modified schmidt games and diophantine approximation with weights abstract we show that the sets of weighted badly approximable vectors in r n are winning sets of certain games which are modifications of games introduced by w m schmidt in 1966 the latter winning property is stable with respect to countable intersections and is shown to imply full hausdorff dimension

numerical presentation of eigenvalue curve veering and mode localization in coupled pendulums to be 0 01 0 01 0 01 0 0 and 0 01 0 the e rst variations and the eigenvalues obtained by the classic method and the improved calculation are given in table 1 the relative errors of the variations are shown in fig 2 the result of the improved calculation is in the very good agreementwith the exact one apartfromthe third mode the e rst variations given by the improved calculation are even better than those with the second variations by the classic method a large modie cation of the truss is considered by multiplying a and b with a factor from 1 through 10 the relative errors of the e rst variations are shown in fig 3 the errors in the e rst mode increase from 0 0827 to 0 8410 whereas those of the classic method change from 0 4159 to 4 1834 the fourth mode shows the biggest difference as the largest error of the improved calculation is 0 0023 but 9 9982 for the classic method conclusions an improved calculation of eigenvalue variations has been presented the eigenvalue approximation from the e rst variation is a rayleigh quotient the variations are different from those in the classic method when there is a variation of b which is the inertiarelated matrix in a structural dynamics although there are more terms in the variations of eigenvalues no extra matrix operation is required thegivenexamplecone rmsthattheaccuracyoftheeigenvalue variations is much better than that obtained with the classic method

non similar normal oscillations in a strongly non linear discrete system abstract the free oscillations of a strongly non linear discrete oscillator are examined by computing its non similar non linear normal modes these are motions represented by curves in the configuration space of the system and they are not encountered in classical linear vibration theory or in existing non linear perturbation techniques the mikhlin manevich asymptotic methodology is used for solving the singular functional equation describing the non similar modes and approximate analytical expressions are derived for an oscillator with weak coupling stiffness and mistuning both localized and non localized modes are detected occurring in small neighborhoods of degenerate and global similar modes of the tuned system when strong coupling is considered only non localized modes were found to exist an interesting result of this work is the detection of mode localization in the tuned periodic system a result with no counterpart in existing theories on linear mode localization as a check of the analytical results numerical integrations of the equations of motion were carried out and the existence of the theoretically predicted non similar modes was verified

behavior of derivatives of eigenvalues and eigenvectors in curve veering and mode localization and their relation to close eigenvalues the problem to measure the phenomena of eigenvalue curve veering and mode localization is addressed in this paper the second derivative of an eigenvalue and the first derivative of an eigenvector are taken as the measures numerically showing curve veering and mode localization based on the measurement close eigenvalues as a key factor for the occurrence of the phenomena are defined two eigenvalues are considered to be close if their difference is small enough to cause the occurrence of the phenomena the curve veering and mode localization can be noted by comparison of the derivatives with a critical value and hence the associated eigenvalues are close weakly coupled springs are given as an example

comments on energy confinement in non periodic structures in this study we investigate the possibility of mode localization occurrence in a non periodic pfluger s column model of a rocket with an intermediate concentrated mass at its middle point we discuss the effects of varying the intermediate mass magnitude and its position and the resulting energy confinement for two cases free vibration analysis and the severity of mode localization are appraised without decoupling the system by considering as a solution basis the fundamental free response or dynamical solution this allows for the reduction of the dimension of the algebraic modal equation that arises from satisfying the boundary and continuity conditions by using the same methodology we also consider the case of a cantilevered pluger s column with rotational stiffness at the middle support instead of an intermediate concentrated mass

modal analysis of a nonlinear periodic structure with cyclic symmetry this paper carries out modal analysis of a nonlinear periodic structure with cyclic symme try the nonlinear normal mode nnm theory is brie y described and a computational algorithm for the nnm computation is presented the results obtained on a simpli ed model of a bladed assembly show that this system possesses a very complicated struc ture of nnms including similar and nonsimilar nnms nonlocalized and localized nnms bifurcating and internally resonant nnms modal interactions that occur without neces sarily having commensurate natural frequencies in the underlying linear system are also discussed

badly approximable points on planar curves and winning abstract for any i j 0 with i j 1 let bad i j denote the set of points x y r 2 such that max q x 1 i q y 1 j c q for some positive constant c c x y and all q n we show that bad i j c is winning in the sense of schmidt games for a large class of planar curves c namely everywhere non degenerate planar curves and straight lines satisfying a natural diophantine condition this strengthens recent results solving a problem of davenport from the sixties in short within the context of davenport s problem the winning statement is best possible furthermore we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of bad i j is winning for two dimensional schmidt games

passive spatial confinement of impulsive responses in coupled nonlinear beams a system of weakly coupled geometrically nonlinear beams is examined a galerkin procedure is used to express the motions of the two beams in terms of their linearized flexural modes transient impulsive excitations are considered and the response of the system is analytically and numerically computed for small values of a coupling nonlinear parameter the vibrational energy injected into the system is proved to localize mainly at the directly forced beam and only a small portion of this energy leaks to the unforced one this passive transient motion confinement is solely due to nonlinear localized modes of the unforced system and it becomes more profound for stronger impulse excitations and as the nonlinearity increases and or the coupling stiffness connecting the two beams decreases

2 dimensional badly approximable vectors and schmidt s game we prove that for any pair s t of nonnegative numbers with s t 1 the set of s t badly approximable vectors in r 2 is 34 p 2 1 winning for schmidt s game

bounded orbits of diagonalizable flows on finite volume quotients of products of sl2 r abstract we prove a number field analogue of a result of c mcmullen that the set of badly approximable numbers is absolute winning we also prove a weighted version we use this to prove an instance of a conjecture of an guan and kleinbock 4 namely let g sl 2 r sl 2 r and be a lattice in g we show that the set of points on g whose orbits under a one parameter ad semisimple subgroup of g are bounded form a hyperplane absolute winning set

free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints abstract an analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges and elastically coupled together at an arbitrary angle the interactions of all four wave groups bending waves out of plane shearing waves in plane longitudinal waves and in plane shearing waves have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses each of the transverse and in plane displacement functions is expressed as the superposition of a two dimensional 2 d fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x y plane as mathematically implied by a 2 d fourier series the unknown expansions coefficients are calculated using the rayleigh ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains numerical examples are presented for several different coupling configurations a good comparison is observed between the current results and the fea models although this study is specifically focused on the coupling of two plates the proposed method can be directly extended to structures consisting of any number of plates

frequency coalescence and mode localization phenomena a geometric theory physical systems the natural frequencies of which depend parametrically on certain quantities may exhibit the phenomenon of frequency coalescence when two or more natural frequencies become equal or the phenomenon of avoided crossings when two or more frequencies approach rapidly and then avoid crossing each other in the process the natural modes undergo significant variations it is shown that both phenomena can be described in a unified manner from the singularities of the mapping from the complex parameter plane into the complex frequency plane in particular the formation of a saddle point in the frequency plane and a branch point in the parameter space is the basis for explaining the properties associated with these phenomena the question of sensitivity of the system response near such singularities is subsequently examined it is shown that the development of a branch point singularity in the complex parameter plane can cause large or even infinite sensitivity of the system about this point depending on the distance of the branch point from the real axis in the parameter space it is further shown that mode localization is also associated with the appearance of a branch point near the real axis of the complex parameter plane which causes the large sensitivity that characterizes the phenomenon the theory is applied to examples of conservative and dissipative systems undergoing free vibrations

numerical computation of wave localization in large disordered beam like lattice trusses wave localization in randomly disordered multiwave structures is investigated the smallest localization factor is of particular interest and is related to the smallest positive lyapunov exponent the numerical algorithm by wolf et al is modified to determine all of the lyapunov exponents the wave localization in a large beamlike lattice truss modeled as an equivalent continuous timoshenko beam is studied any engineering structures are designed to be composed of identically constructed elements assembled end to end to form a spatially periodic structure such as long space antennae or large beamlike lattice trusses used in space solar power stations periodic structures behave like bandpass filters when propagating stress waves if damping is neglected waves are classified as traveling waves that propagate without attenuation and nontraveling or attenuating waves whose amplitudes attenuate as the waves propagate for single wave periodic structures the frequency axis is divided into alternating passbands and stopbands in the frequency passbands the wave is a traveling wave whereas in the frequency stopbands the wave is an attenuating wave for multiwave structures in a given frequency band some of the waves may be traveling waves and the others nontraveling waves however due to defects in manufacture and assembly no structure designed as a periodic structure can be perfectly periodic disorder can occur in the geometry of configuration s and material properties of the structure in a disordered periodic structure wave amplitudes of all waves will be attenuated even those that are traveling waves in the perfectly periodic counterpart this means that the vibrational energy imparted to the structure by an external source cannot propagate to arbitrarily long distances but is instead substantially confined to a region close to the source this phenomenon is known as wave localization in disordered periodic structures it is therefore of great practical importance to study the localization behavior and evaluate the so called localization factors namely the exponential rates at which the amplitudes of the waves propagating in the structures decay the reciprocal of the localization factor gives the localization length which characterizes the distance that the propagating wave extends in the structure as an application the range of damage that is spread in a structure due to impact at some location can be estimated there are d waves in a d wave disordered periodic structure each wave attenuates at a certain exponential rate or corresponds to a certain localization factor which implies that each wave will extend to a certain localization length the smallest localization factor or the largest localization length is of particular interest for multiwave structures since it corresponds to the wave that has potentially the least amount of decay or that carries energy along the multiwave structure farther than any other waves the study of the localization phenomenon has been and remains an active research area in solid state physics after the celebrated work of anderson 1 hodges2 was the first to recognize the relevance of localization theory to dynamical behavior of periodic engineering structures since then there have been several studies on the localization of vibration of structures in a series of publications pierre and dowell 3 pierre 4 and cha and pierre5 studied the localization phenomenon for monocoupled disordered structural systems cai and lin6 developed a perturbation scheme to calculate the localization factor based on a generic periodic structure kissel7 and ariaratnam and xie8 9 used a traveling wave approach to investigate the localization effects in one dimension al periodic engineering structures a transfer matrix formulation including wave transfer matrices was used to model disordered periodic structures furstenberg s theorem10 for products of random matrices was applied to calculate the localization effect as a function of frequency the localization factor was related to the largest lyapunov exponent in these publications only one dimensional or monocoupled periodic structures were considered in which the transfer matrices are of dimension 2 few studies have been done on the localization behavior of multiwave periodic structures kissel11 derived the localization factor as a function of the transmission matrix for multiwave disordered systems using the multiplicative ergodic theorem of oseledec 12 the difficulty in the study of multiwave periodic structures is that the localization factor is related to the smallest positive lyapunov exponent and not the largest lyapunov exponent as in the singlewave case the numerical algorithms used previously for evaluating the largest lyapunov exponent for single wave periodic structures cannot be employed for multiwave structures in this paper a modification of an algorithm due to wolf et al 13 is used to determine all of the lyapunov exponents for randomly disordered multiwave periodic structures the localization factor of a large beamlike lattice truss modeled as an equivalent continuous timoshenko beam is evaluated

on nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation this work presents an investigation on the effect of an initial geometric imperfection wavelength amplitude and degree of localization on the in plane nonlinear natural frequencies veering and mode localization of an elastic euler bernoulli beam resting on a winkler elastic foundation the beam is assumed to be pinned pinned with a linear torsional spring at one end the effect of the axial force induced by mid plane stretching is accounted for in the derivation of the mathematical model due to its known importance and significant effect on the nonlinear dynamic behavior of the beam as it was proved and presented in earlier investigations the governing partial differential equation is discretized using the assumed mode method and the resulting nonlinear temporal equation was solved using the harmonic balance method to obtain results for the nonlinear natural frequencies and mode shapes the results are presented in the form of characteristic curves which show the variations of the nonlinear natural frequencies of the first three modes of vibration for a selected range of physical parameters like torsional spring constant elastic foundation stiffness and amplitude and wavelength of a localized and non localized initial slack

effects of dry friction damping on the occurrence of localized forced vibrations in nearly cyclic structures a multi harmonic balance method is applied to analyze the occurrence of strongly localized forced vibrations in a mistuned cyclic assembly of coupled non linear component structures under traveling wave excitation in addition to considering the effects of the strength of coupling between and disorder among component systems the study also investigates the relative effects of dry friction damping and viscous damping on the occurrence of strong localization results indicate that the degree of localization decreases as the amount of damping increases and that dry friction damped systems are more susceptible to localized vibrations than viscously damped ones this multi degree offreedom system with many dry friction dampers is analyzed by a multi harmonic frequency domain procedure the issues of accuracy and convergence of the method when applied to systems with several strongly non linear elements are discussed this frequency domain solution technique is compared to time domain methods in structural dynamics the sensitivity to disorder is a major concern in the design and analysis of nominally periodic structures made of coupled component systems it is now well known that the ratio of internal coupling strength to disorder strength is the key parameter governing the sensitivity to disorder for small values of this ratio small structural irregularities which break the structure s periodicity may inhibit the propagation of vibration from the source of excitation and confine the vibration modes to a narrow geometric region a phenomenon known as normal mode localization a number of studies have provided analyses of localized free vibrations for a variety of nearly periodic structures such as chains of coupled pendulums l 3 coupled beam assemblies 4 bladed disk assemblies 5 and cyclic assemblies 6 81 all these studies led to the conclusion that the degree of localization of the mode shapes depends primarily upon the ratio of coupling to disorder nonetheless besides coupling and disorder the magnitude and type of damping and the type of excitation must be considered in the study of localized forced vibrations hodges and woodhouse 2 and pierre 4 investigated localized forced vibrations under a local source of excitation for chains of coupled pendulums and coupled beam assemblies respectively kissel 9 also studied wave propagation and weak localization in disordered spring mass chains using a statistical approach wei and pierre 7 demonstrated the occurrence of localized forced vibrations in mistuned assemblies with cyclic symmetry when all component systems are excited harmonically however the role of damping on the strength of localized forced vibrations has not been explored therefore the primary objective of this paper is to investigate the effects of dry friction and viscous damping on the degree of forced vibration localization

vibration confinement in a general beam structure during harmonic excitations abstract in the present paper the green s function is utilized to provide a simple exact and direct analytical method for analysis of a beam structure of generic boundary conditions with attached springs and or finite masses the aim is to confine the vibration in a certain part of the structure the beam equation is based on the timoshenko beam theory with corrections for shear deformation and rotary inertia effects in the analysis the beam is driven by a harmonic external excitation the attached springs are modeled as simple reactions that provide transverse forces to the beam while each added mass provides a transverse force in addition to a moment at its location these forces moments act as secondary forces moments that reduce the response caused by the external force numerical simulations are conducted to find the optimal masses and or springs that confine the vibration in a certain chosen region the results were compared to that obtained from euler bernoulli beam theory beams that are excited by a bi harmonic force as well as dual excitation forces are analyzed in addition the case when the beams are excited near resonances is discussed also a method is proposed to impose a node at any desired location along the structure

the vibration and stability of a friction guided translating string abstract eyelets capstans and cylindrical surfaces are often used in thread fiber and paper handling machinery to guide the axially moving element in addition to providing positional control these guides introduce dry friction forces that alter the vibration and stability characteristics of the system this paper examines the lateral response of a string that slides through an elastically supported dry friction guide exact expressions are derived for the linear response under free and forced conditions solutions for the eigenvalue spectrum exhibit unusual features including multiple divergence instabilities regions of flutter instability and regions of curve veering associated with mode localization a second order perturbation solution is derived to examine the behavior of the eigenvalue spectrum in regions of flutter instability and curve veering the analysis highlights the common features of these two phenomena and suggests ways to minimize vibration by adjusting various design variables the analysis also demonstrates that the eigenvalue loci in regions of flutter instability and curve veering are naturally described by a local hyperbolic approximation

statistical distribution of frequency response in disordered periodic structures a structure designed to be spatially periodic cannot be exactly periodic the departure from exact periodicity is known as disorder and it causes localization of normal modes and additional attenuation of wave motions in the structure not related to damping the present investigation is directed at a possible adverse effect of disorder namely higher structural response near the point at which a dynamic excitation is applied than would occur in a perfectly periodic structure thereby reducing structure safety and reliability a systematic procedure is developed herein for the analysis of such an effect for a generic disordered periodic structure in particular the probability distribution of structural response is obtained by analysis and by monte carlo simulation which is needed both for fundamental understanding of the effect and for predicting the reliability of a system it is shown also that given probability distributions of the disordered parameters of a structure the mean and standard deviation of structural response can be obtained exactly for a damped randomly disordered periodic structure if the number of disordered cell units is not large and approximately if the number is large application of the procedure is illustrated by an example and the results are compared with monte carlo simulations

localization of vibrations by structural irregularity an investigation of the localization phenomenon is presented for disrodered structural systems consisting of weakly coupled component systems emphasis is placed on the development of a perturbation method that allows one to obtain the localized modes of vibration of the disordered system from the modes of the individual subsystems the principal feature of this perturbation method is that it not only considers the variations of the parameters as perturbations but it also treats the small coupling between subsystems as a perturbation such a perturbation analysis is cost effective as compared to a global eigenvalue analysis of the entire system and proves to be very accurate moreover it provides physical insight into the localization phenomenon and allows one to formulate a criterion that predicts the occurrence of strongly localized modes results are obtained for a chain of single degree of freedom coupled oscillators but these can be readilt generalized to deal with chains of multi degree of freedom component systems

vibration isolation from irregularity in a nearly periodic structure theory and measurements this article describes the theory and a simple experiment carried out to demonstrate the phenomenon of anderson localization in an acoustical context this is an effect whereby the propagation of vibration in a structure which is not entirely regular is impeded by the irregularities giving rise on the average to an exponential decay of vibration level away from the driving point even in the absence of any dissipation the structure used in the experiment was a stretched string with masses attached to it this string was studied with regular spacings of the masses and after the masses had been moved in a controlled way to provide a small degree of irregularity in both cases the transmission of energy from end to end of the string was measured as a function of frequency and also the mode shapes in the second and fourth passbands were measured so as to demonstrate the underlying physics of the localization phenomenon in which the individual modes making up each passband change from being extended throughout the structure in the regular case to being localized in specific areas of the structure in the presence of irregularity all measurements yielded satisfactory agreement with the theoretical predictions

statistical characteristics of frequency response localization in nearly periodic systems premature failures or malfunctions often occur in a mechanical system that has periodically repeated subcomponents due to slight irregularities among subcomponents of the periodic system maximal frequency responses of a few subcomponents become often significantly larger than those of other subcomponents these phenomena called frequency response localization need to be analyzed elaborately for reliable designs of the periodic systems in the present study the strength of frequency response localization for the periodic system is defined and the effects of some parameters on the strength of frequency response localization are investigated statistically

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

effects of mode localization on input output directional properties of structures abstract the singular value decomposition svd is employed to study the effects of localization phenomena on input output relationships and power and energy transmission ratios of structures for weakly and strongly coupled systems existence of strong localization of singular vectors and abrupt veering of singular value loci are shown occurrence of strong localization causes abrupt changes in input output directional properties in contrast to eigenvalues singular values do not veer away by introducing a disorder in weakly coupled systems in particular the use of singular values and vectors is computationally advantageous in considering multiple load cases while eigenvalue based analyses give information about the resonance frequencies and vibration modes of a structure singular values of the structure are related to the forced response characteristics and give the dynamic behavior in the frequency domain representative examples in structural dynamics are presented

mode localization in a class of multidegree of freedom nonlinear systems with cyclic symmetry the free oscillations of n degree of freedom dof nonlinear systems with cyclic symmetry and weak coupling between substructures are examined an asymptotic methodology is used to detect localized nonsimilar normal modes i e free periodic motions spatially confined to only a limited number of substructures of the cyclic system it is shown that nonlinear mode localization occurs in the perfectly symmetric weakly coupled structure in contrast to linear mode localization which exists only in the presence of substructure mistuning in addition to the localized modes nonlocalized modes are also found in the weakly coupled system the stability of the identified modes is investigated by means of an approximate two timing averaging methodology and the general theory is applied to the case of a cyclic system with three dof the theoretical results are then verified by direct numerical integrations of the equations of motion

vibration mode localization in two dimensional systems the method of regular perturbation is applied to study vibration mode localization in randomly disordered weakly coupled two dimensional cantilever spring arrays localization factors which characterize the average exponential rates of decay or growth of the amplitudes of vibration are defined in terms of the angles of orientation first order approximate results of the localization factor are obtained using a combined analytical numerical approach the localization factors are symmetric about the cantilever and the horizontal and vertical axes passing through the cantilever at which vibration is originated for the systems under consideration the direction in which vibration is originated corresponds to the smallest localization factor whereas the diagonal directions correspond to the largest rate of decay or growth of the amplitudes of vibration when plotted in the logarithmic scale the vibration modes are of hill shape with the amplitudes of vibration decaying linearly away from the cantilever at which vibration is originated

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

mode localization by structural imperfections in one dimensional continuous systems longitudinal free oscillations of a beam with small axial rigidity continuously restrained by imperfect elastic springs are analyzed it is shown that small imperfections can localize the first modes of vibration in a restricted region of the beam systems with single localized defects are first considered these provide physical insights and make clear that the phenomenon is governed by a turning point mathematical problem analogies with similar problems in quantum mechanics are emphasized periodic or nearly periodic imperfections are then analyzed by applying the floquet theory and finally non periodic imperfections are numerically treated it is shown that when the exponential decay of the solution is strong the modes localize in the neighbourhood of each defect so it is possible to limit oneself to a local analysis the problem of a beam of infinite length with a single imperfection is then solved by applying the asymptotic wkb method an application relative to a parabolic imperfection is analytically developed

mode localization and eigenvalue loci veering phenomena in disordered structures an investigation of the effects of disorder on the modes of vibration of nearly periodic structures is presented it is shown that in structures with close eigenvalues small structural irregularities result in both strong localization of the mode shapes and abrupt veering away or mutual repulsion of the loci of the eigenvalues when these are plotted against a parameter representing the disorder in the system perturbation methods for the eigenvalue problem are applied to predict the occurrence of strong localization and eigenvalue loci veering which are shown to be two manifestations of the same phenomenon also perturbation methods that handle the dramatic effects of small disorder are developed to analyze eigenvalue loci veering and strong localization two representative disordered nearly periodic structures are studied a mistuned assembly of coupled oscillators and a multi span beam with irregular spacing of the supports

the dynamics of large scale arrays of coupled resonators abstract this work describes an analytical framework suitable for the analysis of large scale arrays of coupled resonators including those which feature amplitude and phase dynamics inherent element level parameter variation nonlinearity and or noise in particular this analysis allows for the consideration of coupled systems in which the number of individual resonators is large extending as far as the continuum limit corresponding to an infinite number of resonators moreover this framework permits analytical predictions for the amplitude and phase dynamics of such systems the utility of this analytical methodology is explored through the analysis of a system of n non identical resonators with global coupling including both reactive and dissipative components physically motivated by an electromagnetically transduced microresonator array in addition to the amplitude and phase dynamics the behavior of the system as the number of resonators varies is investigated and the convergence of the discrete system to the infinite n limit is characterized

some stronger forms of fuzzy continuous mapping on fuzzy topological spaces abstract the paper is devoted to the introduction and study of the concepts of certain classes of functions between fuzzy topological spaces each of these functions presents a stronger form of the fuzzy continuous functions these maps have been characterized and investigated mainly in the light of the notions of q neighbourhoods quasi coincidence fuzzy closure and ds neighbourhoods they have also been compared as to their mutual dependence as also with regard to their interactions with certain other fuzzy non continuous functions

probabilistic vibration analysis of nearly periodic structures a disordered structure with weakly coupled subsystems has localized modes with the vibratory energy concentrated in one part of the mode shape the localization of modes may significantly affect the forced vibration response increasing the maximum vibration amplitude dramatically in some cases it is the scope of this paper to study the forced vibration of nearly periodic systems consisting of almost identical substructures and to analyze their free and forced dynamic responses probabilistically it is shown that the sensitivity of the forced response to the degree of localization depends on a combination of the symmetry of the mode which is excited and the phase difference between the forces acting on each substructure these results might explain the range of contrasting conclusions of earlier publications on the effects of forced response on mistuned structures a probabilistic analysis of the free and forced responses of a nearly periodic structure is shown to be useful in the design of structures that are sensitive to the degree of localization the results of the probabilistic analysis are verified using monte carlo simulation

numerical statistical investigation on the dynamics of finitely long nearly periodic chains a numerical statistical analysis of the effects of structural irregularities on the dynamics of finitely long nearly periodic assemblies is presented in particular emphasis is placed on the numerical analyses of the statistical behavior of the exponential decay rate of the vibration amplitude as the system is excited harmonically at one end when the coupling is weak and the localization is strong the localization factor the exponential decay constant for finitely long systems is normally distributed with its standard deviation being inversely proportional to the square root of the distance from the excited end thus for weakly coupled systems with many sites the standard deviation of the localization factor is small compared with its mean making the average decay factor for strong localization a true typical value this allows us to make inferences about the population based on the information contained in the sample for strong coupling where localization is weak on the other hand the probability distribution of the localization factor for finitely long systems is skewed to the right with a long tail with its standard deviation having the same order of magnitude as that of its mean physically this implies that the ensemble average for the localization factor may differ radically from the typical value this serves as a warning against the indiscriminate use of the ensemble mean as a predictor for the behavior of a typical system when the localization is weak

energy confinement in imperfect periodic systems abstract the energy fraction avg is developed as a measure of energy confinement in periodic systems of finite extent based on the response of a system to uniform broadband forcing avg is experimentally measurable but can be expensive to calculate it is shown that a norm of the eigenvector matrix avg is a good approximation for avg when damping is light avg is almost three orders of magnitude faster to calculate than avg which makes detailed monte carlo studies of imperfections practical one dimensional linear chain and cyclic systems of a range of sizes are studied in line with previous research it is found that a periodic system s propensity to confine energy increases with system size it is also found that cyclic systems are less likely to suffer energy confinement than otherwise equivalent linear chain systems

asymptotic analysis of wave propagation along weakly non uniform repetitive systems we consider monochromatic wave propagation along a long finite one dimensional slightly non uniform waveguide whose ends are connected to uniform semi infinite waveguides the non uniformity in the system parameters which is assumed slowly varying and deterministic can be tuned to produce a desired scattered wave field or reflection transmission properties for a broad range of incident wave fields with this objective in mind we obtain an analytic solution for wave propagation along repetitive systems asymptotic in the slowness of the variation of the system parameters we consider systems governed by a second order finite difference equation and apply the wkb method allowing the index variable to be complex this allows complex turning points to be considered the coefficients of the difference equation are represented by their discrete fourier modes for complex turning points we obtain exponentially small reflection a new result in the context of difference equations the asymptotic solution besides revealing how the non uniformity in the parameters affects wave propagation furnishes an analytic expression for the system scattering matrix as a function of the system parameters it also sheds light on the mechanism of localization phenomena for this class of repetitive systems we also compare the asymptotic results with numerical experiments for large finite one dimensional non uniform chains of coupled pendula

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n