Vibrational Convection in the Hele-Shaw Cell. Theory and Experiment

The influence of high-frequency horizontal vibrations on convection in the Hele-Shaw cell located in a uniform gravity field is considered experimentally and theoretically. Nonlinear regimes of vibrational convection in the supercritical region are examined. It is shown that horizontal vibrations (directed toward the wide sides of the cell) decrease the threshold of quasi-equilibrium stability. Regions of existence of one- and two-vortex steady flows are found, and unsteady regular and random regimes of thermal vibrational convection are considered. New random regimes in the Hele-Shaw cell are found, which result from nonlinear interaction of the “lower” modes responsible for the formation of regular supercritical convective regimes. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 40–48, March–April, 2006.     

Energy analysis in a nonlinear hybrid system containing linear and nonlinear subsystems coupled by hereditary element

Energy transfer between subsystems coupled by standard light hereditary element in hybrid system is very important for different engineering applications, especially for dynamical absorption. An analytical study of the energy transfer between coupled linear and nonlinear oscillators in the free vibrations of a viscoelastically connected double-oscillator system as a new hybrid nonlinear system with two and half degrees of freedom is pointed out. The analytical study shows that the viscoelastic–hereditary connection between oscillators causes the appearance of like two-frequency regimes of subsystem's vibrations and that the energy transfer between subsystems appears. The Lyapunov exponents corresponding to each of two eigenmodes of the hybrid system, as well as to the subsystems are obtained and expressed by using energy of the corresponding eigentime components. The Lyapunov exponents are measures of the vibration processes stability in the hybrid system and in component subsystem vibrations. In Honor of Giuseppe Rega and Fabrizio Vestroni on the Occasion of their 60th Birthday.     

Control of a Nonlinear System Using the Saturation Phenomenon

In this paper, a theoretical investigation of nonlinear vibrations of a 2 degrees of freedom system when subjected to saturation is studied. The method has been especially applied to a system that consists of a DC motor with a nonlinear controller and a harmonic forcing voltage. Approximate solutions are sought using the method of multiple scales. It is shown that the closed-loop system exhibits different response regimes. The nature and stability of these regimes are studied and the stability boundaries are obtained. The effects of the initial conditions on the response of the system have also been investigated. Furthermore, the second-order solution is presented and the corresponding results are compared with those of the first-order solution. It is shown that by increasing the amplitude of the excitation voltage, the higher-order term in the solution becomes significant and causes a drift in the response. In order to verify the obtained theoretical results, they are compared with the corresponding numerical results. Good agreement between the two sets of results is observed.     

An investigation on lateral and torsional coupled vibrations of high power density PMSM rotor caused by electromagnetic excitation

Electromagnetic excitation in high power density permanent magnet synchronous motors (PMSMs) due to eccentricity is a significant concern in industry; however, the treatment of lateral and torsional coupled vibrations caused by electromagnetic excitation is rarely addressed, yet it is very important for evaluating the stability of dynamic rotor vibrations. This study focuses on an analytical method for analyzing the stability of coupled lateral/torsional vibrations in rotor systems caused by electromagnetic excitation in a PMSM. An electromechanically coupled lateral/torsional dynamic model of a PMSM Jeffcott rotor is derived using a Lagrange–Maxwell approach. Equilibrium stability was analyzed using a linearized matrix of the equation describing the system. The stability criteria of coupled torsional–lateral motions are provided, and the influences of the electromagnetic and mechanical parameters on mechanical vibration stability and nonlinear behavior were investigated. These results provide better understanding of the nonlinear response of an eccentric PMSM rotor system and are beneficial for controlling and diagnosing eccentricity.     

Vibrations of shallow shells rectangular in the horizontal projection with two freely supported opposite edges

The exact mode shapes of linear vibrations of a shallow shell rectangular in the horizontal projection with two freely supported opposite edges are obtained. These shapes are used to construct a discretemodel of vibrations of a shallow shell in geometrically nonlinear deformation. The harmonic balance method is used to study the free and forced nonlinear vibrations under internal resonance. The Lyapunov stability of the obtained periodic vibrations is analyzed.     

Influence of periodic excitation on self-sustained vibrations of one disk rotors in arbitrary length journals bearings

Interaction of forced and self-sustained vibrations of one disk rotor is described by nonlinear finite-degree-of-freedom dynamical system. The shaft of the rotor is supported by two journal bearings. The combination of the shooting technique and the continuation algorithm is used to study the rotor periodic vibrations. The Floquet multipliers are calculated to analyze periodic vibrations stability. The results of periodic motions analysis are shown on the frequency response. The quasi-periodic motions are investigated. These nonlinear vibrations coexist with the periodic forced vibrations.     

Control of Thermo- and Solutocapillary Flows in FZ Crystal Growth by High-Frequency Vibrations

The paper deals with the numerical investigation of the possibilities to control convective flows in the liquid bridge in zero gravity conditions applying axial vibrations. The surface tension is assumed to be dependent both on the temperature and on the solute concentration. The free surface deformations and the curvature of the phase change surfaces are neglected but pulsational deformations of the free surface are accounted for. The first part of the paper concerns axisymmetric steady flows. The calculations show that the evolution of convective flow with the variation of thermal Marangoni number at a fixed value of the solutal Marangoni number is accompanied by the hysteresis phenomenon, which is related to the existence of two stable steady regimes in a certain parameter range. One of these regimes is thermocapillary dominated, it corresponds to the two-vortex flow, and the other is solutocapillary dominated, it corresponds to the single-vortex flow. Under vibrations, the range of the Marangoni numbers where hysteresis is observed becomes narrower and is shifted to the area of larger values. The second part of the paper concerns the stability of axisymmetric thermo-and solutocapillary flows and the transition to three-dimensional regimes. Significant mutual influence of flows generated by each process on the stability of the other is discovered. Stability maps in the parametric plane for the thermal Marangoni number, the solutal Marangoni number, are obtained for different values of vibration parameters. It is shown, that vibrations exert a stabilizing effect, increasing critical Marangoni numbers for all modes of instability. However, this effect is different for different modes and at high vibration intensity destabilization is possible. Consequently, vibrations can modify the scenario of the transition to the three-dimensional mode.     

Nonlinear problems of the dynamics of elastic shells partialiy filled with a liquid

Theoretical and experimental investigations of the nonlinear vibrations and dynamic stability of thin shells partially filled with a liquid are reviewed. The paper deals with the basic laws governing the dynamic high-deflection deformation of carrying shell structures and the considerable vibrations of the free liquid surface due to the natural, forced, and parametrically excited vibrations of the combined system and also due to impulse loads acting on the carrying object. The nonlinear dynamic interaction of shells with a liquid filler is analyzed with allowance for the wave motions of the free liquid surface. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 3–34, April, 2000.     

Chaos in Acoustic Subspace Raised by the Sommerfeld–Kononenko Effect

This paper deals with vibrations of an infinite plate in contact with an acoustic medium where the plate is subjected to a point excitation by an electric motor of limited power-supply. The whole system is divided into two “exciter - foundation” and “foundation-plate-medium”. In the system “motor-foundation” three classes of steady state regimes are determined: stationary, periodic and chaotic. The vibrations of the plate and the pressure in the acoustic fluid are described for each of these regimes of excitation. For the first class they are periodic functions of time, for the second they are modulated periodic functions, in general with an infinite number of carrying frequencies, the difference between which is constant. For the last class they correspond to chaotic functions. In another mathematical model where the exciter stands directly on an infinite plate (without foundation) it was shown that chaos might occur in the system due to the feedback influence of waves in the infinite hydro-elastic subsystem in the regime of motor shaft rotation. In this case the process of rotation can be approximately described as a solution of the fourth order nonlinear differential equation and may have the same three classes of steady state regimes as the first model. That is the electric motor may generate periodic acoustic waves, modulated waves with an infinite number of frequencies or chaotic acoustic waves in a fluid.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Studying the harmonic vibrations of a cylindrical shell made of a nonlinear elastic piezoelectric material

The paper examines the harmonic vibrations of an infinitely long thin cylindrical shell made of a nonlinear elastic piezoceramic material and subjected to periodic electric loading. Amplitude-frequency characteristics are plotted for different amplitudes of the load. Points of these characteristics are analyzed for stability. The transients occurring while harmonic vibrations attain the steady state are studied __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 101–106, April 2008.     

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 Vibrations of a Cylindrical Shell Containing a Flowing Fluid

The Bogolyubov-Mitropolsky method is used to find approximate periodic solutions to the system of nonlinear equations that describes the large-amplitude vibrations of cylindrical shells interacting with a fluid flow. Three quantitatively different cases are studied: (i) the shell is subject to hydrodynamic pressure and external periodical loading, (ii) the shell executes parametric vibrations due to the pulsation of the fluid velocity, and (iii) the shell experiences both forced and parametric vibrations. For each of these cases, the first-order amplitude-frequency characteristic is derived and stability criteria for stationary vibrations are established__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 75–84, April 2005.     

Analysis on nonlinear dynamics of a deploying composite laminated cantilever plate

This paper presents the analysis on the nonlinear dynamics of a deploying orthotropic composite laminated cantilever rectangular plate subjected to the aerodynamic pressures and the in-plane harmonic excitation. The third-order nonlinear piston theory is employed to model the transverse air pressures. Based on Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the nonlinear governing equations of motion are derived for the deploying composite laminated cantilever rectangular plate. The Galerkin method is utilized to discretize the partial differential governing equations to a two-degree-of-freedom nonlinear system. The two-degree-of-freedom nonlinear system is numerically studied to analyze the stability and nonlinear vibrations of the deploying composite laminated cantilever rectangular plate with the change of the realistic parameters. The influences of different parameters on the stability of the deploying composite laminated cantilever rectangular plate are analyzed. The numerical results show that the deploying velocity and damping coefficient have great effects on the amplitudes of the nonlinear vibrations, which may lead to the jumping phenomenon of the amplitudes for first-order and second-order modes. The increase of the damping coefficient can suppress the increase of the amplitudes of the nonlinear vibration.     

Nonlinear modes of cylindrical panels with complex boundaries. <Emphasis Type="Italic">R</Emphasis>-function method

Nonlinear vibrations of cylindrical panels with complex base are analyzed. The Donnell-Mushtari-Vlasov equations with respect to displacements are used to study vibrations of shallow shell with geometrical nonlinearity. R-function method is applied to satisfy the panel boundary conditions. The Rayleigh-Ritz method is used to obtain the linear vibrations eigenmodes, which contain R-function. The nonlinear vibrations of panel are expanded by using these eigenmodes. The harmonic balance method and nonlinear normal modes are used to study the free nonlinear vibrations.     

Chatter mitigation using the nonlinear tuned vibration absorber

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature for suppressing machine tool vibrations, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the stability properties of the machining operation, while its nonlinear properties are chosen in order to control the bifurcation behavior of the system and guarantee robustness of stable operation. In this study, the NLTVA is applied to turning machining.     

Influence of Initial Geometric Imperfections on the Vibrations and Dynamic Stability of Elastic Shells

The results from studies into the vibrations and dynamic stability of thin elastic shells with initial geometric imperfections are analyzed. The corresponding dynamic problems are solved in both linear and nonlinear formulations. The influence of initial axisymmetric and nonaxisymmetric deflections on natural, forced, parametrically excited, and self-excited vibrations (flutter) is studied. The dynamic buckling of imperfect shells under short-term impulsive loading is examined. Some aspects of experimental investigation into the vibrations of shells with small geometric imperfections (deviations from the design shape) are considered     

Fractional modified Duffing–Rayleigh system and its synchronization

Nonlinear Dynamics - Chaotic vibrations, stability and synchronization are important topics in nonlinear dynamics, and thus are studied in this paper for a new chaotic system with quadratic and...     

Influence of inhomogeneity on the nonlinear stabilization of acoustic vibrations of a heat releasing medium in a confined space

The problem of determining the amplitudes of unstable acoustic vibrations when they are stabilized in a confined heat releasing medium by transfer of energy from the unstable mode to the damped mode as a result of their nonlinear interaction has been considered by Artamonov and Vorob'ev [1]. Their treatment applied to a gas with volume heat release filling a channel of finite length under the assumption that the parameters of the gas in the steady state are constant over the volume. In the present paper an investigation is made in the nonlinear approximation of the stability of a weakly inhomogeneous heat releasing gas that fills a channel of finite length with respect to acoustic vibrations propagating along its axis in the direction of the gradients of the steady parameters. It is shown that the spatial inhomogeneity of the gas leads to breakdown of the resonance of the excited acoustic vibrations, which in turn leads to a higher level of the steady vibrations compared with the case of a spatially homogeneous medium.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.