Inverse problem for a vibrating beam

Summary The problem of inferring the flexural rigidity and density of a beam from its eigenfrequencies is considered for the particular case in which one end is clamped. It is shown that three spectra associated with three sets of boundary conditions at the other end are required in order to insure a unique solution of the inverse problem. Furthermore, it is shown that this data set is equivalent to the information contained in the time history of the displacement and slope of the free end of the beam set in motion by a concentrated impulse.

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Résumé Cet article traite le problème relatif à la déduction des caractéristiques d'une barre à partir de ses fréquences propres. Bien que nos résultats soient généraux, nous considererons seulement le cas où la barre est encastrée à une extrémité. Nous montrerons que trois spèctres associés à trois types de conditions aux limites pour la seconde extrémité sont nécéssaires pour garantir l'unicité de la solution du problème inverse. De plus, nous montrerons que ces trois spèctres sont équivalents à l'information contenue dans les fonctions représentant le déplacement et la pente de l'extrémité libre de la barre mise en mouvement par un choc donné à cette extrémité.

     

The global Cauchy problem for a vibrating beam equation

We consider the global Cauchy problem for an evolution equation which models an Euler-Bernoulli vibrating beam with time dependent elastic modulus under a force linear function of the displacement u, of the slope x_∂u, of and . These two last derivatives are proportional to the bending moment and to the shear respectively. We show results of well-posedness in Sobolev spaces assuming that the coefficient of the shear term has a decay rate |x|^−σ, σ?1, for the position x→±∞ and that all the coefficients of , 1?k?3, satisfy suitable Levi conditions since we allow the elastic modulus to vanish at some time t=t₀.     

Inverse problem for a parabolic equation

Mathematical Notes -     

Nonlinear bending inertia of a vibrating beam

Zusammenfassung Die bei Querschwingungen schlanker Stäbe mit zeitabhängiger Längsbelastung in der Bewegungsdifferentialgleichung auftretenden nichtlinearen Trägheitsterme, herrührend von der Longitudinalbewegung, werden hier im Gegensatz zu früheren Arbeiten mitberücksichtigt. Die sich daraus ergebenden nichtlinearen Korrekturen der Eigenfrequenzen des entsprechenden linearen Systems werden abgeschätzt.     

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.     

Inverse problem for cuts

Let U be an initial segment of closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U, which is the intersection of U and an internal subset of . Suppose A has lower U-density α strictly between 0 and 3/5. We show that either there exists a standard real > 0 and there are sufficiently large x in A such that | (A+A) ∩ [0, 2x]| > (10/3+ ) | A ∩ [0, x]| or A is a large subset of an arithmetic progression of difference greater than 1 or A is a large subset of the union of two arithmetic progressions with the same difference greater than 2 or A is a large subset of the union of three arithmetic progressions with the same difference greater than 4.     

Identification problem for a one‐dimensional vibrating system

We discuss an inverse problem of determining a coefficient matrix and an initial value for a one‐dimensional non‐symmetric hyperbolic system of the first order by means of boundary values over a time interval. Provided that a time interval is sufficiently long and a given initial value satisfies some non‐degeneracy condition, we characterize coefficient matrices and initial values realizing the same boundary values. In the case where the initial value is fixed, we can prove the uniqueness in determining all the components of the coefficient matrices. The proof is based on a transformation formula and spectral properties. Copyright © 2005 John Wiley & Sons, Ltd.     

Sturm theory for the equation of vibrating beam

In this paper we develop an extension of the classical Sturm theory [C. Sturm, Sur une classe d'equations à derivée partielle, J. Math. Pures Appl. 1 (1836) 373-444], to study the oscillation properties for the eigenfunctions of some fourth-order two point boundary value problems on the interval [0,1]. We are mainly interested in the case when these problems have negative eigenvalues induced by the sign of the parameters in the boundary conditions. In particular, we give an asymptotic estimate of the number of zeros in (0,1) of the first eigenfunction in terms of the variation of parameters in the boundary conditions.     

Inverse problem equations for a three-wave quantum system

Analytic properties of the Jost solutions of the auxiliary linear problem of a three-wave system, whose potentials are operator-valued functions, are studied. Creation and annihilation operators of elementary excitations and their bound states are constructed. Singular integral equations which let one reconstruct the local fields from the creation and annihilation operators mentioned are derived.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 53–69, 1986.     

Inverse scattering problem for a wave equation with absorption

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1580–1584, November, 2007.     

Inverse problem for a quadratic pencil of Sturm-Liouville operator

The purpose of this paper is to solve the inverse problem for a quadratic pencil of Sturm-Liouville operator on a finite interval. By using the Hochstadt's method, it is shown that the potential function can be given from two spectras as uniquely. Later, by generalizing this result, the Hochstadt's theorem which is related to the structure of difference for the potential functions is obtained.