geometrically nonlinear free vibrations of simply supported isotropic thin circular plates nonlinear free axisymmetric vibration of simply supported isotropic circular plates is investigated by using the energy method and a multimode approach in plane deformation is included in the formulation lagrange s equations are used to derive the governing equation of motion using the harmonic balance method the equation of motion is converted into a nonlinear algebraic form the numerical iterative method of solution adopted here is the so called linearized updated mode method which permits the authors to obtain accurate results for vibration amplitudes up to three times the plate thickness the percentage of participation of each out of plane basic function to the deflection shape and to the bending stress at the plate centre and of each in plane basic function to the membrane stress at the centre are calculated in order to determine the minimum number of in and out of plane basic functions to be used in order to achieve a good accuracy of the model the nonlinear frequency the nonlinear fundamental mode shape and their associated nonlinear bending and membrane stresses are determined at large amplitudes of vibration the numerical results obtained here are presented and compared with available published results based on various approaches and with the single mode solution the limit of validity of the single mode approach is also investigated

finite element multiple mode approach to nonlinear free vibrations of shallow shells two finite element fe modal formulations for large amplitude free vibration of isotropic and arbitrary laminated composite shallow shells are presented the system equation of motion is formulated first in the physical structural node degrees of freedom dof then the system is transformed into two distinctly different sets of general duffing type modal equations based on 1 modal amplitudes of coupled linear bending and in plane modes where in plane inertia is included in the formulation and 2 modal amplitudes of linear bending modes only where the in plane inertia is neglected multiple modes and the first order transverse shear deformation are considered in the formulations a shallow shell finite element is developed as an extension from the triangular mindlin min3 plate element with the improved shear correction factor by tessler time numerical integration is employed to determine the nonlinear periodic frequency characteristics the inaccuracy in characterizing a shallow shell response with coupled linear bending and in plane modes is demonstrated and discussed by comparing with the fe solution in structural node dof study cases include isotropic and composite panels of different shallow shell geometries

the effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part i simply supported and clamped clamped beams abstract a method for calculating the non linear mode shapes and natural frequencies of fully clamped beams at large vibration amplitudes is presented and results are compared with those of previous studies and of experimental measurements first the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions then the non linear deformation energy is expressed by taking into account the non linear terms due to the axial strain induced by large deflections a set of non linear algebraic equations which reduces to the classical linear eigenvalue problem when non linear terms are neglected is determined through hamilton s principle it is also shown that unless a condition is imposed on the contribution of one mode the solution of this set leads to the linear case consequently in order to obtain a numerical solution for the non linear problem in the neighbourhood of a given mode the contribution of this mode is chosen and those of other modes are calculated in this paper part i of a series of three papers this method is applied to obtain the first three non linear mode shapes of clamped clamped and simply supported beams the results obtained corresponding to the fundamental non linear mode shape are in good agreement with those of a previous theoretical and experimental study in particular high values of increase of beam curvatures are noticed near the clamps causing a highly non linear increase in bending strain with increasing deflection

non linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations this study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations geometric non linearity due to moderately large deflections has been included movable and immovable simply supported plates and immovable clamped plates resting on winkler pasternak and non linear winkler foundations have been considered the von karman type governing equations have been employed harmonic vibrations are assumed and the time t is eliminated by the kantorovich averaging method an orthogonal point collocation method is used for spatial discretization numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non linear period to the linear period of natural vibration the effects of foundation parameters the orthotropic parameter and the edge conditions on the non linear vibration behaviour have been investigated

finite element method for nonlinear free vibrations of composite plates a multimode time domain modal formulation based on the finite element method for large amplitude free vibration of thin composite plates is presented accurate frequency maximum deflection relations can be predicted for the fundamental and the higher nonlinear modes a modal participation is defined and accurate and convergent frequencies can be determined with minimum number of linear modes a procedure for the selection of initial conditions for periodic plate response is presented convergence of frequency with gridwork refinement and number of linear modes is studied the classical single mode elliptic function frequency solutions for simply supported beams and square plates are assessed examples of orthotropic and composite plates are given and the characteristics of nonlinear response are studied

on the dimension of complex responses in nonlinear structural vibrations abstract the ability to accurately model engineering systems under extreme dynamic loads would prove a major breakthrough in many aspects of aerospace mechanical and civil engineering extreme loads frequently induce both nonlinearities and coupling which increase the complexity of the response and the computational cost of finite element models dimension reduction has recently gained traction and promises the ability to distill dynamic responses down to a minimal dimension without sacrificing accuracy in this context the dimensionality of a response is related to the number of modes needed in a reduced order model to accurately simulate the response thus an important step is characterizing the dimensionality of complex nonlinear responses of structures in this work the dimensionality of the nonlinear response of a post buckled beam is investigated significant detail is dedicated to carefully introducing the experiment the verification of a finite element model and the dimensionality estimation algorithm as it is hoped that this system may help serve as a benchmark test case it is shown that with minor modifications the method of false nearest neighbors can quantitatively distinguish between the response dimension of various snap through non snap through random and deterministic loads the state space dimension of the nonlinear system in question increased from 2 to 10 as the system response moved from simple low level harmonic to chaotic snap through beyond the problem studied herein the techniques developed will serve as a prescriptive guide in developing fast and accurate dimensionally reduced models of nonlinear systems and eventually as a tool for adaptive dimension reduction in numerical modeling the results are especially relevant in the aerospace industry for the design of thin structures such as beams panels and shells which are all capable of spatio temporally complex dynamic responses that are difficult and computationally expensive to model

fixed point error analysis of discrete wigner ville distribution the wigner ville distribution wvd has been a powerful signal processing tool for time frequency signal analysis consequently many algorithms have been proposed in the literature for computing the wvd in real time applications however boashash 1987 has proposed and showed that the evaluation of the analytic signal using the time domain approach and involving the hilbert transformer is the most efficient algorithm for real time applications a fixed point error analysis of this algorithm has been carried out the theoretical noise to signal ratio nsr is derived and verified through simulation the results indicate that for this algorithm the nsr increases by 0 5 bit stage whereas for the other algorithms it increases by 1 bit stage

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

the wave equation approach for solving inverse eigenvalue problems of a multi connected region in r3 with robin conditions the asymptotic expansion for t of the trace of the wave kernel email protected t email protected u 1exp email protected 1 2 u where m u u 1 are the eigenvalues of the negative laplacian email protected email protected 3 b 1 x b 2 in the x 1 x 2 x 3 space where i 1 and

nonlinear normal modes in an intrinsic theory of anisotropic beams nonlinear normal modes in nonlinear oscillations of beams are derived from intrinsic equations that is using velocities and strains as primary degrees of freedom displacements and rotations are thus not system states but are instead obtained using the propagation of the local beam material reference frames as in rigid body dynamics it is shown that the intrinsic variables sufce to describe the free vibrations of the beam the approach does not need assumptions in the material properties i e it is valid for general anisotropic behavior or the beam kinematics i e it is based on cosserat s exact geometrical description of the deformable curve furthermore the nonlinear modal equations in intrinsic coordinates are obtained from integrals involving only products of the mode shapes and known coecients using this

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

nonlinear vibrations of shallow shells with complex boundary r functions method and experiments geometrically nonlinear vibrations of shallow circular cylindrical panels with complex shape of the boundary are considered the r functions theory and variational methods are used to study the problem the r functions method rfm allows constructing in analytical form the sequence of basis functions satisfying the given boundary conditions in case of complex shape of the boundary the problem is reduced to a single second order differential equation with quadratic and cubic nonlinear terms the method developed has been initially applied to study free vibrations of shallow circular cylindrical panels with rectangular base for different boundary conditions i clamped edges ii in plane immovable simply supported edges iii classically simply supported edges and iv in plane free simply supported edges then the same approach is applied to a shell with complex shape of the boundary experiments have been conducted on an aluminum panel with complex shape of the boundary in order to identify the nonlinear response of the fundamental mode these experimental results have been compared to numerical results

reduced order models for large amplitude vibrations of shells including in plane inertia non linear normal modes nnms are used in order to derive reduced order models for large amplitude geometrically non linear vibrations of thin shells the main objective of the paper is to compare the accuracy of different truncations using linear and non linear modes in order to predict the response of shells structures subjected to harmonic excitation for an exhaustive comparison three different shell problems have been selected i a doubly curved shallow shell simply supported on a rectangular base ii a circular cylindrical panel with simply supported in plane free edges and iii a simply supported closed circular cylindrical shell in each case the models are derived by using refined shell theories for expressing the strain displacement relationship as a consequence in plane inertia is retained in the formulation reduction to one or two nnms shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases and this limitation is extended to two times the thickness for two of the selected models 2008 elsevier b v all rights reserved

large amplitude axisymmetric vibrations of geometrically imperfect circular plates abstract this paper deals with the effects of geometric imperfections on the large amplitude vibrations of circular plates it is found that geometric imperfections of the order of a fraction of the plate thickness may significantly raise the linear vibration frequencies furthermore such imperfections may even change the inherent non linear hard spring character of the circular plates and cause them to exhibit soft spring behavior the effects of various boundary conditions are examined

dynamics of linear discrete systems connected to local essentially non linear attachments the dynamics of a linear periodic substructure weakly coupled to an essentially non linear attachment are studied the essential non linearizable non linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena e g passive one way irreversible transfer of energy from the substructure to the attachment as a specific application the dynamics of a finite linear chain of coupled oscillators with a non linear end attachment is examined in the absence of damping it is found that the dynamical effect of the non linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain when damping exists energy pumping phenomena are realized in the system it is shown that energy pumping strongly depends on the topological structure of the non linear normal modes nnms of the underlying undamped system this is due to the fact that energy pumping is caused by the excitation of certain damped invariant nnm manifolds that are analytic continuations for weak damping of nnms of the underlying undamped system the bifurcations of the nnms of the undamped system help explain resonance capture cascades in the damped system this is a series of energy pumping phenomena occurring at different frequencies with sudden lower frequency transitions between sequential events the observed multi frequency energy pumping cascades are particularly interesting from a practical point of view since they indicate that non linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure in compatibility with the design objectives

on the θ function of a riemannian manifold with boundary let be a compact riemannian manifold of dimension n with smooth boundary let 1 2 be the eigenvalues of the laplace beltrami operator with the boundary condition n 0 the associated function t n 1 exp n t has an asymptotic expansion of the form 4 t n 2 t a 0 a 1 t 1 2 a 2 t a 3 t 3 2 a 4 t 2 the values of a 0 a 1 are well known

theory of a curved planar waveguide with robin boundary conditions a model of a thin straight strip with a uniformly curved section and with boundary requirements zeroing at the edges a linear superposition of the wave function and its normal derivative robin boundary condition is analyzed theoretically within the framework of the linear schrodinger equation and is applied to the study of the processes in the bent magnetic multilayers superconducting films and metallic ferrite filled waveguides in particular subband thresholds of the straight and curved parts of the film are calculated and analyzed as a function of the robin parameter 1 lambda with lambda being an extrapolation length entering robin boundary condition for the arbitrary robin coefficients which are equal on the opposite interfaces of the strip and for all bend parameters the lowest mode energy of the continuously curved duct is always smaller than its straight counterpart accordingly the bound state below the fundamental propagation threshold of the straight arms always exists as a result of the bend in terms of the superconductivity language it means an increased critical temperature of the curved film compared to its straight counterpart localized level dependence on the parameters of the curve is investigated with its energy decreasing with increasing bend angle and decreasing bend radius conditions of the bound state existence for the different robin parameters on the opposite edges are analyzed too in particular it is shown that the bound state below the first transverse threshold of the straight arm always exists if the inner extrapolation length is not larger than the outer one in the opposite case there is a range of the bend parameters where the curved film cannot trap the wave and form the localized mode for example for the fixed bend radius the bound state emerges from the continuum at some nonzero bend angle that depends on the difference of the two lengths lambda at the opposite interfaces various transport properties of the film such as interference blockade of the current flow at some special energies is also discussed with the special attention being paid to the transformation from the dirichlet to the neumann case as the extrapolation length lambda sweeps the positive axis

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

finite element method for non linear forced vibrations of circular plates geometric non linearities for large amplitude free and forced vibrations of circular plates are investigated in plane displacement and in plane inertia are included in the formulation the finite element method is used an harmonic force matrix for non linear forced vibration analysis is introduced and derived various out of plane and in plane boundary conditions are considered the relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented

higher dimensional inverse problem of the wave equation for a general multi connected bounded domain with a finite number of smooth mixed boundary conditions the spectral function m t j 1 exp it m j 1 2 where m j j 1 are the eigenvalues of the negative laplacian d 3 u 1 3 x u 2 in the x 1 x 2 x 3 space is studied for small t for a variety of bounded domains where

the non linear vibration and dynamic instability of thin shallow shells in this paper the non linear vibration and dynamic instability of thin shallow spherical and conical shells subjected to periodic transverse and in plane loads are investigated the marguerre type dynamic equations used for the analysis of shallow shells when treated by the galerkin method will result in a system of total differential equations in the time functions known as duffing and mathieu equations from which the various kinds of non linear vibration and dynamic instability are determined by using numerical methods numerical results are presented for axisymmetric vibrations and dynamic instabilities of shallow spherical and conical shells with a clamped and b supported edge conditions as numerical examples non linear vibration frequencies and instability regions for shells are determined the effects of static load as well as static snap through buckling on the instability are also investigated

roundoff error analysis of the discrete wigner distribution using fixed point arithmetic the issue of roundoff noise effects in the implementation of the discrete wigner distribution using fixed point arithmetic is addressed the sign magnitude number representation is assumed throughout the analysis the measure of roundoff noise effects in an algorithm is the output noise to signal ratio using a statistical model an analytical expression of the noise to signal ratio is derived as a function of the wordlength b and the transform length n the noise to signal ratio is obtained by evaluating the signal and noise powers at different points in the algorithm then reflecting to the output both signal and noise powers based on the derived noise to signal ratio is is noted that if the transform length is doubled then one additional bit is required in the wordlength to maintain a constant noise to signal ratio it is demonstrated through the software simulations that the predicted noise to signal ratio is a good closed form estimate of the true roundoff error it is also found from the simulation that the wordlength b and the transform length n 2 sup v must satisfy the condition b v or 4

non linear generalization of principal component analysis from a global to a local approach abstract principal component analysis pca also known as proper orthogonal decomposition or karhunen loeve transform is commonly used to reduce the dimensionality of a data set with a large number of interdependent variables pca is the optimal linear transformation with respect to minimizing the mean square reconstruction error but it only considers second order statistics if the data have non linear dependencies an important issue is to develop a technique which takes higher order statistics into account and which can eliminate dependencies not removed by pca recognizing the shortcomings of pca researchers in the field of statistics and neural networks have developed non linear extensions of pca the purpose of this paper is to present a non linear generalization of pca called vqpca this algorithm builds local linear models by combining pca with clustering of the input space this paper concludes by observing from two illustrative examples that vqpca is potentially a more effective tool than conventional pca

a new formulation for the existence and calculation of nonlinear normal modes a new formulation is presented here for the existence and calculation of nonlinear normal modes in undamped nonlinear autonomous mechanical systems as in the linear case an expression is developed for the mode in terms of the amplitude mode shape and frequency with the distinctive feature that the last two quantities are amplitude and total phase dependent the dynamic of the periodic response is defined by a one dimensional nonlinear differential equation governing the total phase motion the period of the oscillations depending only on the amplitude is easily deduced it is established that the frequency and the mode shape provide the solution to a 2 periodic nonlinear eigenvalue problem from which a numerical galerkin procedure is developed for approximating the nonlinear modes the procedure is applied to various mechanical systems with two degrees of freedom

normal mode localization for a two degrees of freedom system with quadratic and cubic non linearities abstract the non similar normal modes of free oscillations of a coupled non linear oscillator are examined so far the study of non linear vibrations has been based on the assumption that the system is admissible this requirement is satisfied when the stiffness of the springs are odd functions of their displacement in this work a two degrees of freedom tuned system is considered with stiffness elements having linear quadratic and cubic non linearities the potential energy function of this system is not symmetric with respect to the origin equilibrium point of the configuration space due to the presence of the quadratic non linearity hence the system considered is no longer admissible a study of the balancing diagrams is performed to determine the degenerate and global similar modes of the system manevich mikhlin asymptotic methodology is used for solving the singular differential equation describing the non similar modes and approximate analytical expressions are derived for this system with weak coupling localized non similar modes are detected in a small neighborhood of degenerate similar modes of the tuned system numerical integration is used to verify theoretically predicted non similar normal modes it is found that these modes pass periodically through a non zero point in the configuration space

eigenvalues of the laplacian for the third boundary values problem an extension to higher dimensions soit r a r b 0 0 2 une coque spherique on suppose connu exactement le spectre de valeurs propres 0 1 2 3 j quand j pour le probleme 3 u 0 dans u r 1 u r a 0 u r 2 u r b 0 ou 1 et 2 sont des constantes positives on determine la geometrie de et les impedances 1 et 2

numerical computation of periodic responses of nonlinear large scale systems by shooting method abstract geometrically nonlinear vibrations of three dimensional elastic structures due to harmonic external excitations are investigated in the frequency domain the material of the structure is assumed to be linearly elastic the equation of motion is derived by the conservation of linear momentum in lagrangian coordinate system and it is discretized into a system of ordinary differential equations by the finite element method the shooting method is used to obtain the periodic solutions a procedure which transforms the initial value problem into a two point boundary value problem for the periodicity condition and then it finds the initial conditions which lead to periodic response is developed and presented for systems of second order ordinary differential equations the elmer software is used for computing the local and global mass and stiffness matrices and the force vector as well for computing the correction of the initial conditions by the shooting method stability of the solutions is studied by the floquet theory sequential continuation method is used to define the prediction for the next point from the frequency response diagram the main goal of the current work is to investigate and present the potential of the proposed numerical methods for the efficient computation of the frequency response functions of large scale nonlinear systems which often result from space discretization of real life engineering applications

non linear vibrations of a clamped circular plate with initial deflection and initial edge displacement part i theory abstract theoretical analyses are presented for axisymmetric non linear vibrations of a clamped circular plate of isotropic materials under uniformly distributed lateral loading with the effect of both initial deflection and initial edge displacement taken into consideration the dynamic analog of the marguerre equations is used and the steady state solutions are obtained by first applying the galerkin method and then the harmonic balance method actual calculations are carried out for the three degree of freedom galerkin system and the frequency response characteristics and typical waveforms are determined under various initial edge displacements including initially buckled cases effects of both initial deflection and initial edge displacement on the static deflection as well as the lower natural frequencies are also clarified

normal modes of a non linear clamped clamped beam abstract non linear modal analysis approach based on invariant manifold method proposed earlier by shaw and pierre journal of sound and vibration 164 85 124 is utilized here to obtain the non linear normal modes of a clamped clamped beam for large amplitude displacements the results obtained for the fundamental normal mode are compared with the corresponding reported experimental and theoretical studies the effects of modal coupling are examined in greater detail the limitation of the present method for analyzing non linear behavior is highlighted

the effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates part ii iterative and explicit analytical solution for non linear coupled transverse and in plane vibrations the objective of this paper is to present a more realistic and complete study of the geometrically non linear free vibrations of clamped immovable circular plates by taking into account the in plane deformation which has not been examined in part i of this series of papers j sound vibration 265 2003 123 the problem is solved by a numerical iterative procedure in order to obtain more accurate results for vibration amplitudes up to twice the plate thickness the numerical results thus obtained are presented and compared with the available published results and with the ones calculated when neglecting the in plane displacement for the first two non linear axisymmetric mode shapes an explicit analytical solution is then presented and its range of validity for the fundamental non linear mode is determined via a detailed comparison with the solution based on the iterative procedure the results obtained by the explicit method show the usefulness of the new approach in comparison with the single mode approach solution frequently used in non linear vibration analysis of structures and are expected to be easy to implement in fatigue models in order to make more realistic and secure predictions of the structural fatigue life

can one hear the dimension of a fractal we consider the spectrum of the laplacian in a bounded open domain of n with a rough boundary i e with possibly non integer dimension and we discuss a conjecture by m v berry generalizing weyl s conjecture then using ideas mark kac developed in his famous study of the drum we give upper and lower bounds for the second term of the expansion of the partition function the main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on minkowski dimensions and on minkowski measures rather than on haussdorff ones

hearing the shape of a general convex domain abstract the spectral function t j 1 exp t j where j j 1 are the eigenvalues of the laplace operator i 1 2 x i 2 in the x 1 x 2 plane is studied for a general convex domain together with an impedance condition on a part of its boundary and another impedance condition on the remaining part of that boundary

stress distribution over plates vibrating at large amplitudes abstract the method of analysis of stress intensity distribution over vibrating plates is presented both the geometrical and physical nonlinearities are taken into account it is showed that with the help of the small parameter method it is possible to obtain the expression for the transverse normal strain in a form which is convenient for analytical transformations the dynamic deflection is expanded into truncated series of eigenmodes and discrete model of the system is derived the backbone curves of thin rectangular immovably simply supported plate vibrations are obtained the stress intensity distribution patterns over the plate are analyzed it is noted that in case of nonlinear vibrations not only the magnitudes of stress intensity depend on the vibration amplitude but also on the pattern of its distribution

vibration of symmetrically laminated rectangular plates considering deformation and rotatory inertia an analytical study was conducted using the galerkin technique to determine the natural frequencies and mode shapes for symmetrically laminated rectangular plates considering the effects of shear deformation and rotatory inertia two different graphite epoxy symmetric plates were used in the analysis and three different boundary conditions for each plate simply supported clamped and two opposite sides clamped two opposite sides simply supported were considered convergence characteristics comparison to classical results and the effects of length to thickness ratios were investigated the finding indicated that as the length to thickness ratios were reduced b h 3q for the first mode shear deformation effects significantly lowered the natural frequencies analysis also showed that rotatory inertia effects were very small convergence characteristics for all three boundary conditions were very good and excellent agreement with the classical solutions was achieved

low dimensional models for the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal decomposition in formulating mathematical models for dynamical systems obtaining a high degree of qualitative correctness i e predictive capability may not be the only objective the model must be useful for its intended application and models of reduced complexity are attractive in many cases where time consuming numerical procedures are required this paper discusses the derivation of discrete low dimensional models for the nonlinear vibration analysis of thin cylindrical shells in order to understand the peculiarities inherent to this class of structural problems the nonlinear vibrations and dynamic stability of a circular cylindrical shell subjected to static and dynamic loads are analyzed this choice is based on the fact that cylindrical shells exhibit a highly nonlinear behavior under both static and dynamic loads geometric nonlinearities due to finite amplitude shell motions are considered by using donnell s nonlinear shallow shell theory a perturbation procedure validated in previous studies is used to derive a general expression for the nonlinear vibration modes and the discretized equations of motion are obtained by the galerkin method using modal expansions for the displacements that

an inverse problem for a general annular bounded domain in r2 with mixed boundary conditions and its physical applications this paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative laplacian in r 2 the asymptotic expansion of the trace of the heat kernel q t u 1 exp t m n where m n n 1 are the eigenvalues of the negative laplacian d 2 k 1 2 x k 2 in the x 1 x 2 plane is studied for short time t of a general annular bounded domain w in r 2 together with its smooth inner boundary w 1 and its smooth outer boundary w 2 where a finite number of dirichlet neumann and robin boundary conditions on the piecewise smooth components c i i 1 m of w 1 and on the piecewise smooth components c i i m 1 n of w 2 such that w 1 i 1 m c i and w 2 i m 1 n c i are considered in this paper one may extract information on the geometry of w by analyzing the asymptotic expansions of q t for short time t some applications of q t for an ideal gas enclosed in w are given thermodynamic quantities of an ideal gas enclosed in w are examined we use an asymptotic expansion for high temperatures to obtain the partition function of an ideal gas showing the leading corrections to the internal energy due to a finite container we show that the ideal gas cannot feel the shape of its container although it can feel some geometrical properties of it

a semi analytical approach to the non linear dynamic response problem of beams at large vibration amplitudes part ii multimode approach to the steady state forced periodic response abstract the semi analytical approach to the non linear dynamic response of beams based on multimode analysis has been presented in part i of this series of papers azrar et al 1999 journal of sound and vibration 224 183 207 1 the mathematical formulation of the problem and single mode analysis have been studied the objective of this paper is to take advantage of applying this semi analytical approach to the large amplitude forced vibrations of beams various types of excitation forces such as harmonic distributed and concentrated loads are considered the governing equation of motion is obtained and can be considered as a multi dimensional form of the duffing equation using the harmonic balance method the equation of motion is converted into non linear algebraic form techniques of solution based on iterative incremental procedures are presented the non linear frequency and the non linear modes are determined at large amplitudes of vibration the basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated the percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution also the percentage contributions of the higher modes to the bending moment near to the clamps are given in order to determine accurately the error introduced in the non linear bending stress estimated by different approximations solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency the non linear deflection shapes in various regions of the solution the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes the numerical results obtained here for the non linear forced response are compared with those from the linear theory with available non linear results based on various approaches and with the single mode analysis

analysis of non linear mode shapes and natural frequencies of continuous damped systems in this paper the aim is to find the non linear mode shapes and natural frequencies for a class of one dimensional continuous damped systems with weak cubic inertia damping and stiffness non linearities this paper presents general formulations for natural frequencies and mode shapes with all non linearity effects initially the non linear system with general boundary conditions is discretized and using a two dimensional manifold the model of cubic non linearities is constructed and the general equation of motion which governs non linear system is derived the method of multiple scales is then used to extend the non linear mode shapes and natural frequencies during this analysis it is realized that when the natural frequencies of the linear system become equal to the natural frequencies of the non linear system a one to one internal resonance will appear also there is a three to one internal resonance which is not dependent on the damping of the system finally general formulations of amplitude for vibrations natural frequencies and mode shapes of the non linear system are obtained in parametric forms thus a non linear problem with some simple integration can be solved the formulations are capable of handling any non linearities in inertia damping stiffness or any combination of them under any arbitrary boundary conditions

nonlinear vibrations of rectangular laminated thin plates using the basic assumption of thin plate theory including nonlinear terms in the von karman sense the free flexural large amplitude vibrations of arbitrarily laminated rectangular plates are studied full mode displacement formulation with the aid of galerkin s approximation is treated for more accurate analyses the duffingtype equation is solved by the method of harmonic balance with sufficient number of terms in time series which is compared with the method of direct numerical integration the convergence of the number of half waves in the x and y directions and that of time series is established the nonlinear frequencies for the simply supported with movable edges and fully clamped with immovable edges are compared with the available solutions in the literature

nonlinear normal modes of structural systems via asymptotic approach abstract an asymptotic approach based on the method of multiple scales is employed to construct the nonlinear normal modes nnm s of self adjoint structural systems with arbitrary linear inertia and elastic stiffness operators general cubic inertia and geometric nonlinearities the methodology employed for constructing the approximate invariant manifolds of individual nnm s away from internal resonances and of the resonant modes near three to one internal resonances attempts to generalize previous studies based on asymptotic techniques the theory is applied to a hinged hinged uniform elastic beam carrying a lumped mass and undergoing axis stretching depending on the lumped mass relative to the beam mass and on its position along the span different classes of nonlinear normal modes and their stability are investigated

nonlinear modes of clarinet like musical instruments the concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments using a modal expansion of the impedance of the instrument and by projecting the equation for the acoustic pressure on the normal modes of the air column a system of second order ordinary differential equations is obtained the equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed the system is treated using an amplitude phase formulation for nonlinear modes where the frequency and damping functions as well as the invariant manifolds in the phase space are unknowns to be determined the formulation gives without explicit integration of the underlying ordinary differential equation access to the transient the limit cycle its period and stability the process is illustrated for a model reduced to three normal modes of the air column

on hearing the shape of rectilinear regions from complete knowledge of the eigenvalues of the negative laplacian on a bounded domain one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function explicit calculations are performed for an equilateral triangular domain with dirichlet or neumann boundary conditions yielding in particular the corner angle terms in three dimensions some applications to eigenvalue problems for an equilateral triangular prism are dealt with including the solid vertex terms

an inverse problem of the wave equation for a general doubly connected region in r2 with a finite number of piecewise smooth robin boundary conditions the spectral distribution m t w 1 exp ite w 1 2 where e w w 1 are the eigenvalues of the negative laplacian d k 1 2 x k 2in the x 1 x 2 plane is studied for a variety of domains where 0

numerical computation of nonlinear normal modes in mechanical engineering this paper reviews the recent advances in computational methods for nonlinear normal modes nnms different algorithms for the computation of undamped and damped nnms are presented and their respective advantages and limitations are discussed the methods are illustrated using various applications ranging from low dimensional weakly nonlinear systems to strongly nonlinear industrial structures

hardening softening behaviour in non linear oscillations of structural systems using non linear normal modes the definition of a non linear normal mode nnm as an invariant manifold in phase space is used in conservative cases it is shown that normal form theory allows one to compute all nnms as well as the attendant dynamics onto the manifolds in a single operation then a single mode motion is studied the aim of the present work is to show that too severe truncature using a single linear mode can lead to erroneous results using single non linear mode motion predicts the correct behaviour hence the nonlinear change of co ordinates allowing one to pass from the linear modal variables to the normal ones linked to the nnms defines a framework to properly truncate non linear vibration pdes two examples are studied a discrete system a mass connected to two springs and a continuous one a linear euler bernoulli beam resting on a non linear elastic foundation for the latter a comparison is given between the developed method and previously published results

some erroneous finite element formulations of non linear vibrations of beams and plates the non linear vibrations of straight beams and flat plates are governed by differential equations which effectively involve a cubic non linearity this paper brings out the source of errors in some finite element formulations of these non linear problems which are based on the introduction of a linearizing function in the expression for the strain energy the magnitude of this error is derived it is explicitly shown that the discrepancy between the results of approximate analytical solutions and those of these finite element solutions is attributable to these errors in the finite element formulations

numerical methods and parallel algorithms for computation of periodic responses of plates numerical methods that compute the nonlinear frequency response curves of plates are presented the equation of motion of the plate is derived by assuming first order shear deformation theory and considering geometrical type of nonlinearity it is discretized by the finite element method using bilinear quadrilateral elements the frequency response curve is computed by shooting and continuation methods parallel implementation of the shooting method is presented and its scalability is investigated

an inverse eigenvalue problem for a general convex domain the underlying problem is to deduce the precise shape of a membrane from the complete knowledge of the eigenvalues i for the lapace operator a cf a ax in the x1x2 plane let d g r be a simply connected bounded domain with a smooth boundary ad let us suppose that the a are the eigenvalues of the impedance problem au k o in d

numerical methods for thermally stressed shallow shell equations abstract we develop efficient and accurate numerical methods to solve a class of shallow shell problems of the von karman type the governing equations form a fourth order coupled system of nonlinear biharmonic equations for the transverse deflection and airy s stress function a second order finite difference discretization with three iterative methods picard newton and trust region dogleg is proposed for the numerical solution of the nonlinear pde system three simple boundary conditions and two application motivated mixed boundary conditions are considered along with the nonlinearity of the system boundary singularities that appear when mixed boundary conditions are specified are the main numerical challenges two approaches that use either a transition function or local corrections are developed to deal with these boundary singularities all the proposed numerical methods are validated using carefully designed numerical tests where expected orders of accuracy and rates of convergence are observed a rough run time performance comparison is also conducted to illustrate the efficiency of our methods as an application of the methods a snap through thermal buckling problem is considered the critical thermal loads of shell buckling with various boundary conditions are numerically calculated and snap through bifurcation curves are also obtained using our numerical methods together with a pseudo arclength continuation method our results are consistent with previous studies

a method for nonlinear modal analysis and synthesis application to harmonically forced and self excited mechanical systems abstract the recently developed generalized fourier galerkin method is complemented by a numerical continuation with respect to the kinetic energy which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes in order to enhance the practicability regarding the investigation of complex large scale systems it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations a novel reduced order model rom is developed for those regimes where internal resonances are absent the approach allows for an accurate approximation of the multi harmonic content of the resonant mode and accounts for the contributions of the off resonant modes in their linearized forms the rom facilitates the efficient analysis of self excited limit cycle oscillations frequency response functions and the direct tracing of forced resonances the rom is equipped with a large parameter space including parameters associated with linear damping and near resonant harmonic forcing terms an important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology this is achieved by selected numerical examples including finite element models of structures with strongly nonlinear non conservative contact constraints

non linear vibration characteristics of clamped laminated shallow shells abstract this paper examines non linear free vibration characteristics of first and second vibration modes of laminated shallow shells with rigidly clamped edges non linear equations of motion for the shells based on the first order shear deformation and classical shell theories are derived by means of hamilton s principle we apply galerkin s procedure to the equations of motion in which eigenvectors for first and second modes of linear vibration obtained by the ritz method are employed as trial functions then simultaneous non linear ordinary differential equations are derived in terms of amplitudes of the first and second vibration modes backbone curves for the first and second vibration modes are solved numerically by the gauss legendre integration method and the shooting method respectively the effects of lamination sequences and transverse shear deformation on the behavior are discussed it is also shown that the motion of the first vibration mode affects the response for the second vibration mode

an inverse problem for a general vibrating annular membrane in r3 with its physical applications further results this paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative laplacian in r 3 the trace of the heat semigroup q t email protected v 1 exp email protected v where m v v 1 are the eigenvalues of the negative laplacian email protected email protected b 1 email protected x b 2 in the x 1 x 2 x 3 space is studied for short time t for a general vibrating annular membrane w in r 3 together with its smooth inner bounding surface s 1 and its smooth outer bounding surface s 2 where a finite number of dirichlet neumann and robin boundary conditions on the piecewise smooth components s i i 1 m of s 1 and on the piecewise smooth components s i i m 1 n of s 2 is considered such that s email protected i 1 ms i and s email protected i m 1 ns i in this paper one may extract information on the geometry of w by analyzing the asymptotic expansions of q t for short time t some applications of q t for an ideal gas enclosed in the general vibrating annular membrane w are given we show that the asymptotic expansion of q t for short time t plays an important role in investigating the influence of the annular region w on the thermodynamic quantities of an ideal gas some applications of q t for an ideal gas enclosed in a compact n dimensional riemannian manifold are also given we show that the ideal gas cannot feel the shape of its container although it can feel some geometrical properties of it

a finite element time domain modal formulation for large amplitude free vibrations of beams and plates abstract a finite element time domain modal fomulation is presented for the large amplitude free vibration of plates the procedure of deriving the non linear modal equations of motion is simple and general accurate frequency maximum deflection relations can be obtained for the fundamental and higher non linear modes the percentage of participation from each linear mode determines the number of modes to be included for accurate frequency results numerical examples on beams and square plates are given and compared with classical continuum analytical solutions a plate of complex planform and boundary conditions is also given to show the versatility of the presented modal formulation

an inverse eigenvalue problem of the wave equation for a multi connected region in r2 together with three different types of boundary conditions the trace of the wave kernel m t email protected w 1exp ite 1 2 w where e w w 1 are the eigenvalues of the negative laplacian email protected email protected 2 k 1 email protected x k 2 in the x 1 x 2 plane is studied for a variety of bounded domains where

identification of coupled non linear modes from free vibration using time frequency representations abstract a new method based on time frequency representations is presented for identifying the non linear modal parameters of a multi degree of freedom non linear lightly damped mechanical system the coupled non linear modes are first introduced it is shown that by using the coupled non linear modes the free response can be expressed as a linear combination of frequency and amplitude modulated components this suggests the use of time frequency methods to characterize these modulation laws and then identify the coupled non linear modes these modulation laws are extracted by using an extended definition of the so called ridges of the continuous gabor transform which enables a more accurate estimation an identification procedure of the coupled non linear modes is proposed and the validity of the whole process is confirmed by performing computer simulations of different types of non linear elastic dynamic systems

a spectral characterization of nonlinear normal modes belgian network dysco dynamical systems control and optimization funded by the interuniversity attraction poles programme initiated by the belgian science policy office

theory and experiments for large amplitude vibrations of circular cylindrical panels with geometric imperfections abstract large amplitude vibrations of circular cylindrical panels open shells subjected to harmonic excitation are numerically and experimentally investigated the donnell nonlinear strain displacement relationships are used to describe the geometric nonlinearity in plane inertia is taken into account specific boundary conditions with zero transverse displacement at the panel edges and free or elastically restrained in plane displacements not previously considered have been introduced in order to model the experimental boundary conditions the nonlinear equations of motion are obtained by the lagrange equations with multi mode approach and are studied by using a code based on the pseudo arclength continuation method two thin circular cylindrical panels of different dimensions and made of stainless steel have been experimentally tested in the laboratory for several excitation amplitudes in order to characterize the nonlinearity the dimensions of the two panels have been chosen in order to have the fundamental mode with one and two circumferential half waves respectively numerical results are able to reproduce the experimental results with high accuracy for both panels the effect of geometric imperfections on the trend of nonlinearity and on natural frequencies is shown convergence of the solution with the number of generalized coordinates is numerically verified

nonlinear vibrations of circular cylindrical panels abstract large amplitude geometrically nonlinear vibrations of circular cylindrical panels with rectangular base simply supported at the four edges and subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated two different nonlinear strain displacement relationships from the donnell s and novozhilov s shell theories are used to calculate the elastic strain energy in plane inertia and geometric imperfections are taken into account the solution is obtained by lagrangian approach the nonlinear equations of motion are studied by using i a code based on arclength continuation method that allows bifurcation analysis and ii direct time integration numerical results are compared to those available in the literature and convergence of the solution is shown interaction of modes having integer ratio among their natural frequencies giving rise to internal resonances is also discussed

non linear vibration of a hinged orthotropic circular plate with a concentric rigid mass abstract non linear free vibration of hinged orthotropic circular plates with a concentric rigid mass at the centre is studied by using the finite element method hamilton s principle is applied to derive the basic non linear partial differential equations and associated boundary conditions for the problem of large amplitude of an orthotropic circular plate the applications of the finite element method to the dynamic problem rely on the use of a variation principle to derive the necessary element property s equations the assembled equations for the plate are formed by summing each of the element equations obtained in consideration of a single element then the boundary conditions are imposed on the vector of nodal field variables so that the appropriate boundary conditions are satisfied the assembled equations form an eigenvalue problem and are solved for the unknown field variables the relations between the fundamental frequencies and the amplitudes of non linear vibrations of the circular orthotropic elastic plate with a rigid core are obtained the results show that the frequency responses of the plate varies with changes of boundary conditions and the ratio between tangential and radial elastic constant

the wave equation approach to an inverse problem for a general multi connected domain in r2 with mixed boundary conditions the spectral function email protected t email protected j 1 exp email protected j 1 2 where m j j 1 are the eigenvalues of the negative laplacian email protected email protected u 1 email protected x u 2 in r 2 is studied for small t for a variety of domains where

semi analytical approach to the non linear dynamic response problem of s s and c c beams at large vibration amplitudes part i general theory and application to the single mode approach to free and forced vibration analysis abstract in a previous series of papers 1 3 a general model based on hamilton s principle and spectral analysis was developed for non linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rectangular homogeneous and composite plates as an introduction to the present work concerned with the forced non linear response of c c and s s beams the above model has been derived using spectral analysis lagrange s equations and the harmonic balance method then the forced case has been examined and the analysis led to a set of non linear partial differential equations which reduces to the classical modal analysis forced response matrix equation when the non linear terms are neglected on the other hand if only one mode is assumed this set reduces to the duffing equation very well known in one mode analyses of non linear systems having cubic non linearities so it appeared sensible to consider such a formulation as the multidimensional duffing equation in order to solve the multidimensional duffing equation in the case of harmonic excitation of beam like structures a method is proposed based on the harmonic balance method and a set of non linear algebraic equations is obtained whose numerical solution leads in each case to the basic function contribution coefficients to the displacement response function these coefficients depend on the excitation frequency and the distribution of the applied forces the frequency response curve obtained here exhibits qualitatively a classical non linear behaviour with multivalued regions in which the jump phenomenon could occur quantitatively the analytical results obtained here without assuming any limitation to the scale of the excitation is identical to that obtained by the multiple scale method which assumes small values of the scaling parameter attention was focused on the assumed one mode in order to improve the results obtained in the case of free vibrations the analytical solution obtained by elliptic functions has been expanded into power series of higher orders using the symbolic manipulation program maple it has been shown that extreme care must be taken in the choice of the polynomial approximation which is valid only in a zone limited by a radius of convergence the use of pade approximants permitted considerable increase in the zone of validity of the solution obtained for very large vibration amplitudes

an energy preserving description of nonlinear beam vibrations in modal coordinates abstract conserved quantities are identified in the equations describing large amplitude free vibrations of beams projected onto their linear normal modes this is achieved by writing the geometrically exact equations of motion in their intrinsic or hamiltonian form before the modal transformation for nonlinear free vibrations about a zero force equilibrium it is shown that the finite dimensional equations of motion in modal coordinates are energy preserving even though they only approximate the total energy of the infinite dimensional system for beams with constant follower forces energy like conserved quantities are also obtained in the finite dimensional equations of motion via casimir functions the duality between space and time variables in the intrinsic description is finally carried over to the definition of a conserved quantity in space which is identified as the local cross sectional power numerical examples are used to illustrate the main results

periodic geometrically nonlinear free vibrations of circular plates abstract the geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi degree of freedom model which is based on thin plate theory and on von karman s nonlinear strain displacement relations the middle plane in plane displacements are included in the formulation and the common axisymmetry restriction is not imposed the equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in plane and transverse displacement fields are given by weighted series of spatial functions these spatial functions are based on hierarchical sets of polynomials which have been successfully used in p version finite elements for beams and rectangular plates and on trigonometric functions employing the harmonic balance method the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method convergence with the number of shape functions and of harmonics is analysed the numerical results obtained are presented and compared with available published results it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom internal resonances are found and the ensuing multimodal oscillations are described

hearing the shape of a general doubly connected domain in r3 with impedance boundary conditions the basic problem in this paper is that of determining the geometry of a general doubly connected domain in r3 together with an impedance condition on its inner bounding surface and another impedance condition on its outer bounding surface from the complete knowledge of the eigenvalues j j 1 for the three dimensional laplacian using the asymptotic expansion of the spectral function t j 1 exp t j for small positive t

physical interpretation of the proper orthogonal modes using the singular value decomposition abstract proper orthogonal decomposition is a statistical pattern analysis technique for finding the dominant structures called the proper orthogonal modes in an ensemble of spatially distributed data while the proper orthogonal modes are obtained through a statistical formulation they can be physically interpreted in the field of structural dynamics the purpose of this paper is thus to provide some insights into the physical interpretation of the proper orthogonal modes using the singular value decomposition

analytical method for vibration of angle ply cylindrical shells having arbitrary edges a theoretical method is presented for solving the free vibration of angle ply laminated cylindrical shells the angle ply laminated shell is macroscopically modeled as a thin shell of general anisotropy by using the classical lamination theory the functional derived from the flugge type shell theory is minimized by following the ritz procedure and arbitrary combinations of boundary conditions at both ends are accommodated by introducing newly developed admissible functions a computer code is made to verify the validity of the present method and the natural frequencies and mode shapes of the laminated cylindrical shells are given for some typical edge conditions numerically

nonlinear dynamic behaviors of clamped laminated shallow shells with one to one internal resonance abstract this paper investigates one to one internal resonance of laminated shallow shells with rigidly clamped edges it is assumed that the natural frequencies 2 and 3 of two asymmetric second and third vibration modes have the relationship 2 3 the displacements are expressed by using eigenvectors for linear vibration modes calculated by the ritz method applying galerkin s procedure to the equation of motion nonlinear differential equations are derived by considering the first vibration mode in addition to the two asymmetric vibration modes quadratic nonlinear terms expressing the interaction between the asymmetric and the first modes appear in the differential equations shooting method is used to obtain the steady state response when the driving frequency is near 2 the dynamic characteristics of the shells with the internal resonance are discussed

on hearing the shape of a general multi connected vibrating membrane in r2 with piecewise smooth positive functions in the robin boundary conditions the asymptotic expansion of the trace of the heat kernel q t email protected j 1 exp email protected j as t 0 has been derived for a variety of domains where l j are the eigenvalues of the negative laplace email protected email protected i 1 email protected email protected x i 2in the x 1 x 2 plane the dependence of q t on the connectivity of domains and the boundary conditions are analyzed particular attention is given for a general multiply connected bounded domain in r 2 together with a finite number of piecewise robin boundary conditions where the coefficients in the boundary conditions are piecewise smooth positive functions some applications of an ideal gas enclosed in the general multiply connected domain are given

strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry abstract dynamic responses of a linear oscillator coupled to a nonlinear energy sink nes under harmonic forcing in the regime of 1 1 1 resonance are investigated primary attention is paid to the detailed investigation of the so called strongly modulated response smr which is not related to the fixed points of average modulation equations of the system essential mass asymmetry allows a global analysis of the responses despite strong nonlinearity it is demonstrated that the strongly modulated response is related to a relaxation type motion and its description in the limit of small mass ratio maybe reduced to the 1d return map of a subset at a fold line of slow invariant manifold the smr exists in the o e vicinity of the exact resonance where e 1 characterizes the mass asymmetry it is also shown that the smr appears in the system as a result of global fold bifurcation of limit cycles and exhibits some properties pertinent to generic 1d nonlinear maps such as period doubling transient responses with finite number of relaxation cycles and subsequent attraction to stable periodic attractor are revealed analytic results are compared to numeric simulations and a good agreement is observed

improved clustered look ahead pipelining algorithm with minimum order augmentation the author compares the overall performance of clustered look ahead cla filters with minimum order augmentation and scattered look ahead sla filters to optimize the domain searching procedure and to improve the numerical performance an improved cla algorithm with minimum order augmentation which is especially beneficial for high q high speed digital filters is proposed this algorithm is optimized in the aspect of minimizing both the augmented pipelining order and undesirable quantization effects simultaneously the proposed cla structure with minimum order augmentation turns out to be very suitable for the applications that require high speed and numerical stability