buckling of stochastically heterogeneous beams using a functional perturbation method abstract the buckling load and its probabilistic nature average and variance of bernoulli beams with stochastic material bending stiffness properties is derived analytically by a new functional perturbation method fpm a buckling shape function is assumed based on the homogeneous solution and additional terms to account for the morphology effects the buckling load in the transcendental equation is treated as a functional of the bending modulus stiffness or compliance field applying a functional perturbation to the above equation the buckling load is found analytically to any desired degree of accuracy as a function of material morphology the fpm is executed using both stiffness and compliance statistical data the impact of each of the two data sources on the solution accuracy is examined showing that compliance based solutions are accurate for small correlation lengths statically indeterminate problems can be treated with no additional effort an example of a simply supported beam is solved in detail comparison with previous studies where stochastic finite element and monte carlo simulation were used showed the relative accuracy and insight capabilities of the method the clamped free case is also studied to demonstrate that symmetry conditions used for homogeneous beams to find the buckling load on the basis of a simply supported case are not valid for heterogeneous beams

robust adaptive boundary control of a flexible marine riser with vessel dynamics in this paper robust adaptive boundary control for a flexible marine riser with vessel dynamics is developed to suppress the riser s vibration to provide an accurate and concise representation of the riser s dynamic behavior the flexible marine riser with vessel dynamics is described by a distributed parameter system with a partial differential equation pde and four ordinary differential equations odes boundary control is proposed at the top boundary of the riser based on lyapunov s direct method to regulate the riser s vibration adaptive control is designed when the system parametric uncertainty exists with the proposed robust adaptive boundary control uniform boundedness under the ocean current disturbance can be achieved the proposed control is implementable with actual instrumentation since all the required signals in the control can be measured by sensors or calculated by a backward difference algorithm the state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters simulations are provided to illustrate the applicability and effectiveness of the proposed control

precise asymptotic of eigenvalues of resonant quasilinear systems in this work we study the sequence of variational eigenvalues of a system of resonant type involving p and q laplacians on rn with a coupling term depending on two parameters and satisfying p q 1 we show that the order of growth of the kth eigenvalue depends on k o k n

a simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint in this paper a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity for various end boundary conditions we transform the governing equation with varying coefficients to linear algebraic equations then a characteristic equation in critical buckling loads will be obtained several examples of estimating buckling loads under typical end supports are discussed by comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams it can be found that our method has fast convergence and the obtained numerical results have high accuracy moreover the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties the effects of gradient parameters on the critical buckling loads are elucidated finally we give an example to illustrate the enhancement of the load carrying capacity of tapered beams for admissible shape profiles with constant volume or weight the proposed method is of benefit to optimum design of beams against buckling in engineering applications

estimate of the axial force in slender beams with unknown boundary conditions using one flexural mode shape abstract this paper presents an experimental procedure for the axial load identification of slender prismatic beams with unknown boundary conditions by making use of one vibration frequency and of five amplitudes of the corresponding mode shape in fact this method does not require the knowledge of the effective length of the beam under examination but only the flexural rigidity and mass per unit length the proposed algorithm was verified by means of many numerical and experimental tests on tie rods having different boundary conditions excellent estimates of the axial forces were obtained finally the influence of the location of the instrumented sections on the estimation of the axial load was analyzed using a simply supported beam model

particularities of newton s method in space frame force determination utilising eigenpair functions abstract redundant space frame dynamics are modelled through the structural eigenproblem which is formulated to account for the geometric stiffness due to self equilibrating membrane forces the resulting mathematical eigenvalues and eigenvectors respectively indicative of natural frequencies and modes of oscillation are then assumed to be continuous functions of a scalar factor upon a normalised set of forces in equilibrium newton s method or sensitivity analysis provides a means for minimising the difference between mathematical and physically observed eigenvalues whereupon the axial forces are inferred from the model issues of convergence and root uniqueness are anticipated eigenvalue coalescence in which eigenvalues transitorily coalesce to permute their arbitrarily numerical ordering is seen to define non smooth eigenvalue functions and hence encumber newton s method mode tracing strategies to overcome this are discussed the existence of various degrees of eigenvalue degeneracy owing to frame cyclic symmetry or otherwise is anticipated in terms of the influence upon newton s method if the degree of frame static redundancy is greater than one and there exist a number of linearly independent force distributions then the required force determination is a multidimensional newton method problems encountered in these manifold dimensions including those arising from frame spatial periodicity and the adversity of newton s method are overcome emphasis is placed upon minimising the dimensionality of newton s method through enforcement of equilibrium constraints illustrative numerical simulations are given and some applications of the method are proposed

on the flexural vibration of cylinders under axial loads numerical and experimental study abstract the flexural vibration of a homogeneous isotropic linearly elastic cylinder of any aspect ratio is analysed in this paper natural frequencies of a cylinder under uniformly distributed axial loads acting on its bases are calculated numerically by the ritz method with terms of power series in the coordinate directions as approximating functions the effect of axial loads on the flexural vibration cannot be described by applying infinitesimal strain theory therefore geometrically nonlinear strain displacement relations with second order terms are considered here the natural frequencies of free free clamped clamped and sliding sliding cylinders subjected to axial loads are calculated using the proposed three dimensional ritz approach and are compared with those obtained with the finite element method and the bernoulli euler theory different experiments with cylinders axially compressed by a hydraulic press are carried out and the experimental results for the lowest flexural frequency are compared with the numerical results an approach based on the ritz formulation is proposed for the flexural vibration of a cylinder between the platens of the press with constraints varying with the intensity of the compression the results show that for low compressions the cylinder behaves similarly to a sliding sliding cylinder whereas for high compressions the cylinder vibrates as a clamped clamped one

a new approach for free vibration of axially functionally graded beams with non uniform cross section this paper studies free vibration of axially functionally graded beams with non uniform cross section a novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density for various end supports including simply supported clamped and free ends we transform the governing equation with varying coefficients to fredholm integral equations natural frequencies can be determined by requiring that the resulting fredholm integral equation has a non trivial solution our method has fast convergence and obtained numerical results have high accuracy the effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non homogeneity moreover fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties the effects of the geometrical and gradient parameters are elucidated the present results are of benefit to optimum design of non homogeneous tapered beam structures

lower bounds for generalized eigenvalues of the quasilinear systems abstract in this paper we establish several new lyapunov type inequalities for two classes of one dimensional quasilinear elliptic systems of resonant type which generalize or improve all related existing ones then we use the lyapunov type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one dimensional quasilinear elliptic system with the dirichlet boundary conditions

isospectral vibrating beams it has long been known that two scaling factors and three spectra corresponding to three different end conditions are required to determine the masses lengths and stiffnesses in a discrete model of a beam in flexural vibration what had not been known was how to find a family of beams that all have the same spectrum under one set of end conditions say those corresponding to a cantilever this paper presents two procedures for finding families of such isospectral beams the first uses shifted qr factorization and yields a four parameter family the second uses toda flow to find another more restricted family

locally distinguishing inputs for systems on lie groups and parallelism this paper is concerned with invariant systems on lie groups a sufficient condition for a piecewise constant input to distinguish between any two distinct initial states contained in some open neighbourhood of the state space is given this condition involves the parallel translation of vectors along curves thus relating observability to parallelism

on lyapunov type inequality for quasilinear systems in this paper we sketch some recent developments in the theory of lyapunov type inequalities and present some new results relating to a quasilinear system special cases of which contain some well known differential equations such as half linear and linear equations our result generalize the lyapunov type inequality given in 18

flexural vibration of axially loaded beams with linear or parabolic taper abstract curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections previous curves have only included axial forces in a very limited number of cases whereas here axial forces within the range of greatest practical importance can always be allowed for the present curves cover 11 combinations of end conditions three types of taper all taper ratios between 0 2 and 1 and all axial forces between p c and 0 6 p c where p c is the critical buckling load for pure compression and can be found from curves provided a list of principal nomenclature is given in appendix 2 the cross sections covered include thin circular ones of constant thickness and thin walled cross sections consisting of constant thickness flat plates which all have their breadths tapering in the same way truncated wedges of varying depth and constant width are also covered the taper is linear between the ends or linear between a maximum value at the centre and equal values at the ends or parabolic with a maximum value at the centre simple examples show how the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1 the curves also illustrate how natural frequencies and buckling loads of thin walled members are altered by changing the amount and type of a member s taper while keeping its wall thickness and mass constant finally it is shown that for the ranges covered by the curves frequencies which separate the first five natural frequencies can be found a priori the theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy exact bernoulli euler dynamic stiffnesses which allowed for axial force effects were used for these uniform components in appendix 1 the effects of using exact timoshenko stiffnesses instead of the bernoulli euler ones are illustrated

exact solution of free vibration of a uniform tensioned beam combined with both lateral and rotational linear sub systems abstract a general exact solution of free vibration of a uniform tensioned beam with any number of both lateral and rotational sub systems is addressed this study is completed in complex domain because of the presence of the non proportional damping caused by the dampers utilized in sub systems the sub systems constructed herein could degenerate to the conventional supports and sub systems and be arbitrarily placed accordingly the solution is also suitable for multi span beams the argument principle method apm which does not need initial values and iteration is employed to solve the characteristic equation in complex domain nevertheless the presence of multi valued square root functions due to the axial loading renders the function non analytic in complex domain a variable substitution method in which the main branch of logarithmic function is chosen is proposed to solve this problem the results obtained are compared with those in literature furthermore several typical case studies were completed and the root loci of the frequency parameters attained are compared with those obtained by the finite element method fem a good agreement is observed which confirms the validity of the presented methodologies in this paper

exploring isospectral spring mass systems with firefly algorithm this paper investigates in line spring mass systems an fixed at one end and free at the other with n degrees of freedom d f the objective is to find feasible in line systems b n that are isospectral to a given system the spring mass systems a n and b n are represented by jacobi matrices an error function is developed with the help of the jacobi matrices a n and b n the problem of finding the isospectral systems is posed as an optimization problem with the aim of minimizing the error function the approach for creating isospectral systems uses the fact that the trace of two isospectral jacobi matrices a n and b n should be identical a modification is made to the diagonal elements of the given jacobi matrix a n to create the isospectral systems the optimization problem is solved using the firefly algorithm augmented by a local search procedure numerical results are obtained and resulting isospectral systems are shown for 4 d f and 10 d f systems

statistical modeling of heterogeneous micro beams statistical characteristics averages and variances of heterogeneous small size beams micro beams composed of micro elements grains of randomly distributed stiffness isotropic is studied only longitudinal variations are considered element to beam size length ratio is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient using standard statistical tools probability density and correlation functions it is shown that for indeterminate cases even average deflections obtained explicitly are load dependent and are not identical with their corresponding homogeneous case bounds for the statistical dispersion of deflections and reaction forces are found analytically for some cases of weak heterogeneity it is shown that dispersion parameters for indeterminate problems can be obtained by superposition using both averages and dispersion data of corresponding determinate cases leading to reciprocal type relations

harmonic frequency spectra of vibrating stepped strings the frequency spectrum of a vibrating string with a step in density is investigated for proportions of length and 1 having different constant densities in corresponding ratio r if the quantity r r 1 is a rational number then the vibrations of the string with fixed ends exhibit harmonic and or odd harmonic frequency sub spectra expressible in simple analytical form if r 1 then the entire frequency spectrum is a pure harmonic series indistinguishable from that of a homogeneous string consideration of the string vibrating in the free fixed configuration serves to remove this coincidence the free free configuration is also dealt with in a similar approach

higher order effective modeling of periodic heterogeneous beams i asymptotic expansion method this paper is concerned with the elastostatic behavior of heterogeneous beams with a cross section and elastic moduli varying periodically along the beam axis by using the two scale asymptotic expansion method the interior solution is formally derived up to an arbitrary desired order in particular this method is shown to constitute a systematic way of improving bernoulli s theory by including higher order terms without any assumption in contrast to timoshenko s theory or other refined beam models moreover the incompatibility between the interior asymptotic expansions and the real boundary conditions is emphasized and the necessity of a specific treatment of edge effects is thus underlined

vibration tailoring of inhomogeneous rod that possesses a trigonometric fundamental mode shape in this study a special class of closed form solutions for inhomogeneous rod is investigated namely the following problem is considered determine the distribution axial rigidity when the material density is given of an inhomogeneous rod so that the postulated fundamental trigonometric mode shape serves as an exact vibration mode in this study the associated semi inverse problem is solved that results in the distributions of axial rigidity that together with a specified law of material density satisfy the governing eigenvalue problem for comparison the obtained closed form solutions are contrasted with approximate solutions based on an appropriate polynomial shapes serving as trial functions in an energy method the obtained results are utilized for vibration tailoring i e construction of the rod with a given natural frequency

the free vibration of a type of tapered beams this paper studies the vibrational characteristics of tapered beams with continuously varying rectangular cross section of depth and breadth proportional to xs and xt respectively where both s and t are arbitrary real numbers for a truncated beam and arbitrary positive numbers for a sharp ended beam and x is the axial co ordinate measured from the sharp end of the beam the bernoulli euler theory of bending is used to describe the motion of the beam a new set of beam functions are developed as the admissible functions which are the complete solution of the tapered beam under an arbitrary static load expanded into a taylor series the eigenfrequency equation is obtained by the rayleigh ritz method the accuracy is assured from the convergency and comparison studies the effect of the location of the taylor series expanding point on the convergency is discussed the analysis shows that the present approach is convergent for arbitrary truncation factor by taking the midpoint of the beam as the expanding point of the taylor series numerical results are tabulated for three different tapered beams with various boundary conditions and truncation factors it is shown that the eigenfrequencies can be obtained with high accuracy by using only a small number of terms of the static beam functions 2000 elsevier science s a all rights reserved

invariant boundary value problems for a fourth order dynamic euler bernoulli beam equation we obtain the complete lie symmetry group classification of the dynamic fourth order euler bernoulli partial differential equation where the elastic modulus the area moment of inertia are constants and the applied load is a nonlinear function in the lie analysis the principal lie algebra which is two dimensional extends in three cases viz the linear the exponential and the general power law for each of the nontrivial cases we determine symmetry reductions to ordinary differential equations which are of order four in only one case related to the power law we are able to have a compatible initial boundary value problem for a clamped end and a free beam for these cases we deduce the corresponding fourth order ordinary differential equations with appropriate boundary conditions we provide an asymptotic solution for the reduced fourth order ordinary differential equation corresponding to a clamped or free beam

can harmonic functions constitute closed form buckling modes of inhomogeneous columns it is uniformly known that the buckling modes of uniform columns are given by trigonometric namely harmonic functions for inhomogeneous columns the buckling modes usually are derived via special functions including bessel and lommel functions recently it was demonstrated that the buckling modes of specific inhomogeneous columns assume a simple polynomial form the question posed in the title of this study therefore naturally arises it is shown that the reply to this query is affirmative four cases of harmonically varying buckling modes are postulated and semi inverse problems are solved that result in the distributions of the flexural rigidity compatible to the preselected modes and to specified axial load distributions in all cases the closed form solutions are obtained for the eigenvalue parameter

the inverse problem for the euler bernoulli beam it has long been known that two scaling factors and three spectra corresponding to three different end conditions are required to determine the cross sectional area a x and second moment of area i x of an euler bernoulli beam what has not been known are the necessary and sufficient conditions on the spectral data which will yield positive functions a x i x such a set of conditions is derived in this paper

a finite beam element for vibration analysis of rotating tapered timoshenko beams a finite beam element for vibration analysis of a rotating tapered beam including shear deformations and rotary inertia is derived in this paper the finite element has four degrees of freedom and accounts for linear tapering in two planes this formulation permits any combination of taper ratios as well as unequal element lengths explicit expressions for the finite element mass and stiffness matrices are derived using the consistent mass approach while accounting for the centrifugal stiffness effects the generalized eigenvalue problem is defined and numerical solutions are generated for a wide range of rotational speed and taper ratios results obtained include the first ten free vibrational modes for both fixed and hinged end conditions comparisons are made whenever possible with exact solutions and with numerical results available in the literature the results display high accuracy when compared with other numerical results

further flexural vibration curves for axially loaded beams with linear or parabolic taper curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections these cross sections complement those of an earlier paper such that the curves in the two papers cover all of the most common types of tapered member the curves presented cover eleven combinations of end conditions three types of taper all taper ratios between 0 3 and 1 and all axial forces between pc and 0 6pc where pc is the critical buckling load for pure compression and can be found from curves provided the results cover truncated wedges which have their depth and width varying identically so that all cross sections are similar in the sense that similar triangles are the cross sections covered also include thin circular ones with thickness proportional to diameter and thin walled cross sections consisting of flat plates which all have their breadths and thicknesses tapering identically the taper is linear between the ends or linear between a maximum value at the centre and equal values at the ends or parabolic with a maximum value at the centre the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1 the curves also illustrate how natural frequencies and buckling loads are altered by changing the amount and type of a member s taper while keeping its mass constant finally it is hown that for the ranges covered by the curves frequencies which separate the first five natural frequencies can be found a priori the theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy exact bernoulli euler dynamic stiffnesses which allowed for axial force effects were used for these uniform components illustrative results show the effects of using exact timoshenko stiffnesses instead of the bernoulli euler ones

eigenvalues of the p laplacian in fractal strings with indefinite weights in this paper we study the spectral counting function of the weighted p laplacian in fractal strings where the weight is allowed to change sign we obtain error estimates related to the interior minkowski dimension of the boundary we also find the asymptotic behavior of eigenvalues

free vibration of rotating tapered beams using the dynamic stiffness method abstract the free bending vibration of rotating tapered beams is investigated by using the dynamic stiffness method the range of problems considered includes beams for which the depth and or width of the cross section vary linearly along the length first the governing differential equation of motion of the rotating tapered beam in free flap bending vibration is derived for the most general case using hamilton s principle allowing for the effects of centrifugal stiffening an arbitrary outboard force and the hub radius term for harmonic oscillation the differential equation is solved for bending displacement by applying the frobenius method of series solution the expressions for bending rotation shear force and bending moment at any cross section of the beam are also obtained in explicit analytical form then the dynamic stiffness matrix is developed by relating the amplitudes of forces and moments to those of the displacements and rotations at the ends of the harmonically vibrating tapered beam next the wittrick williams algorithm is used as a solution technique to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of some illustrative examples a parametric study is carried out to demonstrate the effects of rotational speed taper ratio and hub radius on the results which are discussed and compared with the published ones finally some conclusions are drawn

uniqueness in the determination of unknown coefficients of an euler bernoulli beam equation with observation in an arbitrary small interval of time abstract in the present article we prove an uniqueness result for the non linear problem of identifying the flexural stiffness and density of an euler bernoulli beam using observation of boundary measurements here we show that the knowledge of the displacement and slope of at the free extremity of a clamped free vibrating beam for an arbitrary small interval of time leads to the uniqueness in the identification of its rigidity and density this result extends previous works that showed that the identification was possible when the observation was collected in an unbounded interval of time or when the original dynamic problem for the euler bernoulli beam was linearised

apparently first closed form solutions for inhomogeneous vibrating beams under axial loading closed form solutions are presented for fundamental natural frequencies of inhomogeneous vibrating beams under axially distributed loading the mode shape is postulated as coinciding with the static deflection of the associated homogeneous beam without distributed axial loading then the inverse problem of determining the stiffness and mass density distributions producing the above mode shape is solved to describe these variations the family of polynomial functions is used several sets of boundary conditions are considered it is shown that the natural frequency vanishes when the intensity of the axially distributed loading equals the critical buckling value a linear relationship is established between the square of the natural frequency and the load ratio for all reported sets of boundary conditions in contrast to uniform beams where the exact linear relationship holds for columns with simply supported and or sliding ends

vibrations of non uniform continuous beams under moving loads abstract the dynamic behavior of multi span non uniform beams transversed by a moving load at a constant and variable velocity is investigated the continuous beam is modelled using bernoulli euler beam theory the solution is obtained by using both the modal analysis method and the direct integration method the natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross section along the beam using the exact element method the mode shapes are expressed as infinite polynomial series using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study and the results are compared to previously published results

reconstructing a pure state of a spin s through three stern gerlach measurements consider a spin s prepared in a pure state it is shown that generically the moduli of the spin components along three directions in space determine the state unambiguously these probabilities are accessible experimentally by means of a standard stern gerlach apparatus it is therefore possible to reconstruct a generic pure state on the basis of 6s independent measured intensities

the equivalence problem for the euler bernoulli beam equation via cartan s method we completely solve the local point equivalence problem for the euler bernoulli beam equation using cartan s method of equivalence we obtain five equivalence classes for each equivalence class we establish the necessary and sufficient conditions for similarity and derive a basis of differential invariants as well as operators of invariant differentiation

on quantum state observability and measurement we consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable the system is assumed to be a finite dimensional quantum control system for which we can influence the dynamics by generating all the unitary evolutions in a lie group we investigate to what extent by an appropriate sequence of evolutions and measurements we can obtain information on the initial state of the system we present a system theoretic viewpoint of this problem in that we study the observability of the system in this context we characterize the equivalence classes of indistinguishable states and propose algorithms for state identification

semi inverse method for axially functionally graded beams with an anti symmetric vibration mode abstract in this paper we use the semi inverse method to find the solutions to the dynamic equation of inhomogeneous functionally graded simply supported beams for the anti symmetric mode and the material density or the young s modulus that are both polynomial functions and and have been pre specified we find the young s modulus or the density in polynomial functions and determine the closed from expression for the natural frequency

on the analysis of laterally vibrating slender beams subject to various complicating effects the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain a simple analysis of the free vibration problem of slender beams subject to various complicating effects these effects are the presence of concentrated masses and or concentrated moments of inertia intermediate supports axial loading and non uniformity of cross section such may exist separately or in any combination numerical results are presented for several example beams illustrating the accuracy and versatility of the approach

exact bernoulli euler dynamic stiffness matrix for a range of tapered beams bernoulli euler theory and bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial torsional and flexural vibrations of any beam which is tapered such that a varies as yn and gj and i both vary as y n 2 where a gj and i have their usual meanings y cx l 1 c is a constant such that c 1 l is the length of the beam and x is the distance from one end of the beam numerical checks give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements a procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped this enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members

on local observability for invariant systems on lie groups and coset spaces the purpose of this short paper is to provide a proof of a local observability criterion for invariant systems on lie groups and coset spaces the criterion was originally presented with an ad hoc proof based on lie theoretic arguments the alternative proof presented here shows that the criterion essentially comes from the classical observability results involving the observability codistribution and its properties in this way it unifies the two approaches and extends the application field of the classical results

the inverse problem for the vibrating beam the beam is modelled by a system of rigid rods joined together by rotational springs and with masses at the joints the left end is clamped while the right end is subject to one of a number of conditions it is free pinned sliding or clamped it is shown that the resonant and antiresonant frequencies of the system may be almost completely ordered necessary and sufficient conditions are found that these frequencies must satisfy to correspond to an actual system with positive parameters and two stripping procedures are devised for the determination of these system parameters from a knowledge of certain frequency spectra

apparently first closed form solution for vibrating inhomogeneous beams abstract free vibration of non uniform beams which possess non homogeneous material density and elastic modulus along their axis are studied under various boundary conditions closed form expressions for the fundamental natural frequency are derived it is shown that there is an infinite number of beams that share the same natural frequency moreover it is proved that some coefficients describing the density and elastic modulus functions can be deterministic or random yet remarkably in special circumstances the fundamental natural frequencies turn out to be deterministic quantities extensive numerical analysis is performed to substantiate this seemingly paradoxical finding by the monte carlo method boobnov galerkin method and the finite element method

the riemann zeta function and the one dimensional weyl berry conjecture for fractal drums based on his earlier work on the vibrations of drums with fractal boundary the first author has refined m v berry s conjecture that extended from the smooth to the fractal case h weyl s conjecture for the asymptotics of the eigenvalues of the laplacian on a bounded open subset of w see 16 we solve here in the one dimensional case that is when n 1 this modified weyl berry conjecture we discover in the process some unexpected and intriguing connections between spectral geometry fractal geometry and the riemann zeta function we therefore show that one can hear that is recover from the spectrum not only the minkowski fractal dimension of the boundary as was established previously by the first author but also under the stronger assumptions of the conjecture its minkowski content a fractal analogue of its length we also prove still in dimension one a related conjecture of the first author as well as its converse which characterizes the situation when the error estimates of the aforementioned paper are sharp

vibration frequencies of a rotating tapered beam with end mass abstract the natural frequencies of vibration of a rotating tapered beam with tip mass are investigated explicit expressions for the finite element mass and stiffness matrices are derived by using a consistent mass formulation the beam is assumed to be linearly tapered in two planes the generalized eigenvalue problem is defined and numerical solutions are generated for a wide range of rotational speed and tip mass variations both fixed and hinged end conditions are considered comparisons are made whenever possible with exact solutions and numerical results available in the literature the results display high accuracy when compared with the exact solutions

exact bernoulli euler static stiffness matrix for a range of tapered beam columns the static stiffnesses for torsional and flexural deformation of a tapered beam under axial loading are determined analytically extending the bernoulli euler bessel function approach of banerjee and williams 1985 the derivation of the expressions is explained in detail and numerical results for sample problems are presented in tables and shown to be in excellent agreement with those obtained using stepped beams of 400 or 500 elements

natural frequencies of beams under tensile axial loads abstract the effect of a constant axial tensile force on natural frequencies and mode shapes of a uniform single span beam with different combinations of end conditions is presented numerical observations indicate that the variation of normalized natural frequency parameter with normalized tension parameter is almost the same for clamped pinned and pinned free and similarly for clamped clamped and clamped sliding beams the variation of the sliding free beam is only slightly different from that of the latter pair and the free free beam for pinned pinned pinned sliding and sliding sliding beams this variation may exactly be expressed as 1 u this formula may be used for beams with other types of end constraints when the beam vibrates in a third mode or higher it also gives the upper bound approximation to the fundamental natural frequency of a pinned free beam for beams with other types of boundary conditions this approximation may be expressed as 1 u where the coefficient depends only on the type of the end constraints it is found that when the dimensionless tension parameter u is greater than about 12 then u can be expressed as an analytical function of u where is the dimensionless natural frequency parameter for such a beam in the first few modes the natural frequency is independent of the flexural rigidity and the beam behaves like a string the string solution gives a lower bound approximation to the natural frequency

identifying multiple axial load patterns using measured vibration data abstract multiply redundant frames possess as many linearly independent axial load patterns as their degree of static indeterminacy any state of equilibrium can be described as the linear sum of these axial load patterns as a frame is loaded in the multiple patterns the changes in member axial loads affect the frame frequencies and mode shapes in a complicated way through geometric stiffening by representing this behaviour in a finite element model fem it is possible to measure the dynamic characteristics of a physical frame update the axial loads in the fem until the difference between measured and model frequencies is minimised and thus infer the member axial loads what is updated is the factors on the axial load patterns and this is done iteratively using traditional model updating newton s method thus all member axial loads can be identified using just a set of updating parameters equal in number to the degree of static indeterminacy a numerical simulation and for the first time an implementation on a physical realisation of a multiply redundant frame using measured vibration data are presented herein and show that member loads can be identified to an encouraging degree of accuracy beneficial strategies such as formulating the problem on an orthonormal basis for the axial load patterns and a modal tagging scheme which helps to conserve the strength of the necessary but otherwise typically compromised pairing of mode shapes of the fem to those measured are discussed

estimates for eigenvalues of quasilinear elliptic systems part ii abstract in this paper we find explicit lower bounds for dirichlet eigenvalues of a weighted quasilinear elliptic system of resonant type in terms of the eigenvalues of a single p laplace equation also we obtain asymptotic bounds by studying the spectral counting function which is defined as the number of eigenvalues smaller than a given value

lekhnitskii s classic formula serving as an exact mode shape of simply supported polar orthotropic inhomogeneous circular plates the semi inverse problem of free vibration of simply supported inhomogeneous polar orthotropic plates is studied the mode shape is postulated in the form of the classic formula by lekhnitskii namely the static deflection of the associated uniform polar orthotropic circular plate under uniform loading the ratios of the circumferential rigidity to the radial rigidity are identified as integer numbers so that the candidate mode shapes constitute the polynomial functions the flexural rigidities themselves are also represented by polynomials the semi inverse problem of identifying the coefficients of the flexural rigidities is solved analytically it appears remarkable that for numerous cases the simple method developed in this study provides novel closed form solutions for the design of the polar orthotropic circular plates with pre specified mode shapes

exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates we use reddy s third order plate theory to study buckling and steady state vibrations of a simply supported functionally gradient isotropic polygonal plate resting on a winkler pasternak elastic foundation and subjected to uniform in plane hydrostatic loads young s modulus and the poisson ratio for the material of the plate are assumed to vary only in the thickness direction effects of rotary inertia are considered the problem of determining the critical buckling load or the vibration frequency of the plate is found to be analogous to that of ascertaining the frequency of a membrane clamped at the edges and whose shape coincides with that of the plate the critical buckling load and the vibration frequency are shown to be positive some available results for plates symmetric about the mid plane can be retrieved from the present analysis

a new functional perturbation method for linear non homogeneous materials abstract a functional perturbation method fpm for solving boundary value problems of linear materials with non homogeneous properties is introduced the fpm is based on considering the unknown field such as displacements or temperatures as a functional of the non uniform property i e elastic modulus or thermal conductivity the governing differential equations are expanded functionally by frechet series leading to a set of differential equations with constant coefficients from which the unknown field is found successively to any desirable degree of accuracy a unique property of the fpm is that once the frechet functions are found the solution for any morphology is obtained by direct integration without re solving the differential equation for each case the fpm procedure is outlined first for general linear differential equations with non uniform coefficients then four examples are solved and discussed a 1d tensile loading of a rod with continuously varying and discontinuous moduli beam bending beam deflection on non uniform elastic foundation and a unidirectional heat conduction problem fpm results are compared with the exact if exists or numerical solution the fpm accuracy for the bending problem is also compared to the common rayleigh ritz and galerkin methods it is shown that the fpm is inherently more accurate since the convergence rate of the other methods depends on the arbitrarily chosen shape functions while in the fpm these functions are obtained as generic results of each order of the solution the fpm solution is analytical and is shown to be suitable for large variations in material properties thus a direct insight of each functional perturbation order is possible advantages and limitations of the fpm as compared to other existing methods are discussed in detail

using energy methods to compare linear vibration models abstract in this paper it is proved that the series representation of the solutions of the general linear vibration model with modal damping is valid with respect to the energy norm the partial sums of the series representation of solutions may therefore be used to compare different models parseval type relationships are obtained for the eigenfunction expansions of functions these relationships are used to derive expressions for the relative error in energy for partial sum approximations of functions an example is included where a string model wave equation model and a modified euler bernoulli beam model for the transverse vibration of a steel wire are compared

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

vibration and stability analysis of non uniform timoshenko beams under axial and distributed tangential loads two sets of governing equations for transverse vibration of non uniform timoshenko beam subjected to both axial and tangential loads have been presented in the first set the axial and tangential loads were taken perpendicular to the shearing force i e normal to the cross section inclined at an angle while in the second set the axial force is assumed to be tangential to the axis of the beam column for each case there exist a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending the two coupled second order governing differential equations were combined into one fourth order ordinary differential equation with variable coefficients the parameters of the frequency equation were determined for different boundary conditions the exact fundamental solutions could be found by expressing the coefficients of the reduced differential equation in a polynomial form before applying the frobenius method several illustrative examples of uniform and non uniform beams with various boundary conditions such as clamped supported elastically supported and free end mass and pinned end mass have been presented the stability analysis for the variation of the natural frequencies of the uniform and non uniform beams with the axial force has also been investigated moreover the present work illustrates the frequency behavior of the beam under a tangential load

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

euler bernoulli beams from a symmetry standpoint characterization of equivalent equations abstract we completely solve the equivalence problem for euler bernoulli equation using lie symmetry analysis we show that the quotient of the symmetry lie algebra of the bernoulli equation by the infinite dimensional lie algebra spanned by solution symmetries is a representation of one of the following lie algebras 2 a 1 a 1 a 2 3 a 1 or a 3 3 a 1 each quotient symmetry lie algebra determines an equivalence class of euler bernoulli equations save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters lineal mass density and flexural rigidity for each class we provide a simple representative and we explicitly construct transformations that maps a class member to its representative the maximally symmetric class described by the four dimensional quotient symmetry lie algebra a 3 3 a 1 corresponds to euler bernoulli equations homeomorphic to the uniform one constant lineal mass density and flexural rigidity we rigorously derive some non trivial and non uniform euler bernoulli equations reducible to the uniform unit beam our models extend and emphasize the symmetry flavor of gottlieb s iso spectral beams h p w gottlieb isospectral euler bernoulli beam with continuous density and rigidity functions proc r soc lond ser a math phys eng sci 413 1987 235 250

a closed form solution for the generalized euler problem euler9s problem that of buckling of a uniform column under its own weight is generalized to include columns of variable stiffness it is posed as an inverse problem namely determine the variation of the stiffness along the axis of the column so that the column has the prescribed buckling mode given as a simple polynomial function

inverse buckling problem for inhomogeneous columns abstract this study is motivated by our recent study elishakoff i rollot o 1999 new closed form solutions for buckling of a variable stiffness column by mathematica journal of sound and vibration 224 172 182 which presented new closed form solutions for buckling of columns with variable stiffness the column was simply supported at both ends in the present study we rederive in other means the same solution and moreover obtain new solutions for two other sets of the boundary conditions by posing an inverse buckling problem it is shown that the buckling load is dependent upon a single stiffness coefficient by suitable choice of this parameter the buckling load can be made arbitrarily large

vibration of viscoelastic beams subjected to an eccentric compressive force and a concentrated moving harmonic force abstract the problem of lateral vibration of a beam subjected to an eccentric compressive force and a harmonically varying transverse concentrated moving force is analyzed within the framework of the bernoulli euler beam theory the lagrange equations are used to examine the free vibration characteristics of an axially loaded beam and the dynamic response of a beam subjected to an eccentric compressive force and a moving harmonic concentrated force the constraint conditions of supports are taken into account by using the lagrange multipliers in the study trial function denoting the deflection of the beam is expressed in a polynomial form by using the lagrange equations the problem is reduced to the solution of a system of algebraic equations results of numerical simulations are presented for various combinations of the value of the eccentricity the eccentric compressive force excitation frequency and the constant velocity of the transverse moving harmonic force convergence studies are made the validity of the obtained results is demonstrated by comparing them with exact solutions based on the bernoulli euler beam theory obtained for the special cases of the investigated problem

observability and observers for nonlinear systems this paper deals with observability and observers for general nonlinear systems in the non control affine case we characterize systems that are observable independently of the inputs an exponential observer for these systems is also exhibited

determination of natural frequencies of a beam with varying section properties abstract transverse vibrations of a beam with uniform inertia and uniform area have been studied extensively by scientists and engineers and closed form solutions leading to simple formulae are readily available for estimating the natural frequencies of a beam when subjected to various types of boundary conditions however no such simple formulae exists for beams whose inertia cross sectional area and mass vary along the length in practice the use of beams with varying sectional properties such as inertia area and mass is common and to estimate the natural frequencies of these beams the engineer has to resort to more expensive methods such as the finite element method in this paper a procedure based on the galerkin method is employed to estimate the first few natural frequencies of a beam with varying sectional properties and including the effects due to rotary inertia in a particular instance the numerical values are computed and are verified by using finite element analysis also a formula based on effective inertia and effective area is proposed from which the natural frequencies can be estimated with reasonable accuracy this formula is simple and is similar to that used for a beam with uniform section properties also it enables one to evaluate the rotary inertia effect and the shear effect

model identification for spin networks abstract we consider the problem of determining the unknown parameters of the hamiltonian of a network of spin 1 2 particles in particular we study experiments in which the system is driven by an externally applied electro magnetic field and the expectation value of the total magnetization is measured under appropriate system theoretic assumptions we prove that if it is possible to prepare the system in a known initial state the above experiment allows to identify the parameters of the hamiltonian in the case where the initial state is itself an unknown parameter we characterize all the pairs hamiltonian initial state which give the same value of the magnetization for every form of the driving electro magnetic field the analysis is motivated by recent results on the isospectrality of hamiltonians describing magnetic molecules

on informational completeness of covariant localization observables and wigner coefficients an orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group from this a series of lemmas is derived showing the informational completeness of natural covariant localization operators as well as of the generalized wigner distributions matrix elements of the group some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators the results are applied to special phase space representations of the heisenberg affine and galilei groups

observability of systems on lie groups and coset spaces the purpose of this paper is to study the observability of a class of systems for which the state space is a lie group and the output space is a cosec space the study of this type of system was initiated by brockett siam j control 10 1972 pp 265 284 in this paper brockett s observability results are generalized and necessary and sufficient conditions for observability are obtained effective algorithms are established to verify such conditions finally as an application some disturbance decoupling problems are considered

motion control of drift free left invariant systems on lie groups in this paper we address the constructive controllability problem for drift free left invariant systems on finite dimensional lie groups with fewer controls than state dimension we consider small spl epsiv amplitude low frequency periodically time varying controls and derive average solutions for system behavior we show how the pth order average formula can be used to construct open loop controls for point to point maneuvering of systems which require up to p 1 iterations of lie brackets to satisfy the lie algebra controllability rank condition in the cases p 2 3 we give algorithms for constructing these controls as a function of structure constants that define the control authority i e the actuator capability of the system the algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with o spl epsiv sup p accuracy in general exactly if the lie algebra is nilpotent the methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs

degrees of controllability for quantum systems and application to atomic systems precise definitions for different degrees of controllability for quantum systems are given and necessary and sufficient conditions for each type of controllability are discussed the results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies

isospectral euler bernoulli beams with continuous density and rigidity functions isospectral systems are those that have exactly the same free vibration frequency spectrum with respect to a given boundary condition configuration in this paper seven different classes of inhomogeneous euler bernoulli beams with continuous density and flexural rigidity functions are found that are analytically solvable and that are isospectral with a homogeneous beam in the clamped clamped configuration of these exact solutions one class is isospectral with the homogeneous case in all ten distinct configurations obtainable from clamped sliding and two antiresonant end conditions whereas another class is isospectral in the six configurations with clamped hinged or free end conditions a connection with inverse problems is discussed