free vibration analysis of a timoshenko beam carrying multiple spring mass systems by using the numerical assembly technique from the equation of motion of the bare timoshenko beam without any spring mass systems attached an eigenfunction in terms of four unknown integration constants is obtained the substitution of the eigenfunction into the three compatible equations one force equilibrium equation and one governing equation for the sprung mass 1 n yields a matrix equation of the form b c 0 similarly when the eigenfunction is substituted into the two boundary condition equations at the left end and those at the right end of the beam one obtained bl cl 0 and br cr 0 respectively assembly of the coefficient matrices bl b and br will arrive at the eigen equation b c 0 where the elements of c are composed of the integration constants c i 1 n and i 1 4 and the modal displacements of the sprung mass z for a timoshenko beam carrying n spring mass systems the order of the overall coefficient matrix b is 5n 4 the solutions of b 0 where denotes a determinant give the natural frequencies of the constrained beam carrying multiple spring mass systems and the substitution of each corresponding values of c i into the associated eigenfunction at the attaching points will define the corresponding mode shapes in the existing literature the eigen equation b c 0 was denoted in explicit form and then solved analytically or numerically because of the lengthy explicit mathematical expressions the existing approach becomes impractical if the total number of the spring mass systems is larger than two but any number of spring mass systems will not make trouble for the numerical assembly technique presented in this paper copyright 2001 john wiley sons ltd

gauss and the invention of least squares the most famous priority dispute in the history of statistics is that between gauss and legendre over the discovery of the method of least squares new evidence both documentary and statistical is discussed and an attempt is made to evaluate gauss s claim it is argued though not conclusively that gauss probably possessed the method well before legendre but that he was unsuccessful in communicating it to his contemporaries data on the french meridian arc are presented that could conceivably permit a definitive verification of the claim

exact closed form solution for the vibration modes of the euler bernoulli beam with multiple open cracks abstract in this study exact closed form expressions for the vibration modes of the euler bernoulli beam in the presence of multiple concentrated cracks are presented the proposed expressions are provided explicitly as functions of four integration constants only to be determined by the standard boundary conditions the enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations besides the evaluation of the natural frequencies neither computational work nor recurrence expressions for the vibration modes are required the cracks that are not subjected to the closing phenomenon are modelled as a sequence of dirac s delta generalised functions in the flexural stiffness the eigen mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions which are already accounted for in the adopted flexural stiffness model the vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed form expressions in order to show their efficiency

free vibration analysis of a uniform multi span beam carrying multiple spring mass systems abstract the literature regarding the free vibration analysis of single span beams carrying a number of spring mass systems is plenty but that of multi span beams carrying multiple spring mass systems is fewer thus this paper aims at determining the exact solutions for the natural frequencies and mode shapes of a uniform multi span beam carrying multiple spring mass systems firstly the coefficient matrices for an intermediate pinned support an intermediate spring mass system left end support and right end support of a uniform beam are derived next the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system finally equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes in this paper the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made thus the last solutions are the exact ones the effects of attached spring mass systems on the free vibration characteristics of the 1 4 span beams are studied

principal eigenvalues and the dirichlet problem for fully nonlinear elliptic operators abstract we study uniformly elliptic fully nonlinear equations of the type f d 2 u d u u x f x we show that convex positively 1 homogeneous operators possess two principal eigenvalues and eigenfunctions and study these objects we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues in some cases only one of them are positive finally we obtain an existence result for nonproper isaac s equations

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

global properties of invariant measures abstract we study global regularity properties of invariant measures associated with second order differential operators in r n under suitable conditions we prove global boundedness of the density sobolev regularity a harnack inequality and pointwise upper and lower bounds

on the problem of minimum asymptotic exit rate for stochastically perturbed multi channel dynamical systems we consider the problem of minimizing the asymptotic exit rate with which the controlled diffusion process of a stochastically perturbed multi channel dynamical system exits from a given bounded open domain in particular for a class of admissible bounded linear feedback operators we establish a connection between the asymptotic exit rate with which such a controlled diffusion process exits from the given domain and the asymptotic behavior i e a probabilistic characterization of the principal eigenvalue of the infinitesimal generator which corresponds to the stochastically perturbed dynamical system with zero boundary conditions on the given domain finally we briefly remark on the implication of our result for evaluating the performance of the associated deterministic multi channel dynamical system when such a dynamical system is composed with a set of sub optimal admissible linear feedback operators

a unified matrix polynomial approach to modal identification abstract one important current focus of modal identification is a reformulation of modal parameter estimation algorithms into a single consistent mathematical formulation with a corresponding set of definitions and unifying concepts particularly a matrix polynomial approach is used to unify the presentation with respect to current algorithms such as the least squares complex exponential lsce the polyreference time domain ptd ibrahim time domain itd eigensystem realization algorithm era rational fraction polynomial rfp polyreference frequency domain pfd and the complex mode indication function cmif methods using this unified matrix polynomial approach umpa allows a discussion of the similarities and differences of the commonly used methods the use of least squares ls total least squares tls double least squares dls and singular value decomposition svd methods is discussed in order to take advantage of redundant measurement data eigenvalue and svd transformation methods are utilized to reduce the effective size of the resulting eigenvalue eigenvector problem as well

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

parameter estimation through ignorance dynamical modeling lies at the heart of our understanding of physical systems its role in science is deeper than mere operational forecasting in that it allows us to evaluate the adequacy of the mathematical structure of our models despite the importance of model parameters there is no general method of parameter estimation outside linear systems a relatively simple method of parameter estimation for nonlinear systems is introduced based on variations in the accuracy of probability forecasts it is illustrated on the logistic map the henon map and the 12 dimensional lorenz96 flow and its ability to outperform linear least squares in these systems is explored at various noise levels and sampling rates as expected it is more effective when the forecast error distributions are non gaussian the method selects parameter values by minimizing a proper local skill score for continuous probability forecasts as a function of the parameter values this approach is easier to implement in practice than alternative nonlinear methods based on the geometry of attractors or the ability of the model to shadow the observations direct measures of inadequacy in the model the implied ignorance and the information deficit are introduced

dynamic response of a viscously damped cantilever with a viscous end condition this paper deals with the dynamical analysis of a cantilevered bernoulli euler beam subjected to distributed external viscous damping in span and with a viscous end condition by a single damper in order to evaluate the vibration characteristics of the system a procedure is presented where overdamped and underdamped modes are investigated simultaneously via the dynamic stiffness matrix of the system further the orthogonality conditions which allow the decoupling of the equations of motion in terms of principal coordinates are derived then the complex frequency response function is obtained through a formula which was established for the receptance matrix previously finally the dynamic impulse response function of the system is evaluated in the time domain comparison with the numerical results obtained via a boundary value formulation justifies the approaches used here

physically based dirac s delta functions in the static analysis of multi cracked euler bernoulli and timoshenko beams abstract dirac s delta functions enable simple and effective representations of point loads and singularities in a variety of structural problems leading very often to elegant and otherwise unworkable closed form solutions this is the case of cracked beams under static loads whose theoretical and practical significance has attracted in recent years the interest of many researchers nevertheless analytical formulations currently available for this problem are not completely satisfactory either in terms of computational efficiency when the continuity conditions must be enforced with auxiliary equations or in terms of physical consistency when the singularities in the beam s flexural rigidity are represented with dirac s delta functions having a questionable negative sign these considerations motivate the present study which offers a novel and physically based modelling of slender euler bernoulli beams and short timoshenko beams with any number and severity of cracks conducing in both cases to exact closed form solutions for validation purposes a standard finite element code is used along with two nascent deltas uniform and gaussian density functions to describe a smeared increase in the bending flexibility around the abscissa of the crack

complex modal analysis of rods with viscous damping devices abstract the complex modal analysis of rods equipped with an arbitrary number of viscous damping devices is addressed the following types of damping devices are considered external grounded spring damper attached mass spring damper and internal spring damper within a standard 1d formulation of the vibration problem the theory of generalized functions is used to model axial stress and displacement discontinuities at the locations of the damping devices by using the separate variable approach a simple solution procedure of the motion equation leads to exact closed form expressions of the characteristic equation and eigenfunctions which inherently fulfill the required matching conditions at the locations of the damping devices based on the characteristic equation a closed form sensitivity analysis of the eigensolution is implemented the displacement eigenfunctions exhibit orthogonality conditions they can be used with the complex mode superposition principle to tackle forced vibration problems and in conjunction with the stress eigenfunctions to build the exact dynamic stiffness matrix of the rod for complex modal analysis of truss structures numerical results are discussed for a variety of parameters

a closed form solution of a bernoulli euler beam on a viscoelastic foundation under harmonic line loads fourier transform is used in this paper to solve the problem of steady state response of a beam on a viscoelastic foundation subjected to a harmonic line load the solution is constructed in the form of the convolution of the green function of the beam the theorem of residue is employed to evaluate the generalized integral such that a closed form solution can be achieved all the different combinations of damping and vibration frequency are discussed and analytical solutions are presented as a special case the solution of the beam on a winkler foundation is also discussed the validation of the solution is verified by considering the static solution of the beam and comparing the degraded solution to a well known result the closed form expression of the result can be used to construct algorithms for the inverse problems of non destructive testing of pavement structures using vibration devices

synchronization of coupled self excited elastic beams abstract the behavior of four coupled self excited elastic beams is numerically studied using the finite difference method self excitation is modeled by adding a van der pol damping term coupling based on shared boundary conditions at the roots of the beams is proposed and the influence of the coupling on the dynamics is analyzed time and space correlation coefficients are employed to measure the degree of synchronization among beams through the power spectral density multiple frequencies of the beam tip time series are determined unveiling the complex nature of the beam motions finally the robustness of the proposed behavior to changes in fluid structure interaction and the mass of the beams is determined

free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method abstract the natural frequencies and mode shapes of a uniform cantilever beam carrying any number of concentrated masses were determined by using an analytical and numerical combined method anc method the eigenvalue equation was first derived analytically by using the expansion theorem and then the eigenvalues and eigenvectors were calculated numerically in comparison with the general finite element method fem the anc method has the advantage that there is no necessity to derive the property matrices of each beam element and then to develop the overall ones of the entire beam in comparison with the transfer matrix method tmm the anc method requires no tedious matrix multiplication so that some computer time may be saved the anc method is also better than the pure analytical closed form method since instead of a few special cases a uniform beam carrying any number of point masses of various magnitudes or distributions along the beam length can be easily solved besides by using the anc method there is no difficulty in taking more modes in the mode superposition equation and hence more higher mode natural frequencies of better accuracy may be obtained

on a controlled eigenvalue problem abstract we consider the problem of minimizing the asymptotic exit rate of a diffusion from a bounded domain and derive the associated hamilton jacobi bellman equation this turns out to be a nonlinear eigenvalue problem a verification theorem for optimal control is also provided

on applications of generalized functions to beam bending problems using a mathematical approach this paper seeks an eacient solution to the problem of beams bending under singular loading conditions and having various jump discontinuities for two instances the boundary value problem that describes beam bending cannot be written in the space of classical functions in the first instance the beam is under singular loading conditions such as point forces and moments and in the second instance the dependent variable s and its derivatives have jump discontinuities in the most general case we consider both instances first we study singular loading conditions and present a theorem by which the equivalent distributed force of a general class of singular loading conditions can be found as a consequence of obtaining the equivalent distributed force of a distributed moment we find a mathematical explanation for the corner condition in classical plate theory while plate theory is not the focus of this paper this explanation is interesting then beams with various jump discontinuities are considered when beams have jump discontinuities the form of the governing diaerential equations changes we find the governing diaerential equations in the space of generalized functions it is shown that for euler bernoulli beams with jump discontinuities the operator of the diaerential equation remains unchanged only the force term changes so that delta function and its distributional derivatives appear within it but for timoshenko beams with jump discontinuities in addition to changes in the force terms the operator of one of the governing diaerential equations changes we then propose a new method for solving these equations this method which we term the auxiliary beam method is to solve the governing diaerential equations not in the space of generalized functions but rather to solve them by means of solving equivalent boundary value problems in the space of classical functions the auxiliary beam method reduces the number of diaerential equations and at the same time obviates the need to solve these diaerential equations in the space of generalized functions which can be more diacult 7 2000 elsevier science ltd all rights reserved

modeling and nonlinear parameter estimation with kronecker product representation for coupled oscillators and spatiotemporal systems we focus here on the question of global modeling that is building an ordinary differential equation ode of minimal dimensions which models a given multivariate time dependent data set we give proof of convergence of the given method we will furthermore introduce a new method which adapts the least squares best approximation by a kronecker product representation c f van loan n pitsianis approximation with kronecker products www math tkk fi mhuhtane kronecker ps to analyze underlying structure when it exists as a system of coupled oscillators several examples are presented including diffusively coupled chaotic systems some interesting prospective problems will be briefly discussed in the conclusions

bifurcation and optimal stochastic control abstract two questions concerning bifuraction theory and optimal stochastic control are considered first in a few examples we give the interpretation of a bifurcation in terms of optimal stochastic control next we introduce the analogue of the lowest eigenvalue for the nonlinear operator associated with the hamilton jacobi bellman equations of optimal stochastic control author

closed form solutions for frequency equation and mode shapes of elastically supported euler bernoulli beams abstract this paper is concerned with the problem of free vibration of an euler bernoulli beam supported with an arbitrary number of translational springs of different constant stiffnesses the natural frequencies and mode shapes are calculated by means of the green s function method seven different boundary conditions are considered at both ends of the beam fixed pinned sliding free translational spring rotational spring and combined translational rotational spring each boundary condition on one end is combined with all other boundary conditions on the other yielding a total of 49 possible cases it has been shown in this paper that a green s function for any set of boundary conditions can be represented as a linear combination of function coefficients calculated for homogeneous boundary conditions and also a new form of the general solution is formulated using this principle green s functions are calculated and tabulated for each of the 49 possible cases which in essence closes the question of analytical solutions for this particular kind of problem using green s functions the tabulated solutions were used as the starting point in the derivation of the general formulae and to provide a new insight into the underlying mathematical structure of the solutions formulated with green s functions this can be of practical interest to the end user who wishes to implement them in computer code the validity of the obtained solutions is demonstrated with examples whose solutions are compared to the results in the published literature and to the results obtained with fem

coupled bending torsional dynamic stiffness matrix for beam elements explicit expressions for the coupled bending torsional dynamic stiffness matrix of a uniform beam element are derived in an exact sense by solving the governing differential equation of the beam implementation of the derived dynamic stiffness matrix in a space frame computer program is discussed with particular reference to an established algorithm to enable vibration analysis of coupled systems to be made the application of the theory is demonstrated by an illustrative example wherein the results for a cantilever beam with a substantial amount of coupling between bending and torsion are highlighted the correctness of the theory is confirmed to a high degree of accuracy by computed results and numerical checks

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

a note on the vibrations of restrained beams and rods with point masses abstract this paper is concerned with the approximate determination of the fundamental frequency and first mode shape of a beam or rod to which springs and point masses are attached after formulating the general eigenvalue problem which forms the basis of the analysis the theory is applied to three examples and it is shown that the results are quite satisfactory

use of synchronization and adaptive control in parameter estimation from a time series a technique is introduced for estimating unknown parameters when a time series of only one variable from a multivariate nonlinear dynamical system is given the technique employs a combination of two different control methods linear feedback for synchronizing system variables and adaptive control and is based on dynamic minimization of synchronization error the technique is shown to work even when the unknown parameters appear in the evolution equations of the variables other than the one for which the time series is given the technique not only establishes that explicit detailed information about all system variables and parameters is contained in a scalar time series but also gives a way to extract it out under suitable conditions illustrations are presented for lorenz and rossler systems and a nonlinear dynamical system in plasma physics also it is found that the technique is reasonably stable against noise in the given time series and the estimated value of a parameter fluctuates around the correct value with the error of estimation growing linearly with the noise strength for small noise

multi cracked euler bernoulli beams mathematical modeling and exact solutions localized flexibility models of cracks enable one for simple and effective representation of the behavior of damaged beams and frames important applications such as the determination of closed form solutions and the development of diagnostic methods of analysis have attracted the interest of many researchers in recent years nevertheless certain fundamental questions have not been completely clarified yet one of these issues concerns with the mechanical justification of the macroscopic model of rotational elastic spring commonly used to describe the presence of an open crack in a beam under bending deformation two main analytical formulations have been recently proposed to take into account the singularity generated by the crack the crack is represented by suitable dirac s delta functions either in the beam s flexural rigidity or in the beam s flexural flexibility both approaches require some caution due to mathematical subtleties of the analysis motivated by these considerations in this paper we propose a justification of the rotational elastic spring model of an open crack in a beam in bending deformation we show that this localized flexibility model of a crack is the variational limit of a family of one dimensional beams when the flexural stiffness of these beams tends to zero in an interval centered at the cracked cross section and simultaneously the length of the interval vanishes in a suitable way we also show that the static and dynamic problem for the flexibility model of cracked beam can be formulated within the classical context of the theory of distributions avoiding the hindrances encountered in previous approaches to the problem in addition the proposed treatment leads to a simple and efficient determination of exact closed form solutions of both static and dynamic problems for beams with multiple cracks

dynamic analysis of a multi span uniform beam carrying a number of various concentrated elements abstract this paper employs the numerical assembly method nam to determine the exact frequency response amplitudes of a multiple span beam carrying a number of various concentrated elements and subjected to a harmonic force and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force first the coefficient matrices for the intermediate concentrated elements pinned support applied force left end support and right end support of a beam are derived next the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method fem finally the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix the graph of dynamic response amplitudes versus various exciting frequencies gives the frequency response curve for any point of a multiple span beam carrying a number of various concentrated elements for the case of zero harmonic force the above mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained

derivation of structural design sensitivities from vibration test data abstract structural sensitivities are often required in dynamic analyses of engineering structures such as system identification and control structural modification and optimization many different methods have been developed in the last three decades for the efficient computation of such sensitivities though these existing methods have proven to be very useful tools to structural analysts they are restricted to those cases where accurate analytical or finite element models are available in many practical applications where sensitivities are needed in the solution of troubleshooting problems only limited measured data are available existing methods being inapplicable in this paper a new and effective method has been developed to derive structural design sensitivities both frequency response function sensitivities and eigenvalue and eigenvector sensitivities from limited vibration test data design sensitivities calculated directly from measured data are more accurate than those calculated from analytical or finite element models since structural modelling errors are inevitable due to the complexity of almost all engineering structures the relationship between frequency response function sensitivities and eigenvalue and eigenvector sensitivities has been established this relationship provides the theoretical basis for the experimental determination of eigenvalue and eigenvector sensitivities the effect of residue contribution of out of range modes upon analysis accuracy has been examined detailed numerical assessment based on a turbine bladed disk model as well as experimental investigation of a beam structure have been conducted and the results have demonstrated the practicality of the proposed method

better nonlinear models from noisy data attractors with maximum likelihood a new approach to nonlinear modelling is presented which by incorporating the global behaviour of the model lifts shortcomings of both least squares and total least squares parameter estimates although ubiquitous in practice a least squares approach is fundamentally flawed in that it assumes independent normally distributed ind forecast errors nonlinear models will not yield ind errors even if the noise is ind a new cost function is obtained via the maximum likelihood principle superior results are illustrated both for small data sets and infinitely long data streams

a hierarchical bayes ensemble kalman filter abstract a new ensemble filter that allows for the uncertainty in the prior distribution is proposed and tested the filter relies on the conditional gaussian distribution of the state given the model error and predictability error covariance matrices the latter are treated as random matrices and updated in a hierarchical bayes scheme along with the state the hyper prior distribution of the covariance matrices is assumed to be inverse wishart the new hierarchical bayes ensemble filter hbef assimilates ensemble members as generalized observations and allows ordinary observations to influence the covariances the actual probability distribution of the ensemble members is allowed to be different from the true one an approximation that leads to a practicable analysis algorithm is proposed the new filter is studied in numerical experiments with a doubly stochastic one variable model of truth the model permits the assessment of the variance of the truth and the true filtering error variance at each time instance the hbef is shown to outperform the enkf and the henkf by myrseth and omre 2010 in a wide range of filtering regimes in terms of performance of its primary and secondary filters

statistical methods of parameter estimation for deterministically chaotic time series we discuss the possibility of applying some standard statistical methods the least square method the maximum likelihood method and the method of statistical moments for estimation of parameters to deterministically chaotic low dimensional dynamic system the logistic map containing an observational noise a segmentation fitting maximum likelihood ml method is suggested to estimate the structural parameter of the logistic map along with the initial value x 1 considered as an additional unknown parameter the segmentation fitting method called piece wise ml is similar in spirit but simpler and has smaller bias than the multiple shooting previously proposed comparisons with different previously proposed techniques on simulated numerical examples give favorable results at least for the investigated combinations of sample size n and noise level besides unlike some suggested techniques our method does not require the a priori knowledge of the noise variance we also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed bayesian approaches we note the trade off between the need of using a large number of data points in the ml analysis to decrease the bias to guarantee consistency of the estimation and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition the method of statistical moments for the estimation of the parameter of the logistic map is discussed this method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically not only numerically

a closed form solution of a longitudinal bar with a viscous boundary condition abstract a closed form solution of a longitudinal bar with a viscous boundary condition subjected to point loading is developed in this paper a new series solution is formulated that allows the time and space modes of the beam to decouple this expansion yields explicit eigenvalues and eigenvectors a frequency domain example is presented and the results are compared with finite element solutions of the same problem it is shown that the closed form solution is computationally more efficient than a finite element solution additionally the truncation error at lower frequencies is shown to be extremely small the method is easily implemented and can provide time and frequency domain solutions to this class of problems

free and forced vibrations of a restrained uniform beam carrying an intermediate lumped mass and a rotary inertia abstract the free and forced vibrations of a uniform beam elastically restrained against rotation at one end and against translation at the other end and carrying a lumped mass with rotary inertia and external loading at an arbitrary intermediate point are analyzed the analysis involves solving the exact eigenvalue problem for the base beam to obtain the exact mode shape functions which account for the effects of the elastic restraints at the beam ends these mode shape functions are then used in conjunction with galerkin s method with the lumped mass and rotary inertia treated as applied loads to obtain the approximate eigenvalue parameters and resonance response of the beam mass system parametric studies are carried out to investigate the effects of the stiffness of elastic restraints and location and magnitude of the lumped mass and rotary inertia on the eigenfrequency parameters and resonance response of the beam mass system partial computational results are compared with existing data the agreement is generally good for convenience the results are presented in dimensionless form

on the numerical convergence of discretization methods for the free vibrations of beams with attached inertia elements abstract this paper is concerned with a numerical convergence study of three discretization methods rayleigh ritz galerkin and finite element as applied to the analysis of free bending linear vibrations of a uniform bernoulli euler beam carrying inertia elements at intermediate points a cantilever beam carrying a lumped mass with rotary inertia at an arbitrary intermediate point and another at the beam tip is used as a case study the first five frequency parameters of this beam obtained by using the rayleigh ritz method in conjunction with broadly admissible base beam eigenfunctions galerkin s method in conjunction with strictly admissible base beam eigenfunctions and the classical finite element method are compared with the exact ones over a wide range of values of the system parameters conclusions are drawn regarding the effects of the various system parameters on the accuracy and convergence characteristics of the above three discrete methods and on the mode shapes and the first five frequency parameters of the beam system

eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems increasing the number of degrees of freedom used in finite element analysis for mechanical and structural systems with viscoelastic damping the need to consider the modal truncation problem of viscoelastic systems is more than ever before the higher modes may be unnecessary to obtain in dynamic analysis for engineering applications for viscoelastic systems the modal truncation problem may be more frequently encountered since the nonviscous modes are difficult or even impossible to be found accurately even if a small scaled problem is considered for some eigensolution methods this study aims at eliminating the influence of the higher modes on the frequency responses of viscoelastically damped systems a method is presented by making the equilibrium equations of motion into a subspace equation spanned in terms of the columns of a projection basis obtained by considering the use of the contribution of the lower modes and the first two terms of the neumann expansion of the contribution of the unavailable modes finally three example studies are provided to illustrate the effectiveness of the derived results it is shown that the proposed method can reduce the modal truncation error significantly

an exact generalised function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping abstract this paper concerns the frequency response analysis of beams and plane frames with an arbitrary number of kelvin voigt viscoelastic dampers typical external and internal dampers are considered as grounded translational tuned mass rotational and axial dampers for bending and axial vibrations respectively using the theory of generalised functions within a 1d formulation of equations of motion exact closed form expressions are derived for beam dynamic green s functions and frequency response functions under arbitrary polynomial load for any number of dampers for a plane frame exact global frequency response matrix and load vector are built with size depending only on the number of beam to column nodes for any number of dampers and point polynomial loads along the frame members from the nodal displacement solution the exact frequency response in all frame members is also obtained in closed analytical form numerical applications show many of the advantages of the proposed method

complex modal analysis of a flexural vibrating beam with viscous end conditions abstract the complex mode superposition method for the dynamical analysis of a simply supported beam with two rotational dampers attached at its end is presented first a numerical procedure for the evaluation of complex frequencies and modes of vibration has been worked out second the appropriate orthogonality conditions have been established in order to decouple the equation of motion the complex mode superposition method has then been used for the dynamic analysis in the time and frequency domains finally some numerical applications under impulsive and harmonic transverse accelerations of both ends have been reported in order to illustrate the effects of the dampers on the response amplitude

free vibrations of uniform timoshenko beams with attachments the problem of free transverse vibrations of timoshenko beams with attachments like translational and rotational springs concentrated mass including the moment of inertia linear undamped oscillators and additional supports is considered the frequency equation for the combined system is derived by means of the lagrange multiplier formalism the exact solution of the free vibration problem of the beam without attachments is taken into account for the formulation of the free vibration problem of the combined system numerical examples show the separate or coupling influences of the additional elements on the combined system s frequencies the comparison of results obtained by using the present approach with results of the exact solution indicates a good agreement

an eigenvalue approach to the risk sensitive control problem in near monotone case abstract risk sensitive control problem under near monotonicity condition is considered we prove existence of a solution to the corresponding hamilton jacobi bellman equation by an eigenvalue approach existence of an optimal control has also been proved

an exact modal analysis approach to vibration analysis of structures with mass spring subsystems and rotational joints abstract a novel exact modal analysis approach is presented for vibration analysis of plane continuous structures which are coupled with discrete mass spring subsystems and include elastic rotational joints modelling local flexibility using the theory of generalised functions to handle the discontinuities of the response variables every continuous member with any number of mass spring subsystems and joints is treated as a two node element for which a 6 6 exact dynamic stiffness matrix is obtained in closed form as a result the global dynamic stiffness matrix is built by a standard finite element assembling procedure with size depending only on the number of nonzero nodal degrees of freedom of member to member nodes upon deriving pertinent orthogonality conditions for the modes the system response under arbitrary loads is obtained by modal impulse and modal frequency response functions under the assumption of proportional damping the solutions are exact and can be used as benchmark for classical finite element solutions the approach is formulated for various mass spring subsystems acting in transverse and axial directions relative to every member

dynamic analysis of a uniform cantilever beam carrying a number of elastically mounted point masses with dampers abstract the free and forced vibration analyses of a uniform cantilever beam carrying a number of spring damper mass systems with arbitrary magnitudes and locations were made by means of the analytical and numerical combined method ancm first of all a method was presented to replace each spring damper mass system by a massless equivalent spring damper system with effective spring constant keffand effective damping coefficient ceffso that the ancm is available for the title problem next the equation of motion for the constrained beam with spring damper mass systems attached was derived by using the natural frequencies and normal mode shapes of the unconstrained beam without carrying any attachments incorporating the expansion theorem finally the eigenvalues and the forced vibration responses of the constrained beam were determined by conventional numerical methods to confirm the reliability of the presented theory all the numerical results obtained from the ancm were compared with the corresponding ones obtained from the conventional finite element method fem and good agreement was achieved the influence of the damping magnitude of each spring damper mass system on the eigenvalues and the forced vibration responses of the constrained beam was studied because the order of the overall property matrices for the equation of motion of the constrained beam derived from the ancm is much lower than that from the conventional fem the storing memory and the cpu time required by the ancm are much less than those required by the fem

modal analysis of multi span timoshenko beams connected or supported by resilient joints with damping the present paper proposes an exact modelling and modal analysis method for non uniform multi span beam type structure supported and or connected by resilient joints with damping to this end an exact dynamic matrix for a timoshenko beam element is derived by means of the spatial domain laplace transform a generalized modal analysis method is also proposed and applied to the derivation of frequency response and time response formulas for general beam structures three examples are provided for validating and or illustrating the proposed method in the first numerical example the proposed method is compared with fem the second example deals with a three stepped beam structure supported by joints with damping property in the final example a dynamic analysis of a multi span beam under moving load is demonstrated the numerical study proves that the proposed method is useful for the dynamic analysis of multi span beam type structures

vibration of beams with arbitrary discontinuities and boundary conditions abstract a general solution of the vibration of an euler bernoulli beam with arbitrary type of discontinuity at arbitrary number of locations is presented in this paper to account for the discontinuity term induced by various additional elements on the beam heaviside s function is used to express the modal displacement of the whole beam by a single function this general modal displacement function is then solved by using laplace transformation this general solution consists of four types of basic modal shapes induced by four corresponding types of discontinuity terms at the discontinuity points various discontinuity terms are obtained and expressed by the boundary values of the modal displacement in a recursive way consequently the modal displacement can be determined by examining only the conditions on the boundary in such a way the present solution reduces the vibration of beams with arbitrary discontinuities to the same order of the case without discontinuity point to demonstrate the efficiency and applicability of the present method three application examples are presented calculation example shows that the lead zirconate titanate pzt actuator should be placed as close to the fixed end as possible to achieve the best excitation effect on a cantilever beam a new method to calculate the driving point anti resonance frequency is also proposed numerical results suggest that the variation of driving point anti resonance frequency can be used to determine the location and size of crack in beams due to the generic nature of the solution and the problem the present method can be utilized in smart structures modeling and structural health monitoring of beam type structures

the exact explicit dynamic stiffness matrix of multi cracked euler bernoulli beam and applications to damaged frame structures abstract in this paper the closed form expression of the exact dynamic stiffness matrix of an euler bernoulli beam in the presence of an arbitrary number of concentrated cracks is derived the procedure adopted for the evaluation of the dynamic stiffness matrix is based on the availability of the exact closed form solution of the vibration modes of the multi cracked beam derived by the same authors in a previous paper the knowledge of the exact explicit dynamic stiffness matrix of the multi cracked beam makes the direct evaluation of the exact global dynamic stiffness matrix of damaged frame structures possible furthermore it allows the exact evaluation of the frequencies and the corresponding vibration modes consistent with the distributed parameter model through the application of the well known wittrick williams algorithm some numerical applications relative to the evaluation of frequencies and the corresponding mode shapes of multi cracked framed structure are reported furthermore the closed form solution has been validated by comparing with some exact results available in the literature for a simple single cracked frame finally further new results for a multi cracked frame have been compared with those obtained by a finite element simulation

vibration frequencies of a rotating flexible arm carrying a moving mass abstract a clamped free flexible beam rotating in a horizontal plane and carrying a moving mass is modelled by the euler bernoulli beam theory the equation of motion is derived by hamilton s principle including the effects of centrifugal stiffening arising from the rotation of the beam the motion of the moving mass and the beam is coupled the equation of motion is a coupled non linear partial differential equation where the coupling terms have to be evaluated at the position of the moving mass in order to obtain the mode shapes which account for the motion of the moving mass the solution is discretized into space and time functions and the beam is divided into two separate regions with respect to the moving mass this results in two non homogeneous linear mode shape ordinary differential equations with four boundary one discontinuity and three continuity conditions the power series method is used to solve for the mode shape differential equations a frequency equation is derived giving the relationship between the non dimensional modal frequencies and the four non dimensional parameters i e the moving mass position the moving mass the beam angular velocity and the total moment of inertia about the hub the numerical bisection method is used to solve for the vibration frequencies under different parameters results are presented for the first three modes of vibration

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

on the eigenfrequencies of a cantilever beam with attached tip mass and a spring mass system this paper deals with the derivation of the frequency equation of a special combined dynamic system it consists of a clamped free bernoulli euler beam with a tip mass where a spring mass system is attached to it the derivation of the frequency equation is essentially carried out by means of the lagrange multipliers method frequency equations of some simpler systems are also obtained from the general expression by using limiting processes these equations are then numerically solved for various combinations of physical parameters the comparison with the results from the solution of the corresponding bernoulli euler frequency equations shows that the frequency equations established are very accurate

likelihood and bayesian prediction of chaotic systems abstract there has recently been considerable interest in both applied disciplines and in mathematics as well as in the popular science literature in the areas of nonlinear dynamical systems and chaotic processes by a nonlinear deterministic dynamical system we mean a time series in which starting at some initial condition the values of the series are some fixed nonlinear function of the previous states one of the more intriguing aspects of these models is their propensity for displaying very complex apparently random behavior even when simple models are analyzed a consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system the difficulty in prediction stems from the fact that even the tiniest of errors including computer roundoff in either the specification of the function or the initial condition can lead to huge errors in prediction after a brief review of dynamical systems and the role of probability in dealing with uncertainty a com

on the asymptotic estimates for exit probabilities and minimum exit rates of diffusion processes pertaining to a chain of distributed control systems in this paper we consider a diffusion process pertaining to a distributed control system formed by a chain of n subsystems with n ge 2 where a random perturbation enters only in the first subsystem and is then subsequently transmitted to the other subsystems we also assume that for any ell in 2 ldots n the distributed control system formed by the first ell subsystems i e from the first subsystem up to the ell th subsystem satisfies an appropriate hormander condition as a result the diffusion process is degenerate in the sense that the backward operator associated with it is a degenerate parabolic equation in particular we consider the following two problems i we provide an asymptotic estimate for the exit probability with which the diffusion process corresponding to a particular subsystem exits from a given bounded open domain during a certain time interval the approach for such an asymptotic estimate basically relies on the interpretation of the exit probability

a level set approach for damage identification of continuum structures based on dynamic responses abstract the present paper aims to propose a novel approach for damage identification of continuum structures based on their dynamic performances the main idea is resorting to the level set model which is used to describe the shape and topology of the damage regions implicitly the original natural frequency based inverse problem is thus transferred into a generalized shape optimization problem which can be tackled by solving an evolution type hamilton jacobi equation compared to traditional approaches the distinctive advantage of the proposed approach is that it can deal with damage regions of complex shapes in a convenient way numerical examples demonstrate the effectiveness of the proposed approach

dynamics of multiple cracked prismatic beams with uncertain but bounded depths under deterministic and stochastic loads abstract dynamic response of cracked beams changes with respect to undamaged ones due to the change in mechanical properties moreover usually the crack characteristics crack size crack position applied stress etc are assumed to be known however in practice they possess considerable scatter of uncertainty the aim of this paper is to evaluate the time varying upper and lower bounds of the response of damaged beams with multiple cracks subjected to deterministic or stochastic excitations assuming uncertain crack depths as interval variables in this study the analysis is performed adopting a finite element model for the damaged beam extended to the case where multiple non propagating cracks are presented in order to provide the bounds of the response the authors propose an approach developed in the time domain starting from the derivation of the bounds of the interval eigenvalues evaluated as solution of two appropriate deterministic eigenvalue problems to assess the efficiency and the accuracy of the presented method a damaged cantilever prismatic aluminum beam with multiple uncertain but bounded cracks is examined the bounds of the free end displacement and variance are calculated and compared with the exact solution via combinatorial procedure the proposed results show a high level of accuracy for both increasing amplitude value of interval deviation and increasing number of uncertain parameters

estimation of noise levels for models of chaotic dynamical systems we investigate how far it is possible to identify and separate dynamical noise from measurement noise in observed nonlinear time series using bayesian methods we derive estimates for the two noise levels and find that given a good model of the dynamics these can give accurate results even if the dynamical noise level is orders of magnitude smaller than the measurement noise level whereas a simple calculation of root mean square error badly understates the dynamical noise we argue that this allows better estimates of the underlying dynamical time series and so better predictions of its future and of its fundamental dynamical properties

on euler bernoulli discontinuous beam solutions via uniform beam green s functions abstract the bending problem of euler bernoulli discontinuous beams is dealt with the purpose is to show that uniform beam green s functions can be used to build efficient solutions for beams with internal discontinuities due to along axis constraints and flexural stiffness jumps specifically upon deriving the equilibrium equation in the space of generalized functions first it is seen that the original bending problem may be recast as linear superposition of a principal and an auxiliary bending problem both involving a uniform reference beam and homogeneous boundary conditions then based on the green s functions of the reference beam closed form solutions are developed for the principal beam response while the auxiliary beam response is obtained by solving in general r 2 s algebraic equations written at the discontinuity locations being r the number of discontinuities due to along axis constraints and s the number of flexural stiffness jumps in this manner an appreciable reduction of computational effort is achieved as compared to alternative analytical solutions in the literature

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

density estimates for a random noise propagating through a chain of differential equations we here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d the first one being forced by a non degenerate random noise and the n 1 other ones being degenerate the system formed by the n equations satisfies a suitable hormander condition the second equation feels the noise plugged into the first equation the third equation feels the noise transmitted from the first to the second equation and so on so that the noise propagates one way through the system when the coefficients of the system are lipschitz continuous we show that the density of the solution satisfies gaussian bounds with non diffusive time scales the proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem

on the frequency response function of a damped cantilever simply supported in span and carrying a tip mass abstract this paper deals with the determination of the frequency response function of a cantilevered bernoulli euler beam which is viscously damped by a single damper the beam is simply supported in span and carries a tip mass the frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear systems subjected to linear constraint equations by considering the simple support as a linear constraint imposed on generalized co ordinates the comparison of the numerical results obtained via a boundary value problem formulation justifies the approach used here

indistinguishable states i perfect model scenario an accurate forecast of a nonlinear system will require an accurate estimation of the initial state it is shown that even under the ideal conditions of a perfect model and infinite past observations of a deterministic nonlinear system uncertainty in the observations makes exact state estimation is impossible consistent with the noisy observations there is a set of states indistinguishable from the true state this implies that an accurate forecast must be based on a probability density on the indistinguishable states this paper shows that this density can be calculated by first calculating a maximum likelihood estimate of the state and then an ensemble estimate of the density of states that are indistinguishable from the maximum likelihood state a new method for calculating the maximum likelihood estimate of the true state is presented which allows practical ensemble forecasting even when the recurrence time of the system is long in a subsequent paper the theory and practice described in this paper are extended to an imperfect model scenario 2001 published by elsevier science b v

a new analytical technique for vibration analysis of non proportionally damped beams abstract vibrating linear mechanical systems in particular continuous systems are often modelled considering proportional damping distributions only although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy in this paper an analytical method is given to take into account the effects of a more general viscous damping model referred to as non proportional damping on a class of vibrating continuous systems a state form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form i e complex modes corresponding to non synchronous motions numerical examples are included in order to show the efficiency of the proposed method non proportional damping distributions of different type such as internal and external lumped or distributed viscous damping are tested on non homogeneous euler bernoulli beams in bending vibration with different boundary conditions finally a discussion on root locus diagrams behaviour and on modal damping ratio significance for non proportionally damped systems is presented

small noise asymptotics for invariant densities for a class of diffusions a control theoretic view abstract we consider multi dimensional nondegenerate diffusions with invariant densities with the diffusion matrix scaled by a small 0 the o d e limit corresponding to 0 is assumed to have the origin as its unique globally asymptotically stable equilibrium using control theoretic methods we show that in the 0 limit the invariant density has the form exp w x 2 where the w is characterized as the optimal cost of a deterministic control problem this generalizes an earlier work of sheu extension to multiple equilibria is also given

eigenvalue and dirichlet problem for fully nonlinear operators in non smooth domains we study the maximum principle the existence of eigenvalue and the existence of solution for the dirichlet problem relative to operators which are fully nonlinear elliptic but presenting some singularity or degeneracy which are similar to those of the p laplacian we consider the equations in bounded domains which only satisfy the exterior cone condition

a new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses abstract in theory one may obtain five equations from each attaching point of a spring mass system and two boundary equations from each end of the uniform beam hence for a uniform beam carrying n spring mass systems simultaneous equations of the form b c 0 will be obtained the solutions of b 0 where represents a determinant give the natural frequencies of the constrained beam and the substitution of each corresponding values of c j j 1 4 into the eigenfunction will define the associated mode shapes while the foregoing theory is simple the lengthy explicit mathematical expressions become impractical if the total number of spring mass systems is larger than two for this reason it was applied to do the free vibration analysis of a uniform beam carrying one spring mass system only in the existing literature the purpose of this paper is to present a numerical technique to apply the foregoing theory to obtain the exact solutions for the lowest several natural frequencies and mode shapes of a uniform beam carrying any number of spring mass systems with various boundary conditions to this end each integration constant c vi and each mode displacement z v v 1 n i 1 4 at the attaching point and the two ends of the beam are considered as nodal displacements of a finite beam element and are assigned an appropriate degree of freedom dof hence each associated coefficient matrix will be equivalent to the stiffness matrix of a beam element and the conventional numerical assembly technique for the finite element method fem may be used to determine the overall coefficient matrix b therefore the eigenvalue equation b c 0 is easily obtained

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

on the natural frequencies and mode shapes of a multispan timoshenko beam carrying a number of various concentrated elements abstract the purpose of this paper is to utilize the numerical assembly method nam to determine the exact natural frequencies and mode shapes of the multispan timoshenko beam carrying a number of various concentrated elements including point masses rotary inertias linear springs rotational springs and spring mass systems first the coefficient matrices for an intermediate pinned support an intermediate concentrated element left and right end support of a timoshenko beam are derived next the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method finally the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions respectively the effects of distribution of in span pinned supports and various concentrated elements on the dynamic characteristics of the timoshenko beam are also studied

vibration analysis for a coupled beam sdof system by using the recurrence equation method this paper presents the vibration analysis of a coupled system composed of a beam and a single degree of freedom sdof system the harmonic responses of this kind of system are formulated by means of the recurrence equation method in addition the natural characteristics of the system are also analyzed numerical examples of the undamped system are given and compared with the published results and an excellent correlation shows that the study work in this paper is correct

forced vibration of euler bernoulli beams by means of dynamic green functions abstract a method for determining the dynamic response of prismatic damped euler bernoulli beams subjected to distributed and concentrated loads is presented the method yields exact solutions in closed form and may be used for single and multi span beams single and multi loaded beams and statically determinate and indeterminate beams also green functions for various beams with different homogenous and elastic boundary conditions are given in order to demonstrate the use of the green functions method several examples are given some of the obtained results are compared with those given in the references

analysis of non homogeneous timoshenko beams with generalized damping distributions abstract this paper presents a study on the effects of generalized damping distributions on non homogeneous timoshenko beams on the basis of some fundamentals of modal analysis for damped continuous systems applied to the particular case of the timoshenko beam model the eigenproblem is solved by applying a method combining a state space representation with a transfer matrix technique yielding closed form expressions for the eigenfunctions after validation by means of numerical examples using the finite element method response functions in both the time and the frequency domain are discussed and compared according to different damping distributions