Consideration Of Nonlinear Terms Research Articles

Application of Laplace iteration method to study of nonlinear vibration of laminated composite plates

In this paper, free vibration characteristics of laminated composite plates are investigated. A model is developed for a composite layer based on the consideration of non-linear terms in von-Karman’s non-linear deformation theory. The governing partial equation of motion is reduced to an ordinary non-linear equation and then solved using LIM method. The variation of frequency ratio of the Isotropic and composite plates is brought out considering parameters such as aspect ratio, fiber arrangements (orientation), number of layers, and modal ratios.

Large deflections of rectangular plates

Approximation of large, non-linear deflections of rectangular plates by a Fourier cosine series and evaluation of the resulting energy expressions leads to highly accurate results for maximum plate deflection. Consideration of non-linear terms which arise in the expression for membrane energy leads to a stiffer plate behavior resulting in considerably smaller deflections than can be accounted for by the classical small-deflection theory and by some modern approaches to largedeflection analysis. Results compare favorably to infinite element analysis results. Various boundary conditions are considered.

A simple model for Poincar� self-stresses

The classical theory of elementary charges as originally proposed by Abraham and Lorentz included implicitly a stability problem, which was first discussed by Poincare in a very artificial way. It is shown that the consideration of nonlinear terms in the field Lagrangian densities provides the necessary cohesive forces in a natural way. Some examples of particle models are considered.

Nonequilibrium quantum statistics; Application to the laser

A method is presented for describing a general nonequilibrium system in contact with a reservoir in terms of the correlation functions of its quantized field operators. Equations of motion for these correlation functions are derived for a system of multilevel moving atoms interacting with the radiation field, which interacts in turn with the reservoir system. In an appropriate limit these equations are shown to include the usual rate equations for the level distributions. A simple and rigorous description of the influence of a cavity and an optical pump is derived and other types of reservoir coupling are briefly discussed. This description is then applied to a model of a gas laser. The breakdown of the linear theory at the usual lasing threshold suggests consideration of nonlinear terms, which are developed in an expansion in the field strength. Using only the first nonlinear term we find the equations of Lamb by examining the stability of pure modes of the radiation field. An analysis of the incoherent part of the field shows, however, that the existence of a pure mode is precluded by the presence of spontaneous emission of radiation. Nevertheless a value of the linewidth can be plausibly derived from this calculation and is found to be half the Townes value with correction terms. A further discussion of the laser is presented in which the presence of a pure mode is not assumed. It is shown that while a simple incoherent perturbation theory gives divergences, a more self-consistent calculation taking partial coherence into account gives the same results made plausible by pure mode theory.

Influence of viscosity and thermal conductivity on propagation of sound impulses

The first investigation of sound wave attenuation in a viscous gas is due to Stokes, the effect of thermal conductivity was taken into account later by Kirchhoff. A quite complete presentation of their results is available in the classic monograph of Lord Rayleigh [1], From the very beginning Stokes, Just as Kirchhoff, used the linear equations of acoustics as the starting point, and not the exact equations which describe the motion of continuous media.In his studies of propagation of olane sound impulses in an ideal gas (without viscosity and thermal conductivity) Crussard showed [2] that asymptotic relationships of shock wave decay at large distances from the location of their origin are different from those for accoustic waves. Correct derivation of these relationships is not possible without consideration of non-linear terms in equations of gas dynamics. Extension of Crussard's theory to cylindrical and spherical shock waves was given by Landau [3], Khristianovich [4], Sedov [5] and Whitham [6] using different methods.It follows from the paper of Taylor [7] that the structure of weak shock waves is determined basically by corrective processes, related to the non-linear nature of Navier-Stokes equations, and by dissipation of energy at the expense of viscosity and thermal conductivity of real media. Therefore it appeared natural that both factors mentioned will influence the propagation of sound impulses to the same extent. This point of view was expressed by Lighthill [8]. He made a detailed analysis of this concept using plane motion as his example.The decay of perturbations in cylindrical and spherical sound impulses is examined below. It turns out that the structure of waves and asymptotic relationships of their decay when time t → ∞ are related to effects of viscosity and thermal conductivity. At this stage of the process, consideration of nonlinear terms in the Navier-Stokes equations may be neglected because their influence on the formation of the flow field is negligibly small. Variation of all gas parameters within the impulses occurs smoothly, shock waves are absent. Conversely, the motion of shock waves, as long as their width is much smaller than the general length of the wave, is determined by nonlinear convective terms of equations of gas dynamics.When t → ∞ the change of the maximum value of the excess pressure in N-waves, with consideration of viscosity and thermal conductivity, is inversely proportional to ∼ for motions with axial symmetry and to t2 for cenrally symmetric motions. The assertion of Lighthill [8] that asymptotic relationships of decay of perturbations must be exponential, turned out to be incorrect; the excess pressure varies according to an exponential law only in periodic sound waves with fixed wave length [1]. The conclusions obtained are based on a generalization of analysis of short waves carried out by Khristianovich [4] for unsteady one-dimensional motion of an ideal gas.