0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Schilhansl, M. J., 1958, “Bending Frequency of a Rotating Cantilever Beam,” ASME J. Appl. Mech., 25, pp. 28–30.
Yoo, H. H., and Shin, S. H., 1998, “Vibration Analysis of Rotating Cantilever Beams,” J. Sound Vib., 212(5), pp. 807–828. [CrossRef]
Oh, S. Y., and Librescu, L., 2003, “Effects of Pretwist and Presetting on Coupled Bending Vibrations of Rotating Thin-Walled Composite Beams,” Int. J. Solids Struct., 40(5), pp. 1203–1224. [CrossRef]
Ansari, M., Esmailzadeh, E., and Jalili, N., 2011, “Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations,” ASME J. Vib. Acoust., 133(4), p. 041003. [CrossRef]
Turhan, Ö., and Bulut, G., 2009, “On Nonlinear Vibrations of a Rotating Beam,” J. Sound Vib., 322(1–2), pp. 314–335. [CrossRef]
Zhu, W. D., and Mote, C. D., Jr., 1997, “Dynamic Modeling and Optimal Control of Rotating Euler-Bernoulli Beams,” ASME J. Dyn. Syst. Meas. Control, 119(4), pp. 802–808. [CrossRef]
Abolghasemi, M., and Jalali, M. A., 2003, “Attractors of a Rotating Viscoelastic Beam,” Int. J. Nonlinear Mech., 38(5), pp. 739–751. [CrossRef]
Larsen, J. W., and Nielsen, S. R. K., 2006, “Non-Linear Dynamics of Wind Turbine Wings,” Int. J. Nonlinear Mech., 41(5), pp. 629–643. [CrossRef]
Larsen, J. W., and Nielsen, S. R. K., 2007, “Nonlinear Parametric Instability of Wind Turbine Wings,” J. Sound Vib., 299(1–2), pp. 64–82. [CrossRef]
Yang, J. B., Jiang, L. J., and Chen, D. C., 2004, “Dynamic Modelling and Control of a Rotating Euler-Bernoulli Beam,” J. Sound Vib., 274(3–5), pp. 863–875. [CrossRef]
Wang, F. X., and Zhang, W., 2012, “Stability Analysis of a Nonlinear Rotating Blade With Torsional Vibrations,” J. Sound Vib., 331(26), pp. 5755–5773. [CrossRef]
Rao, J. S., and Carnegie, W., 1972, “Non-Linear Vibrations of Rotating Cantilever Blades Treated by the Ritz Averaging Process,” Aeronaut. J., 76, pp. 566–569.
Peshek, E., Pierre, C., and Shaw, S. W., 2002, “Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes,” ASME J. Vib. Acoust., 124(2), pp. 229–236. [CrossRef]
Inoue, T., Ishida, Y., and Kiyohara, T., 2012, “Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out of Plane Vibration of the Elastic Blade),” ASME J. Vib. Acoust., 134(3), pp. 229–236. [CrossRef]
Chopra, I., and Dugundji, J., 1979, “Non-Linear Dynamic Response of a Wind Turbine Blade,” J. Sound Vib., 63(2), pp. 265–286. [CrossRef]
Lau, S. L., and Cheung, Y. K., 1981, “Amplitude Incremental Variational Principle for Nonlinear Structural Vibrations,” ASME J. Appl. Mech., 48(4), pp. 959–964. [CrossRef]
Lau, S. L., and Cheung, Y. K., 1982, “Incremental Time-Space Finite Strip Method for Nonlinear Structural Vibrations,” Earthquake Eng. Struct. Dyn., 10(2), pp. 239–253. [CrossRef]
Lau, S. L., Cheung, Y. K., and Wu, S. Y., 1982, “A Variable Parameter Incrementation Method for Dynamic Instability of Linear and Nonlinear Elastic Systems,” ASME J. Appl. Mech., 49(4), pp. 849–853. [CrossRef]
Xu, G. Y., and Zhu, W. D., 2010, “Nonlinear and Time-Varying Dynamics of High-Dimensional Models of a Translating Beam With a Stationary Load Subsystem,” ASME J. Vib. Acoust., 132(6), p. 061012. [CrossRef]
Zhu, W. D., Ren, H., and Xiao, C., 2011, “A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application to Elevator Traveling and Compensation Cables,” ASME J. Appl. Mech., 78(4), p. 041017. [CrossRef]
Cheung, Y. K., Chen, S. H., and Lau, S. L., 1990, “Application of the Incremental Harmonic Balance Method to Cubic Non-Linearity Systems,” J. Sound Vib., 140(2), pp. 273–286. [CrossRef]
Friedmann, P., Hammond, C. E., and Woo, T. H., 1977, “Efficient Numerical Treatment of Periodic Systems With Application to Stability Problems,” Int. J. Numer. Methods Eng., 11(7), pp. 1117–1136. [CrossRef]
Hsu, C. S., 1972, “Impulsive Parametric Excitation: Theory,” ASME J. Appl. Mech., 39(2), pp. 551–558. [CrossRef]
Hsu, C. S., 1974, “On Approximating a General Linear Periodic System,” J. Math. Anal. Appl., 45(1), pp. 234–251. [CrossRef]
Hsu, C. S., and Cheng, W. H., 1973, “Applications of the Theory of Impulsive Parametric Excitation and New Treatments of General Parametric Excitation Problems,” ASME J. Appl. Mech., 40(1), pp. 78–86. [CrossRef]
Huang, J. L., Su, R. K. L., and Chen, S. H., 2009, “Precise Hsu's Method for Analyzing the Stability of Periodic Solutions of Multi-Degrees-of-Freedom Systems With Cubic Nonlinearity,” Comput. Struct., 87(23–24), pp. 1624–1630. [CrossRef]
Zhong, W. X., and Williams, F. W., 1994, “A Precise Time Step Integration Method,” J. Mech. Eng. Sci., 208(6), pp. 427–430. [CrossRef]
Nayfeh, A. H., and Balachandran, B., 1994, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, Wiley, New York, pp. 205–208.

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In