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Research Papers

Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load

[+] Author and Article Information
J. L. Huang

Department of Applied Mechanics
and Engineering,
Sun Yat-sen University,
Guangzhou 510275, China
e-mail: huangjl@mail.sysu.edu.cn

W. D. Zhu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
P.O. Box 137,
Harbin 150001, China
Department of Mechanical Engineering,
University of Maryland--Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 20, 2013; final manuscript received July 16, 2014; accepted manuscript posted July 21, 2014; published online August 14, 2014. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(10), 101007 (Aug 14, 2014) (20 pages) Paper No: JAM-13-1479; doi: 10.1115/1.4028046 History: Received November 20, 2013; Revised July 16, 2014; Accepted July 21, 2014

Nonlinear dynamic responses of an Euler–Bernoulli beam attached to a rotating rigid hub with a constant angular velocity under the gravity load are investigated. The slope angle of the centroid line of the beam is used to describe its motion, and the nonlinear integro-partial differential equation that governs the motion of the rotating hub-beam system is derived using Hamilton's principle. Spatially discretized governing equations are derived using Lagrange's equations based on discretized expressions of kinetic and potential energies of the system, yielding a set of second-order nonlinear ordinary differential equations with combined parametric and forced harmonic excitations due to the gravity load. The incremental harmonic balance (IHB) method is used to solve for periodic responses of a high-dimensional model of the system for which convergence is reached and its period-doubling bifurcations. The multivariable Floquet theory along with the precise Hsu's method is used to investigate the stability of the periodic responses. Phase portraits and bifurcation points obtained from the IHB method agree very well with those from numerical integration.

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References

Figures

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Fig. 1

Schematic of a rotating hub-beam system under the gravity load

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Fig. 2

Free vibration response curves of a rotating hub-beam system ω/ω¯1~A11 for various angular velocities Ω

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Fig. 3

Nonlinear frequency response curves of the rotating hub-beam system in Sec. 6.3 with g = 0.2 and ξ = 0.01, calculated using different numbers of included trial functions in Eq. (30): (a) Ω~A11 and (b) Ω~A21

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Fig. 5

Nonlinear frequency response curves of period-2 and period-4 solutions with r0 = 0.0909, g = 0.2, ξ = 0.01, and n = 5: (a) Ω~A11/4; (b) Ω~A11/2; and (c) Ω~A13/4

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Fig. 4

Nonlinear frequency response curves of the rotating hub-beam system in Sec. 6.3 with r0 = 0.0909, g = 0.2, ξ = 0.01, and n = 5: (a) Ω~A11, and (b) Ω~A21; enlarged views of two zones highlighted in (a) are shown in (c) and (d), and an enlarged view of a highlighted zone in (d) is shown in (e)

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Fig. 6

Quasi-periodic response for Ω = 2.8163: ((a) and (b)) time histories; ((c) and (d)) Fourier spectra; ((e) and (f)) phase plane diagrams; and ((g) and (h)) Poincaré sections

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Fig. 7

Phase plane diagrams of some stable period-1, period-2, and period-4 solutions with r0 = 0.0909, g = 0.2, ξ = 0.01, and n = 5: ((a) and (b)) a period-1 solution with Ω = 2.4783; ((c) and (d)) a period-2 solution with Ω = 2.4708; and ((e) and (f)) a period-4 solution with Ω = 2.4643. Solid lines, the IHB method; and small circles, numerical integration.

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Fig. 8

Fourier spectra of the stable period-1, period-2, and period-4 solutions in Fig. 6 with r0 = 0.0909, g = 0.2, ξ = 0.01, and n = 5: (a) and (b) the period-1 solution with Ω = 2.4783; (c) and (d) the period-2 solution with Ω = 2.4708; and (e) and (f) the period-4 solution with Ω = 2.4643

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