comments on non linear formulations for travelling string and beam problems abstract kinematic aspects of non linear travelling structure problems are reviewed and clarified in this paper the description of the motion of a structure can be based on two different formulations eulerian or mixed eulerian lagrangian the mixed formulation leads to simpler equations but it requires comprehensive restrictions these restrictions and the formulation itself are considered and explained the inertia terms of the equations of motion are derived and are shown to be the same as those in the earlier papers

persistence of gaussian processes non summable correlations suppose the auto correlations of real valued centered gaussian process z cdot are non negative and decay as rho s t for some rho cdot regularly varying at infinity of order alpha in 1 0 with i rho t int 0 t rho s ds its primitive we show that the persistence probabilities decay rate of log mathbb p sup t in 0 t z t 0 is precisely of order t i rho t log i rho t thereby closing the gap between the lower and upper bounds of newell and rosenblatt ann math stat 33 1306 1313 1962 which stood as such for over fifty years we demonstrate its usefulness by sharpening recent results of sakagawa adv appl probab 47 146 163 2015 about the dependence on d of such persistence decay for the langevin dynamics of certain nabla phi interface models on mathbb z d

the numerical study of the nonlinear dynamics of a light axially moving string abstract the vibration characteristics of a light axially moving band are investigated by a numerical study in the subcritical and supercritical speed ranges equations of motion for the geometrically nonlinear axially moving string are formulated using hamilton s principle and discretized by the finite element method the periodic nonlinear problem for the string is solved by the fourier galerkin newton fgn method the nonlinear dynamic behaviour of an axially moving band is examined through the dependences between the fundamental frequency axial velocity and vibration amplitude resulting from nonlinear free vibration in general the behaviour of the nonlinear axially moving string is similar to that of a nonlinear beam the most notable differences being the fact that the string does not undergo bifurcation from a straight configuration to curved equilibrium states in a supercritical transport speed regime nor does the equilibrium state remain straight

the transient amplitude of the viscoelastic travelling string an integral constitutive law abstract the problem considered is an initially stressed viscoelastic string subjected to steady state and harmonic variation of axial motion the string material is assumed to be constituted by the hereditary integral type the partial differential integral equation of motion is derived first then by applying galerkin s method the governing equation is reduced to a set of second order non linear differential integral equations which are solved by finite difference numerical integration procedures the viscoelastic string with a linear spring dashpot model is considered as the voigt element in series with a spring three parameter model the qualitative aspect of parametric excitation due to the non uniform travelling velocity of a viscoelastic string is investigated finally the results are compared with those obtained by a constitutive law of a differential type method the effects of elastic and viscoelastic parameters constant and non constant transport speeds the wave propagation speed ratio and the non linear term on the transient amplitudes are also investigated

static elastica formulations of a pine conveying fluid an elastic pipe in an equilibrium configuration of arbitrary large deflection discharging fluid from its end experiences static centrifugal and frictional drag forces along its complete length these are however entirely equivalent to an end follower force of magnitude av 2 this equivalence is examined in detail by using the intrinsic field equations which are suitable for closed form solutions in terms of elliptic integrals once the pipe moves it also experiences gyroscopic coriolis forces along its length but these are not considered in this static examination it is shown in detail how a discharging pipe with end forces and moments is statically equivalent to a beam or strut with the same end forces and moments plus the reversed momentum vector av 2 it is seen that a cantilevered pipe with a free end can have no statical equilibrium states at all at either large or small deflections while pipes with constrained ends have large static deflections identical to those of the equivalent struts

supercritical stability of an axially moving beam part ii vibration and stability analyses abstract this paper focuses on the stability of axially moving beam like materials e g belts bands paper and webs which translate at speeds near to and above the so called critical speed stability limit in the companion paper a theoretical model for an axially moving beam was presented which accounted for geometrically non linear beam deflections and the initial beam curvature generated by supporting wheels and pulleys in that paper analysis of steady response revealed that the beam possesses multiple non trivial equilibrium states when translating at supercritical speeds the equations of motion are presently linearized about these equilibria and their stability is predicted from the eigenvalue problem for free response asymptotic and numerical solutions to the eigenvalue problem are presented for the respective cases of small and arbitrary equilibrium curvature the solutions illustrate that the translating beam has multiple stable equilibrium states in the supercritical speed regime the solutions confirm that the critical speed behavior for axially moving materials is extremely sensitive to system imperfections such as initial curvature

supercritical speed stability of the trivial equilibrium of an axially moving string on an elastic foundation the stability of an axially moving string supported by a discrete or distributed elastic foundation is examined analytically particular attention is directed at the distribution of the critical speeds and identifying the divergence instability of the trivial equilibrium the elastically supported string shows unique stability behavior that is considerably different from unsupported axially moving string in particular any elastic foundation discrete or distributed leads to multiple critical speeds and a single region of divergence instability above the first critical speed whereas the unsupported string has one critical speed and is stable at all supercritical speeds additionally the elastically supported string critical speeds are bounded above and the maximum critical speed is the upper bound of the divergent speed region the analysis draws on the self adjoint eigenvalue problem for the critical speeds and a perturbation analysis about the critical speeds neither numerical solution nor spatial discretization which can produce substantially incorrect results are required the system falls in the class of dispersive gyroscopic continua and its behavior provides a useful comparison for general gyroscopic systems the analytical findings also serve as a benchmark for approximate methods applied to gyroscopic continua

vibration of an axially moving string with geometric non linearity and translating acceleration the vibration of an axially moving string is studied when the string has geometric non linearity and translating acceleration based upon the von karman strain theory the equations of motion are derived considering the longitudinal and transverse deflections the equation for the longitudinal vibration is linear and uncoupled while the equation for the transverse vibration is non linear and coupled between the longitudinal and transverse deflections these equations are discretized by the galerkin method after they are transformed into the variational equations i e the weak forms so that the admissible and comparison functions can be used for the bases of the longitudinal and transverse deflections respectively with the discretized equations the natural frequencies the time histories of the deflections and the distributions of the deflection and stress are investigated in addition comparisons between the results of linear and non linear theories are provided

on the dynamic properties of axially moving systems abstract the objective of the present paper is a deep analysis of some recent numerical and experimental results regarding the complex dynamics of axially moving systems such important mechanical systems exhibit interesting dynamic behaviors homoclinic orbits sub harmonic responses amplitude modulations and chaos these dynamics have been obtained numerically and in some cases have been experimentally observed using recent techniques of the non linear time series analysis the response of axially moving systems has been studied for a large variety of test cases the correlation dimension of the time series which is deeply related to the minimal dimension of a system able to reproduce the dynamics is estimated lyapunov exponents are evaluated in order to quantify the response regularity the present work gives a contribution towards understanding the complex dynamics observed both in conservative and dissipative systems the dynamical phenomena are analyzed within the unified framework of the non linear time series analysis in the case of experimental data the new non linear filtering techniques based on the embedding techniques have been applied to reduce high noise when classical techniques give bad results

first passage times of levy flights coexisting with subdiffusion we investigate both analytically and numerically the first passage time fpt problem in one dimension for anomalous diffusion processes in which l evy flights and subdiffusion coexist we analyze the fpt for three subclasses of l evy stable motions i symmetric l evy motions characterized by l evy index ensuremath mu 0l ensuremath mu l2 and skewness parameter ensuremath beta 0 ii one sided l evy motions with ensuremath mu 0l ensuremath mu l1 and skewness ensuremath beta 1 and iii two sided skewed l evy motions the extreme case 1l ensuremath mu l2 and skewness ensuremath beta ensuremath 1 in all three cases the waiting times between successive jumps are heavy tailed with index ensuremath alpha we show that in all three cases the fpt distributions are power laws our findings extend earlier studies on fpts of l evy flights by considering the interplay between long rests and the l evy long jumps

on semi markov processes and their kolmogorov s integro differential equations abstract semi markov processes are a generalization of markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution this paper provides an integro differential form of the kolmogorov s backward equations for a large class of homogeneous semi markov processes having the form of an abstract volterra integro differential equation an equivalent evolutionary differential form of the equations is also provided fractional equations in the time variable are a particular case of our analysis weak limits of semi markov processes are also considered and their corresponding integro differential kolmogorov s equations are identified

spectral analysis for the transverse vibration of an axially moving timoshenko beam the use of frequency dependent spectral element matrix or exact dynamic stiffness matrix in structural dynamics is known to provide very accurate solutions while reducing the number of degrees of freedom to resolve the computational and cost problems thus in the present paper the spectral element model is formulated for the axially moving timoshenko beam under a uniform axial tension the high accuracy of the present spectral element is then verified by comparing its solutions with the conventional finite element solutions and exact analytical solutions the effects of the axial velocity and axial tension on the vibration characteristics the dispersion relation and the stability of a moving timoshenko beam are analytically and numerically investigated

supercritical stability of an axially moving beam part i model and equilibrium analysis axially moving material problems consider the dynamic response vibration and stability of long slender members which are in a state of translation this study focuses on the response of axially moving beam like elements at translation speeds that exceed the classical critical speed stability limit a non linear model for an axially moving beam is derived that accounts for the initial beam curvature induced by supporting pulleys or wheels presently the model is used to determine steady responses at critical and supercritical translation speeds the properties of the equilibrium problem are examined using an approximate linear solution and an exact non linear solution the deficiency of the linear solution is illustrated by its inability to capture essential features of the equilibrium problem particularly near and above the critical speed in this high speed region the translating beam undergoes large overall buckling deformations leading to multiple and bifurcated equilibrium states the stability of the equilibria is assessed in part ii

dynamic modeling of beams with non material deformation dependent boundary conditions abstract in conventional problems of structural mechanics both kinematic boundary conditions and external forces are prescribed at fixed material points that are known in advance if however a structure may move relative to its supports the position of the imposed constraint relations generally changes in the course of motion a class of problems which inherently exhibits this particular type of non material boundary conditions is that of axially moving continua despite varying in time the positions of the supports relative to the material points of the body have usually assumed to be known a priori throughout the deformation process in previous investigations this requirement is abandoned in the present paper where the dynamic behavior of a structure is studied which may move freely relative to one of its supports as a consequence the position of such a non material boundary relative to the structure does not only change in time but also depends on the current state of deformation of the body the variational formulation of the equilibrium relations of a slender beam that may undergo large deformations is presented to this end a theory based on reissner s geometrically exact relations for the plane deformation of beams is adopted in which shear deformation is neglected for the sake of brevity before a finite element scheme is developed a deformation dependent transformation of the beam s material coordinate is introduced by which the varying positions of the constraint relations are mapped onto fixed points with respect to the new non material coordinate by means of this transformation additional convective terms emerge from the virtual work of the inertia forces whose symmetry properties turn out to be different from what has previously been presented in the literature in order to obtain approximate solutions a finite element discretization utilizing absolute nodal displacements as coordinates is subsequently used in characteristic numerical examples which give an insight into the complex dynamic behavior of problems of this type on the one hand the free vibrations of a statically pre deformed beam are investigated on the other hand an extended version of the sliding beam problem is studied

a frequency domain based numeric analytical method for non linear dynamical systems abstract in this paper a multiharmonic balancing technique is used to develop certain algorithms to determine periodic orbits of non liner dynamical systems with external parametric and self excitations essentially in this method the non linear differential equations are transformed into a set of non linear algebraic equations in terms of the fourier coefficients of the periodic solutions which are solved by using the newton raphson technique the method is developed such that both fast fourier transform and discrete fourier transform algorithms can be used it is capable of treating all types of non linearities and higher dimensional systems the stability of periodic orbits is investigated by obtaining the monodromy matrix a path following algorithm based on the predictor corrector method is also presented to enable the bifurcation analysis the prediction is done with a cubic extrapolation technique with an arc length incrementation while the correction is done with the use of the least square minimisation technique the under determined system of equations is solved by singular value decomposition the suitability of the method is demonstrated by obtaining the bifurcational behaviour of rolling contact vibrations modelled by hertz contact law

dynamic stiffness matrix development and free vibration analysis of a moving beam abstract the dynamic stiffness matrix of a moving bernoulli euler beam is developed and used to investigate its free flexural vibration characteristics in order to develop the dynamic stiffness matrix it is necessary to derive and solve the governing differential equation of motion of the moving beam in closed analytical form the solution is then used to obtain the general expressions for both responses and loads boundary conditions are applied to determine the constants in the general solution leading to the formation of the frequency dependent dynamic stiffness matrix of the moving beam relating the amplitudes of the harmonically varying loads to those of the corresponding responses the application of the resulting dynamic stiffness matrix using the wittrick williams algorithm is demonstrated by some illustrative examples numerical results for both simply supported and fixed fixed end conditions of the beam are discussed and wherever possible some are compared with those available in the literature

stokes flow in curved channels in this paper the problem of stokes flow in curved channels is solved by applying the matched eigenfunction expansion method the flow region is partitioned into a union of sectorial subregions then within each subregion the stream function is represented by an expansion of papkovich fadle eigenfunctions the coefficients in these expansions are obtained by requiring weak c 3 continuity of the stream function across common interfaces biorthogonality conditions in cylindrical coordinates are utilized

infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations abstract the transverse vibrations of simply supported axially moving euler bernoulli beams are investigated the beam has a time varying axial velocity with viscous damping traveling beam eigenfunctions with infinite number of modes are considered approximate analytical solutions are sought using the method of multiple scales a perturbation technique a detailed analysis of the resonances in which upto four modes of vibration involved are performed stability analysis is treated for each type of resonance approximate stability borders are given for the resonances involving only two modes for higher number of modes involved in a resonance sample numerical examples are presented for stabilities

divergence and flutter instabilities of nanobeams in moving state accounting for surface and shear effects abstract transverse vibrations of elastically rested moving beam like nanostructures accounting for surface effect are of high concern the role of nonlocality on the free dynamic response of moving nanobeams has been revealed in recent years nevertheless the influence of the surface energy on the mechanical behavior of such elements has not been explained yet in this paper equations of motion of rested nanoscaled beams in the moving state are derived carefully via surface energetic shear deformable beam models subsequently the transverse vibrations of the nanostructure are evaluated using galerkin based assumed mode method the explicit expressions of divergence velocities are obtained analytically and these are successfully verified with the results of a numerical approach the roles of crucial parameters on the first divergence velocity are addressed in some detail additionally the stable and unstable regions are determined systematically and the influence of both surface energy and shear energy on the stability of moving nanostructure is discussed

a computation method for nonlinear vibration of axially accelerating viscoelastic strings a numerical algorithm is proposed for computing nonlinear vibration of axially accelerating viscoelastic strings based on independent functions the variational principle is used to discretize the governing equation into a set of differential algebraic equations numerical examples are presented

stochastic beam equations under random dynamic loads the fourth order partial differential equation representing beams under random loading is considered a general solution for this equation is obtained using the eigenfunction and variation of parameters techniques also the average and the variance of the beam deflection shear and bending moment are obtained the load is divided into a deterministic function and a randomly perturbed function representing the expected error in the deterministic load the general closed form solution is obtained in stochastic integral terms some important statistical moments of the solution process are computed and illustrated

a consistent corotational formulation for the nonlinear dynamic analysis of sliding beams abstract this paper presents a consistent corotational formulation for the geometric nonlinear dynamic analysis of 2d sliding beams compared with the works of previously published papers the same cubic shape functions are used to derive the elastic force vector and the inertia force vector consequently the consistency of the element is ensured the shape functions are used to describe the local displacements to establish an element independent framework moreover all kinds of standard elements can be embedded within this framework therefore the presented method is more versatile than previous approaches to consider the shear deformation the sliding beam a system of changing mass is discretized using a fixed number of variable domain interdependent interpolation elements iie in addition the nonlinear axial strain and the rotary inertia are also considered in this paper the nonlinear motion equations are derived by using the extended hamilton s principle and solved by combining the newton raphson method and the hilber hughes taylor hht method furthermore the closed form expressions of the iterative tangent matrix and the residual force vector are obtained three classic examples are given to verify the high accuracy and efficiency of this formulation by comparing the results with those of commercial software and published papers the simulation results also show that the shear deformation and the rotary inertia cannot be neglected for the large rotation and high frequency problem

the energetics and the stability of axially moving kirchhoff strings l the energetics and the stability of free transverse vibration of an axially moving string are investigated based on the kirchhoff nonlinear model the model is derived in a physically meaningful manner the time rate of the total mechanical energy associated with the vibration is calculated to show that the energy is not conserved it is proved that there exists a conserved quantity that remains a constant during the nonlinear vibration the conserved quantity is applied to verify the lyapunov stability of the axially moving kirchhoff string

on semi infinite spectral elements for poisson problems with re entrant boundary singularities abstract a spectral element method is described which enables poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi infinite rectangular regions two choices of trial functions are considered namely the eigenfunctions of the differential operator and chebyshev polynomials the coefficients in the series expansions are obtained by imposing weak c1 matching conditions across element interfaces singularities at re entrant corners are treated by a post processing technique which makes use of the known asymptotic behaviour of the solution at the singular point accurate approximations are obtained with few degrees of freedom

galerkin methods for natural frequencies of high speed axially moving beams in this paper natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges in the supercritical transport speed regime the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions the governing equations of coupled planar is reduced to two nonlinear models of transverse vibration for motion about each bifurcated solution those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform the natural frequencies are investigated for the beams via the galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary differential equations under the simple support boundary numerical results indicate that the nonlinear coefficient has little effects on the natural frequency and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro partial differential equation yields results quantitatively closer to those of the coupled equations

weak convergence of positive self similar markov processes and overshoots of levy processes using lamperti s relationship between levy processes and positive self similar markov processes pssmp we study the weak convergence of the law x of a pssmp starting at x 0 in the skorohod space of cadlag paths when x tends to 0 to do so we first give conditions which allow us to construct a cadlag markov process x 0 starting from 0 which stays positive and verifies the scaling property then we establish necessary and sufficient conditions for the laws x to converge weakly to the law of x 0 as x goes to 0 in particular this answers a question raised by lamperti z wahrsch verw gebiete 22 1972 205 225 about the feller property for pssmp at x 0

transverse vibration of an axially accelerating string abstract the transverse vibration of an axially accelerating string is investigated the equation of motion is developed using hamilton s principle the resulting partial differential equations are discretized using galerkin s method assuming the axial velocity to be periodic a stability analysis is performed using floquet theory one two three four six and eight term series approximations are considered in the galerkin s method the one term approximation leads to a mathieu equation the solution of which is well known the numerical results for one term are compared with the analytical solution for the mathieu equation and they are in full agreement the two term approximation leads to gyroscopically coupled equations and the solutions differ significantly from that of the one term approximation whereas the three term approximation solutions look similar to the one term approximation solutions the analysis is carried out for higher order even approximations and the solutions are in qualitative agreement the two and four term approximation solutions are compared with the analytical results from hsu s method and are in reasonable agreement the results show that instabilities occur at much higher amplitudes and frequencies of the periodic axial velocity than that of typical devices such as tape machines and band saws

lie group theory and analytical solutions for the axially accelerating string problem abstract transverse vibrations of a string moving with time dependent velocity v t have been investigated analytical solutions of the problem are found using the systematic approach of lie group theory group classification with respect to the arbitrary velocity function has been performed using a newly developed technique of equivalence transformations from the symmetries of the partial differential equation the method for deriving exact solutions for the arbitrary velocity case is shown special cases of interest such as constant velocity constant acceleration harmonically varying velocity and exponentially decaying velocity are investigated in detail finally for a simply supported strip approximate solutions are presented for the exponentially decaying and harmonically varying cases

a finite element formulation for sliding beams part i we use the updated lagrangian and the co rotational finite element methods to obtain solutions for geometrically non linear flexible sliding beams finite element formulations are normally carried out for fixed domains since the sliding beam is a system of changing mass first we discretize the system by introducing a variable domain beam element and model the sliding beam by a fixed number of elements with changing length second we transform the system governing equations of motion to a fixed domain and use conventional finite elements fixed size and number to discretize the system then our investigation is followed by a comparison between two formulations finally we use the co rotational method in conjunction with a variable domain beam element to obtain the discretized system equations to do so we consider the beam to slide with respect to a fixed channel and later we consider a formulation in which the beam remains at rest and the channel slides with a prescribed velocity we show that both formulations end up with identical discretized equations of motion 1998 john wiley sons ltd

principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law the steady state transverse responses and the stability of an axially accelerating viscoelastic string are investigated the governing equation is derived from the eulerian equation of motion of a continuum which leads to the mote model for transverse motion the kirchhoff model is derived from the mote model by replacing the tension with the averaged tension over the string the method of multiple scales is applied to the two models in the case of principal parametric resonance closed form expressions of the amplitudes and the existence conditions of steady state periodical responses are presented the lyapunov linearized stability theory is employed to demonstrate that the first second non trivial steady state response is always stable unstable numerical calculations show that the two models are qualitatively the same but quantitatively different numerical results are also presented to highlight the effects of the mean axial speed the axial speed fluctuation amplitude and the viscoelastic parameters

extinction time of non markovian self similar processes persistence annihilation of jumps and the frechet distribution we start by providing an explicit characterization and analytical properties including the persistence phenomena of the distribution of the extinction time mathbb t of a class of non markovian self similar stochastic processes with two sided jumps that we introduce as a stochastic time change of markovian self similar processes for a suitably chosen time change we observe for classes with two sided jumps the following surprising facts on the one hand all the mathbb t s within a class have the same law which we identify in a simple form for all classes and reduces in the spectrally positive case to the frechet distribution on the other hand each of its distribution corresponds to the law of an extinction time of a single markov process without positive jumps leaving the interpretation that the time change has annihilated the effect of positive jumps the example of the non markovian processes associated to levy stable processes is detailed

dynamics of flexible sliding beams non linear analysis part i formulation abstract equations of motion for the geometrically non linear analysis of flexible sliding beams deployed or retrieved through a rigid channel are derived through an extension of hamilton s principle based on the assumptions of euler bernoulli beam theory the equations of motion account for small strains but large rotations also provided is an alternative formulation wherein by superposition of a prescribed axial velocity the beam is brought to rest and the channel assumes the prescribed axial velocity the consistency of the two formulations is shown through an appropriate transformation of the governing equations to a fixed domain the fixed domain provides a very convenient frame work for numerical solutions of the equations of motion discretization procedures using galerkin s method and numerical examples involving large amplitude vibrations of the flexible sliding beam are presented in part ii

law of the absorption time of some positive self similar markov processes let x be a spectrally negative self similar markov process with 0 as an absorbing state in this paper we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density we provide a power series and a contour integral representation of this density then by means of probabilistic arguments we deduce some interesting analytical properties satisfied by these functions which include for instance several types of hypergeometric functions we also give several characterizations of the kesten s constant appearing in the study of the asymptotic tail distribution of the absorbtion time we end the paper by detailing some known and new examples in particular we offer an alternative proof of the recent result obtained by bernyk dalang and peskir ann probab 36 2008 1777 1789 regarding the law of the maximum of spectrally positive levy stable processes

log convex solutions to the functional equationf x 1 g x f x γ type functions abstract in this paper we discuss log convex solutionsf r r to the functional equation with initial condition given by f x 1 g x f x forx 0 andf 1 1 whereg r r our main result a generalization of the bohr mollerup artin classical characterization of the gamma function is that ifgis eventually log concave with the property that for eachw 0 g x w g x 1 asx then has a unique eventually log convex solution determined by the formula f x lim n g n g 1 g x n g n x g x forx 0 a functionfarising thus is called a type function two examples of which are the gamma function and theq gamma function q 0 r r to certain functional equations of the type f x f x 1 m f x m 1 m h x 1 the backdrop of type functions serves to place classical theory into perspective throughout the concepts of log convexity and log concavity play central roles

transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law abstract this paper deals with the transverse vibration of an initially stressed moving viscoelastic string obeying a fractional differentiation constitutive law the governing equation is derived from newtonian second law of motion and reduced to a set of non linear differential integral equations based on galerkin s truncation a numerical approach is proposed to solve numerically the differential integral equation through developing an approximate expression of the fractional derivatives involved some numerical examples are presented to highlight the effects of viscoelastic parameters and frequencies of parametric excitations on the transient responses of the axially moving string

convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh canonical edge problems for the biharmonic equation can be solved by separating variables the eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of r c smith 1952 we study the reduced system which is governed by a vector ordinary differential equation a solution of the biharmonic problem governed by a partial differential equation can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem the theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f y with values in 2 can be represented in terms of eigenvectors of the reduced problem this paper adds new theorems and conjectures to the theory we extend smith s generalization to fourth order problems of the methods introduced by titchmarsh 1946 to study eigenfunction expansions associated with second order problems we use this method to prove that if f y f1 y f2y 1 y 1 f y e c1 1 1 f e l2 1 1 then the series expressing f y converges uniformly to f y in the open interval 1 1 uniformly in 1 1 if f1 1 0 and in any case to 0 f2 1 f1 1 at y 1 this is unlike fourier series which converge to the mean value of the periodic extension of a function the series exhibits a gibbs phenomenon near the end points of discontinuity when f1 1 0

on the boundary conditions for axially moving beams axially moving beam typed structures are of technical importance and present in a wide class of engineering problem as the axial speed of a beam may significantly affect the dynamic characteristics of the structure even at a low velocity it is important to accurately predict the dynamic characteristics and stability of such structures in most previous studies the net energy flux through the left end and right end boundaries of the finite beam over two simple supports has been implicitly assumed to be zero by completely ignoring the effects of its left incoming and right outgoing semi infinite beam parts or by applying fixed boundary conditions for its longitudinal vibration which seems to be very non realistic from the physical point of view thus this paper investigates the effects of the continuously incoming and outgoing semi infinite beam parts on the dynamic characteristics and stability of an axially moving beam by using the spectral element method the spectral element model is formulated from the equations of motion derived by using the hamilton s principle extended for the systems of changing mass it is numerically shown that the effects of the continuously incoming and outgoing semi infinite beam parts should be taken into account for further accurate prediction of the dynamic characteristics and stability for such axially moving beams

stability analysis of an axially accelerating string the dynamic response of an axially accelerating string is investigated the time dependent velocity is assumed to vary harmonically about a constant mean velocity approximate analytical solutions are sought using two different approaches in the first approach the equations are discretized first and then the method of multiple scales is applied to the resulting equations in the second approach the method of multiple scales is applied directly to the partial differential system principal parametric resonances and combination resonances are investigated in detail stability boundaries are determined analytically it is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system or when the frequency is close to the sum of any two natural frequencies when the velocity variation frequency is close to zero or to the difference of two natural frequencies however no instabilities are detected up to the first order of perturbation numerical results are presented for a band saw and a threadline problem

hitting times for gaussian processes this is the publisher s version also available electronically from http www jstor org stable 25450587 seq 1 page scan tab contents

complex dynamics of high speed axially moving systems in this paper the dynamic response of a simply supported travelling beam subjected to a transverse load is investigated in the super critical speed range the well known axially moving beam theory is considered and a simple viscous damping mechanism has been introduced the displacement field is expanded in a series of the buckling modes a sine series and different techniques have been used in analyzing the response of the dynamical system periodic oscillations are studied by means of continuation techniques while non stationary dynamics are investigated through direct simulations a comparison with the literature and a convergence test on the series expansion are performed a sample case of a physical beam is developed and numerical results are presented concerning bifurcation analysis and stability and direct simulations of global postcritical dynamics a complex scenario of alternate regular and chaotic motions is found in a large range of the main parameters

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt