On the equations of non-linear vibrations

The purpose of this short paper is to show that equations of non-linear vibrations whose linear parts involve two coefficient matrices can be simplified so that the linear portions become uncoupled. Equivalence transformations are utilized in the simplification, which can substantially streamline any subsequent analysis.     

On numerical solution to fractional non-linear oscillatory equations

In this article, the multi-step differential transform method (MsDTM) is applied to give approximate solutions of nonlinear ordinary differential equation such as fractional-non-linear oscillatory and vibration equations. The results indicate that the method is very effective and sufficient for solving nonlinear differential equations of fractional order.     

On the exact solution to certain non-linear partial differential equations

This paper, based on the theory of stratifications, gives a brand-new classification of partial differential equations. To be dedicated to my beloved teacher, H. Cartan, academician of French Academy of Sciences, on the occasion of his 90th birthday. This paper was partially supported by the National Natural Science Foundation of China     

On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media

Some equivalence conditions are formulated for non-linear models of polymer melts and solutions that are analogous to known conditions for three-constant linear rheological equations. The resulting model is analysed in simple shear and elongational flows. The kinematics of finite elastoviscous strains is considered in an appendix.     

On the incremental equations in non-linear elasticity — I. Membrane theory

A new formulation of the equations of membrane theory in non-linear elasticity is described. It is based on the consistent use of certain conjugate variables averaged through the (undeformed) thickness of the thin shell which the membrane approximates. The deformation gradient is taken as the basic measure of deformation, and its average value as the membrane measure of deformation. It is shown that the average elastic strain energy can be regarded as a function of the average deformation gradient to within an error which is of the second order in a certain small parameter. Moreover, to the same order, the average strain energy is a potential function for the average nominal stress. This means that the averages of the conjugate variables (nominal stress and deformation gradient) are also conjugate.In terms of the average conjugate variables, the membrane equilibrium equations are obtained by averaging from the equilibrium equations of the full three-dimensional theory. Discussion of the order of magnitude of the errors involved in the membrane approximation is a feature of the analysis.The corresponding incremental equations are also derived as a prelude to their application in certain bifurcation problems. One such problem is examined in the companion paper (Part II) in which results for thick shells and membranes are compared.     

Nonholonomic systems with non-linear constraint equations

The development of a form of Lagrange's equations applicable with nonholonomic systems with non-linear constraint equations is presented and discussed. The analysis is based upon, and is an extension of. a method developed by the authors for nonholonomic systems with linear constraint equations in the generalized coordinate derivatives. The method is illustrated with the problem of the “balancing pole”.     

On the derivation of some non-linear evolution equations and their progressive wave solutions

In the present work, utilizing the reductive perturbation method, the non-linear equations of a prestressed viscoelastic thick tube filled with a viscous fluid are examined in the longwave approximation and some evolution equations and their modified forms are derived. The analytical solution of some of these equations are obtained and it is shown that for perturbed cases, the wave amplitude and the phase velocity decay in the time parameter.     

On order reduction of non-linear equations of mechanics and mathematical physics,new integrable equations and exact solutions

Some classes of non-linear equations of mechanics and mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where a first-order partial derivative is taken as a new independent variable and a second-order partial derivative is taken as the new dependent variable. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes, Euler, and boundary layer) and deriving exact solutions to these equations. Associated Bäcklund transformations are constructed for evolution equations of general form (special cases include Burgers, Korteweg-de Vries, and many other non-linear equations of mathematical physics). A number of new integrable non-linear equations, inclusive of the generalized Calogero equation, are considered.     

Parametric stability of non-linear time delay equations

The stability behaviour of delay differential equations describing chatter instability in machining operations has been studied. A transcendental characteristic equation characterizing the onset of the instability is analysed. It is found that increasing the feed rate increases stable machining regions as the depth of cut is continuously increased near a critical value. Furthermore, the non-linear behaviour occurring when stable machining is lost has been shown to be super- and subcritical bifurcations without the assumption of small time delay between successive tool cuts.     

Forces associated with non-linear non-holonomic constraint equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by non-holonomic equations that are inherently non-linear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known non-holonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as non-holonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor first to form the former, larger set and subsequently perform matrix multiplications.     

On non-linear stability in unconstrained non-linear elasticity

A method of obtaining a full three-dimensional non-linear Hadamard stability analysis of inhomogeneous deformations of arbitrary, unconstrained, hyperelastic materials is presented. The analysis is an extension of that given by Chen and Haughton (Proc. Roy. Soc. London A 459 (2003) 137) for two-dimensional incompressible problems. The process that we present replaces the second variation condition expressed as an integral involving a quadratic in three arbitrary perturbations, with an equivalent sixth-order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of a thick-walled spherical shell. The present analysis provides a simpler alternative approach to bifurcation problems approached by using the incremental equations of non-linear elasticity.     

On non-linear periodic differential equations and associated point mappings. Application to stability problems

In this paper a method is given allowing the determination of the lower degree terms of a point mapping (i.e. recurrent relationship) associated to a non-linear differential equation with time periodic coefficients. This permits the study of some resonance phenomena. An example of application is given.     

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.     

Solitary wavepackets from oscillatory non-linear equations with damping

In this paper, an infinite family of solutions describing solitary wave packets with a finite number of nodes is presented. These structures arise from the study of damping in the framework of non-linear ordinary differential equations with oscillatory behaviour. Usually one expects to find effects of this kind in physical systems described by a set of partial differential equations. The standard argument is that the non-linear term acts against the dispersive flux and this balance explains the appearance of solitary waves. Here we show that the non-linear oscillatory behaviour can also balance the effect of damping in special cases. The theory used to discriminate among the various possibilities is plain Painlevé analysis. Several physical applications are briefly discussed.     

Multiple scale perturbation for second-order non-linear differential equations

Asymptotic solutions of a class of second-order non-linear differential equations with variable coefficients are studied. For large values of the parameter, the differential equations are of the singular-perturbation type and approximations are constructed by the generalized method of multiple scales.     

Post-buckling behavior of columns with non-linear constitutive equations

The post-buckling behavior of a column made of a material with cubic constitutive equation, σ = E₁? + E₂?³, is unstable for a range of values of E₂. In these cases, the imperfection sensitivity is qualitatively described using catastrophe theory. A numerical method is given to compute the post-buckling deflections.