Showing papers in "International Journal of Non-linear Mechanics in 1984"

Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations

Abstract: The statistical moments of a non-linear system responding to random excitations are governed by an infinite hierarchy of equations; therefore, suitable closure schemes are needed to compute the more important lower order moments approximately. One easily implemented and versatile scheme is to set the cumulants of response variables higher than a given order to zero. This is applied to three non-linear oscillators with very different dynamic properties, and with Gaussian white noises acting as external and/or parametric excitations. It is found that the accuracy of computed second moments can be improved greatly by extending from the second order closure (Gaussian closure) to the fourth order closure, and that further refinement is unnecessary for practical purposes. Treatment of nonstationary transient response is also illustrated.

Planar non-linear free vibrations of an elastic cable

Abstract: Continuum non-linear equations of free motion of a heavy elastic cable about a deformed initial configuration are developed. Referring to an assumed mode technique one ordinary equation for the cable planar motion is obtained via a Galerkin procedure, an approximate solution of which is pursued through a perturbation method. Suitable nondimensional results are presented for the vibrations in the first symmetric mode with different values of the cable properties. Which procedure is the proper one to account consistently for the non-linear kinematical relations of the cable in one ordinary equation of motion is discussed.

Lagrangian description and incremental formulation in the non-linear theory of thin shells

Abstract: Equations of equilibrium and four geometric and static boundary conditions are constructed for an entirely Lagrangian non-linear theory of shells. In the case of a linearly-elastic material and conservative surface and boundary loadings the shell relations are derivable as stationarity conditions of the Hu-Washizu free functional. For the geometrically non-linear first-approximation theory of elastic shells several consistently simplified versions of the shell equations are discussed. Several sets of equations for teories of shells undergoing moderate or large/small rotations are presented. The majority of the simplified versions allow an exact variational formulation using a Hu-Washizu free functional. A unified theory of superposition of non-linear deformations in thin shells is outlined and two equivalent incremental formulations of shell equations in the total Lagrangian and the updated Lagrangian descriptions are given.

Certain inverse solutions of a non-Newtonian fluid

Abstract: Solutions for the equations of motion of an incompressible second grade fluid are obtained by employing semi-inverse methods in which we assume certain geometrical or kinematical properties of the fields. Specifically the problems studied in viscous fluids by Jeffery, Hamel and Gortler and Wieghardt, etc. are considered in a second grade fluid and the results for stream lines, velocities and pressure distribution are compared in the two cases.

A perturbation method for certain non-linear oscillators

Abstract: We present a perturbation method for the analysis of single degree of freedom non-linear oscillation phenomena governed by an equation of motion containing a parameter ϵ which need not be small. The approach is to define a new parameter α = α ( ϵ ) in such a way that asymptotic solutions in power series in α converge more quickly than do the standard perturbation expansions in power series in ϵ. Phenomena considered are free vibration of strongly non-linear conservative oscillators and steady state response of strongly non-linear oscillators subject to weak harmonic excitation.

A note on integration of the Emden-Fowler equation

Abstract: Using a simple change of variables, the Emden-Fowler equation, ( x v + α y ′)′ + ax v y n = 0 is shown to be integrable provided that either of the constraints ( v + α − 1) n = 3 − α + v or ( v + α − 1) n = 3 − 2 α − v is satisfied. Every integrable case generates a one parameter family of integrable Emden-Fowler equations.

Coupled flutter and divergence bifurcation of a double pendulum

Abstract: The loss of stability of the equilibrium position of a double pendulum with follower force loading and elastic end support is studied. At a special parameter combination the linearized system is characterized by a zero root and a pure imaginary pair of eigenvalues. Therefore, the stability problem is a complicated critical case in the sense of Liapunov and requires a non-linear analysis. A complete post-bifurcation investigation of the coupled divergence and flutter motions is given by means of centre manifold theory, and bifurcation diagrams. Among the different types of motions even the appearance of chaotic behavior is shown.

Radiation instability in a stratified shear flow

Abstract: The paper deals with a new mechanism of instability in stratified fluid caused by internal wave radiation from a shear layer. This radiation instability may be explained in terms of the negativeenergy wave concept. These waves are able to grow due to radiation losses. The dispersion relation and non-linear stage of the process are analyzed through a model with tangential velocity discontinuity. The non-linear description is based on a long-wave approximation leading to an evolution equation of the Korteweg-de Vries type with certain additional terms responsible for instability as well as dissipation due to turbulence. Approximate solutions of the equation are obtained which, in particular, describe the evolution of solitons including explosive growth and approaching the steady state.

Finite deformation and stability behaviour of spherical inflatables under axi-symmetric concentrated loads

Abstract: The large deflection and stability behaviour of spherical inflatables subjected to a concentrated load applied at the apex and undergoing various degrees of wrinkling is analyzed. It is shown that for low profile inflatables, the behaviour of such structures under axi-symmetric concentrated loads is non-linear, without instability or limit point characteristics. On the other hand for high-profile membranes with height/span ratios greater than 0.5 and undergoing deflections equal to or greater than the initial height or radius of curvature of the structure, “snap through” behaviour is established. Ultimate loads for the totally wrinkled membranes are also established. The wrinkled region is considered in an Eulerian description satisfying equilibrium equations and the Gauss-Codazzi relations. The equations are solved numerically and the results are presented in non-dimensional form in a number of figures.

A field method and its application to the theory of vibrations

Abstract: The problem of integration of the differential equations of motion of a nonconservative dynamical system is replaced by an equivalent problem of finding a complete integral of a quasi-linear partial differential equation of the first order. In the second part, these complete integrals are combined with the two time scales perturbation method in the study of non-linear oscillatory motions.

A kinetic model for rapid flows of granular materials

Abstract: A kinetic model for rapid flows of granular material is considered. An evolution equation for the first order distribution function is developed which includes the effects of intergranular friction force in an average sense. The collision operator is approximated by the BGK relaxation model. The fundamental equations of motion including an equation for dynamics of the fluctuation energy are derived and discussed. The gravity and the plane Couette flows of granular materials are treated as examples of the applications of the present theory.

Some energetic properties of smooth solutions in rate-type viscoelasticity

Abstract: A free energy function compatible with the second law of thermodynamics is constructed for the semilinear rate-type viscoelasticity. Under physically acceptable conditions this energy function is a positive and convex function of stress and strain. It is shown that for a class of one dimensional initial-boundary value problems, the total energy at any time is bounded by the sum of the total energy of the initial data and the energy supplied to the body by nonvanishing body forces. A Maxwell type viscosity approach to non-linear elasticity for an isolated body is also discussed. The stability in total energy and uniqueness of the smooth solutions of some initial-boundary value problems is discussed for the one dimensional case as well as for the three dimensional case of small deformations.

Integrating factors and conservation laws for nonconservative dynamical systems

Abstract: A general approach to the construction of conservation laws for classical nonconservative dynamical systems is presented. The conservation laws are constructed by finding corresponding integrating factors for the equations of motion. Necessary conditions for existence of the conservation laws are studied in detail. A connection between an a priori known conservation law and the corresponding integrating factors is established. The theory is applied to two particular problems.

SH waves due to a point source in an inhomogeneous medium

Abstract: The propagation of SH waves due to a point source in an homogeneous medium lying over an inhomogeneous substratum has been investigated in this paper. The Green's function technique developed by Ghosh has been used to solve the problem.

A unilateral contact problem with the heavy elastica

Abstract: A heavy inextensible elastic beam of infinite length and lying on a rigid foundation, is loaded by a concentrated force directed opposite to the gravity field. As a result a region of non-contact develops. This contact problem, in which finite deflections are considered, leads to a free boundary value problem for a system of non-linear ordinary differential equations. This system is discretized by means of finite-differences, and a Newton-Raphson method is applied to solve the nonlinear equations. In this type of problems the matrix has not a band structure and this hampers the use of available fast numerical procedures. However, through the use of an artificial devise this difficulty could be circumvented. Numerical results are given and their accuracy is discussed, in particular for large values of the external force.

On the problem of vibrations of non-linear-elastic electroconductive plates in transverse and longitudinal magnetic fields

Abstract: Non-linear cylindrical-form vibrations of an electroconductive plate-strip in longitudinal and transverse magnetic fields are considered. For the plate material a non-linear relationship between stress and deformation is accepted. On the basis of the known hypotheses and assumptions the general system is reduced to one equation for the normal displacement of the plate. Using the Bubnov-Galerkin method and the method of asymptotic integration, the amplitude-frequency characteristics of vibrations are defined. On the basis of numerical analysis different degrees of the influence of various factors (non-linearity, magnetic field, conductivity, etc.) on the nature of vibrations are compared. Qualitative and quantitative differences between the nature of vibrations in longitudinal and transverse magnetic fields are revealed.

Dynamic stability for a simply supported beam under periodic axial excitation

Abstract: 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 1 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.

Group analysis and constitutive laws for fluid filled elastic tubes

Abstract: The group properties are investigated for the non-linear mathematical model describing thin walled elastic tubes filled with incompressible fluid. The analysis developed for determining the generators of the group provides a mathematical approach for characterizing classes of constitutive laws for internal area, outflow and viscous retarding force. Thus integration of a system of (non-linear) ordinary differential equations gives rise to several classes of invariant solutions for non-linear arterial flow problems. Some features of these solutions are also discussed.

Large deplection of corrugated circular plate with a plane central region under the action of concentrated loads at the center

Abstract: This paper solves the problem of large deflections for corrugated circular plates with a plane central region under the action of concentrated loads at the center by means of the non-linear bending theory for anisotropic circular plates. Using the modified iteration method, the characteristic relation of the plate is obtained. This formula may be applied directly to design of elastic elements of measuring instruments.

Dynamical modelling of wind-induced vibrations of overhead lines☆

Abstract: A dynamical model is presented for the wind-induced vibrations of overhead lines due to Karman vortex shedding Because of the complexity of the excitation mechanism the model for the continuous structure is based on an appropriate non-linear model of a circular rigid cylinder that is oscillating transversally in a flowing fluid The exciting non-linear forces acting on the cylinder are deduced from known experimental data It is shown that the observed well-known “lock-in”-effect can be described in satisfactory manner by this model The transition from the rigid body model to the continuous, flexible structure leads to a non-linear boundary value problem Approximate solutions are derived by perturbation theory The model is able to predict the observed vibration frequency and the corresponding vibration mode (depending on the wind velocity) as well as the observed vibration amplitude An important consequence of the model consists in the self-restricting behaviour of the aerodynamic forces On this basis, a new conception of the vortex shedding mechanism on a vibrating cable will be presented

Non-linear axisymmetric static analysis of orthotropic thin annular plates

Abstract: This investigation deals with the geometrically non-linear axisymmetric static analysis of elastic orthotropic thin annular plates subjected to uniformly distributed and ring loads. Non-linear differential equations in terms of the transverse displacement w and stress function ψ have been employed. Both w and ψ are expanded in finite power series and the orthogonal point collocation method has been used to obtain discretised algebraic equations from the governing differential equations. Results have been presented for annular plates with and without a rigid plug in the hole; simply supported, clamped and free outer edge conditions; three orthotropic parameters β; two annular ratios and two loading conditions. Results have also been presented illustrating the effect of an elastic rotational constraint at the edge and the effect of a prescribed inplane displacement of the edge.

Free vibrations of a loaded rubber string

Abstract: The problem of the finite amplitude, free vertical oscillatory motion of a massattached to a neo-Hookean rubberlike string is solved exactly in terms of elementary functions and the Heuman lambda function, which is related to the elliptic integral of the third kind. Hence, the period of the oscillations for the various possible motions may be computed from tables of values of the complete lambda function. It is shown that the results differ significantly from those obtained elsewhere for a string having linearly elastic behavior; and all of the results are described graphically.

Parameter and state identification in non-linearizable uncertain systems

Abstract: A dynamical process is modelled by a system of non-linearizable ordinary differential equations with uncertain but bounded state variables and variable parameters. When stochastic identification is not feasible (no assumptions upon random parameters, single run control, etc), the "worst case" design is required. To avoid this penalty, we propose to extend the Liapunov design technique of building adaptive (on-line) identifiers, so far developed for linear systems with constant parameters. The standard study of stability of an error-equation is replaced by investigating convergence to diagonal set in the Cartesian product of state-parameter spaces of the model and the identifier. We also attempt to stabilize the model. Conditions for the above are introduced together with proposing suitable Liapunov functions. The method is illustrated on two examples with wide applicability: a damped Hamiltonian system and the non-linear oscillator.

Large deflections of bars bent through frictionless supports

Abstract: A procedure enabling the numerically exact solution of the deflected shape of a bar bent through frictionless supports is presented. It is shown that the axial force at any point is proportional to the square of the bending moment, and based on this property, coefficients relating end rotations and end moments for a single member are presented. A non-linear analysis using Newton-Raphson iteration is then adopted for solving the overall problem.

On the non-linear buckling and post-buckling analysis of thin elastic shells☆

Abstract: In this paper the non-linear stability and post-buckling analysis of elastic structures is considered in the frame of a geometrically non-linear shell theory with moderate rotations. Using a total Lagrangian description variational statements as well as associated sets of shell equations including boundary conditions are derived to determine the fundamental equilibrium path, critical points of snap-through or bifurcation buckling and also the post-buckling deformations. To present these equations in a compact form a unified operator description is introduced. It also allows one to prove some important properties, which are needed to construct appropriate approximation procedures like finite element methods. A shell example is calculated numerically by using a simple Rayleigh-Ritz approximation. It is shown that for a two-parameter loading the collection of critical snap-through buckling points is a catastrophe of the ‘cusp’ type.

On the energy transfer equation for weakly interacting waves

Abstract: The derivation of the transfer equation based on analysis of the equations for spectral semi-invariant and not invoking equations for realization of the random wave field is presented Uniformly valid asymptotic expansions for the third and the fourth spectral semi-invariant are constructed using the multiple scale method and the matched asymptotic expansion method This approach makes it possible to investigate the boundary layer in a neighbourhood of the resonant surface where intensive growth in time of the third spectral semi-invariant occurs This boundary layer defines the form of the transfer equations An analogous boundary layer for the fourth spectral semiinvariant and its influence on the second and the third spectral semi-invariants are also investigated

Dynamic shakedown of elastic-plastic solids for a set of alternative loading histories

Abstract: The paper deals with dynamic shakedown of an elastic-perfectly plastic solid body subjected to a loading history which is unknown but is allowed to belong to a given set of loading histories. In the hypothesis of a piecewise linear convex set, a sufficient shakedown theorem is given and a bounding principle for the plastic work produced is formulated in terms of the dynamic elastic responses to a discrete set of loading histories. The solution of a minimization problem gives the most stringent bound which also proves to possess a local character, i.e., it regards the plastic work density at any point.

On rate-type viscoelasticity and the second law of thermodynamics

Abstract: We deduce an energy identity which must be satisfied by the smooth solutions of the system of equations governing the dynamics of body with quasilinear rate-type constitutive equation. We give conditions when a unique energy function exists for rate-type viscoelasticity. In the semilinear case we give the conditions when a unique, positive and convex energy function exists and we obtain estimates in energy for the smooth solutions of initial-boundary value problems. A viscoelastic approach to nonlinear elasticity is discussed. Finally, an example shows that the second law of thermodynamics does not imply stability.

Optimal location of additional supports for plastic cylindrical shells subjected to impulsive loading

Abstract: Necessary optimality conditions for location of the additional supports, restricting the dynamic response of a rigid perfectly plastic cylindrical shell subjected to an initial transverse velocity, are deduced. Incorporating the concept of the mode-form motions the problem is treated as a self-conjugate problem of the optimal control theory with distributed parameters. We use the control technique, developed previously for the case in which the state variables (the bending moment, the shear force, the deflection and its rate) together with these derivatives, being continuous in certain subdomains, may have discontinuities on the boundaries of these subregions. A particular case. associated with the uniform initial impulse is examined in detail.

Non-linear convection in a porous medium with internal heat sources

Abstract: The paper studies non-linear thermal convection in a horizontal porous layer of fluid with nearly insulating boundaries and in the presence of internal heat sources. Square and hexagonal cells are found to be the only possible stable convection cells. Finite amplitude instability could exist for some particular forms of an internal heat source Q. For a uniform Q, the preferred flow pattern is that of hexagons for amplitude e smaller than some critical value ec, while both squares and hexagonal cells are stable for e ⩾ ec. The convective motion is downward at the hexagonal cell's centers. For a non-uniform Q, the qualitative features of thermal convection depend on the actual form Q. In particular, a non-uniform Q can increase or decrease the cell's size and the critical Rayleigh number at the onset of convection, and stabilize or destabilize the convective motion in the form of hexagonal cells with either upward or downward motion at the cell's centers.