Showing papers in "Nonlinear Dynamics in 2003"

A study of the nonlinear response of a resonant micro-beam to an electric actuation

Abstract: An investigation into the response of a resonant microbeam to anelectric actuation is presented. A nonlinear model is used to accountfor the mid-plane stretching, a DC electrostatic force, and an ACharmonic force. Design parameters are included in the model by lumpingthem into nondimensional parameters. A perturbation method, the methodof multiple scales, is used to obtain two first-order nonlinearordinary-differential equations that describe the modulation of theamplitude and phase of the response and its stability. The model and theresults obtained by the perturbation analysis are validated by comparingthem with published experimental results. The case of three-to-oneinternal resonance is treated. The effect of the design parameters on the dynamic responses isdiscussed. The results show that increasing the axial force improves thelinear characteristics of the resonance frequency and decreases theundesirable frequency shift produced by the nonlinearities. In contrast,increasing the mid-plane stretching has the reverse effect. Moreover,the DC electrostatic load is found to affect the qualitative andquantitative nature of the frequency-response curves, resulting ineither a softening or a hardening behavior. The results also show thatan inaccurate representation of the system nonlinearities may lead to anerroneous prediction of the frequency response.

Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results

Abstract: We study theoretically and experimentally the effect that anonlinear energy sink (NES) has on the steady state dynamics of a weaklycoupled system. The NES possesses essentially nonlinear(nonlinearizable) stiffness nonlinearity of the third degree. We findthat, in contrast to the classical linear vibration absorber, the NES iscapable of absorbing steady state vibration energy from the linearoscillator over a relatively broad frequency range. This results inlocalization of the steady state vibration in the NES, away from thedirectly forced subsystem. For a forward frequency sweep the localizedbranch of steady state motions is suddenly eliminated by a jump to alinearized low-amplitude motion, whereas, for a backward frequency sweepa reverse jump occurs. The difference in the frequencies of the twojumps introduces a nonlinear hysteresis loop. This work extends to thesteady state case of earlier transient passive energy pumping results.The notion of passively transferring vibration energy to an a prioridetermined NES, weakly attached to a main structure, is novel. The useof nonlinear energy sinks for passively absorbing energy from a linearmain structure can form the basis of relatively simple and modularvibration and shock isolation designs of mechanical systems.

Description of Elastic Forces in Absolute Nodal Coordinate Formulation

Abstract: The objective of this paper is to investigate the accuracy of the elastic force models that can be used in the absolute nodal coordinate finite element formulation. This study focuses on the description of the elastic forces in three-dimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for a three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces in the beam element. The improvement proposal is verified using several numerical examples that show that the proposed elastic force model of the beam element agrees with the analytical results as well as with the solutions obtained using existing finite element formulation. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that a beam element based on the absolute nodal coordinate formulation relaxes the assumption of a rigid cross-section and is capable of representing a distortional deformation of the cross-section. The numerical results also imply that the beam element does not suffer from the phenomenon called shear locking.

The Vectorial Parameterization of Rotation

Abstract: The parameterization of rotation is the subject of continuous research and development in many theoretical and applied fields of mechanics, such as rigid body, structural, and multibody dynamics, robotics, spacecraft attitude dynamics, navigation, image processing, and so on. This paper introduces the vectorial parameterization of rotation, a class of parameterization techniques encompassing many formulations independently developed to date for the analysis of rotational motion. The exponential map of rotation, the Rodrigues, Cayley, Gibbs, Wiener, and Milenkovic parameterization all are special cases of the vectorial parameterization. This generalization parameterization sheds additional light on the fundamental properties of these techniques, pointing out the similarities in their formal structure and showing their inter-relationships. Although presented in a compact manner, all of the formulae needed for a complete implementation of the vectorial parameterization of rotation are included in this paper.

Sway Reduction on Container Cranes Using Delayed Feedback Controller

Abstract: Traditionally, container cranes are modeled as a simple pendulum, witheither a flexible or a rigid hoisting cable, and a lumped mass at theend of that cable. In the case of large container cranes, the actualconfiguration of the hoisting mechanism is significantly different. Itconsists typically of an arrangement of four hoisting cables, which arehoisted from four different points on the trolley and attached on theload side to four points on a spreader bar used to lift containers.Thus, the dynamics of the actual container-crane hoisting assembly isdifferent from that of a simple pendulum. A controller design based onthe actual model is more likely to result in an improved response. Inthis work, a mathematical model of the actual container crane isdeveloped. Then, a simplified version of this model is used to calculatethe gain and delay for the delay controller developed earlier. Numericalsimulations are performed by applying the delay controller to the fullnonlinear model of the container crane.

Formulation of Three-Dimensional Joint Constraints Using the Absolute Nodal Coordinates

Abstract: A wide variety of mechanical and structural multibody systems consist ofvery flexible components subject to kinematic constraints. The widelyused floating frame of reference formulation that employs linear modelsto describe the local deformation leads to a highly nonlinear expressionfor the inertia forces and can be applied to only small deformationproblems. This paper is concerned with the formulation and computerimplementation of spatial joint constraints and forces using the largedeformation absolute nodal coordinate formulation. Unlike the floatingframe of reference formulation that employs a mixed set of absolutereference and local elastic coordinates, in the absolute nodalcoordinate formulation, global displacement and slope coordinates areused. The nonlinear kinematic constraint equations and generalized forceexpressions are expressed in terms of the absolute global displacementsand slopes. In particular, a new formulation for the sliding jointbetween two very flexible bodies is developed. A surface parameter isintroduced as an additional new variable in order to facilitate theformulation of this sliding joint. The constraint and force expressionsdeveloped in this paper are also expressed in terms of generalizedCholesky coordinates that lead to an identity inertia matrix. Severalexamples are presented in order to demonstrate the use of theformulations developed in the paper.

Nonlinear Dynamic Analysis of MEMS Switches by Nonlinear Modal Analysis

Abstract: A nonlinear modal analysis approach based on the invariant manifoldmethod proposed earlier by Boivin et al. [10] is applied in this paperto perform the dynamic analysis of a micro switch. The micro switch ismodeled as a clamped-clamped microbeam subjected to a transverseelectrostatic force. Two kinds of nonlinearities are encountered in thenonlinear system: geometric nonlinearity of the microbeam associatedwith large deflection, and nonlinear coupling between two energydomains. Using Galerkin method, the nonlinear partial differentialgoverning equation is decoupled into a set of nonlinear ordinarydifferential equations. Based on the invariant manifold method, theassociated nonlinear modal shapes, and modal motion governing equationsare obtained. The equation of motion restricted to these manifolds,which provide the dynamics of the associated normal modes, are solved bythe approach of nonlinear normal forms. Nonlinearities and the pull-inphenomena are examined. The numerical results are compared with thoseobtained from the finite difference method. The estimate for the pull-involtage of the micro device is also presented.

Nonclassical Responses of Oscillators with Hysteresis

Abstract: The responses and codimension-one bifurcations in Masing-type and Bouc-Wen hysteretic oscillators are investigated. The pertinent state space is formulated for each system and the periodic orbits are sought as the fixed points of an appropriate Poincare map. The implemented path-following scheme is a pseudo-arclength algorithm based on arclength parameterization. The eigenvalues of the Jacobian of the map, calculated via a finite-difference scheme, are used to ascertain the stability and bifurcations of the periodic steady-state solutions. Frequency-response curves for various excitation levels are constructed considering representative hysteresis loop shapes generated with the two models in the primary and superharmonic frequency ranges. In addition to known behaviors, a rich class of solutions and bifurcations, mostly unexpected for hysteretic oscillators - including jump phenomena, symmetry-breaking, complete period-doubling cascades, fold, and secondary Hopf - is found. Complex (mode-locked) periodic and nonperiodic responses are also investigated thereby allowing to draw a more comprehensive picture of the dynamical behavior exhibited by these systems.

Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic, Fractional Derivative, Generalized Friction Model

Abstract: In presenting a nonlinear dynamic model of a rubber vibration isolator, the quasistatic and dynamic motion influences on the force response are investigated within the time and frequency domain. It is found that the dynamic stiffness at the frequency of a harmonic displacement excitation, superimposed upon the long term isolator response, is strongly dependent on static precompression, dynamic amplitude and frequency. The problems of simultaneously modelling the elastic, viscoelastic and friction forces are removed by additively splitting them, modelling the elastic force response by a nonlinear, shape factor based approach, displaying results that agree with those of a neo-Hookean hyperelastic isolator at a long term precompression. The viscoelastic force is modeled by a fractional derivative element, while the friction force governs from a generalized friction element displaying a smoothed Coulomb force. A harmonic displacement excitation is shown to result in a force response containing the excitation frequency and its every other higher-order harmonic, while using a linearized elastic force response model, whereas all higher-order harmonics are present for the fully nonlinear case. It is furthermore found that the dynamic stiffness magnitude increases with static precompression and frequency, while decreasing with dynamic excitation amplitude-eventually increasing at the highest amplitudes due to nonlinear elastic effects-with its loss angle displaying a maximum at an intermediate amplitude. Finally, the dynamic stiffness at a static precompression, using a linearized elastic force response model, is shown to agree with the fully nonlinear model except at the highest dynamic amplitudes.

An Investigation of Internal Resonance in Aeroelastic Systems

Abstract: Although the study of internal resonance in mechanical systems has been given significant consideration, minimal attention has been given to internal resonance for systems which consider the presence of aerodynamic forces. Herein, the investigators examine the possible existence of internal resonances, and the related nonlinear pathologies that such responses may have, for an aeroelastic system which possesses nonlinear aerodynamic loads. Evidence of internal resonance is presented for specific classes of aeroelastic systems, and such adverse response indicates nonlinearities may lead to aeroelastic instabilities that are not predicted by traditional (linear) approaches.

Optimal Control of Deployment of a Tethered Subsatellite

Abstract: One of the most important operations during a tethered satellite system mission is the deployment of a subsatellite from a space ship. We restrict tothe simple but practically important case that the system ismoving on a circular orbit around the Earth. The main problem duringdeployment due to gravity gradient is that the two satellites do not move along the straight radial relative equilibrium position which is stable for a tether of constant length. Instead, deploymentleads to an unstable motion with respect to the radial relativeequilibrium configuration. Therefore we introduce an optimal control strategy using theMaximum Principle to achieve a force controlled deployment of the tethered subsatellite from the radial relative equilibrium position close to the space ship to the radial relative equilibrium position far away from the space ship.

Optimal Control of Nonregular Dynamics in a Duffing Oscillator

Abstract: A method for controlling nonlinear dynamics and chaos previouslydeveloped by the authors is applied to the classical Duffing oscillator.The method, which consists in choosing the best shape of externalperiodic excitations permitting to avoid the transverse intersection ofthe stable and unstable manifolds of the hilltop saddle, is firstillustrated and then applied by using the Melnikov method foranalytically detecting homoclinic bifurcations. Attention is focused onoptimal excitations with a finite number of superharmonics, because theyare theoretically performant and easy to reproduce. Extensive numericalinvestigations aimed at confirming the theoretical predictions andchecking the effectiveness of the method are performed. In particular,the elimination of the homoclinic tangency and the regularization offractal basins of attraction are numerically verified. The reduction ofthe erosion of the basins of attraction is also investigated in detail,and the paper ends with a study of the effects of control on delayingcross-well chaotic attractors.

Symmetries of Itô and Stratonovich Dynamical Systems and Their Conserved Quantities

Abstract: A new definition for the symmetries of Ito and Stratonovich dynamicalsystem is given. Determining systems of symmetries for Ito andStratonovich systems have been obtained, and their relation has beendiscussed. It has been shown that some of the Lie point symmetries ofthe Fokker–Planck equation can be constructed using the symmetries ofIto dynamical systems. Conserved quantities can be found from thesymmetries of stochastic dynamical systems which do not arise from aHamiltonian. The results have been applied to an example.

Fractional Derivative Viscoelasticity at Large Deformations

Abstract: A time domain viscoelastic model for large three-dimensional responses underisothermal conditions is presented. Internal variables with fractional orderevolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the timedependency of many polymeric materials can be described using relatively fewparameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does nothold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code forstructural dynamics. The algorithm calculates the fractional derivatives by means of the Grunwald–Lubich approach.Analytical and numerical calculations of the constitutive response in the nonlinearregime are presented and compared. The dynamicstructural response of a viscoelastic bar as well as the quasi-static response of athick walled tube are computed, including both geometrically and materiallynonlinear effects. Moreover, it isshown that by applying relatively small load magnitudes, the responses ofthe linear viscoelastic model are recovered.

Optimization of Vehicle Suspension Systems for Improved Comfort of Road Vehicles Using Flexible Multibody Dynamics

Abstract: Methods that account for the flexibility of multibody systems extend the range of applications to areas such as flexible robots, precision machinery, vehicle dynamics or space satellites. The method proposed here for flexible multibody models allows for the representation of complex-shaped bodies using general finite-element discretizations which deform during the dynamic loading of the system, while the gross rigid body motion of these bodies is still captured using fixed-body coordinate frames. Components of the system for which the deformations are relatively unimportant are represented with rigid bodies. This method is applied to a road vehicle where flexibility plays an important role in its ride and handling dynamic behavior. Therefore, for the study of the limit behavior of the vehicles, the use of flexible multibody models is of high importance. The design process of these vehicles, very often based on intuition and experience, can be greatly enhanced through the use of generalized optimization techniques concurrently with multibody codes. The use of sparse matrix system solvers and modal superposition, to reduce the number of flexible coordinates, in a computer simulation, assures a fast and reliable analysis tool for the optimization process. The optimum design of the vehicle is achieved through the use of an optimization algorithm with finite-differencesensitivities, where the characteristics of the vehicle components are the design variables on which appropriate constraints are imposed. The ride optimization is achieved by finding the optimum of a ride index that results from a metric that accounts for the acceleration in several key points in the vehicle properly weighted in face of their importance for the comfort of the occupant. Simulations with different road profiles are performed for different speeds to account for diverse ride situations. The results are presented and discussed in view of the different methods usedwith emphasis on models and algorithms.

Large amplitude three-dimensional free vibrations of inclined sagged elastic cables

Abstract: The nonlinear characteristics in the large amplitude three-dimensionalfree vibrations of inclined sagged elastic cables are investigated. Amodel formulation which is not limited to cables having smallsag-to-span ratios and takes into account the axial deformation effectis considered. Based on a multi-degree-of-freedom cable model, a finitedifference discretization is employed within a numerical solution of thegoverning equations of three-dimensional coupled motion. Variousnumerical examples of arbitrarily inclined sagged cables with initialout-of-plane or in-plane motions are carried out for the case of aspecified end tension. The major findings consist of highlighting theextent of two-and three-dimensional nonlinear couplings, the occurrenceof nonlinear dynamic tensions, and the meaningfulness of modaltransition phenomena ensuing from the activation of various internalresonance conditions. The influence of cable inclination on thenonlinear dynamic behavior is also evaluated. Comprehensive discussionand comparison of large amplitude free vibrations of horizontal andinclined sagged cables are presented.

Describing Rigid-Flexible Multibody Systems Using Absolute Coordinates

Abstract: This paper deals with the dynamic description of interconnected rigid and flexible bodies. The absolute nodal coordinate formulation is used to describe the motion of flexible bodies and natural coordinates are used to describe the motion of the rigid bodies. The absolute nodal coordinate formulation is a nonincremental finite element procedure, especially suitable for the dynamic analysis of flexible bodies exhibiting rigid body motion and large deformations. Nodal coordinates, which include global position vectors and global slopes, are all defined in a global inertial coordinate system. The advantages of using the absolute nodal coordinate formulation include constancy in the mass matrix and the need for only a minimal set of nonlinear constraint equations when connecting different flexible bodies with kinematic joints. When bodies within the system can be considered rigid, the above-mentioned advantages of the equations of motion can be preserved, provided natural coordinates are used. In the natural coordinate method, the coordinates used to describe rigid bodies include global position vectors of basic points and global unit vectors. As occurs in absolute nodal coordinate formulation, rotational coordinates are avoided and the mass matrix is also constant. This paper provides computer implementation of this formulation that uses absolute coordinates for general two-dimensional multibody systems. The constraint equations needed to define kinematic joints between different bodies can be linear or nonlinear. The linear constraint equations, which include those needed to define rigid connections and revolute joints, are used to define constant connectivity matrices that reduce the size of the system coordinates. These constant connectivity matrices are also used to obtain the mass matrix and generalized forces of the system. However, the nonlinear constraint equations that account for sliding joints require the use of the Lagrange multipliers technique. Numerical examples are provided and compared to the results of other existing formulations.

On the Transfer of Energy between Widely Spaced Modes in Structures

Abstract: An experimental and theoretical study of the response of aflexible cantilever beam to an external harmonic excitation nearthe beam's third natural frequency is presented. For a certain range ofthe excitation frequency, we observed experimentally that the responseincludes a large contribution due to the first mode of the beamaccompanied by a slow modulation of the amplitude and phase of the thirdmode. In addition, we noted that the energy transfer between the thirdand first modes is very much dependent upon the closeness of themodulation (or Hopf bifurcation) frequency to the first-mode naturalfrequency. In earlier studies by Nayfeh and coworkers, the modulationfrequency was close to the first-mode natural frequency, and thereforelarge first-mode swaying was observed. But for higher forcingamplitudes, the present experiments show that the modulation frequencytends to shift away from the first-mode natural frequency, andsubsequently very little swaying is observed. We also developed areduced-order analytical model by discretizing the integralpartial-differential equation of motion, derived by Crespo daSilva and Glenn, using the Galerkin procedure with a four-modeapproximation. The reduced-order model demonstrates the energy transferfrom the third mode to the first mode.

Large Oscillations of a Thin Cantilever Beam: Physical Experiments and Simulation Using the Absolute Nodal Coordinate Formulation

Abstract: Many papers have studied computer-aided simulations of elastic bodies undergoing large deflections and large deformations. But there have not been many attempts to check the validity of the numerical formulations used in these studies. The main aim of this paper is to demonstrate the validity of one such numerical formulation, the absolute nodal coordinate formulation (ANCF), by comparing the results it generates with the results of real experiments. Large oscillations of a thin cantilever beam are studied in this paper to numerically model the beam, which also accounts for the effects of an attached end-point weight and damping forces. The experiments were carried out using a high-speed camera and a data acquisition system.

Nonlinear Parametric Vibrations of an Elastic Structure with a Rectangular Liquid Tank

Abstract: The nonlinear coupled vibrations of an elastic structure and liquidsloshing in a rectangular tank partially filled with liquid, are investigated.The structure on which the liquid tank is attached is vertically subjected to a sinusoidal excitation when the natural frequency of the structure is equal to twicethe natural frequency of one of the sloshing modes. In the theoretical analysis, the modal equations are derivedby taking nonlinear fluid force into account. Responses of the structure and the liquid surface are presented asresonance curves using the harmonic balance method. From this theoreticalanalysis the following predictions are obtained: (a) due to the nonlinearity of the fluid force, harmonic oscillations appear in the structure, while subharmonic oscillations occur on the liquid surface; (b) the shapes of the resonance curves markedly change depending on the liquid level; and (c) when the tuning condition is slightly deviated, amplitudemodulated motions and chaotic oscillations appear during a certain range of the excitation frequency. These were qualitatively in agreement with the experimental results.

Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

Abstract: The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter e, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe e3-order in the 1:3 case, they already arise at thee2-order in the 1:2 case. Thus, by truncating theanalysis at the e3-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.

Identification of Nonlinear and Viscoelastic Properties of Flexible Polyurethane Foam

Abstract: Analysis of the steady-state response of a polyurethane foam and masssystem to harmonic excitation is presented. The foam's uni-directionaldynamic behavior is modeled by using nonlinear stiffness, linearviscoelastic and velocity proportional damping components. Therelaxation kernel for the viscoelastic model is assumed to be a sum ofexponentials. The harmonic balance method is used to develop one- andtwo-term approximations to periodic solutions, and the equationsdeveloped are utilized for system identification. The identificationprocess is based on least-squares minimization of a sub-optimal costfunction that uses response data at various excitation frequencies andamplitudes. The effects of frequency range, spacing and amplitudes ofthe harmonic input on the results of the model parameter estimation arediscussed. The identification procedure is applied to measurements ofthe steady-state response of a base-excited foam-mass system. Estimatesof the system parameters at different levels of compression and inputamplitudes are thus determined. The choice of model-order and thefeasibility of describing the system behavior at several inputamplitudes with a single set of parameters are also addressed.

Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method

Abstract: This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.

Centrifugal Pendulum Vibration Absorbers: An Experimental and Theoretical Investigation

Abstract: This paper presents results from tests completed on a rotor system fitted with pendulum-type centrifugal torsional vibration absorbers. A review of the associated theoretical background is also given and the experimental and theoretical results are compared andcontrasted. An overview of the test apparatus is provided and itsunique features are discussed. To the best knowledge of the authors,this is the first time that a systematic study of the dynamic behaviorof torsional vibration absorbers has been undertaken in a controlledenvironment.

Nonlinear System Identification of Multi-Degree-of-Freedom Systems

Abstract: A nonlinear system identification methodology based on theprinciple of harmonic balance is extended tomulti-degree-of-freedom systems. The methodology, called HarmonicBalance Nonlinearity IDentification (HBNID), is then used toidentify two theoretical two-degree-of-freedom models and anexperimental single-degree-of freedom system. The three modelsand experiments deal with self-excited motions of afluid-structure system with a subcritical Hopf bifurcation. Theperformance of HBNID in capturing the stable and unstable limitcycles in the global bifurcation behavior of these systems is alsostudied. It is found that if the model structure is well known,HBNID performs well in capturing the unknown parameters. If themodel structure is not well known, however, HBNID captures thestable limit cycle but not the unstable limit cycle.

Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow

Abstract: In this paper chaotic behavior of nonlinear viscoelastic panels in asupersonic flow is investigated. The governing equations, based on vonKaarman's large deflection theory of isotropic flat plates, areconsidered with viscoelastic structural damping of Kelvin's modelincluded. Quasi-steady aerodynamic panel loadings are determined usingpiston theory. The effect of constant axial loading in the panel middlesurface and static pressure differential have also been included in thegoverning equation. The panel nonlinear partial differential equation istransformed into a set of nonlinear ordinary differential equationsthrough a Galerkin approach. The resulting system of equations is solvedthrough the fourth and fifth-order Runge–Kutta–Fehlberg (RKF-45)integration method. Static (divergence) and Hopf (flutter) bifurcationboundaries are presented for various levels of viscoelastic structuraldamping. Despite the deterministic nature of the system of equations,the dynamic panel response can become random-like. Chaotic analysis isperformed using several conventional criteria. Results are indicative ofthe important influence of structural damping on the domain of chaoticregion.

Bifurcations of a Thin Plate-Strip Excited Transversally and Axially

Abstract: The complex vibrations and bifurcations of plates modeled as systemswith infinite degrees-of-freedom are considered. Both theBubnov–Galerkin with high-order approximations and finite differencemethods with approximation O(h4)are applied. In addition, the calculation ofthe Lyapunov exponents of the system is performed, and the results arecompared to those derived by Bennetin's method. Some examples of newnonlinear phenomena exhibited by the considered systems are reported.

2:2:1 resonance in the quasiperiodic mathieu equation

Abstract: In this work, we investigate regions of stability in the vicinity of 2:2:1 resonance in the quasiperiodic Mathieu equation d 2 x dt 2 + (δ + e cos t + eµ cos(1 + e�)t)x = 0, using two successive perturbation methods. The parameters e and µ are assumed to be small. The parameter e serves for deriving the corresponding slow flow differential system and µ serves to implement a second perturb- ation analysis on the slow flow system near its proper resonance. This strategy allows us to obtain analytical expressions for the transition curves in the resonant quasiperiodic Mathieu equation. We compare the analytical results with those of direct numerical integration. This work has application to parametrically excited systems in which there are two periodic drivers, each with frequency close to twice the frequency of the unforced system.

First-passage time of Duffing oscillator under combined harmonic and white-noise excitations

Abstract: The first-passage time of Duffing oscillator under combined harmonic and white-noise excitations is studied. The equation of motion of the system is first reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. Numerical results for two resonant cases with several sets of parameter values are obtained and the analytical results are verified by using those from digital simulation.

A Gluing Algorithm for Distributed Simulation of Multibody Systems

Abstract: A new gluing algorithm is presented that can be used tocouple distributed subsystem models fordynamics simulation of mechanical systems. Using this gluingalgorithm, subsystem models can be analyzed attheir distributed locations, using their own independent solvers,and on their own platforms. The gluing algorithmdeveloped relies only on information available at the subsysteminterfaces. This not only enables efficientintegration of subsystem models, but also engenders modelsecurity by limiting model access only to the exposedinterface information. These features make the algorithm suitablefor a real and practical distributed simulationenvironment.