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Research Papers

Nonlinear and Time-Varying Dynamics of High-Dimensional Models of a Translating Beam With a Stationary Load Subsystem

[+] Author and Article Information
G. Y. Xu

 Eigen Corporate Office, 13366 Grass Valley Avenue, Grass Valley, CA 95945

W. D. Zhu1

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250

1

Corresponding author.

J. Vib. Acoust 132(6), 061012 (Oct 12, 2010) (17 pages) doi:10.1115/1.4000464 History: Received June 12, 2008; Revised June 08, 2009; Published October 12, 2010; Online October 12, 2010

Nonlinear vibration and dynamic stability analyses of distributed structural systems have often been conducted for their low-dimensional spatially discretized models, and the results obtained from the low-dimensional models may not accurately represent the behaviors of the distributed systems. In this work the incremental harmonic balance method is used to handle a variety of problems pertaining to determining periodic solutions of high-dimensional models of distributed structural systems. The methodology is demonstrated on a translating tensioned beam with a stationary load subsystem and some related systems. With sufficient numbers of included trial functions and harmonic terms, convergent and accurate results are obtained in all the cases. The effect of nonlinearities due to the vibration-dependent friction force between the translating beam and the stationary load subsystem, which results from nonproportionality of the load parameters, decreases as the number of included trial functions increases. A low-dimensional spatially discretized model of the nonlinear distributed system can yield quantitatively and qualitatively inaccurate predictions. The methodology can be applied to other nonlinear and/or time-varying distributed structural systems.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic of a fixed-fixed translating beam with a stationary load subsystem

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Figure 2

Frequency response of a fixed-fixed translating beam without a stationary load subsystem, with (a) five, (b) ten, and (c) 30 included trial functions; the amplitudes shown here are Bi=ai12+bi12(i=1,2,3)

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Figure 10

Parametric instability regions of a pinned-pinned translating beam with v=vf sin(ωft) by using (a) one, (b) two, and (c) five included trial functions; the shaded instability regions are obtained using Floquet theory and the boundary curves of the simple parametric instability regions are obtained using the IHB method

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Figure 11

Principal parametric instability region of the first vibration mode of a pinned-pinned translating beam with v=v0+vf sin(ωft) by using different numbers of included trial functions: (a) v0=0 and (b) v0=3.0 m/s

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Figure 12

Principal simple parametric instability regions of the first three vibration modes of a pinned-pinned translating beam with v=3.0+vf sin(ωft) m/s by using ten included trial functions: without a stationary load subsystem (solid lines), and with a stationary load subsystem whose parameters are proportional (dashed lines)

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Figure 13

Periodic solutions of a pinned-pinned translating beam with a stationary load subsystem whose parameters are nonproportional by assuming b11=0.001,0.1,1 for the three left curves and a11=0.001,0.1,1 for the three right curves

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Figure 14

Frequency responses of a pinned-pinned translating beam with v=v0+vf sin(ωft) under the boundary excitation by using 20 included trial functions: (a) v0=3 m/s and vf=0, and (b) v0=vf=3 m/s; the amplitudes shown here are Bi=ai12+bi12 (i=1,2,3), and ωf is varied from 5 to 30 rad/s with an increment of 0.1 rad/s

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Figure 15

The vibration amplitude of a pinned-pinned translating beam at x=a with a stationary load subsystem whose parameters are (a) proportional and (b) nonproportional, under combined parametric and boundary excitations, as a function of excitation frequency

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Figure 16

A segment of the translating beam in Fig. 1 whose end points correspond to the Lagrangian variables s1 and s2

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Figure 17

An arbitrary segment of a translating beam which is (a) unforced, (b) subjected to a tension T, and (c) subjected to both the tension T and the bending moment M

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Figure 3

The vibration amplitude of a fixed-fixed translating beam with a stationary load subsystem at x=a as a function of excitation frequency; different numbers of included trial functions are used

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Figure 4

Comparison of the transient response from direct numerical integration (thin solid line) and the steady-state response from the IHB method (thick solid line) for the 10DOF spatially discretized model of a fixed-fixed translating beam with a stationary load subsystem

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Figure 5

Steady-state responses of the system for Fig. 4 with different numbers of included trial functions

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Figure 6

The vibration amplitude of the beam at x=a versus the location of the stationary load subsystem; different numbers of included trial functions are used

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Figure 7

Parametric instability regions in the (ωf/ω0,vf) parameter plane for the single-DOF model of a translating string with constant tension and a sinusoidally varying velocity; the shaded instability regions are obtained using Floquet theory and their boundary curves are obtained using the IHB method

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Figure 8

Principal instability region boundaries for the single-DOF model of a translating string with constant tension and a sinusoidally varying velocity, determined using different numbers of harmonic terms for the generalized coordinate

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Figure 9

Parametric instability regions in the (ωf/ω0,vf) parameter plane for the single-DOF model of a translating string with variable tension and a sinusoidally varying velocity; the shaded instability regions are obtained using Floquet theory and their boundary curves are obtained using the IHB method

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