limit cycle oscillation of missile control fin with structural non linearity non linear aeroelastic characteristics of a deployable missile control fin with structural non linearity are investigated a deployable missile control fin is modelled as a two dimensional typical section model doublet point method is used for the calculation of supersonic unsteady aerodynamic forces and aerodynamic forces are approximated by using the minimum state approximation for non linear flutter analysis structural non linearity is represented by an asymmetric bilinear spring and is linearized by using the describing function method the linear and non linear flutter analyses indicate that the flutter characteristics are significantly dependent on the frequency ratio from the non linear flutter analysis various types of limit cycle oscillations are observed in a wide range of air speeds below or above the linear divergent flutter boundary the non linear flutter characteristics and the non linear aeroelastic responses are investigated

the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

bilinear modal representations for reduced order modeling of localized piecewise linear oscillators abstract a novel reduced order modeling method for vibration problems of elastic structures with localized piecewise linearity is proposed the focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity such as cracked structures and delaminated plates the modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions first it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced order model that can well capture the nonlinear normal modes next it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions these linear modes are referred to as bi linear modes and are selected by an elaborate methodology which utilizes certain similarities between the bi linear modes and approximations for the dominant proper orthogonal modes these approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models the proposed method is compared with traditional reduced order modeling methods such as component mode synthesis and its advantages are discussed forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology

limit cycle stability reversal near a hopf bifurcation with aeroelastic applications abstract the objective of this paper is to present an analytical numerical analysis of the phenomenon of limit cycle stability reversal from unstable to stable and vice versa a singular perturbation technique the method of the normal form in the asymptotic expansion version is utilized the number of equations is then reduced to a minimal set for which the results are in good agreement with those from the original equations this minimal set is determined by the amplitude of the points a concept closely related to the small divisors in the kam theory this set is larger than that corresponding to the zero real part eigenvalues center manifold theorem the method is applied to a specific problem an aeroelastic section with cubic free play non linearities where the parameter is the flight speed numerical studies have been performed to show the dependence of the hopf bifurcation characteristics upon the structural and geometric properties of the wing section plots depicting amplitudes and frequency versus flight speed are presented

studies in numerical nonlinear instability ii a new look at u t uu x 0 abstract it is shown that in general leap frog schemes any particular unstable solution behaves as an attractor of other solutions for a leap frog discretization of u t uu x 0 a particular kind of unstable solution is constructed which generically attracts any other solution estimates of the overflow time are presented and related to the notions of stability threshold and restricted stability

bifurcation analysis of airfoils in incompressible flow abstract bifurcations of airfoils with non linear pitching stiffness in incompressible flow are investigated the harmonic balance method which is more practical is adopted for the analysis and numerical integration results are used as standards of checking analytical solutions calculated by asymptotic expansion method and averaging method with the aid of computer algebra have confirmed the feasibility of the harmonic balance method

analysis of forced bilinear oscillators and the application to cracked beam dynamics a new closed form solution based on the use of two square wave functions to model stiffness change is proposed for bilinear oscillators under low frequency excitation the solution procedure is extended to cases with bilinear forcing function which usually occur in the dynamics of damaged structures the proposed solution provides the spectral pattern and the magnitude of each harmonic component for a damaged rectangular beam according to the size and location of the crack the predicted spectral pattern and the magnitude of each harmonic component for a simply supported beam containing a fatigue crack are validated by numerical simulation experimental work has not been done the results show that the proposed solution may be used to predict efficiently spectral patterns of damaged structures while avoiding lengthy computation such a procedure may be developed into an attractive on line diagnostic tool

analysis of limit cycle flutter of an airfoil in incompressible flow both experimental and theoretical analyses are perfomed to investigate several types of self excited oscillations of a two dimensional wing model with non linear pitching stiffiness free play non linearity receives especial attention in the low speed wind tunnel test of the wing model with free play in pitch an interesting case of double limit cycle flutter is found subsequent harmonic balance analyses ascertain not only these two sustained oscillations but also two other unstable limit cycles a digital simulation method is also used for flutter analysis the theoretical results agree well with the experimental data

an extension of the lax richtmyer theory we give a rigorous proof of the validity of the lax equivalence theorem when the true solution at timet and its approximation lie in different spaces thus modelling the practical situation where the former is a function of the space variables and the latter is only defined at grid points

numerical stability for nonlinear evolution equations the paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations brezis and pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by crandall and liggett by using the approximation theorem of brezis and pazy we show the n stability of the abstract nonlinear discrete problem for the implicit euler method motivated by the rational approximation methods for linear semigroups we propose a more general time discretisation method and prove its n stability as well

nonlinear stability and convergence of finite difference methods for the good boussinesq equation the good boussinesq equationu tt u xxxx u xx u 2 xx has recently been found to possess an interesting soliton interaction mechanism in this paper we study the nonlinear stability and the convergence of some simple finite difference schemes for the numerical solution of problems involving the good boussinesq equation numerical experimentas are also reported

subharmonic bifurcation analysis of wing with store flutter abstract an approximate method is developed to analyze the subharmonic bifurcation of a wing with a store aeroelastic system of which the junction stiffness is piecewise linear the key point is to treat the store as a forced oscillator under harmonic excitation exerted through the junction from the wing in limit cycle flutter states then the criteria with respect to excitation frequency can be applied to predict subharmonic bifurcation both numerical integration and wind tunnel tests show that the approximate approach is feasible

a stability result for sectorial operators in branch spaces in this paper the author considers approximations of holomorphic semigroups on general banach spaces based on a acceptable rational functions a general stability theorem is proved it covers rational functions r z with r infty 1 and can be applied even to variable step cases in particular the question of the text l p stability p ne 2 of crank nicolson schemes for parabolic initial value problems is settled

an iterated maps approach for dynamic ratchetting in sdof hysteretic damping systems dynamic ratchetting is a phenomenon in which plastic deformation increases in successive cycles in the present paper first various characteristics of dynamic ratchetting are demonstrated based on numerical simulation second iterated maps are developed as a tool for the investigation of dynamic ratchetting in a single degree of freedom sdof system when an elasto perfectly plastic model is employed to represent hysteretic damping a piecewise linear solution can be obtained and used to develop iterated maps a stability investigation in the iterated maps shows that dynamic ratchetting is developed under a stable cycling in this stable cycling dynamic ratchetting occurs when an excitation function loses antiperiodicity shift symmetry

the hilbert phenomenon in chaotic motions abstract it is proved that chaotic dynamical systems have d 1 singularity due to such singularity and its periodicity in hausdorff metric spaces a chaotic dynamical system induces a hilbert iterated system which was discovered at the end of the 19th century this implies that the most puzzling discovery in nature in the 20th century can be put into deep correspondence with that in thoughts in the 19th century all this is verified by the numerical experiments of chaotic vibrations of a beam and a tension slack oscillator

stability and nonlinear dynamics of a vibration system with oblique collisions abstract two moving rigid bodies can contact with each other as the clearance between them is exceeded supposing that the contacting areas are two parallel planes with an oblique angle to the moving direction periodic responses of the system with oblique collision are determined by applying poincare mapping impulse momentum theorem and poisson s hypothesis in this paper first the relations of the body velocities before and after collisions are derived upon an instantaneous collision hypothesis then the existence condition and the stability criterion of the period n motion with single collision are deduced by using the poincare mapping and the floquet theory last numerical simulations are carried out and bifurcation diagrams poincare maps and phase portraits are adopted to prove the stability analysis and illustrate rich dynamical behaviors of the system over a rather broad range of frequency of the external excitation transitions from the single collision period n motion to chaotic one through three ways period doubling bifurcation hopf bifurcation and abruptly transition are revealed

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

hopf and two multiple semi stable limit cycle bifurcations of a restrained plate subjected to subsonic flow abstract this paper is mainly about the stabilities and bifurcations of a cantilevered plate with nonlinear motion constraints in an axial subsonic flow the galerkin method is employed to discretize the governing partial differential equation the fixed points and their stabilities are presented in a parameter space based on qualitative analysis and numerical studies the system loses stability by flutter and undergoes limit cycle oscillations after instability due to the nonlinearity the stabilities of the limit cycle oscillations are addressed on the basis of the equivalent linearized method the type of hopf bifurcation subcritical or supercritical is dependent on the location of the nonlinear motion constraints interestingly for some certain states the hopf bifurcations are both subcritical and supercritical the two multiple semi stable limit cycle bifurcation which is due to the extreme point of the flutter curve is also determined results of the numerical integration method sufficiently support the analytical method presented in this paper

the corresponding phenomena of mechanical and electronic impact oscillator a mechanical impact oscillator and a diode rectifier circuit of an impact oscillator are presented time series and phase portrait analysis are performed for both oscillators the results show that the proposed electronic impact oscillator is in agreement with the mechanical impact oscillator for varying the driving frequency a fixed harmonic input force can generate a periodic or non periodic impact oscillation the phenomena of bifurcation are also observed in both impact oscillators

double impact orbits of periodically forced impact oscillators we consider a periodically forced oscillator which impacts against a rigid constraint the forcing is both sinusoidal and non sinusoidal and is made up of a combination of the first and second harmonics of a fourier series a simple restitution law is used to describe the impact this paper centres around the analysis of double impact periodic solutions that is motions which repeat after every second impact solutions for these double impact orbits are found including non physical orbits and unstable orbits behaviour typical of nonlinear dynamical systems such as period doubling bifurcations are detected and grazing bifurcations which are unique to discontinuous dynamical systems are also observed a combination of analytical and numerical techniques are used to uncover the behaviour of the system at odd multiples of the natural frequency where numerical experiments tell us that stable double impact periodic solutions exist the persistence of these stable double impact orbits are also examined as parameters are varied we conclude by presenting some numerical calculations of two more general cases of impacting systems

on riemannian spaces with conformal symmetries or a tool for the study of generalized kaluza klein theories the riemann geometry of a space with conformal symmetries is written in terms of intrinsic objects defined from the action of the symmetries its application in the study of generalized kaluza klein theories is discussed

on the formulation of the positive energy theorem in kaluza klein theories the positive energy theorem is formulated in the context of kaluza klein theories different cases are considered including the situation in which no symmetry is assumed this work offers a new technique for stability considerations in kaluza klein theories

type i periodic motions for nonlinear impact oscillators abstract in this paper a general class of nonlinear impact oscillators is considered for subharmonic bifurcation this system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation and its unperturbed system possesses a pair of homoclinic cycles via the identification given by the impact law and three separate families of periodic orbits inside and outside the homoclinic cycles by discussing the subharmonic orbits inside the homoclinic cycles the subharmonic melnikov method established for smooth dynamical systems is extended to be applicable to the non smooth system

low dimensional maps for piecewise smooth oscillators abstract dynamics of the piecewise smooth nonlinear oscillators is considered for which general methodology of reducing multidimensional flows to low dimensional maps is proposed this includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution the global map is comprised of local maps which are constructed in the smooth sub regions of phase space to construct this low dimensional map it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory the dimension reduction is directly related to the dimension of the chosen boundary and the lower its dimension is the larger dimension reduction can be achieved full details are given for a drifting impact oscillator where the five dimensional flow is reduced to one dimensional 1d approximate analytical map first an exact two dimensional map has been formulated and analysed a further reduction to 1d approximate map is introduced and discussed standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos and co existence of multiple attractors accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters

a general equivalence theorem in the theory of discretization methods the lax richtmyer theorem is extended to work in the framework of stetter s theory of discretizations the new result applies to both initial and boundary value problems discretized by finite elements finite differences etc several examples are given together with a comparison with other available equivalence theorems the proof relies on a generalized banach steinhaus theorem 1 introduction in this paper we extend the classical lax richtmyer equivalence theorem 6 so as to cover in a simple way not only initial value problems but also boundary value problems mixed problems etc our theory relies on a generalized banach steinhaus theorem 9 and works essentially in the framework of stetter 13 this set up employs restriction operators to compare the true and discretized solutions as distinct to those theories which use prolongation operators one of the oldest prolongation theories is probably that of aubin summarized in 16 our main result is given in section 2 sections 3 and 4 are devoted to examples and counterexamples the former are meant to show the scope of our result and include the galerkin method for boundary value elliptic problems and semidiscrete and fully discrete schemes for initial value problems the counterexamples prove that the present hypotheses cannot be dispensed with in particular we show that a method which is consistent and convergent for all data in a banach space may be unstable the final section contains a comparison with other available equivalence theorems

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

stability of rational multistep approximations of holomorphic semigroups in this paper we prove the stability of semidiscretizations in time of holomorphic semigroups in banach spaces by means of a stable rational multistep methods no assumptions on the method other than a stability are required our result is applicable in the maximum norm analysis of parabolic problems

accurate numerical integration of state space models for aeroelastic systems with free play results show the importance of accurately locating the switching point between linear subdomains when numerically integrating a piecewise linear system of equations henon s method locates the switching point to a high degree of accuracy in one integration step while eliminating the need for a specified tolerance

the mapping dynamics of periodic motions for a three piecewise linear system under a periodic excitation in this paper the mapping dynamics of periodic motions for a three piecewise linear system under a periodic excitation is developed and the mapping structures for specified periodic motions are constructed based on the mapping structures an analytical prediction of all possible stable and unstable periodic motions is given the symmetry for the stable asymmetrical periodic motions of such a system is observed however the unstable periodic motions do not have such symmetry the methodology presented in this paper is applicable to other non smooth systems such as friction induced vibration impact oscillator and power control systems in addition the mapping dynamics provides a useful and efficient tool for the co existence of periodic motions and or chaotic motions in nonlinear dynamical systems

bifurcation analysis of an impact oscillator with a one sided elastic constraint near grazing in this paper a linear oscillator undergoing impact with a secondary elastic support is studied experimentally and semi analytically for near grazing conditions the experimentally observed bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces smooth as well as nonsmooth bifurcations are observed and the resulting atypical bifurcations are explained often as an interplay between them in order to understand the observed bifurcation scenarios a global analysis is required due to the influence of stable and unstable orbits which are born in distant bifurcations but become important at near grazing conditions the good degree of correspondence between experiment and theory fully justifies the modelling approach

effect of smoothing piecewise linear oscillators on their stability predictions the effects of smoothing non smooth function on estimation of dynamical stability of the periodic response determined in the frequency domain are considered in this paper for that purpose a simple single degree of freedom system with piecewise linear force displacement relationship subjected to a harmonic force excitation is analyzed stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the floquet liapounov theorem results obtained are verified in the time domain by the method of piecing the exact solutions and runge kutta integration routine

parametric flutter margin method for aeroservoelastic stability analysis a new parametric flutter margin method for linear and nonlinear stability analysis of aeroservoelastic systems is presented the method is based on frequency response calculations with the system s

asymmetric type ii periodic motions for nonlinear impact oscillators abstract in this paper a general class of nonlinear impact oscillators is considered for type ii periodic motions this system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation the unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law by approximating the poincare map to o e directly a general method of melnikov type for detecting the existence of asymmetric type ii subharmonic orbits outside the homoclinic cycles is presented

estimation of clearances and impact forces using vibroimpact response sinusoidal excitation abstract a novel clearance estimation approach has been developed for a randomly excited vibroimpact system the clearance estimates based on this approach were obtained using data from the mechanical analogue of an impact oscillator and simulation using a digital computer and they reasonably agree with the actual clearances

regions of stability equivalence theorems and courant friedichs lewy condition the celebrated cfl condition for discretizations of hyperbolic pdes is shown to be equivalent to some results of jeltsch and nevanlinna concerning regions of stability ofk step m stage linear methods for the integration of odes we characterize the methods for the numerical integration of the model equation ut ux which are weakly stable when the mesh ratio takes the maximum value allowed by the cfl condition we provide new equivalence theorems between stability and convergence which improve on the classical results

on the positive energy theorem involving mass and electromagnetic charges a new theorem relating mass and charges is deduced which can be applied to more general physical systems than those covered by the theorem of gibbons and hull 1

on the stability of variable stepsize rational approximations of holomorphic semigroups we consider variable stepsize time approximations of holomorphic semigroups on general banach spaces for strongly a o acceptable rational functions a general stability theorem is proved which does not impose any constraint on the ratios between stepsizes

non linear modal analysis of structural components subjected to unilateral constraints in this paper we present a detailed numerical study of the non linear dynamics in structural components under unilateral contact constraints here the unilateral term characterises the constitutive law of the restoring force in the constraints as they only sustain elastic reactions in one direction either compressive or tensile thus the non differentiability of the contact law at the discontinuity point is the only source of non linearity in our approach the discrete lattice method dlm is used to treat the continuous system as a piecewise linear model thus the trajectory of each node in the discrete model would be a sequence of smooth solutions with the switching times between them the application of the one step integration scheme allows us to detect the occurrence of contact i e the instants that the lattice nodes cross the discontinuity boundary and consequently update the active constraints we also consider embedding the bisection algorithm into the time integration procedure to localise the instants at which the nodes cross the boundary and minimise the accumulative error subsequently the resulting unconditionally stable integration scheme is utilised as the modelling tool in combination with the shooting technique to perform a novel non smooth modal analysis in analogy with the smooth non linear systems the evolution of non smooth periodic motions is presented in the frequency stiffness plots we apply our method to obtain non linear normal modes nnms for a number of representative problems including a bar obstacle system a beam substrate system and a granular chain with tensionless interactions these numerical examples demonstrate the efficiency of the solution procedure to trace the family of energy independent non linear modes across the range of contact stiffnesses moreover the stability analysis of the modes on the plot backbone reveal that they may become unstable due to the interaction with the higher modes or bifurcation of a new nnm with a period equal to the integer multiple of the main mode period our results also indicate that the bilinear formula can accurately predict the non linear frequencies only if the corresponding mode exhibits a smooth character regardless of the commutativity conditions of the system stiffness matrix however it is obvious that the assumption of smooth bilinear behaviour for non linear modes is not generally valid this highlights the importance of the present numerical framework for the computation of non smooth resonance frequencies

dynamics of a two degree of freedom periodically forced system with a rigid stop diversity and evolution of periodic impact motions abstract a two degree of freedom periodically forced system with a clearance is considered the correlation between dynamic performance and system parameters is studied to find their feasible matching law the fundamental group of impact motions is defined which have the excitation period and differ by the number p of impacts the generation mechanism of complete and incomplete chattering impact vibration of the system is studied a series of singular points between any two adjacent fundamental impact motions are found at which real grazing and bare grazing bifurcation boundaries of one of them saddle node and period doubling bifurcation boundaries of the other alternately intersect and create two types of transition regions hysteresis and tongue shaped regions subharmonic impact motions are found to regularly appear in the tongue shaped regions based on the sampling ranges of parameters dynamics of the system is studied with emphasis on the influence of system parameters on impact velocities existence regions and correlative distribution of single impact periodic motions by multi parameter and multi performance co simulation analysis

frequency response characteristics of a multi degree of freedom system with clearances abstract in an earlier study the frequency response characteristics of a single degree of freedom system with a clearance non linearity were studied the current study an extension of the earlier work is concerned with the frequency response characteristics of a multi degree of freedom system with clearances the method of harmonic balance is used to develop approximate analytical solutions of the undamped equations of motion of a multi degree of freedom system composed of three coupled non linear oscillators for primary resonances and a harmonic excitation general formulations are presented which can be used to study both the existence and the stability of the solutions these general formulations are used to discuss a number of modal spacing and modal coupling issues an analysis methodology for multi degree of freedom systems is introduced and illustrated by two special but practical cases a strongly non linear system and a weakly non linear system the results of the analysis of the special cases are validated by using analog simulation it is shown that an analysis of the special cases of the general multi degree of freedom non linear formulation can provide not only an improved understanding of the dynamic behavior of non linear systems but also can be used as the basis for the development of simplified approximate solutions the limitations of the current study and areas for further research are discussed

non linear real and complex modes of conewise linear systems abstract conewise piecewise linear non gyroscopic and gyroscopic autonomous systems are studied in detail first the general properties of such systems are examined then the non linear real and complex modes are obtained for both non gyroscopic and gyroscopic cases by means of analytical or numerical tools several interesting characteristics of non linear modes are found and compared to those of linear modes accurate analytical estimates of non linear mode periods are formulated for both non gyroscopic and gyroscopic systems the validity of such estimates is discussed in relation to a non linear mode bifurcation phenomenon which is located by using both characteristic multipliers and poincare mapping

structural non linearities and the nature of the classic flutter instability abstract the paper presents an investigation of the subcritical supercritical nature of the flutter hopf bifurcation of a two degree of freedom with linear quadratic and cubic restoring forces under certain conditions the instability gives rise to stable limit cycle oscillations whereas unstable periodic orbits appear for other conditions providing a potentially dangerous non linear instability mechanism in regimes for which linear analyses predict stability the analytic approach employed herein allows for a more thorough investigation of the high dimensional parameter space than that afforded by others experimental and computational efforts counterexamples to previously held conjectures are readily found favorable comparisons are made to predictions based on less rigorous describing function or harmonic balance techniques

convergence analysis of one step schemes in the method of lines we present an expository account of some fundamental results concerning the analysis of one step schemes for semidiscretizations of evolutionary problems in partial differential equations in the paper the emphasis lies in the interplay between the stability and convergence properties of the fully discrete scheme and those of the ordinary differential equations solver much attention is paid to the phenomenon of order reduction

numerical scheme of locating the periodic response of non smooth non autonomous systems of high dimension abstract the paper presents a numerical scheme of locating the periodic response of high dimensional systems with non smooth characters such as clearances constraints frictions the basic idea comes from the fact that an asymptotic stable periodic response of the system attracts its transient response and it may be extracted from the transient response somehow the paper thus suggests a novel iteration scheme to extrapolate the periodic response from a short time history of the transient response by curve fitting compared with the current schemes such as shooting incremental harmonic balancing and their improved forms the present scheme makes full use of the information within the transient response and the system characters so that the computational efficiency has increased by an order and the numerical convergence depending on the initial iteration has greatly improved

deflection of rectangular orthotropic plates under uniform load an experiment was performed to determine the deflected shape of symmetrically supported orthotropic plates under uniform load the experimental results were then compared with the mathematical model for shape and deflection a symmetric clamped simply supported boundary condition case was selected because it has a larger difference in peak to overall plate deflection compared with a plate that is simply supported on all sides

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

stability concepts and their applications the stability is one of the most basic requirement for the numerical model which is mostly elaborated for the linear problems in this paper we analyze the stability notions for the nonlinear problems we show that in case of consistency both the n stability and k stability notions guarantee the convergence moreover by using the n stability we prove the convergence of the centralized crank nicolson method for the periodic initial value transport equation the k stability is applied for the investigation of the forward euler method and the method for the cauchy problem with lipschitzian right side

non linear dynamics of a two dimensional airfoil by incremental harmonic balance method periodic oscillations and bifurcations of a two dimensional airfoil in plunge and pitching motions with cubic pitching stiffness in incompressible flow is investigated using the incremental harmonic balance ihb method the bifurcations are obtained with the parametric continuation technique and the stability of the periodic motions is investigated using the floquet theory the autonomous non linear system of the airfoil undergoes initial hopf bifurcation leading to limit cycle oscillation as the airspeed parameter is increased further increase in the airspeed causes symmetry breaking saddle node and period doubling bifurcations leading to chaos the frequency of the limit cycle oscillation is also determined in the ihb method

numerical solutions for nonlinear free surface flows by finite element methods numerical computations for the water wave problem have been extensively studied but the corresponding numerical analysis has not been worked out because of nonlinearity of free surface equation we will discuss numerical stability and error estimates of an algorithm which uses finite element galerkin method to solve the water wave problem in order to do that we employ an extended lax richtmyer equivalence theorem

bifurcation analysis of a two degree of freedom aeroelastic system with hysteresis structural nonlinearity by a perturbation incremental method abstract a perturbation incremental pi method is presented for the computation continuation and bifurcation analysis of limit cycle oscillations lco of a two degree of freedom aeroelastic system containing a freeplay structural nonlinearity both stable and unstable lcos can be calculated to any desired degree of accuracy and their stabilities are determined by the floquet theory thus the present method is capable of detecting complex aeroelastic responses such as periodic motion with harmonics period doubling pd saddle node bifurcation neimark sacker bifurcation and the coexistence of limit cycles emanating branch from a pd bifurcation can be constructed this method can also be applied to any piecewise linear systems

stability and dynamics of a harmonically excited elastic perfectly plastic oscillator this paper deals with the stability and the dynamics of a harmonically excited elastic perfectly plastic oscillator the hysteretical system is written as a non smooth forced autonomous system it is shown that the dimension of the phase space can be reduced using adapted variables free vibrations of such a system are then considered for the damped system the extended direct method of liapounov is applied to this non smooth mechanical system and the asymptotic stability of the origin is proven in the new phase space the forced vibration of such an oscillator is treated by numerical approach by using the time locating techniques the stability of the limit cycles is analytically investigated with a perturbation approach the boundary between elastoplastic shakedown and alternating plasticity is given in closed form it is shown that this boundary corresponds to a bifurcation boundary for the undamped system period doubling bifurcation finally the equivalent damping of this hysteretic system is characterized from dynamical properties

stability and bifurcation analysis for oscillators with motion limiting constraints abstract the stability and bifurcation analysis is presented for a class of non linear single degree of freedom vibrating systems excited by harmonic forcing the common characteristic of these systems is that their damping and stiffness parameters can change abruptly at certain displacement values the most general n periodic steady state solutions with an arbitrary number of contacts per response cycle are analyzed it is shown that when the damping is positive the change of stability of the steady state response cannot happen through a hopf bifurcation

an iterative procedure for nonlinear flutter analysis an iterative procedure in the frequency domain is presented for flutter analysis of large dynamic systems with multiple structural nonlinearities the major components of the procedure are the describing function approach for system linearization a structural dynamics modification method for shifting system mode shapes and frequencies and a complex eigenvalue algorithm for solution of the flutter equation the purpose of the procedure is to achieve alignment of the oscillatory amplitude in each nonlinear spring with the describing function prediction of stiffness before computing the final stability characteristics the result is a system tuned to the flutter frequency at the time of instability to support the development and validation of the procedure several describing functions are formulated and a quantitative measure of the errors in each is presented validation of the iterative method is accomplished through examples involving dynamic systems of increasing complexity coupled with various representations of the unsteady aerodynamic forces both numerical simulations and experimental data are used to compare with the iterative predictions in the cases studied the agreement is good to excellent with the method accurately predicting the amplitude of a limit cycle flutter as well as the initial disturbance required to produce flutter

bifurcations and chaos in a system with impacts abstract a vibro impact system is considered a body bounces on a flat horizontal surface of the vibro bench hopf bifurcations of the vibro impact system in two kinds of strong resonance cases 03 1 and 04 1 are investigated a poincare map of the system is established the period 1 single impact motion of the system and its stability are studied by analytical methods and hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined a center manifold theorem technique is applied to reduce the poincare map to a two dimensional one which is put into normal form by theory of normal forms by the theory of hopf and subharmonic bifurcations of fixed points in r 2 strong resonance dynamic behavior of the vibro impact system near points of resonance is analyzed in the resonance case of 03 1 the system generally exhibits unstable period 3 three impact motion in the resonance case of 04 1 the system can exhibit stable period 4 four impact motion and quasi periodic motion the theoretical analyses are verified by numerical solutions routes to chaos in two kinds of strong resonance cases considered are obtained by numerical simulations quasi periodic impacts in non resonance and weak resonance cases doubling periodic bifurcations and chaotic motions are also stated briefly by numerical analyses

torus doubling bifurcations and strange nonchaotic attractors in a vibro impact system whether strange nonchaotic attractors snas can occur typically in dynamical systems other than quasiperiodically driven systems has been an open problem here we show that the snas can be induced by the interruption of torus doubling bifurcation in a periodically driven vibro impact system it is found that the creation of snas occurs due to the collision of doubled torus with some unstable periodic orbits based on the poincare map technique two types of codimension 3 bifurcations are represented and typical torus doubling bifurcations can occur in the neighborhood of these bifurcation points we use the singular continuous fourier spectrum and its scaling to verify the strangeness nondifferentiability of the attractors the implicitly transcendental map is used to calculate the largest lyapunov exponent by qr based algorithm which ensures that the underlying dynamics with nonpositive lyapunov exponents is nonchaotic the associated mechanism is described for the creation of snas through detecting unstable periodic orbits from poincare map series

continuous and discontinuous grazing bifurcations in impacting oscillators this paper seeks to formulate conditions for the persistence of a local attractor in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system a local analysis based on the discontinuity mapping approach is employed to derive a normal form description of the dynamics near the grazing trajectory in agreement with previous studies of grazing periodic trajectories it is found that the catastrophic loss of a local attractor and strong instability characteristic of grazing bifurcations is directly associated with the repeated application of a square root term that appears to lowest order in the normal form expansion specifically it is found that the square root term is absent in the description of the dynamics normal to a quasiperiodic trajectory covering a co dimension one invariant torus resulting in a piecewise linear description of the normal dynamics and at most a weak instability in contrast for co dimension two or higher the square root term is generically present in the normal dynamics here however the quasiperiodicity of the grazing motion implies that there is no upper limit to the time between impacts on nearby trajectories suggesting the persistence of a local attractor for some interval about the parameter value corresponding to grazing the results of the analysis are illustrated through a series of model examples

power optimization in random wireless networks in this paper we analyze the problem of power control in large random wireless networks that are obtained by erasing a finite fraction of nodes from a regular d dimensional lattice of n transmit receive pairs in this model which has the important feature of a minimum distance between transmitter nodes we find that when the network is infinite power control is always feasible below a positive critical value of the users signal to interference plus noise ratio sinr target drawing on tools and ideas from statistical physics we show how this problem can be mapped to the anderson impurity model for diffusion in random media in this way by employing the so called coherent potential approximation method we calculate the average power in the system and its variance for 1 d and 2 d networks this approach is equivalent to traditional techniques from random matrix theory and is in excellent agreement with the numerical simulations however it fails to predict when power control becomes infeasible in this regard even though infinitely large systems are always unstable beyond a critical value of the users sinr target finite systems remain stable with high probability even beyond this critical sinr threshold we calculate this probability by analyzing the density of low lying eigenvalues of an associated random schrodinger operator and we show that the network can exceed this critical sinr threshold by at least mathop mathcal o nolimits log n 2 d before undergoing a phase transition to the unstable regime finally using the same techniques we also calculate the tails of the distribution of transmit power in the system and the rate of convergence of the foschini miljanic power control algorithm in the presence of random erasures

transonic limit cycle oscillation analysis using aerodynamic describing functions and superposition principle limit cycle oscillation for an airfoil in transonic air flow considering aerodynamic nonlinearity is studied transonic nonlinear aerodynamic forces are calculated by solving euler equations or navier stokes equations with respect to harmonic airfoil plunging pitching motion and then the aerodynamic describing functions and the superposition principle are used to build an equivalent linearized aerodynamic model in the frequency domain hence the transonic limit cycle oscillation solutions can be obtained by the frequency domain flutter analysis method the procedures for building the aerodynamic describing function and obtaining the limit cycle oscillation solution are described in detail four examples are adopted to verify the accuracy of the present method the isogai wing model with naca 64a010 airfoil is used as the first example to verify the accuracy of the transonic linear flutter solution of the present method another naca 64a010 airfoil model with different structural parameters is then adopt