Dynamics of a system exhibiting the global bifurcation of a limit cycle at infinity

This paper concerns the dynamics of a class of non-linear oscillators of the form:

$\text{x″ + x ? ?x′(1?ax}{}^2\text{?bx′}{}^2\text{) = 0}$

. The non-linear term contains two parameters a and b which may be varied to give the Rayleigh and Van der Pol differential equations as special cases.The existence and approximation of limit cycles in this system are investigated using the Poincare-Bendixson theorem and the Lindstedt perturbation method. Analysis of the system at infinity is used to study the global bifurcation through which the limit cycle is created from four saddle-saddle connections between equilibrium points at infinity. Center manifold theory is used to determine the stability of the equilibrium points at infinity. Numerical integration is used to verify the analytical results.It is shown that an arbitrarily small perturbation to the damping term of the Rayleigh equation results in points close to the stable limit cycle escaping to infinity.     

Geometric stability criteria for certain non-linear time varying systems

Explicit geometric criteria for the exponential stability of a non-linear feedback system (consisting of a time-invariant block G in the forward path and a non-linear time varying gain φ.k(t) in the feedback path) are presented when φ(.) belongs to certain classes of non-linear functions. The resulting bound on [$\text{(}\text{dk}\text{dt}\text{)}\text{k}$] is less stringent than those found in the literature.     

Subharmonic steady vibrations of a non-linear damper with two degrees of freedom and viscous damping

The steady-state. $\text{1}\text{3}$ subharmonic vibrations of a dynamic damper (or vibration absorber) with two degrees of freedom, sinusoidal forcing function and internal viscous damping. are presented. The study of these oscillations leads to the determination of suitable “form functions” of the solutions, by following a methodology recently introduced by Nocilla for studying the harmonic vibrations of non-linear systems with one and two degrees of freedom. The proposed theory. which is valid even if the non-linearity is large, gives satisfactory results in all the cases in which the subharmonic component is predominant in the steady-state oscillation of the system.     

Linear methods in problems of non-linear differential equations with a small parameter

Methods have been considered for deriving asymptotical formulas for the systems of the type

$\text{ε}{}^{\mathrm{p}}\text{d}\text{x}{}_{\mathrm{k}}\text{d}\text{t}\text{= f}{}_{\mathrm{k}}\text{(x) + ε}\text{f}\text{?}{}_{\mathrm{k}}\text{(x) + …}$

by constructing an analog of the Schrödinger perturbation theory of the linear operator

$\text{∑}\text{k}\text{[f}{}_{\mathrm{k}}\text{(x) + ε}\text{f}\text{?}{}_{\mathrm{k}}\text{(x)]}\text{?F}\text{?x}{}_{\mathrm{k}}\text{= A}{}_{\mathrm{o}}\text{F + εA}{}_1\text{F.}$

These methods can be extended to some classes of partial differential equations, in particular, to Whitham's non-linear theory.     

Surface waves created by low-frequency magnetic fields

This paper analyses the effects of a low frequency A.C. magnetic field on the free surface of a liquid metal. The action of the vertical and uniform magnetic field is twofold. First it creates forced standing surface waves which generally exhibit symmetry related to that of the container; second it triggers non-symmetric free surface instabilities superimposed on the forced regime. A previous paper considered the case of a circular cylindrical tank where axisymmetric forced standing waves caused an electric current perturbation which then excited non-axisymmetric waves at a critical A.C. field intensity. Nonlinear interaction between the symmetric and non-symmetric modes was not taken into account. The present work treats the problem from a more general standpoint. Equilibrium perturbations are developed systematically to order $\text{N}{}^2$ (where $\text{N}$ is the magnetic interaction parameter) and at this level of approximation we also need to consider nonlinear mode interactions and electromagnetic damping. The theory applies to tanks of arbitrary shape and the $\text{O}\text{(}\text{N}\text{)}$ irrotational motion may be described by the torsion function for the particular pool cross-section. For circular and annular tanks we then derive a system of coupled Mathieu–Hill equations for the time-development of non-symmetric surface modes. Two main types of parametric resonance are predicted, namely the single or combination mode, and the particular type observed may depend on the geometry of the tank. Results of the stability analysis are confirmed by experimental work carried out in mercury pools.     

A class of complete integrals of plane eikonal equations

A class of complete integrals of the plane eikonal equation

$\text{|}\text{grad}\text{u|}{}^2\text{= f(x,y)}$

for harmonic u(x,y) is determined by using complex variables. The case in which z = 0 is a singular point of the analytic function whose real part is log f is also treated. Illustrative examples are given.     

Exact solution of the non-linear differential equation RR̈ + 32R2 − AR−4 + B = 0

The non-linear equation $\text{R}\text{R}\text{?}\text{+}\text{3}\text{2}\text{R}{}^2\text{- AR}{}^{\mathrm{?4}}\text{+ B = 0}$ is shown to represent simply periodic motion with a minimum at R₁ and a maximum at R₁R₀ or a maximum at R₁ and a minimum at R₁R₀^?1. R₀ is a function of the ratio $\text{A}\text{B}$ and is greater than 1 for $\text{A}\text{B}$ > 1 and less than 1 for $\text{A}\text{B}$ > 1. The period of the motion satisfies the simple relation T(R₀^?1) = R₀^?1T(R₀). The exact solution to the above equation is represented in terms of elliptic integrals of the first and second kinds and a simple algebraic function.     

An approximate explicit constitutive relation for non-flowing granular materials

An approximate stress-strain relation is derived for a granular material composed of spherical elastic granules in contact. The material is assumed to be statistically homogeneous so that the effective stress tensor can be obtained by a volume average. The resulting stress-strain relation is markedly non-linear and begins with the term $\text{∥ε∥}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, where ε is the classical infinitesimal strain tensor. Some simple deformation fields are worked out.     

Plastic flow and dislocation structure at small strains in OFHC copper deformed in tension,torsion and combined tension-torsion

OFHC copper specimens of 39 μm grain size were deformed to small strains (up to 8%) in tension, torsion and combined tension-torsion at 300 K and the resulting dislocation structures, distributions and densities were determined using transmission electron microscopy. Employing the von Mises yield criterion and the plastic-work hypothesis good agreement was obtained for the three testing conditions for (i) equivalent stress $\text{}\text{?}$gs vs equivalent strain $\text{}\text{?}$g3^p curves, (ii) the dislocation structure, distribution and density ρ as a function of $\text{}\text{?}$g3^p, and (iii) $\text{}\text{?}$gs as a function of $\text{ρ}\text{1}\text{2}$. Furthermore, upon comparing the $\text{}\text{?}$gs vs $\text{ρ}\text{1}\text{2}$ curve for polycrystalline copper with the τ_RSS vs $\text{ρ}\text{1}\text{2}$ curve for single crystals, an average Taylor factor M= (σ/τ_RSS) of approximately 3.2 was obtained, which is in good accord with that predicted theoretically for FCC metals. Almost equally good correlations for the stressstrain curves and for the dislocation density were obtained on the basis of maximum shear stress τ_max and maximum shear strain γ^p_max as on the basis of $\text{}\text{?}$gs and $\text{}\text{?}$g3^P. Therefore, the present results do not permit a positive decision on the question whether the dislocation density correlates better with $\text{}\text{?}$gs and $\text{}\text{?}$g3^P or with τ_max and γ^P_max.A single test in which the direction of straining in torsion was reversed yielded a density and distribution of dislocations (and a corresponding value of $\text{}\text{?}$gs) equivalent to those that developed at a smaller strain in unidirectional straining.     

Dynamic stability for a simply supported beam under periodic axial excitation

This paper studies the dynamic stability for a simply supported straight beam under periodic axial excitation by using the averaging method and the Routh-Hurwitz stability criteria. By considering the first two modes coupled, we discuss the effect of the stability-instability region and the amplitudes of vibration. Furthermore, by studying the principal parametric resonance i.e. subharmonic order $\text{1}\text{2}$, we investigate the effect of the amplitude of the main system by various kinds of non-linearities of the subsystem. Finally, by obtaining the transient results, we describe the beat phenomenon, and harmonic oscillation.