Periodic solution of a second order, autonomous, nonlinear system

There exist a sufficient condition for the existence of at least one periodic solution for a type of second order autonomous ordinary differential equations. The correctness of the condition has been pointed out by Schauder's fixed point theorem. In order to indicate the validity of the assumptions made, two illustrative examples, showing its application in the nonlinear vibration and relaxation oscillation are presented.     

A second order solution of the non-linear equations of conical shells

A second order asymptotic solution to the Donnell type non-linear equations of elastic homogeneous conical shells is presented. Closed form solutions for the displacements and stress resultants including twelve constants of integration are obtained by considering the lateral displacement to be of the order of kt where k is a geometric parameter and t is the shell thickness. Terms of order up tot/r in comparison to unity are omitted. The solutions are valid for asymmetric slowly varying edge or surface loads.     

The stability of zero solution of second order system of variable coefficients with four non-linear terms

This paper discusses the following non-linear systems of second order coefficients: This paper is the generalization of works [1] and [2].     

A second order stationary solution of the three dimensional wave diffraction potential

A theory on the second order wave diffraction by a three dimensional body fixed in a regular sea has been developed in the present paper. By regarding the sinusoidal disturb potential as a stationary solution of an initial value problem, and using Laplace transformation method and Tauberian theorem, the boundary value problems of stationary solution of the first and second order diffraction potential have been derived in this paper. Furthermore, the explicit solution of the second order stationary diffraction potential has been obtained with the method of the double Fourier transformation. It is found that the asymptotic behaviour of the second order stationary solution at far field is dependent on two wave systems, the first is “free wave”, travelling independently of the first order wave system, the other is “phase locked waves”, which accompany the first order waves. At the same time, the radiation conditions of the second order diffraction problems are derived. We also find that one can still pursue a steady state formulation with the inclusion of Rayleigh damping. Finally, as an example, the second order wave forces upon a fixed vertical circular cylinder have been calculated, and the numerical results agree well with the experimental data.     

Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution

This work investigates the nonlinear longitudinal forced vibrations of a bar with hysteretic dynamics typical of rocks and man-made geomaterials. The material properties are described by a constitutive relation that includes energy dissipation effects consistently with hysteretic dynamics. The harmonic balance method and a perturbation technique that uses the material's modulus defect as perturbation parameter are employed to solve the equation of motion. The spatial dependence of the modulus defect, which is disregarded in earlier solutions of this problem, is duly accounted for. According to this model, the resonance frequency shift is proportional to the amplitude of the excitation, while the inverse of the quality factor increases as the square root of the latter. Further, the spatial variation of the modulus defect is shown to affect the modulation of the fundamental component along the bar. The nonlinear spectral components are of odd order, with amplitude proportional to the maximum value of the modulus defect and to the square of the excitation's amplitude, and decreases non-monotonically with increasing harmonic order. The motivation for this work is twofold. Analytical models may improve our understanding of the dynamics of hysteretic materials in general, and of the mutual interaction of material defects in particular. Secondly, this work establishes a bench-mark result on a mathematically simple hysteretic system against which alternative mathematical approaches, possibly concerning material with complex constitutive relations, could be tested.     

The domain of convergence for the iterative solution of nonlinear second order boundary value problems

The domain of convergence for the so called Picard iteration scheme ([6] and [7]) of nonlinear, two-point, second order boundary value problems is determined. The general theory is then applied to the study of two well known examples with special boundary conditions. It is explained how the domain of convergence depends on the Lipschitz constants which are involved in the problem.     

Stability of a linear second order system under random parametric excitation

Summary The stability of a linear second order system with small, random, Gaussian parametric excitation will be investigated. It will be shown that if the spectrum of the parameter variation is continuous near twice the natural frequency of the system, stability in mean square is only dependent upon the value of the spectral density at this part of the spectrum.

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Übersicht Es wird die Stabilität eines linearen Systems zweiter Ordnung mit kleiner, zufälliger, gaußischer Parametererregung untersucht. Dabei zeigt sich, daß die Stabilität im quadratischen Mittel nur von dem Wert der Spektraldichte in der Nähe der doppelten Eigenfrequenz des Systems abhängt, sofern das Spektrum der Parametererregung in diesem Bereich stetig ist.

     

A second order solution for the distortion of large scale turbulence around a 2-D obstacle

The problem of large scale (d/L « 1, where d is the dimension of the dimension of the body and L is the integral length scale of turbulence) three dimensional turbulence approaching an arbitrary two dimensional body is analysed following Hunt's (1973) theory. The analysis, based on Rapid Distortion Theory, incorporates both blocking and weak distortion of the velocity field caused by the obstacle. Besides obvious engineering applications, the solution illustrates the application of RDT for inhomogeneous turbulence problems.     

Evaluation of the second order elastic moduli

The expressions of the apparent linear elastic moduli and their first and second derivatives, with respect to hydrostatic pressure, are obtained according to the second order elasticity theory. As a particular case when the material is hyperelastic, formulae of the first derivatives of the linear elastic moduli reduce to those obtained by Seeger and Buck.     

Numerical solution of integrodifferential equations with convolution integrals

Archive of Applied Mechanics - A numerical solution is presented for the solution of integrodifferential equations involving convolution integrals. These equations can be viewed as a generalization...     

Approximate analytical solution in slow-fast system based on modified multi-scale method

A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.     

Numerical solution of incompressible flows by discrete singular convolution

A discrete singular convolution (DSC) solver is developed for treating incompressible flows. Three different two‐dimensional benchmark problems, the Taylor problem, the driven cavity flow, and a periodic shear layer flow, are utilized to test the accuracy, to explore the reliability and to demonstrate the efficiency of the present approach. Solution of extremely high accuracy is attained in the analytically solvable Taylor problem. The results of treating the other problems are in excellent agreement with those in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

The general solution of second order effects in elastic half-space acted upon by a non-uniform shear load

I.IntroductionInfiniteelasticitytheorythemathematicalequationsgoverningthedeformationofanisotropiccompressibleelasticmaterialarehighlynonlinear.Asaresult,theexactsolutionsoftheboundaryvalueproblemshavebeenpossibleonlyinsomerestrictedcasesand,oftenapproxim…     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

An analytical approximated solution and numerical simulations of a non-ideal system with saturation phenomenon

Nonlinear Dynamics - Nowadays, researches about non-ideal problems have been increased considerably in technical-scientific community. Nonlinear problems have been widely studied due to their...     

Exact analytical solution of the MHD Jeffery-Hamel flow problem

This article shows that the well known nonlinear boundary value problem namely MHD Jeffery-Hamel flow problem, investigated in recent years by many numerical and semi-analytical approximative methods, is exactly solvable and furthermore, gives analytical exact solution in the implicit form for further physical interpretation.     

Singularly perturbed phenomena of semilinear second order systems

In this paper we consider singular perturbed phenomena of semilinear second ordersystems, under appropriate assumptions, the existence and asymptotic behavior asε→0⁺ of solution of vector boundary value problem are proved by constructing specialinvariant regions in which solutions display so-called boundary layer phenomena andangular layer phenomena.     

The transient response of second order non-linear equations

A method is presented for calculating the transient response of second order non-linear differential equations. The increase in accuracy over such methods as the Kryloff-Bogoliuboff technique, physically speaking, is due to the inclusion of the phase-lag effects caused by the linear damping term in the equation. Mathematically the increase in accuracy is obtained by an appropriate choice of the zero-th order terms for the dependent variable and its first derivative. This inclusion requires little more calculation effort than that required in the Kryloff-Bogoliuoff technique. Additionally, this method may be iterated to obtain solutions of increased accuracy ; the next higher order solution being carried out in detail in the text. Examples are presented to show the increase in accuracy of the present method over the Kryloff-Bogoliuboff technique.