Showing papers in "International Journal of Non-linear Mechanics in 2008"
TL;DR: In this paper, a similarity transform was used to reduce the Navier-Stokes equations to a set of non-linear ordinary differential equations, which are then integrated numerically.
Abstract: The stagnation flow towards a shrinking sheet is studied. A similarity transform reduces the Navier–Stokes equations to a set of non-linear ordinary differential equations which are then integrated numerically. Both two-dimensional and axisymmetric stagnation flows are considered. It is found that solutions do not exist for larger shrinking rates and may be non-unique in the two-dimensional case. The non-alignment of the stagnation flow and the shrinking sheet complicates the flow structure. Convective heat transfer decreases with the shrinking rate due to an increase in boundary layer thickness.
610 citations
TL;DR: In this paper, the peristaltic flow of Jeffrey fluid in an asymmetric channel is studied under long wavelength and low Reynolds number assumptions, where the fluid is electrically conducting by a transverse magnetic field.
Abstract: The peristaltic flow of a Jeffrey fluid in an asymmetric channel is studied under long wavelength and low Reynolds number assumptions. The fluid is electrically conducting by a transverse magnetic field. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The expressions for stream function, axial velocity and axial pressure gradient have been obtained. The effects of various emerging parameters on the flow characteristics are shown and discussed with the help of graphs. The pumping characteristics, axial pressure gradient and trapping phenomenon have been studied. Comparison of various wave forms (namely sinusoidal, triangular, square and trapezoidal) on the flow is discussed.
285 citations
TL;DR: In this article, non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions using elliptic integral solutions.
Abstract: Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.
150 citations
TL;DR: In this article, an analysis is carried out to study the unsteady magnetohydrodynamic (MHD) two-dimensional boundary layer flow of a second grade viscoelastic fluid over an oscillatory stretching surface, which is induced due to an infinite elastic sheet which is stretched back and forth in its own plane.
Abstract: An analysis is carried out to study the unsteady magnetohydrodynamic (MHD) two-dimensional boundary layer flow of a second grade viscoelastic fluid over an oscillatory stretching surface. The flow is induced due to an infinite elastic sheet which is stretched back and forth in its own plane. For the investigated problem, the governing equations are reduced to a non-linear partial differential equation by means of similarity transformations. This equation is solved both by a newly developed analytic technique, namely homotopy analysis method (HAM) and by a numerical method employing the finite difference scheme, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The results obtained by means of both methods are then compared and show an excellent agreement. The effects of various parameters like visco-elastic parameter, the Hartman number and the relative frequency amplitude of the oscillatory sheet to the stretching rate on the velocity field are graphically illustrated and analysed. The values of wall shear stress for these parameters are also tabulated and discussed.
101 citations
TL;DR: In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied and the system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well.
Abstract: In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincare section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.
98 citations
TL;DR: In this paper, the criticality of Hopf bifurcation depends on the feed rate of the turning process and is determined not only by the rotation of the workpiece, but also by the vibrations of the tool.
Abstract: In this paper the non-linear dynamics of a state-dependent delay model of the turning process is analyzed. The size of the regenerative delay is determined not only by the rotation of the workpiece, but also by the vibrations of the tool. A numerical continuation technique is developed that can be used to follow the periodic orbits of a system with implicitly defined state-dependent delays. The numerical analysis of the model reveals that the criticality of the Hopf bifurcation depends on the feed rate. This is in contrast to simpler constant delay models where the criticality does not change. For small feed rates, subcritical Hopf bifurcations are found, similar to the constant delay models. In this case, periodic orbits coexist with the stable stationary cutting state and so there is the potential for large amplitude chatter and bistability. For large feed rates, the Hopf bifurcation becomes supercritical for a range of spindle speeds. In this case, stable periodic orbits instead coexist with the unstable stationary cutting state, removing the possibility of large amplitude chatter. Thus, the state-dependent delay in the model has a kind of stabilizing effect, since the supercritical case is more favorable from a practical viewpoint than the subcritical one.
97 citations
TL;DR: In this article, the authors consider the problem of bulging or necking of an infinite thin-walled hyperelastic tube that is inflated by an internal pressure, with the axial stretch at infinity maintained at unity, and they present a simple procedure that can be used to derive the bifurcation condition and to determine the nearcritical behaviour analytically.
Abstract: We consider the problem of bulging, or necking, of an infinite thin-walled hyperelastic tube that is inflated by an internal pressure, with the axial stretch at infinity maintained at unity. We present a simple procedure that can be used to derive the bifurcation condition and to determine the near-critical behaviour analytically. It is shown that there is a bifurcation with zero mode number and that the associated axial variation of near-critical bifurcated configurations is governed by a first-order differential equation that admits a locally bulging or necking solution. This result suggests that the corresponding bifurcation pressure can be identified with the so-called initiation pressure which featured in recent experimental studies. This is supported by good agreement between our theoretical predictions and one set of experimental data. It is also shown that the Gent material model can support both bulging and necking solutions whereas the Varga and Ogden material models can only support bulging solutions. Relevance of the present method to the study of non-linear wave propagation in a fluid-filled distensible tube is also discussed.
90 citations
TL;DR: Simulation results show that the unscented Kalman filter (UKF) is efficient and effective for the real-time state estimation and parameter identification of highly non-linear hysteretic systems with degradation and pinching.
Abstract: Many civil and mechanical structures exhibit hysteresis with degradation and/or pinching when subject to severe cyclic loadings such as earthquakes, wind, or sea waves. The modeling and identification of non-linear hysteretic systems with degradation and pinching is therefore a practical problem encountered in the engineering mechanics field. On-line identification of degrading and pinching hysteretic systems is quite a challenging problem because of its complexity. A recently developed technique, the unscented Kalman filter (UKF) which is capable of handling any functional non-linearity, is applied to the on-line parametric system identification of hysteretic differential models with degradation and pinching. Simulation results show that the UKF is efficient and effective for the real-time state estimation and parameter identification of highly non-linear hysteretic systems with degradation and pinching.
83 citations
TL;DR: In this article, the dynamic evolution of the rod from an initially straight state, through a buckled state in the approximate form of a helix, through the dynamic collapse of this helix into a near-planar loop with one site of self-contact, and the subsequent intertwining of this loop with multiple sites of self contact.
Abstract: Twisted marine cables on the sea floor can form highly contorted three-dimensional loops that resemble tangles. Such tangles or ‘hockles’ are topologically equivalent to the plectomenes that form in supercoiled DNA molecules. The dynamic evolution of these intertwined loops is studied herein using a computational rod model that explicitly accounts for dynamic self-contact. Numerical solutions are presented for an illustrative example of a long rod subjected to increasing twist at one end. The solutions reveal the dynamic evolution of the rod from an initially straight state, through a buckled state in the approximate form of a helix, through the dynamic collapse of this helix into a near-planar loop with one site of self-contact, and the subsequent intertwining of this loop with multiple sites of self-contact. This evolution is controlled by the dynamic conversion of torsional strain energy to bending strain energy or, alternatively, by the dynamic conversion of twist (Tw) to writhe (Wr).
81 citations
TL;DR: In this paper, a new approach is used to study the global dynamics of regenerative metal cutting in turning, where the cut surface is modeled using a partial differential equation coupled, via boundary conditions, to an ordinary differential equation (ODE) modeling the dynamics of the cutting tool.
Abstract: A new approach is used to study the global dynamics of regenerative metal cutting in turning. The cut surface is modeled using a partial differential equation (PDE) coupled, via boundary conditions, to an ordinary differential equation (ODE) modeling the dynamics of the cutting tool. This approach automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Taylor's 3/4 power law model for the cutting force is adopted. Lower dimensional ODE approximations are obtained for the combined tool–workpiece model using Galerkin projections, and a bifurcation diagram computed. The unstable solution branch off the subcritical Hopf bifurcation meets the stable branch involving self-interrupted dynamics in a turning point bifurcation. The tool displacement at that turning point is estimated, which helps identify cutting parameter ranges where loss of stability leads to much larger self-interrupted motions than in some other ranges. Numerical bounds are also obtained on the parameter values which guarantee global stability of steady-state cutting, i.e., parameter values for which there exist neither unstable periodic motions nor self-interrupted motions about the stable equilibrium.
78 citations
TL;DR: In this article, the authors investigated the non-linear in-plane buckling of pin-ended shallow circular arches with elastic end rotational restraints under a central concentrated load.
Abstract: This paper investigates the non-linear in-plane buckling of pin-ended shallow circular arches with elastic end rotational restraints under a central concentrated load. A virtual work method is used to establish both the non-linear equilibrium equations and the buckling equilibrium equations. Analytical solutions for the non-linear in-plane symmetric snap-through and antisymmetric bifurcation buckling loads are obtained. It is found that the effects of the stiffness of the end rotational restraints on the buckling loads, and on the buckling and postbuckling behaviour of arches, are significant. The buckling loads increase with an increase of the stiffness of the rotational restraints. The values of the arch slenderness that delineate its snap-through and bifurcation buckling modes, and that define the conditions of buckling and of no buckling for the arch, increase with an increase of the stiffness of the rotational end restraints.
TL;DR: In this paper, the primary resonance response of an asymmetric Duffing oscillator with no linear stiffness term and with hardening characteristic is investigated, and an approximate solution corresponding to the steady-state response is sought by applying the harmonic balance method.
Abstract: The primary resonance response of an asymmetric Duffing oscillator with no linear stiffness term and with hardening characteristic is investigated in this paper. An approximate solution corresponding to the steady-state response is sought by applying the harmonic balance method. Its stability is also studied. It is found that different shapes of frequency–response curves can exist. Multiple-valued solutions, indicating the occurrence of jump phenomena, are observed analytically and confirmed numerically. The influence of the system parameters on the primary resonance response is also examined.
TL;DR: In this article, the steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder in a stream flowing vertically upwards is numerically studied for both cases of heated and cooled cylinders.
Abstract: The steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder in a stream flowing vertically upwards is numerically studied for both cases of heated and cooled cylinders. The governing partial differential equations are transformed into dimensionless forms using an appropriate transformation and then solved numerically using the Keller-box method. The comparison between the solutions obtained and those for a Newtonian fluid is found to be very good. Effects of the mixed convection and elasticity parameters on the skin friction and heat transfer coefficients for a fluid having the Prandtl number equal to one are also discussed. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point. However, for a sufficiently cold cylinder there will not be a boundary layer.
TL;DR: In this article, a new transient non-linear elastic wave spectroscopy (TNEWS) is presented for the detection and localization of a scattered zone (damage) in a composite plate.
Abstract: To reduce the costs related to maintenance of aircraft structures, there is the need to develop new robust, accurate and reliable damage detection methods. A possible answer to this problem is offered by newly developed non-linear acoustic/ultrasonic techniques, which monitor the non-linear elastic wave propagation behaviour introduced by damage, to detect its presence and location. In this paper, a new transient non-linear elastic wave spectroscopy (TNEWS) is presented for the detection and localization of a scattered zone (damage) in a composite plate. The TNEWS analyses the uncorrelations between two structural dynamic responses generated by two different pulse excitation amplitudes by using a time–frequency coherence function. A numerical validation of the proposed method is presented. Damage was introduced and modelled using a multi-scale material constitutive model (Preisach–Mayergoyz space). The developed technique identified in a clear manner the faulted zone, showing its robustness to locate and characterize non-linear sources in composite materials
TL;DR: In this article, it is shown that for any material or structural model expressible as a Masing model, there exists a unique parallel-series (displacement-based) Iwan system that characterizes that model as a function of displacement history.
Abstract: It is shown that for any material or structural model expressible as a Masing model, there exists a unique parallel-series (displacement-based) Iwan system that characterizes that model as a function of displacement history. This poses advantages both in terms of more convenient force evaluation in arbitrary deformation histories as well as in terms of model inversion. Characterization as an Iwan system is demonstrated through the inversion of the Ramberg–Osgood model, a force(stress)-based material model that is not explicitly invertible. An implication of the inversion process is that direct, rigorous comparisons of different Masing models, regardless of the ability to invert their constitutive relationship, can be achieved through the comparison of their associated Iwan distribution densities.
TL;DR: In this paper, the equations of motion of a third grade fluid between micro-parallel plates are derived and made dimensionless, and approximate analytical solutions are obtained by perturbation techniques.
Abstract: Electro-osmotic flow of a third grade fluid between micro-parallel plates is considered. The equations of motion are derived and made dimensionless. Approximate analytical solutions are obtained by perturbation techniques. Constant viscosity and temperature dependent viscosity (Reynolds model) cases are treated separately. Numerical solutions of the equations are also obtained. Influences of non-Newtonian parameter, Joule heating effect, viscosity index and electro-kinetic effect on the velocity and temperature profiles are shown. Approximate and numerical solutions are contrasted.
TL;DR: In this article, a two-plane automatic balancing device for rigid rotors is presented, where ball bearings, which are free to travel around a race, are used to eliminate imbalance due to shaft eccentricity or misalignment.
Abstract: We present an analysis of a two-plane automatic balancing device for rigid rotors. Ball bearings, which are free to travel around a race, are used to eliminate imbalance due to shaft eccentricity or misalignment. The rotating frame is used to derive autonomous equations of motion and the symmetry breaking bifurcations of this system are investigated. Stability diagrams in various parameter planes show the coexistence of a stable balanced state with other less desirable dynamics.
TL;DR: In this paper, the authors investigated how the character of the Poincare map changes over a large parameter range as the system is driven from a non-impacting orbit to an impacting orbit.
Abstract: Soft impacting mechanical systems—where the impacting surface is cushioned with a spring–damper support—are common in engineering. Mathematically such systems come under the description of switching dynamical systems, where the dynamics toggle between two (or more) sets of differential equations, determined by switching conditions. It has been shown that the Poincare map of such a system would have a power of 1/2 (the so-called square-root singularity) if the vector fields at the two sides of the switching manifold differ, and a power of 3/2 if they are the same. These results were obtained by concentrating on the leading order terms in a Taylor expansion of the zero-time discontinuity map, and are true in the immediate neighbourhood of a grazing orbit. In this paper we investigate how the character of the two-dimensional map changes over a large parameter range as the system is driven from a non-impacting orbit to an impacting orbit. This study leads to vital conclusions regarding the character of the normal form of the map not only in the immediate vicinity of the grazing orbit, but also away from it, as dependent on the system parameters. We obtain these characteristics by experiment and by simulation.
TL;DR: In this paper, the Laplace transform method was used to obtain exact solutions to the second Stokes problem for Newtonian fluids, and these solutions, presented as a sum of the steady-state and transient solutions are in accordance with the previous results.
Abstract: New and simpler exact solutions corresponding to the second problem of Stokes for Newtonian fluids are established by the Laplace transform method. These solutions, presented as a sum of the steady-state and transient solutions are in accordance with the previous results (see Figs. 1–4). The amplitudes of the wall shear stresses corresponding to the cosine and sine oscillations are almost identical, except for a small initial time interval. The time required to attain the steady-state for the cosine oscillations of the boundary is smaller than that for the sine oscillations of the boundary. This time decreases if the frequency of the velocity of the boundary increases.
TL;DR: It is shown that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented.
Abstract: Non-linear feedback control provides an effective methodology for vibration mitigation in non-linear dynamic systems. However, within digital circuits, actuation mechanisms, filters, and controller processing time, intrinsic time-delays unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If not well-studied, these inherent and compounding delays may inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering controllers’ performance ineffective. In this work, we present a comprehensive investigation of the effect of time delays on the non-linear control of parametrically excited cantilever beams. More specifically, we examine three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback. Utilizing the method of multiple scales, we derive the modulation equations that govern the non-linear dynamics of the beam. These equations are then utilized to investigate the effect of time delays on the stability, amplitude, and frequency–response behavior. We show that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented.
TL;DR: In this article, the authors proposed two different approaches to detect cracks in metallic samples in coupling elastic wave spectroscopy (NEWS) and time reversal (TR) process, which can be used to either increase the stress on the retrofocusing position or to retrofocuse elastic waves on the defect.
Abstract: In non-destructive evaluation, non-linear elastic wave spectroscopy (NEWS) methods represent powerful tools to explore damaged zones in a sample. The combination of these methods with time reversal (TR) process can be used to either increase the stress on the retrofocusing position or to retrofocuse elastic waves on the defect, when only the non-linear components of the received signal are time reversed. In this paper, we propose two different approaches to detect cracks in metallic samples in coupling NEWS methods and TR process. The first one, NEWS-TR, is defined by sending back only the non-linear components which are preliminary time reversed. Two different techniques to filter non-linear components have been numerically studied: classical harmonic filter and pulse inversion. In these two cases, performances of retrofocusing on different defect positions are analyzed and compared. The second approach, TR-NEWS, consists in measuring the non-linear signature on the focal spot. An experimental study of parametric interaction between two reversed signals ( f 1 and f 2 ) is led. Measurements of components at f 2 - f 1 and f 2 + f 1 around a crack are performed and discussed.
TL;DR: In this article, it is shown that by combining the elastic energy localization of time reversal (TR) with non-linear elastic wave spectroscopy (NEWS), one can isolate nonlinear scatterers in solids.
Abstract: Non-linear elastic wave spectroscopy (NEWS) has been shown to exhibit a high degree of sensitivity to both distributed and isolated nonlinear scatterers in solids. In the case of an isolated non-linear scatterer such as a crack, by combining the elastic energy localization of time reversal (TR) with NEWS, it is shown that one can isolate non-linear scatterers in solids. The experiments reviewed here present two distinct methods of combining TR and NEWS for this purpose. The techniques each have there own advantages and disadvantages, with respect to each other and other non-linear methods, which are discussed. 2008 Published by Elsevier Ltd.
TL;DR: In this paper, the steady two-dimensional stagnation point flow of a second-grade fluid with slip was examined, where the fluid impinges on the wall either orthogonally or obliquely.
Abstract: The steady two-dimensional stagnation-point flow of a second-grade fluid with slip is examined. The fluid impinges on the wall either orthogonally or obliquely. Numerical solutions are obtained using a quasi-linearization technique.
TL;DR: A computational rod model that accounts for non-homogeneous and discontinuous changes in stiffness to support the analysis of DNA looping is contributed and it is demonstrated that while moderate stiffness variations have only modest influence on LacR-mediatedDNA looping, highly localized softening may substantially reduce the energetic cost of looping and profoundly affect loop geometry.
Abstract: DNA molecules may form loops in response to binding with regulatory proteins that control the expression of genes. While DNA looping is a widely accepted gene regulatory mechanism, basic questions regarding the mechanics of the looping process remain open. The present paper contributes a computational rod model that accounts for non-homogeneous and discontinuous changes in stiffness to support the analysis of DNA looping. We pursue this objective in two steps. First, we illustrate the effects of non-homogeneous stiffness on the looping of generic rods under pure torsion. Results computed for this idealized case support our intuition that elastic deformation and strain energy localize in ‘soft’ regions and that equilibrium bifurcations are sensitive to non-homogenous stiffness. Next, we extend the formulation to describe the combined bending, torsion and compression induced on DNA by the LacR protein. We demonstrate that while moderate stiffness variations have only modest influence on LacR-mediated DNA looping, highly localized softening (e.g., ‘kinkable’ or ‘melted’ subdomains) may substantially reduce the energetic cost of looping and profoundly affect loop geometry.
TL;DR: In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained, which involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution.
Abstract: In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained. The solution involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution. Multiple solution branches are observed. The effects of mass transfer and wall stretching are analyzed. Interesting velocity profiles including velocity overshoot and reversal flows are observed in the presence of both mass transfer and wall stretching. These solutions greatly enrich the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.
TL;DR: In this paper, exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed, where the modified Darcy's law is used in the flow modelling.
Abstract: Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.
TL;DR: In this paper, an experimental study and mathematical modelling of newly designed vibro-impact moling rig are presented, which is based on electro-mechanical interactions of a conductor with an oscillating magnetic field.
Abstract: In this paper experimental study and mathematical modelling of newly designed vibro-impact moling rig are presented. The design is based on electro-mechanical interactions of a conductor with an oscillating magnetic field. The rig consists of a metal bar placed within a solenoid which is connected to an RLC circuit, and an obstacle block positioned nearby. Both the solenoid and the block are attached to a base board. Externally supplied alternating voltage causes the bar to oscillate and hit the block resulting in the forward motion of the base board mimicking a mole penetration through the soil. By varying the excitation voltage and the capacitance in the circuit, a variety of system responses can be obtained. In the paper the rig design and experimental procedure are explained in detail, and the mathematical modelling of the rig is described. Then the obtained coupled electro-mechanical equations of motion are integrated numerically, and a comparison between experimental results and numerical predictions is presented.
TL;DR: In this article, a rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models, where a structure is subjected to a uniformly distributed axial and a thrust force.
Abstract: A rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models. Two important loading scenarios were considered, where a structure is subjected to a uniformly distributed axial and a thrust force. These loads are to mimic the main forces acting on an offshore riser, for which an analytical methodology has been developed and applied. In particular, non-linear normal modes (NNMs) and non-linear multi-modes (NMMs) have been constructed by using the method of multiple scales. This is to effectively analyse the transversal vibration responses by monitoring the modal responses and mode interactions. The developed analytical models have been crosschecked against the results from FEM simulation. The FEM model having 26 elements and 77 degrees-of-freedom gave similar results as the low-dimensional (one degree-of-freedom) non-linear oscillator, which was developed by constructing a so-called invariant manifold. The comparisons of the dynamical responses were made in terms of time histories, phase portraits and mode shapes.
TL;DR: In this paper, a triaxial constitutive law for concrete within the framework of isotropic damage combined with plasticity is proposed, which is complemented with a regularization method based on the crack band approach.
Abstract: A triaxial constitutive law for concrete within the framework of isotropic damage combined with plasticity is proposed in this paper. It covers typical characteristics of concrete like non-linear uniaxial compression and tension, bi- and triaxial failure criteria and dilatancy with a unified strain-based approach. Thus, this model is quite simple and especially suitable for strain-driven methods like common finite elements. It is complemented with a regularization method based on the crack band approach. A further issue is discussed with procedures for the model parameter determination for a wide range of concrete grades. The application of the model is demonstrated with typical benchmark tests for plain concrete.
TL;DR: In this paper, a sector p-element is presented for the large amplitude free vibration analysis of laminated composite annular sector plates, where the effects of out-of-plane shear deformations, rotary inertia, and geometric non-linearity are taken into account.
Abstract: A sector p-element is presented for the large amplitude free vibration analysis of laminated composite annular sector plates. The effects of out-of-plane shear deformations, rotary inertia, and geometric non-linearity are taken into account. The shape functions are derived from the shifted Legendre orthogonal polynomials. The element stiffness and mass matrices are integrated analytically with the aid of symbolic computing. The method consists of modeling the annular sector plate as one element. The accuracy of the solution is improved simply by increasing the polynomial order. The time-dependent coefficients are described by a truncated Fourier series. The equations of free motion are obtained using the harmonic balance method and solved by the linearized updated mode method. Results for the linear and non-linear frequencies of clamped laminated composite annular sector plates are obtained. The case of a clamped isotropic annular sector plate is also shown. The linear frequencies are found to converge rapidly downwards as the polynomial order is increased. Comparisons of the linear frequencies with published results show excellent agreement. The effects of sector angle, inner-to-outer radius ratio, thickness-to-outer radius ratio, moduli ratio, number of plies, and layup sequence on the backbone curves are also investigated. It is shown that the hardening behavior increases or decreases depending on geometric and lamination parameters.