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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

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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