplitude to the power 3/2.     

Approximate Analytical Solution of the Nonlinear Diffusion Equation for Arbitrary Boundary Conditions

A general approximation for the solution of the one-dimensional nonlinear diffusion equation is presented. It applies to arbitrary soil properties and boundary conditions. The approximation becomes more accurate when the soil-water diffusivity approaches a delta function, yet the result is still very accurate for constant diffusivity suggesting that the present formulation is a reliable one. Three examples are given where the method is applied, for a constant water content at the surface, when a saturated zone exists and for a time-dependent surface flux.     

Nonlinear and chaotic oscillations of a constrained cantilevered pipe conveying fluid: A full nonlinear analysis

In this paper, the planar dynamics of a nonlinearly constrained pipe conveying fluid is examined numerically, by considering the full nonlinear equation of motions and a refined trilinear-spring model for the impact constraints—completing the circle of several studies on the subject. The effect of varying system parameters is investigated for the two-degree-of-freedom (N=2) model of the system, followed by less extensive similar investigations forN=3 and 4. Phase portraits, bifurcation diagrams, power spectra and Lyapunov exponents are presented for a selected set of system parameters, showing some rather interesting, and sometimes unexpected, results. The numerical results are compared with experimental ones obtained previously. It is found that in the parameter space that includesN, there exists a subspace wherein excellent qualitative, and reasonably good (N=2) to excellent (N=4) quantitative agreement with experiment. In the latter case, excellent agreement is not only obtained in the threshold flow velocities (u) for the key bifurcations, but the inclusion of the nonlinear terms improves agreement with experiment in terms of amplitudes of motion and by capturing features of behaviour not hitherto predicted by theory.     

Analytical asymptotic solution of the problem of the nonlinear oscillations of a thick charged jet of viscous fluid

On the basis of second-order analytical asymptotic calculations the problem of the nonlinear oscillation of a uniformly charged jet of an electrically conductive viscous fluid is solved with account for the radial fluid velocity distribution in the jet. It is found that energy from the initially excited wave is gradually transmitted to a wave with double the wave number excited due to nonlinear interaction in the viscous jet, whereas in an ideal fluid the nonlinear correction amplitude is formed instantaneously at the initial instant.     

Tangential oscillations of a differentially rotating spherical fluid layer

The natural harmonic oscillations of a differentially rotating fluid layer under the action of a potential force are considered. The rise in the layer level is assumed to be negligible. The oscillations satisfy an ordinary second-order differential equation with singular coefficients that depend on the spatial coordinate. This equation is solved by the method of local separation of singularities based on the use of the properties of the Fuchs series for a bounded solution. Various laws for the latitude dependence of the angular rate of ocean rotation and the effect of these laws on the problem spectrum are considered. An equation is obtained for the streamlines of the oscillations investigated. Two cases in which the latitude dependence of the base flow velocity coincides with the real dependence for a celestial body are considered and the corresponding modes are found.     

NONLINEAR THEORY OF MULTILAYER SANDWICH SHELLS AND ITS APPLICATION (Ⅰ)──GENERAL THEORY

In this paper, a nonlinear theory is given for multilayer sandwich shells undergoing small strains and moderate rotations. Then a simplified theory for the shells undergoing moderate or moderate/small rotations are obtained.     

Three-dimensional oscillations of a cantilever pipe conveying fluid

The aim of the study described in this paper is to investigate the two-dimensional (2-D) and three-dimensional (3-D) flutter of cantilevered pipes conveying fluid. Specifically, by means of a complete set of non-linear equations of motion, two questions are addressed: (i) whether for a system losing stability by either 2-D or 3-D flutter the motion remains of the same type as the flow velocity is increased substantially beyond the Hopf bifurcation precipitating the flutter; (ii) whether the bifurcational behaviour of a horizontal system and a vertical one (sufficiently long for gravity to have an important effect on the dynamics) are substantially similar. Stability maps and tables are used to delineate areas in a flow velocity versus mass parameter plane where 2-D or 3-D motions occur, and limit-cycle motions are illustrated by phase-plane plots, PSDs and cross-sectional diagrams showing whether the motion is circular (3-D) or planar (2-D).     

On nonlinear oscillations of a thin bar

Summary Based on one of the simplest mathematical model of a solid, nonlinear interactions between waves in a rectilinear bar are investigated, in order to reveal and display a number of dynamic properties inherent not only to the bar, but also to most weakly nonlinear mechanical systems with internal resonances. The presence of internal resonances in the bar is twofold. Firstly, there exists a slow periodic energy exchange between the longitudinal and the two quasi-harmonic bending waves involved in the resonant triad due to the phase matching, secondly, triple-frequency envelope solitons can be created from the resonant triad with the same modal state. The paper investigates the evolution of waves in the bar with the aim to classify the elementary type of wave triplet resonant interactions and define their existence and coesistence areas.The research described here has been made possible in part by Grant N R9B000 from the International Science Foundation. The authors would like to thank Professor G.A. Maugin for having sent copies of his papers, in particular [23], as well as for his permanent interest in our work.     

Nonlinear Capillary Oscillations of a Volumetrically Charged Dielectric Drop

For the axisymmetric capillary oscillations of a charged dielectric fluid drop an expression describing the shape of the generating surface of the drop as a function of time is obtained in the quadratic approximation in the amplitude of an arbitrary initial deformation of its spherical equilibrium shape. It is shown that in contrast to a perfectly conducting charged drop there is no displacement of the drop charge center during oscillation and, hence, such a drop cannot be a source of dipole electromagnetic radiation like a conducting drop in the quadratic approximation.     

Surface waves induced by external periodic pressure in a fluid with an uneven bottom

The behavior of waves generated by periodic pressure on the free surface is considered within the linear shallow-water theory. The fluid depth is a piecewise-constant function, which implies the presence of a finite-size bottom trench or elevation. For an arbitrary shape of bottom unevenness, the solution of the problem reduces to a system of integral boundary equations. Manifestation of wave-guiding properties of bottom unevenness is illustrated by an example of an extended rectangular elevation.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 70–77, January–February, 2005.     

Nonlinear vibration and stability analyses on the basis of a kinetic stability theory for arbitrarily curved rods

Summary After formulating nonlinear kinetic stability equations for curved and twisted rods, these are used for the solution of static and kinetic lateral buckling problems as well as for linear and nonlinear stability analysis of parametrically excited vibrations of circular arches. Accepted for publication 26 May 1997     

Nonlinear Oscillations of a Nonresonant Cable under In-Plane Excitation with a Longitudinal Control

The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.     

Effect of gravity on nonlinear oscillations of a horizontal,immovable-end beam

A critical problem in designing large structures for space applications, such as space stations and parabolic antennas, is the limitation of testing these structures and their substructures on earth. These structures will exhibit very high flexibilities due to the small loads expected to be encountered in orbit. It has been reported in the literature that the gravitational sag effect under dead weight is of extreme importance during ground tests of space-station structural components [1–4]. An investigation of a horizontal, pinned-pinned beam with complete axial restraint and undergoing large-amplitude oscillations about the statically deflected position is presented here. This paper presents a solution for the frequency-amplitude relationship of the nonlinear free oscillations of a horizontal, immovable-end beam under the influence of gravity.The governing equation of motion used for the analysis is the Bernoulli-Euler type modified to include the effects of mid-plane stretching and gravity. Boundary conditions are simply supported such that at both ends there is no bending moment and no transverse and axial displacements. These boundary conditions give rise to an initial tension in the statically deflected shape. The displacement function consists of an assumed space mode using a simple sine function and unknown amplitude which is a function of time. This assumption provides for satisfaction of the boundary conditions and leads to an ordinary differential equation which is nonlinear, containing both quadratic and cubic functions of the amplitude. The perturbation method of multiple scales is used to provide an approximate solution for the fundamental frequency-amplitude relationship.Since the beam is initially deflected the small-amplitude fundamental natural frequency always increases relative to the free vibration situation provided in zero gravity. The nonlinear equation provides for interactions between frequency and amplitude in that both hardening and softening effects arise. The coefficient of the quadratic term in the nonlinear equation arises from the static (dead load) portion of the deflection. This quadratic term, depending upon its magnitude, introduces a softening effect that overcomes the hardening term (due to initial axial tension developed by deflection) for large slenderness ratios.For very large slender, immovable-end beams, the fundamental natural frequency is greater than that of beams without axial constraints undergoing small amplitude oscillations. This phenomenon is attributed to the stiffening effect of the statically-induced axial tension. However, the stiffening effect of axial tension in beams with slenderness ratios greater than approximately 392 undergoing large-amplitude symmetric-mode oscillations is overpowered by the presence of gravitational loading.Nomenclature A amplitude of the first harmonic - A ₁ cross-sectional area of beam - a(t) vibratory amplitude - E Young's modulus - g acceleration due to gravity - g ₁,g ₂,g ₃ constants defined in equations (8) - I area moment of inertia of cross-section - L length of beam - N axial tension force induced by gravitational loading     

Special class of solutions of the kinetic equation of a bubbly fluid

A new class of solutions is constructed for the kinetic model of bubble motion in a perfect fluid proposed by Russo and Smereka. These solutions are characterized by a linear relationship between the Riemann integral invariants. Using the expressions following from this relationship, the construction of solutions in the special class is reduced to the integration of a hyperbolic system of two differential equations with two independent variables. Exact solutions in the class of simple waves are obtained, and their physical interpretation is given.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 33–43, March–April, 2005.     

Nonlinear vibration analysis of viscoelastic plates based on a refined Timoshenko theory

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 120–131, May 2006.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. I: Primary resonance

Nonlinear coupling between torsional and both in-plane and out-of-plane flexural motion is examined for inextensional beams (or beam-like structures) whose torsional and flexural eigenfrequencies are of the same order. The analysis presented here is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities, and account for torsional dynamics. Response characteristics, including stability, are determined for cantilever beams subjected to a lateral periodic excitation. The beam's response in the presence of a one-to-one internal resonance involving a torsional frequency and an in-plane bending frequency is investigated in detail.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. II: Combination resonance

In Part I of this work nonlinear coupling between torsional motion and both in-plane and out-of-plane flexural motion was examined for inextensional beams in the presence of a one-to-one internal resonance. Here the nonlinear response of the system considered in Part I is investigated for the case of an internal combination resonance involving modes associated with bending in two directions and torsion. The analysis presented is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities and account for torsional dynamics.     

Simulation of parametric ship roll and hull twist oscillations

A new mechanical model for simulating both the ship oscillations and the induced twisting of the hull in the case of longitudinal seas is presented. Particular attention is given to the onset of parametric rolling, which may result from non-linearly coupled heave-pitch-roll motions. It is shown that in these sea conditions the phenomenon of twisting is likely to occur under a mechanism similar to that of parametric rolling.