On the Uniqueness of the Solution of an Inverse Spectral Problem

Akhtyamov, A. M.

doi:10.1023/b:dieq.0000011278.62330.ba

Cited by 3 publications

(4 citation statements)

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“…The problems of diagnosing the fastening of strings, membranes, and plates have been studied previously [12][13][14][15][16][17][18][19]. For conduits, however, the problem formulated here is apparently considered for the first time.…”

Section: Introductionmentioning

confidence: 99%

“…For conduits, however, the problem formulated here is apparently considered for the first time. In addition, unlike in [12][13][14][15][16][17][18][19], in the present work, four rather than two boundary conditions are sought, which significantly complicates the problem and requires the use of different methods for its solution.Problems of calculating the eigenfrequencies of flexural vibrations of conduits were investigated in [20,21]. However, the inverse problem -determining the boundary conditions from eigenfrequencies -was not studied in these papers.…”

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Vibration-proof conduit fastening

Akhtyamov

Сафина

2008

J Appl Mech Tech Phys

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It is shown that the problem of determining the type and parameters of conduit end fastening from the eigenfrequency spectrum has a dual solution. A method of solving this problem is developed. Some examples are given. Introduction.Conduits are important elements of the fuel systems of cars, tractors, ships, planes, etc. Their vibrations often cause drumming, leading to discomfort for crew members and passengers. This is due to the fact that the frequency spectra of conduit vibrations are sometimes in a range hazardous to human health. To change the conduit vibration frequencies, it is not always reasonable to change the conduit length or attach concentrated masses. Therefore, to produce comfort conditions for passengers, it is required to determine the types of conduit fastening that provide the necessary (safe) range of conduit vibration frequencies. This refers not only to the fundamental vibration mode but also overtones. This problem is related to issues of noise suppression [1-3], acoustic diagnostics, [4][5][6][7][8][9] and the theory of inverse problems of mathematical physics [10,11].The goal of the present work was to determine the fastening parameters of a conduit filled with a fluid from the eigenfrequencies of its flexural vibrations. The problems of diagnosing the fastening of strings, membranes, and plates have been studied previously [12][13][14][15][16][17][18][19]. For conduits, however, the problem formulated here is apparently considered for the first time. In addition, unlike in [12][13][14][15][16][17][18][19], in the present work, four rather than two boundary conditions are sought, which significantly complicates the problem and requires the use of different methods for its solution.Problems of calculating the eigenfrequencies of flexural vibrations of conduits were investigated in [20,21]. However, the inverse problem -determining the boundary conditions from eigenfrequencies -was not studied in these papers. In addition, in [20,21], only approximate methods (for example, the Galerkin and Rayleigh-Ritz methods) were considered, which are unsuitable for the solution of the problem formulated.1. Primal Problem. The small free vibrations of a conduit filled with a fluid (which is incompressible) is described by the following equation [20] (see also [21, pp. 193-196]):

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Vibration-proof conduit fastening

Akhtyamov

Сафина

2008

J Appl Mech Tech Phys

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“…The question arises whether one would be able to detect boundary conditions, using finite number eigenvalues. The following papers give and substantiate a positive answer to this question for several cases (see [1], [2], [3], [4], [5]). In this paper we continue these researches.…”

Section: Introductionmentioning

confidence: 99%

On the Well-possedness of the Problem of Reconstruction of Non-separate Boundary Conditions

Akhtyamov,

Mouftakhov,

Teicher

et al. 2008

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“…But it turns out that we can speak of duality in the solution of this problem. Here we observe an analogy with the problem of determining the rigidity coefficients of springs for elastic fixing of a string [8]: the rigidity coefficients of the springs are determined by the natural frequencies uniquely up to permutations of the springs. …”

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On a method for determining the fixing conditions for a rectangular plate from the natural frequencies

et al. 2007

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Abstract-We prove the duality of solutions for the problem of determining the boundary conditions on two opposite sides of a rectangular plate from the frequency spectrum of its bending vibrations. A method for determining the boundary conditions on two opposite sides of a rectangular plate from nine natural frequencies is obtained. The results of numerical experiments justifying the theoretical conclusions of the paper are presented. Rectangular plates are widely used in various technical fields. They serve as printed circuit boards and header plates, bridging plates, aircraft and ship skin, and parts of various mechanical structures [1][2][3][4]. If the plate fixing cannot be inspected visually, then one can use the natural bending vibration frequencies to find faults in the plate fixing. For circular and annular plates, methods for testing the plate fixing were found in [5][6][7], where it was shown that the type of fixing of a circular or annular plate can be determined uniquely from the natural bending vibration frequencies. The following question arises: Is it possible to determine the type of fixing of a rectangular plate on two opposite sides of the plate from the natural bending vibration frequencies if the other two sides are simply supported? Since the opposite sides of the plate are equivalent to each other, a plate with "rigid restraint-free edge" fixing will sound exactly the same as a plate with "free edge-rigid restraint" fixing. Hence we cannot say that the type of fixing of a rectangular plate on two opposite sides can be uniquely determined from its natural bending vibration frequencies. But it turns out that we can speak of duality in the solution of this problem. Here we observe an analogy with the problem of determining the rigidity coefficients of springs for elastic fixing of a string [8]: the rigidity coefficients of the springs are determined by the natural frequencies uniquely up to permutations of the springs. DOI: 10.3103/S00256544070101161. DIRECT PROBLEM The problem of identifying the boundary conditions can be categorized as an inverse spectral problem (an inverse eigenvalue problem) [9]. Prior to stating the inverse spectral problem, we recall how to state the direct problem. If the thickness h of a homogeneous plate is constant (the cylindrical rigidity D is constant), then the equation of free vibrations of the rectangular plate has the form (e.g., see [1]

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On the Uniqueness of the Solution of an Inverse Spectral Problem

Cited by 3 publications

References 2 publications

Vibration-proof conduit fastening

Vibration-proof conduit fastening

On the Well-possedness of the Problem of Reconstruction of Non-separate Boundary Conditions

On a method for determining the fixing conditions for a rectangular plate from the natural frequencies

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