Simplified nonlinear theory of the dielectric loaded rectangular Cerenkov maser

To rapidly and accurately investigate the performance of the dielectric loaded rectangular Cerenkov maser, a simplified nonlinear theory is proposed, in which the variations of wave amplitude and wave phase are determined by two coupled first-order differential equations. Through combining with the relativistic equation of motion and adopting the forward wave assumption, the evolutions of the forward wave power, the power growth rate, the axial wave number, the accumulated phase offset, and the information of the particle movement can be obtained in a single-pass calculation. For an illustrative example, this method is used to study the influences of the beam current, the gap distance between the beam and the dielectric surface, and the momentum spread on the forward wave. The variations of the saturated power and the saturation length with the working frequency for the beams with different momentum spreads have also been studied. The result shows that the beam wave interaction is very sensitive to the electron beam state. To further verify this simplified theory, a comparison with the result produced from a rigorous method is also provided, we find that the evolution curves of the forward wave power predicted by the two methods exhibit excellent agreement. In practical applications, the developed theory can be used for the design and analysis of the rectangular Cerenkov maser.

THE RE-EXAMINATION OF THE WEAKLY NONLINEAR THEORY OF HYDRODYNAMIC STABILITY

The weakly nonlinear theory has been widely applied in the problem of hydrodynamic stability and also in other fields. However, although its application has been successful for some problems, yet, for other problems, the results obtainedhre not satisfactory, especially for problems like transition or the evolution of the vortex in the free shear flow, for which the goal of the theoretical investigation is not seeking for a steady state, but predicting an evolutional process. In this paper, we shall examine the reason for the unsuccessfulness and suggest ways for its amendment.

Monitoring the change in horizontal stress with multi-wave time-lapse seismic response based on nonlinear elasticity theory

Monitoring the change in horizontal stress from the geophysical data is a tough challenge, and it has a crucial impact on broad practical scenarios which involve reservoir exploration and development, carbon dioxide (CO_(2)) injection and storage, shallow surface prospecting and deep-earth structure description. The change in in-situ stress induced by hydrocarbon production and localized tectonic movements causes the changes in rock mechanic properties (e.g. wave velocities, density and anisotropy) and further causes the changes in seismic amplitudes, phases and travel times. In this study, the nonlinear elasticity theory that regards the rock skeleton (solid phase) and pore fluid as an effective whole is used to characterize the effect of horizontal principal stress on rock overall elastic properties and the stress-dependent anisotropy parameters are therefore formulated. Then the approximate P-wave, SV-wave and SH-wave angle-dependent reflection coefficient equations for the horizontal-stress-induced anisotropic media are proposed. It is shown that, on the different reflectors, the stress-induced relative changes in reflectivities (i.e., relative difference) of elastic parameters (i.e., P- and S-wave velocities and density) are much less than the changes in contrasts of anisotropy parameters. Therefore, the effects of stress change on the reflectivities of three elastic parameters are reasonably neglected to further propose an AVO inversion approach incorporating P-, SH- and SV-wave information to estimate the change in horizontal principal stress from the corresponding time-lapse seismic data. Compared with the existing methods, our method eliminates the need for man-made rock-physical or fitting parameters, providing more stable predictive power. 1D test illustrates that the estimated result from time-lapse P-wave reflection data shows the most reasonable agreement with the real model, while the estimated result from SH-wave reflection data shows the largest bias. 2D test illustrates the feasibility of the proposed inversion method for estimating the change in horizontal stress from P-wave time-lapse seismic data.

Linear and Nonlinear Stokes Waves Theory: Numerical Hydrodynamic and Energy Studies

The increase of wave energy in electricity production is an objective shared by many countries to meet growing demand and global warming. To analyze devices capable of converting the energy of sea waves into electrical energy, it is important to master the various theories of gravity waves and generation. We will in our work consider a numerical waves tank for an amplitude A=0.5, a wavelength λ=0.25 , an average height He=10 and a Froude number fixed at 1 × 105. Numerical wave channel analysis is used to reproduce the natural phenomenon of wave propagation in an experimental model. Wave makers are usually used to generate waves in the channel. In theory, the influence of an incident wave can be considered, as in the case of our study. In this study, the evolution of the hydrodynamic parameters and the energy transported in one wavelength can be determined by calculation. A change of variable will be done in this work to facilitate the writing of the boundary conditions at the free surface and at the bottom. The nonlinear Stokes theory will be studied in this case in order to provide hydrodynamic solutions through the Navier-Stokes equations to finally deduce the energetic results. To do this, the finite difference method will be used for the hydrodynamic results such as the velocity potential and the free surface elevation and the trapezium method of Newton for the energetic results. Thus, we will determine the energetic potential according to the decrease in the slope of the tank. To do this, we will take as values of beta representing the inverse of the slope of the tank, β=100, β=105, β=110 and β=105. .

Nonlinear dynamics of the reversed shear Alfvén eigenmode in burning plasmas

In a tokamak fusion reactor operated at steady state,the equilibrium magnetic field is likely to have reversed shear in the core region,as the noninductive bootstrap current profile generally peaks off-axis.The reversed shear Alfvén eigenmode(RSAE)as a unique branch of the shear Alfvén wave in this equilibrium,can exist with a broad spectrum in wavenumber and frequency,and be resonantly driven unstable by energetic particles(EP).After briefly discussing the RSAE linear properties in burning plasma condition,we review several key topics of the nonlinear dynamics for the RSAE through both wave-EP resonance and wave-wave coupling channels,and illustrate their potentially important role in reactor-scale fusion plasmas.By means of simplified hybrid MHD-kinetic simulations,the RSAEs are shown to have typically broad phase space resonance structure with both circulating and trapped EP,as results of weak/vanishing magnetic shear and relatively low frequency.Through the route of wave-EP nonlinearity,the dominant saturation mechanism is mainly due to the transported resonant EP radially decoupling with the localized RSAE mode structure,and the resultant EP transport generally has a convective feature.The saturated RSAEs also undergo various nonlinear couplings with other collective oscillations.Two typical routes as parametric decay and modulational instability are studied using nonlinear gyrokinetic theory,and applied to the scenario of spontaneous excitation by a finite amplitude pump RSAE.Multiple RSAEs could naturally couple and induce the spectral energy cascade into a low frequency Alfvénic mode,which may effectively transfer the EP energy to fuel ions via collisionless Landau damping.Moreover,zero frequency zonal field structure could be spontaneously excited by modulation of the pump RSAE envelope,and may also lead to saturation of the pump RSAE by both scattering into stable domain and local distortion of the continuum structure.

Nonlinear Kinetic Theory and Pulse Interactions in Phase Transition

The kinetics of nucleation of phase transition is a phenomenal theory.Some new technologies of preparation of nanomaterials,for example,by shock wave and by electropulsing,are pulse interactions.Based on the known nonlinear theories of phase transition,the nonlinear kinetics of phase transition is discussed,and a soliton-like model is proposed. This mathematical method can not only explain the basic characteristics of pulse interactions and suddenness of phase transition, and possesses a consistency of mechanism for nucleation and growth.

A Nonlinear Theory of Elastic Plates without Using Kirchhoff-Love Assumptions and Its Application

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Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube （SWCNT） subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.

Nonlinearity analysis of piezoelectric micromachined ultrasonic transducers based on couple stress theory

This paper studies the static deformation behavior of a piezoelectric micromachined ultrasonic transducer （PMUT） actuated by a strong external electric field. The transducer membrane consists of a piezoelectric layer, a passive layer and two electrode layers. The nonlinearities of the piezoelectric layer caused by electrostriction under a strong electric field are analyzed. Because the thickness of the transducer membrane is on the microscale, the size dependence of the deformation behavior is evaluated using the couple stress theory. The results show that the optimal ratio of the top electrode diameter and the membrane diameter is around 0.674. It is also found that this optimal value does not depend on any other parameters if the thicknesses of the two electrodes are negligible compared with those of the piezo- electric and passive layers. In addition, the nonlinearities of the piezoelectric layer will become stronger along with the increase of the electric field, which means that softening of the membrane stiffness occurs when a strong external electric field is applied. Meanwhile, the optimal thickness ratio for the passive layer and the piezoelectric layer is not equal to 1.0 which is usually adopted by previous researchers. Because there exists size dependence of membrane deforma-tion, the optimal value of this thickness ratio needs to be greater than 1.0 on the microscale.

SOME EXTENDED RESULTS OF“SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

GENERALIZED THEORY OF NONLINEAR AND UNSTEADY MECHANICS AND APPLICATIONS TO PARTICLE PHYSICS

In this paper we consider that the momentum of a free particle motion withhigh-level speed presenting nonlinear effects may be expanded by using Laurent seriesand then obtain the complete expression of nonlinear and unsteady momentum. These nonlinear and unsieady phenoniena of high-level speed may further expand to the theory of kinematics and it may be determined by Fredholm's integral equation of the first kind. In addition, according to the nonlinear and unsteady momentum obtained the relations of the nonlinear mechanics equations .work and energy, mass and energymay be derived.Finaly .this paper also calculates those experimental results which done in particle physics for mu-mesons u±and fast neutrons n, these results are in agreement with data perfectly.

Investigation of Properties of Motion of Superconductive Electrons in Superconductors by Nonlinear Quantum Mechanical Theory

The properties and rules of motion of superconductive electrons in steady and time-dependent non-equilibrium states of superconductors are studied by using the Ginzberg-Landau （GL） equations and nonlinear quantum theory. In the absence of external fields, the superconductive electrons move in the solitons with certain energy and velocity in a uniform system, The superconductive electron is still a soliton under action of an electromagnetic field, but its amplitude, phase and shape are changed. Thus we conclude that superconductivity is a result of motion of soliton of superconductive electrons. Since soliton has the feature of motion for retaining its energy and form, thus a permanent current occurs in superconductor. From these solutions of GL equations under action of an electromagnetic field, we gain the structure of vortex lines-magnetic flux lines observed experimentally in type-Ⅱ superconductors. In the time-dependent nonequilibrium states of superconductor, the motions of superconductive electrons exhibit still the soliton features, but the shape and amplitude have changed. In an invariant electric-field, it moves in a constant acceleration. In the medium with dissipation, the superconductive electron behaves still like a soliton, although its form, amplitude, and velocity are altered. Thus we have to convince that the superconductive electron is essentially a soliton in both non-equilibrium and equilibrium superconductors.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Experimental,Numerical,and Analytical Studies on the Bending of Mechanically Lined Pipe

Mechanically lined pipe(MLP)is often used for offshore oil and gas transport because of its low cost and corrosion resistance.During installation and operation,the pipe may undergo severe bending deformation,which causes the liner to separate from the outer pipe and buckles,affecting the stability of the whole line.In this paper,the buckling response of MLP subjected to bending is investigated to clarify its bending characteristics by employing both experiments,numerical simulation,as theoretical methods.Two types of MLPs were manufactured with GB 45 carbon steel(SLP)and Al 6061(ALP)used as the outer pipe material,respectively.The hydraulic expansion and bending experiments of small-scale MLPs are conducted.In addition to the ovalized shape of the cross-section for the SLP specimens,the copper liner was found to wrinkle on the compressive side.In contrast,the liner of ALP remains intact without developing any wrinkling and collapse mode.In addition,a dedicated numerical framework and theoretical models were also established.It was found both the manufacturing and bending responses of the MLP can be well reproduced,and the predicted maximum moment and critical curvatures are in good agreement with the experimental results.

Finite Element Analysis to Two-Dimensional Nonlinear Sloshing Problems

A two-dimensional nonlinear sloshing problem is analyzed by means of the fully nonlinear theory and time domain second order theory of water waves. Liquid sloshing in a rectangular container Subjected to a horizontal excitation is simulated by the finite element method. Comparisons between the two theories are made based on their numerical results. It is found that good agreement is obtained for the case of small amplitude oscillation and obvious differences occur for large amplitude excitation. Even though, the second order solution can still exhibit typical nonlinear features of nonlinear wave and can be used instead of the fully nonlinear theory.

Research on nonlinear R／S method and its application in earthquake prediction

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The wave-corpuscle properties of microscopic particlesin the nonlinear quantum-mechanical systems

We debate first the properties of quantum mechanics and its difficulties and the reasons resulting in these diffuculties and its direction of development. The fundamental principles of nonlinear quantum mechanics are proposed and established based on these shortcomings of quantum mechanics and real motions and interactions of microscopic particles and backgound field in physical systems. Subsequently, the motion laws and wave-corpuscle duality of microscopic particles described by nonlinear Schr?dinger equation are studied completely in detail using these elementary principles and theories. Concretely speaking, we investigate the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the mechanism and rules of particle collision and the uncertainty relation of particle’s momentum and position, and so on. We obtained that the microscopic particles obey the classical rules of collision of motion and satisfy the minimum uncertainty relation of position and momentum, etc. From these studies we see clearly that the moved rules and features of microscopic particle in nonlinear quantum mechanics is different from those in linear quantum mechanics. Therefore, nolinear quantum mechanics is a necessary result of development of quantum mechanics and represents correctly the properties of microscopic particles in nonlinear systems, which can solve difficulties and problems disputed for about a century by scientists in linear quantum mechanics field.

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.

Nonlinear free vibration of piezoelectric cylindrical nanoshells

The nonlinear vibration characteristics of the piezoelectric circular cylindrical nanoshells resting on an elastic foundation are analyzed. The small scale effect and thermo-electro-mechanical loading are taken into account. Based on the nonlocal elasticity theory and Donnell's nonlinear shell theory, the nonlinear governing equations and the corresponding boundary conditions are derived by employing Hamilton's principle. Then,the Galerkin method is used to transform the governing equations into a set of ordinary differential equations, and subsequently, the multiple-scale method is used to obtain an approximate analytical solution. Finally, an extensive parametric study is conducted to examine the effects of the nonlocal parameter, the external electric potential, the temperature rise, and the Winkler-Pasternak foundation parameters on the nonlinear vibration characteristics of circular cylindrical piezoelectric nanoshells.

THE APPLICATION OF NONLINEAR GAUGE MATHOD TO THE ANALYSIS OF LOCAL FINITE DEFORMATION IN THE NECKING OF CYLINDRICAL BAR

Localized deformation and instability is the focal point of research in mechanics. The most typical problem is the plastic analysis of cylindrical bar neckingand shear band under uniaxial tension. Traditional elasto-plastic mechanics of infinitesimal deformation can not solve this problem successfully. In this paper, on the basis of S(strain) -R(rotation) decomposition theorem, the authors obtain the localstrain distribution and progressive state of axial symmetric finite deformation of cylindrical bar under uniaxial tension adopting nonlinear gauge approximate method and computer modelling technique.