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TECHNICAL PAPERS

Multiscale, Multiphenomena Modeling and Simulation at the Nanoscale: On Constructing Reduced-Order Models for Nonlinear Dynamical Systems With Many Degrees-of-Freedom

[+] Author and Article Information
E. H. Dowell

Director of the Center for Nonlinear and Complex Systems, Dean Emeritus, Pratt School of Engineering

D. Tang

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300

J. Appl. Mech 70(3), 328-338 (Jun 11, 2003) (11 pages) doi:10.1115/1.1558079 History: Received December 09, 2001; Revised September 25, 2002; Online June 11, 2003
Copyright © 2003 by ASME
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References

Rudd,  R. E., and Broughton,  J. Q., 1999, “Atomistic Simulation of MEMS Resonators Through the Coupling of Length Scales,” J. Model. Sim. Microsyst., 1 , p. 29.
Rudd,  R. E., and Broughton,  J. Q., 2000, “Concurrent Coupling of Length Scales in Solid State Systems,” Phys. Status Solidi B, 217(1), pp. 251–281.
Dowell,  E. H., and Hall,  K. C., 2001, “Modeling of Fluid-Structure Interaction,” Annu. Rev. Fluid Mech., 33, pp. 445–490.
Bolotin, V. V., 1963, Nonconservative Problems of the Elastic Theory of Stability, Pergamon Press, New York.
Dowell, E. H., 1975, Aeroelasticity of Plates and Shells, Kluwer, Dordrecht, The Netherlands.
Dowell, E. H., and Ilgamov, M., 1988, Studies in Nonlinear Aeroelasticity, Springer-Verlag, New York.
Pescheck,  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(10–11), pp. 1085–1097 (also see references therein to the earlier literature on nonlinear normal modes).
Lyon, R. H., and De Jong, R. G., 1995, Theory and Applications of Statistical Energy Analysis, Butterworth-Heinemann, Boston.
Dowell,  E. H., and Tang,  D. M., 1998, “The High Frequency Response of a Plate Carrying a Concentrated Mass/Spring System,” J. Sound Vib., 213(5), pp. 843–864 (also see references therein to the earlier literature on asymptotic modal analysis).
Craig, R. R, 1981, Structural Dynamics: An Introduction to Computer Methods, John Wiley and Sons, New York, Chap. 19 (Professor Craig is one of the pioneers in component mode analysis and this book provides a very readable introduction to the fundamental concepts).
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Figures

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A one-dimensional, discrete spring-mass system
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Natural frequencies of first two modes for various total number of particles, Nb, in system (b). Note: Natural frequencies are scaled by multiplication by Nb so that a finite asymptote is reached as Nb→∞, corresponding to a continuum limit.
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Schematic diagram of macromolecular chain model with AFM measurement system
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Nonlinear interatomic force versus the interatomic separation, r/σ
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Dynamic response of the macromolecular chain for A0/σ=3.5 and μ=0.01
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Time history at free end for A0/σ=5 and μ=0.01
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RMS amplitude of the macromolecular chain
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Total rms error versus excitation amplitude, A0/σ for different excitation frequency, μ=0.01, 0.05, and 0.2
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Deflection response between last two atoms, rI/σ=(xI−xI−1)/σ, versus τ for several different A0/σ and μ
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Nondimensional natural frequencies of the system (a) versus the eigenmodes number
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Dynamic response of the macromolecular chain using reduced-order model with and without the quasi-static correction (QSC) for A0/σ=0.1 and μ=0.01
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Total rms error versus eigenmodes, for A0/σ=0.1 and μ=0.01, using the reduced-order model with and without quasi-static correction
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Dynamic response of the macromolecular chain using the reduced-order model with quasi-static correction (QSC) for A0/σ=0.1 and μ=0.05
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RMS amplitude using the reduced-order model with quasi-static correction (QSC) for A0/σ=0.075 and μ=0.2
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Total rms error versus eigenmodes for different excitation frequency, μ=0.01, 0.05, and 0.2, using the reduced-order model with quasi-static correction. (a) For smaller excitation amplitude (weak local nonlinearities) and (b) for larger excitation amplitude (strong local nonlinearities).

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