Nonlinear Structural Dynamics Research Articles

Convergence of the Galerkin Method for Nonlinear Dynamics of the Continuous Structure

The Galerkin truncation method is a powerful method for nonlinear dynamics analysis, and has been widely used to discretize the spatial differential operator. Due to more and more new fields of application, the research interest on the Galerkin method is still high today. In this paper, research on the convergence of Galerkin method for nonlinear dynamics of the continuous structure is thoroughly reviewed. At the beginning, the Galerkin method is briefly introduced. Then, the paper reviews the application of the truncation method. This paper also sums up the comparative study on the Galerkin method with other methods, such as the finite difference method (FDM), the finite element method (FEM), and the multiple time scales method. In the investigations concerning the convergence of the Galerkin method, this paper summarizes recent studies on nonlinear dynamics of the axially moving systems, the continua on the nonlinear foundation, and the belt-pulley systems. Finally, the truncation terms of Galerkin method for the continuous structure's nonlinear dynamics analysis is suggested for the future research applications.

Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas

A general theoretical framework for investigating the nonlinear dynamics of phase space zonal structures is presented in this work. It is then, more specifically, applied to the limit where the nonlinear evolution time scale is smaller or comparable to the wave–particle trapping period. In this limit, both theoretical and numerical simulation studies show that nonadiabatic frequency chirping and phase locking could lead to secular resonant particle transport on meso- or macro-scales. The interplay between mode structures and resonant particles then provides the crucial ingredient to properly understand and analyze the nonlinear dynamics of Alfvén wave instabilities excited by nonperturbative energetic particles in burning fusion plasmas. Analogies with autoresonance in nonlinear dynamics and with superradiance in free-electron lasers are also briefly discussed.

A Comparative Study on Optimal Structural Dynamics Using Wavelet Functions

Wavelet solution techniques have become the focus of interest among researchers in different disciplines of science and technology. In this paper, implementation of two different wavelet basis functions has been comparatively considered for dynamic analysis of structures. For this aim, computational technique is developed by using free scale of simple Haar wavelet, initially. Later, complex and continuous Chebyshev wavelet basis functions are presented to improve the time history analysis of structures. Free-scaled Chebyshev coefficient matrix and operation of integration are derived to directly approximate displacements of the corresponding system. In addition, stability of responses has been investigated for the proposed algorithm of discrete Haar wavelet compared against continuous Chebyshev wavelet. To demonstrate the validity of the wavelet-based algorithms, aforesaid schemes have been extended to the linear and nonlinear structural dynamics. The effectiveness of free-scaled Chebyshev wavelet has been compared with simple Haar wavelet and two common integration methods. It is deduced that either indirect method proposed for discrete Haar wavelet or direct approach for continuous Chebyshev wavelet is unconditionally stable. Finally, it is concluded that numerical solution is highly benefited by the least computation time involved and high accuracy of response, particularly using low scale of complex Chebyshev wavelet.

Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

This work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's “Lagrangian ingredients''---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy proper orthogonal decomposition and reduced-basis sparsification. Results for a parameterized truss-structure problem demonstrate the practical importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.

Non-linear incidental dynamics of frame structures

A simulation of failures on responsible elements is only one form of the extreme structural behavior analysis. By understanding the dynamic behavior in incidental situations, it is possible to make a special structural design from the point of the largest axial force, stress and redundancy. The numerical realization of one such simulation analysis was performed using FEM in this paper. The boundary parameters of transient analysis, such as overall structural damping coefficient, load accelerations, time of load fall and internal forces in the responsible structural elements, were determined on the basis of the dynamic experimental parameters. The structure eigenfrequencies were determined in modal analysis. In the study, the basic incidental models were set. The models were identified by many years of monitoring incidental situations and the most frequent human errors in work with heavy structures. The combined load models of structure are defined in the paper since the incidents simply arise as consequences of cumulative errors and failures. A feature of a combined model is that the single incident causes the next incident (consecutive timing) as well as that other simple dynamic actions are simultaneous. The structure was observed in three typical load positions taken from the crane passport (range-load). The obtained dynamic responses indicate the degree of structural sensitivity depending on the character of incident. The dynamic coefficient KD was adopted as a parameter for the evaluation of structural sensitivity.

Transient Responses in Nonlinear Dynamics of Structures

Innumerable mechanical structures including vibrating profiles, sandwich plates, platforms, thin uniperiodic shells, laminated windshells, isotropic and orthotropic platesor even multibody systems exhibit complex dynamic behavior. Analysis of the corresponding phenomena involves interesting mathematical and experimental methods that are sometimes used in parallel. Mostly, a dynamical transient response of any of the enumerated complex structures leads to the unpredictable states that need to be described and predicted. For this reason, the Special Issue extends selected papers presented during the international conference on “Dynamical Systems-Theory and Applications” hold in December 2–5, 2013 in Lodź, Poland. Main areas of modern experimental and numerical analysis taken into consideration by authors of these papers could be mentioned: bifurcations and chaos in dynamical systems, stability of dynamical systems, original numerical methods of vibration analysis, non-smooth systems, engineering systems and differential equations, control in dynamical systems, asymptotic methods in nonlinear dynamics,vibrations of lumped and continuous systems,dynamics in life sciences and bioengineering. A brief description of contents of this Special Issue follows.

Editorial Dynamics and Control of Aerospace and Vertical Transportation Systems

This Special Issue presents a selection of papers initially presented at the 11th International Conference on Vibration Problems (ICOVP-2013), held from 9 to 12 September 2013 in Lisbon, Portugal. The main topics of this Special Issue are linear and, mainly, nonlinear dynamics, chaos and control of systems and structures and their applications in different field of science and engineering. According to the goal of the Special Issue, the selected contributions are divided into three major parts: “Vibration Problems in Vertical Transportation Systems”, “Nonlinear Dynamics, Chaos and Control of Elastic Structures” and “New Strategies and Challenges for Aerospace and Ocean Structures Dynamics and Control”.

Cardiovascular fluid‐structure interaction: A partitioned approach utilizing the p‐FEM

AbstractIn this paper, we propose a technique for simulating the fluid‐structure interaction in blood vessels. A partitioned approach is used, which allows for an independent discretization of the fluid domain and the structural domain. We choose the finite volume method to solve the Navier‐Stokes equations and the p‐version of the finite element method (p‐FEM) to solve the equations of geometrically nonlinear structural dynamics. The solution strategy can be seen as a first approach towards a comprehensive study of the hemodynamics in vascular substitutes with the goal of improving their long term functionality. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Galerkin‐based multi‐scale time integration for nonlinear structural dynamics

AbstractThis paper deals with a GALERKIN‐based multi‐scale time integration of a viscoelastic rope model. Using HAMILTON's dynamical formulation, NEWTON's equation of motion as a second‐order partial differential equation is transformed into two coupled first order partial differential equations in time. The considered finite viscoelastic deformations are described by means of a deformation‐like internal variable determined by a first order ordinary differential equation in time. The corresponding multi‐scale time‐integration is based on a PETROV‐GALERKIN approximation of all time evolution equations, leading to a new family of time stepping schemes with different accuracy orders in the state variables. The resulting nonlinear algebraic time evolution equations are solved by a multi‐level NEWTON‐RAPHSON method. Realizing this transient numerical simulation, we also demonstrates a parallelized solution of the viscous evolution equation in CUDA©. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling of fluid–structure interaction for simulating vortex-induced vibration of flexible riser: finite difference method combined with wake oscillator model

This paper proposes a numerical simulation method for the dynamic motion of a flexible riser pipe undergoing vortex-induced vibration (VIV). The method is based on a finite difference scheme for solving nonlinear structural dynamics of the pipe and wake oscillator model for quantifying vortex-induced forces acting on the pipe, the combination of which can offer a very efficient and stable computation. To investigate the accuracy of the method, we performed simulations of the VIV of riser pipes under uniform flow and sheared flow conditions; and then compared obtained results with experiments of preceding works. We consequently confirmed that the present method can simulate a couple of important aspects of the VIV of the pipes: frequency, mode shape, and amplitude of displacement of cross-flow displacement.

Optimal performance-based design of non-linear stochastic dynamical RC structures subject to stationary wind excitation

Performance-based design, or performance-based engineering (PBE), is currently well accepted as a proper methodology for assessing risk and designing facilities which can be subject to continuous levels of damage caused by extreme responses under various hazards of varying magnitudes. However, the difficulties in assessing probabilities associated with different hazard and performance levels, especially when nonlinearity is considered in dynamically excited structural systems, have been a limiting factor for incorporating PBE into design optimization. This paper advances the state-of-the-art by incorporating PBE into the optimal design of non-linear/hysteretic stochastic dynamical systems. The approach combines a statistical linearization technique with time-variant reliability analysis concepts, in order to formulate a total expected life-cycle cost optimization problem. As a numerical example, reinforced concrete buildings modeled as MDOF Bouc–Wen hysteretic systems subjected to wind excitation are studied. Optimal transversal stiffness of the buildings columns are obtained both for the linear and the nonlinear cases, as well as for various design life values. Optimal stiffness values determined herein consider the initial costs but also expected losses over the lifetime of the structure, for several wind hazard magnitudes and displacement response levels of the structure.

Grey-box identification of a non-linear solar array structure using cubic splines

Most identification methods in non-linear structural dynamics assume in advance a mathematical model of the non-linearities. This is however possible in specific situations only, since non-linear effects may be caused by numerous phenomena and a priori knowledge is generally limited. The present paper investigates the usefulness of piecewise third-order polynomials, termed cubic splines, to identify the complex non-linear dynamics of solar arrays in their stowed configuration. The estimation of the model parameters is achieved using the frequency-domain non-linear subspace identification (FNSI) method. A distinct advantage of the FNSI approach is its capability to calculate accurately a large number of parameters, while maintaining an acceptable computational burden. This makes tractable the use of cubic splines to represent non-linearity in real-life mechanical systems, as the dimensionality of the inverse problem is known to increase dramatically in this case. The experimental structure of interest consists of two parallel aluminium plates assembled with bolted connections. This application is challenging because of the existence of impacts between the two plates at high excitation amplitude, and of the activation of complicated stiffness and damping mechanisms within the bolted connections.

A lattice Boltzmann model for multiphase flows interacting with deformable bodies

In this paper, a numerical model to simulate a multiphase flow interacting with deformable solid bodies is proposed. The fluid domain is modeled through the lattice Boltzmann method and the Shan–Chen model is adopted to handle the multiphase feature. The interaction of the flow with immersed solid bodies is accounted for by using the Immersed Boundary method. Corotational beam finite elements are used to model the deformable bodies and non-linear structure dynamics is predicted through the Time Discontinuous Galerkin method. A numerical campaign is carried out in order to assess the effectiveness and accuracy of the proposed modeling by involving different scenarios. In particular, the model is validated by performing the bubble test and by comparing present results with the ones from a numerical commercial software. Moreover, the properties in terms of convergence are discussed. In addition, the effectiveness of the proposed methodology is evaluated by computing the error in terms of the energy that is artificially introduced in the system at the fluid–solid interface. Present findings show that the proposed approach is robust, accurate and suitable of being applied to a lot of practical applications involving the interaction between multiphase flows and deformable solid bodies.

Applications of hybrid time–frequency methods in nonlinear structural dynamics

This paper presents a study on methods which may be used to compute the nonlinear response of systems whose linear properties are determined in the frequency or Laplace domain. Typically, this kind of situation may arise in soil–structure and fluid–structure interaction problems. In particular three methods are investigated: (a) the hybrid time–frequency method, (b) the computation of the convolution integral which requires an inverse Fourier or Laplace transform of the system’s transfer function, and (c) the identification of an equivalent system defined in the time domain which may be solved with classical time integration methods. These methods are illustrated by their application to some simple, one degree of freedom, non-linear systems and their advantages and drawbacks are highlighted.

Nonlinear Dynamical Analysis for the Cable Excited with Parametric and Forced Excitation

Considering the deck vibration effect on the cable in cable-stayed bridge, using nonlinear structure dynamics theory, the nonlinear dynamical equation for the stayed cable excited with deck vibration is proposed. Research shows that the vertical vibration of the deck has a combined parametric and forced excitation effect on the cable when the angle of the cable is taken into consideration. Using multiscale method, the 1/2 principle parametric resonance is studied and the bifurcation equation is obtained. Despite the parameters analysis, the bifurcation characters of the dynamical system are studied. At last, by means of numerical method and software MATHMATIC, the effect rules of system parameters to the dynamical behavior of the system are studied, and some useful conclusions are obtained.

Computationally efficient multi-time-step method for partitioned time integration of highly nonlinear structural dynamics

An efficient and accurate method for solving large-scale problems in non-linear structural dynamics is presented. The method uses dual-Schur domain decomposition to divide a large finite element mesh into a number of smaller subdomains, which are solved independently using a suitable mesh-size and time-step to capture the local spatial and temporal scales of the problem. Continuity of the solution between subdomains is enforced by Lagrange multipliers. It is shown that the proposed method is stable, accurate and computationally more efficient than using a uniform time-step for the entire mesh. Numerical examples are presented to illustrate and corroborate these properties.

On Increasing the Efficiency of Penetration of an Electromagnetic Wave into a Dense Plasma Layer

UDC 533.9+537.8 We study a method for increasing the efficiency of penetration of an intense electromagnetic wave into a layer of dense plasma. The method is based on the phenomenon of spatio-temporal synchronism of the nonlinear dynamic structures, which are excited in the layer, with the modulated wave incident on the layer. It has been established that in the case of a relatively weakly overdense plasma, the efficiency of constant-amplitude wave penetration into the plasma layer is determined by a single nonlinear parameter, namely, the ratio of the intensity of the incident radiation to the degree of plasma super-criticality, and increases with an increase in this parameter. It is found that a modulation of the amplitude of an incident wave, whose period is a multiple of the soliton generation period, can lead to a significant increase in the penetration efficiency. It is well known that the interface between two transparent media can efficiently reflect the radiation which is incident on it. An example of this phenomenon in optics can be the effect of total internal reflection, on which the operation of many optical devices is based. When the ionosphere is studied, researchers make extensive use of ionosphere sounding with radio waves of the radio-frequency (RF) band and analysis of the parameters of the signal reflected from the dense plasma of the F layer [1–3]. However, in the case of strong radiation, one should allow for changes in the properties of the matter in the presence of the electromagnetic field in it [4–7]. For example, due to the striction decrease in the plasma density in a strong electromagnetic field, an incident wave can penetrate partially into the depth of the dense plasma layer under certain conditions [8–10]. In the ionosphere plasma, which is characterized by the density that increases smoothly on the scale of the radio wavelength, the striction self-action consists in the deformation of the spatial distribution of the plasma density and a shift of the wave reflection point [11, 12], formation of field solitons and cavitons (holes) of plasma density and nonstationarity of the reflection, which manifests itself, in particular, in the effects of “superreflection” and “afteraction” [2, 13–19]. A homogeneous layer of overdense plasma with a finite thickness can become transparent for intense radiation due to soliton excitation by the incident wave at the front “illuminated” boundary of the layer and radiation of energy of these solitons at the rear “dark” boundary of the layer [20, 21]. Similar effects of light penetration in the form of jets excited by light beams, which are incident on the interface boundary at angles exceeding the angle of total internal reflection, are observed in nonlinear optical media [22, 23]. However, the problem about the methods of increasing the efficiency of penetration of intense radiation beams into a medium being opaque for low-amplitude waves and their passage through finite-width layers has not been properly examined adequately so far. In this work, we use the example of a layer with homogeneous dense nonlinear plasma irradiated by a TE mode to find the coefficient of penetration of the incident wave into the layer as a function of the plasma density and amplitude and demonstrate the methods of increasing the efficiency of this penetration.

Experimental and numerical characterization of the structural dynamics of flapping beams

The nonlinear structural dynamics of a slender beam in flapping motion is examined both experimentally and computationally. In the experiments the periodic flapping motion is imposed on the clamped edge of the cantilever beam using a 4-bar crank-and-rocker mechanism. Aluminum beams with nominal dimensions of 150mm×25mm×0.4mm are tested in air over a range of flapping frequencies up to 1.3 times the linear first modal frequency at two different flapping amplitudes, 15° and 30°. The response of the beam is characterized experimentally through bending strain and tip displacement data obtained from foil strain gage and high-speed camera, respectively. It was determined that for the particular combination of beam specimen (dimensions, material properties) and forcing parameters investigated, all experimental responses were periodic. The frequency response curves based upon the experimental bending strain data reveal a secondary superharmonic peak in addition to the primary resonance peak. As the flapping frequency is increased, the response of the beam is observed to change from symmetric (with respect to equilibrium position) periodic vibrations with a period equal to the flapping period to asymmetric periodic vibrations with higher harmonic content featuring local oscillations in the time histories. Experimental tip displacement results show that the beam spends more time during stroke reversals when the flapping frequency is near primary and secondary resonance regions. In addition to experiment, numerical simulations are performed using two-node, isoparametric degenerate-continuum based geometrically nonlinear beam elements. The HHT-α version of the Newmark finite difference scheme is used to discretize the problem in time and a linear viscous damping model is assumed. Overall the numerical simulations agree well with the experiments and capture most of the nonlinear dynamical features of the beam response. It is, however, found that in resonance regions the simulation overpredicts response magnitudes, possibly due to the use of the linear damping model and linear elastic constitutive model. Additional numerical simulations of the beam tip response reveal dynamics which include periodic, asymmetric periodic, quasi-periodic and aperiodic motions.

Reduced-order computational model in nonlinear structural dynamics for structures having numerous local elastic modes in the low-frequency range. Application to fuel assemblies

We are interested in the construction of a reduced-order computational model for nonlinear complex dynamical structures which are characterized by the presence of numerous local elastic modes in the low-frequency band. This high modal density makes the use of the classical modal analysis method not suitable. Therefore the reduced-order computational model is constructed using a basis of a space of global displacements, which is constructed a priori and which allows the nonlinear dynamical response of the structure observed on the stiff part to be predicted with a good accuracy. The methodology is applied to a complex industrial structure which is made up of a row of seven fuel assemblies with possibility of collisions between grids and which is submitted to a seismic loading.

Performance of an implicit time integration scheme in the analysis of wave propagations

In earlier work an effective implicit time integration scheme was proposed for the finite element solution of nonlinear dynamic problems [1,2]. The method, referred to as the Bathe method, was shown to possess unusual stability and accuracy characteristics for the solution of problems in linear and nonlinear structural dynamics [1–3]. In this paper we study the dispersion properties of the method, in comparison to those of the widely used Newmark trapezoidal rule, and show that the desired characteristics of the Bathe method for structural dynamics are also seen, and are very important, in the solution of wave propagation problems. A dispersion analysis is given and problems are solved to illustrate the capabilities of the scheme for the solution of wave propagation problems.