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Research Papers

Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics

[+] Author and Article Information
Themistoklis P. Sapsis

Department of Mechanical Engineering,Massachusetts Institute of Technologysapsis@mit.edu

Alexander F. Vakakis

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801avakakis@illinois.edu

J. Comput. Nonlinear Dynam 6(1), 011014 (Oct 07, 2010) (10 pages) doi:10.1115/1.4002337 History: Received July 28, 2009; Revised November 16, 2009; Published October 07, 2010; Online October 07, 2010

We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.

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Figures

Grahic Jump Location
Figure 7

Subharmonic responses computed by solving the NLBVP 14 for k=1 (blue curves) and the original forced strongly nonlinear oscillator 1 (black curves) for ε=0.001; the subharmonic responses depicted in (a) and (b) correspond to the periodic orbits (a) and (c) in the bifurcation diagram of Fig. 3, respectively

Grahic Jump Location
Figure 6

Surfaces SX′(pY′,pX) and SY(pY′,pX) of the NLBVPs 14, k=3, and 14, k=4, for B¯=0.671; the zero contours of SX′(pY′,pX) are denoted by black lines and of SY(pY′,pX) by red lines for the online version and grey lines for the printed version

Grahic Jump Location
Figure 5

Bifurcation diagrams of the solutions of the NLBVP 14 for 1:1 resonance and k=2; on the left, we depict some characteristic periodic orbits of Eq. 12 corresponding to points marked in the bifurcation diagram

Grahic Jump Location
Figure 4

Surfaces SX′(pY′,pX) and SY(pY′,pX) of the NLBVP 14 for k=2 and (a) B¯=0.671 and (b) B¯=0.101; the lower plots show the zero contours of SX′(pY′,pX) (black lines) and SY(pY′,pX) (red lines for the online version and grey lines for the printed version)

Grahic Jump Location
Figure 3

Bifurcation diagrams of the solutions of the NLBVP 14 for 1:1 resonance and k=1; on the left, we depict some characteristic periodic orbits of 12 corresponding to points marked in the bifurcation diagram

Grahic Jump Location
Figure 2

Surfaces SX′(pY′,pX) and SY(pY′,pX) of the NLBVP 14 for k=1 and (a) B¯=0.671 and (b) B¯=0.101; the lower plots show the zero contours of SX′(pY′,pX) (black lines) and SY(pY′,pX) (red lines for the online version and grey for the printed version)

Grahic Jump Location
Figure 8

Subharmonic responses computed by solving the NLBVP 14 for k=2 (blue curves) and the original forced strongly nonlinear oscillator 1 (black curves) for ε=0.001; the subharmonic responses depicted in (a) and (b) correspond to the periodic orbits (b) and (h) in the bifurcation diagram of Fig. 5, respectively

Grahic Jump Location
Figure 1

Nonsmooth variables τ(η̂) and e(η̂)

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