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

asymmetric non linear forced vibrations of free edge circular plates part 1 theory in this article a detailed study of the forced asymmetric non linear vibrations of circular plates with a free edge is presented the dynamic analogue of the von karman equations is used to establish the governing equations the plate displacement at a given point is expanded on the linear natural modes the forcing is harmonic with a frequency close to the natural frequency omega n of one asymmetric mode of the plate thus the vibration is governed by the two degenerated modes corresponding to omega n which are one to one internally resonant an approximate analytical solution using the method of multiple scales is presented attention is focused on the case where one configuration which is not directly excited by the load gets energy through non linear coupling with the other configuration slight imperfections of the plate are taken into account experimental validations are presented in the second part of this paper

simpler polynomial solutions in stochastic feedback control this paper is an exact parallel to the solution of the stochastic feedback control problem in discrete time given in kucera s book 1979 the difference is that here it is shown that usually it is only necessary to solve one rather than two diophantine equations the resulting reduction in dimension will be particularly important in large scale systems

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

free vibration analysis of a flat plate using the hierarchical finite element method abstract the hierarchical finite element method is used to determined the natural frequencies and modes of a flat rectangular plate ten different boundary conditions including free edges and point supports are considered in this paper extensive results are presented for each case including the variation of frequency with the aspect ratio and the poisson ratio and these are shown to be in very good agreement with the work of other investigators this confirms both the applicability and accuracy of solution of the hfem to problem of this type

optimal multivariable lqg control using a single diophantine equation the general solution of the optimal lqg problem using the polynomial equation approach involves the solution of a couple of two sided matrix polynomial equations the conditions under which a single one sided equation which is related to the original two the implied diophantine equation yields the unique optimal controller are derived in this paper the result has considerable relevance to aspects such as computation and self tuning control applications

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

the harmonic balance method with arc length continuation in rotor stator contact problems there are a variety of abnormal running conditions in rotating machinery which lead to rotor stator interaction dynamics which in turn can cause a rich mixture of effects associated with rub related phenomena these effects manifest themselves in the occurrence of multiple solutions for steady state vibration response scenarios including amplitude jumps during rotor acceleration and vibration responses at different multiple frequencies of excitation forces such as unbalance this paper describes a numerical algorithm based on the harmonic balance method to calculate the periodic response of a non linear system under periodic excitation the algorithm also calculates the stability of the periodic solutions found marks turning and bifurcation points and follows a solution branch over varying system parameters via arc length continuation

polynomial equations for the linear mmse state estimation the linear minimum mean square error mmse state estimation problem is solved via spectral factorization and a pair of bilateral diophantine equations detectability and or stability requirements in the riccati based solution context are expressed in terms of the stability of the greatest common right and left divisors of polynomial matrices

normal modes of vibration for non linear continuous systems a definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non linear partial differential equations the definition for normal modes is given in terms of the dynamics on two dimensional invariant manifolds in the system phase space these manifolds are a continuation of the planes which represent the well known normal modes of the linearized system a local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds the procedure also provides the non linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion in this paper the general theory is described for the application of vibrations of continuous media but it can easily be extended to other situations in order to demonstrate the power of the approach and its unique procedural aspects three examples involving beam vibrations are worked out in detail the examples are conservative simply supported beams the first demonstrates the methodology for a linear beam model the second is a beam on a non linear elastic foundation and the third example is a beam with non linear torsional springs attached at each end in these examples the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure

computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background abstract a general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady state is presented the equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis the multi harmonic solution of motion equation is written in a form of truncated fourier series next the galerkin ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation the matrix amplitude equation as well as the associated tangent matrix are given in an explicit form the stability of steady state solution is discussed by using the floquet theory the numerical algorithm and an example application are described in a companion paper by lewandowski lewandowski r computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and numerical examples international journal of solids and structures in preparation

optimal multivariable regulation with disturbance measurement feedforward abstract the problem of optimal stochastic regulation based on output feedback and disturbance measurement feedforward for multivariable systems is solved using the polynomial equation approach two independent couples of polynomial matrix equations are derived which yield the feedback and the feedforward parts of the regulator

polynomial optimization of stochastic discrete time control for unstable plants the matrix polynomial method of optimization of linear multivariable discrete time control for unstable plants with stationary stochastic inputs is derived in a constructive manner new sufficient conditions are obtained and kucera s second diophantine equation is found directly from the requirement that the system be stable although the second equation must be satisfied it is proved that it need not be used in the optimization when a certain coprimeness exists

continuous time lq regulator design by polynomial equations abstract the deterministic continuous time lq output regulation problem is solved by polynomial equations as an alternative to the usual riccati equation approach in particular it is shown that riccati based and polynomial methods are fully conceptually equivalent in steady state lq regulation

non linear free vibrations of beams by the finite element and continuation methods abstract a computational method is presented for determining backbone curves of arbitrary geometry as an example beam vibration is considered the beam response is expanded into a truncated fourier series with respect to time after starting from the integral of action the variational approach and the finite element method are used to formulate the non linear eigenvalue problem the continuation method is adopted to solve the resulting non linear eigenvalue problem and to obtain the non linear frequencies and modes of vibration numerical results for various beams are presented and compared with available results to demonstrate the accuracy and applicability of the method moreover the bifurcation points on some beams backbone curves are found and reported for the first time

finite element analysis of three dimensional vibrations of thick circular and annular plates abstract three dimensional vibrations of thick circular and annular plates are analyzed by a finite element method which with a properly assumed set of displacement field is different from the traditional 3 d finite element analysis and is reduced to a sequence of 2 d analyses one for each circumferential wave number the present approach has several unique features 1 it can obtain vibration frequencies which are comparable to or as accurate as those by other three dimensional approaches whenever comparisons between them are possible and yet the formulation of the present method is simpler and its application is straightforward 2 different simply supported boundary conditions can be exactly imposed with the present approach the latter feature may be quite difficult if not impossible for other methods in the present analysis vibration frequencies of a circular and an annular plate under different combinations of boundary conditions wave numbers and finite element meshes are calculated vibration characteristics are examined and some typical mode shapes are shown graphically then the results of some example problems by the present method are compared to those available in the literature to show its validity and accuracy

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

non linear interactions in asymmetric vibrations of a circular plate abstract in order to investigate the non linear asymmetric vibrations of a clamped circular plate on an elastic foundation the primary resonances of a plate with an internal resonance in which the natural frequencies of two asymmetric modes are commensurable are considered the response is expressed as an expansion in terms of the linear free oscillation modes and its amplitude is considered to be small but finite the method of multiple scales is used to reduce the non linear governing equations to a system of autonomous ordinary differential equations for amplitude and phase variables for a numerical example the case of internal resonance a commensurable relationship between natural frequencies 32 3 11 where the first subscript refers to the number of nodal diameters and the second subscript the number of nodal circles including boundary is considered when the frequency of excitation is near 11 there exist at most five stable steady state responses four of them are superpositions of traveling wave components and one is a superposition of standing wave components the result shows the interaction between modes corresponding to 11 and 32 by showing non vanishing amplitudes of the mode not directly excited when the frequency of excitation is near 32 similarly the interaction between modes is shown to exist all of the responses with non vanishing amplitudes of modes excited indirectly however turn out to be unstable which is a peculiar phenomenon

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

maximization of the vibration amplitude and bifurcation analysis of nonlinear systems using the constrained optimization shooting method abstract an original method for calculating the maximum vibration amplitude of periodic solutions of nonlinear systems is presented the problem of determining the worst maximum vibration is transformed into a nonlinear optimization problem the shooting method and the floquet theory are selected to construct the general nonlinear equality and inequality constraints the resulting constrained maximization problem is then solved by using the multistart algorithm finally the effectiveness and ability of the proposed approach are illustrated through two numerical examples numerical examples show that the proposed method can give results with higher accuracy as compared with numerical results obtained by a parameter continuation method and the ability of the present method is also demonstrated

on the polynomial equations for the mimo lq stochastic regulator the role played by three polynomial equations in the lq stochastic regulation problem is discussed the emphasis is on establishing conditions under which the lq regulator can be obtained via the solution of a single uncoupled diophantine equation nevertheless the authors do not claim that there is any advantage in solving a single uncoupled equation in place of two coupled diophantine equations

a comparison of classical and high dimensional harmonic balance approaches for a duffing oscillator the present study focuses on a novel harmonic balance formulation which is much easier to implement than the standard classical harmonic balance method for complex nonlinear mathematical models and algorithms both harmonic balance approaches are applied to duffing s oscillator to demonstrate the advantages and disadvantages of the two approaches a fundamental understanding of the difference between these two methods is achieved and the properties of each method are analyzed in detail

non linear free periodic vibrations of open cylindrical shallow shells this paper is concerned with the non linear free periodic vibrations of thin open cylindrical and shallow shells vibrating in the geometrically non linear regime a multi degree of freedom model with hierarchical basis functions is adopted and the principle of the virtual work is used to define the time domain equations of motion these equations are transformed into the frequency domain by the harmonic balance method and are finally solved by an arc length continuation method shells of different thicknesses and of different curvature radius are analysed and the variation of the non linear natural frequencies of these shells with the vibration amplitude are investigated in some detail the variation of the mode shapes with the vibration amplitude is demonstrated it is found that both softening and hardening spring effects occur and that the number of couplings between vibration modes is rather large in undamped shells

on the free vibration of completely free open cylindrically curved isotropic shell panels abstract theh pversion of the finite element method is used to furnish a detailed study of the vibration characteristics of completely free open cylindrically curved isotropic shell panels convergence studies have been carried out to ascertain the most appropriate mix ofh refinement andp enrichment necessary to give a reasonable solution accuracy this approach has been corroborated by comparing results from the present methodology with the work of other investigators and the results forthcoming from a proprietary finite element package extensive results have been presented for panels with aspect ratios of 0 5 1 and 2 the curvature of which is gradually increased from that of a flat plate at one extreme to a complete although unjoined cylinder at the other curves of frequency versus included angle curvature have been obtained for the first 20 modes of each of these panels and the following observations have been made i the majority of the low frequency modes are virtually independent of shell curvature and ii the modes that are heavily influenced by curvature effects involve motion that is confined to the lengthwise edge regions of the panel

modal interaction in circular plates abstract the dynamic analogue of the von karman equations in polar form is used to study the symmetric response of a circular plate to a harmonic external excitation the method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the phases and amplitudes of oscillation for the case of an internal or autoparametric resonance the results show that the internal resonance is responsible for coupling of the modes involved and that the excited mode is not necessarily the dominant one furthermore the results show that for a choice of parameters the multi mode response loses its stability through a hopf bifurcation resulting in periodically or chaotically modulated motions of the plate

a high order purely frequency based harmonic balance formulation for continuation of periodic solutions combining the harmonic balance method hbm and a continuation method is a well known technique to follow the periodic solutions of dynamical systems when a control parameter is varied however since deriving the algebraic system containing the fourier coefficients can be a highly cumbersome procedure the classical hbm is often limited to polynomial quadratic and cubic nonlinearities and or a few harmonics several variations on the classical hbm such as the incremental hbm or the alternating frequency time domain hbm have been presented in the literature to overcome this shortcoming here we present an alternative approach that can be applied to a very large class of dynamical systems autonomous or forced with smooth equations the main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the hbm once the equations have been rendered quadratic it becomes obvious to derive the algebraic system and solve it by the so called asymptotic numerical method anm continuation technique several classical examples are presented to illustrate the use of this numerical approach

acoustic impulse responses for nonuniform media abstract the acoustic impulse responses of a family of nonuniform lossless one dimensional media are determined the nonuniform portions of the media are confined to layers of either finite or infinite thicknesses the acoustic impedances of the nonuniform layers as functions of travel time are assumed known the impulse responses are determined by evaluating the fourier transforms of their reflection coefficients using infinite series expansions and contour integrations

paper implicit and explicit lqg self tuning controllers the design of optimal controllers for use in self tuning systems is considered linear quadratic gaussian lqg controllers are widely used elsewhere but have not until recently been applied in self tuning systems except in minimum output variance forms the lqg controllers offer a guarantee of stability when the plant parameters are known which is particularly useful for nonminimum phase systems the explicit lqg self tuning controllers introduced in the following are relatively simple to implement the first version of an implicit lqg self tuning controller is also introduced the magnitude of the transport delay terms need not be known a priori and integral action may be introduced easily if required the desired closed loop poles of the system can be prespecified

new modified multi level residue harmonic balance method for solving nonlinearly vibrating double beam problem abstract in this study a new modified multi level residue harmonic balance method is presented and adopted to investigate the forced nonlinear vibrations of axially loaded double beams although numerous nonlinear beam or linear double beam problems have been tackled and solved there have been few studies of this nonlinear double beam problem the geometric nonlinear formulations for a double beam model are developed the main advantage of the proposed method is that a set of decoupled nonlinear algebraic equations is generated at each solution level this heavily reduces the computational effort compared with solving the coupled nonlinear algebraic equations generated in the classical harmonic balance method the proposed method can generate the higher level nonlinear solutions that are neglected by the previous modified harmonic balance method the results from the proposed method agree reasonably well with those from the classical harmonic balance method the effects of damping axial force and excitation magnitude on the nonlinear vibrational behaviour are examined

hierarchical finite element analyses of geometrically non linear vibration of beams and plane frames geometrically non linear vibrations of beams and plane frameworks are analyzed by the hierarchical finite element method hfem two main points are of interest the first is to compare polynomials trigonometric functions and beam eigenfunctions as displacement shape functions for beam hierarchical finite elements the second is to examine the suitability of the hfem for time domain non linear analyses

implied polynomial matrix equations in multivariable stochastic optimal control abstract this paper reports recent work in the theoretical development of the polynomial equation approach to the optimization of multivariable control systems the algebraic properties of the polynomial matrix equations which define the optimal controller are investigated and new results concerned with the numerical solvability of the equations are derived

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

vibrations of completely free shallow shells of rectangular planform abstract although much has been written about the free vibrations of rectangular plates having completely free boundaries very little has appeared for the case when the plates have curvature i e shallow shells a solution of the problem is presented here for shells having arbitrary but constant curvature the ritz method is used with displacement functions assumed in the form of polynomials convergence studies were made to determine the number of terms required for reasonable solution accuracy numerical results were obtained for the frequencies and mode shapes of three types of shells circular cylindrical spherical and hyperbolic paraboidal and these are compared with those of a flat plate

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

diophantine equations in control a survey abstract this survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems these are linear equations in a ring and result from a fractional representation of the systems involved the cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant one can then choose in principle the best controllers for various applications these ideas evolved from early attempts to use polynomial equations in the design of discrete time linear systems by now they have been extended to continuous time infinite dimensional time varying and non linear systems

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

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

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

extension of linear quadratic optimal control theory for mixed backgrounds the problem of finding an optimal controller while both deterministic and stochastic signals are present in the feedback system is addressed a general framework is developed to solve the above problem and to unify conventional frequency domain lq and lqg optimal control theories both the system transient and steady state behaviours are considered and the exact tracking of the arbitrary reference signal is guaranteed the decomposition of a mixed signal into the deterministic and stochastic parts gives a chance to investigate the effect of the weighting factors which were specified in the cost function on the compromise between transient and steady state system performance the solution which characterizes the structure of the optimal controller results in a set of independent diophantine equations instead of a set of coupled ones moreover the number of equations can be reduced under weak conditions finally both transient command tracking and steady state noise rejection capabilities can be op

coupled polynomial equations for lq control synthesis and an algorithm for solution the polynomial equation approach can be used to solve a variety of important control problems the approach leads to the requirement to solve linear polynomial equations we present a new algorithm for the numerical solution of a general coupled pair of linear polynomial equations the algorithm is based upon the method of polynomial reductions theoretically solving the coupled equations together enhances the numerical robustness of the equation solution the algorithm finds a specific and unique minimal degree solution of the equations the general couple of equations studied is representative of a number of problems in linear control theory and the minimal degree solution usually corresponds to an optimizing solution of the control problem

on the polynomial equations for the continuous time lq optimization problem this paper discusses the role and the relationships among three coupled diophantine polynomial equations involved in the solution of the optimal continuous time lq regulation problem the emphasis is on establishing conditions under which the lq regulator can be obtained via the solution of a single uncoupled diophantine equation this analysis is motivated by the fact that although recent papers have clarified the matter for the discrete time domain some subtle differences exist in the continuous time domain as a consequence the extension of the discrete time results to the continuous time context requires a deeper analysis which is the main aim of this paper

corrected solvability conditions for non linear asymmetric vibrations of a circular plate an investigation into non linear asymmetric vibrations of a clamped circular plate under a harmonic excitation is made we re examined a primary resonance studied by sridhar mook and nayfeh in which the frequency of excitation is near the natural frequency of an asymmetric mode of the plate we corrected their solvability conditions and found that in the absence of internal resonance the steady state response can have not only the form of standing wave but also the form of travelling wave which is a remarkable contrast to their conclusion

convergence of numerical inversion methods for discontinuous impedance profiles we study numerical methods for the one dimensional inverse problem of reflection seismology an impulsive plane wave in pressure is applied to the surface of a piecewise smooth stratified elastic half space and the resulting particle velocity is measured at the surface the characteristic impedance of the medium is to be recovered as a function of travel time we consider two methods for the numerical solution of this inverse problem a discretization of the gopinath sondhi integral equation and a downward continuation method both methods are exact for so called goupillaud layered media both use second order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium a formally second order accurate equation based on the theory of geometrical acoustics is used to recover the impedance by analyzing the discretization of the gopinath sondhi integral equation we prove that if the impulse response is measured by sampling the step response or

lqg predictive optimal control for adaptive applications abstract the lqg optimal control philosophy is extended so that use may be made of future set point or reference signal information the tracking error is defined in terms of an ideal response model so that a model following capability may be introduced the cost function may also be augmented by the addition of prediction error terms the controller was derived for adaptive control applications and has many of the desirable features of the new predictive control philosophy however there are fewer variables to choose and guaranteed stability properties are obtained for all weightings or prediction intervals the self tuning controller briefly described is almost as simple as existing lqg industrial self tuners

direct and inverse scattering in the time domain via invariant imbedding equations this paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain which is applied here to the case of a one dimensional lossless medium the method is based on an integro differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques an example of an exact solution to this equation is given numerical schemes for solving the direct and inverse problems are derived by discretizing this equation the behavior of the resulting algorithms is then tested on several examples the results suggest that the inversion method is stable in the presence of additive noise