On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

In this paper an initial-boundary value problem for a weakly nonlinear plate equation with a quadratic nonlinearity will be studied. This initial-boundary value problem can be regarded as a simple model describing free oscillations of a simply supported square plate on an elastic foundation. It is assumed that the foundation has a different behavior for compression and for expansion. An approximation for the solution of the initial-boundary value problem will be constructed using a two-timescales perturbation method. The existence and uniqueness of the solution of the problem will be proved. Also the asymptotic validity of the constructed approximations will be shown on long timescales. For specific parameter values, it turns out that complicated internal resonances occur.     

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.     

On Approximations of First Integrals for a System of Weakly Nonlinear,Coupled Harmonic Oscillators

In this paper a system of weakly nonlinear, coupled harmonic oscillatorswill be studied. It will be shown that the recently developedperturbation method based on integrating vectors can be used toapproximate first integrals and periodic solutions. To show how thisperturbation method works the method is applied to a system of weaklynonlinear, coupled harmonic oscillators with 1:3 and 3:1 internalresonances. Not only approximations of first integrals will be given,but it will also be shown how, in a rather efficient way, the existenceand stability of time-periodic solutions can be obtained from theseapproximations. In addition some bifurcation diagrams for a set ofvalues of the parameters will be presented.     

Nonlinear Vibrations of a Simply Supported Rectangular Metallic Plate Subjected to Transverse Harmonic Excitation in the Presence of a One-to-One Internal Resonance

The nonlinear response of rectangular and square metallic plates subjectto transverse harmonic excitations is studied. The nonlinearitiesoriginate from the use of Von Kármán strains. The method of multiplescales is used to solve the system of differential equationsapproximately. Frequency response curves are presented for both squareand rectangular plates for primary resonance of either mode in thepresence of a one-to-one internal resonance. Stability of steady statesolutions is investigated. Bifurcation points and their types arediscussed.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

弹性地基上四边自由矩形厚板非线性大挠度弯曲

基于Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｌｉｎ一阶剪切变形板理论,采用摄动－Ｇａｌｅｒｋｉｎ混合法,给出双参数弹性地基上四边自由矩形中厚板在对称分布局部荷载作用下的大挠度弯曲渐近解,满足全部自由边界条件和控制方程,同时讨论弹性地基刚度系数对自由矩形厚板大挠度弯曲的影响。     

Nonstationary Vibrations of a String with Time-Varying Length and a Mass-Spring Attached at the Lower End

The purpose of this paper is to study the nonstationary vibration of a string with time-varying length and a mass-spring system attached at the lower end. The string is hung vertically and excited sinusoidally by a horizontal displacement at its upper end. The mass is supported by a guide spring horizontally and has two-degrees-of-freedom, vertical and horizontal. It is shown analytically that axial velocity of the string influences the peak amplitude of the string vibration at the passage through resonances. Moreover, it is shown numerically that the amplitudes of both the string and the mass vibrations depend on the sign of the axial velocity, when the natural frequency of the mass-spring system is close to the frequency of the excitation. The above two theoretical results are confirmed experimentally with a simple experimental setup.     

On the simulation of vibrations of industrial gear drives (Complex interactions of physics,mathematics, numerics,and experiments)

Summary In the design stage of a gear drive with spur or helical gears, there is need for a reliable simulation of the vibrations, mainly because of their sound generation. This topic is discussed in view of problems addressed in the subtitle. Enclosure Methods were instrumental in the discovery of paradoxes in the interchange of simulation and computation. These paradoxes are partly due to the employment of standard numerical methods in conjunction with certain commercial compilers. The paper is confined to the case of the absence of imperfections of all kinds in the gear drive.This paper is dedicated to Professor Dr-Ing. Dr-Ing. e.h. Gustav Niemann, former Professor of Machine Elements at the Technical University of Munich, in view of his original and essential contributions to the science of gears.     

On the Resonant Generation of Weakly Nonlinear Stokes Waves in Regions with Fast Varying Topography and Free Surface Current

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｉｎｔｅｒａｃｔｉｏｎｏｆｓｕｒｆａｃｅｗａｔｅｒｗａｖｅｓｗｉｔｈａｍｂｉｅｎｔｃｕｒｒｅｎｔｓａｎｄｕｎｄｕｌａｔｉｎｇｓｅａｂｅｄｔｏｐｏｇｒａｐｈｙｉｓｏｆｆｕｎｄａｍｅｎｔａｌｉｍｐｏｒｔａｎｃｅｔｏｃｏａｓｔａｌｅｎｇｉｎｅｅｒｓａｎｄｓｅｄｉｍｅｎｔｏｌｏｇｉｓｔｓ.Ｆｏｒｅｘａｍｐｌｅ,ｔｈｅｒｅｓｏｎａｎｔｇｅｎｅｒａｔｉｏｎｏｆｓｕｒｆａｃｅｗａｖｅｓｉｎａｃｕｒｒｅｎｔｏｖｅｒｏｔｈｅｒｔｉｄａｌｌｙｏｒｗａｖｅｉｎｄｕｃｅｄｂｅｄｆｏｒｍｓ,ｓ…     

On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

On aspects of damping for a vertical beam with a tuned mass damper at the top

In this paper, the wind-induced, horizontal vibrations of a vertical Euler–Bernoulli beam will be considered. At the top of the beam, a tuned mass damper (TMD) has been installed. The horizontal vibrations can be described by an initial-boundary value problem. Perturbation methods will be applied to construct approximations of the solutions of the initial-boundary value problem, and it will be shown that the TMD uniformly damps the oscillation modes of the beam. In the analysis, it will be assumed that damping, wind-force, and gravity effects are small but not negligible.     

The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter

This paper studies the necessary and sufficient condition of uniformly convergent difference scheme for the elliptic-parabolic partial differential equation with a small parameter.Communicated by Lin Zong-chi.     

A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

In this article, we present a higher‐order finite volume method with a ‘Modified Implicit Pressure Explicit Saturation’ (MIMPES) formulation to model the 2D incompressible and immiscible two‐phase flow of oil and water in heterogeneous and anisotropic porous media. We used a median‐dual vertex‐centered finite volume method with an edge‐based data structure to discretize both, the elliptic pressure and the hyperbolic saturation equations. In the classical IMPES approach, first, the pressure equation is solved implicitly from an initial saturation distribution; then, the velocity field is computed explicitly from the pressure field, and finally, the saturation equation is solved explicitly. This saturation field is then used to re‐compute the pressure field, and the process follows until the end of the simulation is reached. Because of the explicit solution of the saturation equation, severe time restrictions are imposed on the simulation. In order to circumvent this problem, an edge‐based implementation of the MIMPES method of Hurtado and co‐workers was developed. In the MIMPES approach, the pressure equation is solved, and the velocity field is computed less frequently than the saturation field, using the fact that, usually, the velocity field varies slowly throughout the simulation. The solution of the pressure equation is performed using a modification of Crumpton's two‐step approach, which was designed to handle material discontinuity properly. The saturation equation is solved explicitly using an edge‐based implementation of a modified second‐order monotonic upstream scheme for conservation laws type method. Some examples are presented in order to validate the proposed formulation. Our results match quite well with others found in literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear Vibration Control of a System with Dry Friction and Viscous Damping Using the Saturation Phenomenon

Application of saturation to provide active nonlinear vibration control was introduced not long ago. Saturation occurs when two natural frequencies of a system with quadratic nonlinearities are in a ratio of around 2:1 and the system is excited at a frequency near its higher natural frequency. Under these conditions, there is a small upper limit for the high-frequency response and the rest of the input energy is channeled to the low-frequency mode. In this way, the vibration of one of the degrees of freedom of a coupled 2 degrees of freedom system is attenuated. In the present paper, the effect of dry friction on the response of a system that implements this vibration absorber is discussed. The system is basically a plant with a permanent magnet DC (PMDC) motor excited by a harmonic forcing term and coupled with a quadratic nonlinear controller. The absorber is built in electric circuitry and takes advantage of the saturation phenomenon. The method of multiple scales is used to find approximate solutions. Various response regimes of the closed-loop system as well as the stability of these regimes are studied and the stability boundaries are obtained. Especial attention is paid on the effect of dry friction on the stability boundaries. It is shown that while dry friction tends to shrink the stable region in some parts, it enlarges other parts of the stable region. To verify the theoretical results, they have been compared with numerical solution and good agreement between the two is observed.This work was done while the authors were associated with the Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran.     

On the Bending, Vibration and Stability of Laminated Rectangular Plates with Transversely Isotropic Layers

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On the Reynolds-number independence of mixed convection in a vertical channel subjected to asymmetric wall temperatures with and without flow reversal

The problem of mixed convection in a vertical channel with asymmetric wall temperatures including situations of flow reversal is studied numerically. The SIMPLER algorithm with a staggered grid system is employed to solve the corresponding numerical equations formulated by the finite-volume method. A second-order upwind scheme is used to model the convective term, and a suitable grid distribution is introduced. The ranges of the parameters studied are 0 r_t 1, 1 Re 1000, and 0 Gr/Re 500.

The numerical results, with the streamwise coordinate scaled by the Reynolds number (Re), show that solutions for the velocity and temperature fields are independent of the Reynolds number when Re 50, even in the presence of flow reversal. These solutions, however, are dependent on r_t and Gr/Re. Subsequently, correlations are proposed for the bulk temperature distribution and the local Nusselt numbers along the hot wall and the cold wall.     

On the regularization method of the first kind of Fredholm integral equation with a complex kernel and its application

The regularized integrodifferential equation for the first kind of Fredholm integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-demensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given. Project supported by the National Natural Science Foundation, of China     

Reconstructing the transient,dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever-present throughout the fields of science and engineering. While the prediction of the transient dynamics of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dissipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to reconstruct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions are in excellent agreement with exact trajectories of displacement and velocity determined via numerical integration of the governing equation. A wide range of system parameters and initial conditions are utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory.