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Research Papers

Experimental Analysis of a Vertical and Flexible Cylinder in Water: Response to Top Motion Excitation and Parametric Resonance

[+] Author and Article Information
Guilherme R. Franzini

Department of Structural Engineering
and Geotechnics,
Escola Politécnica,
University of São Paulo–Brazil,
Av. Prof. Luciano Gualberto
travessa 3 n° 380-05508-010,
São Paulo, SP, Brazil

Celso P. Pesce, Rafael Salles

Offshore Mechanics Laboratory (LMO),
Escola Politécnica,
University of São Paulo–Brazil,
Av. Prof. Luciano Gualberto
travessa 3 n° 380-05508-010,
São Paulo, SP, Brazil

Rodolfo T. Gonçalves, André L. C. Fujarra

Numerical Offshore Tank (TPN),
Escola Politécnica,
University of São Paulo–Brazil,
Av. Prof. Luciano Gualberto
travessa 3 n° 380-05508-010,
São Paulo, SP, Brazil

Pedro Mendes

Center of Research and Development,
Petrobras,
Av. Horácio Macedo,
950, 21.941-915,
Rio de Janeiro, RJ, Brazil

Notice that this particular time scaling leads to the definition of the Mathieu parameters, δ and ε, such that the resulting Strutt diagram has the vertex of the first unstable region located at δ = 0.25.

Notice that the only nonlinear term preserved in this mathematical model is that related to the hydrodynamic damping. Other nonlinear terms could be considered, as those related to stretching, see, e.g., Ref. [20].

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 1, 2014; final manuscript received November 25, 2014; published online January 27, 2015. Assoc. Editor: Paul C.-P. Chao.

J. Vib. Acoust 137(3), 031010 (Jun 01, 2015) (12 pages) Paper No: VIB-14-1119; doi: 10.1115/1.4029265 History: Received April 01, 2014; Revised November 25, 2014; Online January 27, 2015

Experiments with a vertical, flexible, and submerged cylinder were carried out to investigate fundamental aspects of risers dynamics subjected to harmonic excitation at the top. The flexible model was designed aiming a high level of dynamic similarity with a real riser. Vertical motion, with amplitude of 1% of the unstretched length, was imposed with a device driven by a servomotor. Responses to distinct exciting frequency ratios were investigated, namely, ft:fN,1 = 1:3; 1:1; 2:1, and 3:1. Cartesian coordinates of 43 monitored points positioned all along the span were experimentally acquired by using an optical tracking system. A simple Galerkin's projection applied for modal decomposition, combined with standard Mathieu chart analysis, led to the identification of parametric resonances. A curious finding is that the Mathieu instability may simultaneously occur in more than one mode, leading to interesting dynamic behaviors, also revealed through standard power spectra analysis and displacement scalograms.

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References

Figures

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Fig. 1

Strutt diagram obtained from Eq. (1) and the secondary bifurcation curve in the δε plane. Focus on the first instability region. Extracted from Ref. [6].

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Fig. 5

Example of reconstruction of the elastica. Original data and reconstruction with four modes: ft:fN,1 = 3:1.

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Fig. 4

Spanwise distribution of PSD. Free decay tests in still water.

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Fig. 3

Sketch of the experimental setup

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Fig. 2

Pictures of the experimental arrangement. (a) General view of the experimental setup, (b) submerged cameras, and (c) servomotor device.

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Fig. 6

Scalograms x*(z*, t). Only 10 s of acquisition are shown. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 7

PSD spectra Sx(z*, f). (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 9

Modal amplitude time series u1(t)/D and corresponding PSD. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 10

First mode phase portraits u·1/(ωdD) versus u1(t)/D. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 11

Modal amplitude time series u2(t)/D and corresponding PSD. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 8

Strutt diagrams. Red (triangle): ft:fN,1 = 1:1; green (circle): ft:fN,1 = 2:1; blue (square): ft:fN,1 = 3:1. Results for the condition ft:fN,1 = 1:3 are within the stable region and are not shown. (a) First Eigen mode, (b) second Eigen mode, and (c) third Eigen mode.

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Fig. 12

Second mode phase portraits u·2/(ωdD) versus u2(t)/D. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 13

Modal amplitude time series u3(t)/D and corresponding PSD. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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Fig. 14

Third mode phase portraits u·3/(ωdD) versus u3(t)/D. (a) ft:fN,1 = 1:3, (b) ft:fN,1 = 1:1, (c) ft:fN,1 = 2:1, and (d) ft:fN,1 = 3:1.

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