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Journal of Applied Mechanics | Volume 74 | Issue 2 | TECHNICAL PAPERS
TECHNICAL PAPERS

Quasi-Periodic Response Regimes of Linear Oscillator Coupled to Nonlinear Energy Sink Under Periodic Forcing

[+] Author and Article Information
O. V. Gendelman1

Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israelovgend@tx.technion.ac.il

Yu. Starosvetsky

Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israelstryuli@techunix.technion.ac.il

1

Author to whom correspondence should be addressed.

J. Appl. Mech 74(2), 325-331 (Feb 09, 2006) (7 pages) doi:10.1115/1.2198546 History: Received May 18, 2005; Revised February 09, 2006
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Quasi-periodic response of a linear oscillator attached to nonlinear energy sink with relatively small mass under external sinusoidal forcing in a vicinity of main (1:1) resonance is studied analytically and numerically. It is shown that the quasi-periodic response is exhibited in well-defined amplitude-frequency range of the external force. Two qualitatively different regimes of the quasi-periodic response are revealed. The first appears as a result of linear instability of the steady-state regime of the oscillations. The second one occurs due to interaction of the dynamical flow with invariant manifold of damped-forced nonlinear normal mode of the system, resulting in hysteretic motion of the flow in the vicinity of this mode. Parameters of external forcing giving rise to the quasi-periodic response are predicted by means of simplified analytic model. The model also allows predicting that the stable quasi-periodic regimes appear for certain range of damping coefficient. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.
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Grahic Jump Location
+Zones for weakly and strongly quasi-periodic responses at the plane of parameters. Weakly quasi-periodic responses can exist within the red boundary (Eq. 13), family 23 of the strongly quasi-periodic attractors exists above blue curve, family 24—above green curve.
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Zones for weakly and strongly quasi-periodic responses at the plane of parameters. Weakly quasi-periodic responses can exist within the red boundary (Eq. 13), family 23 of the strongly quasi-periodic attractors exists above blue curve, family 24—above green curve.
Figure 1

Zones for weakly and strongly quasi-periodic responses at the plane of parameters. Weakly quasi-periodic responses can exist within the red boundary (Eq. 13), family 23 of the strongly quasi-periodic attractors exists above blue curve, family 24—above green curve.

Grahic Jump Location
+(a) Steady-state response of system 1 for A=0.225, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15, and (b). Strongly quasi-periodic response of system 1 for A=0.225, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
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(a) Steady-state response of system 1 for A=0.225, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15, and (b). Strongly quasi-periodic response of system 1 for A=0.225, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
Figure 2

(a) Steady-state response of system 1 for A=0.225, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15, and (b). Strongly quasi-periodic response of system 1 for A=0.225, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.

Grahic Jump Location
+(a) weakly quasi-periodic response of system 1 for A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15, and (b) strongly quasi-periodic response of system 1 for A=0.24, λ=0.2, ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
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(a) weakly quasi-periodic response of system 1 for A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15, and (b) strongly quasi-periodic response of system 1 for A=0.24, λ=0.2, ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
Figure 3

(a) weakly quasi-periodic response of system 1 for A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15, and (b) strongly quasi-periodic response of system 1 for A=0.24, λ=0.2, ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.

Grahic Jump Location
+(a) Steady-state response of system 1 for A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=−0.052, dy1∕dt(0)=0.187, y2(0)=−1, dy2∕dt(0)=0, and (b) strongly quasi-periodic response of system 1 for A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
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(a) Steady-state response of system 1 for A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=−0.052, dy1∕dt(0)=0.187, y2(0)=−1, dy2∕dt(0)=0, and (b) strongly quasi-periodic response of system 1 for A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
Figure 4

(a) Steady-state response of system 1 for A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=−0.052, dy1∕dt(0)=0.187, y2(0)=−1, dy2∕dt(0)=0, and (b) strongly quasi-periodic response of system 1 for A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.

Grahic Jump Location
+Fast Fourier transform of the response for parameter values A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=−0.052, dy1∕dt(0)=0.187, y2(0)=−1, and dy2∕dt(0)=0.
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Fast Fourier transform of the response for parameter values A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=−0.052, dy1∕dt(0)=0.187, y2(0)=−1, and dy2∕dt(0)=0.
Figure 5

Fast Fourier transform of the response for parameter values A=1, λ=0.2, and ε=0.05. Initial conditions are y1(0)=−0.052, dy1∕dt(0)=0.187, y2(0)=−1, and dy2∕dt(0)=0.

Grahic Jump Location
+Fast Fourier transform of the response for parameter values A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15.
View Large  |  Save Figure  |  Download Slide (.ppt)
Fast Fourier transform of the response for parameter values A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15.
Figure 6

Fast Fourier transform of the response for parameter values A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0.29, dy1∕dt(0)=0.25, y2(0)=0, and dy2∕dt(0)=−0.15.

Grahic Jump Location
+Fast Fourier transform of the response for parameter values A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
View Large  |  Save Figure  |  Download Slide (.ppt)
Fast Fourier transform of the response for parameter values A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.
Figure 7

Fast Fourier transform of the response for parameter values A=0.24, λ=0.2, and ε=0.05. Initial conditions are y1(0)=0, dy1∕dt(0)=0, y2(0)=0, and dy2∕dt(0)=0.

Grahic Jump Location
+Direct numeric simulation of system 1 for A=0.225, λ=0.2, and ε=0.05 (thin line) and modulation shape obtained from solution of Eq. 19 (thick line). The time scale is shifted to zero and scaled by factor ε.
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Direct numeric simulation of system 1 for A=0.225, λ=0.2, and ε=0.05 (thin line) and modulation shape obtained from solution of Eq. 19 (thick line). The time scale is shifted to zero and scaled by factor ε.
Figure 8

Direct numeric simulation of system 1 for A=0.225, λ=0.2, and ε=0.05 (thin line) and modulation shape obtained from solution of Eq. 19 (thick line). The time scale is shifted to zero and scaled by factor ε.

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