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Research Papers

Vibration Analysis of Flexible Rotating Rings Using a Spectral Element Formulation

[+] Author and Article Information
Danilo Beli

Department of Computational Mechanics,
Faculty of Mechanical Engineering,
University of Campinas,
Rua Mendeleyev, 200,
Cidade Universitária Zeferino Vaz,
Campinas, São Paulo 13083-860, Brazil
e-mail: dbeli@fem.unicamp.br

Priscilla Brandão Silva

Department of Computational Mechanics,
Faculty of Mechanical Engineering,
University of Campinas,
Rua Mendeleyev, 200,
Cidade Universitária Zeferino Vaz,
Campinas, São Paulo 13083-860, Brazil
e-mail: pbrandaos@fem.unicamp.br

José R. de França Arruda

Mem. ASME
Department of Computational Mechanics,
Faculty of Mechanical Engineering,
University of Campinas,
Rua Mendeleyev, 200,
Cidade Universitária Zeferino Vaz,
Campinas, São Paulo 13083-860, Brazil
e-mail: arruda@fem.unicamp.br

In this paper, the frequency ranges of analysis considered are: low-frequency range from 0 to 100 Hz, mid-frequency range from 100 Hz to 1 kHz and high-frequency range from 1 kHz to 10 kHz.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 21, 2014; final manuscript received February 6, 2015; published online March 12, 2015. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 137(4), 041003 (Aug 01, 2015) (11 pages) Paper No: VIB-14-1311; doi: 10.1115/1.4029828 History: Received August 21, 2014; Revised February 06, 2015; Online March 12, 2015

In this work, the forced response of rotating rings exhibiting nonperiodic and periodic variations in material properties and geometry is assessed by means of the spectral element method (SEM). Based on the Euler–Bernoulli beam theory, a spectral element for a planar rotating ring is derived. This spectral element allows the investigation of the effects of structural damping, internal pressure and elastic foundations in the harmonic response of rotating rings. The dynamic response of rotating rings including periodic imperfections that lead to band gap effects is addressed. The spectral element formulation provides exact solutions within the range of validity of the applied theory using a reduced number of degrees-of-freedom. Thus, it contributes to reducing the computational time. It also provides a straightforward way to solve structural dynamics problems including arbitrary boundary conditions and discontinuities. The proposed formulation is validated by comparison with analytical solutions, which are available only for uniform homogenous rings.

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References

Figures

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Fig. 1

Differential element of rotating ring supported on elastic foundation, gray shaded area, and under internal pressure

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Fig. 2

Spectral element of a rotating ring including spectral nodal forces and wave propagation convention

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Fig. 3

SE model of a rotating ring

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Fig. 4

Spectrum relations for a stationary ring ((a) and (b)) and for a rotating ring with Ω = 50 rad/s ((c) and (d)): positive wavenumbers (---) and negative wavenumbers (o)

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Fig. 5

FRF of the rotating ring with Ω = 50 rad/s: SEM (---) and analytical solution (x)

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Fig. 6

Campbell diagram for single force: SEM (grayscale map) and analytical solution (o)

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Fig. 7

Configuration of the single applied force at subsequent time instances

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Fig. 8

Real and imaginary parts of wavenumbers with Ω = 50 rad/s for: internal pressure effect ((a) and (d)) with p0=1 bar (x) and p0 = 0 (---), elastic foundation effect ((b) and (e)) with kw = ku = 104 N/m3 (x) and kw = ku = 0 (---), structural damping effect ((c) and (f)) with η = 0.01(x) and η = 0 (---)

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Fig. 9

FRF with Ω = 50 rad/s for analysis of (a) internal pressure with p0 = 1 bar, (b) elastic foundation with kw = ku = 104 N/m3, and (c) structural damping with η = 0.01: SEM (-x-), analytical solution (vertical lines) and SEM with p0 = kw = ku = η = 0 (—) for comparison

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Fig. 10

Operational frequencies versus (a) internal pressure, (b) elastic foundation stiffness, and (c) structural damping for Ω = 0 (—), Ω = 50 rad/s (x) and Ω = 100 rad/s (o)

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Fig. 11

Models of nonuniform and nonhomogeneous rotating ring with: (a) local imperfection, (b) material periodicity, (c) geometric periodicity, and (d) elastic foundation periodicity

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Fig. 12

Campbell diagrams for a ring (a) with and (b) without local imperfection

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Fig. 13

Forced response with Ω = 50 rad/s of (a) uniform and homogenous equivalent ring, and rings including periodic imperfections: (b) material periodicity, (c) geometric periodicity, and (d) elastic foundation periodicity

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Fig. 14

Campbell diagrams for (a) uniform and homogenous equivalent ring, and rings including periodic imperfections: (b) material periodicity, (c) geometric periodicity, and (d) elastic foundation periodicity

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