computations and evaluations of higher order theories for free vibration analysis of beams abstract this paper deals with higher order theories for the analysis of free vibration of beam structures refined theories are implemented by the application of the unified formulation by the first author which allows one to introduce any order expansions of the displacement unknowns over the beam sections the selection of the most appropriate theory is made by using a so called axiomatic asymptotic approach which permits one to retain only those terms of the displacement expansion which have been established to be significant with respect to an assigned control parameter the finite element method is used to provide numerical solutions various beam sections as well as boundary conditions are considered depending on the vibration modes bending torsion etc quite different theories are selected in general the number of the effective terms of the resultant theories is much lower than the full expansion case amount the nature of these terms can differ very much as different beam geometries and boundary conditions are considered it has been concluded that the method proposed appears to be suitable and convenient to establish the most appropriate beam theory for a given problem it leads in fact to the cheapest computational model for a given accuracy

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

dynamics of elastic piezoelectric two layer beams using spectral element method abstract it is important to predict the dynamic characteristics of a piezoelectrically actuated beam very accurately for successful vibration controls it has been well recognized that spectral elements provide very accurate solutions for such simple structures as beams thus this paper introduces a spectral element method sem and a spectral element based modal analysis method mam for elastic piezoelectric two layer beams the axial bending coupled equations of motion are derived first by using hamilton s principle and the spectral element matrix is formulated from the spectrally formulated exact eigenfunctions of the coupled governing equations for mam the orthogonality of the eigenfunctions i e natural modes is proved present solution approaches are verified by comparing their results with the conventional fem results it is shown that the results by mam and fem converge to those by sem as the number of superposed natural modes and the number of discretized finite elements are increased respectively it is also shown that as the thickness of piezoelectric layer vanishes the axial bending coupled problems are decoupled to yield the solutions for two independent problems the pure axial motion problem and the pure bending motion problem

boundary integral equations for recovery of design sensitivities in shape optimization a new formulation for obtaining design sensitivities in shape optimization has been developed the formulation is based on a direct application of the material derivative concept to the appropriate boundary integral equations for displacements and stresses in an elastic solid as a check on accuracy the approach was applied to a uniformly loaded infinite plate containing an elliptical hole in this case the availability of an analytical solution made it possible to calculate errors exactly furthermore a wide range of stress states could be considered simply by varying the aspect ratio of the elliptical hole for convenience the design sensitivities were calculated with respect to changes in the major axis numerical convergence was established by comparing results based on four successive boundary meshes in general the predictions for both displacement and stress sensitivities were remarkably accurate

forced vibrations of triply coupled periodically and elastically supported finite open section channels abstract an exact analytical method is presented for the analysis of forced vibrations of uniform open section single and multi bay periodic channels the centre of gravity and the shear centre of the channel cross sections do not coincide and hence the flexural vibrations in two mutually perpendicular directions and the torsional vibrations are all coupled the ends of the channels and the periodic intermediate supports are modelled with springs having finite flexural and torsional stiffnesses single point force excitation has been considered throughout the study although the developed method is also capable of dealing with multi point excitation the channels are assumed to be of euler bernoulli type beams the study also takes the effects of cross sectional warping into consideration the developed method is suitable for structurally damped analysis and in addition to yielding forced vibration characteristics it also straightforwardly reveals the free vibration properties like the mode shapes

a bem sensitivity formulation for three dimensional active noise control summary in this paper a novel boundary element formulation for sensitivity analysis of local active noise control in a three dimensional field is presented the formulation is based on the helmholtz differential equation and is valid for internal as well as for scattering wave propagation problems the primary noise is attenuated within an enclosure called control volume by adding a control source modelled as an object with a vibrating surface both the optimum position of the control volume and the optimum location orientation of the secondary source are determined by minimisation of a suitable cost function the sensitivities are determined by implicit differentiation a hierarchical adaptive cross approximation approach in conjunction with the generalised minimal residual solver is implemented to accelerate the convergence four numerical examples are presented to demonstrate the accuracy and efficiency of the proposed formulations copyright 2012 john wiley sons ltd

a boundary integral equation method for an inverse problem related to crack detection this paper discusses an application of a boundary integral equation method biem to an inverse problem of determining the shape and the location of cracks by boundary measurements suppose that a given body contains an interior crack the shape and the location of which are unknown on the exterior boundary of this body one carries out measurements which are interpreted mathematically as prescribing dirichlet data and measuring the corresponding neumann data or vice versa for a field governed by laplace s equation the inverse problem considered here attempts to determine the geometry of the crack from these experimental data we propose to solve this problem by minimizing the error of a certain boundary integral equation bie the process of this minimization however is shown to require solutions of certain are proposed several 2d and 3d numerical examples are given in order to test the performance of the present method

coupled bending torsional dynamic stiffness matrix of an axially loaded timoshenko beam element abstract analytical expressions for the coupled bending torsional dynamic stiffness matrix terms of an axially loaded uniform timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element the symbolic computing package reduce has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form for check purposes numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications stiffnesses obtained from both methods agreed with each other to machine accuracy application of the developed theory is discussed with particular reference to an established algorithm the influence of axial force shear deformation and rotatory inertia on the natural frequencies of a bending torsion coupled beam with cantilever end conditions is demonstrated by numerical results such results are not generally available in the literature therefore results obtained by partially restricting the present theory are compared with the existing literature wherever possible the results indicate that the method is accurate and efficient

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

dynamic behaviour of beams and rectangular plates under moving loads abstract a theory concerning the dynamic response of finite elastic structures rayleigh beams and plates having arbitrary end supports and under an arbitrary number of moving masses is developed the versatile solution technique presented is based on modified generalized finite integral transforms and the modified struble s method various analytical results are presented numerical examples are given and the results compare favourably with existing ones the efficiency of the solution technique is demonstrated and discussed

spectral element modeling for the beams treated with active constrained layer damping abstract this paper introduces a spectrally formulated finite element for the beams with active constrained layer damping acld the spectral acld beam element is formulated from exact wave solutions of a set of fully coupled dynamic equations of motion the fully coupled dynamic equations of motion are derived by using hamilton s principle and they include the axial motion and rotary inertia of the viscoelastic layer the dynamic responses obtained by the experiment spectral element method and the finite element method are compared to evaluate the validity and accuracy of the present spectral acld beam element model the control performances of an actively controlled acld beam predicted by using the present spectral element model and the conventional finite element model are also compared the spectral acld beam element model is found to provide very reliable results when compared with the conventional finite element model

exact dynamic and static stiffness matrices of shear deformable thin walled beam columns abstract total potential energy of non symmetric thin walled beam columns in the general form is presented by introducing the displacement field based on semitangential rotations and deriving transformation equations between displacement and force parameters defined at the arbitrary axis and the centroid shear center axis respectively next governing equations and force deformation relations are derived from the total potential energy for a shear deformable uniform beam element and a system of linear eigenproblem with non symmetric matrices is constructed based on 14 displacement parameters and then explicit expressions for displacement parameters are derived and exact dynamic stiffness matrices are determined using force deformatin relationships in addition the modified numerical method to eliminate multiple zero eigenvalues and to evaluate the exact static stiffness matrix is developed for spatial stability analysis finally in order to demonstrate the validity and the accuracy of this study the spatially coupled natural frequencies and buckling loads are evaluated and compared with analytical solutions or results analyzed by thin walled beam elements and abaqus s shell elements

coupled vibrations of beams an exact dynamic element stiffness matrix a uniform linearly elastic beam element with non coinciding centres of geometry shear and mass is studied under stationary harmonic end excitation the euler bernoulli saint venant theory is applied thus the effect of warping is not taken into account the frequency dependent 12 12 element stiffness matrix is established by use of an exact method the translational and rotational displacement functions are represented as sums real of complex exponential terms where the complex exponents are numerically found built up structures containing beam elements of the described type can be analysed with ease and certainty using existing library subroutines

stiffness matrices for flexural torsional lateral buckling and vibration analysis of thin walled beam abstract based on the power series method the static and dynamic stiffness matrices for the flexural torsional buckling and free vibration analysis of thin walled beam with non symmetric cross section subjected to linearly variable axial force are newly presented additionally the static stiffness matrix for the lateral buckling analysis of non symmetric beam is presented for the first time for this the elastic strain energy the potential energy considering the second order terms of finite rotations and the kinetic energy for thin walled beam with non symmetric cross section are introduced then equations of motion and force deformation relations are derived from the energy principle explicit expressions for displacement parameters are derived based on power series expansions of displacement components finally the static and dynamic element stiffness matrices are determined using force deformation relationships in order to verify the accuracy of this study the numerical solutions are presented and compared with the finite element solutions using the hermitian beam elements and abaqus s shell elements

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

a bem sensitivity and shape identification analysis for acoustic scattering in fluid solid problems in this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems the formulation developed utilizes the boundary integral equation of the helmholtz equation for the external problem and the cauchy navier equation for the internal elastic problem the sensitivities are obtained by the implicit differentiation technique examples are presented to demonstrate the accuracy of the proposed formulations 1998 john wiley sons ltd

on coupled bending and torsional vibration of uniform beams abstract it is known that the bernoulli euler theory of beam flexure may be combined with the saint venant theory of torsion to describe the coupled antisymmetric free vibration of uniform beams having a plane of symmetry 1 the principal modes may be expressed in terms of tabulated functions this paper extends the theory to allow for warping of the beam cross section for this can make a large difference to results for thin walled beams of open section

shape sensitivity analysis of sequential structural acoustic problems using fem and bem abstract a shape design sensitivity formulation for structural acoustic problems using sequential finite element and boundary element methods is presented frequency response analysis is used to obtain the dynamic behavior of the structure while boundary element analysis is used to solve for the pressure response of the acoustic domain it is shown that the adjoint method which takes the reverse direction to response analysis provides a very efficient way of sensitivity calculation in addition it has been shown that the adjoint equation for the shape design problem is the same as that of the sizing design problem the only difference is the numerical integration that evaluates the sensitivity coefficient the combination of the semi analytical method for the structure and the analytical differentiation method for the acoustic cavity yields a very practical approach for the shape design sensitivity formulation the accuracy of the sensitivity information is compared with the analytical sensitivity as well as the sensitivity calculated using the finite difference method

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

beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation a numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods the radius of convergence of eigenvalue perturbation methods based on taylor series is known to decrease when eigenvalues are getting closer to each other this phenomenon knwon as veering in structural dynamics is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective when this degeneracy referred to as exceptional point ep is close to the real axis the veering becomes stronger the main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity to do so two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter this yields a new robust eigenvalue reconstruction scheme which is compared to taylor and puiseux series in all cases theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model this system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter computationally efficient the proposed algorithm could also be relevant for actual numerical models of large size arising from other applications involving parametric eigenvalue problems e g waveguides rotating machinery or instability problems such as squeal or flutter

conforming boundary elements in plane elasticity for shape design sensitivity the structural design sensitivity analysis of a two dimensional continuum using conforming continuous boundary elements is investigated implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices a singular formulation of the boundary element method is used implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis the behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated a rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices thus avoiding the problems associated with their numerical integration a formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented the design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results numerical data comparing the performance of conforming and non conforming formulations in the calculation of design sensitivities are also presented the accuracy of the present results is demonstrated through comparisons with existing analytical results

determination of nonideal beam boundary conditions a spectral element approach a spectral element approach to determine the nonideal or unknown structural boundary conditions of beams is introduced the nonideal structural boundary conditions are represented by the frequency dependent effective boundary springs transverse springs and rotational springs the effective boundary spring constants are then determined from the measured frequency response functions in conjunction with the spectral element method experiments are conducted for the one end supported and the two end supported beams to verify the present approach of boundary conditions identification the analytical predictions obtained by using the identified boundary conditions are very close to the experiment measurements

dynamic behaviour of structures in large frequency range by continuous element methods abstract the continuous element method is presented in the context of the harmonic response of beam assemblies a general formulation is described from the displacement solution of the elementary problem a direct computation of elementary dynamic stiffness matrices is presented in the present formulation distributed loadings are taken into account in the case of more complex geometries for which many coupling phenomena occur an explicit formulation is no more conceivable in this case a numerical approach is presented this approach allows an algorithmic computation of exact dynamic stiffness matrices this method called numerical continuous element allows one to consider the coupled vibrations of curved beams and those of helical beams the validation of this numerical method is achieved by comparisons with the harmonic response of various beams obtained by a finite element approach finally a comparison between eigenfrequencies obtained experimentally and numerically for a straight beam and a helical beam has been made to evaluate the performances of the method

multi parameter perturbation methods for the eigensolution sensitivity analysis of nearly resonant non defective multi degree of freedom systems the dynamic behavior of structural systems may be strongly characterized by the occurrence of multiple internal resonances for particular combinations of the mechanical parameters the linear models governing these resonant or nearly resonant systems tend to exhibit high sensitivity of the eigenvalues and eigenvectors to small parameter modifications this pathological condition is recognized as a source of relevant phenomena such as frequency veering and mode localization or hybridization the paper presents the generalization of uniformly valid perturbation methods to perform eigensolution sensitivity analyses in multi degree of freedom hamiltonian systems with a generic number of close eigenvalues the leading idea is to treat systematically nearly resonant systems as multi parameter perturbations of a perfectly resonant non defective though a priori unknown reference system given a single nearly resonant system a multi parameter perturbation method is presented to achieve a two fold objective first identify a close resonant system suited to serve as a starting point for sensitivity analyses inverse problem second to approximate asymptotically the eigensolution of all the nearly resonant systems which may arise from its generic perturbation direct problem the conditions of existence and uniqueness of the inverse problem solution are discussed the direct problem solution is analyzed with a focus on the eigensolution sensitivity to parameter perturbations with different physical meanings such as a slight geometric disorder or weak elastic coupling in periodic structures finally the procedure is verified on a prototypical structural system describing the section dynamics of a suspended bridge

triply coupled vibrations of thin walled open cross section beams including rotary inertia effects an exact analytical method is presented for predicting the undamped natural frequencies of beams with thin walled open cross sections having no axis of symmetry the governing differential equations give a characteristic equation of the 12th order with real coefficients the roots are found numerically and the exact boundary conditions are considered especially for free ends to obtain natural frequencies the simpler cases of neglecting cross sectional warping and or rotary inertia are also dealt with it is seen that when the effect of rotary inertia is neglected significant errors incur for some boundary conditions cross section thicknesses and mode numbers this is more profound when the warping effect is taken into account

application of least square method to arbitrary order problems with separated boundary conditions in this paper differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied in fact it will be shown that for every e 0 there exists an approximate solution v e of b spline functions such that the corresponding least square error is less than e 0 and also v e satisfies the exact boundary conditions some examples are given and the optimal errors are obtained for the sake of comparison

exact dynamic stiffness matrix of a three dimensional shear beam with doubly asymmetric cross section abstract the dynamic member stiffness matrix of a three dimensional shear beam with doubly asymmetric cross section is derived exactly from the governing sixth order differential equation of motion such a formulation accounts for the uniform distribution of mass in the member and necessitates the solution of a transcendental eigenvalue problem this is achieved using the wittrick williams algorithm where the necessary parameters are developed using a generalised procedure an example is given to clarify the theory together with a small parametric study that indicates when lateral torsional coupling may safely be ignored the work also holds considerable potential in its application to the approximate analysis of asymmetric multi storey three dimensional frame structures

dynamic behavior and modal control of beams under moving mass abstract the constitutive equation of an euler bernoulli beam under the excitation of moving mass is considered the dynamics of the uncontrolled system is governed by a linear self adjoint partial differential equation a dirac delta function is used to describe the position of the moving mass along the beam and its inertial effects an approximate formulation to the problem is obtained by limiting the inertial effect of the moving mass merely to the vertical component of acceleration having defined a critical velocity in terms of the fundamental period and span of the beam it is shown that for smaller velocities the approximate and exact approaches to the problem almost coincide since the defined critical velocity is fairly large compared to those in practical cases the approximate approach can effectively be used for a wide range of problems there is however a slight variation in critical velocity depending on the weight of the moving mass on the other hand it is shown that the effect of higher vibrational modes is not negligible for certain velocity ranges finally a linear classical optimal control algorithm with a time varying gain matrix with displacement velocity feedback is used to control the response of the beam the efficiency of the control algorithm in suppressing the response of the system under the effect of moving mass with different number of controlled modes and actuators is investigated

approximate solutions to boundary value problems of higher order by the modified decomposition method abstract in this paper we present an efficient numerical algorithm for approximate solutions of higher order boundary value problems with two point boundary conditions a modified form of the adomian decomposition method will be implemented to construct such solutions the approach provides the solution in the form of a rapidly convergent series the analysis is accompanied by numerical examples the obtained results demonstrate reliability and efficiency of the proposed scheme

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

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

dynamic stiffness matrix of a rectangular plate for the general case the dynamic stiffness matrix of a rectangular plate for the most general case is developed by solving the bi harmonic equation and finally casting the solution in terms of the force displacement relationship of the freely vibrating plate essentially the frequency dependent dynamic stiffness matrix of the plate when all its sides are free is derived making it possible to achieve exact solution for free vibration of plates or plate assemblies with any boundary conditions previous research on the dynamic stiffness formulation of a plate was restricted to the special case when the two opposite sides of the plate are simply supported this restriction is quite severe and made the general purpose application of the dynamic stiffness method impossible the theory developed in this paper overcomes this long lasting restriction the research carried out here is basically fundamental in that the bi harmonic equation which governs the free vibratory motion of a plate in harmonic oscillation is solved in an exact sense leading to the development of the dynamic stiffness method it is significant that the ingeniously sought solution presented in this paper is completely general covering all possible cases of elastic deformations of the plate the wittrick williams algorithm is applied to the ensuing dynamic stiffness matrix to provide solutions for some representative problems a carefully selected sample of mode shapes is also presented

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

optimal piezoelectric control of a plate subject to time dependent boundary moments and forcing function for vibration damping active vibration control problem for a rectangular plate subject to moment boundary conditions and forcing function is solved by means of a maximum principle the control is exercised by a patch actuator and the solution is formulated using an adjoint variable leading to a coupled boundary initial terminal value problem the application of the maximum principle yields the optimal control expression as well as the explicit solution for a simply supported plate the objective functional to be minimized is defined as a quadratic functional of displacement and velocity and also includes a penalty in terms of control voltage applied to the piezoelectric patch actuator the penalty term limits the amount of control energy spent during the control process numerical results are presented to assess the effect of the optimal control algorithm

stochastic vibration of axially loaded monosymmetric timoshenko thin walled beam abstract an analytical method is presented to perform the flexure torsion coupled stochastic response analysis of monosymmetric axially loaded timoshenko thin walled beam subjected to various kinds of concentrated and distributed stochastic excitations with stationary and ergodic properties the effects of warping stiffness axial force shear deformation and rotary inertia are included in the present formulations first the damped general governing differential equations of motion of axially loaded timoshenko thin walled beam are developed and the free vibration analysis is performed once the natural frequencies and mode shapes are obtained mode superposition method in conjunction with receptance method is used to compute the mean square displacement response of the axially loaded thin walled beam finally the method is illustrated by its application to two test examples to investigate the effects of warping stiffness axial force shear deformation and rotary inertia on the stochastic response of the thin walled beams

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

flaw identification using the boundary element method in this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two dimensional structures an introduction to inverse analysis is given with special reference to methods of flaw identification along with a brief review of the optimization methods employed both the standard boundary element and the dual boundary element method are presented with the dual boundary element method proposed as the basis for the new formulation the flaw identification method is presented along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location

three dimensional asymptotic approach to inhomogeneous and laminated piezoelectric plates an asymptotic theory is developed for anisotropic inhomogeneous and laminated piezoelectric plates on the basis of three dimensional linear piezoelectricity the inhomogeneity is assumed in the thickness direction and includes the important piezoelectric laminates as a special case through asymptotic expansions the resulting two dimensional differential equations are of the same form for each order with different nonhomogeneous terms being determined systematically by preceding order solutions the governing equations of the leading order when degenerated to pure elasticity are shown to be the same as those for equivalent classical thin elastic plates the proposed methodology is illustrated by considering a rectangular piezoelectric plate subject to both mechanical and electric loadings with its edges simply supported and grounded a three dimensional solution for the fully electromechanically coupled problem is obtained by successively solving the two dimensional field equations from the leading order to higher orders excellent agreement is observed with established results and new results are presented from which significant physical insights are discussed

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

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

vibrations of a mindlin plate subjected to a pair of inertial loads moving in opposite directions abstract a mindlin plate subjected to a pair of inertial loads traveling at a constant high speed in opposite directions along arbitrary trajectory straight or curved is presented the masses represent vehicles passing a bridge or track plates a numerical solution is obtained using the space time finite element method since it allows a clear and simple derivation of the characteristic matrices of the time stepping procedure the transition from one spatial finite element to another must be energetically consistent in the case of the moving inertial load the classical time integration schemes are methodologically difficult since we consider the dirac delta term with a moving argument the proposed numerical approach provides the correct definition of force equilibrium in the time interval the given approach closes the problem of the numerical analysis of vibration of a structure subjected to inertial loads moving arbitrarily with acceleration the results obtained for a massless and an inertial load traveling over a mindlin plate at various speeds are compared with benchmark results obtained for a kirchhoff plate the pair of inertial forces traveling in opposite directions causes displacements and stresses more than twice as large as their corresponding quantities observed for the passage of a single mass

application of active piezoelectric patches in controlling the dynamic response of a thin rectangular plate under a moving mass abstract the governing differential equation of motion for an undamped thin rectangular plate with a number of bonded piezoelectric patches on its surface and arbitrary boundary conditions is derived using hamilton s principle a moving mass traveling on an arbitrary trajectory acts as an external excitation for the system the effect of the moving mass inertia is considered using all the out of plane translational acceleration components the method of eigenfunction expansion is used to transform the equation of motion into a number of coupled ordinary differential equations a classical closed loop optimal control algorithm is employed to suppress the dynamic response of the system determining the required voltage of each piezoactuator at any time interval in a numerical example for a simply supported square plate under two different loading paths the effect of the mass velocity and mass weight of the moving load on the dynamic behavior of the uncontrolled system is investigated the results show that depending on the path of the moving mass the inertia effect is very important causing different behaviors of the system in addition the number of vibrational modes involved in determining the dynamic response of the system is crucial the inertia effect is more important for an orbiting mass loading case compared to the case in which the moving mass is traversing the plate on a straight line a number of equally spaced piezo patches are used on the lower surface of the plate to control the displacement of the center point of the plate the implemented control mechanism proves to be very efficient in suppressing the near resonant dynamic response of the system requiring fairly low levels of voltage for each patch increasing the area of the employed piezo patches would reduce the required maximum voltage for controlling the response of the system

beam bending torsion dynamic stiffness method for calculation of exact vibration modes abstract the exact dynamic stiffness matrix is derived for a straight and uniform beam element whose elastic and inertial axes are not coincident elementary bending torsion beam theory is used and bending translation is restricted to one direction the element matrix can be used in the dynamic stiffness method for calculation of exact natural frequencies mode shapes and generalized masses for planar assemblages of connected bending torsion beams the dynamic stiffness method is outlined and details pertinent to the bending torsion beam element are given an example of a three beam assemblage representing an airplane wing is posed and exact numerical solutions are tabulated for natural frequencies and generalized masses solutions calculated by the finite element method are also tabulated for comparison

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

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

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

beam element matrices derived from vlasov s theory of open thin walled elastic beams a uniform beam element of open thin walled cross section is studied under stationary harmonic end excitation an exact dynamic transcendentally frequency dependent 14 14 element stiffness matrix is derived from vlasov s coupled differential equations special attention is paid to the computational problems arising when coefficients vanish in these equations because of symmetric cross section zero warping stiffness etc the dynamic element stiffness matrix is established via a generalized linear eigenvalue problem and a system of linear algebraic equations with complex matrices a static stiffness matrix is also derived and the associated consistent mass and geometric stiffness matrices are given modal masses are evaluated a fortran program and a numerical example are included

comments on curve veering in eigenvalue problems abstract the dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci when two loci approach each other they often cross or abruptly diverge the latter case called curve veering has been observed in approximate solutions associated with discretized models the influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions the existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem veering is then examined in a general eigenvalue problem criteria are established to distinguish veerings from crossings in both continuous and discretized models the application of the criteria is illustrated by examples

spatial stability and free vibration of shear flexible thin walled elastic beams ii numerical approach in a companion paper 1 equations of motion and closed form solutions for spatial stability and free vibration analysis of shear flexible thin walled elastic beams were analytically derived from the linearized hellinger reissner principle in this paper elastic and geometric stiffness matrices and consistent mass matrix for finite element analysis are evaluated by using isoparametric and hermitian interpolation polynomials isoparametric interpolation functions with 2 3 and 4 nodes per element are utilized in isoparametric beam elements and in hermitian beam elements the third and fifth order hermitian polynomials including shear deformation effects are newly derived and applied for the calculation of element matrices in order to verify the validity of the finite element formulation both analytic and numerical solutions for spatial buckling and free vibration problems including shear effects are presented and compared

free vibration of functionally graded beams and frameworks using the dynamic stiffness method the free vibration analysis of functionally graded beams fgbs and frameworks containing fgbs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form the usually adopted rule that the material properties of the fgb vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration the differential equations are solved in closed analytical form when the free vibratory motion is harmonic the dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam next the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation finally the wittrick williams algorithm is applied as solution technique to solve the free vibration problems of fgbs with uniform cross section stepped fgbs and frameworks consisting of fgbs some numerical results are validated against published results but in the absence of published results for frameworks containing fgbs consistency checks on the reliability of results are performed the paper closes with discussion of results and conclusions

exact natural frequencies of a three dimensional shear torsion beam with doubly asymmetric cross section using a two dimensional approach abstract this paper presents an exact analytical approach to the calculation of the natural frequencies of structures comprising contiguous three dimensional shear torsion beams with doubly asymmetric cross section such component members have the unusual theoretical property that they allow for coupled torsional and shearing deformation but not bending deformation initially exact dynamic member stiffness matrices exact finite elements are developed for planar shear and torsional motion these matrices can be combined in the usual way to model stepped and continuous beams whose uncoupled frequencies can then be determined exactly it is then shown how the corresponding coupled frequencies can be established easily from the uncoupled values through an exact relationship this enables coupled three dimensional vibration problems to be solved very efficiently using a two dimensional approach the paper is concluded with an example that clarifies the theory together with a parametric study that enables guidance to be given as to when lateral torsional coupling may safely be ignored

exact dynamic stiffness elements based on one dimensional higher order theories for free vibration analysis of solid and thin walled structures in this paper an exact dynamic stiffness formulation using one dimensional 1d higher order theories is presented and subsequently used to investigate the free vibration characteristics of solid and thin walled structures higher order kinematic fields are developed using the carrera unified formulation which allows for straightforward implementation of any order theory without the need for ad hoc formulations classical beam theories euler bernoulli and timoshenko are also captured from the formulation as degenerate cases the principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions an exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses the explicit terms of the dynamic stiffness matrices are also presented the resulting dynamic stiffness matrix is used with particular reference to the wittrick williams algorithm to carry out the free vibration analysis of solid and thin walled structures the accuracy of the theory is confirmed both by published literature and by extensive finite element solutions using the commercial code msc nastran

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

spectral element modeling for extended timoshenko beams periodic lattice structures such as the large space lattice structures and carbon nanotubes may take the extension transverse shear bending coupled vibrations which can be well represented by the extended timoshenko beam theory in this paper the spectrally formulated finite element model simply spectral element model has been developed for extended timoshenko beams and applied to some typical periodic lattice structures such as the armchair carbon nanotube the periodic plane truss and the periodic space lattice beam

asymptotic governing equation for wave propagation along weakly non uniform euler bernoulli beams abstract non uniformity in beams arises either from manufacturing imperfections or by design and can have a singular impact on the qualitative properties of the vibratory response of the beam to describe the mechanism causing such large changes on the dynamics of the beam we derived asymptotically a simpler equation in the form ss q s s 0 the coefficient function q s is given by equation 52 herein in terms of the beam flexural rigidity the mass per unit length and the tensile force applied to the beam the equation is asymptotic to the non uniformity of the beam but under certain restrictions namely of having constant tension and a constant product of the beam mass per unit length and flexural rigidity it is an exact governing equation for wave propagation along bernoulli euler beams and it has a helmholtz like form the behavior of the equation is systematically explored and illustrated through numerical results

boundary element implicit differentiation equations for design sensitivities of axisymmetric structures abstract design sensitivity analysis of axisymmetric elastic media is formulated using boundary elements the kernels for sensitivity matrices are obtained through implicit differentiation of the corresponding boundary element elasticity kernels the singular terms are obtained by applying the boundary displacements and tractions and their respective sensitivities for the rigid body motion mode and the inflation mode the equations for the recovery of sensitivities of axisymmetric boundary stresses are presented as a check on accuracy the approach is applied to a series of examples for which analytical elasticity solutions are available the predictions for both displacement and stress sensitivities are accurate additional examples are provided to demonstrate the versatility of the present approach

coupled flexural torsional vibration of beams in the presence of static axial loads and end moments abstract the problem of coupled flexural torsional vibration of a deep rectangular beam in the presence of a static axial load and an end moment is studied closed form analytical solutions are obtained for simply supported boundary conditions numerical results are obtained for the coupled frequencies and mode shapes in terms of the location of axes of rotation of the cross section for different values of the load and the geometry parameters the results show that the predominantly flexural frequencies of coupled flexural torsional vibration can be obtained as uncoupled flexural frequencies of an equivalent beam column by defining an effective axial load and that by defining an equivalent moment parameter the presentation of the results for the predominantly flexural mode can be made in a form independent of the slenderness of the beam in the depth direction