TY - JOUR
T1 - Transitions from strongly to weakly nonlinear motions of damped nonlinear oscillators
AU - Salenger, G.
AU - Vakakis, A. F.
AU - Gendelman, Oleg
AU - Manevitch, Leonid
AU - Andrianov, Igor
N1 - Funding Information:
This work was supported in part by an NSF International Supplement to NSF Young Investigator Award CMS-94-57750 (A. F. Vakakis); Dr. Devendra Garg and Ms. Cassandra Dudca are the Grant monitors. Additional financial support was provided by the Russian Foundation of Basic Research through Grant No. 98-03-33366a (L. Manevitch). The authors are grateful to these funding agencies for their support.
PY - 1999
Y1 - 1999
N2 - We construct analytical approximations for the transition from strongly nonlinear, early-time oscillations to weakly nonlinear, late-time motions of single degree of freedom, damped, nonlinear oscillators. Two methods are developed. The first relies on (a) derivation of an analytic solution for the initial value problem of an exactly integrable damped system, (b) development of separate early- and late-time approximations to the damped motion using the integrable solution, and (c) patching of the two approximations in the time domain by imposing continuity conditions on the composite solution at the point of matching. The second approach relies on a multiple-scales application of the method of nonsmooth transformations first developed by Pilipchuck, but complemented with a corrected frequency-amplitude relation. This improved relation is obtained by developing two separate frequency-amplitude asymptotic expansions in the frequency-amplitude plane, that are valid for large and small amplitudes, respectively, and then matching them using two-point diagonal Pade approximants. Comparisons between analytical approximations and numerical results validate the two approaches developed.
AB - We construct analytical approximations for the transition from strongly nonlinear, early-time oscillations to weakly nonlinear, late-time motions of single degree of freedom, damped, nonlinear oscillators. Two methods are developed. The first relies on (a) derivation of an analytic solution for the initial value problem of an exactly integrable damped system, (b) development of separate early- and late-time approximations to the damped motion using the integrable solution, and (c) patching of the two approximations in the time domain by imposing continuity conditions on the composite solution at the point of matching. The second approach relies on a multiple-scales application of the method of nonsmooth transformations first developed by Pilipchuck, but complemented with a corrected frequency-amplitude relation. This improved relation is obtained by developing two separate frequency-amplitude asymptotic expansions in the frequency-amplitude plane, that are valid for large and small amplitudes, respectively, and then matching them using two-point diagonal Pade approximants. Comparisons between analytical approximations and numerical results validate the two approaches developed.
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U2 - 10.1023/A:1008354208466
DO - 10.1023/A:1008354208466
M3 - Article
AN - SCOPUS:0032595376
SN - 0924-090X
VL - 20
SP - 99
EP - 114
JO - Nonlinear Dynamics
JF - Nonlinear Dynamics
IS - 2
ER -