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

similarity solutions for unsteady free convection flow from a continuous moving vertical surface the transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid the thermal boundary layer induced within a vertical semi infinite layer of boussinseq fluid by a constant heated bounding plate the application of two parameter groups reduces the number of independent variables by two and consequently the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions the obtained ordinary differential equations are solved analytically for the temperature and numerically for the velocity using the shooting method effect of prandtl number on the thermal boundary layer and velocity boundary layer are studied and plotted in curves

analysis of forced bilinear oscillators and the application to cracked beam dynamics a new closed form solution based on the use of two square wave functions to model stiffness change is proposed for bilinear oscillators under low frequency excitation the solution procedure is extended to cases with bilinear forcing function which usually occur in the dynamics of damaged structures the proposed solution provides the spectral pattern and the magnitude of each harmonic component for a damaged rectangular beam according to the size and location of the crack the predicted spectral pattern and the magnitude of each harmonic component for a simply supported beam containing a fatigue crack are validated by numerical simulation experimental work has not been done the results show that the proposed solution may be used to predict efficiently spectral patterns of damaged structures while avoiding lengthy computation such a procedure may be developed into an attractive on line diagnostic tool

on some matrix inequalities the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2sj ab sj a a b b j 1 2 for any matrices a b we first give new proofs of this inequality and its equivalent form then we use it to prove the following trace inequality let a0 be a positive definite matrix and a1 ak be positive semidefinite matrices then tr j 1k i 0jai 2aj tra0 1

influence of centerline extensibility on the in plane free vibrations of loaded circular arches free vibrations of circular arches about a prestressed static equilibrium state are investigated the equations of motion for the small amplitude free vibrations about the initial equilibrium state attained under distributed tangential and normal loading derived in a previous paper and specialized to constant curvature are used particular attention is given to the influence of centerline stretching during the vibratory motion the vanishing of the loaded free vibration frequencies signals static buckling and the load intensities at which the frequencies go to zero correspond to the static buckling loads free vibrations with no initial loading is treated as a special case of the problem considered it is shown that exact numerical values for the frequencies can be obtained in this case in general the presence of distributed initial loading precludes the possibility of obtaining closed form analytical solutions barring a few simple loading situations such as uniform constant directional pressure as the governing differential equations have variable coefficients in the present work numerical results are obtained for uniform arches subjected to gravity loading the galerkin method with polynomial trial functions that satisfy the geometric boundary conditions is employed in obtaining accurate values of free vibration frequencies convergence of the method with increasing number of terms in the assumed expansion representing the true solution is demonstrated arches with pinned and clamped end support conditions are considered considerably different frequencies are found depending upon whether one includes or ignores the influence of centerline stretching during the vibratory motion

periodic motions and resonances of impact oscillators abstract bilinear oscillators the oscillators whose springs have different stiffnesses in compression and tension model a wide range of phenomena a limiting case of bilinear oscillator with infinite stiffness in compression the impact oscillator is studied here we investigate a special set of impact times the eigenset which corresponds to the solution of the homogeneous equation i e the oscillator without the driving force we found that this set and its subsets are stable with respect to variation of initial conditions furthermore amongst all periodic sets of impact times with the period commensurate with the period of driving force the eigenset is the only one which can support resonances in particular the multi harmonic resonances other resonances should produce non periodic sets of impact times this funding indicates that the usual simplifying assumption e g s w shaw p j holmes a periodically forced piecewise linear oscillator journal of sound and vibration 90 1983 129 155 that the times between impacts are commensurate with the period of the driving force does not always hold we showed that for the first sub harmonic resonance the resonance achieved on a half frequency of the main resonance the set of impact times is asymptotically close to the eigenset the envelope of the oscillations in this resonance increases as a square root of time opposite to the linear increase characteristic of multi harmonic resonances

exact solution of in plane vibrations of circular arches with account taken of axial extension transverse shear and rotatory inertia effects exact solution of free in plane vibrations of circular arches of uniform cross section is given by considering axial extension transverse shear and rotatory inertia effects in contrast with kirchhoff s beam theory the restrictions of perpendicular cross section and inextensible arc length are removed the principal axes of the cross section are assumed to coincide with the principal normal and binormal vectors of the centerline of the beam a solution procedure is applied to obtain the fundamental matrix for various end conditions natural frequencies and mode shapes are given in figures and tables

structure of resonances and formation of stationary points in symmetrical chains of bilinear oscillators abstract dynamics of strongly nonlinear systems can in many cases be modelled by bilinear oscillators which are the oscillators whose springs have different stiffnesses in compression and tension this underpins the analysis of a wide range of phenomena from oscillations of fragmented structures connections and mooring lines to deformation of geological media single bilinear oscillators were studied previously and the presence of multiple resonances both super and sub harmonic was found less attention was paid to systems of multiple bilinear oscillators that describe many natural and engineering processes such as for example the behaviour of fragmented solids here we fill this gap concentrating on the simplest case 1d symmetrical chains of bilinear oscillators we show that the presence and structure of resonances in a symmetric chain of bilinear oscillators with fixed ends depends upon the number of oscillating masses two elementary chains act as the basic ones a single mass bilinear chain a mass connected to the fixed points by two bilinear springs that behaves as a linear oscillator with a single resonance and a two mass chain that is a coupled bilinear oscillator two masses connected by three bilinear springs the latter has multiple resonances we demonstrate that longer chains either do not have resonances or get decomposed in the resonance into either the single mass or two mass elementary chains with stationary masses in between the resonance frequencies are inherited from the basic chains of decomposition we show that if the number of masses is odd the chain can be decomposed into the single mass bilinear chains separated by stationary masses it then inherits the resonances of the single mass bilinear chain the chains with the number of masses minus 2 divisible by 3 can be decomposed into the two mass bilinear chains separated by stationary masses and inherit the resonances of the two mass chains the chains whose lengths satisfy both criteria such as chains with 5 11 17 masses allow both types of resonances

commutators and accretive operators abstract in this paper singular values of commutators of hilbert space operators are estimated to this aim the accretivity of a transform of the operators is applied some recent results of kittaneh f kittaneh singular value inequalities for commutators of hilbert space operators linear algebra appl 430 2009 2362 2367 are extended

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

commutator inequalities for hilbert space operators we prove singular value inequalities for positive operators some of these inequalities generalize recent results for commutators due to bhatia kitttaneh kittaneh and wang du applications of our results are given

the frobenius norm and the commutator in an earlier paper we conjectured an inequality for the frobenius norm of the commutator of two matrices this conjecture was recently proved by seak weng vong and xiao qing jin and independently also by zhiqin lu we here give a completely different proof of this inequality prove some related results and embark on the corresponding question for unitarily invariant norms

singular value inequalities for commutators of hilbert space operators abstract we prove several singular value inequalities for commutators of hilbert space operators it is shown among other inequalities that if a b and x are operators on a complex separable hilbert space such that a and b are positive and x is compact then the singular values of ax xb are dominated by those of max a b x x where is the usual operator norm

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

maximum response statistics of mdof linear structures excited by non stationary random processes the paper deals with the problem of predicting the maximum response statistics of multi degree of freedom mdof linear structures subjected to non stationary non white noises the extension of two different censored closures of gumbel type originally proposed by the authors for the response of single degree of freedom oscillators it is presented the improvement associated with the introduction in the closure of a consistent censorship factor accounting for the response bandwidth it is pointed out simple and effective step by step procedures are formulated and described in details numerical applications on a realistic 25 storey moment resisting frame along with comparisons with classical approximations and monte carlo simulations are also included

proof of bottcher and wenzel s conjecture on commutator norms for 3 by 3 matrices abstract bottcher and wenzel conjectured that the frobenius norm of the commutator of two real matrices of frobenius norm 1 is at most 2 and proved this for 2 by 2 matrices in this paper the conjecture is proved for 3 by 3 matrices

flutter and divergence instability of the multi cracked cantilever beam column for conservative systems instability can occur only by divergence and the presence of damage can produce both a reduction of the buckling loads and modification of the corresponding mode shapes depending on the positions and intensities of the damage distribution for nonconservative systems instability is found to occur by divergence flutter or both characterised by multiple stable and unstable ranges of the loads whose boundary can be altered by the damage distribution this paper focuses on the stability behaviour of multi cracked cantilever euler beam column subjected to conservative or nonconservative axial loads the exact flutter and divergence critical loads are obtained by means of the exact closed form solution of the multi cracked beam column derived by the authors in a previous paper the extensive numerical applications reported in the paper aimed at evaluating the influence of several damage scenarios for different values of the degree of nonconservativeness it is shown how the presence of damage can strongly modify the ranges of divergence and flutter critical loads of the corresponding undamaged cantilever column which has been the subject of several papers starting from the pfluger paradoxical results

vibration analysis of multiple cracked non uniform beams this paper presents the energy based method for the vibration identification of non uniform euler bernoulli beams having multiple open cracks the method includes significant modifications for the energy based method presented by yang et al crack identification in vibrating beams using the energy method journal of sound and vibration 244 2 2001 339 357 the distribution of the energy consumed is determined by taking into account not only the strain change at the cracked beam surface as in general but also the considerable effect of the stress field caused by the angular displacement of the beam due to bending the rayleigh ritz approximation method is used in the analysis the method is adapted to the cases of multiple cracks with an approach based on the definition of strain disturbance variation along the beam examples are presented on cantilever beams having different truncation factors when the results are compared with a commercial finite element program and with the results of zheng and fan natural frequencies of a non uniform beam with multiple cracks via modified fourier series journal of sound and vibration 242 4 2001 701 717 good agreements are obtained the effects of truncation factors and positions of cracks on the natural frequency ratios are presented in graphics

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

multiplication of distributions in one dimension possible approaches and applications to δ function and its derivatives we introduce a new class of multiplications of distributions in one dimension merging two different regularizations of distributions some of the features of these multiplications are discussed in detail we use our theory to study a number of examples involving products between dirac delta functions and its successive derivatives

closed form solutions of a multi cracked circular arch under static loads abstract generalised functions have been widely adopted in structural mechanics to treat singularities of beam like structures however due to the curved geometry that couples axial and transversal displacements their use has never been explored for curved beams in this paper the capability of distributions of leading to closed form exact solutions for multi cracked circular arch is shown the exact closed form solution of a circular euler arch in presence of any number of discontinuities due to concentrated damage and subjected to an arbitrary distribution of static loads is obtained damage under the form of concentrated cracks has been modeled through the widely adopted and validated equivalent elastic hinge concept and has been introduced in the governing differential equations by making use of dirac s delta functions the resulting nontrivial generalised six order differential equations have been derived and solved in closed form independently of the number of along arch concentrated cracks the solution is expressed as a function of six integration constants only in which the damage positions and intensities are given data appearing explicitly in the solution expression this latter aspect constitutes a fundamental aid towards the resolution of the static damage inverse identification problem the results have been validated through some comparisons with finite element numerical simulations examples referred to multi cracked euler arches with different boundary conditions damage and load scenarios are presented

inequalities for commutators of positive operators abstract it is shown that if a b and x are operators on a complex separable hilbert space such that a and b are compact and positive then the singular values of the generalized commutator a x x b are dominated by those of x a b where is the usual operator norm consequently for every unitarily invariant norm we have a x x b x a b it is also shown that if a and b are positive and x is compact then a x x b max a b x for every unitarily invariant norm moreover if x is positive then the singular values of the commutator a x x a are dominated by those of 1 2 a x x consequently a x x a 1 2 a x x for every unitarily invariant norm for the usual operator norm these norm inequalities hold without the compactness conditions and in this case the first two norm inequalities are the same our inequalities include and improve upon earlier inequalities proved in this context and they seem natural enough and applicable to be widely useful

in plane vibration and stability of shallow circular arches subjected to axial forces natural frequencies and buckling loads of a simply supported shallow circular arch with sufficiently small depth to radius of curvature ratio h r 1 subjected to initial axial tensile and or compressive forces are analysed by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a one dimensional higher order arch theory for in plane vibration problems of shallow circular arches is derived through hamilton s principle several sets of truncated approximate theories which can take into account the effects of both shear deformations with depth changes and rotary inertia are applied to solve the eigenvalue problems of an elastic arch convergence properties of the natural frequency and the buckling load of simply supported shallow circular arches are examined in detail the present approximate theories can predict the natural frequencies and buckling loads of shallow circular arches with small length to depth ratio l h more accurately compared with previously published results

crack identification in a stepped beam carrying a rigid disk abstract in this study a simple algorithm based on a mathematical model is proposed to identify crack location and depth in a stepped cantilever euler bernoulli beam carrying a rigid disk at its tip the mathematical model that describes the lateral vibration of the beam is derived using the assumed mode method that coalesces with the lagrange s equation a massless torsional spring whose stiffness depend on the severity of the crack is used as a crack model using this crack modeling method combined with the assumed mode method the crack effect is introduced to the system flexibility as global additional structural flexibility for the assumed mode method the mode shapes for two uniform beams connected by a massless torsional spring simulating the cracked beam are adopted as trial functions the proposed identification algorithm utilizes the first three natural frequencies shift of the beam caused by a crack to estimate its location and depth in addition the proposed mathematical model is used to illustrate the effect of the crack depth and its location on the dynamic characteristics of the system using the commercial finite element fe software ansys 8 0 three dimensional finite element analysis fea is carried out to show the accuracy of the derived mathematical model and to demonstrate the reliability of the proposed crack identification algorithm the analysis showed consistency with the assumed mode results it showed that the error in concurrent prediction of crack depth and its location using the proposed algorithm is about 10

inequalities for sums and direct sums of hilbert space operators abstract we prove several singular value inequalities and norm inequalities involving sums and direct sums of hilbert space operators it is shown among other inequalities that if x and y are compact operators then the singular values of x y 2 are dominated by those of x y applications of these inequalities are also given

group solution of a time dependent chemical convective process the time dependent progress of a chemical reaction over a flat vertical plate is here considered the problem is solved using the two parameter group method which reduces the number of independent by one and leads to a set of nonlinear ordinary differential equation the behavior of the process is numerically investigated for the chemical reaction order n 1 and different schmidt numbers as the problem shows a singularity at n 1 the nonlinear system of ordinary differential equation resulting from the transformation of the problem are analytically solved through the perturbation method the velocity and concentration of chemicals based on the analytical and numerical solutions are presented compared and discussed

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

closed form solutions of euler bernoulli beams with singularities abstract the problem of the integration of the static governing equations of the uniform euler bernoulli beam with discontinuities is studied in particular two types of discontinuities have been considered flexural stiffness and slope discontinuities both the above mentioned discontinuities have been modeled as singularities of the flexural stiffness by means of superimposition of suitable distributions generalized functions to a uniform one dimensional field closed form solutions of governing differential equation requiring the knowledge of the boundary conditions only are proposed and no continuity conditions are enforced at intermediate cross sections where discontinuities are shown the continuity conditions are in fact embedded in the flexural stiffness model and are automatically accounted for by the proposed integration procedure finally the proposed closed form solution for the cases of slope discontinuity is compared with the solution of a beam having an internal hinge with rotational spring reproducing the slope discontinuity

norm inequalities for commutators of positive operators and applications let x y and z be operators on a hilbert space such that x and z are positive it is shown that left xy yz right leqslant rm max left left x right left z right right left y right applications of this commutator inequality are given

singular values norms and commutators abstract let a i b i and x i i 1 n be bounded linear operators on a separable hilbert space such that x i is compact for i 1 n it is shown that the singular values of i 1 n a i x i b i are dominated by those of i 1 n a i b i i 1 n x i where is the usual operator norm among other applications of this inequality we prove that if a and b are self adjoint operators such that a 1 a a 2 and b 1 b b 2 for some real numbers a 1 a 2 b 1 and b 2 and if x is compact then the singular values of the generalized commutator ax xb are dominated by those of max b 2 a 1 a 2 b 1 x x this inequality proves a recent conjecture concerning the singular values of commutators several inequalities for norms of commutators are also given

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

analytical and numerical results for vibration analysis of multi stepped and multi damaged circular arches this paper investigates the in plane linear dynamic behaviour of multi stepped and multi damaged circular arches under different boundary conditions cracked cross sections are modelled as massless elastic rotational hinges in damaged configuration cracks can be located both at the interface between two adjacent portions as well as inside the portion itself for each arch portion bounded by two cracks the differential equations of motion have been established considering axial extension transverse shear effects and rotatory inertia the equilibrium equations of arch portions are combined in the coupled fundamental system in terms of radial displacement tangential displacement and rotation analytical and numerical solutions for multi stepped arches in undamaged as well as in damaged configurations are proposed the analytical solution is based on the euler characteristic exponent procedure involving the roots of characteristic polynomials while the numerical method is focused on the generalized differential quadrature gdq method and the generalized differential quadrature element gdqe technique numerical results are shown in terms of the first 10 analytical and numerical frequencies of multi stepped and multi damaged arches with different boundary conditions finally convergence and stability characteristics of the gdqe procedure are investigated the convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical procedure is very good

exact solution of the multi cracked euler bernoulli column abstract the use of distributions generalized functions is a powerful tool to treat singularities in structural mechanics and besides providing a mathematical modelling their capability of leading to closed form exact solutions is shown in this paper in particular the problem of stability of the uniform euler bernoulli column in presence of multiple concentrated cracks subjected to an axial compression load under general boundary conditions is tackled concentrated cracks are modelled by means of dirac s delta distributions an integration procedure of the fourth order differential governing equation which is not allowed by the classical distribution theory is proposed the exact buckling mode solution of the column as functions of four integration constants and the corresponding exact buckling load equation for any number position and intensity of the cracks are presented as an example a parametric study of the multi cracked simply supported and clamped clamped euler bernoulli columns is presented

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

generalized solutions to a linear discontinuous differential equation in this paper we construct generalized solutions weak solutions to a linear discontinuous differential equation which appears as mathematical model in mechanics and for which usual methods are not quite adequate the result is applied to the equation of an elastic beam on a winkler type of foundation

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

group solution for unsteady free convection flow from a vertical moving plate subjected to constant heat flux the problem of heat and mass transfer in an unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid is investigated for constant heat flux using the group theoretical method the nonlinear coupled partial differential equation governing the flow and the boundary conditions are transformed to a system of ordinary differential equations with appropriate boundary conditions the obtained ordinary differential equations are solved numerically using the shooting method the effect of prandlt number on the velocity and temperature of the boundary layer is plotted in curves a comparison with previous work is presented

vibration of a beam with a breathing crack a continuous cracked beam vibration theory is used for the prediction of changes in transverse vibration of a simply supported beam with a breathing crack the equation of motion and the boundary conditions of the cracked beam considered as a one dimensional continuum were used the eigenfrequency changes due to a breathing edge crack are shown to depend on the bi linear character of the system the associated linear problems are solved over their respective domains of definition and the solutions are matched through the initial conditions the changes in vibration frequencies for a fatigue breathing crack are smaller than the ones caused by open cracks the method has been tested for the evaluation of the lowest natural frequency of lateral vibration for beams with a single edge breathing crack experimental results from aluminium beams with fatigue cracks are used for comparison with the analytical results

meshfree co rotational formulation for two dimensional continua in this paper a meshfree co rotational formulation for two dimensional continua is proposed in a co rotational formulation the motion of a body is separated into rigid motion and strain producing deformation traditionally this has been done in the setting of finite elements for beams and shell type elements in the present work every node in a meshfree discretized domain has its own co rotating coordinate system three key ingredients are established in order to apply the co rotational formulation i the relationship between global and local variables ii the angle of rotation of a typical co rotating coordinate system and iii a variationally consistent tangent stiffness matrix an algorithm for the co rotational formulation based on load control is provided maximum entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations numerical examples are presented to demonstrate the validity of the meshfree co rotational formulation copyright 2009 john wiley sons ltd

out of plane dynamic analysis of beams with arbitrarily varying curvature and cross section by dynamic stiffness matrix method abstract the first known dynamic stiffness matrix for noncircular curved beams with variable cross section is developed with which an exact solution of the out of plane free vibration of this type of beam is derived by using the laplace transform technique and the developed dynamic stiffness matrix and equivalent nodal force vector the highly accurate dynamic responses including the stress resultants of the curved beams subjected to various types of loading can be easily obtained the dynamic stiffness matrix and equivalent nodal force vector are derived based on the general series solution of the differential equations for the out of plane motion of the curved beams with arbitrary shapes and cross sections the validity of the present solution for free vibration is demonstrated through comparison with published data the accuracy of the present solution for transient response is also confirmed through comparison with the modal superposition solution for a simply supported circular beam subjected to a moving load with the proposed solution both the free vibration and forced vibration of non uniform parabolic curved beams with various ratios of rise to span are carried out nondimensional frequency parameters for the first five modes are presented in graphic form over a range of rise to span ratios 0 05 h l 0 75 with different variations of the cross section dynamic responses of the fixed fixed parabolic curved beam subjected to a rectangular pulse are also presented for different rise to span ratios

group analysis for natural convection from a vertical plate the steady laminar natural convection of a fluid having chemical reaction of order n past a semi infinite vertical plate is considered the solution of the problem by means of one parameter group method reduces the number of independent variables by one leading to a system of nonlinear ordinary differential equations two different similarity transformations are found in each case the set of differential equations are solved numerically using runge kutta and the shooting method for each transformation different schmidt numbers and chemical reaction orders are tested

how big can the commutator of two matrices be and how big is it typically abstract numerical experiments show that the frobenius norm of the commutator of two large matrices typically clusters sharply around a certain value which moreover is much smaller than one would predict the purpose of this paper is to give a rigorous foundation of this phenomenon we also discuss the question how big the frobenius norm of commutator of the two matrices can actually be

flutter and divergence instability of nonconservative beams and plates for a conservative system the only possible instability is of divergence type for a nonconservative system however instability can be of divergence flutter or both depending on the amount of nonconservativeness in this paper we study the instability of a cantilevered beam and a simply supported plate subjected to a combination of fixed and follower forces a nonconservative parameter is introduced to provide all possible combinations of these forces for the beam instability changes from divergence to flutter at a critical value of this parameter for values of the parameter above the critical value the flutter instability remains as the only instability pattern in contrast to implications in the literature for the plate the instability is governed by flutter for a certain range of the nonconservative parameter even though divergence instability still exists the range depends on the geometry aspect ratio and material property poisson s ratio of the plate

free vibration of an edge cracked beam with a dugdale barenblatt cohesive zone abstract this paper presents a theoretical investigation into the effect of crack plane plasticity on the vibration of a transversely edge cracked beam the crack plane is presumed to be loaded statically in either bending or shear such that dugdale barenblatt type cohesive zones are activated in the crack plane ahead of the crack tip then the beam undergoes small amplitude vibrations about this state euler bernoulli beam theory is used for the vibration analysis in which an infinitesimally thin beam element surrounding the crack plane is replaced with a bending and a shear line spring the line springs by nature account for the discontinuities in slope and deflection respectively across the crack plane previous studies of the elastic problem no cohesive zones calculate the compliances of the line springs from tabulated linear elastic fracture mechanics solutions and the compliances depend only on the flexural rigidity of the beam and on the crack length to beam depth ratio here the compliance of the line spring for the mode loaded statically in the nonlinear range is calculated using the boundary element method bem in an iterative nonlinear analysis and the compliance depends nonlinearly on the load across the crack plane this nonlinear load deformation relation is linearized about the applied static load level and the resulting compliance used in the vibration analysis among other results for mode i cohesive zones there is a strong reduction of the fundamental frequency as both load and crack length increase and as yield strength decreases the results show that there is some potential for designing a non destructive material characterization technique which would use the changes in frequency to infer the properties of material behavior laws in the cohesive zone

detection of damage in beams subjected to diffused cracking this paper addresses the problem of identifying structural damage affecting one zone of a beam using measured frequencies the beam model has a zone in which the stiffness is lower than the undamaged value damage is defined by three parameters position extension and degree which need to be identified in the inverse problem the solution of the direct problem is first obtained and the peculiarities of damage detection are examined two different procedures for damage identification are proposed which use frequency measurements and take advantage of the peculiarities of the problem the first procedure is based on the characteristic equation error and the second on the comparison between analytical and experimental frequency values the identifiability and ill conditioning properties are discussed by referring to cases with pseudo experimental data

absolute versus relative formulations of the moving oscillator problem in the general framework of the bridge vehicle dynamic interaction the so called moving oscillator problem is revisited in order to provide a deeper insight into some analytical and physical aspects not specifically analyzed in previous investigations without lack of generality the case of a stream of moving oscillators crossing a simply supported beam with arbitrary time law is considered the formulations in terms of both absolute and relative displacements of the moving oscillators are critically reviewed and compared and alternative sets of differential equations with time dependent coefficients are derived the study enlightens both theoretically and numerically that impulsive contributions to the dynamic response appear in the relative displacement formulation at the time instants in which each vehicle enters or exits the bridge it is demonstrated that such contributions somehow hidden in the absolute displacement formulation may have a significant influence on the vibration of the moving oscillators and thus cannot be a priori neglected in the analysis it is also shown that the analytical and computational difficulties associated with these additional impulses make preferable the use of the absolute displacement formulation far from being restricted to the case of simply supported beams these findings are valid for any type of bridge structure which induces a discontinuity in the slope of the road profile experienced by the vehicles

free vibrations of spatial timoshenko arches abstract this paper addresses the evaluation of the exact natural frequencies and vibration modes of structures obtained by assemblage of plane circular arched timoshenko beams the exact dynamic stiffness matrix of the single circular arch in which both the in plane and out of plane motions are taken into account is derived in an useful dimensionless form by revisiting the mathematical approach already adopted by howson and jemah 1999 18 for the in plane and the out of plan natural frequencies of curved timoshenko beams the knowledge of the exact dynamic stiffness matrix of the single arch makes the direct evaluation of the exact global dynamic stiffness matrix of spatial arch structures possible furthermore it allows the exact evaluation of the frequencies and the corresponding vibration modes for the distributed parameter model through the application of the wittrick and williams algorithm consistently with the dimensionless form proposed in the derivation of the equations of motion and the dynamic stiffness matrix an original and extensive parametric analysis on the in plane and out of plane dynamic behaviour of the single arch for a wide range of structural and geometrical dimensionless parameters has been performed moreover some numerical applications relative to the evaluation of exact frequencies and the corresponding mode shapes in spatial arched structures are reported the exact solution has been numerically validated by comparing the results with those obtained by a refined finite element simulation

on nonuniform euler bernoulli and timoshenko beams with jump discontinuities application of distribution theory in this article bending of nonuniform euler bernoulli and timoshenko beams with jump discontinuities in the slope deflection and mechanical properties are studied the governing equations are obtained in the space of generalized functions and the expression of its governing differential equations in terms of a single displacement function and a single rotation function is shown always to be possible in contrast for a nonuniform euler bernoulli beam with jump discontinuities in slope and deflection and abrupt changes in flexural stiffness the governing equation can be written in terms of a single displacement function only under certain conditions it is observed that for most discontinuous nonuniform euler bernoulli beams we cannot write the governing differential equation in terms of a single displacement function usually if there are n discontinuity points on a nonuniform euler bernoulli beam n 1 displacement functions appear in the governing equilibrium equation

damage detection from changes in curvature mode shapes abstract a damage in a structure alters its dynamic characteristics the change is characterized by changes in the eigenparameters i e natural frequency damping values and the mode shapes associated with each natural frequency considerable effort has been spent in obtaining a relationship between the changes in eigenparameters the damage location and the damage size most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage in this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure by using a cantilever and a simply supported analytical beam model it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure the changes in the curvature mode shapes increase with increasing size of damage this information can be used to obtain the amount of damage in the structure finite element analysis was used to obtain the displacement mode shapes of the two models by using a central difference approximation curvature mode shapes were then calculated from the displacement mode shapes