In-Plane and Out-of-Plane Free Vibrations of Curved Beams With Variable Sections

Kawakami, Masahide; Sakiyama, T.; Matsuda, Hiroshi; Morita, Chihiro

doi:10.1006/jsvi.1995.0531

Cited by 83 publications

(29 citation statements)

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“…Kawakami i in. [4] rozwiązali zagadnienie własne, stosując dyskretną funkcję Greena. Liu i Wu [5] do analizy zagadnienia własnego zastosowali uogólnioną zasadę kwadratur róż-nicowych, przyjmując założenie o braku odkształcalności osiowej.…”

Section: Wprowadzenieunclassified

Eigenproblem of non prismatic circular arch solution using Chebyshev series

Ruta¹,

Meissner²

2013

JCEEA

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Przedmiotem analizy jest zagadnienie własne luku kołowego o zmiennym przekroju, opisane według teorii Bernoulliego-Eulera. Problem jest rozwiązywany z wykorzystaniem metody aproksymacyjnej, w której do aproksymacji wykorzystuje się szeregi wielomianów Czebyszewa I rodzaju. Zastosowana w pracy metoda jest oparta na ogólnym twierdzeniu opisującym związki rekurencyjne dla równań róż-niczkowych o zmiennych współczynnikach. Metoda ta prowadzi do wyznaczenia nieskończonego układu równań algebraicznych, którego współczynniki są określo-ne zamkniętymi formułami analitycznymi. Formuły te w sposób jawny zależą od wyrazów szeregów, w które rozwinięto zmienne współczynniki wyjściowych rów-nań różniczkowych. Otrzymana w ten sposób ogólna postać równań algebraicznych pozwala na rozwiązanie analizowanego zagadnienia dla dowolnych geometrycznych parametrów łuku, takich jak: krzywizna, zmienne pole i zmienny moment bezwładności przekroju czy gęstość łuku. Do analitycznych formuł opisują-cych współczynniki układu równań algebraicznych wystarczy bowiem podstawić współczynniki szeregów opisujących parametry materiałowe i geometryczne łuku. W celu weryfikacji poprawności oraz skuteczności otrzymanego algorytmu uzyskane prezentowaną w pracy metodą częstości i formy własne porównano z wynikami uzyskanymi metodą elementów skończonych. Obliczenia wykonano programem Cosmos/M, stosując do aproksymacji elementy belkowe 3D o liniowo zmiennym przekroju. W celu oceny różnicy między formami własnymi wyznaczono dla nich standardowy indeks MAC (Modal Assurance Criterion). Otrzymane rezultaty potwierdziły poprawność oraz skuteczność omawianej w pracy metody.Słowa kluczowe: zagadnienie własne, łuk niepryzmatyczny, szeregi Czebyszewa 1 Autor do korespondencji: Piotr Ruta,

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Section: Wprowadzenieunclassified

Eigenproblem of non prismatic circular arch solution using Chebyshev series

Ruta¹,

Meissner²

2013

JCEEA

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“…The above-mentioned values of moment of inertias for the cross-section area are calculated with the classical formulas used by some papers [7][8][9], I x = ab 3 /12, I y = a 3 b/12 and J = I x + I y , rather than by the formulas given by Equations (3a) and (3b). This is one of the main reasons that the numerical results of one report may be slightly different from those of the other report as shown in Reference [20].…”

Section: Natural Frequencies Of a Curved Beammentioning

confidence: 99%

A new approach for displacement functions of a curved Timoshenko beam element in motions normal to its initial plane

Chiang

2005

Int. J. Numer. Meth. Engng

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SUMMARYIn existing literature, either analytical methods or numerical methods, the formulations for free vibration analysis of circularly curved beams normal to its initial plane are somewhat complicated, particularly if the effects of both shear deformation (SD) and rotary inertia (RI) are considered. It is hoped that the simple approach presented in this paper may improve the above-mentioned drawback of the existing techniques. First, the three functions for axial (or normal to plane) displacement and rotational angles about radial and circumferential (or tangential) axes of a curved beam element were assumed. Since each function consists of six integration constants, one has 18 unknown constants for the three assumed displacement functions. Next, from the last three displacement functions, the three force-displacement differential equations and the three static equilibrium equations for the arc element, one obtained three polynomial expressions. Equating to zero the coefficients of the terms in each of the last three expressions, respectively, one obtained 17 simultaneous equations as functions of the 18 unknown constants. Excluding the five dependent ones among the last 17 equations, one obtained 12 independent simultaneous equations. Solving the last 12 independent equations, one obtained a unique solution in terms of six unknown constants. Finally, imposing the six boundary conditions at the two ends of an arc element, one determined the last six unknown constants and completely defined the three displacement functions. By means of the last displacement functions, one may calculate the shape functions, stiffness matrix, mass matrix and external loading vector for each arc element and then perform the free and forced vibration analyses of the entire curved beam. Good agreement between the results of this paper and those of the existing literature confirms the reliability of the presented theory.

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“…For the complicated problems of many architectural and structural implementations, curved beams with variable cross-sections are generally main parts, such that beams can be used not only in the design of rib, curved continuous bridge, and ship, but also in gear, pump, turbine and so on. Kawakami et al [1] present an approximate method to study the analysis for the inplane and out-of-plane free vibration of horizontally curved beams with arbitrary shapes and variable cross-sections. It is stated that the characteristic equation for free vibration can be derived by applying the Green function, which is obtained as a discrete type solution of differential equations governing the flexural behavior of the curved beam under the action of a concentrated load.…”

Section: Introductionmentioning

confidence: 99%

Out‐of‐Plane Vibration of Curved Uniform and Tapered Beams with Additional Mass

Özyiğit

Yetmez

Uzun

2017

Mathematical Problems in Engineering

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As there is a gap in literature about out-of-plane vibrations of curved and variable cross-sectioned beams, the aim of this study is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. Out-ofplane free vibration of curved uniform and tapered beams with additional mass is also investigated. Finite element method is used for all analyses. Curvature type is assumed to be circular. For the different boundary conditions, natural frequencies of both symmetrical and unsymmetrical tapered beams are given together with that of uniform tapered beam. Bending, torsional, and rotary inertia effects are considered with respect to no-shear effect. Variations of natural frequencies with additional mass and the mass location are examined. Results are given in tabular form. It is concluded that (i) for the uniform tapered beam there is a good agreement between the results of this study and that of literature and (ii) for the symmetrical curved tapered beam there is also a good agreement between the results of this study and that of a finite element model by using MSC.Marc. Results of out-of-plane free vibration of symmetrically tapered beams for specified boundary conditions are addressed.

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In-Plane and Out-of-Plane Free Vibrations of Curved Beams With Variable Sections

Cited by 83 publications

References 0 publications

Eigenproblem of non prismatic circular arch solution using Chebyshev series

Eigenproblem of non prismatic circular arch solution using Chebyshev series

A new approach for displacement functions of a curved Timoshenko beam element in motions normal to its initial plane

Out‐of‐Plane Vibration of Curved Uniform and Tapered Beams with Additional Mass

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