Showing papers in "International Journal of Non-linear Mechanics in 2003"

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

Abstract: The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.

Non-linear analysis of functionally graded plates under transverse and in-plane loads

Abstract: Large deflection and postbuckling responses of functionally graded rectangular plates under transverse and in-plane loads are investigated by using a semi-analytical approach. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on the classical plate theory, accounting for the plate-foundation interaction effects by a two-parameter model (Pasternak-type), from which Winkler elastic foundation can be treated as a limiting case. A perturbation technique in conjunction with one-dimensional differential quadrature approximation and Galerkin procedure are employed in the present analysis. The numerical illustrations concern the large deflection and postbuckling behavior of functional graded plates with two pairs of constituent materials. Effects played by volume fraction, the character of boundary conditions, plate aspect ratio, foundation stiffness, initial compressive stress as well as initial transverse pressure are studied.

Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems

Abstract: Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.

Non-linear constitutive relations for magnetostrictive materials

Abstract: In this paper, non-linear deformation behavior of magnetostrictive materials is studied and three magnetoelastic coupling constitutive models are developed. The standard square (SS) constitutive model is developed by means of truncating the polynomial expansion of the Gibbs free energy. The hyperbolic tangent (HT) constitutive equations, which involve a hyperbolic tangent magnetic-field dependence, are proposed to model the magnetic-field-induced strain saturation of magnetostrictive materials in the region of intense magnetic fields. A new model based on density of domain switching (DDS) is established in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation. In this model, it is assumed that the relation between density of domain switching, defined by the quantity of magnetic domains switched by per unit magnetic field and magnetic field can be described by a density function with normal distribution. The moduli in these constitutive models can be determined by a material function that is proposed to describe the dependence of the peak piezomagnetic coefficient on the compressive pre-stress for one-dimensional cases based on the experimental results published. The accuracy of the non-linear constitutive relations is evaluated by comparing the theoretical values with experimental results of a Terfenol-D rod operated under both compressive pre-stress and bias magnetic field. Results indicate that the SS constitutive equations can accurately predict the experimental results under a low or moderate magnetic field while the HT model can, to some extent, reflect the trend of saturation of magnetostrictive strain under a high magnetic field. The model based on DDS, which is more effective in simulating the experimental curves, can capture the main characteristics of the mechanism of magnetoelastic coupling deformation of a Terfenol-D rod, such as the notable dependence of magnetoelastic response on external stress and the saturation of magnetostrictive strain under intense magnetic fields. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed.

A new exact solution for the flow of a Maxwell fluid past an infinite plate

Abstract: A new exact solution corresponding to the flow of a Maxwell fluid over a suddenly moved flat plate is determined. This solution is in all accordance with a previous one and for λ→0 it goes to the well-known solution for Navier–Stokes fluids.

Buckling analysis of cylindrical shells with random geometric imperfections

Abstract: In this paper the effect of random geometric imperfections on the limit loads of isotropic, thin-walled, cylindrical shells under deterministic axial compression is presented. Therefore, a concept for the numerical prediction of the large scatter in the limit load observed in experiments using direct Monte Carlo simulation technique in context with the Finite Element method is introduced. Geometric imperfections are modeled as a two dimensional, Gaussian stochastic process with prescribed second moment characteristics based on a data bank of measured imperfections. (The initial imperfection data bank at the Delft University of Technology, Part 1. Technical Report LR-290, Department of Aerospace Engineering, Delft University of Technology). In order to generate realizations of geometric imperfections, the estimated covariance kernel is decomposed into an orthogonal series in terms of eigenfunctions with corresponding uncorrelated Gaussian random variables, known as the Karhunen–Loeve expansion. For the determination of the limit load a geometrically non-linear static analysis is carried out using the general purpose code STAGS (STructural Analysis of General Shells, user manual, LMSC P032594, version 3.0, Lockheed Martin Missiles and Space Co., Inc., Palo Alto, CA, USA). As a result of the direct Monte Carlo simulation, second moment characteristics of the limit load are presented. The numerically predicted statistics of the limit load coincide reasonably well with the actual observations, particularly in view of the limited data available, which is reflected in the statistical estimators.

Analytical approximations for stick–slip vibration amplitudes

Abstract: The classical “mass-on-moving-belt” model for describing friction-induced vibrations is considered, with a friction law describing friction forces that first decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions, the amplitudes, and the base frequencies of friction-induced stick–slip and pure-slip oscillations. For stick–slip oscillations, this is accomplished by using perturbation analysis for the finite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. The results are illustrated and tested by time-series, phase plots and amplitude response diagrams, which compare very favorably with results obtained by numerical simulation of the equation of motion, as long as the difference in static and kinetic friction is not too large.

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

Abstract: The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.

The first problem of Stokes for an Oldroyd-B fluid

Abstract: The velocity field and the associated tangential tension corresponding to the flow of an Oldroyd-B fluid over a suddenly moved flat plate are determined. The well-known solutions for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second-grade one, appear as limiting cases of our solutions. Finally, some comparative diagrams concerning the velocity and tension profiles are presented for different values of the material constants.

An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude

Abstract: The currently developed analytic technique known as the homotopy analysis method is employed to propose a new approach for free oscillations of positively damped systems with algebraically decaying amplitude. In contrast to perturbation techniques, this approach is valid even for damped systems without any small/large parameters. Besides, unlike other analytic techniques, this approach itself provides us with a convenient way to adjust and control convergence of approximation series. Some typical examples are employed to illustrate its validity, effectiveness and flexibility.

Diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet

Abstract: The influence of reaction rate on the transfer of chemically reactive species in the laminar visco-elastic fluid flow immersed in a porous medium over a stretching sheet is considered. The flow is caused solely by the linearly stretching sheet and the reactive species is emitted from this sheet and undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. A similarity transformation is introduced, which reduces the concentration conservation equation to an ordinary differential equation. An exact analytical solution due to Siddappa and Abel (Z. Angew. Math. Phys. 36 (1985) 890) is adopted for velocity, where as the concentration equation is obtained numerically for higher-order reactions. The numerical computations show that the effect of destructive chemical reaction is to reduce the thickness of concentration boundary layer and increase the mass transfer rate from the sheet to the surrounding fluid. This effect is more effective for zero- and first-order reaction than second- and third-order reactions.

Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback

Abstract: We investigate the primary resonance of an externally excited van der Pol oscillator under state feedback control with a time delay. By means of the asymptotic perturbation method, two slow-flow equations on the amplitude and phase of the oscillator are obtained and external excitation–response and frequency–response curves are shown. We discuss how vibration control and high amplitude response suppression can be performed with appropriate time delay and feedback gains. Moreover, energy considerations are used in order to investigate existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-period modulated motion for the van der Pol oscillator. We demonstrate that appropriate choices for the feedback gains and the time delay can exclude the possibility of modulated motion and reduce the amplitude peak of the primary resonance. Analytical results are verified with numerical simulations.

Fragility analysis of bridges under ground motion with spatial variation

Abstract: Seismic ground motion can vary significantly over distances comparable to the length of a majority of highway bridges on multiple supports. This paper presents results of fragility analysis of highway bridges under ground motion with spatial variation. Ground motion time histories are artificially generated with different amplitudes, phases, as well as frequency contents at different support locations. Monte Carlo simulation is performed to study dynamic responses of an example multi-span bridge under these ground motions. The effect of spatial variation on the seismic response is systematically examined and the resulting fragility curves are compared with those under identical support ground motion. This study shows that ductility demands for the bridge columns can be underestimated if the bridge is analyzed using identical support ground motions rather than differential support ground motions. Fragility curves are developed as functions of different measures of ground motion intensity including peak ground acceleration, peak ground velocity, spectral acceleration, spectral velocity and spectral intensity. This study represents a first attempt to develop fragility curves under spatially varying ground motion and provides information useful for improvement of the current seismic design codes so as to account for the effects of spatial variation in the seismic design of long-span bridges.

On unsteady motions of a second-order fluid over a plane wall

Abstract: In this paper, three types of unsteady flows of second-order fluids are considered, namely, flow caused by impulsive motion of a flat plate, flow induced by a constantly accelerating plane and flow imposed by a flat plate that applies a constant tangential stress to the fluid. The previous attempts made regarding these problems, by using the Laplace transform, have failed. In this paper, the sine and the cosine transforms are used to solve these problems and exact solutions for the velocity distributions are found in terms of definite integrals. It is shown that these exact solutions satisfy the initial and the boundary conditions and the governing equation.

Vibration control for a flexible-link robot arm with deflection feedback

Abstract: The use of flexible links in a robot inevitably causes the elastic deflection and vibration of the endpoint of the robot during high-speed operations. The deflection and vibration will tend to degrade the positioning performance of the robot. In this paper, an optical sensing system consisting of a laser diode and a position sensitive detector is introduced for the real-time measurement of the dynamic deflection. Utilising a non-linear, coupled and measurement-based dynamic system model, a Lyapunov-type controller based on the deflection feedback is then proposed to damp out the tip oscillations and regulate the endpoint of the flexible robot. Experimental tests are conducted for a flexible one-link robot arm with a payload mass at the tip. The results demonstrate the effectiveness of the proposed measuring and control schemes.

Implications of cubic physical/aerodynamic non-linearities on the character of the flutter instability boundary

Abstract: The present paper deals with a study of the benign and catastrophic characters of the flutter instability boundary of 2-D lifting surfaces in a supersonic flow field. The objectives of this work are: (i) to contribute to a better understanding of the implications of aerodynamic and physical non-linearities on the character of the flutter boundary and (ii), to outline the effects exerted in the same respect by some important parameters of the aeroelastic system. With the aim of addressing this problem, the method based on the First Liapunov Quantity is used to study the bifurcational behavior of the aeroelastic system in the vicinity of the flutter boundary. The expected outcomes of this study are: (a) to greatly enhance the scope and reliability of the aeroelastic analysis and design criteria of advanced aircraft and, (b) to provide a theoretical basis for the analysis of more complex non-linear aeroelastic systems.

The effect of tangential elasticity of the contact layer between stator and rotor in travelling wave ultrasonic motors

Abstract: In travelling wave ultrasonic motors the elliptical motion of material points of the stator drives the rotor due to frictional mechanisms. The motor characteristic strongly depends on the mechanical properties of the components stator, rotor and contact layer. In order to predict the motor behaviour, a model for the contact between stator and rotor has been developed. The goal of the present paper is to point out the importance of the tangential elasticity of the contact layer which is responsible for the formation of stick zones and also for the amount of friction losses and overall efficiency. Therefore a comparison with a model with a contact layer rigid in tangential direction is given. Based on a visco-elastic foundation model for the contact layer, torque–speed curves as well as torque–efficiency curves are computed. Experimental investigations for identification of parameters, check of assumptions and model validation are carried out. Finally, the model is used to show the results of parameter variations for normal force, vibration amplitude and modulus of elasticity of the contact layer.

Constitutive relations for the interaction force in multicomponent particulate flows

Abstract: In the mechanics of multiphase (or multicomponent) mixtures, one of the outstanding issues is the formulation of constitutive relations for the interaction force. In this paper, we give a brief review of the various relations proposed for this interaction force. The review is tilted toward presenting the works of those who have used the mixture theory (or the theory of interacting continua) to derive or to propose a relationship for the interaction (or diffusive) force. We propose a constitutive relation which is general and frame-indifferent and thus suitable for use in many flow conditions. At the end, we provide an alternative approach for finding the drag force on a particle in a particulate mixture. This approach has been used in the non-Newtonian fluid mechanics to find the drag force on surfaces.

Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions

Abstract: The effect of non-linear magnetic forces on the non-linear response of the shaft is examined for the case of superharmonic resonance in this paper. It is shown that the steady-state superharmonic periodic solutions lose their stability by either saddle-node or Hopf bifurcations. The system exhibits many typical characteristics of the behavior of non-linear dynamical systems such as multiple coexisting solutions, jump phenomenon, and sensitive dependence on initial conditions. The effects of the feedback gains and imbalance eccentricity on the non-linear response of the system are studied. Finally, numerical simulations are performed to verify the analytical predictions.

Exact flow of a third-grade fluid on a porous wall

Abstract: The flow of a third-grade fluid occupying the space over a wall is studied. At the surface of the wall suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a non-linear partial differential equation. The resulting equation is analysed analytically using Lie group methods.

Spectral simulation and shock absorber identification

Abstract: In vehicle dynamics shock absorbers are used for the optimization of driving comfort and driving safety. Therefore, it is necessary to identify characteristics of shock absorbers under real conditions. This paper introduces the use of hardware-in-the-loop simulations for the identification of shock absorbers involving stochastic models of the road roughness. For this purpose a dynamic hydraulic test stand is used replacing the classical mechanical test stands which allow only sinusoidal excitation. For the Monte Carlo simulation with a real shock absorber in the loop, the random excitation of ground roughness is generated using a modified spectral representation method based on the famous contributions of Shinozuka. Motion and force of the shock absorber are measured and fed back to the Monte Carlo simulation of a car model in real time. The characteristic of the shock absorber is identified using the classical least squares method and a correlation-based method. A piecewise linear model for the characteristic relating the damping force and the velocity of the piston is applied for the shock absorber identification.

Sufficient conditions for strong ellipticity for a class of anisotropic materials

Abstract: We provide sufficient conditions for strong ellipticity for a general class of anisotropic hyperelastic materials. This general class includes as a subclass transversely isotropic materials. Our sufficient conditions require that the first partial derivatives of the reduced-stored energy function satisfy some simple inequalities and that the second partial derivatives satisfy a convexity condition. We also characterize a restricted type of strong ellipticity for a subclass of transversely isotropic materials undergoing pure homogeneous deformations. We apply our results to a model of soft tissue from the biomechanics literature.

Exact stationary probability density functions for non-linear systems under Poisson white noise excitation

Abstract: The paper presents exact stationary probability density functions for systems under Poisson white noise excitation Two different solution methods are outlined In the first one, a class of non-linear systems is determined whose state vector is a memoryless transformation of the state vector of a linear system The second method considers the generalized Fokker–Planck (Kolmogorov-forward) equation Non-linear system functions are identified such that the stationary solution of the system admits a prescribed stationary probability density function Both methods make use of the stochastic integro-differential equations approach This approach seems to have some computational advantages for the determination of exact stationary probability density functions when compared to the stochastic differential equations approach

Imperfection sensitivity of curved panels under combined compression and shear

Abstract: The initial buckling load of curved panels under compressive loads is substantially reduced by the existence of imperfections, in particular geometric imperfections. It is therefore essential that these imperfections are considered in analysing components which incorporate such panels in order to accurately predict their buckling behaviour. Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections, however, to obtain accurate results information is required on the exact size and shape of the imperfection to be modelled. In most cases this data is not available. It is therefore generally recommended that the imperfections are modelled on the first eigenmode and have an amplitude selected according to the manufacturing procedure. This paper presents the effects of varying imperfection shape and amplitude on the buckling and postbuckling behaviour of one specific case, a curved panel under combined shear and compression, to test the accuracy of such recommendations.

Non-isothermal oscillations of pseudoelastic devices

Abstract: The non-linear dynamic response of a pseudoelastic oscillator embedded in a convective environment is studied taking into account the temperature variations induced, during oscillations, by the latent heat of transformation and by the heat exchange with the surroundings. The asymptotic periodic response under harmonic excitation is characterized by frequency–response curves in terms of maximum displacement, maximum and mean temperature. The periodic thermomechanical response is computed by a multi-component harmonic balance method implemented within a continuation algorithm that enables to trace out multivalued frequency–response curves. The accuracy of the results is checked by comparison with the results of the numerical integration of the basic equations governing the dynamics of the system. The response is investigated for various excitation amplitude levels and in various material parameters ranges. The resulting picture of the mechanical response shows, in some cases, features similar to other hysteretic oscillators, while, in other cases, points out peculiar behaviors. It turns out that the temperature variations induced by the phase transformations influence the mechanical response and that the results obtained under the simplifying assumption of isothermal behavior can be rather different from those obtained in a fully thermomechanical setting.

Post-buckling analysis of slender elastic rods subjected to terminal forces

Abstract: This paper presents formulation and solutions for the elastica of slender rods subjected to axial terminal forces and boundary conditions assumed hinged and elastically restrained with a rotational spring. The set of five first-order non-linear ordinary differential equations with boundary conditions specified at both ends constitutes a complex two-point boundary value problem. Solutions for buckling, initial post-buckling (perturbation), large loads (asymptotic) and numerical integration are developed. Results are presented in non-dimensional graphs for a range of rotational spring stiffness, tuning the analysis from double-hinged to hinged–built-in rods.

Analytical approximation to large-amplitude oscillation of a non-linear conservative system

Abstract: This paper deals with non-linear oscillation of a conservative system having inertia and static non-linearities. By combining the linearization of the governing equation with the method of harmonic balance, we establish analytical approximate solutions for the non-linear oscillations of the system. Unlike the classical harmonic balance method, linearization is performed prior to proceeding with harmonic balancing, thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations. Hence, we are able to establish analytical approximate formulas for the exact frequency and periodic solution. These analytical approximate formulas show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation.

Large free vibrations of a beam carrying a moving mass

Abstract: The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution. In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.

The Rayleigh-Stokes-problem for a fluid of Maxwellian type

Abstract: The velocity fields corresponding to an incompressible fluid of Maxwellian type subjected to a linear flow on an infinite flat plate and within an infinite edge are determined by means of the Fourier sine transforms. They are in close proximity of those of a second grade fluid. The well known solutions for a Navier–Stokes fluid appear as a limiting case of our solutions.

Stokes’ first problem for a Rivlin–Ericksen fluid of second grade in a porous half-space

Abstract: Stokes’ first problem for a Rivlin–Ericksen fluid of second grade in a porous half-space is considered under isothermal conditions. Laplace transform techniques are used to determine the exact solution, temporal limits, small-time expressions, and displacement thickness. In addition, special/limiting cases are noted, energy aspects are covered, and numerical results are presented graphically. Most significantly, it is shown that the flow suffers a jump discontinuity on start-up, that due to this jump a nonpositive steady-state development time can result, and that for a special case of the material constants the flow instantly attains its steady-state configuration.