×

zbMATH — the first resource for mathematics

A novel perturbation expansion method for coupled system of acoustics and structures. (English) Zbl 1220.74018
Summary: We formulate a coupled vibration problem between a structure and an acoustic field in a mathematically rigorous fashion. A typical example of the structure is a car body that can be modeled by a cluster of thin plates. The problem leads to a nonstandard eigenvalue problem in a function space. Furthermore, we introduce a coupling strength parameter \(\varepsilon \) as a multiplier applied to the nondiagonal coupling terms. A natural interpretation of this parameter is given. We represent an eigenpair for the coupled system by a perturbation series with respect to \(\varepsilon \) which enable us to express the eigenpair for the coupled case by those for the decoupled case. We prove that the series consists only of even order terms of \(\varepsilon \). In some practical applications, by using this perturbation series, it would become unnecessary to perform time consuming computations to get coupled eigenvalues, and hence the present results obtained by the perturbation analysis might have considerable engineering importance. We confirm the adequacy of this perturbation analysis by investigating some numerical examples. The results are given for a two-dimensional coupled problem. Without loss of generality, the same operator theoretical approach applies to the eigenvalue problems of the coupled vibration problem in a more general three-dimensional acoustic region with a plate for a part of its boundary.

MSC:
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
76Q05 Hydro- and aero-acoustics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Li, D.; Kako, T.; Hagiwara, I., Development of coupled structural-acoustic eigenpair expression from decoupled eigenpair (1st report, induction by finite perturbation series), Trans. Japan society of mech. engin. (part C), 63, 3446-3453, (1997)
[2] Li, D.; Kako, T., Finite element approximation of eigenvalue problem for a coupled vibration between acoustic field and plate, J. comput. math., 15, 3, 265-278, (1997) · Zbl 0891.73062
[3] Craggs, A.; Stead, G., Sound transmission between enclosures—A study using plate and acoustic finite elements, Acoustica, 35, 89-98, (1976) · Zbl 0318.76058
[4] Babuska, I.; Osborn, J.E., Eigenvalue problem, (), 683-692
[5] Kato, T., Perturbation theory for linear operators, (1966), Springer-Verlag Amsterdam · Zbl 0148.12601
[6] Kotukue, W.; Ma, Z.; Hagiwara, I., Sensitivity solution for the model frequency response by omitting high and low terms, Trans. JSIAM, 4, 141-164, (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.