investigation of natural frequencies and mode shapes of buckled beams the linear modes of vibration of buckled beams are investigated analytically and experimentally assuming a static buckled shape corresponding to the nth buckling mode an exact solution is obtained for the linear modes and associated frequencies of initially buckled beams with fixed fixed fixed hinged and hinged hinged boundary conditions the analytical solution for a first mode buckled beam is validated experimentally for the fixed fixed case the analytically obtained natural frequencies are in excellent agreement with those obtained experimentally these exact linear modes provide a basis from which to study the nonlinear vibrations of buckled beams

determination of the mutual radiation resistances of a rectangular plate and their impact on the radiated sound power abstract the mutual radiation resistances resulting from cross modal coupling have been mostly excluded from sound power calculations on the belief or premise that either for some reasons they have an insignificant impact on the power radiation or consideration of them can be an enormous computational burden in this paper the characteristics of the mutual radiation resistances are investigated of a simply supported rectangular plate it is shown that once the self radiation resistances or the so called modal radiation efficiencies have been calculated the mutual radiation resistances can be readily obtained at virtually no cost of equal importance because no approximation is made the present formulation is fully accurate in the entire frequency range an asymptotic expression is also derived for large modal wavenumbers numerical results are presented to illustrate the spectral characteristics of the mutual radiation resistances and the corresponding impact on the radiated sound power

subharmonic and homoclinic bifurcations in a parametrically forced pendulum abstract depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the melnikov method and of averaging methods it is shown that as a parameter is varied repeated resonances of successively higher periods occur culminating in homoclinic orbits according to the theorem of smale homoclinic bifurcation is the source of the unstable chaotic motions observed for some selected parameter sets the theoretical predictions are tested by numerical calculations very good agreement is found between analytical and numerical results

dynamic analysis of generally supported beams using fourier series the applicability of fourier series to the dynamic analysis of beams having arbitrary boundary conditions is explored for the simplicity of presenting the analysis procedure beams having zero end displacement are used in the discussion numerical results for free vibration and dynamic response based on the present analysis procedure are compared with existing results for natural frequencies and dynamic behavior obtained by the conventional modal analysis these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is seemingly simpler and more straightforward than the conventional modal analysis

on the linear normal modes of planar pre stressed curved beams the natural frequencies and mode shapes of planar shear undeformable beams around their curved pre stressed post buckling configurations are investigated neglecting rotary inertia effects two mechanical models are considered depending on the assumed boundary conditions in the buckling and post buckling phases with the first model the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end a sliding hinge in the second case the beam is assumed inextensible in the pre stressed phase same boundary conditions as above whereas it is extensible in the subsequent free linear dynamic phase because the sliding hinged boundary is changed into a stationary hinged end linear vibrations are governed by partial differential equations with non constant coefficients and the solutions for the frequencies and mode shapes are found employing two approximate approaches a fully numerical method based on a finite element formulation and a semi analytical method based on a weak formulation galerkin method the main results are compared and a close agreement in the outcomes is found the leading mechanical differences in the linear normal modes of the two pre stressed curved beam models are discussed

an innovative eigenvalue problem solver for free vibration of euler bernoulli beam by using the adomian decomposition method this paper deals with free vibration problems of euler bernoulli beam under various supporting conditions the technique we have used is based on applying the adomian decomposition method adm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literature these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simple and more straightforward than the other modal analysis

simplicity transformations for three way arrays with symmetric slices and applications to tucker 3 models with sparse core arrays abstract tucker three way pca and candecomp parafac are two well known methods of generalizing principal component analysis to three way data candecomp parafac yields component matrices a e g for subjects or objects b e g for variables and c e g for occasions that are typically unique up to jointly permuting and rescaling columns tucker 3 analysis on the other hand has full transformational freedom that is the fit does not change when a b and c are postmultiplied by nonsingular transformation matrices provided that the inverse transformations are applied to the so called core array g this freedom of transformation can be used to create a simple structure in a b c and or in g this paper deals with the latter possibility exclusively it revolves around the question of how a core array or in fact any three way array can be transformed to have a maximum number of zero elements direct applications are in tucker 3 analysis where simplicity of the core may facilitate the interpretation of a tucker 3 solution and in constrained tucker 3 analysis where hypotheses involving sparse cores are taken into account in the latter cases it is important to know what degree of sparseness can be attained as a tautology by using the transformational freedom in addition simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three way arrays so far a number of simplicity results have been attained pertaining to arrays sampled randomly from continuous distributions these results do not apply to three way arrays with symmetric slices in one direction the present paper offers a number of simplicity results for arrays with symmetric slices of order 2 2 3 3 and 4 4 some generalizations to higher orders are also discussed as a mathematical application the problem of determining the typical rank of 4 3 3 and 5 3 3 arrays with symmetric slices will be revisited using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one that can be attained almost surely for such arrays the issue of maximal simplicity of the targets to be presented will be addressed either by formal proofs or by relying on simulation results

solution of a mixed parity nonlinear oscillator harmonic balance abstract the method of harmonic balance is used to calculate first order approximations to the periodic solutions of a mixed parity nonlinear oscillator first the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction then the two auxiliary equations where the restoring force functions are odd are solved by using a first harmonic term without a constant therefore we obtain the two approximate solutions to the mixed parity nonlinear oscillator one is expressed in terms of the exact amplitude in the negative direction the other in terms of the approximate amplitude these solutions are more accurate than the second approximate solution obtained by the lindstedt poincare method for large amplitudes

prebending effects upon the vibrational modes of thermally prestressed planar beams abstract in this paper the linear vibrations of thermally prestressed beams are studied including the effects of predisplacement due to prebending and initial imperfections only low prestressed states far from the buckling stage are considered one of the motivations of this paper being the issue of taking into account climatic temperature effects in damage detection based on modal techniques a brief general review is first presented in order to give some theoretical and physical insights upon structural vibrations superimposed on an initial static state both the total lagrangian and the updated lagrangian formulations are applied to a planar euler bernouli beam under the assumption of small prestrains and large predisplacements the governing equilibrium equations are solved using a finite element method some illustrative numerical results are given the model is then validated through experiments inside a climatic chamber it is concluded that in addition to the axial prestress the presence of prebending is also likely to have a significant effect upon some eigenfrequencies even in the case of rather small predisplacements

response of infinite length rods and beams with periodically varying area abstract this paper develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically the conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal horizontal spatial variable the displacement field is written as a series expansion using a periodic form for the horizontal wavenumber the area and moment of inertia expressions are each expanded into a fourier series these are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients thus yielding the displacement of the system an example problem of both a rod and beam are analyzed for three different geometrical shapes the solutions to both problems are compared to results from finite element analysis for validation dispersion curves of the systems are shown graphically convergence of the series solutions is illustrated and discussed

free vibrations of a non uniform beam with general elastically restrained boundary conditions abstract a simple and efficient method is presented to study the problem of free vibration of a non uniform bernoulli euler beam with general elastic restraints at boundary points with one set of particularly chosen fundamental solutions of the system the frequency equation is derived and expressed in concise form if the closed form fundamental solutions of the system are not available the approximate fundamental solutions can be obtained through a newly developed algorithm which is shown to be efficient convenient and accurate finally several limiting cases of the general system are examined and some examples are presented to illustrate the validity and accuracy of the proposed method

numerical experiments on free and forced vibrations of beams of non uniform cross section the objective of this paper is to obtain an approximate solution of the title problem by using rayleigh s optimization technique when the mechanical system and its excitation in the case where it is subject to a p 0 cos t type loading is symmetric it is shown that if one takes into account the symmetry of the system the calculated eigenvalues are in general more accurate than those which would result if the complete structural system were analyzed it is also assumed that the beam is subject to an axial force and results for the buckling problem are also presented the analytically determined eigenvalues are compared with values calculated by using the finite element method in several instances

exact solutions for the non conservative elastic stability of non uniform columns abstract the exact characteristic equations for the non conservative elastic stability of general elastic restrained non uniform columns governed by a fourth order ordinary differential equation with arbitrarily varying polynomial coefficients are derived the results are compared with those in the existing literature in addition the influence of the boundary conditions taper ratio and variation of linearly distributed tangential forces on the non conservative elastic stability of two kinds of tapered columns subjected to linearly distributed tangential forces is investigated it is shown that the parameter e of the linearly distributed tangential forces g x g 0 1 ex l has great influence on the instability mechanism for the clamped clamped columns

generalized fourier series and limit cycles of generalized van der pol oscillators abstract a very simple generalization using jacobian elliptic functions of the usual fourier series appropriate for non linear systems is used to study the first order approximate solution of the generalized van der pol oscillators x ax 2bx 3 z 3 z 2 x 2 z 1 x 4 dot x 0 a generalized harmonic balance method is used to determine the limit cycles the cases a 0 b 0 and a 0 are considered in some detail for given values of the parameters zi the values of a and b for which limit cycles exist are found as functions of m numerical values for the radius frequency and energy of the limit cycles are given the presence of zero one two three or four limit cycles depends on the value of the parameters of the equation stability bifurcations of fixed points and the limit cycle stability are studied qualitatively

nonlinear vibration of buckled beams some exact solutions in this paper exact solutions are obtained for the dynamics of buckled beams with different types of end conditions the differential equation for the vibrations of buckled beams with restrictions on axial stretch is nonlinear using the modes of the corresponding linear problem which readily satisfy the boundary conditions the natural frequencies are obtained for the nonlinear problem by using jacobi elliptic functions analytical solutions for the dynamics of buckled beams with various rotational restraints are presented

vibration and buckling analysis of rectangular plates with nonlinear elastic end restraints against rotation abstract the fundamental natural frequencies and buckling loads of a rectangular plate with nonlinearly rotational restraints are obtained by using the finite element technique if the rotational springs are unsymmetric the iterative scheme must be employed to acquire the solutions of the nonlinear problem the values which describe the free vibration and stability behaviour of the plate will increase when either the parameters of rotational spring or the initial rotational angles increase incidentally it can be concluded that these results grown nonlinearly with respect to either the linear or nonlinear rotational spring constants finally both the frequency and stability parameters are evaluated for several boundary conditions which are quite useful in engineering analysis and design

elastic stability of non uniform columns abstract a simple and efficient method is proposed to investigate the elastic stability of three different tapered columns subjected to uniformly distributed follower forces the influences of the boundary conditions and taper ratio on critical buckling loads are investigated the critical buckling loads of columns of rectangular cross section with constant depth and linearly varied width t 1 constant width and linearly varied depth t 2 and double taper t 3 are investigated among the three different non uniform columns considered taper ratio has the greatest influence on the critical buckling load of column t 3 and the lowest influence on that of column t 1 the types of instability mechanisms for hinged hinged and cantilever non uniform columns are divergence and flutter respectively however for clamped hinged and clamped clamped non uniform columns the type of instability mechanism for column t 1 is divergence while that for columns t 2 and t 3 is divergence only when the taper ratio of the columns is greater than certain critical values and flutter for the rest value of taper ratio when the type of instability mechanism changes from divergence to flutter there is a finite jump for the critical buckling load the influence of taper ratio on the elastic stability of cantilever column t 3 is very sensitive for small values of the taper ratio and there also exist some discontinieties in the critical buckling loads of flutter instability for a hinged hinged non uniform column t 2 or t 3 with a rotational spring at the left end of the column when the taper ratio is less than the critical value the instability mechanism changes from divergence to flutter as the rotational spring constant is increased for a clamped elastically supported non uniform column when the taper ratio is greater than the critical value the instability mechanism changes from flutter to divergence as the translational spring constant is increased

analysis of a beam column system under varying axial forces of elliptic type the exact solution of lame s equation abstract we investigate the dynamical response of a beam column system with hinged ends subjected to an axial pulsating force of elliptic type it is shown that the resulting equation is of the form d 2 y 1 d 2 y 1 d 1 d 2 cn 2 k 2 0 which is the well known lame equation higher transcendental functions bateman manuscript project edited by mcgraw hill new york vol 3 in this paper we obtain the general exact solution of this equation that reveals stable behavior of the beam column system if the assigned initial conditions are of the form y1 0 y10 and y 1 0 0 it is also found that at a certain value of the modulus of the elliptic force the lateral vibrational frequency is independent of the material properties of the beam column system

periodic solutions of strongly quadratic non linear oscillators by the elliptic lindstedt poincare method abstract the elliptic lindstedt poincare method is used employed to study the periodic solutions of quadratic strongly non linear oscillators of the form x c 1 x c 2 x 2 e f x x in which the jacobian elliptic functions are employed instead of the usual circular functions in the classical lindstedt poincare method the generalized van de pol equation with f x x 0 1 x 2 x 2 is studied in detail comparisons are made with the solutions obtained by using the lindstedt poincare method and runge kutta method to show the efficiency of the present method

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

analysis of buckled and pre bent fixed end columns used as vibration isolators the use of a buckled or pre bent column with fixed ends as a vibration isolator is analyzed the column is designed to have a high axial stiffness under the weight that it supports so that the static displacement of the weight is not excessive and then to have a low stiffness during excitation the base of the column is assumed to have an axial motion which is simple harmonic or a linear combination of two simple harmonic functions the column is modeled as an elastica first the equilibrium shape under the supported weight is determined then small steady state vibrations about the equilibrium configuration are obtained numerically using a shooting method the inertia of the supported weight and the transverse and axial inertias of the column are included the axial displacement transmissibility is computed and the effects of external and internal damping column stiffness supported weight and initial curvature are investigated for the two frequency excitation the effects of the relative amplitudes and frequencies of the excitation components are considered

a case of extreme simplicity of the core matrix in three mode principal components analysis in three mode principal components analysis thep q r core matrixg can be transformed to simple structure before it is interpreted it is well known that whenp qr g can be transformed to the identity matrix which implies that all elements become equal to values specified a priori in the present paper it is shown that whenp qr 1 g can be transformed to have nearly all elements equal to values spectified a priori a cllsed form solution for this transformation is offered theoretical and practical implications of this simple structure transformation ofg are discussed

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

improvement of a krylov bogoliubov method that uses jacobi elliptic functions abstract an improved version of a krylov bogoliubov method that gives the approximate solution of the non linear cubic oscillator x c 1 x c 3 x 3 ef x x dot 0 in terms of jacobi elliptic functions is described compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual krylov bogoliubov method which gives the approximate solution in terms of circular functions these expressions are especially simple for quasilinear c3 0 and quasi pure cubic c1 0 oscillators two types of cubic oscillators have been used as examples the linear damped oscillator f x x dot x dot and the van der pol oscillator f x x dot x 2 x dot the approximate solutions of these quasilinear and quasi pure cubic oscillators are simple and accurate the influence of the non linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied

the typical rank of tall three way arrays the rank of a three way array refers to the smallest number of rank one arrays outer products of three vectors that generate the array as their sum it is also the number of components required for a full decomposition of a three way array by candecomp parafac the typical rank of a three way array refers to the rank a three way array has almost surely the present paper deals with typical rank and generalizes existing results on the typical rank ofi j k arrays withk 2 to a particular class of arrays withk 2 it is shown that the typical rank isi when the array is tall in the sense thatjk j i jk in addition typical rank results are given for the case wherei equalsjk j

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

improvement to the averaging method using the jacobian elliptic function the averaging method was improved by using the jacobian elliptic sine sn cosine cn and delta dn functions as generating solutions in order to obtain a highly accurate periodic solution for a strongly nonlinear dynamical system the proposed method can be applied to a relatively general nonlinear system based on the single degree of freedom duffing equation two methods of stability analysis were applied and evaluated for the approximate solutions obtained by the proposed method the approximate solutions obtained by the proposed method for practical examples were compared to the solutions obtained by the shooting method the results confirmed that the proposed method provides a more accurate solution and more accurate stability analysis results than those obtained by the conventional averaging method that uses trigonometric functions as the generating solution

vibration analysis of rectangular plates with general elastic boundary supports abstract in this investigation the rayleigh ritz method is used to determine the modal characteristics of a rectangular plate with general elastic supports alone its edges each of the admissible functions here is composed of a trigonometric function and an arbitrary continuous function that is introduced to ensure the sufficient smoothness of the so called residual displacement function at the edges as a result a drastic improvement of the convergence can be expected of the solution expressed as a series expansion in terms of the admissible functions perhaps more importantly this study has developed a general approach for deriving a complete set of admissible functions that can be universally applied to various boundary conditions several numerical examples are given to demonstrate the accuracy and convergence of the current solution

investigation on non linear vibration in arched beam for bridges construction via agm method analyzing and modeling the vibrational behavior of arched bridges during earthquake in order to decrease the exerted damages to the structure is a very hard task to do in the present paper this item has been performed analytically for the first time due to the importance of arched bridges building as a great structure in the human being civilization and its specifications such as transferring vertical loads to its arcs and the lack of bending moments and shearing forces this case study is devoted to this special issue therefore the nonlinear vibration of arched bridges has been modeled and simulated by an arched beam with harmonic vertical loads and its behavior has been investigated by analyzing a nonlinear partial differential equation governing on the system it is notable that the procedure has been done analytically by agm akbari ganji s method it has found that angular frequency is increased significantly by escalating the amount of the center outlet furthermore comparisons have been made between the obtained results by numerical method rkf 45 and agm in order to assess the scientific validity obtained results indicate the precision of the achieved answers by agm method

prediction of fundamental frequencies of stressed spring hinged tapered beams introduction i nterstages thrust frames payload adapters etc of rocket missile structural systems are often con gured with beams attached to end rings the beams are joined to the end rings and based on the joint con guration the ends of the beams are usually idealized using end rotational constraints the stiffness parameter value of which is xed based on the quality of the joint these beams during their service conditions are subjected to compressive loads further the vibration behavior of these beams especially the fundamental frequency of prediction is most important for evaluation of the transmissibilityof gravitationalaccelerationloads due to the ight dynamicenvironment to savemass thesebeams are often designedas taperedbeams which are near optimumfor engineering purposes quick prediction and information about the fundamental frequency of such beams under initial stress is of signi cant importance during their design phase the nite elementmethod is widely used to estimate the fundamental frequency of beams with initial stresses 1 i 5 end rotationalconstraints are considered in refs 6 10 though the nite elementmethod is versatile closed formsolution isveryelegantandyieldsquickresults in thisnote a simple formula is derived to predict the fundamental frequency of stressed tapered beamswith end rotationalrestraintswhen its stress free frequencyis known it is assumed that the mode shapes for the buckling and free vibration are the same as that of the initially stressed vibration of the tapered beam a set of numerical results for tapered beams with end rotational restraints having symmetric linear variation in depth and constant breadth are generated for various values of rotational stiffness parameters using the formula the nite element method the beam is idealized using 20 tapered elements is employed to verify these results the detailsof the formulationand the numerical results are presented

three mode orthomax rotation factor analysis and principal components analysis pca are often followed by an orthomax rotation to rotate a loading matrix to simple structure the simple structure is usually defined in terms of the simplicity of the columns of the loading matrix in three mode pca rotational freedom of the so called core a three way array relating components for the three different modes can be used similarly to find a simple structure of the core simple structure of the core can be defined with respect to all three modes simultaneously possibly with different emphases on the different modes the present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously computationally this approach relies on repeated two way orthomax applied to supermatrices containing the frontal lateral or horizontal slabs respectively the procedure is illustrated by means of a number of exemplary analyses as a by product application of the three mode orthomax procedures to two way arrays is shown to reveal interesting relations with and interpretations of existing two way simple structure rotation techniques

free vibrations of beams with general boundary conditions a simple and unified approach is presented for the vibration analysis of a generally supported beam the flexural displacement of the beam is sought as the linear combination of a fourier series and an auxiliary polynomial function the polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives as a result not only is it always possible to expand the displacement in a fourier series for beams with any boundary conditions but also the solution converges at a much faster speed the reliability and robustness of the proposed technique are demonstrated through numerical examples

a study of a nonlinear vibration isolator with a quasi zero stiffness characteristic a vibration isolator consisting of a vertical linear spring and two nonlinear pre stressed oblique springs is considered in this paper the system has both geometrical and physical nonlinearity firstly a static analysis is carried out the softening parameter leading to quasi zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry pre stress and the stiffness of the springs the optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone it also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position for these values a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities it includes considering the possibilities of the appearance of period doubling bifurcation and its development into chaotic motion for this purpose approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots poincare maps and lyapunov exponents are used finally the frequency at which the first period doubling bifurcation appears is found and the effect of damping on this frequency determined

simplicity and typical rank results for three way arrays matrices can be diagonalized by singular vectors or when they are symmetric by eigenvectors pairs of square matrices often admit simultaneous diagonalization and always admit block wise simultaneous diagonalization generalizing these possibilities to more than two non square matrices leads to methods of simplifying three way arrays by nonsingular transformations such transformations have direct applications in tucker pca for three way arrays where transforming the core array to simplicity is allowed without loss of fit simplifying arrays also facilitates the study of array rank the typical rank of a three way array is the smallest number of rank one arrays that have the array as their sum when the array is generated by random sampling from a continuous distribution in some applications the core array of tucker pca is constrained to have a vast majority of zero elements both simplicity and typical rank results can be applied to distinguish constrained tucker pca models from tautologies an update of typical rank results over the real number field is given in the form of two tables

vibration of beams with generally restrained boundary conditions using fourier series this work presents a method to find accurate vibration frequencies of beams with generally restrained boundary conditions using fourier series the suggested method is very convenient to find an accurate frequency parameter for beams with not only classical boundary conditions but also non classical boundary conditions restrained by rotational and translational springs numerical results for various degenerate cases are compared with existing results for natural frequency obtained by conventional analysis the results indicate that the present method can be used very effectively in the design of beams with various supporting conditions

a generalization of the method of harmonic balance a procedure is provided for generalizing the method of harmonic balance to obtain higher order approximations to the periodic solutions of differential equations for two systems one a conservative system and the other nonconservative the procedure is currently being applied to investigate the possible solution behaviors of singular nonlinear oscillators where the usual perturbation methods do not work

an approximate method for one dimensional structures with strong nonlinear and nonhomogenous boundary conditions abstract there is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non homogenous boundary conditions this paper proposes such a method a distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions the total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions the field solution can be obtained in terms of the modal expansion the modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations which are solved with the harmonic balance method two examples are presented to study the convergence and accuracy of the solution the first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right the wave method and the differential quadrature method dqm are compared with the proposed method and are found in good agreement the second example is a beam with nonlinear torsional boundary absorbers the method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method dqem the proposed method is also compared with the multi scale method msm and shows an advantage to deal with strong nonlinearities the proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions

multi domain simulations of the time dependent navier stokes equations benchmark error analysis of some nesting procedures this paper presents a benchmark error analysis of various approaches for treating multiple domain calculations within an anelastic finite difference model one way and two way interactive nesting errors with and without temporal refinement are evaluated the two way interactive nesting approach is one where solutions between fine and coarse grid domains are matched through the simple post insertion of data on the other hand the equations can be matched by using the pressure defect correction approach it is shown that for the present model the two way interactive nesting method gives identical results to multi domain solutions using the pressure defect correction approach the present results indicate that in this type of analastic framework a priori matching of the equations is equivalent to the a posteriori matching of the solutions this result is attributed to the inflexible nature of the neumann boundary conditions on the fine mesh pressure which need to be specified from the coarse mesh since a large number of meterological models employ the hydrostatic assumption it is of interest to know of nesting errors attributable to this approximation the results presented indicate essentially equivalent error levels for both the hydrostatic and nonhydrostatic systems of equations for the present case of airflow over an isolated mountain it is shown how nesting technology can be used in a virtual sense to reduce the central memory requirements for large array sized numerical simulations nesting can be used in this sense to decompose the maximum memory working space required without affecting the results

a modified elliptic lindstedt poincare method for certain strongly non linear oscillators abstract a modified elliptic lindstedt poincare method is presented for the steady state analysis of strongly non linear oscillators of the form x c 1 x c 3 x 3 ef x x in which a new parameter e is employed such that the value of is always small regardless of the magnitude of the original parameter e therefore the above strongly non linear oscillators with large parameter e is transformed into a small parameter system with respect to

a fully multidimensional positive definite advection transport algorithm with small implicit diffusion in numerical modeling of physical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions numerical schemes of second or higher order accuracy can produce negative values in the solution due to the dispersive ripples lower order schemes such as the donor cell or lax friedrichs or higher order schemes with zeroth order diffusion added produce no ripples but suffer from excessive implicit diffusion in the last ten years a possible resolution of this dilemma has been developed in the form of hybrid schemes in which the advective fluxes are given as a weighted average of a first order positive definite scheme s fluxes and a higher order scheme s fluxes the difference in determination of the weights in the calculation of the average advective fluxes has led to different hybrid schemes two main hybrid type schemes have been developed one the so called flux corrected transport fct method was originated by boris and book i 31 and generalized by zalesak 141 the other was developed by harten and zwas 6 9 in the form of the self adjusting hybrid schemes sahs method both methods were constructed to deal effectively with shocks and contact discontinuities solutions of the advection transport equation obtained by using fct or sahs maintain positive definiteness of the initial condition and as can be seen from presented tests zalesak 141 harten 6 be very accurate unfortunately application of these methods to the modeling of complex multidimensional hydrodynamical systems like atmospheric phenomena is rather limited due to the excessive computer time required furthermore in many hydrodynamical systems

the multidimensional positive definite advection transport algorithm further development and applications abstract this paper presents further generalizations of the advection algorithm described in j comput phys 54 1984 325 time dependent velocity fields and a generalized form of the continuity equation are considered applicability of the algorithm to the diffusion equation and to the transport of nonpositive scalars is also discussed theoretical considerations are illustrated through numerical tests and applications to some particular geophysical fluid dynamics problems

transforming three way arrays to maximal simplicity transforming the core array in tucker three way component analysis to simplicity is an intriguing way of revealing structures in between standard tucker three way pca where the core array is unconstrained and candecomp parafac where the core array has a generalized diagonal form for certain classes of arrays transformations to simplicity that is transformations that produce a large number of zeros can be obtained explicitly by solving sets of linear equations the present paper extends these results first a method is offered to simplifyj j 2 arrays next it is shown that the transformation that simplifies ani j k array can be used to also simplify the complementary arrays of order jk i j k of orderi ik j k and of orderi j ij k finally the question of what constitutes the maximal simplicity for arrays the maximal number of zero elements will be considered it is shown that cases of extreme simplicity considered in the past are in fact cases of maximal simplicity

an exact series solution for calculating the eigenfrequencies of orthotropic plates with completely free boundary abstract an analysis for solving boundary value problems in structural mechanics which was proposed by wang and lin 1996 journal of sound and vibration 196 285 293 1999 journal of applied mechanics 66 380 387 1 2 has been extended to the calculation of the eigenfrequencies of an orthotropic plate under all free boundary conditions the convergence of the series solution is assured and the procedure leads to pointwise exact solutions the calculated eigenfrequencies have been verified by a different approach and indicate that the present method is simple and effective

geometrically exact planar beams with initial pre stress and large curvature static configurations natural frequencies and mode shapes abstract within this paper an analytical formulation is provided and used to determine the natural frequencies and mode shapes of a planar beam with initial pre stress and large variable curvature the static configuration mode shapes and natural frequencies of the pre stressed beam are obtained by using geometrically exact euler bernoulli beam theory the beam is assumed to be not shear deformable and inextensible because of its slenderness and uniform closed cross section as well as the boundary conditions under consideration the static configuration and the modal information are validated with experimental data and compared to results obtained from nonlinear finite element analysis software in addition to the modal analysis about general static configurations special consideration is given to an initially straight beam that is deformed into semi circular and circular static configurations for these special circular cases the partial differential equation of motion is reduced to a sixth order differential equation with constant coefficients and solutions of this system are examined this work can serve as a basis for studying slender structures with large curvatures