natural frequencies of structures with interval parameters abstract this paper deals with the evaluation of the lower and upper bounds of the natural frequencies of structures with uncertain but bounded parameters the solution of the generalized interval eigenvalue problem is pursued by taking into account the actual variability and dependencies of uncertain structural parameters affecting the mass and stiffness matrices to this aim interval uncertainties are handled by applying the improved interval analysis via extra unitary interval eui recently introduced by the first two authors by associating an eui to each uncertain but bounded parameter the cases of mass and stiffness matrices affected by fully disjoint completely or partially coincident uncertainties are considered then based on sensitivity analysis it is shown that the bounds of the interval eigenvalues can be evaluated as solution of two appropriate deterministic eigenvalue problems without requiring any combinatorial procedure if the eigenvalues are monotonic functions of the uncertain parameters then the exact bounds are obtained the accuracy of the proposed method is demonstrated by numerical results concerning truss and beam structures with material and or geometrical uncertainties

global existence of harmonic maps in minkowski space on considere la construction d une classe d applications harmoniques classiques globales de l espace de minkowski m 4 dans une variete de riemann complete a n dimensions

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

eigenvalue bounds of structures with uncertain but bounded parameters many analysis and design problems in engineering and science involve uncertainty to varying degrees this paper is concerned with the structural vibration problem involving uncertain material or geometric parameters specified as bounds on these parameters this produces interval stiffness and mass matrices and the problem is transformed into a generalized interval eigenvalue problem in interval mathematics however tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem often the stiffness and mass matrices can be formed as a non negative decomposition in the uncertain structural parameters in this case the eigenvalue bounds may be obtained from the parameter vertex solutions even more efficiently using interval extension from interval mathematics the generalized interval eigenvalue problem may be divided into two generalized eigenvalue problems for real symmetric matrix pairs the parameter vertex solution algorithm is compared with deif s solution the eigenvalue inclusion principle and the interval perturbation method in numerical examples

nonprobabilistic convex theoretic modeling of scatter in material properties nonprobabilistic convex modeling of uncertain material properties for viscoelastic structures is developed in this paper in particular the problem of forced vibrations of viscoelastic beams is studied first the analytic solution by inman is generalized for a deterministic set of variables describing material properties next these variables are treated as varying in a solid ball in the four dimensional space thus modeling the scatter in material properties the least favorable response needed for the design of the structure is determined

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

structured pseudospectra and random eigenvalues problems in vibrating systems this paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and propagation in structural dynamics different types of pseudospectra of matrices and matrix polynomials are explained particular emphasis is given to structured pseudospectra for matrix polynomials which offer a deterministic way of dealing with uncertainties for structural dynamic systems the pseudospectra analysis is compared with the results from monte carlo simulations of uncertain discrete systems two illustrative example problems one with probabilistic uncertainty with various types of statistical distributions and the other with interval type of uncertainty are studied in detail excellent agreement is found between the pseudospectra results and monte carlo simulation results

stochastic interval analysis of natural frequency and mode shape of structures with uncertainties abstract in this paper natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach expressions for the mean value and variance of natural frequencies and mode shapes are derived by using perturbation method and random interval moment method the bounds of these probabilistic characteristics are then determined by interval arithmetic two examples are given first to illustrate the feasibility of the presented method and the results are verified by monte carlo simulations the presented approach is also applicable to solve pure random and pure interval problems this capability is demonstrated in the third and fourth examples through the comparisons with the peer research outcomes

extending bauer s corollary to fractional derivatives we comment on the method of dreisigmeyer and young j phys a math gen 368297 to model nonconservative systems with fractional derivatives it was previously hoped that using fractional derivatives in an action would allow us to derive a single retarded equation of motion using a variational principle it is proven that under certain reasonable assumptions the method of dreisigmeyer and young fails

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

an interval algorithm for uncertain dynamic stability analysis abstract analysis on the stability of dynamic systems is a very important field in structural elasticity theory for possible engineering situations where the system parameters are uncertain but bounded the propagation from the initial uncertainties of parameters to the consequent uncertainty of dynamic stability of structure is elaborately investigated by interval analysis for the first time the interval dynamic stability issue is studied through the column structure as a generalized interval eigenvalue problem and an analytic theorem is rigorously derived and numerically verified for solving the generalized interval eigenvalue problem moreover to avoid the exaggeration on actual bounds of solution an effective approach is established to limit the dependency phenomenon in the interval dynamic stability analysis the interval relationships of the excitation parameter and the critical load frequency with five types of system parameters are further bridged according to the interval arithmetic to propose a three stage interval scheme for evaluating the effects of interval system parameters on the dynamic stability of structures numerical studies demonstrate that the uncertainty of the constant part of load influences the boundaries of principal instability region much more than that of the variable part of load within all structural parameters the uncertainty of column length has the most effect on the boundaries which could make the critical frequency of load be magnified about ten times the uncertainty of parameter in particular if all the system parameters are interval with an identical uncertainty degree the uncertainty of parameter would be propagated in the system of dynamic stability and enlarged as high as 20 times the uncertainty of parameter consequently the impacts of parameter uncertainties on the dynamic stability of structures are fairly significant

the time response of structures with bounded parameters and interval initial conditions uncertainty plays an important role in the performance of structures in this paper we focus on the dynamic response of structures with bounded parameters and interval initial conditions and present a new method to determine the supremum and infimum of the time response the method is based on the vertex solution theorem for the first order deviation of the dynamic response from its central value and avoids interval extension problems present in current methods where the length of the interval increases significantly due to the intermediate calculations the method is more accurate than existing perturbation methods and provides tighter bounds on the response the approach neglects the second order terms in the equation of motion and care should be exercised when the parameter variations are large the other advantage of this method is its ability to solve problems with uncertainties in the initial conditions

on harmonic maps from r1 1 into hilbert loop groups motivated by the works of gu pure appl math 33 727 1980 ginibre and velo ann phys 142 393 1982 on classical solutions of a two dimensional model the global existence of harmonic maps from the minkowski plane r1 1 into hilbert loop groups is proved the construction of the harmonic maps strongly depends on some geometric properties of the loop groups

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

high dimensional model representation method for fuzzy structural dynamics abstract uncertainty propagation in multi parameter complex structures possess significant computational challenges this paper investigates the possibility of using the high dimensional model representation hdmr approach when uncertain system parameters are modeled using fuzzy variables in particular the application of hdmr is proposed for fuzzy finite element analysis of linear dynamical systems the hdmr expansion is an efficient formulation for high dimensional mapping in complex systems if the higher order variable correlations are weak thereby permitting the input output relationship behavior to be captured by the terms of low order the computational effort to determine the expansion functions using the cut method scales polynomically with the number of variables rather than exponentially this logic is based on the fundamental assumption underlying the hdmr representation that only low order correlations among the input variables are likely to have significant impacts upon the outputs for most high dimensional complex systems the proposed method is first illustrated for multi parameter nonlinear mathematical test functions with fuzzy variables the method is then integrated with a commercial finite element software adina modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach in the numerical examples triangular membership functions have been used and the results have been validated against direct monte carlo simulations it is shown that using the proposed hdmr approach the number of finite element function calls can be reduced without significantly compromising the accuracy

computation of bounds for eigenvalues of structures with interval parameters in this paper the computation of eigenvalue bounds for generalized interval eigenvalue problem is considered two algorithms based on the properties of continuous functions are developed for evaluating upper and lower eigenvalue bounds of structures with interval parameters the method can provide the tightest bounds within a given precision numerical examples illustrate the effectiveness of the proposed method

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

a spectral approach for fuzzy uncertainty propagation in finite element analysis abstract uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge this paper proposes a polynomial chaos type spectral approach based on orthogonal function expansion a fuzzy variable is represented as a set of interval variables via the membership function the interval variables are further transformed into the standard interval 1 1 smooth nonlinear functions of standard interval variables are projected in the basis of legendre polynomials by exploiting its orthogonal properties over the interval 1 1 the coefficients associated with the basis functions are obtained by a galerkin type of error minimisation the method is first illustrated using scalar functions of multiple fuzzy variables later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables the response of such systems can be expressed in the complete basis of multivariate legendre polynomials the coefficients obtained by galerkin type of error minimisation can be calculated from the solution of an extended set of linear algebraic equations an eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way a numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results

on harmonic maps into gauge groups we continue the work of the paper j math phys 37 4076 1996 on the existence of globally defined harmonic maps from the minkowski plane r1 1 into an infinite dimensional hilbert lie group we prove that the cauchy problem for harmonic maps from r1 1 into hilbert loop groups hs lg can be solved globally for all remaining cases 12 s 34 and we obtain similar results for harmonic maps from r1 1 into certain hilbert lie gauge groups hs sn g n 2 these answer the two questions remaining from the above paper

nonconservative lagrangian mechanics a generalized function approach we reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle in particular a formalism is developed that allows the inclusion of fractional derivatives this is done within the lagrangian framework by treating the action as a volterra series it is then possible to derive two equations of motion one of these is an advanced equation and the other is retarded

a deterministic method to predict the effect of unknown but bounded elastic moduli on the buckling of composite structures abstract a new non probabilistic method is developed in this study to predict the variability in buckling loads of composite plates and shells stemming from the unavoidable scatter in elastic moduli the available measurements of elastic moduli are fitted by the four dimensional uncertainty box and appropriate ellipsoid for the specific cases considered the maximum buckling load of the composite plate exceeds by 11 the value of the buckling load based on the calculation using the nominal values of elastic moduli in the case of the cylindrical shell an appropriate exceedance constitutes 9

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

on a nonlinear distributed order fractional differential equation abstract we study the existence and the uniqueness of mild and classical solutions for a class of equations of the form y 2 t 0 2 y t d f y t such equations arise in distributed derivatives models of viscoelasticity and system identification theory we also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations

safety margin analysis of buckling for structures with unknown but bounded uncertainties abstract a safety margin of buckling for structures with unknown but bounded uncertainties is proposed in this paper the perturbation matrixes consisting of the elastic stiffness matrix and the geometric stiffness matrix are introduced to derive the concept of safety margin the concept of safety margin of buckling is first proposed to quantify the influence of uncertain parameters on structural buckling based on the spectrum norm and the singular value analysis the mathematical basis of the safety margin is expounded furthermore to obtain a more accurate estimation of safety margin the correction factor for the safety margin is introduced through the interval extension of interval analysis two engineering examples compared with traditional interval analysis method are utilized to demonstrate the validity of the proposed safety margin and numerical results indicate that the safety margin can provide a relatively accurate index to indicate the allowable perturbation range of the perturbation matrix of a structure under given loads

natural frequency intervals for vibrating systems with polytopic uncertainty we consider the problem of locating the natural frequencies of uncertain systems whose describing matrices are functions of an unknown parameter vector which is included in an assigned bounding set we face what we call the weak frequency interval detection problem of determining the smallest interval which includes all possible frequencies we show that if the system matrices depend affinely on the parameter vector whose bounding set is a compact polyhedron then this problem requires the solution of a finite number of eigenvalue problems associated with the vertices of such a polyhedron unfortunately detecting the intervals associated with all the natural frequencies strong frequency interval detection problem cannot rely on this property so that one must resort to monte carlo methods or numerical optimization to find them we show that the strong version is solvable exploring the vertices only under some stronger assumptions in the case in which the uncertainty bounding set is not defined by linear inequalities not even the extremal frequencies can be associated with the vertices of the admissibility domain then again numerical approach is necessary unless we accept to merge the original system in a larger one of an affine nature finally we present as an application the study of structures with uncertain mass distribution