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TECHNICAL PAPERS

Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes*

[+] Author and Article Information
Eric Pesheck

Mechanical Dynamics, Inc., Ann Arbor, MI 48105email: epesh@adams.com

Christophe Pierre

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109email: pierre@umich.edu

Steven W. Shaw

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226email: shaw@egr.msu.edu

J. Vib. Acoust 124(2), 229-236 (Mar 26, 2002) (8 pages) doi:10.1115/1.1426071 History: Received November 01, 1999; Revised August 01, 2001; Online March 26, 2002
Copyright © 2002 by ASME
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References

Friedmann,  P. P., 1977, “Recent Developments in Rotary-Wing Aeroelasticity,” J. Aircr., 14(11), pp. 1027–1041.
Hodges,  D. H., Hopkins,  A. S., and Kunz,  D. L., 1989, “Analysis of Structures with Rotating, Flexible Substructures Applied to Rotorcraft Aeroelasticity,” AIAA J., 27(2), pp. 192–200.
Kosmatka,  J. B., and Friedmann,  P. P., 1989, “Vibration Analysis of Composite Turbopropellers Using a Nonlinear Beam-Type Finite-Element Approach,” AIAA J., 27(11), pp. 1606–1614.
Hodges, D. H., and Dowell, E. H., 1974, “Nonlinear Equations of Motion for the Elastic Bending and Torsion of Twisted Nonuniform Rotor Blades,” Technical Report TN D-7818, NASA.
Crespo Da Silva,  M., and Hodges,  D. H., 1986, “Nonlinear Flexure and Torsion of Rotating Beams, With Application to Helicopter Rotor Blades—I. Formulation,” Vertica, 10(2), pp. 151–169.
Simo,  J. C., and Vu-Quoc,  L., 1988, “On the Dynamics in Space of Rods Undergoing Large Motions—A Geometrically Exact Approach,” Comput. Methods Appl. Mech. Eng., 66, pp. 125–161.
Wright,  A. D., Smith,  C. E., Thresher,  R. W., and Wang,  J. L. C., 1982, “Vibration Modes of Centrifugally Stiffened Beams,” ASME J. Appl. Mech., 49, pp. 197–202.
Du,  H., Lim,  M. K., and Liew,  K. M., 1994, “A Power Series Solution for Vibration of a Rotating Timoshenko Beam,” J. Sound Vib., 175(4), pp. 505–523.
Naguleswaran,  S., 1994, “Lateral Vibration of a Centrifugally Tensioned Uniform Euler-Bernoulli Beam,” J. Sound Vib., 176(5), pp. 613–624.
Yoo,  H. H., and Shin,  S. H., 1998, “Vibration Analysis of Rotating Cantilever Beams,” J. Sound Vib., 212(5), pp. 807–828.
Pesheck, E., Pierre, C., Shaw, S. W., and Boivin, N., 2001, “Nonlinear Modal Analysis of Structural Systems Using Multi-mode Invariant Manifolds,” Nonlinear Dyn., in press.
Shaw,  S. W., and Pierre,  C., 1993, “Normal Modes for Non-Linear Vibratory Systems,” J. Sound Vib., 164(1), pp. 85–124.
Shaw,  S. W., and Pierre,  C., 1994, “Normal Modes of Vibration for Non-Linear Continuous Systems,” J. Sound Vib., 169(3), pp. 319–347.
Meirovitch, L., 1967, Analytical Methods in Vibrations, MacMillan Publishing Co., Inc.
Shaw,  S. W., Pierre,  C., and Pesheck,  E., 1999, “Modal Analysis-Based Reduced-Order Models for Nonlinear Structures—An Invariant Manifold Approach,” Shock Vib. Dig., 31(1), pp. 3–16.
Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. V., Pilipchuck, V. N., and Zevin, A. A., 1996, Normal Modes and Localization in Nonlinear Systems, John Wiley & Sons.
Apiwattanalunggarn, P., Shaw, S. W., Pierre, C., and Jiang, D., 2001, “Finite-Element-Based Nonlinear Modal Reduction of a Rotating Blade with Large-Amplitude Motion,” Nonlinear Dynamics, to appear.
Pesheck, E., Pierre, C., and Shaw, S. W., 2001, “A New Galerkin-Based Approach for Accurate Nonlinear Normal Modes Through Invariant Manifolds,” J. Sound Vib., in press.
Pesheck,  E., Pierre,  C., and Shaw,  S. W., 2001, “Accurate Reduced-Order Models for a Simple Rotor Blade Model Using Nonlinear Normal Modes,” Math. Comput. Modell., 33, pp. 1085–1097.

Figures

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Rotating beam system, Ω=constant
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Axial deflection, ud(x), due to a static transverse deflection, w(x), for various values of Na; (a) Na=1, (b) Na=3, (c) Na=6, (d) Na=12
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Transverse dynamics initiated in the second transverse linear mode, shown at the beam tip, for various values of Na
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Axial dynamics due to initial conditions in the second linear transverse mode, shown at the beam tip, for various values of Na
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Transverse dynamics initiated in the second transverse linear mode, shown at the beam tip, for various values of Nt
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First NNM frequency versus number of linear modes. Here, Na=Nt, and c1(0)=1.0, corresponding to an end deflection of w(L)≈0.2 m
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Transverse and axial deflections, w(x,t) and u(x,t), for a quarter-period of motion in the third nonlinear mode. The deflections are shown at a set of equal time intervals spaced throughout the first quarter-period of the response. The dashed line denotes the static deflection, us(x), and the top curve for w(x) corresponds to the bottom u(x) curve, which occur at t=0. Note that the beam starts with the maximum transverse deflection and the lowest axial deflection, and transitions to zero transverse deflection and u(x,t)≈us(x) at the quarter-period.
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Response frequency as a function of modal amplitude for several one-mode models, as well as the reference solution, for the first three transverse modes
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Transverse deflection, w(L,t), initiated on a two-mode (first and second transverse) nonlinear manifold with 3ωt,1≈ωt,2, for the reference (18-DOF) and various reduced (2-DOF) models
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Modal deflection of the ninth transverse mode, c9(t), initiated on a two-mode (first and second transverse) nonlinear manifold with 3ωt,1≈ωt,2 as predicted by simulation, and reconstructed using the constraint equations, Eq. (14)
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Transverse deflection, w(L,t), initiated on a two-mode (fourth transverse and first axial) nonlinear manifold with 2ωt,4≈ωa,1, for the reference (18-DOF) and various reduced (2-DOF) models

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