Transitions from Strongly to Weakly Nonlinear Motions of Damped Nonlinear Oscillators |
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Authors: | Salenger G Vakakis A F Gendelman Oleg Manevitch Leonid Andrianov Igor |
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Institution: | (1) Department of Mechanical & Industrial Engineering, University of Illinois, 1206 W. Green Street, Urbana, IL, 61801, U.S.A;(2) Institute of Chemical Physics, Russian Academy of Science, Kosygin Str. 4, 117977 Moscow, Russia;(3) Department of Mathematics, Prydneprovie State Academy of Civil Engineering and Architecture, 320000 Dnepropetrovsk, Ukraine |
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Abstract: | 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 Padé approximants. Comparisons between analytical approximations and numerical results validate the two approaches developed |
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Keywords: | damped nonlinear systems integrability Padé approximations |
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