Showing papers in "International Journal of Non-linear Mechanics in 1999"

Variational iteration method – a kind of non-linear analytical technique: some examples

Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.

An explicit, totally analytic approximate solution for Blasius’ viscous flow problems

Abstract: By means of using an operator A to denote non-linear differential equations in general, we first give a systematic description of a new kind of analytic technique for non-linear problems, namely the homotopy analysis method (HAM). Secondly, we generally discuss the convergence of the related approximate solution sequences and show that, as long as the approximate solution sequence given by the HAM is convergent, it must converge to one solution of the non-linear problem under consideration. Besides, we illustrate that even though a non-linear problem has one and only one solution, the sole solution might have an infinite number of expressions. Finally, to show the validity of the HAM, we apply it to give an explicit, purely analytic solution of the 2D laminar viscous flow over a semi-infinite flat plate. This explicit analytic solution is valid in the whole region η=[0, +∞) and can give, the first time in history (to our knowledge), an analytic value f ″(0)=0.33206 , which agrees very well with Howarth’s numerical result. This verifies the validity and great potential of the proposed homotopy analysis method as a new kind of powerful analytic tool.

Flow and heat transfer in a second grade fluid over a stretching sheet

Abstract: An analysis is carried out to study the flow and heat transfer characteristics in a second grade fluid over a stretching sheet with prescribed surface temperature including the effects of frictional heating, internal heat generation or absorption, and work due to deformation. In order to solve the fourth-order non-linear differential equation, associated with the flow problem, a fourth boundary condition is augmented and a proper sign for the normal stress modulus is used. It is observed that for a physical flow problem the solution is unique. The solutions for the temperature and the heat transfer characteristics are obtained numerically and presented by a table and graphs. Furthermore, it is shown that the heat flow is always from the stretching sheet to the fluid.

Multiple resonances in suspended cables: direct versus reduced-order models

Abstract: We apply two analytical approaches to construct asymptotic models for the non-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and a two-to-one internal resonance with the first symmetric out-of-plane mode. First, we apply the method of multiple scales directly to the governing two integral-partial-differential equations and associated boundary conditions. Reconstitution of the solvability conditions at second and third orders leads to a system of four coupled non-linear complex-valued equations describing the modulation of the amplitudes and phases of the interacting modes. The homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problem are needed to make the reconstituted modulation equations derivable from a Lagrangian. However, this procedure leads to an indeterminacy, indicating a likely inconsistency with this specific application of the method of multiple scales. Then, we apply the method to a four-degree-of-freedom Galerkin discretized model obtained using the pertinent excited eigenmodes. Again, the homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problems are required to make the reconstituted modulation equations derivable from a Lagrangian. Frequency–response curves obtained using the two generated asymptotic models, for a specific choice of the arbitrary constant appearing in both models, show different qualitative as well as quantitative predictions for some classes of motions. The effects of an inconsistent reconstitution in the direct approach are also investigated.

Non-linear vibration of a travelling beam

Abstract: Free and forced responses of a travelling slender beam including the non-linear terms are studied. A simple method of calculating the non-linear, non-stationary, complex normal mode is presented. The hardening characteristic and intercoupling of linear modes during the non-linear modal vibration are clearly brought out. The response of the beam, excited by a point harmonic load, is calculated using the non-linear normal modes. A stability analysis is provided to delineate the stable and unstable roots of the response.

Periodic flows of a non-Newtonian fluid between two parallel plates

Abstract: Exact solutions for the flows of a non-Newtonian fluid between two infinite parallel plates are obtained by using the theory of Fourier transform. The flows discussed are generated by periodic oscillations of one of the plates. Some interesting flows caused by certain special oscillations are also studied.

Non-linear inplane deformation and buckling of rings and high arches

Abstract: A non-linear theory is presented for stretching and inplane-bending of isotropic beams which have constant initial curvature and lie in their plane of symmetry. For the kinematics, the geometrically exact one-dimensional (1-D) measures of deformation are specialized for small strain. The 1-D constitutive law is developed in terms of these measures via an asymptotically correct dimensional reduction of the geometrically non-linear 3-D elasticity under the assumptions of comparable magnitudes of initial radius of curvature and wavelength of deformation, small strain, and small ratio of cross-sectional diameter to initial radius of curvature ( h/R ). The 1-D constitutive law contains an asymptotically correct refinement of O (h/R) beyond the usual stretching and bending strain energies which, for doubly symmetric cross sections, reduces to a stretch–bending elastic coupling term that depends on the initial radius of curvature and Poisson’s ratio. As illustrations, the theory is applied to inplane deformation and buckling of rings and high arches. In spite of a very simple final expression for the second variation of the total potential, it is shown that the only restriction on the validity of the buckling analysis is that the prebuckling strain remains small. Although the term added in the refined theory does not affect the buckling loads, it is shown that non-trivial prebuckling displacements, curvature, and bending moment of high arches are impossible to calculate accurately without this term.

Elastica solution for the hygrothermal buckling of a beam

Abstract: An elliptic integral solution for the post-buckling response of a linear-elastic and hygrothermal beam fully restrained against axial expansion is presented Whereas in the classical solutions the extension of the beam can be neglected, a well-posed formulation of the title problem must include the extension The solution for the limiting case of a string is presented The present solution shows that the magnitude of the compressive axial load is a maximum at the onset of buckling and decreases as the potential for free expansion is increased; this is in contrast to the approximate solutions found in the literature

Non-classical effects in non-linear analysis of pretwisted anisotropic strips

Abstract: The literature on classical analysis of anisotropic beams assumes that all 1D “moment strain” measures (i.e. twist and bending curvatures) are of the same order of magnitude, resulting in a linear cross-sectional analysis. The present paper treats the situation in which one or more of the 1D moment strain measures may be larger than the other(s), resulting in a non-linear cross-sectional analysis. This type of non-classical analysis is needed, for example, in problems where the trapeze effect is important, such as in rotor blades. As a precursor to complicated non-linear sectional analysis of arbitrary cross sections, a non-linear sectional analysis is presented for an anisotropic strip with small pretwist, based on the dimensional reduction of laminated shell theory to a non-linear one-dimensional theory using the variationalasymptotic method. Results obtained from this strip-beam analysis are compared with available theoretical and experimental results for a problem in which the trapeze effect is important. In order to demonstrate the usage of the results in the analysis of structures made of an arbitrary geometrical combination of pretwisted generally anisotropic strips, a closed-form expression is derived for the torsional buckling of a column with a cruciform cross section.

Self-similar shocks in a dusty gas

Abstract: A group theoretic method is used to establish the entire class of self-similar solutions to the problem of shock wave propagation through a dusty gas. Necessary conditions for the existence of similarity solutions for shocks of arbitrary strength as well as for strong shocks are obtained. It is found that the problem admits a self-similar solution only when the ambient medium ahead of the shock is of uniform density. Collapse of imploding cylindrical and spherical shocks is worked out in detail to investigate as to how the shock involution is influenced by the mass concentration of solid particles in the medium, the ratio of the density of solid particles to that of initial density of the medium, the relative specific heat and the amplification mechanism of the flow convergence.

Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltonian systems

Abstract: A new procedure for analyzing the stochastic Hopf bifurcation of quasi-non-integrable-Hamiltonian systems is proposed. A quasi-non-integrable-Hamiltonian system is first reduced to an one-dimensional Ito stochastic differential equation for the averaged Hamiltonian by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. Then the relationship between the qualitative behavior of the stationary probability density of the averaged Hamiltonian and the sample behaviors of the one-dimensional diffusion process of the averaged Hamiltonian near the two boundaries is established. Thus, the stochastic Hopf bifurcation of the original system is determined approximately by examining the sample behaviors of the averaged Hamiltonian near the two boundaries. Two examples are given to illustrate and test the proposed procedure.

Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts

Abstract: This work aims in obtaining the transient response of an overhung rotor undergoing vibro-impacts due to a defective bearing An overhung rotor clamped on one end, with a flywheel on the other and impacts occurring in between, due to a bearing with clearance, is considered The variation of this system, popularly known as the Jeffcot rotor, has been considered in previous works, but there, the system has been reduced to a single degree of freedom for ease of analysis In this work the system is modeled as a continuous rotor including gyroscopic effects and the governing partial differential equations are set up and numerically solved The method of assumed models is used to discretize the system in order to solve the partial differential equations (PDE) by partially decoupling them and solving numerically These partially decoupled equations are more accurate and less time consuming than the ones produced by finite elements or other numerical schemes The most important step in the success of this method is the selection of suitable modes for decoupling the system It is not simply enough to select orthonormal modes for decoupling the PDEs, but care must be taken to select the modes as close to the actual system as possible Using this method numerical experiments are run and representative results are presented The different numerical issues involved are also discussed An experimental setup was also built to run experiments and validate the results In the setup a defective bearing is introduced at the flywheel end of the shaft to create radial impacts on the shaft Laser sensor non-contact probes are used to measure the displacement of the shaft a specified locations Experimental observations show satisfactory qualitative agreement when compared to the numerical integrations

Asymptotic treatment of the trapeze effect in finite element cross-sectional analysis of composite beams

Abstract: Early analyses for numerical cross-sectional analysis of anisotropic beams were based on linear elasticity theory. For more general treatment of non-linear phenomena, asymptotic formulations can serve as the basis for the numerical method, say a finite element method. When based on geometrically non-linear elasticity theory, an intermediate result of such analyses is frequently the splitting of the non-linear 3-D problem into a linear 2-D analysis of the cross section and a non-linear 1-D analysis along the beam. Thus, the published work to date cannot treat the so-called “trapeze effect”, because it stems from non-linearity of the cross-sectional analysis. Herein, a non-linear numerical cross-sectional analysis is presented, based on the variational-asymptotic method and capable of treating cross sections of arbitrary geometry and generally anisotropic material. This type of analysis is particularly important in rotating structures, such as helicopter rotor blades. Results from this model are compared with those available in the literature, both theoretical and experimental.

Internal symmetry in bilinear elastoplasticity

Abstract: Internal symmetry of a constitutive model of bilinear elastoplasticity (i.e. linear elasticity combined with linear kinematic hardening–softening plasticity) is investigated. First the model is analyzed and synthesized so that a two-phase two-stage linear representation of the constitutive model is obtained. The underlying structure of the representation is found to be Minkowski spacetime, in which the augmented active states admit of a Lorentz group of transformations in the on phase. The kinematic rule of the model renders the transformation group inhomogeneous, resulting in a larger group—the proper orthochronous Poincare group.

Adaptive control of flow-induced oscillations including vortex effects

Abstract: Flow-induced vibrations constitute important design criteria for most offshore structures as well as for many other structures subjected to flow-induced forces. Both the main structural elements as well as supporting structural members such as guyed cables must be designed to withstand such oscillations. It is well established that in the process of vibrations induced by flow, vortices form around the body which initiate oscillations in a direction transverse to the general direction of motion. In this paper, Morison’s equation is used to represent the interaction between the flow field and the structure, complemented by terms including the vortex dynamics effect. In order to mitigate against extreme vibration conditions, here, the implementation of active control is proposed to stabilize the motion of a structure immersed in a flow field. Specifically, a tuned mass damper is attached to the structure and adaptive control is utilized for moving the mass along a particular path while allowing for uncertainty in the various hydrodynamic coefficients. The proposed procedure have two distinct beneficial results. The first one being to control the vibration of the structure, and the second one is the estimation of the hydrodynamic coefficients and the validation/calibration of Morison’s equation model for the flow-induced forces.

Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles

Abstract: In this paper, we consider the unsteady equations that govern two- and three-dimensional flows of a perfect gas. We explicitly characterize various classes of exact solutions by introducing some invertible transformations suggested by the invariance with respect to Lie groups of point symmetries and using suitable transformations known in literature as substitution principles.

Oscillations of a pendulum with a periodically varying length and a model of swing

Abstract: Qualitative analysis of a pendulum with a periodically varying length is conducted. It is proved that there are two periodic solutions having a prescribed amplitude A

Natural convection over a vertical wavy frustum of a cone

Abstract: The effects of wavy surfaces on natural convection over a vertical frustum of a cone is studied in this paper. We consider the boundary-layer regime where the Grashof number Gr is very large and assume that the wavy surface have O(1) amplitude and wavelength. The transformed boundary layer equations are solved numerically using the Keller-box method. Detailed results for the local Nusselt number and wall temperature are presented for a selection of parameter sets consisting of the wavy surface amplitude, half cone angle and Prandtl number.

Asymptotic analysis of the non-linear behavior of long anisotropic tubes

Abstract: An asymptotically correct beam model is obtained for a long, thin-walled, circular tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. By virtue of its special geometry certain small parameters cause unusual non-linear phenomena, such as the Brazier effect, to be exhibited. The model is constructed without ad hoc approximations from 3D elasticity by deriving its strain energy functional in terms of generalized 1D strains corresponding to extension, bending, and torsion. Large displacement and rotation are allowed but strain is assumed to be small. Closed-form expressions are provided for the 3D non-linear warping and stress fields, the 1D non-linear stiffness matrix and the bending moment–curvature relationship. In bending, failure could be caused by limit-moment instability, local buckling or material failure of a ply. A procedure to determine the failure load is provided based on the non-linear response, neglecting micro-mechanical failure modes, post-failure behavior, and hygrothermal effects. Asymptotic considerations lead to the neglect of local shell interlaminar and transverse shear stresses for the thin-walled configuration. Results of the theory are illustrated for a few symmetric, antisymmetric angle-ply and unsymmetric layups and show that some previously published theories are not asymptotically correct.

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

Abstract: The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.

Oberbeck convection flow of a couple stress fluid through a vertical porous stratum

Abstract: The flow and heat transfer characteristics of Oberbeck convection of a couple stress fluid in a vertical porous stratum is investigated. The perturbation method of solution is obtained in terms of buoyancy parameter N valid for small values of N . This limitation is relaxed through numerical solutions using the finite difference technique with an error of 0.1×10 -7 . The effect of increase in the values of temperature difference between the plates, permeability parameter and couple stress parameter on velocity, temperature, mass flow rate, skin friction and rate of heat transfer are reported. A new achievement is explored to analyse the flow for strong, weak and comparable porosity with the couple stress parameter. It is noted that both the porous parameter and the couple stress parameter suppress the flow. Higher-temperature difference is required to achieve the mass flow rate equivalent to that of viscous flow.

Homoclinic and heteroclinic bifurcations in the non-linear dynamics of a beam resting on an elastic substrate

Abstract: The non-linear dynamics of a slender “elastica”, fixed at its base and free at the top, resting on an elastic substrate, axially loaded and subjected to periodic excitation, has been analyzed. Taking into account the non-linear inertial terms, the single-mode dynamics of the systems is governed by a Duffing equation with fifth-order non-linearities. In the considered range of parameters, two qualitatively different phase portraits exist. When the axial load p is less than the Eulerian critical value, there are three centers and two saddles (with the related stable and unstable manifolds). After the pitchfork bifurcation, the two saddles and the middle center coalesce in an unique new saddle which has a pair of symmetric homoclinic solutions. Melnikov criteria on the chaotic dynamics of the system are derived on the basis of analytical expressions for the homoclinic and the heteroclinic orbits. They involve transverse intersections of the stable and unstable manifolds that represent the starting point for a subsequent route to a chaotic dynamics. Numerical simulations which aim to show some effects of the global bifurcations on the actual dynamics are presented.

Second-order approximation for chaotic responses of a harmonically excited spring–pendulum system

Abstract: The influence of a higher-order approximation on chaotic responses of a weakly non-linear multi-degree-of-freedom system is investigated. The specific system examined is a harmonically excited spring–pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions. By the method of multiple scales the original system is reduced to a second-order approximate system. The long-term behaviors of both systems are compared by examining the largest Lyapunov exponents. It is observed that the second-order approximation gives better qualitative agreement with the original system than the first-order approximation does.

Fluid flow behaviour in a curved annular conduit

Abstract: Numerical solution has been obtained of the equations of motion of a viscous incompressible fluid in a curved annular conduit with circular cross-section. These equations are approximated by finite-difference equations which are of the second-order accuracy with respect to the grid sizes. The computed results are presented for the range 96⩽ D ⩽8000, where D is the Dean number of the flow and for various sizes of core radii, the limiting cases of a very small and a very large core being also studied. It is shown that in the case of a small core radius, the variation of the Dean number affects significantly the flow properties, situation which is not observed when the core radius is large.

On the evolution of deterministic non-periodic behavior of an airfoil

Abstract: This paper describes some interesting transitional behavior in the limit cycle response of a typical section airfoil with a loosely connected flap subject to a fluid flow. The freeplay non-linearity associated with the flap connection has considerable practical importance since wear and maintenance problems inevitably accompany moving mechanical parts. Piecewise linear systems have been studied within the non-linear dynamics community for some time. However, application to a relatively complicated, fluid–structure interaction problem such as this shows the ubiquity of a number of characteristically non-linear features. Special attention is focused on almost- or quasi-periodic behavior, and the correpsonding stability transitions. Brief reference is made to some prior experimental work which suggests the extent to which these responses can be realized in wind tunnel testing.

Flow of a Johnson–Segalman fluid between rotating co-axial cylinders with and without suction

Abstract: The Johnson–Segalman fluid has a non-monotone relationship between the shear stress and velocity gradient in simple shear flows for a certain range of material parameters resulting in solutions with discontinuous velocity gradients for planar and cylindrical Poiseuille flow. This has been used to explain the phenomenon of “Spurt”. Rao and Rajagopal [ Acta Mechania , to be published] have shown that the addition of suction necessarily smoothens the solutions for planar Poiseuille and cylindrical Poiseuille flows. Here we study the effect of a different geometry on the flow of a Johnson–Segalman fluid. The problems of cylindrical Couette flow, cylindrical Couette flow with suction (or injection) and Hamel flow are studied and it is found that the boundary condition can have an interesting effect on the regularity of the solution. The presence of suction increases the regularity of the solution, i.e. solutions with discontinuous velocity gradients are not possible.

Stability and asymmetric vibrations of pressurized compressible hyperelastic cylindrical shells

Abstract: Cylindrical shells of arbitrary wall thickness subjected to uniform radial tensile or compressive dead-load traction are investigated. The material of the shell is assumed to be homogeneous, isotropic, compressible and hyperelastic. The stability of the finitely deformed state and small, free, radial vibrations about this state are investigated using the theory of small deformations superposed on large elastic deformations. The governing equations are solved numerically using both the multiple shooting method and the finite element method. For the finite element method the commercial program ABAQUS is used. 1 The loss of stability occurs when the motions cease to be periodic. The effects of several geometric and material properties on the stress and the deformation fields are investigated.

Non-linear oscillations of circular plates near a critical speed resonance

Abstract: The non-linear response of an axisymmetric, thin elastic circular plate subject to a constant, space-fixed transverse force and rotating near a critical speed of an asymmetric mode, is analyzed. A small-stretch, moderate-rotation plate theory of Nowinski [J. Appl. Mech. (1964) 72–78], leading to von Karman-type field equations is used. This leads to non-linear modal interactions of a pair of 1–1 internally resonant, asymmetric modes which are studied through first-order averaging. The resulting amplitude equations represent a system whose O(2) symmetry is broken by a resonant rotating force. The non-linear coupling of the modes induces steady-state solutions that have no apparent evolution from any previous linear analyses of this problem. For undamped disks, the analysis of the averaged Hamiltonian predicts two codimension-two bifurcations that give rise to three sets of doubly degenerate, one-dimensional manifolds of steady mixed wave motions. On the addition of the smallest damping, the branches of the backward travelling waves with equal modal content become isolated, and it is proved that these are the only steady motions possible. A simple experiment is used to confirm the analytical predictions.

Conserved quantities from non-Noether symmetries without alternative Lagrangians

Abstract: A new method is presented for obtaining constants of the motion from a non-Noether symmetry, which does not require the existence of an alternative Lagrangian. Comparison is made with a known method of generating conserved quantities without alternative Lagrangians, in which the conserved quantities arise as invariants of a certain 2 N ×2 N matrix (for N -dimensional systems).

Lorentz group SOo(5, 1) for perfect elastoplasticity with large deformation and a consistency numerical scheme

Abstract: How to effectively deal with non-linearity and accurately fulfill the consistency condition is essential for modeling and computing in plasticity. Utilizing the concepts of two phases and homogeneous coordinates, we obtain a linear representation of a constitutive model of perfect elastoplasticity with large deformation, and, furthermore, a linear irreducible representation, which contains a five-order spin tensor. The underlying vector space is found to be the projective realization of a composite space resulting from a surgery on Minkowski spacetime M 5+1. The irreducible representation in the vector space admits of the projective proper orthochronous Lorentz group PSOo(5, 1) in the on (or elastoplastic) phase and the special Euclidean group SE(5) in the off (or elastic) phase. The input path dictates symmetry switching between the two groups. Based on such symmetry a numerical scheme is devised which preserves the consistency condition for every time step. The consistency scheme is shown to be stabler, more accurate, and more efficient than the current numerical schemes developed directly based upon the model itself, because the new scheme preserves the internal symmetry SOo(5, 1) of the model in the on phase so as to locate the stress point automatically on the yield surface at each time step without iterations at all.