Showing papers in "International Journal of Non-linear Mechanics in 2000"

A coupling method of a homotopy technique and a perturbation technique for non-linear problems

Abstract: In this paper, a coupling method of a homotopy technique and a perturbation technique is proposed to solve non-linear problems. In contrast to the traditional perturbation methods, the proposed method does not require a small parameter in the equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter p∈[0, 1] is constructed, and the imbedding parameter is considered as a “small parameter”. So the proposed method can take full advantage of the traditional perturbation methods. Some examples are given. The results reveal that the new method is very effective and simple.

Two-dimensional blood flow through tapered arteries under stenotic conditions

Abstract: A mathematical model of non-linear two-dimensional blood flow in tapered arteries in the presence of stenosis is developed. An improved shape of the time-variant overlapping stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the in vivo situation. The vascular wall deformability is taken to be elastic while the flowing blood contained in it is treated to be Newtonian. The non-linear terms appearing in the Navier–Stokes equations governing blood flow and the instantaneous taper angle are accounted for. The present analytical treatment bears the potential to calculate both the axial and the radial velocity profiles with low computational complexity by exploiting the appropriate boundary conditions and the input pressure gradient arising from the normal functioning of the heart. The computed results are found to converge at a high rate with the tolerance of ∼10−14 and agree well with the corresponding existing data. An extensive quantitative analysis is performed through numerical computations of the desired quantities presented graphically at the end of the paper which help estimating the effects of tapering, the wall motion, the stenosis and the pulsatile pressure gradient on the flow characteristics of blood and thereby the applicability of the present model is established.

A note on an unsteady flow of a viscous fluid due to an oscillating plane wall

Abstract: The flow of a viscous fluid produced by a plane boundary moving in its own plane with a sinusoidal variation of velocity is considered. An analytical solution describing the flow at small and large times after the start of the boundary is obtained by the Laplace transform method. The solution gives not only the steady solution but also the transient solution. The time required to attain steady flows for the cosine oscillation of the boundary is one-half cycle and it is a full cycle for the sine oscillation of the boundary.

Non-linear free vibration of isotropic plates with internal resonance

Abstract: The geometrically non-linear free vibration of thin isotropic plates is investigated using the hierarchical finite element method (HFEM). Von Karman's non-linear strain–displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion are developed by applying the principle of virtual work and the harmonic balance method (HBM), and the solutions are determined using a continuation method. The convergence properties of the HFEM and of the HBM are analyzed. Internal resonances are discovered. The variation of the plate's mode shape with the amplitude of vibration and during the period of vibration is demonstrated.

Finite deformation beam models and triality theory in dynamical post-buckling analysis☆

Abstract: Two new finitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola–Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.

Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems

Abstract: In this paper, we first review results of integral manifolds of singularly perturbed non-linear differential equations. We then outline the basic elements of the integral manifold method in the context of control system design, namely, the existence of an integral manifold, its attractivity, and stability of the equilibrium while the dynamics are restricted to the manifold. Toward this end, we use the composite Lyapunov method and propose a new exponential stability result which gives, as a by-product, an explicit range of the small parameter for which exponential stability is guaranteed. The results are applied to the control problem of multibody systems with rigid links and flexible joints in which the inverse of joint stiffness plays the role of the small parameter. The proposed controller is a composite control law that consists of a fast component, as well as a slow component that was designed based on the integral manifold approach. We show that the proposed composite controller has the following properties: (i) it enables the exact characterization and computation of an integral manifold, (ii) it makes the manifold exponentially attractive, and (iii) it forces the dynamics of the reduced flexible system on the integral manifold to coincide with the dynamics of the corresponding rigid system (i.e. the one obtained by making stiffness very large) implying that any control law that stabilizes the rigid system would stabilize the dynamics of the flexible system on the manifold. We finally present a detailed stability analysis and give an explicit range of the joint stiffness, in terms of system parameters and controller gains, for which the established exponential stability is guaranteed.

Dynamic analysis of piecewise linear oscillators with time periodic coefficients

Abstract: A new method is presented for locating periodic steady-state response of piecewise linear dynamical systems with time periodic coefficients. As an example mechanical model, a gear-pair system with backlash is examined, under the action of a constant torque. Originally, some useful insight is gained on the dynamics by investigating the response of a weakly non-linear Mathieu–Duffing oscillator, subjected to a constant external load. The information obtained is then used in seeking approximate periodic solutions of the piecewise linear system. These solutions are determined by developing a new analytical method, which combines elements from approaches applied to piecewise linear systems with constant coefficients as well as from classical perturbation techniques applied to systems with time varying coefficients. The existence analysis is complemented by appropriate stability analyses, for all the possible types of the located periodic motions. In the second part of the work, this analysis is employed and numerical results are obtained. Namely, a series of typical response diagrams is first presented, illustrating the effect of the variable stiffness, the damping and the constant load parameters on the gear-pair response. Moreover, results obtained by direct integration of the equation of motion are finally presented, showing that the system examined can also exhibit more complicated or irregular dynamics.

Exponential closure method for some randomly excited non-linear systems

Abstract: The probability density function (PDF) of the responses of non-linear stochastic system excited by white noise is approximated with the exponential function of polynomial in state variables. Special measure is taken to satisfy FPK equation in the weak sense of integration with the assumed PDF. Gaussian closure method is a special case of the proposed method. Examples are given to show the application of the method to the systems with additive random excitations and those with both additive and multiplicative random excitations. The PDFs obtained with the proposed method and conventional Gaussian closure method are compared with obtainable exact ones. Numerical results showed that the PDFs obtained with the proposed method can be very close to the exact ones regardless of the degree of system non-linearity. In some cases, even exact solution can be obtained with the proposed method.

Internal symmetry in the constitutive model of perfect elastoplasticity

Abstract: Internal symmetry in the constitutive model of perfect elastoplasticity is investigated here. Using homogeneous coordinates, we convert the non-linear model to a linear system X = AX . In this way the inherent symmetry in the constitutive model of perfect elastoplasticity (in the on phase) is brought out. The underlying structure is found to be the cone of Minkowski spacetime M n+1 on which the proper orthochronous Lorentz group SO 0 (n, 1) left acts. When the plasticity mechanism is shut off by the input path, the internal symmetry is switched to a translation group T(n) acting on the closed disc D n of Euclidean space E n . Based on the group properties a Cayley transformation is developed, which updates the stress points to be automatically on the yield surface at every time increment. These results (and their generalizations to more sophisticated models) are essential for computational plasticity. As an example, the results calculated using the group-preserving scheme and the exact constitutive solutions for a rectilinear path are compared.

Mode jumping in the buckling of struts and plates: a comparative study

Abstract: A detailed investigation of the phenomenon of mode jumping in compressed struts on stiffening foundations and elastic plates of varying lengths is performed, with emphasis on the effects of altering boundary conditions. The variety of possible modal interactions is presented in a concise form using the parameter space of Arnol'd tongues, borrowed from non-linear dynamical systems theory. For the strut system, a full range of end conditions from simply supported to clamped is examined. For the plate, simply supported and clamped flexural conditions along both long (unloaded) and short (loaded) edges are considered, together with in-plane conditions ranging from free to pull in, to fully restrained. For each system, simply supported end conditions are found to provide protection against early mode jumping in a so-called “safety envelope”, but this is eroded as the end conditions are systematically altered from simply supported to clamped. For the plate system, mode jumping is induced at an earlier stage in the loading process by restricting the long (unloaded) edges against in-plane movement, but is delayed by clamping the same edges against rotation.

Helical buckling of a tube in an inclined wellbore

Abstract: In this paper, helical buckling of an elastic tube in an inclined well bore under the action of its own weight and a compressive force applied at its upper end is studied. The tube is considered to be supported by hinges with the upper end free to rotate with respect to its axis. It can also move freely in the axial direction. The tube buckles initially with a sinusoidal mode. In the post-buckling state, the shape of the tube can evolve to be helical. Our interest is to find the critical force for helix formation. In this study, the buckling force for helix formation is expressed as a Rayleigh's quotient, and then solved by Rayleigh's principle using the Rayleigh–Ritz technique.

A further study on the postbuckling of extensible elastic rods

Abstract: The postbuckling of extensible elastic rods is studied using non-linear geometric models. Accordingly the kinematics and equilibrium are stated. Nine different strain–stress relationships are analyzed. The classical Strength of Materials approach is compared and discussed with other eight constitutive laws stated with Lagrangian and Eulerian descriptions. The well-known Cauchy and Green methods in Continuum Mechanics are alternatively employed. Four of the approaches are worked out until an explicit solution of the secondary equilibrium path is obtained. The analysis is applicable to small strain problems. The linearized problem is presented for all the laws together with numerical results for rods with various values of the extensibility parameter. The secondary equilibrium paths are numerically evaluated to illustrate the degree of discrepancy. A specific example that displays unexpected unstable behavior is shown. Both critical loads and postbuckling curves are coincident when the theoretical problem of an inextensible rod is solved. It is shown that even when small strains are addressed, the extensibility influence gives rise to disagreement of the postbuckling response when using the different alternatives.

Vibration of a continuous system with clearance and motion constraints

Abstract: The present study generalises previous research work on the dynamics of discrete oscillators with piecewise linear characteristics and investigates the response of a continuous model system with clearance and motion-limiting constraints. More specifically, in the first part of this work, an analysis is presented for determining exact periodic response of a periodically excited deformable rod, whose motion is constrained by a flexible obstacle. This methodology is based on the exact solution form obtained within response intervals where the parameters remain constant and the system behaviour is governed by a linear equation of motion. The unknowns of the problem are subsequently determined by imposing an appropriate set of periodicity and matching conditions. The analytical part is complemented by a suitable method for determining the stability properties of the located periodic motions. In the second part of the study, the analysis is applied to several example cases in order to investigate the effect of the parameters on the system dynamics. Special emphasis is placed on comparing these results with results obtained for similar but rigid rods. Finally, direct integration of the equation of motion for selected parameter combinations reveals the existence of motions, which are more complicated than the periodic motions determined analytically.

Chaotic motion of a gyrostat with asymmetric rotor

Abstract: In this paper, we discuss an asymmetric gyrostat with a rotor to which a mass point is attached, this breaks the symmetry and perturbs the integrable system periodically. We take this system as a Euler–Poinsot motion perturbed by a small periodic excitation, and apply the Melnikov method to determine the intersection of the stable and unstable manifold of the system's hyperbolic point, since, this is the cause of chaos. We also manifest the chaotic motion in angular momentum space by the Poincare surface of section. The stability about the rotating axis is also showed in the portrait of the Poincare surfaces of section.

Periodic attractors of complex damped non-linear systems

Abstract: The aim of the present paper is to investigate the dynamics of a class of complex damped non-linear systems described by the equation (∗) z +ω 2 z+e z f(z, z , z , z )P( Ω t)=0, where z(t)=x(t)+ i y(t), i = −1 , the bar denotes the complex conjugate and e is a small positive parameter. The periodic attractors of Eq. (∗) are important in the study of these systems, since they represent stationary or repeatable behavior. This equation appears in several fields of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. In the numerical investigation of this work we used the indicatrix method which has been introduced and extended in our previous studies to study the existence of the periodic attractors of our systems. To illustrate these periodic attractors we constructed Poincare plots at some of the parameter values which are obtained by the indicatrix method for the case ω 2 ≅ 1 4 , f=|z| 2 and P( Ω t)= sin 2t as an example. Our recent method which is based on the generalized averaging method is used to obtain approximate analytical solutions of Eq. (∗) and investigate the stability properties of the solutions. We compared the analytical results of our example with the numerical results and excellent agreement is found.

Non-linear dynamic stability characteristics of elastic plates subjected to periodic in-plane load

Abstract: Using a C1 QUAD-8 shear-flexible plate element, based on a new kind of kinematics which allows one to exactly ensure the continuity conditions for displacements and stresses at the interfaces between the layers in the laminates, the non-linear instability behaviour of plates subjected to periodic in-plane load has been studied. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in-plane and rotary inertia effects. Primarily, an attempt is made here to understand the geometrically non-linear parametric instability characteristics of isotropic and composite plates through a finite element formulation with dynamic response analysis. The non-linear governing equations obtained here are solved using the Newmark integration scheme coupled with a modified Newton–Raphson iteration procedure. The analysis brings out various characteristic features of the phenomenon, which are known from experiments, i.e. existence of beats, their dependency on the forcing frequency, the influence of initial conditions and load amplitudes, and the typical character of vibrations in the different regions.

Non-linear multidimensional singular integral equations in two-dimensional fluid mechanics

Abstract: A general two-dimensional fluid mechanics representation analysis is introduced for the determination of the velocity field around a NACA airfoil in an unsteady flow. The problem is reduced to the solution of a non-linear multidimensional singular integral equation, when the form of the source and the vortex strength distribution is dependent on the history of the vorticity and source distribution on the NACA airfoil surface. An application is finally given to the determination of the velocity field around the blades of the special structure of a vertical axis wind turbine, by assuming constant source distribution.

Resonance behavior of viscoelastic fluids in Poiseuille flow and application to flow enhancement

Abstract: Resonance-like phenomena in axisymmetric Poiseuille flows of viscoelastic fluids are studied. It is shown that instantaneous flow velocities of the upper convected Maxwell fluid with a spectrum of relaxation times drastically increase at certain frequencies of the oscillating pressure gradient. This effect becomes more pronounced as the tube radius gets smaller. Non-linear constitutive structures are also considered to show that the most effective mean flow rate enhancement for pulsating flows can be reached at the resonance frequencies.

Buckling of an elastic beam with added high-frequency excitation

Abstract: A non-linear buckling analysis of an elastic beam subjected to an axial static force and high-frequency axial excitation is performed. A Galerkin beam discretization is applied and the method of direct partition of motion is used to obtain a set of autonomous model equations governing the slow averaged behavior. Adding high-frequency excitation increases the buckling load, but stable buckled equilibria may co-exist with the stabilized straight position. The influence of an imperfection in the system is discussed and so is the effect of modal truncation in the discretization.

Wave features and group analysis for an axi-symmetric model of a dusty gas

Abstract: The governing equations of a dusty gas with cylindrical or spherical symmetry are considered. Using the commuting properties of the invariance groups of the system, we are able to characterize appropriate “canonical” variables allowing to reduce the equations to autonomous form. In this way, we characterize particular exact solutions and the propagation of weak discontinuities is considered in such non-constant states.

Non-linear normal modes for systems with discrete symmetry

Abstract: Normal modes in linear mechanical systems with a discrete symmetry group in their equilibrium state can be classified by irreducible representations (irreps) of this group. In non-linear dynamical systems, excitation of a given mode spreads to a number of other modes associated with different irreps, and this collection of modes was called a “bush” of modes in previous papers. There are some special cases where, because of symmetry restrictions , a bush is “irreducible” — it contains modes associated with a single irrep only. We looked for all irreducible bushes of vibrational modes for N -particle mechanical systems with the symmetry of any of the 230 space groups and, for the case of analytical potentials, found that there exist only 19 classes of such bushes. As a result, all modal subspaces to which symmetry determined similar non-linear normal modes (introduced by Rosenberg) belong, were found, as well as all analytical mechanical systems whose dynamics, with a certain mode being initially excited, strictly reduces to only one resonance subspace corresponding to a single irrep. We found that the dimensionality of such resonance subspaces does not exceed four.

Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material

Abstract: In this paper the axial shear deformation of a thick-walled right circular cylindrical tube of compressible isotropic elastic material is considered. Under the assumption that the deformation is isochoric the resulting two governing ordinary differential equations of equilibrium are combined to yield a necessary condition on the strain-energy function for the material to admit pure axial shear (or anti-plane shear). For any strain-energy function satisfying this condition further conditions are required to guarantee the existence of solutions. Such conditions are discussed and used to obtain explicit solutions for several forms of strain-energy function. Existing results are recovered as a special case and new solutions are obtained for some specific strain-energy functions. It is then shown how the results may be used to generate solutions for the axial shear problem in incompressible materials.

Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Liénard systems

Abstract: A parametrically excited Lienard system is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Two coupled equations for the amplitude and the phase of solutions are derived. Their fixed points correspond to limit cycles for the Lienard system and we determine stability of steady-state response as well as response-parametric excitation and response-frequency curves. We use the Poincare–Bendixson theorem, the Dulac's criterion and energy considerations to study existence and characteristics of limit cycles of the two coupled equations. A limit cycle corresponds to a modulated motion in the Lienard system. We show that modulated motion can also be obtained for very low values of the parametric excitation and construct an approximate analytic solution. Moreover, we observe an unusual infinite-period homoclinic bifurcation, because in certain cases due to the symmetry of the two coupled equations two stable limit cycles approach a saddle point and merge to form a greater stable limit cycle. Subsequently, this limit cycle and another unstable limit cycle coalesce and annihilate through a saddle-node bifurcation. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.

On the integration methods of non-holonomic dynamics

Abstract: Some generalizations of the integration methods of the holonomic systems to the non-holonomic systems are given. The generalizations include that of the Whittaker method, the Poisson theory, the field method, the Noether symmetry and the Lie symmetry.

Finite amplitude long-wave instability of power-law liquid films

Abstract: This paper investigates the non-linear long-wave stability of power-law liquid films flow down an inclined plane. The method of long-wave theory is first used to derive a non-linear evolution equation of film thickness. After linearizing the non-linear evolution equation, the method of normal mode is applied to study its linear stability. Then the method of perturbation with multiple scales is used to solve this non-linear equation. The results reveal that the system will be more unstable when power-law exponent n decreases. Near the neutral stable state, the subcritical instability and explosive solution are possible at small n, and the supercritical and unconditional stable region exist only when n exceeds a certain value. Also, decreasing the magnitude of n will increase the dimensional wave speed of the unstable mode.

A Jordan algebra and dynamic system with associator as vector field

Abstract: Dynamic system defined on a vector space V which possesses one or more constraint is found to be likely described by an associator equations system x =[ y , z , u ]≔ y · zu − u · zy . The underlying algebraic structure of such dynamic system is a Jordan algebra with non-associativity, and the resulting associator of this algebra can generate a vector field, which includes one conservative force and one dissipative force. Under certain conditions on the triplet y, z and u, the system may be refreshed as a generalized Hamiltonian system with singular non-canonical metric, or a metric system with degenerate Riemannian metric. For this new formulation, some examples are explored to demonstrate its usefulness, and then the possibility to describe the non-linear dissipative phenomena of physical systems is suggested.

On the control of programmed motion of a rigid body containing moving masses

Abstract: This paper is devoted to the study of the asymptotic stability of programmed motion of a rigid body, with the help of internal moving masses using the Lyapunov function. The adopted method is based on the choice of the programmed servo-control forces such that the rigid body performs the programmed motion, and the stabilizing servo-control forces of this motion. The equilibrium position of the rigid body which occurs at the principal axes of the body coincides with the inertial axes is proved to be asymptotically stable as a special case of the studied problem. The servo-control forces ensuring that it is btained. The advantages of this study are the exactness of the equations of motion and that the servo-control forces are derived as exactly non-linear functions of phase coordinates of the system and are not approximated as linear.

Influence of strain hardening on continuous plate roll-bending process

Abstract: Influence of material strain hardening on the mechanics of steady continuous roll- and edge-bending mode in the four-roll plate bending process was investigated. An analytical method to solve the governing differential equations for large deflection of elastoplastic thin plate with any general strain hardening law of material was developed. Numerical results for the physical quantities of (i) the shift of plate contact angle with rolls, (ii) the forces applied on rolls and (iii) the position setup of side roll, etc. were obtained and shown graphically as a function of the desired finish curvature of plate made of linear strain hardening material. Comparison of the results with perfectly plastic material is also made.

Buckling and post-buckling of a ferromagnetic beam-plate induced by magnetoelastic interactions

Abstract: Buckling and post-buckling behavior of a soft ferromagnetic beam-plate with unmovable simple supports under an applied magnetic field is analyzed by taking into account the non-linear effect of large deflections based on the von Karman model of plates. The study shows that buckling is proceeded by bending only if the applied magnetic field forms an incident angle with the normal of the plate. The characteristics of buckling and post-buckling are explored by numerical analysis. In particular, the numerical solution shows that, when the applied magnetic field is oblique, the plate snapps from a whole wave configuration into a half-wave configuration as buckling/snapping occurs, and that the magnetoelastic buckling strength increases with the increasing oblique angle.

Statistic moments of the total energy of potential systems and application to equivalent non-linearization

Abstract: In this paper some properties of the total energy moments of potential systems, subjected to external white noise processes, are shown. Potential systems with a polynomial form of energy-dependent damping have been considered. It is shown that the analytical relations between the statistical moments of the energy associated with such systems can be obtained with the aid of the standard Ito calculus. Furthermore, it is shown that, for the stationary case, these analytical relations are very useful for the application of the equivalent non-linearization technique.