probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

exponential stability of variable coefficients rayleigh beams under boundary feedback controls a riesz basis approach in this paper we study the boundary stabilizing feedback control problem of rayleigh beams that have non homogeneous spatial parameters we show that no matter how non homogeneous the rayleigh beam is as long as it has positive mass density stiffness and mass moment of inertia it can always be exponentially stabilized when the control parameters are properly chosen the main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a riesz basis in the state hilbert space as a by product a conjecture in guo j optim theory appl 112 3 2002 529 is answered

disturbance estimator based output feedback exponential stabilization for euler bernoulli beam equation with boundary control in this paper we are concerned with the output feedback exponential stabilization problem for a system plant described by a one dimensional euler bernoulli beam equation the measurements are only the displacement and the angular velocity at the right end an infinite dimensional estimator is designed to estimate the disturbance with the estimated disturbance we propose a state observer that is exponentially convergent to the original system then design two different kinds of stabilizing controllers one is based on the velocity feedback the other is based on the angular velocity feedback in both cases by adopting the riesz basis approach the exponential stability of the closed loop systems is built with guaranteeing that all internal systems are uniformly bounded the numerical experiments are carried out to illustrate the theoretical results

transversal waves in beams via the network simulation method abstract in this paper the network simulation method nsm is proposed as a tool for solving the transverse motion of thin beams resulting from a bending action the reliability and robustness of the proposed technique is demonstrated with two examples in which a pinned pinned beam and a clamped mass load beam are respectively submitted to different flexural forces the nsm is a numerical technique for the solution of linear and non linear problems which are defined by a mathematical model assuming that the electrical variables of voltage and current are equivalent respectively to the displacement and the spatial variation of the displacement a network model for each volume element is designed such that electrical equations are formally equivalent to the spatial discretised equations of the mechanical model the whole network model including the devices associated with the boundary conditions is solved by the numerical computer code pspice the numerical results have been compared with those obtained by the mode superposition method the agreement between them is impressive the temporal behaviour of each quantity involved in the problem is obtained without transform complications since the whole study has been conducted in the time domain with the possibility of obtaining the corresponding spectral content at the end of calculation the main contribution of the present paper therefore is to employ the nsm numerical method in order to arrive at solutions in the time domain of dynamic mechanical problems one must also emphasize that nsm has not previously been used by other authors to solve physical problems like the ones presented here

dynamics of timoshenko beam on linear and nonlinear foundation phase relations significance of the second spectrum stability abstract some controversial and unclear issues in the timoshenko beam tb theory are analyzed with emphasis on features of two branches of frequencies taking into account the elastic foundation creates additional opportunities for comparative analysis of both the branches relationship between the second branch modes and so called thickness shear mode is discussed phase relations between the deformation and force factors in beam for both the branches are established which generalize the previously established relations it is shown that the elastic foundation can change these relations for the first branch based on this analysis the significance of the second spectrum is assessed and peculiarities of dispersion relations for frequency phase and group velocities are explained free transverse waves in infinitely long beam on nonlinear cubic foundation are studied using the multiple scales method the obtained nonlinear schrodinger equation enables us to check stability of the harmonic waves in infinite beam and to determine diapasons of their modulational instability

exponential stability for a class of boundary conditions on a euler bernoulli beam subject to disturbances via boundary control abstract this paper outlines the procedure for applying a sliding mode backstepping boundary control technique to a euler bernoulli beam subject to unknown bounded disturbances on the boundary for four different types of boundary conditions referred to in this paper as a class of boundary conditions these boundary conditions include all combinations of pinned and sliding types with control and disturbances on one boundary i e pinned pinned pinned sliding sliding pinned and sliding sliding with input on the second boundary the technique was developed for the specific case of a pinned pinned beam in the literature and in this paper the technique is generalized to the degree that it is possible to cover all four of the aforementioned cases furthermore a proof of the fully arbitrary exponential stability of the closed loop system is provided in this paper to this point only asymptotic stability was shown for the closed loop system for one case the technique outlined in this paper provides for a class of boundary conditions a controller that exponentially stabilizes the vibrations in a beam in the presence of bounded unknown disturbances the beam displacement returns to the origin and its decay is bounded by a known exponential decay function after a known finite time reaching phase

identification of unknown temporal and spatial load distributions in a vibrating euler bernoulli beam from dirichlet boundary measured data source identification problems in a system governed by euler bernoulli beam equation x u t t r x u x x x x f x h t x t 0 l 0 t from available boundary observation measured data namely from measured slope t u x 0 t at x 0 are considered we propose a new approach to identifying the unknown temporal h t and spatial f x loads this novel approach is based on weak solution theory for pdes and quasi solution method for inverse problems combined with the adjoint method it allows to construct not only a mathematical theory of inverse source problems for euler bernoulli beam but also an effective numerical algorithm for reconstruction of unknown loads introducing the input output operators h t u x 0 t h and f t u x 0 t f t 0 t we show that both operators are compact based on this result and general regularization theory we prove an existence of unique solutions of the regularized normal equations i h and i f then we develop the adjoint problem approach to prove frechet differentiability of the corresponding cost functionals and lipschitz continuity of the frechet gradients derived explicit gradient formulas via the adjoint problem solution and known load allow use of gradient type convergent iterative algorithms results of numerical simulations for benchmark problems illustrate robustness and high accuracy of the algorithm based on the proposed approach

theoretical analysis of transient waves in a simply supported timoshenko beam by ray and normal mode methods abstract the dynamic transient responses of a simply supported timoshenko beam subjected to an impact force are investigated by two theoretical approaches ray and normal mode methods the mathematical methodology proposed in this study for the ray method enable us to construct the solution for the interior source problem and to extend to solve the complicated problem for the multi span of the timoshenko beam numerical results based on these two approaches are compared the comparison in this study indicates that the normal mode method is more computationally efficient than the ray method except for very short time after the impact the long time transient responses are easily calculated using the normal mode method it is shown that the average long time transient response converges to the corresponding static value the timoshenko beam theory is more accurate than the bernoulli euler beam theory because it includes shear and rotary inertia this study also provides the slender ratio for which the bernoulli euler beam can be used for the transient response analysis of the displacement moreover the resonant frequencies obtained from finite element calculation based on the three dimensional model are compared with the results calculated using the timoshenko beam and bernoulli euler beam theories it is noted in this study that the resonant frequency can be accurately determined by the timoshenko beam theory if the slender ratio is larger than 100 and by the bernoulli euler beam theory if the slender ratio is larger than 400

numerical approximations of exact controllability problems by optimal control problems for parabolic differential equations we consider the exact controllability problems for systems governed by linear parabolic differential equations we prove that the solution of an associated optimal control problem converges to the solution of the exact controllability problem the numerical solution of the exact controllability problem is then obtained by solving the optimal control problem numerically numerical experiments are presented

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow the present paper is devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow this model has been developed in the flight systems research center of the university of california at los angeles the model is governed by a system of two coupled integro differential equations and a two parameter family of boundary conditions modelling the action of self straining actuators the differential parts of these equations form a coupled linear hyperbolic system the integral parts are of the convolution type the system of equations of motion is equivalent to a single operator evolution convolution equation in the energy space the laplace transform of the solution of this equation can be represented in terms of the so called generalized resolvent operator which is an operator valued function of the spectral parameter this generalized resolvent operator is a finite meromorphic function on the complex plane having a branch cut along the negative real semi axis its poles are precisely the aeroelastic modes which are the main object of interest in the present paper the dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author this generator is a non self adjoint operator in the energy space with a purely discrete spectrum in the aforementioned series of papers it was shown that the spectrum consists of two branches and the precise spectral asymptotics with respect to the eigenvalue number was derived the asymptotic approximations for the mode shapes have also been obtained based on the asymptotic results it has been proved that the set of the generalized eigenvectors of the dynamics generator forms a riesz basis in the energy space in the present paper we consider the entire integro differential system which governs the model namely we investigate the properties of the integral convolution type part of the original system we show in particular that the set of the aeroelastic modes is asymptotically close to the discrete spectrum of the dynamics generator corresponding to the differential part the results of this paper will be important for the reconstruction of the solution of the original initial boundary value problem from its laplace transform and for the analysis of the flutter phenomenon in the forthcoming work

the distributions of the determinant of fixed trace ensembles of real symmetric and of hermitian random matrices probability densities of the determinant are obtained for fixed trace ensembles of n x n hermitian and real symmetric matrices from the exact determinant densities of the gaussian orthogonal ensemble goe and of the gaussian unitary ensemble gue respectively

modelling of floating slab tracks with continuous slabs under oscillating moving loads this paper discusses modelling of floating slab tracks on rigid foundations the model consists of an upper euler bernoulli beam to account for both the rails and a lower euler bernoulli beam to account for the slab there are two continuous resilient layers in the model one to account for railpads between the rails and the slab and one to account for slab bearings underneath the slab the fourier transformation method is used to calculate displacements of such a track under an oscillating moving load these results are used to calculate the cut on frequencies and critical velocity of the track the work in this paper also demonstrates some basic concepts in the context of vibration of infinite systems including dispersion curves critical velocities load velocity lines and coupling of systems in the wavenumber frequency domain the paper offers a complete treatment of the issues important to the analysis of a double beam system not elsewhere available in the published literature

the fixed trace β hermite ensemble of random matrices and the low temperature distribution of the determinant of an n n β hermite matrix the hermite ensemble he of tridiagonal n n random matrices of dumitriu and edelman 2002 j math phys 43 5830 is a continuum of ensembles in which the reciprocal of the temperature in the 2d electrostatic interpretation of the eigenvalue characteristics can take any value the eigenvalue distributions coincide with those of the classical gaussian ensembles goe gue gse for 1 2 4 a fixed trace hermite ensemble fthe is defined from the he and is used to extend the spherical ensembles of classical symmetries to spherical ensembles at low temperature when for a fixed value of n the asymptotic distributions of reduced determinants dn of random n n h and fth matrices are shown to be standard gaussians accordingly the fluctuations of the potential at the origin ln dn have a generalized gumbel distribution at low temperature for large n and large a ln n variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions

exact boundary controllability of a hybrid system of elasticity a hybrid control system is presented an elastic beam governed by a partial differential equation linked to a rigid body which is governed by an ordinary differential equation and to which control forces and torques are applied the entire system elastic beam plus rigid body is proved to be exactly controllable by smooth open loop controllers applied to the rigid body only and in arbitrarily short durations this system is modeled as a two dimensional space structure

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

gevrey s and trace regularity of a semigroup associated with beam equation and non monotone boundary conditions abstract the main aim of the paper is to present a technique that allows to infer wellposedness trace and gevrey s regularity of hyperbolic like pde s with non monotone boundary conditions the lack of monotonicity prevents applicability of the known semigroup methods in this paper we show how recently developed tools of microlocal analysis d tataru a priori estimates of carleman s type in domains with boundary j math pure appl 73 1994 353 387 combined with some spectral theory can be used successfully in order to obtain the needed inequalities the method will be illustrated on a simple example of beam equation with non monotone boundary conditions

mathematical models of flageolet harmonics on stringed instruments flageolet is a common technique to elicit harmonics on stringed instruments like guitars pianos and the violin family the bowed or plucked string is subdivided by a slight touch of the finger the paper discusses appropriate linear wave equations which model the flageolet phenomenon the standard second order wave equation fails because the resulting dirichlet boundary condition at the finger uncouples the two parts of the string and produces tones different from the flageolet we include and discuss fourth order corrections due to string stiffness as a possible source for the flageolet phenomenon

lyapunov approach to output feedback stabilization for the euler bernoulli beam equation with boundary input disturbance we propose a boundary output feedback control law for a one dimensional euler bernoulli beam equation with general external disturbance entering the control end a galerkin approximation scheme is constructed to show the existence of solution to the closed loop system the exponential stability of the closed loop system is obtained by the lyapunov functional method numerical simulations are presented for illustration

four branch vibrational spectrum of double walled carbon nanotube model asymptotic distribution of the discrete spectrum eigenvalues for a recently developed model of a double walled carbon nanotube is presented in this paper the corresponding initial boundary value problem is reduced to an evolution equation whose dynamics generator is a non self adjoint matrix differential operator with a purely discrete spectrum it is shown in the paper that the entire spectrum asymptotically splits into four spectral branches asymptotic representation is derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity to prove the results a two step procedure involving the construction of the left and right reflection matrices has been used

transient wave analysis of a cantilever timoshenko beam subjected to impact loading by laplace transform and normal mode methods abstract this study applies two analytical approaches laplace transform and normal mode methods to investigate the dynamic transient response of a cantilever timoshenko beam subjected to impact forces explicit solutions for the normal mode method and the laplace transform method are presented the durbin method is used to perform the laplace inverse transformation and numerical results based on these two approaches are compared the comparison indicates that the normal mode method is more efficient than the laplace transform method in the transient response analysis of a cantilever timoshenko beam whereas the laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi span timoshenko beam furthermore a three dimensional finite element cantilever beam model is implemented the results are compared with the transient responses for displacement normal stress shear stress and the resonant frequencies of a timoshenko beam and bernoulli euler beam theories the transient displacement response for a cantilever beam can be appropriately evaluated using the timoshenko beam theory if the slender ratio is greater than 10 or using the bernoulli euler beam theory if the slender ratio is greater than 100 moreover the resonant frequency of a cantilever beam can be accurately determined by the timoshenko beam theory if the slender ratio is greater than 100 or by the bernoulli euler beam theory if the slender ratio is greater than 400

non linear coupled transverse and axial vibration of a compliant structure part 1 formulation and free vibration a compliant tower in the ocean environment is modelled as a beam undergoing coupled transverse and axial motion the equations of motion are non linear and coupled and are derived here the beam is assumed to be supported by a linear elastic torsional spring at one end and with a point mass at the other end such a model is representative of numerous applications the equations of motion and boundary conditions are obtained using hamilton s variational principle it is assumed that strains are small but the rotation is moderate compared to the strain so that the equations of motion for the axial and transverse motion are non linearly coupled the free response in vacua and the free response in water are considered in particular the fluid forces the added mass and drag forces are modelled using a semi empirical morison equation the resulting non linear coupled partial differential equations are solved numerically using the finite difference approach in part 2 of this work various forced responses are studied

variational derivation of governing differential equations for truncated version of bresse timoshenko beams abstract in this paper we provide a variational derivation of modified bresse timoshenko equations this process leads to the additional term to so called truncated set of bresse timoshenko equations this new term is associated with the modified slope inertia the two sets of truncated versions of the bresse timoshenko equations are contrasted with each other as well as with the original bresse timoshenko equations on the example of the a beam that is simply supported at its both ends and b the cantilever beam with end mass it is concluded that application of either truncated sets of equations is advantageous over the original bresse timoshenko equations as far as two truncated sets of equations are concerned the variationally derived set appears to be preferable

quasicontinuum based multiscale approaches for plate like beam lattices experiencing in plane and out of plane deformation the quasicontinuum qc method is a multiscale approach that aims to reduce the computational cost of discrete lattice computations the method incorporates smallscale local lattice phenomena e g a single lattice defect in macroscale simulations since the method works directly and only on the beam lattice qc frameworks do not require the construction and calibration of an accompanying continuum model e g a cosserat micropolar description furthermore no coupling procedures are required between the regions of interest in which the beam lattice is fully resolved and coarse domains in which the lattice is effectively homogenized hence the method is relatively straightforward to implement and calibrate in this contribution four variants of the qc method are investigated for their use for planar beam lattices which can also experience out of plane deformation the different frameworks are compared to the direct lattice computations for three truly multiscale test cases in which a single lattice defect is present in an otherwise perfectly regular beam lattice

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

vibration of composite beams with deformable shear connection mathematical modelling and numerical solution using general purpose software for two point boundary value problems abstract in this paper we develop from first principles a linear one dimensional mathematical model for the dynamic response of planar composite beams with deformable shear connection in which the inertias associated with the axial motion of the components and with the bending rotations of their cross sections are consistently taken into account to the best of our knowledge the paper by auclair et al j sound vib 366 2016 230 247 is the only work which heretofore considered these two effects however we point out and correct a flaw in the derivations of these authors to solve the eigenproblem defining the natural vibration frequencies of the composite beam and the corresponding vibration modes we advocate an untried approach the eigenproblem is converted into a standard inhomogeneous boundary value problem which can be solved using freely available state of the art mathematical software for benchmarking purposes the formulation of a conventional displacement based conforming finite element is also briefly addressed the effect of rotational and axial inertias on the natural vibration frequencies is found to be marginal and considerably less pronounced than the predictions of auclair and coworkers in disagreement with the findings of these authors it appears legitimate to ignore the above mentioned effect in a full dynamic analysis of composite beams at least within the practical range of span to depth ratios

the controllability of systems governed by parabolic differential equations abstract the main result of this paper is a new weaker condition for the exact controllability of linear parabolic partial differential equations the result is derived through an examination of a related optimal control problem it is shown that if the terminal state belongs to the sobolev space of functions having two square integrable derivatives then the parabolic equation is exactly controllable

material point method enhanced by modified gradient of shape function a numerical scheme of computing quantities involving gradients of shape functions is introduced for the material point method mpm so that the quantities are continuous as material points move across cell boundaries the noise and instability caused by cell crossing of the material points are then eliminated in this scheme the formulas used to compute these quantities can be expressed in the same forms as in the original material point method but with the gradient of the shape function modified for one dimensional cases the gradient of the shape function used in the generalized interpolation material point gimp method is a special case of the modified gradient if the characteristic function of a material point is introduced the characteristic function of a material point is not otherwise needed in this scheme therefore difficulties in tracking its evolution are avoided although the support of the modified gradient of a shape function is enlarged from the cell containing the material point to also include the immediate neighbor cells all the non local effects of a material point can be accounted for by two consecutive local operations therefore this scheme can be used in calculations with unstructured grids this scheme is proved to satisfy mass and momentum conservations exactly the error in energy conservation is shown to be second order on both spatial and temporal discretizations although the error in energy conservation is the same order as that in the original material point method numerical examples show that this scheme has significantly better energy conservation properties than those of the original material point method

distribution of the determinant of a random real symmetric matrix from the gaussian orthogonal ensemble random matrix theory finds considerable use in various branches of physics notably in nuclear physics quantum chaology and for investigating hamiltonians of disordered and strongly interacting quantum systems 1 3 and references therein among the ensembles of n3n random matrices hn three gaussian ensembles have been studied extensively 1 3 and are still being investigated their probability densities are proportional to exp 2tr hn where tr means trace matrices are real symmetric for the gaussian orthogonal ensemble goe hermitian for the gaussian unitary ensemble gue and quaternion self dual for the gaussian symplectic ensemble many characteristics both exact at finite n and asymptotic at large n of the distributions of eigenvalues of n3n random matrices of various symmetries are known for the gaussian ensembles 1 3 the properties of eigenvalues of random matrix ensembles rme s can be interpreted in two dimensions 2d from the equilibrium characteristics of a gas of n identical point charges on a line 1 10 often referred to as a log gas 9 10 which interact via a logarithmic coulomb potential and are confined by an external potential the external potential is harmonic in the case of gaussian ensembles as can be seen by rewriting eq 1 below as a boltzmann factor at a temperature 1 b that depends only on the symmetry of the considered ensemble with b51 2 and 4 for the orthogonal unitary and symplectic ensembles respectively 1 3 the problems of the distribution and associated fluctuations of linear statistics in rme s 1 10 and references therein are of interest for instance in the study of conductance fluctuations in mesoscopic conductors the linear statistics problem deals with physical quantities f which are given by sums f5 k51 n f l k over the eigenvalues l k of a random matrix where f x may depend nonlinearly on x general arguments have been used to predict that the distribution of any linear statistic is gaussian and independent of the confining potential in the scaled asymptotic limit n provided its variance is finite 5 10 the potential fluctuations have been investigated in 2d and 3d classical charged systems 11 12 and more recently for circular ensembles by baker and forrester who further studied the dipole moment statistic 9 within the log gas interpretation the statistic of the potential at the origin is obtained for f x 52lnuxu that is for f5 2lnudu it may thus be derived directly from the distribution of the determinant d of hn we notice further that the partition functions of log gases whose confining harmonic potential includes a supplementary logarithmic contribution are by products of the derivation of the determinant distributions of gaussian ensembles see eqs 10 12 19 and 26 for b51 the rme s associated with the latter log gases belong to the family of generalized gaussian ensembles 13 14

external and internal coupling effects of rotor s bending and torsional vibrations under unbalances abstract external and internal bending torsion coupling effects of a rotor system with comprehensive unbalances are studied by analytical analysis and numerical simulations based on lagrangian approach a full degree of freedom dynamic model of a jeffcott rotor is developed the harmonic balance method and the floquet theory are combined to analyze the stability of the system equations numerical simulations are conducted to observe the bending torsion coupling effects in the formulation of rotordynamic model two bending torsion coupling patterns external coupling and internal coupling are suggested by analytical analysis it is concluded that the periodic solution of the system is asymptotically stable from numerical simulations three bending torsion coupling effects are observed in three cases under static unbalance synchronous torsional response is observed which is the result of external coupling under unbalanced force under dynamic unbalance two time synchronous frequency torsional response is observed which is the result of internal coupling under unbalanced moment under comprehensive unbalance synchronous and two time synchronous frequency torsional components are observed which are the results of both external and internal couplings under unbalanced force and moment these observations agree with the analytical analysis it is believed that these observed phenomena should make sense in the dynamical design and fault diagnostics of a rotor system

modes for the coupled timoshenko model with a restrained end abstract the modes of the second order timoshenko system for the displacement and rotation of a fixed beam with a restrained end at the left are formulated in terms of a fundamental spatial response this is done without decoupling the system into fourth order scalar equations the restrained end leads to time space boundary conditions which introduce the frequency as a parameter into the system of equations for determining the modes these equations involve first order derivatives and consequently the modes are determined by solving a non conservative differential system this modal differential equation is discussed in terms of a fundamental matrix response it is determined by applying a closed formula that was obtained by the first author and involves the characteristic polynomial of the modal differential equation

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

the second spectrum of timoshenko beam theory further assessment a review of contributions and views on the second spectrum of timoshenko beam theory tbt over the past two decades together with some new results are presented it is shown that the timoshenko frequency equation factorises not solely for hinged hinged end conditions as is often claimed but also for guided guided and guided hinged these new cases may be regarded as portions of a multi span hinged hinged beam a higher derivative lagrangian that leads directly to the well known fourth order timoshenko beam equation is reviewed a simple relationship between the so called ostrogradski energy and the mechanical energy is derived for hinged hinged end conditions it is shown that the ostrogradski energy is positive for the first spectrum but negative for the second within some branches of physics this would be sufficient to conclude that the second spectrum is unphysical a numerical example presented by levinson and cooke is re examined using both tbt and exact plane stress elastodynamic theory agreement is excellent for the first spectrum however the second spectrum predictions are not in consistent agreement with any single mode of vibration for long wavelength it is very close to the second asymmetric mode but as wavelength shortens it becomes closer to the second symmetric then the third asymmetric modes the conclusion remains unchanged the second spectrum predictions of tbt should be disregarded

flexural waves on narrow plates flexural wave speeds on beams or plates depend upon the bending stiffnesses which differ by a well known factor depending on the poisson s ratio a quantitative analysis of a plate of finite lateral width displays the plate to beam transition and permits asymptotic analysis that shows the dependence on the width orthotropic plates are analyzed using both the kirchhoff and kirchhoff rayleigh theories and isotropic plates are considered for mindlin s theory with and without rotational inertia a frequency dependent young s modulus for beams or strips of finite width is suggested although the form of the correction to the modulus is not unique and depends on the theory used the sign of the correction for the kirchhoff theory is opposite to that for the mindlin theory these results indicate that the different plate and beam theories can produce quite distinct behavior this divergence in predictions is further illustrated by comparison of the speeds for antisymmetric flexural modes on narrow plates th

a new rectangular beam theory abstract a new theory for beams of rectangular cross section which includes warping of the cross sections is presented in the present work by satisfying the shear free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required it is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in timoshenko beam theory provided that for the timoshenko equation the shear coefficient is taken to be 5 6 this value lies within the range of values 0 822 0 870 appearing in the literature for the beam of rectangular cross section results for two typical static examples are given for both the new theory and timoshenko beam theory these results are compared with the solutions of the comparable problems in the linear theory of elasticity for the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while timoshenko beam theory underestimates the shear correction term by 20 for the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

large rectangular random matrices models with a multiplet of field variables arranged into rectangular matrices in the limit of infinite dimensions of the matrices are studied in zero dimensional space where the problem is a combinatorial one a closed solution is given that improves the one previously known in arbitrary space dimension a symmetry is described that connects rectangular models with vector models

stability of a flexible structure with destabilizing boundary conditions the euler bernoulli beam model with non dissipative boundary conditions of feedback control type is investigated components of the two dimensional input vector are shear and moment at the right end and components of the observation vector are time derivatives of displacement and slope at the right end the codiagonal matrix depending on two control parameters relates input and observation the paper contains five results first asymptotic approximation for eigenmodes is derived second the main identity is established it provides a relation between mode shapes of two systems one with non zero control parameters and the other one with zero control parameters third when one control parameter is positive and the other one is zero the main identity yields stability of all eigenmodes though the system is non dissipative fourth the stability of eigenmodes is extended to the case when one control parameter is positive and the other one is sufficiently small finally existence and properties of deadbeat modes are investigated

exponential stabilization of a rayleigh beam using collocated control we consider a hinged elastic beam described by the rayleigh beam equation on the interval 0 pi we assume the presence of two sensors one measures the angular velocity of the beam at a point xi epsiv 0 pi and the other measures the bending curvature of the beam at the same point if xi 0 or xi pi then the second output is not needed the corresponding operator semigroup is unitary on a suitable hilbert state space these two measurements are advantageous because they make the open loop system exactly observable regardless of the point xi we design the actuators and the feedback law in order to exponentially stabilize this system using the theory of collocated static output feedback developed in our recent paper we design the actuators such that they are collocated meaning that b c where b is the control operator and c is the observation operator it turns out that if xi epsiv 0 pi then the actuators cause a discontinuity of the bending exactly at this is the price in this example of having collocated actuators and sensors this obliges us to use an extension of to define the output signal in terms of the left and right limit of the bending at xi we prove that for all static output feedback gains in a suitable finite range the closed loop system is well posed and exponentially stable this follows from the general theory in our paper whose main points are recalled here

the second frequency spectrum of timoshenko beams abstract a second spectrum of frequencies was reported in early analytical work on the vibrations of timoshenko beams however in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded in fact from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies in this paper a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately guidelines are provided which help to identify and classify the frequencies into two separate spectra

non dissipative boundary feedback for rayleigh and timoshenko beams abstract we show that a non dissipative feedback that has been shown in the literature to exponentially stabilize an euler bernoulli beam makes a rayleigh beam and a timoshenko beam unstable

stabilisation of timoshenko beam system with delay in the boundary control in this paper we consider the exponential stabilisation problem of a timoshenko beam with delay in boundary control suppose that the controller outputs of the forms 1u1 t 1u1 t and 2u2 t 2u2 t where u1 t and u2 t are the inputs of boundary controllers in the past most of the stabilisation results are required j j 0 j 1 2 in the present paper we shall give a new dynamic feedback control law that makes the system exponential stabilisation 0 provided that j j j 1 2 the exponential stabilisation result is proved via test of exact observability of the system

on the c0 semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam in this paper we show that a linear unbounded operator associated with an euler bernoulli beam equation under shear boundary feedback generates a c0 semig roup in the underlyingstate hilbert space this provides an answer to a longtime unsolved problem due to the lack of dissipativity for the operator the main steps are a careful estimation of the green s function and the verification of the riesz basis property for the generalized eigenfunctions as a consequence we show that this semigroup is differentiable and exponentially stable which is in sharp contrast to the properties possessed by most feedback controlled beams based on a passive design principle 2005 elsevier b v all rights reserved

lie and noether symmetries for a class of fourth order emden fowler equations a group classification of a fourth order ordinary differential equation is carried out the noether symmetries are considered and some first integrals are established solutions for special lane emden systems are also obtained from the invariant solutions of the investigated equation

comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis by using the duffing oscillator as a case study this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the nonlinear output frequency response functions nofrfs are one of the solutions obtained using the harmonic balance method hbm a comparison of the performances of the two methods shows that the hbm can capture the well known jump phenomenon but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components although the nofrfs cannot capture the jump phenomenon the method has few computational restrictions for the nonlinear damping systems the nofrfs can give better predictions for all the harmonic components in the system response than the hbm even when the damping system is strongly nonlinear

vibration analysis of periodic cellular solids based on an effective couple stress continuum model abstract cellular solids are usually treated as homogeneous continuums with effective properties nevertheless these mechanical properties depend strongly on the ratio of the specimen size to the cell size these size effects may be accounted for according to preliminary static analysis of effective continuums based on couple stress theory in this paper an effective dynamic continuum model based on couple stress theory is proposed to analyze the behavior of free vibrations of periodic cellular solids in this continuum model the effective mechanical constants of the effective continuum are deduced by an equivalent energy method the cellular solid structure is then replaced with the equivalent couple stress continuum with same overall dimension and shape moreover the finite element formulation of the couple stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis the eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method a conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids

active disturbance rejection control for rejecting boundary disturbance from multidimensional kirchhoff plate via boundary control in this paper an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional kirchhoff plate via boundary control the disturbance and control input are assumed to be matched the active disturbance rejection control approach is adopted for developing the algorithm a state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time dependent test functions the proposed control law cancels the disturbance using its estimated value all subsystems in the closed loop are shown to be asymptotically stable simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

a simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method a size dependent finite element model is developed nonlocal elasticity theory is used finite element solution is presented for buckling of microtubules the effect of elastic matrix surrounded by microtubules is modeled a simple nonlocal beam model is proposed to study buckling response of protein microtubules the size effect for buckling model of microtubules is considered by using the nonlocal continuum theory finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability the influence of the small length scale on the buckling value is examined for different geometric parameters the effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

nonlinear random matrix statistics symmetric functions and hyperdeterminants nonlinear statistics i e statistics of permanents on the eigenvalues of invariant random matrix models are considered for the three dyson s symmetry classes 1 2 4 general formulas in terms of hyperdeterminants are found for 2 for specific cases and all s more computationally efficient results are obtained based on symmetric functions expansions as an application we consider the case of quantum transport in chaotic cavities extending results from savin et al 2008 phys rev b 77 125332

a series expansion approach to interpreting the spectra of the timoshenko beam abstract the timoshenko beam model results in two fourth order partial differential equations in time and space consequently solving the boundary value problem yields two independent sequences of natural frequencies and two corresponding sequences of mode shapes a particular natural frequency and its corresponding mode shape describe one particular solution to the boundary value problem of the timoshenko beam from an eigenfunction expansion sense all these possible solutions have to be considered in the complete series expansion of the solution however the question of whether these two independent sequences of natural frequencies implies the existence of two distinct spectra of frequencies has been a long standing topic of debate and hitherto has not been resolved completely the object of this paper is to provide answers to some of the issues raised by this debate in this context the complete solution in a series form to the timoshenko beam is investigated and it is shown for the first time that a particular mode shape of the solution is naturally expressed by an ordered pair of characteristic values rather than a single characteristic value this representation facilitates the progressive ordering of all the natural frequencies of the system and their respective mode shapes in a single set and eliminates the remaining argument for the two spectra interpretation

parametric resonance voltage response of electrostatically actuated micro electro mechanical systems cantilever resonators abstract this paper investigates the parametric resonance voltage response of nonlinear parametrically actuated micro electro mechanical systems mems cantilever resonators a soft ac voltage of frequency near natural frequency is applied between the resonator and a parallel ground plate this produces an electrostatic force that leads the structure into parametric resonance the model consists of an euler bernoulli thin cantilever under the actuation of electrostatic force to include fringe effect and damping force two methods of investigation are used namely the method of multiple scales mms and reduced order model rom method rom convergence of the voltage response and the limitation of mms to small to moderate amplitudes with respect to the gap gap amplitudes are reported mms predicts accurately both hopf supercritical and supercritical bifurcation voltages however mms overestimates the large gap amplitudes of the resonator and misses completely or overestimates the saddle node bifurcation occurring at large gap amplitudes rom produces valid results for small and or large gap amplitudes for a sufficient number of terms vibration modes as the voltage is swept up at constant frequency the resonator maintains zero amplitude until reaches the subcritical hopf bifurcation voltage where it loses stability and jumps up to large gap amplitudes next the gap amplitude decreases until it reaches the supercritical hopf bifurcation point and after that the gap amplitude remains zero for the voltage range considered in this work as the voltage is swept down at constant frequency the zero gap amplitude of the resonator starts increasing continuously after reaching the supercritical hopf bifurcation voltage until it reaches the saddle node bifurcation voltage when a sudden jump to zero gap amplitude occurs effects of frequency damping and fringe parameters on the voltage response show that 1 the supercritical hopf bifurcation is shifted to lower voltage values with the increase of any of the mentioned parameters 2 the subcritical hopf bifurcation is shifted to larger voltage values with the increase of damping shifted to lower voltage values with the increase of the fringe parameter and not significantly altered by the change in frequency 3 the saddle node bifurcation voltage decreases with the increase of frequency and damping and decrease of fringe parameter and 4 the saddle node bifurcation gap amplitude decreases with the increase of frequency and damping and it is not significantly altered by the change of the fringe parameter

probability density of the determinant of a random hermitian matrix the probability density function for the determinant of a nn random hermitian matrix taken from the gaussian unitary ensemble is calculated it is found to be a meijer g function or a linear combination of two meijer g functions depending on the parity of n the integer moments of this probability density are also given

riesz basis for strongly continuous groups given a hilbert space and the generator of a strongly continuous group on this hilbert space if the eigenvalues of the generator have a uniform gap and if the span of the corresponding eigenvectors is dense then these eigenvectors form a riesz basis or unconditional basis of the hilbert space furthermore we show that none of the conditions can be weakened

matching experimental and three dimensional numerical models for structural vibration problems with uncertainties abstract the simulation model which examines the dynamic behavior of real structures needs to address the impact of uncertainty in both geometry and material parameters this article investigates three dimensional finite element models for structural dynamics problems with respect to both model and parameter uncertainties the parameter uncertainties are determined via laboratory measurements on several beam like samples the parameters are then considered as random variables to the finite element model for exploring the uncertainty effects on the quality of the model outputs i e natural frequencies the accuracy of the output predictions from the model is compared with the experimental results to this end the non contact experimental modal analysis is conducted to identify the natural frequency of the samples the results show a good agreement compared with experimental data furthermore it is demonstrated that geometrical uncertainties have more influence on the natural frequencies compared to material parameters and material uncertainties are about two times higher than geometrical uncertainties this gives valuable insights for improving the finite element model due to various parameter ranges required in a modeling process involving uncertainty

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

lyapunov stability analysis of a string equation coupled with an ordinary differential system this paper considers the stability problem of a linear time invariant system in feedback with a string equation a new lyapunov functional is proposed using augmented states that enriches and encompasses the classical functionals of the literature it results in tractable stability conditions expressed in terms of linear matrix inequalities this methodology follows from the application of the bessel inequality to the projections over the legendre polynomials numerical examples illustrate the potential of our approach through three scenario a stable ode perturbed by the pde an unstable open loop ode and an unstable closed loop ode stabilized by the pde

exponential stability of uniform euler bernoulli beams with non collocated boundary controllers a b s t r a c t we study the stability of a robot system composed of two euler bernoulli beams with noncollocated controllers by the detailed spectral analysis we prove that the asymptotical spectra of the system are distributed in the complex left half plane and there is a sequence of the generalized eigenfunctions that forms a riesz basis in the energy space since there exist at most finitely many spectral points of the system in the right half plane to obtain the exponential stability we show that one can choose suitable feedback gains such that all eigenvalues of the system are located in the left half plane hence the riesz basis property ensures that the system is exponentially stable finally we give some simulation for spectra of the system

a paradigm of ill posedness with respect to time delays the authors present a simple general model of an infinite dimensional control system which is stabilized by a velocity feedback but then destabilized by small or large time delays in the feedback this model is in some sense generic for boundary stabilized conservative hyperbolic linear partial differential equations

probability density of determinants of random matrices in this brief paper the probability density of a random real complex and quaternion determinant is rederived using the singular values the behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

non ambiguous recovery of biot poroelastic parameters of cellular panels using ultrasonicwaves abstract the inverse problem of the recovery of the poroelastic parameters of open cell soft plastic foam panels is solved by employing transmitted ultrasonic waves usw and the biot johnson koplik champoux allard bjkca model it is shown by constructing the objective functional given by the total square of the difference between predictions from the bjkca interaction model and experimental data obtained with transmitted usw that the inverse problem is ill posed since the functional exhibits several local minima and maxima in order to solve this problem which is beyond the capability of most off the shelf iterative nonlinear least squares optimization algorithms such as the levenberg marquadt or nelder mead simplex methods simple strategies are developed the recovered acoustic parameters are compared with those obtained using simpler interaction models and a method employing asymptotic phase velocity of the transmitted usw the retrieved elastic moduli are validated by solving an inverse vibration spectroscopy problem with data obtained from beam like specimens cut from the panels using an equivalent solid elastodynamic model as estimator the phase velocities are reconstructed using computed measured resonance frequencies and a time frequency decomposition of transient waves induced in the beam specimen these confirm that the elastic parameters recovered using vibration are valid over the frequency range ofstudy

riesz basis property and exponential stability of controlled euler bernoulli beam equations with variable coefficients this paper studies the basis property and the stability of a distributed system described by a nonuniform euler bernoulli beam equation under linear boundary feedback control it is shown that there is a sequence of generalized eigenfunctions of the system which forms a riesz basis for the state hilbert space the asymptotic distribution of eigenvalues the spectrum determined growth condition and the exponential stability are concluded the results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam

a gradient elasticity theory for second grade materials and higher order inertia abstract second grade elastic materials featured by a free energy depending on the strain and the strain gradient and a kinetic energy depending on the velocity and the velocity gradient are addressed an inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics the state variables include the body momentum and the surface momentum related to the velocity in a nonstandard way as well as the concomitant mass accelerations and inertial forces which do intervene into the motion equations and into the force boundary conditions the boundary traction is the sum of two parts i e the cauchy traction and the gurtin murdoch traction whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface as known from the principles of surface mechanics whereby the gurtin murdoch traction incorporating the inertial surface force plays the role of applied surfacial force density the body s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it a feature that makes it possible to exploit the traction cauchy theorem within second grade materials this means that a second grade material is formed up by two sub systems that is the bulk material operating as a classical cauchy continuum and the thin boundary layer operating as a gurtin murdoch material surface the classical linear and angular momentum theorems are suitably extended for higher order inertia from which the local motion equations and the moment equilibrium equations stress symmetry can be derived for an isotropic material featured by four constants i e the lame constants and two length scale parameters aifantis model the dynamic evolution problem is characterized by a hamilton type variational principle and a solution uniqueness theorem closed form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature the paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics

implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity abstract uniform exponential decay of solution is established for the elastodynamic system of elasticity using boundary feedback control energy dissipation is introduced via linear velocity feedbacks acting through a portion of the boundary as traction forces two primary goals are achieved first these results are proven without the imposition of strong geometric assumptions on the controlled portion of the boundary thus extending earlier work which required that the domain be star shaped second the feedback is only a function of velocity as opposed to also containing the tangential derivative of the displacement resulting in a physically viable feedback proof is based on the multiplier method and relies critically on sharp trace estimates for the tangential derivative of the displacement on the boundary as well as on unique continuation results for the corresponding static system

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams

stability of a nonuniform rayleigh beam with indefinite damping abstract this is a continuation of our earlier work j m wang g q xu s p yung exponential stability for variable coefficients rayleigh beams under boundary feedback control a riesz basis approach systems control lett 51 1 2004 33 50 on the study of a nonhomogeneous rayleigh beam and this time the stabilization is achieved via an internal damping instead of the boundary feedbacks we continue to address a conjecture of guo basis property of a rayleigh beam with boundary stabilization j optim theory appl 112 3 2002 529 547 in this paper and demonstrate how the damping term can affect the decay rate asymptotically by a detailed spectral analysis we obtain a necessary condition for the stability and establish the riesz basis property as well as the spectrum determined growth condition for the system furthermore when the damping is indefinite we provide a condition on how negative the damping can be without destroying the exponential stability

a forward approach to numerical data assimilation variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena most notably in weather prediction and are usually solved via an ill posed optimal control problem for the initial state at the time of the first available measurements an alternative forward approach focuses on computation of the final state after this interval which is just as suitable for prediction purposes and is well posed without additional regularization specifically it is possible to compute projections of the unknown final state on all elements of an orthonormal basis which theoretically allows for the complete reconstruction of the final state in this paper an efficient numerical method for linear evolution equations of diffusive type is presented and convergence of the numerical approximation based on a discontinuous galerkin discretization is proved the key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition numerical examples for a scalar convection diffusion equation in two and three dimensions show the effectiveness of the method

bifurcation analysis of coupled lateral torsional vibrations of rotor systems abstract this paper presents a numerical method to analyze the bifurcation of coupled lateral torsional vibrations of rotor systems based on a hamiltonian approach a three degree of freedom dynamic model of a rotor is derived nonlinear ordinary differential equations are derived from the dynamic model the stability of the equilibrium and linear normal modes lnms are analyzed using a linearized matrix of the system equation for bifurcation analysis of the periodic orbits a nonlinear normal modes nnms computation algorithm is performed using multiple shooting methods and pseudo arclength continuation multiple shooting points are continued from lnms near equilibrium bifurcation points of the nnms are detected from the stability change of the periodic orbits during the continuation the proposed stability analysis an nnms computation of coupled lateral torsional vibration is demonstrated using two different rotor models a system with strong eccentricity and a system with weak eccentricity