Showing papers in "International Journal of Non-linear Mechanics in 2010"

The dynamic behavior of MEMS arch resonators actuated electrically

Abstract: In this paper, we investigate the dynamic behavior of clamped–clamped micromachined arches when actuated by a small DC electrostatic load superimposed to an AC harmonic load. A Galerkin-based reduced-order model is derived and utilized to simulate the static behavior and the eigenvalue problem under the DC load actuation. The natural frequencies and mode shapes of the arch are calculated for various values of DC voltages and initial rises. In addition, the dynamic behavior of the arch under the actuation of a DC load superimposed to an AC harmonic load is investigated. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response of the arch due to DC and small AC loads. Results of the perturbation method are compared with those obtained by numerically integrating the reduced-order model equations. The non-linear resonance frequency and the effective non-linearity of the arch are calculated as a function of the initial rise and the DC and AC loads. The results show locally softening-type behavior for the resonance frequency for all DC and AC loads as well as the initial rise of the arch.

A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells

Abstract: A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Karman type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Karman type for vibration amplitudes of about two times the shell thickness for the studied case.

An integral approach for large deflection cantilever beams

Abstract: A new integral approach is proposed to solve the large deflection cantilever beam problems. By using the moment integral treatment, this approach can be applied to problems of complex load and varying beam properties. This versatile approach generally requires only simple numerical techniques thus is easy for application. Treatment for typical loading and beam property conditions are presented to demonstrate the capability of this approach.

Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates

Abstract: The dynamic behavior of partially delaminated at the skin/core interface sandwich plates with flexible cores is studied. The commercial finite element code ABAQUS is used to calculate natural frequencies and mode shapes of the sandwich plates containing a debonding zone. The influence of the debonding size, debonding location and types of debonding on the modal parameters of damaged sandwich plates with various boundary conditions is investigated. The results of dynamic analysis illustrated that they can be useful for analyzing practical problems related to the non-destructive damage detection of partially debonded sandwich plates.

Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array

Abstract: In this paper, the three-dimensional (3-D) non-linear dynamics of a cantilevered pipe conveying fluid, constrained by arrays of four springs attached at a point along its length is investigated. In the theoretical analysis, the 3-D equations are discretized via Galerkin's technique. The resulting coupled non-linear differential equations are solved numerically using a finite difference method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincare maps for five different spring configurations. Interesting dynamical phenomena, such as 2-D or 3-D flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were performed for the cases studied theoretically, and good qualitative and quantitative agreement was observed. The experimental behaviour is illustrated by video clips (electronic annexes). The effect of the number of beam modes in the Galerkin discretization on accuracy of the results and on convergence of the numerical solutions is discussed.

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Abstract: Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency–response curves, stability, and bifurcation points of the system.

Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface

Abstract: This paper presents the application of the second law analysis of thermodynamics to viscoelastic magnetohydrodynamic flow over a stretching surface. The velocity and temperature profiles are obtained analytically using the Kummer's functions and used to compute the entropy generation number. The effects of the magnetic parameter, the Prandtl number, the heat source/heat sink parameter and the surface temperature parameter on velocity and temperature profiles are presented. The influences of the same parameters, the Hartmann number, the dimensionless group parameter and the Reynolds number on the entropy generation are also discussed.

Mixed convection heat transfer over a non-linear stretching surface with variable fluid properties

Abstract: This article presents a numerical solution for the steady two-dimensional mixed convection MHD flow of an electrically conducting viscous fluid over a vertical stretching sheet, in its own plane. The stretching velocity and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent fluid properties, namely, the fluid viscosity and the thermal conductivity are assumed to vary, respectively, as an inverse function of the temperature and a linear function of the temperature. A generalized similarity transformation is introduced to study the influence of temperature dependent fluid properties. The transformed boundary layer equations are solved numerically, using a finite difference scheme known as Keller Box method, for several sets of values of the physical parameters, namely, the stretching parameter, the temperature dependent viscosity parameter, the magnetic parameter, the mixed convection parameter, the temperature dependent thermal conductivity parameter and the Prandtl number. The numerical results thus obtained for the flow and heat transfer characteristics reveal many interesting behaviors. These behaviors warrant further study of the effects of the physical parameters on the flow and heat transfer characteristics. Here it may be noted that, in the case of the classical Navier–Stokes fluid flowing past a horizontal stretching sheet, McLeod and Rajagopal (1987) [42] showed that there exist an unique solution to the problem. This may not be true in the present case. Hence we would like to explore the non-uniqueness of the solution and present the findings in the subsequent paper.

The non-linear dynamic response of the euler-bernoulli beam with an arbitrary number of switching cracks

Abstract: In this study the non-linear dynamic response of the Euler–Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.

Cutting process of composite materials: An experimental study

Abstract: This paper focuses on experimental research of milling process of the epoxide-polymer matrix composite reinforced carbon fibers (EPMC—carbon composite). An influence of two control parameters, namely feed and rotational speed, on cutting forces is investigated. The experiment is conducted on a CNC machine with feed rate ranging from 200 to 720 mm/min and rotational speed from 2000 to 8000 rpm. The experimental time series are analysed by means of the delay coordinates method in order to find stable cutting regions and to recognize the kind of behaviour. Using this information, a new model for the cutting forces is proposed that can be used to build a new regenerative vibration model for EPMC milling.

Three dimensional nonlinear dynamics of submerged, extensible catenary pipes conveying fluid and subjected to end-imposed excitations

Abstract: The complete 3D nonlinear dynamic problem of an extensible, submerged catenary pipe conveying fluid is considered. For describing the dynamics of the system, the Newtonian derivation procedure is followed. The flow inside the pipe is considered inviscid, irrotational and incompressible with constant velocity along the complete length of the pipe. The hydrodynamic effects are taken into account through the nonlinear drag forces and the added inertia due to the hydrodynamic mass. Following the Newtonian derivation, the dynamics of the pipe element and the fluid element are considered separately and the final governing set is derived combining the equations of inertia equilibrium. The system is solved using an efficient, second-order accurate, finite differences numerical scheme. The effect of the steady flow inside the pipe on both the in-plane and the out-of-plane vibrations is assessed with the aid of numerical simulations using as a model a relatively small-sag submerged catenary. The output signals of the time varying components that govern the dynamics of the pipe, have been properly processed to assess the effect of the internal flow on both the in-plane and the out-of-plane vibrations.

Output frequency properties of nonlinear systems

Abstract: Nonlinear systems usually have complicated output frequencies For the class of Volterra systems, some interesting properties of the output frequencies are studied in this paper These properties show theoretically the periodicity of the output super-harmonic and inter-modulation frequencies and clearly demonstrate the mechanism of the interaction between different output harmonics incurred by different input nonlinearities in system output spectrum These new results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities Examples and discussions are given to illustrate these new results

Efficient solution of a vibration equation involving fractional derivatives

Abstract: Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980s, and they grow more and more popular nowadays. However, their efficient numerical calculation is non-trivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.

Comparative studies of localized buckling in sandwich struts with different core bending models

Abstract: Analytical models with geometric non-linearities accounting for interactions between local and global instability modes leading to localized buckling in sandwich struts are formulated. For the core material response, two increasingly sophisticated bending models are compared against each other: Timoshenko beam theory (TBT) and Reddy–Bickford beam theory (RBT). Numerical solutions of the analytical models are validated with the commercial finite element code ABAQUS. It is found that there is a small but significant difference in the critical load between the two models and that the previously obtained solution slightly underestimates the linear buckling strength. More importantly, it is found that the RBT model predicts the onset of interactive buckling before the TBT model and, according to the results from the finite element study, matches the actual behaviour of a strut in both its initial and advanced post-buckling states with excellent correlation.

Constitutive model for special granular structures

Abstract: This work deals with novel “smart systems” that are based on granular materials. These systems consist of mechanical components, such as a beam or bar, enclosed in a tight flexible sleeve that is filled with a granular material. When air is pumped out of the sleeve, the resulting underpressure causes the compression of the granules making the system more rigid and changing its damping characteristics. This allows for quick, easy, and inexpensive control of the damping and stiffness of such a system. This paper presents the experimental results of uniaxial tensile tests of a sleeve filled with different granular materials. These results support the modeling of the mechanics of such a system with the Chaboche viscoplastic constitutive law. The experiments provide the quantitative and functional dependence of the model parameters on the underpressure, which acts as the control variable. The highly non-linear dependence of the system's fundamental mechanical properties on the underpressure is described and discussed. This basic work opens the way to applications in mechanical systems where effective control of vibrations or noise is important.

Boundary layer flow and heat transfer past a stretching plate with variable thermal conductivity

Abstract: In the present paper, the boundary layer flow of viscous incompressible fluid over a stretching plate has been considered to solve heat flow problem with variable thermal conductivity. First, using similarity transformation, the components of velocity have been obtained. Then, the heat flow problem has been considered in two ways: (i) prescribed surface temperature (PST), and (ii) prescribed stretching plate heat flux (PHF) in case of variable thermal conductivity. Due to variable thermal conductivity, temperature profile has its two part—one mean temperature and other temperature profile induced due to variable thermal conductivity. The related results have been discussed with the help of graphs.

Passivity-based control for a flexible spacecraft in the presence of disturbances

Abstract: The attitude regulation control problem for flexible spacecraft is investigated in this paper. Two extended PD+variable structure controllers are proposed using passivity-based control technique instead of sliding mode control approach. The first controller is a basic one, while the second one is an extension of the first one which relaxes the bound requirement for the external disturbances. In the presence of model uncertainties and external disturbances, both controllers presented in this research can make the flexible spacecraft UGAS (uniformly globally asymptotically stable). By virtue of related analysis tools, stability of the proposed controllers is proven theoretically. Numerical simulations are also included to demonstrate the performance of the developed controllers.

Adaptive lag synchronization in coupled chaotic systems with unidirectional delay feedback

Abstract: Lag synchronization in unidirectionally coupled chaotic systems is investigated in this paper. Based on the invariance principle of differential equations, a new adaptive delay feedback scheme is proposed to realize the lag synchronization effectively in the coupled chaotic systems. As an example, numerical simulations for the coupled Hindmarsh–Rose (HR) neuron models are conducted, which is in good agreement with the theoretical analysis. More interestingly, it is found that there is a fine U-shaped structure in the lag synchronization curve for the HR neuron model. Furthermore, lag synchronization and the corresponding U-shaped structure are robust against the small mismatch of parameters and noisy disturbances.

Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations

Abstract: The resonant resonance response of a single-degree-of-freedom non-linear vibro-impact oscillator, with cubic non-linearity items, to combined deterministic harmonic and random excitations is investigated The method of multiple scales is used to derive the equations of modulation of amplitude and phase The effects of damping, detuning, and intensity of random excitations are analyzed by means of perturbation and stochastic averaging method The theoretical analyses verified by numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle Under certain conditions, impact system may have two steady-state responses One is a non-impact response, and the other is either an impact one or a non-impact one

Global and local dynamics of drifting oscillator for different contact force models

Abstract: Accurate description of the contact force between two impacting solid bodies is important in various engineering applications. This paper examines the effects of contact force models on the global and local dynamics of a drifting oscillator. Three contact force models are considered in this study, namely the Kelvin–Voigt (KV), the Hertz stiffness (HS) and the non-linear contact stiffness and damping (NSD) models. The non-linear analysis of the considered system shows that the local and global dynamic behaviour depend on the choice of contact force model. The short-term (local) dynamic response for the system are almost identical for all the three models. However, results for the long-term (global) dynamics of the system depend on the choice of contact force model; while the results for HS model are topologically similar to those of the KV model, the corresponding results for NSD are different from the other two models.

A gradient model of light-induced bending in photochromic liquid crystal elastomer and its nonlinear behaviors

Abstract: Photochromic liquid crystal elastomer was recently reported to be able to deform largely and bend under illumination. In this paper, considering the opto-chemical process and the nematic–isotropic phase transition, we introduce the light and temperature into the constitutive relation of the liquid crystal elastomers, and propose a model for the light-induced bending. The dynamic deflection curve equation of the light-induced bending is derived based on the Hamilton principle. In the equation, the effect of light is introduced as an effective optical bending moment, which is caused by the inhomogeneous light-induced strain and Young's modulus. Several simulation examples are given to show the light-induced bending under different boundary conditions and various illumination or temperature controlling. Under the condition of deep nematic phase and weak enough illumination, the approximate analytical expression of the effective moment and the stress distribution can be obtained. Rich nonlinear behaviors are found in this model. The effective moment is a non-monotonic function of time, thickness ratio, and light intensity when the thickness ratio is not very large. The stress distribution through the thickness is nonlinear with two or three zero-stress planes.

Inelastic seismic demand of low-rise buildings with soil-flexibility

Abstract: Relying on the ductile behaviour of structures during earthquake, building codes introduce response reduction factors ( R ) to reduce design forces in earthquake resistant design. However, applicability of such factors has not been systematically explored for low-rise buildings with stiff periods. Present study is an attempt to address this issue. Both elasto-plastic and degrading hysteresis behaviour for lateral load-resisting structural elements are considered herein, while sub-soil is idealized as linear and elasto-plastic in parallel. The study recognizes that inelastic response for short period systems is very sensitive to R and may be phenomenally amplified even for small R due to soil–structure interaction implying restrictive applicability of dual-design philosophy. Limited study on the plan-asymmetric low-rise buildings depicts that inelastic response of the asymmetric structure relative to its symmetric counterpart is not appreciably influenced due to soil–structure interaction (SSI). The study also confirms that equivalent single story model characterized by the lowest period rather than the fundamental one of the real system tends to yield conservative estimation of inelastic demand at least for the short-period systems.

Identification and parameter estimation of a bilinear oscillator using Volterra series with harmonic probing

Abstract: Identification of non-linear systems is mainly limited to polynomial form non-linearities. Among the non-polynomial forms, bilinear oscillator constitutes an important class of non-linear systems and it has been used for modeling of various physical systems, particularly for structural elements with a breathing crack. An identification procedure is presented here for the class of bilinear oscillator, using higher order FRFs derived from Volterra series under harmonic excitation. The procedure addresses the problem of both; identification of the non-linearity structure as well as estimation of the bilinear parameter, which can be correlated to the crack severity and structural degradation. The procedure is illustrated with numerical simulation and the estimation results indicate that even a weakly bilinear state introduced by a small crack size can be accurately identified and measured.

Identification of time-varying hysteretic structures using wavelet multiresolution analysis

Abstract: In this paper, a wavelet multiresolution technique is proposed to identify time-varying properties of hysteretic structures It is well known that arbitrary transient functions can be effectively and accurately approximated using wavelet multiresolution expansions due to wavelet's good time–frequency localization property By decomposing the time-varying parameters with wavelet multiresolution expansion, a time-varying parametric identification problem can be transformed into a time-invariant non-parametric one The identification in the time-invariant wavelet multiresolution domain can be achieved by choosing a wavelet basis function and performing a suitable parameter estimation technique Since wavelet representation of arbitrary signal uses only a small number of terms, the orthogonal forward regression algorithm can be adopted for significant term selection and parameter estimation Single and multiple degrees of freedom Bouc–Wen hysteretic structures with gradual and abrupt varying properties are used to illustrate the proposed approach Results show that the wavelet multiresolution technique can identify and track the time-varying hysteretic parameters quite accurately The effect of measurement noise is also studied It is found that the presence of noise would affect more on the damping ratios and the Bouc–Wen parameters but less on the equivalent stiffness coefficients

On the contact of a spherical membrane enclosing a fluid with rigid parallel planes

Abstract: The mechanical response of an inflated spherical membrane–fluid structure in contact with rigid parallel planes is studied. The membrane is assumed to be a two-dimensional non-linear elastic and isotropic structure, while no assumption is imposed on the fluid. A numerical procedure is employed to compute the equilibrium configurations of the membrane–fluid structure. This study provides information regarding the contact force, stress distribution and pressure in the membrane and in the enclosed fluid, respectively. It was observed that a transition between unwrinkled to partially wrinkled configurations of the membrane occurs subjected to the loading conditions. Further investigation of the wrinkled configurations is presented.

Vibration reduction of the impact system by an SMA restraint: numerical studies

Abstract: The dynamic behaviour of an impact oscillator with a shape memory alloy (SMA) restraint is modelled and analyzed. This impact oscillator has the secondary support made from an SMA and the thermomechanical description of the SMA element follows the formulation proposed by Bernardini et al. [1,2]. The thermo-mechanical coupling terms included in the energy balance equation allow to undertake the non-isothermal analysis. Due to the mechanical characteristics of the SMA element and the non-smooth nature of the impacts, five different modes of operation can be distinguished. The undertaken numerical investigations suggest that the system can exhibit complex dynamic responses, which if appropriately controlled can be used for vibration reduction. A comparison with an equivalent elastic oscillator is made. It is found out that the low amplitude regimes are not affected by the SMA element. On contrary, for the large amplitude responses, a significant vibration reduction may be achieved due to the phase transformation hysteresis loop. Various bifurcation scenarios are constructed and the influence of the SMA element is discussed. In particular, the analysis of the frequency and amplitude variations of the external excitation is given and the parameter ranges where the vibration reduction is possible are identified.

On the effective stiffness of plates made of hyperelastic materials with initial stresses

Abstract: Within the framework of the direct approach to the plate theory we consider the infinitesimal deformations of a plate made of hyperelastic materials taking into account the non-homogeneously distributed initial stresses. Here we consider the plate as a material surface with 5 degrees of freedom (3 translations and 2 rotations). Starting from the equations of the non-linear elastic body and describing the small deformations superposed on the finite deformation we present the two-dimensional constitutive equations for a plate. The influence of initial stresses in the bulk material on the plate behavior is considered.

Energy pumping for mechanical systems involving non-smooth Saint-Venant terms

Abstract: This paper deals with energy transfer from initial one degree of freedom system including non-smooth terms of friction (Saint-Venant) type to an auxiliary mass via an adapted non-linear coupling. We describe design of auxiliary degree of freedom and non-linear smooth coupling to the initial one degree of freedom system. First the auxiliary system is designed as if the initial one degree of freedom system was simply elastic. Then we study numerically the transfer for the non-smooth system in the cases of free transient oscillations or forced oscillations under periodic external forcing. Efficiency of energy pumping is discussed.

Stability of an internally damped non-ideal flexible spinning shaft

Abstract: The stability domain of an internally damped flexible spinning shaft, which is driven by a non-ideal source, is studied in this paper. It is found that the higher transverse modes may become unstable before the lower ones under certain conditions. In particular, we find the entrainment of the shaft spinning speed at specific values corresponding to the lowest stability threshold among all transverse modes. Moreover, the steady state amplitude of the transverse vibrations when the shaft spinning speed is stuck at the stability threshold is determined analytically. The analytical results thus obtained are validated with numerical simulations.

Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

Abstract: This paper addresses the development of several alternative novel hybrid/multi-field variational formulations of the geometrically exact three-dimensional elastostatic beam boundary-value problem. In the framework of the complementary energy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of stresses only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value problem. Both weak and linearized weak forms of the principles are presented. The main features of the principles are highlighted, giving special emphasis to their relationships from both theoretical and computational standpoints.