implications of time delayed feedback control on limit cycle oscillation of a two dimensional supersonic lifting surface in this paper the problem of implications of time delay feedback control of a two dimensional supersonic lifting surface on the flutter boundary and on its character that is benign or catastrophic is addressed in this context the structural and aerodynamic nonlinearities are included in the aeroelastic governing equations the model and the associated theory are developed for linear and nonlinear plunging and pitching full state proportional and velocity feedback controls center manifold reduction and normal form theory are applied to investigate the stability in the post flutter flight speed regimes numerical simulations are carried out to determine the implications of time delay in the considered controls but are restricted to the cases of proportional feedback control and no structural damping

modal control of vibration in rotating machines and other generally damped systems second order matrix equations arise in the description of real dynamical systems traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices a regrettable consequence of this approach is the discarding of residual off diagonal terms in the modal damping matrix this has particular importance for systems containing skew symmetry in the damping matrix which is entirely discarded in the modal damping matrix in this paper a method to utilise modal control using the decoupled second order matrix equations involving non classical damping is proposed an example of modal control successfully applied to a rotating system is presented in which the system damping matrix contains skew symmetric components

complex component mode synthesis method using hybrid coordinates for generally damped systems with local nonlinearities abstract in numerous engineering structures with local nonlinearities the nonlinear effects are usually localized and most parts of the structures behave linearly considering this a reduced order modelling technique for generally damped systems with localized nonlinearities is proposed firstly the linear parts with non proportional damping of the system is separated to form one component which can be formulated in state space and transformed into the modal coordinates using its linear vibration complex modes the residual flexibility matrix which can be formulated by using a set of equivalent higher order complex modes has been developed to improve the accuracy by capturing the effects of the neglected higher order complex modes secondly the rest regions of the system form another component behaving nonlinearly which is kept in its original form to adopt the entire set of nonlinear terms next the second order differential synthesis equation can be obtained by using compatibility conditions at the junctions in order to calculate dynamic response of the system the aforementioned synthesis equation needs to transform into the first order differential equation because of the singularity of the equivalent mass matrix finally two numerical examples are given to illustrate the effectiveness and the efficiency of the presented method

effect of nonlinear impact damping on the frequency response of a torsional system with clearance in this article we examine a particular nonlinear impact damping model that has been well accepted by the scientific community first we consider the damping element with a linear spring and examine the nonlinear frequency response characteristics of a torsional system second the impact damping is examined along with the clearance nonlinearity including the backlash problem three semi analytical methods namely the describing function method multi term harmonic balance method and stochastic linearization method are utilized to investigate the effect of impact damping feasibility of such methods is confirmed by comparing predictions with results yielded by a numerical method the differences between impact damping and linear viscous damping mechanism are studied in terms of frequency responses time histories and the minimum number of harmonics required to simulate the behavior given sinusoidal excitations furthermore the effect of impact damping on sub and super harmonic responses and quasi periodic regimes are examined

nonstationary regimes in a duffing oscillator subject to biharmonic forcing near a primary resonance an analytical investigation of nonstationary processes in a duffing oscillator subject to biharmonic forcing under conditions of primary resonance is carried out the earlier developed methodology of limiting phase trajectories lpts for studying highly nonstationary regimes characterized by intense energy exchanges between the different degrees of freedom is successfully applied to the system under investigation two distinct types of lpt trajectories are described in the first part of the study conditions for the recurrent transitions in time from one type of lpt to another were derived in the first part of the analysis corresponding to the undamped case an approximation of the lpt related to the higher amplitude of oscillations was derived using nonsmooth transformations an analysis carried out in the study has revealed the necessary and sufficient conditions for excitation of relaxation oscillations exhibited by a lightly damped system it was also demonstrated that the mechanism of relaxations may be approximated and explained by the methodology of lpts characterized by strong energy exchanges between the coupled oscillators or alternatively a single oscillator and an external source of energy the results of analytical approximations and numerical simulations are observed to be in quite satisfactory agreement

free and forced vibration analysis of a nonlinear system with cyclic symmetry application to a simplified model abstract this work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity through the use of the harmonic balance method hbm in order to study the influence of nonlinearity due to the large deflection of blades a simplified model has been developed this approach leads to a system of linearly coupled second order nonlinear differential equations in which nonlinearity appears via cubic terms periodic solutions in both the free and forced cases are sought by applying hbm coupled with an arc length continuation method solution stability has been investigated using floquet s theorem in addition to featuring similar and nonsimilar nonlinear modes the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes in the forced case these nonlinear modes give rise to a complex dynamic behavior many bifurcations can take place thus leading to strong or weak localization that may or may not be stable in this study special attention has been paid to the influence of excitation on dynamic responses several cases of excitation have been analyzed herein localized excitation and low engine order excitation in the case of low engine order excitation sensitivity of the response to a perturbation of this excitation type has been investigated and it has been shown that for a localized or sufficiently detuned excitation several solutions can coexist some of which are represented by closed curves in the frequency amplitude domain these various solutions overlap when increasing the force amplitude leading to forced nonlinear localization because closed curves are not tied up with the basic nonlinear solution they can easily be overlooked in this study they have been calculated using a sequential continuation with the force amplitude as a parameter

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

asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow we study theoretically passive suppression of aeroelastic instabilities of a rigid wing in subsonic flow with an essentially nonlinear attachment the analysis is performed by constructing a reduced order model applying complexification averaging and slow fast partition of the dynamics the resulting slow flow dynamics is then analyzed by singular perturbation we fully recover computational and experimental results for this system reported in previous works and prove the existence of trivial nontrivial stable attractors as well as relaxation oscillations in the slow flow dynamics these in turn correspond to complete partial instability suppression and periodic or quasi periodic strongly modulated responses in the full order dynamics moreover we demonstrate the existence of shilnikov bifurcations in the dynamics the analysis of the slow flow dynamics is confirmed by numerical simulations of the full order system the methodology developed in this work can be used in a predictive capacity when ap

analytic treatment of a system with a vibro impact nonlinear energy sink abstract this communication reports on an approximate analytic procedure for analysis of transient damped response in a system comprising a primary linear oscillator and vibro impact nonlinear energy sink nes multiple scale expansion reduces the problem in the first approximation to low dimensional vibro impact system with forcing and damping the latter problem yields to direct analytic approach the results computed by this way fairly correlate with direct numeric simulations the analysis takes advantage of small ratio of the nes and the primary masses but does not require the restitution coefficient to be close to unity the proposed approach paves the way for efficient design of systems with vibro impact ness

cascades of subharmonic stationary states in strongly non linear driven planar systems the dynamics of a one degree of freedom oscillator with arbitrary polynomial non linearity subjected to an external periodic excitation is studied the sequences cascades of harmonic and subharmonic stationary solutions to the equation of motion are obtained by using the harmonic balance approximation adapted for arbitrary truncation numbers powers of non linearity and orders of subharmonics a scheme for investigating the stability of the harmonic balance stationary solutions of such a general form is developed on the basis of the floquet theorem besides establishing the stable unstable nature of a stationary solution its stability analysis allows the regions of parameters to be obtained where symmetry breaking and period doubling bifurcations occur thus for period doubling cascades each unstable stationary solution is used as a base solution for finding a subsequent stationary state in a cascade the procedure is repeated until this stationary state becomes stable provided that a stable solution can finally be achieved the proposed technique is used to calculate the sequences of subharmonic stationary states in driven hardening duffing s oscillator the existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time difference method with initial conditions being supplied by the harmonic balance approximation

model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds abstract a methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented the approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations due to the existence of external and parametric periodic excitations however the geometry of the manifold varies with time as a result the manifold is constructed in terms of temporal and dominant state variables the governing partial differential equation pde for the manifold is nonlinear and contains time varying coefficients an approximate technique to find solution of this pde using a multivariable taylor fourier series is suggested it is shown that in certain cases it is possible to obtain various reducibility conditions in a closed form the case of time periodic systems is handled through the use of lyapunov floquet l f transformation application of the l f transformation produces a dynamically equivalent system in which the linear part of the system is time invariant however the nonlinear terms get multiplied by a truncated fourier series containing multiple parametric excitation frequencies this warrants some structural changes in the proposed manifold but the solution procedure remains the same two examples namely a 2 dof mass spring damper system and an inverted pendulum with periodic loads are used to illustrate applications of the technique for systems with constant and periodic coefficients respectively results show that the dynamics of these 2 dof systems can be accurately approximated by equivalent 1 dof systems using the proposed methodology

nonlinear model identification and spectral submanifolds for multi degree of freedom mechanical vibrations in a nonlinear oscillatory system spectral submanifolds ssms are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium amplitude frequency plots of the dynamics on ssms provide the classic backbone curves sought in experimental nonlinear model identification we develop here a methodology to compute analytically both the shape of ssms and their corresponding backbone curves from a data assimilating model fitted to experimental vibration signals this model identification utilizes taken s delay embedding theorem as well as a least square fit to the taylor expansion of the sampling map associated with that embedding the ssms are then constructed for the sampling map using the parametrization method for invariant manifolds which assumes that the manifold is an embedding of rather than a graph over a spectral subspace using examples of both synthetic and real experimental data we demonstrate that this approach reproduces backbone curves with high accuracy

dynamics of forced system with vibro impact energy sink abstract the paper treats forced response of primary linear oscillator with vibro impact energy sink this system exhibits some features of dynamics which resemble forced systems with other types of nonlinear energy sinks such as steady state and strongly modulated responses however the differences are crucial in the system with vibro impact sink the strongly modulated response consists of randomly distributed periods of resonant and non resonant motion this salient feature allows us to identify this type of dynamic behavior as chaotic strongly modulated response csmr it is demonstrated that the csmr exists due to special structure of a slow invariant manifold sim the latter is derived in a course of a multiple scale analysis of the system in the considered system this manifold has only one stable and one unstable branch this feature defines new class of universality for the nonlinear energy sinks very different physical system with topologically similar sim the oscillator with rotational energy sink also exhibits csmrs in the system with the vibro impact sink such responses are observed even for very low level of the external forcing this feature makes such system viable for possible energy harvesting applications

a new galerkin based approach for accurate non linear normal modes through invariant manifolds a method for producing accurate reduced order models of non linear vibratory systems is presented based on the invariant manifold description of non linear normal modes nnm this approach makes use of polar co ordinates to obtain equations which govern the geometry of the invariant manifold these equations are discretized through a series expansion and galerkin projection over a chosen amplitude and phase domain yielding non linear equations in the expansion coefficients these equations when solved numerically yield an invariant manifold which is accurate to the degree of the expansion and devoid of the limitations which plague typical asymptotic solutions such galerkin based solutions may be used to generate accurate reduced order models for large amplitude strongly non linear motions this procedure is illustrated using two non linear examples a two degree of freedom oscillator and a finite element beam model the solution convergence and manifold geometry are discussed and the resultant reduced order models are shown to possess exceptional accuracy over large amplitude ranges this approach allows the full potential of the invariant manifold formulation to be reached and is suitably general for application to a wide variety of non linear systems

forced periodic vibration of unsymmetric piecewise linear systems the forced steady response of a single degree of freedom system involving a large non linearity represented by unsymmetric piecewise linear stiffness is determined by a harmonic balance newton raphson method with the application of the fast fourier transformation fft algorithm all possible subharmonic harmonic and superharmonic responses are sought the responses of two systems involving subharmonic and superharmonic dominant motions are calculated by the newly developed method and compared with previously published findings the results obtained by using this method reveal the details of the response of the systems more efficiently than previous methods

isolated sub harmonic resonance branch in the frequency response of an oscillator with slight asymmetry in the clearance a family of isolated sub harmonic branches in nonlinear frequency response of piecewise linear system is examined in this paper given their peculiarity they could be easily overlooked in frequency response studies in numerical integration or laboratory experiments when the quasi static frequency sweeping technique is employed an increase in the viscous damping ratio could shrink the isolated branches and ultimately the sub harmonic resonance may vanish asymmetry does not appear to be the key factor that generates the isolated branches rather the relationship between the linear mean operating and transition point seems to dictate their occurrence

response bounds for complex systems with a localised and uncertain nonlinearity predicting the vibration response of complex nonlinear structures is a significant challenge the response may involve many modes of the structure nonlinearity precludes the use of efficient techniques developed for linear systems and there is often uncertainty associated with the nonlinear law even to the extent that its functional form is not always known this paper builds on a recently developed method for handling this class of problem in a novel way the method exploits the fact that nonlinearities are often spatially localised and seeks the best and worst case system response with respect to a chosen metric by regarding the internal nonlinear force as an independent excitation to the underlying linear system constraints are used to capture what is thought to be known about the nonlinearity without needing to specify a particular law this paper focuses on the case of systems with a single point nonlinearity but with arbitrarily complex underlying linear dynamics driven by a sinusoidal force excitation semi analytic upper and lower bounds are proposed for root mean square response metrics subject to constraints which specify that the nonlinearity should be a combination of a passive b displacement limited and or c force saturating the concept of equivalent linear bounds is also introduced for cases where the response metric is thought to be dominated by the same frequency as the input the bounds corresponding to a passive and displacement limited nonlinearity are compared with monte carlo experimental and numerical results from an impacting beam test rig the bounds corresponding to a passive and force saturating nonlinearity are compared with numerical results for a friction damped beam the global upper and lower bounds are satisfied for all input frequencies but are generally found to be rather conservative the equivalent linear bounds show remarkably good agreement for predicting the range of root mean square velocity responses finally the principle of maximum entropy is used to estimate the response distributions which was found to give surprisingly good agreement with experimental and numerical data

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

reduced order models for large amplitude vibrations of shells including in plane inertia non linear normal modes nnms are used in order to derive reduced order models for large amplitude geometrically non linear vibrations of thin shells the main objective of the paper is to compare the accuracy of different truncations using linear and non linear modes in order to predict the response of shells structures subjected to harmonic excitation for an exhaustive comparison three different shell problems have been selected i a doubly curved shallow shell simply supported on a rectangular base ii a circular cylindrical panel with simply supported in plane free edges and iii a simply supported closed circular cylindrical shell in each case the models are derived by using refined shell theories for expressing the strain displacement relationship as a consequence in plane inertia is retained in the formulation reduction to one or two nnms shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases and this limitation is extended to two times the thickness for two of the selected models 2008 elsevier b v all rights reserved

dynamics of linear discrete systems connected to local essentially non linear attachments the dynamics of a linear periodic substructure weakly coupled to an essentially non linear attachment are studied the essential non linearizable non linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena e g passive one way irreversible transfer of energy from the substructure to the attachment as a specific application the dynamics of a finite linear chain of coupled oscillators with a non linear end attachment is examined in the absence of damping it is found that the dynamical effect of the non linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain when damping exists energy pumping phenomena are realized in the system it is shown that energy pumping strongly depends on the topological structure of the non linear normal modes nnms of the underlying undamped system this is due to the fact that energy pumping is caused by the excitation of certain damped invariant nnm manifolds that are analytic continuations for weak damping of nnms of the underlying undamped system the bifurcations of the nnms of the undamped system help explain resonance capture cascades in the damped system this is a series of energy pumping phenomena occurring at different frequencies with sudden lower frequency transitions between sequential events the observed multi frequency energy pumping cascades are particularly interesting from a practical point of view since they indicate that non linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure in compatibility with the design objectives

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

the harmonic balance method with arc length continuation in rotor stator contact problems there are a variety of abnormal running conditions in rotating machinery which lead to rotor stator interaction dynamics which in turn can cause a rich mixture of effects associated with rub related phenomena these effects manifest themselves in the occurrence of multiple solutions for steady state vibration response scenarios including amplitude jumps during rotor acceleration and vibration responses at different multiple frequencies of excitation forces such as unbalance this paper describes a numerical algorithm based on the harmonic balance method to calculate the periodic response of a non linear system under periodic excitation the algorithm also calculates the stability of the periodic solutions found marks turning and bifurcation points and follows a solution branch over varying system parameters via arc length continuation

the influence of damping on the limit cycles for a self exciting mechanism the objective of the present work is to investigate qualitative aspects of mode coupling instability of self excited friction induced oscillations in the presence of structural damping and a cubic nonlinearity in a previous study sinou and jezequel demonstrated the effects of structural damping in determining stable and unstable zones and indicated that the damping ratio of the coupling modes may be a key factor for avoiding bad design in this paper we propose to complete this study by examining the influence of structural damping on limit cycle amplitudes in order to achieve a complete design including not only the evolution of stable unstable areas but also the evolution of limit cycle amplitudes as functions of the structural damping and nonlinear system parameter for the sake of simplicity a two degree of freedom minimal model is constructed and analysed to examine the effects of structural damping and cubic nonlinearity on the limit cycles the nonlinear behaviour and stable limit cycle amplitudes are determined through a complex nonlinear modal analysis which makes use of the nonlinear unstable mode governing the nonlinear dynamics of structural systems in unstable areas based on this nonlinear modal approach we will produce and explain qualitative and quantitative results of limit cycle evolutions in the presence of structural damping and nonlinearity

the sato tate conjecture for hilbert modular forms we prove the sato tate conjecture for hilbert modular forms more precisely we prove the natural generalisation of the sato tate conjecture for regular algebraic cuspidal automorphic representations of gl2 af f a totally real field which are not of cm type the argument is based on the potential automorphy techniques developed by taylor et al but makes use of automorphy lifting theorems over ramified fields together with a topological argument with local deformation rings in particular we give a new proof of the conjecture for modular forms which does not make use of potential automorphy theorems for non ordinary n dimensional galois representations

bifurcations of self excitation regimes in a van der pol oscillator with a nonlinear energy sink abstract the paper investigates regimes of self excitation in a van der pol oscillator with an attached nonlinear energy sink nes initial equations are reduced by averaging to a 3d system the small relative mass of the nes justifies analysis of this averaged system as singularly perturbed with two slow and one super slow variable such an approach in turn provides a complete analytic description of possible response regimes in addition to almost unperturbed limit cycle oscillations lcos the system can exhibit complete elimination of self excitation small amplitude lcos as well as excitation of a quasiperiodic strongly modulated response smr in the space of parameters the latter can be approached by three distinct bifurcation mechanisms canard explosion shil nikov bifurcation and heteroclinic bifurcation some of the above oscillatory regimes can co exist for the same values of the system parameters in this case it is possible to establish the basins of attraction for the co existing regimes direct numeric simulations demonstrate good coincidence with the analytic predictions

normal modes of vibration for non linear continuous systems a definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non linear partial differential equations the definition for normal modes is given in terms of the dynamics on two dimensional invariant manifolds in the system phase space these manifolds are a continuation of the planes which represent the well known normal modes of the linearized system a local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds the procedure also provides the non linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion in this paper the general theory is described for the application of vibrations of continuous media but it can easily be extended to other situations in order to demonstrate the power of the approach and its unique procedural aspects three examples involving beam vibrations are worked out in detail the examples are conservative simply supported beams the first demonstrates the methodology for a linear beam model the second is a beam on a non linear elastic foundation and the third example is a beam with non linear torsional springs attached at each end in these examples the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure

an analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures this paper considers isolated responses in nonlinear systems both in terms of isolas in the forced responses and isolated backbone curves i e the unforced undamped responses as isolated responses are disconnected from other response branches reliably predicting their existence poses a significant challenge firstly it is shown that breaking the symmetry of a two mass nonlinear oscillator can lead to the breaking of a bifurcation on the backbone curves generating an isolated backbone it is then shown how an energy based analytical method may be used to compute the points at which the forced responses cross the backbone curves at resonance and how this may be used as a tool for finding isolas in the forced responses this is firstly demonstrated for a symmetric system where an isola envelops the secondary backbone curves which emerge from a bifurcation next an asymmetric configuration of the system is considered and it is shown how isolas may envelop a primary backbone curve i e one that is connected directly to the zero amplitude solution as well as the isolated backbone curve this is achieved by using the energy based method to determine the relationship between the external forcing amplitude and the positions of the crossing points of the forced response along with predicting the existence of the isolas this technique also reveals the nature of the responses thus simplifying the process of finding isolas using numerical continuation

bifurcation analysis of aeroelastic systems with hysteresis by incremental harmonic balance method abstract the aeroelastic system of an airfoil with a hysteresis nonlinearity in the pitching degree of freedom was investigated by the incremental harmonic balance ihb method in the procedure of deducing the ihb iterative scheme a method of undetermined coefficient was extended to cope with the problem of expanding the hysteresis nonlinearity numerical examples show that the limit cycle oscillations lcos obtained by the ihb are in nice agreement with the numerical results bifurcations of the attained lcos with respect to the flow speed the preload and the freeplay of the hysteresis nonlinearity are analyzed in detail with the help of the floquet theory symmetry breaking saddle node bifurcation are detected as the flow speed varying

efficient model order reduction for dynamic systems with local nonlinearities abstract in the nonlinear structural analysis the nonlinear effects are commonly localized and the rest of the structure behaves in a linear manner considering this fact this research work proposes a harmonic balance solution in order to determine the nonlinear response of the structures the solution is simplified by using an exact dynamic reduction along with the modal expansion technique this novel approach which is applicable to both discrete and continuous systems converts the system equations of motion in each harmonic to a small set of nonlinear algebraic equations the full set of system equations is reduced to a discrete system with a few generalized degrees of freedom dofs confined to the localized nonlinear regions the resultant reduced order model is shown to be accurate enough for determining the periodic response to demonstrate the capability of the proposed method numerical case studies for continuous and discrete systems including systems with internal resonance have been studied and the outcomes are validated with benchmark studies in addition the method is applied in the identification process of an experimental test setup with unknown frictional support parameters and the results are presented and discussed

a real decoupled method and free interface component mode synthesis methods for generally damped systems this paper reports on the development of a new transformation method in contrast to most existing mode transformation methods in which the first order state space equation of the damped vibration system is transformed into a decoupled form with complex coefficient matrices using the decoupled method presented in this paper the equation of the damped system can be decomposed into a decoupled equation with real coefficient matrices two new free interface component mode synthesis methods are also presented the equivalent full mode matrix of the damped structure is used to capture the effects of the higher order modes additionally this work modifies the compatibility conditions at the junctions that are employed in most of the previous component mode synthesis methods for generally damped systems the first component mode synthesis method is performed in complex space whereas the second method can be applied in real space because the coefficient matrices of the coupled equation constructed by the second component mode synthesis method are all real valued the solution of the eigenproblem for this coupled equation can be performed in real space as well additionally numerical examples demonstrate the accuracy and validity of these two component mode synthesis methods

model reduction to spectral submanifolds and forced response calculation in high dimensional mechanical systems abstract we show how spectral submanifold ssm theory can be used to extract forced response curves without any numerical simulation in high degree of freedom periodically forced mechanical systems we use multivariate recurrence relations to construct the ssms achieving a major speed up relative to earlier autonomous ssm algorithms the increase in computational efficiency promises to close the current gap between studying lower dimensional academic examples and analyzing larger systems obtained from finite element modeling as we illustrate on two different discretized damped forced beam models using the exact reduction procedure via ssms for obtaining forced response curves we further demonstrate speed gains of several orders in magnitude relative to the available state of the art continuation packages while retaining accuracy

dynamics and energy exchanges between a linear oscillator and a nonlinear absorber with local and global potentials abstract the dynamical behavior of a two degree of freedom system made up of a linear oscillator and a coupled nonlinear energy sink with nonlinear global and local potentials is studied the nonlinear global potential of the energy sink performs direct interactions with the linear oscillator while its local potential depends only on its own behavior during vibratory energy exchanges between two oscillators a time multiple scale method around 1 1 1 resonance is used to detect slow invariant manifold of the system its equilibrium and singular points detected equilibrium points permit us to predict periodic regime s while singular points can lead the system to strongly modulated responses characterized by persistent bifurcations several possible scenarios occurring during these strongly modulated regimes are highlighted all analytical predictions are compared with those which are obtained by direct numerical integration of system equations

large amplitude non linear normal modes of piecewise linear systems a numerical method for constructing non linear normal modes nnms for piecewise linear autonomous systems is presented these nnms are based on the concept of invariant manifolds and are obtained using a galerkin based solution of the invariant manifold s non linear partial differential equations the accuracy of the constructed non linear modes is checked by the comparison of the motion on the invariant manifold to the exact solution in both time and frequency domains it is found that this construction approach can accurately capture the nnms over a wide range of amplitudes including those with strong non linear effects several interesting dynamic characteristics of the non linear modal motion are found and compared to those of linear modes a two degree of freedom example is used to illustrate the technique the existence stability and bifurcations of the nnms for this example are investigated

maximization of the vibration amplitude and bifurcation analysis of nonlinear systems using the constrained optimization shooting method abstract an original method for calculating the maximum vibration amplitude of periodic solutions of nonlinear systems is presented the problem of determining the worst maximum vibration is transformed into a nonlinear optimization problem the shooting method and the floquet theory are selected to construct the general nonlinear equality and inequality constraints the resulting constrained maximization problem is then solved by using the multistart algorithm finally the effectiveness and ability of the proposed approach are illustrated through two numerical examples numerical examples show that the proposed method can give results with higher accuracy as compared with numerical results obtained by a parameter continuation method and the ability of the present method is also demonstrated

dynamics of a piecewise non linear system subject to dual harmonic excitation using parametric continuation abstract the dynamic behavior of piecewise linear and or non linear vibratory systems subject to a single harmonic excitation has been recently analyzed by using a one or multi term harmonic balance or galerkin s method piecewise linear techniques analog simulation and direct numerical integration digital simulation instead this paper proposes to utilize the technique of parametriccontinuationto study the steady state response and global dynamics of a two degree of freedom piecewise non linear system with backlash or multi valued springs and impact damping the physical system is under the influence of a mean load and is subject to a dual harmonic external excitation where the second frequency is either twice or three times the fundamental excitation frequency the effects of various parameters such as impact damping mean and alternating loads the piecewise stiffness ratios and the relative phase between the excitation harmonics on the system response are thoroughly examined system or excitation parameter regimes that exhibit sub harmonic super harmonic quasi periodic and chaotic solutions are obtained efficiently and systematically by using the proposed scheme this investigation also clarifies the differences between viscous and non linear impact damping models which have been proposed earlier in the literature for studying clearance non linearities limited experimental data validate our modelling strategy finally selected predictions from this technique match very well with a multi term harmonic balance method and with direct numerical integration wherever applicable

homotopy analysis method for limit cycle flutter of airfoils an analytical approximate method for non linear problems namely the homotopy analysis method is for the first time employed to propose a new approach for the flutter system of a two dimensional airfoil with a cubic structural nonlinearity the frequencies and amplitudes of limit cycle flutters are described as maclaurin series of an embedding parameter a series of algebraic equations in the priori unknown coefficients of the series are derived all the equations are linear except the first one this provides us with a simple iteration scheme to seek the solutions to any desired accuracy numerical examples are presented to illustrate the validity and efficiency of the proposed approach the approximations of the frequencies and amplitudes of the limit cycles are obtained very accurately furthermore it is proved the frequencies are independent of the coefficient of the cubic nonlinearity while the amplitudes are in inverse proportion to the square root of the coefficient

the harmonic balance method for bifurcation analysis of large scale nonlinear mechanical systems abstract the harmonic balance hb method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems the objective of this paper is to exploit the method for bifurcation analysis i e for the detection and tracking of bifurcations of nonlinear systems to this end an algorithm that combines the computation of the floquet exponents with bordering techniques is developed a new procedure for the tracking of neimark sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed the hb method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device

residue harmonic balance for two degree of freedom airfoils with cubic structural nonlinearity a residue harmonic balance method is extended to determine the approximations and bifurcations for a coupled airfoil systemwith cubic structural nonlinearity analytically the approximate solutions to any desired accuracy can be obtained easily by solving a set of linear algebraic equations in each step the strongly cubic nonlinear pitching and plunging stiffness terms in the two degree of freedom self excited equations of airfoil motions are considered the approximations to the angular frequency and limit cycle are analytically found the results show that the secondorder approximations obtained by the proposed technique are in excellent agreement with the numerical integration results the existence of cubic nonlinearity in the stiffness termdoes not affect the linearflutter speed of airfoilmotion a great deal however the responses are different not only after supercritical bifurcation flutter but also before the linear flutter speed in the subcritical bifurcation furthermore the supercritical and subcritical hopf bifurcations of three nonlinear cases of airfoil motions are analyzed taking the flow speed as the bifurcation parameter

determination of nonlinear normal modes of a planar nonlinear system with a constraint condition abstract to demonstrate in what a way nonlinear normal modes may influence or be used to predict system responses this paper derives analytically the nonlinear normal modes with a constraint condition from the free vibration equation of the non conservative nonlinear subsystem of a piecewise smooth system since the nonlinear subsystem describes the planar motion of an isotropic rotor with cross coupling stiffness in contact with its stator the nonlinear normal modes reflecting the two potential whirling directions within a plane possess both positive and negative modal frequencies the results of this paper show that the stable and the unstable nonlinear normal modes affect the responses of the whole piecewise smooth system in different ways moreover the nonlinear normal modes can also be used to predict some important characteristics of the self excited backward whirl motion of the piecewise smooth system

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

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

relaxation oscillations in r3 abstract the existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically it is shown that near a singular periodic orbit a return map can be defined which has a one dimensional slow manifold with a stable invariant foliation under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter additionally the existence of an invariant torus is proved for the periodically forced van der pol oscillator the analysis is based on methods from geometric singular perturbation theory the blow up method is used to analyze the dynamics near the fold curves

nonlinear normal modes for damped geometrically nonlinear systems application to reduced order modelling of harmonically forced structures in order to build efficient reduced order models roms for geometrically nonlinear vibrations of thin structures a normal form procedure is computed for a general class of nonlinear oscillators with quadratic and cubic nonlinearities the linear perturbation brought by considering a modal viscous damping term is especially addressed in the formulation a special attention is focused on how all the linear modal damping terms are gathered together in order to define a precise decay of energy onto the invariant manifolds also defined as nonlinear normal modes nnms then this time independent formulation is used to reduce the dynamics governing the oscillations of a structure excited by an external harmonic force the validity of the proposed roms is systematically discussed and compared with other available methods in particular it is shown that large values of the modal damping of the slave modes may change the type of nonlinearity hardening softening behaviour of the directly excited master mode two examples are used to illustrate the main features of the method a two degrees of freedom dof system allows presentation of the main results through a simple example then a water filled circular cylindrical shell with external resonant forcing is considered in order to show the ability of the method to substantially reduce the dynamics of a continuous structure 2006 elsevier ltd all rights reserved

application of the centre manifold theory in non linear aeroelasticity abstract in this study a frequency relation for limit cycle oscillations of a two degree of freedom aeroelastic system with structural non linearities represented by cubic restoring spring forces is derived the centre manifold theory is applied to reduce the original system of nine dimensional first order ordinary differential equations to a governing system in two dimensions near the bifurcation point the principle of normal form is used to simplify the non linear terms of the lower dimensional system using the frequency relation and the amplitude frequency relationships derived from a previous study limit cycle oscillations lcos for self excited systems can be predicted analytically the mathematical technique proposed here has been applied to investigate lco near a hopf bifurcation for an aeroelastic system with cubic restoring forces not only that an excellent agreement is obtained compared to the numerical results from solving the original system of eight non linear differential equations by runge kutta time integration scheme but we also demonstrate that the use of a mathematical approach leads to a better understanding of non linear aeroelasticity

an algorithm for response and stability of large order non linear systems application to rotor systems abstract a numerical algorithm to calculate the periodic response stability and bifurcations of a periodically excited non conservative multi degree of freedom mdof system with strong local non linearities is presented first the given large order system is reduced using a fixed interface component mode synthesis procedure cms in which the degrees of freedom associated with non linear elements are retained in the physical co ordinates while all others whose number far exceeds the number of non linear dof are transformed to modal co ordinates and reduced using real mode cms a shooting and continuation method is then applied to the reduced system to solve for the periodic response floquet stability theory is used to calculate stability and bifurcations of the periodic response the algorithm is applied to study the response to imbalance stability and bifurcations of a 24 dof flexible rotor supported on journal bearings the results indicate that the proposed algorithm though approximate can yield very accurate information about dynamic behavior of large order non linear systems even with few numbers of retained component modes the algorithm which imposes less demand on computer time and memory is believed to be of considerable potential in analyzing a variety of practical problems

numerical computation of nonlinear normal modes in mechanical engineering this paper reviews the recent advances in computational methods for nonlinear normal modes nnms different algorithms for the computation of undamped and damped nnms are presented and their respective advantages and limitations are discussed the methods are illustrated using various applications ranging from low dimensional weakly nonlinear systems to strongly nonlinear industrial structures

application of the arc length method in nonlinear frequency response several iterative numerical techniques have been developed to solve nonlinear structural problems and some of these methods are capable to trace complex paths in the space load displacement one of those most popular procedures is the arc length method of crisfield which possesses the capability to overcome inflection points without having the necessity of determining them a great similarity exists between curves of the nonlinear load displacement path obtained with the arc length method and curves of the frequency response of nonlinear dynamic systems both curves present limit points with snap back and snap through phenomena this work consists of the description and the application of the arc length method to solve a system of nonlinear equations obtaining as a result the nonlinear frequency response the analysis employs the concept of the describing functions where the fundamental harmonic component is considered the most relevant and for some cases can be considered an approximation to the effect of all harmonics some examples involving a cubic stiffness and a gap nonlinearity are employed to illustrate the methodology

strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry abstract dynamic responses of a linear oscillator coupled to a nonlinear energy sink nes under harmonic forcing in the regime of 1 1 1 resonance are investigated primary attention is paid to the detailed investigation of the so called strongly modulated response smr which is not related to the fixed points of average modulation equations of the system essential mass asymmetry allows a global analysis of the responses despite strong nonlinearity it is demonstrated that the strongly modulated response is related to a relaxation type motion and its description in the limit of small mass ratio maybe reduced to the 1d return map of a subset at a fold line of slow invariant manifold the smr exists in the o e vicinity of the exact resonance where e 1 characterizes the mass asymmetry it is also shown that the smr appears in the system as a result of global fold bifurcation of limit cycles and exhibits some properties pertinent to generic 1d nonlinear maps such as period doubling transient responses with finite number of relaxation cycles and subsequent attraction to stable periodic attractor are revealed analytic results are compared to numeric simulations and a good agreement is observed

super and sub harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method abstract this article presents a new multi term harmonic balance method hbm for nonlinear frequency response calculations of a torsional sub system containing a clearance type nonlinearity the ability of the simplified subsystem to capture the salient behavior of the larger system is verified by the comparison of results to experimental data unlike previous analytical and numerical methods the proposed hbm includes adaptive arc length continuation and stability calculation capabilities to find periodic solutions in multi valued nonlinear frequency response regimes as well as to improve convergence essential steps of the proposed hbm calculations are introduced and it is validated by comparing time and frequency domain predictions with those yielded by numerical solutions experimental studies or analog simulations for several examples then nonlinear frequency response characteristics of an oscillator with clearance nonlinearity are examined with focus on super and sub harmonics we also explore some issues that are not fully resolved in the literature for instance the effect of mean operating point is examined for 0 and 0 18 0 25 cases where is the stiffness ratio of the piecewise linear elastic function in addition the number of harmonic terms that must be included in the hbm response calculations given sinusoidal excitation has been investigated finally some simple analytical predictions for super and sub harmonic resonances are presented

a dual craig bampton method for dynamic substructuring a novel component mode synthesis method for dynamic analysis of structures is presented it is based on free interface vibration modes and residual flexibility components although the ingredients are the same as in previously published procedures e g macneal or rubin our method is fundamentally different in that it assembles the substructures using interface forces dual assembly and enforces only weak interface compatibility the new formulation is based on a fully consistent reduction approach and the reduced matrices so obtained are exactly dual to the craig bampton reduced matrices the new free interface substructuring method proposed here is thus more natural then classical free interface synthesis procedures and leads to simpler reduced matrices we illustrate the efficiency of the dual craig bampton approach for reducing the dynamical representation on a three dimensional frame

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

nonlinear normal modes for vibratory systems under harmonic excitation this paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to harmonic excitation the approach is an extension of the nonlinear normal mode nnm formulation previously developed by the authors for free oscillations wherein an auxiliary system that models the excitation is used to augment the equations of motion in this manner the excitation is simply treated as an additional system state yielding a system with an extra degree of freedom dof whose response is known a reduced order model for the forced system is then determined by the usual nnm procedure and an efficient galerkin based solution method is used to numerically construct the attendant invariant manifolds the technique is illustrated by determining the frequency response for a simple 2 dof mass spring system with cubic nonlinearities and for a discretized beam model with 12 dof the results show that this method provides very accurate responses over a range of frequencies near resonances

eigenanalysis of unsteady flow about airfoils cascades and wings a general technique for constructing reduced order models of unsteady aerodynamic flows about two dimensional isolated airfoils cascades of airfoils and three dimensional wings is developed the starting point is a the domain computational model of the unsteady small disturbance flow for illustration purposes we apply the technique to an unsteady incompressible vortex lattice model the eigenmodes of the system which may be thought of as aerodynamic states are computed and subsequently used to construct computationally efficient reduced order models of the unsteady flowfield only a handful of the most dominant eigenmodes are retained in the reduced order model the effect of the remaining eigenmodes is included approximately using a static correction technique an important advantage of the present method is that once the eigenmode information has been computed reduced order models can be constructed for any number of arbitrary modes of airfold motion very inexpensively numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method finally we show how the reduced order model may be incorporated into an aeroelastic flutter model

a method for nonlinear modal analysis and synthesis application to harmonically forced and self excited mechanical systems abstract the recently developed generalized fourier galerkin method is complemented by a numerical continuation with respect to the kinetic energy which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes in order to enhance the practicability regarding the investigation of complex large scale systems it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations a novel reduced order model rom is developed for those regimes where internal resonances are absent the approach allows for an accurate approximation of the multi harmonic content of the resonant mode and accounts for the contributions of the off resonant modes in their linearized forms the rom facilitates the efficient analysis of self excited limit cycle oscillations frequency response functions and the direct tracing of forced resonances the rom is equipped with a large parameter space including parameters associated with linear damping and near resonant harmonic forcing terms an important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology this is achieved by selected numerical examples including finite element models of structures with strongly nonlinear non conservative contact constraints

one to one resonant double hopf bifurcation in aeroelastic oscillators with tuned mass dampers the effects of a tuned added mass on the aeroelastic stability of a single degree of freedom bluff body exposed to a steady flow are investigated the model captures the essential aspects of the behaviour of flexible structures equipped with tuned mass dampers undergoing galloping oscillations the system exhibits simple as well double hopf bifurcations of non resonant and 1 1 resonant type postcritical behaviour of the system in the neighbourhood of the 1 1 resonant type bifurcation is investigated employing the multiple scale method a second order bifurcation equation in the complex amplitude of motion is obtained analytical solutions are used to describe the bifurcation scenario in the cases of both undercritical and supercritical aerodynamic behaviour of the bluff body the effectiveness of the tuned mass damper even in the postcritical range is proved

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

non linear normal modes and invariant manifolds small amplitude motions of dynamic systems structural fluid control etc about an equilibrium state are modeled by linear differential equations which have constant coefficients these are typically obtained by a taylor series expansion of the forces about the equilibrium point under quite general circumstances these equations admit a set of special solutions called normal mode motions in which each system component moves with the same frequency and with a fixed ratio amongst the displacements of the components for a conservative system for a non conservative system all displacements and velocities are linearly related to a single displacement velocity pair

non linear dynamic interactions of a jeffcott rotor with preloaded snubber ring a two degrees of freedom model of a jeffcott rotor with a preloaded snubber ring subjected to out of balance excitation has been developed the purely impact interactions have been investigated the rotor makes intermittent contacts with the preloaded snubber ring and as a consequence it can be in one of five different contact regimes which boundaries have been found analytically the current location of the snubber ring has been determined using the principle of the minimum elastic energy in the snubber ring consequently a non linear piecewise smooth dynamical system has been obtained and studied numerically the results in form of bifurcation diagrams phase portraits and poincare maps show significant differences for the cases with and without preloading

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

dynamic responses and mitigation of limit cycle oscillations in van der pol duffing oscillator with nonlinear energy sink the paper considers dynamics of van der pol duffing vdpd oscillator with attached nonlinear energy sink due to a cubic nonlinearity of the vdpd oscillator a frequency of oscillations near the unstable origin strongly differs from the frequency of limit cycle oscillations lco the paper demonstrates that despite the strong nonlinearity of the model system one can efficiently describe the dynamics with a combination of averaging and multiple scales methods global structure of possible response regimes is revealed it is also demonstrated that the nonlinear energy sink can efficiently control and mitigate the undesired lcos in this system

a component based hybrid reduced basis finite element method for solid mechanics with local nonlinearities abstract the scrbe static condensation reduced basis element method is a component to system model order reduction approach for efficient many query and real time treatment of linear partial differential equations characterized by many spatially distributed constitutive geometry and topology parameters in this paper we incorporate the scrbe approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities in particular we exploit a linear nonlinear domain decomposition to develop a hybrid formulation we consider a scrbe approximation over the assumed predominant part of the domain associated to a linear elasticity model we revert to a full finite element fe approximation over the part of the domain associated to the locally nonlinear model we adapt the scrbe port training procedures to anticipate the behavior of the field over the linear nonlinear interface and hence ensure an accurate solution over the entire domain we choose a globally conforming approximation which exploits the intrinsic port structure of the scrbe method and yields a decoupled non invasive formulation of the respective linear and nonlinear blocks of the residual vector and newton jacobian matrix we present numerical results for several local nonlinearities elastic contact with and without friction plastic contact which demonstrate substantial savings relative to standard fe approximation over the entire domain in particular in the regime in which the linear part of the domain represents the majority of the fe degrees of freedom

damping synthesis using complex substructure modes and a hermitian system representation modal synthesis techniques have long been used to evaluate the natural frequencies and mode shapes of systems for which modal characteristics of the various components have been determined either experimentally or numerically however little attention has been given in the prediction of damping levels of the total structure from damping information obtained experimentally usually in the form of modal damping ratios and complex or real mode shapes the purpose of this paper is to present such a method to demonstrate its use on a simple example and to discuss two numerical aspects related to its numerical implementation hermitian matrices and use of transformations of the displacements only rather than transformations twice the size incorporating velocities and displacements only the method of fixed constraint modes is presented herein and damping at the joints of the substructures is not considered

a fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions a new algorithm is proposed to combine the split frequency harmonic balance method sf hbm with arc length continuation alc for accurate tracing of the frequency response of oscillators with non expansible nonlinearities alc is incorporated into the sf hbm in a two stage procedure stage i involves finding a reasonably accurate response frequency and solution using a relatively large number of low frequency harmonics this step is achieved using the sf hbm in conjunction with alc stage ii uses the sf hbm to obtain a very accurate solution at the frequency obtained in stage i to guarantee rapid path tracing the frequency axis is appropriately subdivided this gives high chance of success in finding a globally optimum set of harmonic coefficients when approaching a turning point however arc lengths are adaptively reduced to obtain a very accurate solution the combined procedure is tested on three hardening stiffness examples a duffing model an oscillator with non expansible stiffness and single harmonic forcing and an oscillator with non expansible stiffness and multiple harmonic forcing the results show that for non expansible nonlinearities and multiple harmonic forcing the proposed algorithm is capable of tracing out frequency response functions with high accuracy and efficiency

non linear frequency response characteristics of an impact pair abstract the method of harmonic balance is used to develop approximate analytical solutions to the equations of motion of a single impact pair including a general discussion of the existence and stability of these solutions non linear frequency response characteristics are obtained for a single frequency excitation and primary resonance no subharmonic or superharmonic response and the results are validated using an analog computer simulation it has been found that the use of digital simulation numerical integration may lead to inaccurate results over certain frequency ranges it is also shown that although the shapes of the force displacement relationship for clearance non linearity and the classic cubic non linearity are similar the resulting non linear frequency response curves can be quite different various issues related to the clearance non linearity problem are discussed

a split frequency harmonic balance method for nonlinear oscillators with multi harmonic forcing abstract a new harmonic balance method hbm is presented for accurately computing the periodic responses of a nonlinear sdof oscillator with multi harmonic forcing and non expansible nonlinearities the presence of multi harmonic forcing requires a large number of solution harmonics with a substantial increase in computational demand for either the conventional or the incremental hbm in this method the oscillator equation error is first defined in terms of two functions originally proposed for obtaining free vibration periods in r e mickens iteration procedure for determining approximate solutions to nonlinear oscillator equations journal of sound and vibration 116 1987 185 187 and more recently r e mickens a generalised iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillations journal of sound and vibration 287 2005 1045 1051 a fourier series solution is assumed in which the total number of harmonics is fixed by the chosen discrete time interval this series is split into two partial sums nominally associated with either low frequency or high frequency harmonics by exploiting a convergence property of the equation error functions the total solution is obtained in a new iterative scheme in which the low frequency components are computed via a conventional hbm using a small number of algebraic equations whereas the high frequency components are obtained in a separate step by updating by gradually increasing the number of harmonics in the low frequency group the equation error can be progressively reduced efficient use is made of fft based algebraic equation generation which allows an important class of non expansible nonlinearities to be handled the proposed method is tested on a duffing type oscillator and an oscillator with a non expansible 7th power stiffness term where in both cases up to 24 component multi harmonic forcing is applied as a comparison a conventional hbm is also used on the duffing model in which the algebraic equations are generated in symbolic form to totally avoid errors from entering the formulation through complicated expansion of the cubic stiffness term as in i senjanovic harmonic analysis of nonlinear oscillations of cubic dynamical systems journal of ship research 38 3 1994 225 238 and in a raghothama s narayanan periodic response and chaos in nonlinear systems with parametric excitation and time delay nonlinear dynamics 27 2002 341 365 the paper shows that in obtaining period 1 solutions the computational accuracy and efficiency of the proposed method is very good

curves of genus 2 with group of automorphisms isomorphic to d_8 or d_ 12 the classification of curves of genus 2 over an algebraically closed field was studied by clebsch and bolza using invariants of binary sextic forms and completed by igusa with the computation of the corresponding three dimensional moduli variety m 2 the locus of curves with group of automorphisms isomorphic to one of the dihedral groups d 8 or d 12 is a one dimensional subvariety in this paper we classify these curves over an arbitrary perfect field k of characteristic chark 2 in the ds case and char k 2 3 in the d 12 case we first parameterize the k isomorphism classes of curves defined over k by the k rational points of a quasi affine one dimensional subvariety of m 2 then for every curve c k representing a point in that variety we compute all of its k twists which is equivalent to the computation of the cohomology set h 1 g k aut c the classification is always performed by explicitly describing the objects involved the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of gl 2 k in particular we give two generic hyperelliptic equations depending on several parameters of k that by specialization produce all curves in every k isomorphism class

forced vibrations of a torsional oscillator with coulomb friction under a periodically varying normal load the forced vibration response of a single degree of freedom torsional system with coulomb friction under a periodically varying normal load is studied first an enhanced multi term harmonic balance method is developed to calculate nonlinear responses directly in frequency domain this should be in general applicable to periodically varying nonlinear systems second a pulse width modulated normal load is approximated by a truncated fourier series with a reasonable number of harmonic components and utilized for case studies finally the effects of duty ratio on nonlinear frequency responses are examined