In-plane vibrational analysis of rotating curved beam with elastically restrained root

Lee, Sen Yung; Sheu, Jer‐Jia; Lin, Shueei-Muh

doi:10.1016/j.jsv.2008.02.011

Cited by 22 publications

(13 citation statements)

References 24 publications

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“…w pracach Chidamparam i Leiss [1] oraz Lee i in. [2]. W publikacji [1] problem rozwiązano analitycznie z uwzględnieniem i pominięciem odkształcalności osiowej.…”

Section: Wprowadzenieunclassified

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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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“…w pracach Chidamparam i Leiss [1] oraz Lee i in. [2]. W publikacji [1] problem rozwiązano analitycznie z uwzględnieniem i pominięciem odkształcalności osiowej.…”

Section: Wprowadzenieunclassified

“…W pracy Lee i in. [2] fundamentalne rozwiązanie układu równań różniczkowych wyznaczono metodą szeregów potęgowych. Problem drgań swobodnych łuków o zmiennym przekroju rozwiązali różnymi metodami m.in.…”

Section: Wprowadzenieunclassified

Eigenproblem of non prismatic circular arch solution using Chebyshev series

Ruta¹,

Meissner²

2013

JCEEA

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“…Most of this literature concerns planar systems in which only in-plane or out-of-plane vibrations occur or spatial systems with separated in-plane and out-of-plane vibrations (vibrations are separated when the "generalized" cross-sectional moment of deviation equals zero). The problem of the free vibration of prismatic arches was examined by, among others, Chidamparam and Leissa [2] and Lee et al [3]. In Chidamparam and Leissa's paper [2] the problem was solved analytically with and without axial deformability taken into account.…”

Section: Introductionmentioning

confidence: 99%

“…Circular arches with different opening angles were considered, but only to determine the natural frequencies. In paper [3] by Lee et al the natural vibrations of a rotating curved beam with elastically restrained root were analysed. The differential equations describing the problem were derived using the Hamilton principle.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Spatial Vibration of Nonprismatic Arches by Means of Recurrence Relations for the Coefficients of the Chebyshev Series Expansion of the Solution

Ruta

Meissner

2018

Mathematical Problems in Engineering

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The problem of spatial vibrations, both aperiodically forced and free vibrations, of an arch with an arbitrary distribution of material and geometric parameters is considered. Approximation with Chebyshev series was used to solve a conjugated system of partial differential equations describing the problem. The system of differential equations was solved using an algorithm generating a recursive infinite system of equations, developed by S. Paszkowski in “Numerical applications of Chebyshev polynomials” (in Polish), Warsaw PWN, 1975. Since the coefficients of the obtained system of equations are defined by closed analytical formulas they can be directly used to solve any nonprismatic arch, without it being necessary to solve again the considered problem. The algorithm is highly accurate; i.e., already at a small approximation base it yields results agreeing with exact analytical solutions (obviously for problems in the case of which such solutions can be derived). In order to demonstrate this the eigenfrequencies and eigenvectors obtained for a circular prismatic arch were compared with their precise values determined from the exact analytical solutions. The results yielded by the proposed method were also compared with the results obtained by other methods and by other authors. As an illustration, the proposed method was used to solve a more complex problem, i.e., the problem of the free and aperiodically forced vibrations of a nonprismatic arch with its axis described by a catenary curve. In the example the effect of the lack of cross-sectional symmetry of the arch on the form of the system’s spatial free and forced vibrations was analysed.

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“…The in-plane free vibration of a rotating curved beam with an elastically restrained root was studied by Lee et al [14] and the effects of the arc angle, rotational speed, hub radius, and the root spring constant on the natural frequencies were investigated. Abolghasemi and Jalali [15] obtained the attractors of a rotating viscoelastic beam using perturbation technique and they have found that stable limit-cycles exist for the system both in the presence and in the absence of structural damping.…”

Section: Introductionmentioning

confidence: 99%

Non-linear vibration of variable speed rotating viscoelastic beams

Younesian¹,

Esmailzadeh²

2009

Nonlinear Dyn

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Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton's principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.

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In-plane vibrational analysis of rotating curved beam with elastically restrained root

Cited by 22 publications

References 24 publications

Eigenproblem of non prismatic circular arch solution using Chebyshev series

Eigenproblem of non prismatic circular arch solution using Chebyshev series

Analysis of the Spatial Vibration of Nonprismatic Arches by Means of Recurrence Relations for the Coefficients of the Chebyshev Series Expansion of the Solution

Non-linear vibration of variable speed rotating viscoelastic beams

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