Showing papers in "International Journal of Non-linear Mechanics in 2014"

Non-linear vibrations of shells: A literature review from 2003 to 2013

Abstract: The present literature review focuses on geometrically non-linear free and forced vibrations of shells made of traditional and advanced materials. Flat and imperfect plates and membranes are excluded. Closed shells and curved panels made of isotropic, laminated composite, piezoelectric, functionally graded and hyperelastic materials are reviewed and great attention is given to non-linear vibrations of shells subjected to normal and in-plane excitations. Theoretical, numerical and experimental studies dealing with particular dynamical problems involving parametric vibrations, stability, dynamic buckling, non-stationary vibrations and chaotic vibrations are also addressed. Moreover, several original aspects of non-linear vibrations of shells and panels, including (i) fluid–structure interactions, (ii) geometric imperfections, (iii) effect of geometry and boundary conditions, (iv) thermal loads, (v) electrical loads and (vi) reduced-order models and their accuracy including perturbation techniques, proper orthogonal decomposition, non-linear normal modes and meshless methods are reviewed in depth.

Interface damage and its effect on vibrations of slab track under temperature and vehicle dynamic loads

Abstract: This paper presents a three-dimensional finite element model to investigate the interface damage occurred between prefabricated slab and CA (cement asphalt) mortar layer in the China Railway Track System (CRTS-II) slab track system. In the finite element model, a cohesive zone model with a non-linear constitutive law is introduced and utilized to model the damage, cracking and delamination at the interface. Combining with the temperature field database obtained from the three-dimensional transient heat transfer analysis, the interface damage evolution as a result of temperature change is analyzed. A three-dimensional coupled dynamic model of a vehicle and the slab track is then established to calculate the varying rail-supporting forces which are utilized as the inputs to the finite element model. The non-linearities of the wheel–rail contact geometry, the wheel–rail normal contact force and the wheel–rail tangential creep force are taken into account in the model. Setting the maximum interface damaged state calculated under temperature change as the initial condition, the interface damage evolution and its influence on the dynamic response of the slab track are investigated under the joint action of the temperature change and vehicle dynamic load. The analysis indicates that the proposed model is capable of predicting the initiation and propagation of cracks at the interface. The prefabricated slab presents lateral warping, resulting in severe interface damage on both the sides of the slab track along the longitudinal direction during temperature drop process, while the interface damage level does not change significantly under vehicle dynamic loads. The interface damage has great effects on the dynamic responses of the slab track.

Non-linear dynamic stability of piezoelectric functionally graded carbon nanotube-reinforced composite plates with initial geometric imperfection

Abstract: This paper deals with non-linear dynamic stability of initially imperfect piezoelectric functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates under a combined thermal and electrical loadings and interaction of parametric and external resonance. The excitation, which derives from harmonically varying actuators voltage, results in both external and parametric excitation. The governing equations of the piezoelectric CNTRC plates are derived based on first order shear deformation plate theory (FSDT) and von Karman geometric non-linearity. The material properties of FG-CNTRC plate are assumed to be graded in the thickness direction. The single-walled carbon nanotubes (SWCNTs) are assumed aligned, straight and a uniform layout. The linear buckling and vibration behavior of perfect and imperfect plates are obtained in the first step. Then, Galerkin's method is employed to derive the non-linear governing equations of the problem with quadratic and cubic non-linearities associated with mid-plane stretching. Periodic solutions and their stability are determined by using the harmonic balance method with simply supported boundary conditions. The effect of the applied voltage, temperature change, plate geometry, imperfection, the volume fraction and distribution pattern of the SWCNTs on the parametric resonance, in particular the positions and sizes of the instability regions of the smart CNTRC plates as well as amplitude of steady state vibration are investigated through a detailed parametric study.

Non-linear analysis of functionally graded microbeams using Eringen׳s non-local differential model

Abstract: The primary objective of this paper is two-fold: (1) to formulate the governing equations of the Euler–Bernoulli and Timoshenko beams that account for (a) two-constituent material variation through beam thickness, (b) small strains but moderate displacements and rotations, and (c) material length scales based on Eringen׳s non-local differential model; and (2) develop the non-linear finite element models of beam theories with aforementioned features and obtain numerical results for static bending. The principle of virtual displacements is used to derive the non-linear equations governing functionally graded beams with Eringen׳s non-local constitutive models for both the Euler–Bernoulli and Timoshenko beam theories. A power-law model is used for the variation of the material properties of the two constituent materials. Finite element models of the resulting equations are developed and numerical results are presented for pinned–pinned and clamped–clamped boundary conditions, showing the effect of the non-local parameter and the power-law index on deflections and stresses.

Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed

Abstract: Non-linear vibration of viscoelastic pipes conveying fluid around curved equilibrium due to the supercritical flow is investigated with the emphasis on steady-state response in external and internal resonances. The governing equation, a non-linear integro-partial-differential equation, is truncated into a perturbed gyroscopic system via the Galerkin method. The method of multiple scales is applied to establish the solvability condition in the first primary resonance and the 2:1 internal resonance. The approximate analytical expressions are derived for the frequency–amplitude curves of the steady-state responses. The stabilities of the steady-state responses are determined. The generation and the vanishing of a double-jumping phenomenon on the frequency–amplitude curves are examined. The analytical results are supported by the numerical integration results.

Crashworthiness design of foam-filled bitubal structures with uncertainty

Abstract: Structural optimization has been widely used to improve the crashworthiness of foam-filled thin-walled structures. However, majority of the existing optimization studies to date have not considered uncertainties for simplication. Its associated risk is that a deterministic optimization might deteriorate its optimality and/or violate design constraints when being present in uncertain environment. In this study, a multiobjective robust design optimization (MORDO) method is adopted to explore the design of foam-filled bitubal structures. To reduce the computational burden of highly-non-linear crash analysis, adaptive Kriging models are employed in the optimization process. In this strategy, sequential sampling points are generated over the design space and Kriging models are refitted in an iterative fashion. Based on the Kriging models, the multiobjective particle swarm optimization (MOPSO) algorithm is employed to perform the optimization, integrated with Monte Carlo simulation and descriptive sampling technique. The results demonstrate that the proposed method is capable of improving the robustness of Pareto solutions within the prescribed minimum requirements of reliability. Moreover, the influence of varying the emphasis on mean and standard deviation components is also analyzed, which can provide decision-makers with insightful design information.

Physically and geometrically non-linear vibrations of thin rectangular plates

Abstract: Static deflection as well as free and forced large-amplitude vibrations of thin rectangular rubber plates under uniformly distributed pressure are investigated. Both physical, through a neo-Hookean constitutive law to describe the non-linear elastic deformation of the material, and geometrical non-linearities are accounted for. The deflections of a thin initially flat plate are described by the von Karman non-linear plate theory. A method for building a local model, which approximates the plate behavior around a deformed configuration, is proposed. This local model takes the form of a system of ordinary differential equations with quadratic and cubic non-linearities. The corresponding results are compared to the exact solution and are found to be accurate. Two models reflecting both physical and geometrical non-linearities and geometrical non-linearities only are compared. It is found that the sensitivity of the deflection to the physically induced non-linearities at moderate strains is significant.

Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory

Abstract: Based on the nonlocal continuum theory, the nonlinear primary resonance of nano beam with the axial initial load is investigated. The amplitude–frequency response for the primary resonance is derived with the multiple scale method and the stability is analyzed. The nonlinear primary resonance of nano beam is discussed with the influences of small scale effect, axial initial load, mode number, Winkler foundation modulus and the ratio of the length to the diameter. From the results, the typical hardening nonlinearity can be observed. Moreover, some significant and interesting nonlinear phenomena can be found for the primary resonance of nano beam. This work is expected to be useful for the design and analysis for the nonlinear dynamic behaviors of structures at nano scales.

Non-linear rapid heating of FGM beams

Abstract: Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) beams are analyzed in this research. All thermomechanical properties of the beam are assumed to be temperature and position dependent. Beam is subjected to thermal shock on the ceramic-rich surface whereas the metal-rich one is kept at reference temperature or thermally insulated. The one-dimensional transient heat conduction equation is established and solved via a hybrid iterative central finite difference method and Crank–Nicolson method. Total functional of the beam is obtained under the assumptions of uncoupled thermoelasticity laws, first order beam theory, and the von Karman type geometrical non-linearity. The conventional multi-term p-Ritz method appropriate for arbitrary in-plane and out-of-plane boundary conditions is applied to the total functional of the system which results in the matrix representation of the equations of motion. Non-linear coupled equations of motion are solved via the iterative Newton–Raphson method accompanied with the β-Newmark time approximation technique. Numerical results are well validated with the available results for the case of isotropic homogeneous beams. Some parametric studies are conducted to examine the influences of beam geometry, material composition, temperature dependency, in-plane and out-of-plane mechanical and thermal boundary conditions. It is shown that, thermally induced vibrations indeed exist especially for the case of sufficiently thin beams.

Non-linear static bending and forced vibrations of rectangular plates retaining non-linearities in rotations and thickness deformation

Abstract: Geometrically non-linear static bending and forced vibrations of rectangular plates are studied allowing full non-linear terms associated with Green–Lagrange strain–displacement relations, second-order thickness stretching, third-order shear deformation and rotary inertia by using seven independent parameters to describe the shell kinematics. In particular, in addition to non-linearities in membrane and transverse deflection, non-linear terms associated with rotations and thickness deformation parameters are also included. In order to obtain the governing equations of motion, the three-dimensional constitutive equations are used, removing the assumption of zero transverse normal strain. The boundary conditions of the plate are assumed to be simply supported immovable and the equations of motion are derived by using a Lagrangian approach. The numerical solutions are obtained by using pseudo arc-length continuation and collocation scheme. In order to compare the non-linear static response, another analysis has also been carried out by using the finite element code ANSYS and three-dimensional solid modeling. Results reveal that the new theory with full geometric non-linearities provides significant accuracy improvement for rotational and thickness deformation parameters, and, unlike other shear deformation theories, predicts the correct thickness stretching along the plate.

A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations

Abstract: In this study we use a general thermodynamic framework which appeals to the criterion of maximal rate of entropy production to obtain popular models due to Darcy and Brinkman and their generalizations, to describe flow of fluids through porous solids. Such a thermodynamic approach has been used with great success to describe various classes of material response and here we demonstrate its use within the context of mixture theory to obtain the classical models for the flow of fluids through porous media and more general models which are all consistent with the second law of thermodynamics.

Influence of surface energy on the non-linear pull-in instability of nano-switches

Abstract: This paper investigates the pull-in instability of a nano-switch under electrostatic and intermolecular Casimir forces. The analysis is based on the geometrically non-linear Euler–Bernoulli beam theory with consideration of the surface energy. Through differential quadrature method (DQM), the pull-in voltages of the nano-switch are obtained. Results show that the effect of surface energy and geometrically non-linear deformation on the pull-in voltage depends on the length, height and initial gap of the nano-switch. In addition, the effect of intermolecular Casimir force on the pull-in voltage weakens as the initial gap increases.

Reduced order model of parametric resonance of electrostatically actuated MEMS cantilever resonators

Abstract: This paper deals with parametric resonance of microelectromechanical (MEMS) cantilever resonators under soft damping, and soft alternating current (AC) electrostatic actuation to include fringing effect. A comparison between the Reduced Order Model (ROM) method and the Method of Multiple Scales (MMS) for both small and large amplitudes is reported. The actuation is parametric non-linear. It includes non-linear terms with periodic coefficients. The AC frequency is near resonator׳s natural frequency. The amplitude frequency response is investigated using ROM. Damping, voltage, and fringe effects on the response are also reported. It is showed that five terms ROM accurately predicts the behavior of the resonator at all amplitudes, while MMS is accurate only for small amplitudes.

Direct evaluation of the equivalent linear damping for TLCD systems in random vibration for pre-design purposes

Abstract: Passive control of structural vibrations has received in recent years a great attention from researchers concerned with vibration control. Several types of devices have been proposed in order to reduce the dynamic responses of different kinds of structural systems. Among them, the Tuned Liquid Column Damper (TLCD) proved to be very effective in reducing vibration of structures. Since the increasing use of TLCDs in practical realizations, this paper aims at developing an approximate formulation, by means of a statistical linearization technique, able to estimate the parameters of a structure equipped with a TLCD subjected to random loads for pre-design purposes. Moreover, it is shown that results obtained via Monte Carlo approach on the non-linear system are in good agreement with those obtained by means of the proposed formulation.

Non-linear in-plane stability analysis of FGM circular shallow arches under central concentrated force

Abstract: A comprehensive study on the non-linear in-plane stability behavior of shallow arches made of functionally graded materials (FGMs) is presented in this work. Simply supported–simply supported (S–S) and clamped–clamped (C–C) boundary conditions are considered as two types of well-known symmetric boundary conditions for this analysis. The arch is subjected to a central concentrated force and material dispersion is according to the power law distribution. For this aim, the classical single layer theory is adjusted to approximate the displacement field through the arch. Kinematical relations are reduced to suitable ones for shallow arches. Static version of the virtual displacement principle is used to obtaining the governing equations and the complete set of boundary conditions. In the presence of the highly non-linear behavior of shallow arches under central concentrated force, buckling analysis is preformed in the presence of pre-buckling deformations. Existence of secondary equilibrium paths for shallow arches is studied and stability behavior of FGM shallow arches is classified into non-linear bending, full snap-through, bifurcation from post-snap path, and bifurcation. Also, multiple snap-to-state condition is investigated for FGM shallow arches. Results are presented as primary equilibrium paths and effect of material dispersion, geometrical characteristics, and boundary conditions on the stability behavior of shallow arches under central concentrated force is studied.

Effects of stiffness and load imperfection on the isolation performance of a high-static-low-dynamic-stiffness non-linear isolator under base displacement excitation

Abstract: The dynamic characteristics of a high-static-low-dynamic-stiffness (HSLDS) non-linear isolator, built by combining an Euler beams formed negative stiffness corrector and a traditional linear isolator are investigated. The system imperfection caused by detuning of the stiffness and the load are considered. The frequency response curves (FRCs) for the non-linear isolator with load and stiffness imperfection under base displacement excitation are calculated by using Harmonic Balance Method (HBM). The stability is studied by the Floquet theory. The effect of the imperfection on the dynamic response as well as the displacement transmissibility of the non-linear isolator is explored and discussed. The results show that both the stiffness and load imperfection can affect the performance of the non-linear isolator significantly. The HSLDS isolator with load and stiffness imperfection for different base excitation amplitudes can demonstrate pure softening, softening-to-hardening and pure hardening characteristics with multi-valued solutions. The occurrence of jump phenomena is observed and explained by the stiffness variation. When the isolator is subjected to load imperfection, keeping a small positive stiffness, rather than making the minimum dynamic stiffness as zero, is preferred in order to obtain the best isolation performance. The performance of the non-linear isolator can outperform the linear one provided that the base displacement excitation is not too large. The non-linear system may undergo unbounded responses for very large base displacement excitation.

On approximating the effective bandwidth of bi-stable energy harvesters

Abstract: This paper aims to establish an analytical framework to define the effective bandwidth of bi-stable vibratory energy harvesters possessing a symmetric quartic potential function. To achieve this goal, the method of multiple scales is utilized to construct analytical solutions describing the amplitude and stability of the intra- and inter-well dynamics of the harvester. Using these solutions, critical bifurcations in the parameters׳ space are identified and used to define an effective frequency bandwidth of the harvester. The influence of three critical design parameters, namely the time constant ratio (ratio between the time constant of the harvesting circuit and the period of the mechanical system), the electromechanical coupling, and the shape of the potential function, on the effective frequency bandwidth is analyzed. It is shown that, while the time constant ratio has very little influence on the effective bandwidth of the harvester, increasing the electromechanical coupling and/or designing the potential function with deeper potential wells serve to shrink the effective bandwidth for a given level of excitation. In general, it is also observed that narrowing of the effective bandwidth is accompanied by an increase in the electric output further highlighting the competing nature of these two desired objectives.

New generalized and functional separable solutions to non-linear delay reaction–diffusion equations

Abstract: We present a number of new generalized separable, functional separable, periodic and antiperiodic exact solutions to non-linear delay reaction–diffusion equations of the form u t = ku xx + F ( u , w ) , where u = u ( x , t ) and w = u ( x , t − τ ) , with τ being the delay time. The generalized separable solutions are sought in the form u = ∑ n = 1 N Φ n ( x ) Ψ n ( t ) , with the functions Φ n ( x ) and Ψ n ( t ) to be determined in the analysis. Most of the equations considered contain one or two arbitrary functions of a single argument or one arbitrary function of two arguments of special form. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for non-linear delay PDEs. Some results are extended to non-linear delay partial differential equations of any order.

Lateral vibration attenuation of a rotor under mass eccentricity force using non-linear energy sink

Abstract: This paper studies the efficiency of a number of smooth non-linear energy sinks (NESs) on the vibration attenuation of a rotor system under mass eccentricity force. The non-linear energy sinks have a linear damping, linear stiffness and a cubic stiffness. To reduce the number of equations of motion, the modal coordinates and complex transformations are used. For analytical solution, the Multiple Scales-Harmonic Balance Method (MSHBM) is used. The most important parameters to examine NES efficiency is the range of happening of the SMR in the detuning parameter range. For different parameters of the system, the SMR, the WMR, the low amplitude periodic motion, and the high amplitude periodic motion happen in the system. It is shown that, the linear stiffness of the NES is canceled out by the stiffness which is created by the centrifugal force and therefore, by changing the linear stiffness of the NESs, the collection of the NESs to the desired rotational speed of the rotor can be tuned. It should be noted that after this cancellation, the remained stiffness is essentially non-linear (non-linearizable) stiffness. In addition, when the external force reaches its medium magnitude, the area of the occurrence of the SMR in the domain of the system parameters would be larger and the collection of the NESs demonstrates an impressive effect.

Vibratory synchronization and coupling dynamic characteristics of multiple unbalanced rotors on a mass-spring rigid base

Abstract: The investigation of a generalization of the model on vibratory synchronization for multiple unbalanced rotors (URs) is considered in present work. Based on the previous publications and using the average method of small parameters, the dimensionless coupling equation of multiple URs is constructed, and the criterions of synchronization and stability in the simplified form for multiple URs are given. Taking three counting-rotating URs for example, the coupling dynamic characteristics and synchronization regime of the system are discussed numerically. Then a vibrating synchronization bedstand corresponding to the dynamical model used in numerical discussion is set up, a more detailed discussion and parameter study in experiment are provided, through the analyses on experimental data such as phase differences, rotational velocities and synchronization quality, it is shown that the experimental results are in approximate or good agreement with the numerical/theoretical results, which validates the validity of the model and approach.

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

Abstract: A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness stretching, higher-order shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in the in-plane and transverse displacements and are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. The theory uses the three-dimensional constitutive equations and does not need the introduction of traction/compression free hypothesis at the shell inner and outer surfaces. The traction/compression free condition is introduced only to obtain a simplified model with six parameters instead of eight. The third-order thickness stretching theory is applied to cross-ply symmetrically laminated circular cylindrical shells complete around the circumference and simply supported at both ends. Geometrically non-linear forced vibrations are studied by using the present theory and results are compared to those obtained by using a refined higher-order shear deformation non-linear shell theory, which neglects thickness stretching, and to results obtained by using first-order and second-order thickness stretching theories. Results obtained by using the reduced third-order thickness stretching model with six parameters are also presented and compared.

Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading

Abstract: Non-linear bending analysis of tapered functionally graded (FG) beam subjected to thermal and mechanical load with general boundary condition is studied. The governing equations are derived and a discussion is made about the possibility of obtaining analytical solution. In the case of no axial force along the beam, a closed form solution is presented for the problem. For the general case with axial force, the Galerkin technique is employed to overcome the shortcoming of the analytical solution. Moreover, the Generalized Differential Quadrature (GDQ) method is also implemented to discretize and solve the governing equations in the general form and validate the results obtained from two other methods. The effects of various thermal and mechanical loading on the nonlinear bending of tapered FG beam are investigated by implementing different analytical and numerical approaches.

Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: Diversity and parameter matching of periodic-impact motions

Abstract: A two-degree-of-freedom system with a clearance and subjected to harmonic excitation is considered The correlative relationship and matching law between dynamic performance and system parameters are studied by multi-parameter and multi-performance co-simulation analysis Two key parameters of the system, the exciting frequency ω and clearance δ, are emphasized to reveal the influence of the main factors on dynamic performance of the system Diversity and evolution of periodic impact motions are analyzed The fundamental group of impact motions is defined, which have the period of exciting force and differ by the numbers p and q of impacts occurring at the left and right constraints of the clearance The occurrence mechanism of chattering-impact vibration of the system is studied As the clearance δ is small or small enough, the transition from 1–p–p to 1–(p+1)–(p+1) motion (the fundamental group of motions, p≥1) basically goes through the processes as follows: pitchfork bifurcation of symmetric 1–p–p motion, period-doubling bifurcation of asymmetric 1–p–p motion, non-periodic or chaotic motions caused by a succession of period-doubling bifurcations, symmetric 1–(p+1)–(p+1) motion generated by a degeneration of chaos As for slightly large clearance, a series of grazing bifurcations of periodic symmetrical impact motions occur with decreasing the exciting frequency so that the number p of impacts of the fundamental group of motions increases two by two As p becomes big enough, the incomplete chattering-impact motion will appear which exhibits a chattering sequence in an excitation period followed by a finite sequence of impacts with successively reduced velocity and reaches the non-sticking region Finally, the complete chattering-impact motion with sticking will occur with decreasing the exciting frequency ω up to the sliding bifurcation boundary A series of singular points on the boundaries between existence regions of any adjacent symmetrical impact motions with fundamental period are found, ie, two different saddle-node bifurcation boundaries of one of them, real-grazing and bare-grazing bifurcation boundaries of the other alternately and mutually cross themselves at the points of intersection and create inevitably two types of transition regions: narrow hysteresis and small tongue-shaped regions A series of zones of regular periodic and subharmonic impact motions are found to exist in the tongue-shaped regions Based on the sampling ranges of parameters, the influence of dynamic parameters on impact velocities, existence regions and correlative distribution of different types of periodic-impact motions of the system is emphatically analyzed

Effective bond length of FRP stiffeners

Abstract: The problem of an elastic bar bonded to an elastic half space and pulled at one end is considered to model the performance of FRP strips glued to concrete or masonry substrates. If the bond is perfect, stress singularities at both bar-extremities do appear. These can be removed by assuming cohesive contact forces a la Baranblatt that annihilate the stress intensity factor. We show that the presence of such cohesive zones is crucial to predict the experimentally measured effective bond length (EBL), i.e., the bond length beyond which no apparent increase of strength is attained. In particular, it is the cohesive zone at the loaded end of the stiffener, rather than that at the free end, that governs the phenomenon because the EBL coincides with the maximal length of such a zone. The proposed approach provides better estimates than formulas proposed in technical standards.

Non-linear theories of beams and plates accounting for moderate rotations and material length scales

Abstract: The primary objective of this paper is to formulate the governing equations of shear deformable beams and plates that account for moderate rotations and microstructural material length scales. This is done using two different approaches: (1) a modified von Karman non-linear theory with modified couple stress model and (2) a gradient elasticity theory of fully constrained finitely deforming hyperelastic cosserat continuum where the directors are constrained to rotate with the body rotation. Such theories would be useful in determining the response of elastic continua, for example, consisting of embedded stiff short fibers or inclusions and that accounts for certain longer range interactions. Unlike a conventional approach based on postulating additional balance laws or ad hoc addition of terms to the strain energy functional, the approaches presented here extend existing ideas to thermodynamically consistent models. Two major ideas introduced are: (1) inclusion of the same order terms in the strain–displacement relations as those in the conventional von Karman non-linear strains and (2) the use of the polar decomposition theorem as a constraint and a representation for finite rotations in terms of displacement gradients for large deformation beam and plate theories. Classical couple stress theory is recovered for small strains from the ideas expressed in (1) and (2). As a part of this development, an overview of Eringen׳s non-local, Mindlin׳s modified couple stress theory, and the gradient elasticity theory of Srinivasa–Reddy is presented.

Dynamics of a time delayed Duffing oscillator

Abstract: This paper focuses on the primary resonance of the time delayed Duffing oscillator solved by means of the multiple scales method. The second order approximation is used because the first approximation applied in the literature does not exhibit specific phenomena of the studied system. The Duffing system is investigated in two aspects. First of all, how delay displacement feedback influences the primary resonance of the classical Duffing oscillator in order to control the system and, on the other hand, how the external harmonic force influences vibrations of the system with time delay which is essential in applications, e.g. in machining. Stability of solutions and their bifurcations caused by the system parameters are shown. Especially, coexistence of possible stable and unstable solutions is investigated versus time delay and external excitation parameters. The selected analytical results are compared with numerical simulations. The model presented in the paper is considered as a general problem of non-linear oscillator with time delay but specific attention is paid to cutting process control.

Extended models of non-linear waves in liquid with gas bubbles

Abstract: In this work we generalize the models for non-linear waves in a gas–liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for non-linear waves We also take into consideration high order terms with respect to the small parameter Two new non-linear differential equations are derived for long weakly non-linear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above-mentioned physical properties One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on non-linear waves propagation The other equation is the perturbation of the Burgers–Korteweg–de Vries equation and corresponds to main influence of dispersion on non-linear waves propagation

Double rolling isolation systems: A mathematical model and experimental validation

Abstract: Rolling isolation systems (RISs) protect fragile building contents from earthquake hazards by decoupling horizontal floor motions from the horizontal responses of the isolated object. The RISs in use today have displacement capacities of about 20 cm. This displacement capacity can be increased by stacking two systems. This paper presents and evaluates a complete non-linear model of the coupled dynamics of double RISs. The model is derived through the fundamental form of Lagrange׳s equation and involves the non-holonomic constraints of spheres rolling between non-parallel surfaces. The derivation requires the use of two translating and rotating reference frames. The proposed model is validated through comparisons between experimentally measured and numerically predicted time histories and peak response quantities—total acceleration and relative displacement. The effects of the initial conditions, the mass of the isolated object, and the amplitude and period of the disturbance on the system׳s performance are assessed.

Curvature dependent surface energy for a free standing monolayer graphene: Some closed form solutions of the non-linear theory

Abstract: Continuum modeling of a free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specifications of the components of the shift vector that act as an auxiliary variable. The field equations are then the equations ruling the shift vector, together with momentum and moment of momentum equations. We present an analysis of simple loading histories such as axial, biaxial tension/compression and simple shear for a range of problems of increasing difficulty. We start by laying down the equations of a simplified model which can still capture bending effects. Initially, we ignore out of plane deformations. For this case, we solve analytically the equations ruling the auxiliary variables in terms of the shift vector; these equations are algebraic when the loading is specified. As a next step, still working on the simplified model, out-of-plane deformations are taken into account and the equations complicate dramatically. We describe how wrinkling/buckling can be introduced into the model and apply the Cauchy–Kowalevski theorem to get existence and uniqueness in terms of the shift vector for some characteristic cases. Finally, for the treatment of the most general problem, we classify the equations of momentum and give conditions for the Cauchy–Kowalevski theorem to apply.

Sensitivity in the structural behavior of shallow arches

Abstract: It is well established that certain structural buckling problems are extremely sensitive to small changes in configuration: geometric imperfections, load application, symmetry, boundary conditions, etc. This paper considers the behavior of a very shallow arch under lateral point loading, and specifically under the influence of changes in the thermal environment. In some ways the system under study is especially sensitive since small changes influence whether the arch ‘snaps-through’ or not. The experimental results provide insight into the challenges of understanding the behavior of these types of structural components in a practical, and thus necessarily imperfect, situation. The focus is on static loading or at least quasi-static loading, in which loading occurs on a slow time scale. This study also acts as a back-drop for studying the dynamic behavior of shallow arches, an area of concern in the context of aerospace structural components.