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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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References

1
Gendelman, O. V., 2001, “Transition of Energy to Nonlinear Localized Mode in Highly Asymmetric System of Nonlinear Oscillators,” Nonlinear Dyn.  [CrossRef], 25 , pp. 237–253.
2
Gendelman, O. V., Vakakis, A. F., Manevitch, L. I., and McCloskey, R., 2001, “Energy Pumping in Nonlinear Mechanical Oscillators. I: Dynamics of the Underlying Hamiltonian System,” ASME J. Appl. Mech.  [CrossRef], 68 (1), pp. 34–41.
3
Vakakis, A. F., and Gendelman, O. V., 2001, “Energy Pumping in Nonlinear Mechanical Oscillators. II: Resonance Capture,” ASME J. Appl. Mech.  [CrossRef], 68 (1), pp. 42–48.
4
Vakakis, A. F., 2001, “Inducing Passive Nonlinear Energy Sinks in Linear Vibrating Systems,” ASME J. Vibr. Acoust.  [CrossRef], 123 (3), pp. 324–332.
5
Vakakis, A. F., Manevitch, L. I., Gendelman, O., and Bergman, L., 2003, “Dynamics of Linear Discrete Systems Connected to Local Essentially Nonlinear Attachments,” J. Sound Vib.  [CrossRef], 264 , pp. 559–577.
6
Georgiadis, F., Vakakis, A. F., McFarland, M., and Bergman, L., 2003, “Shock Isolation Through Passive Energy Pumping in a System With Piecewise Linear Stiffnesses,” "Proceedings of DETC.03 ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference", Chicago, September 2–6, ASME, New York, ASME Paper No. VIB-48490, CD-ROM.
7
Gendelman, O. V., 2004, “Bifurcations of Nonlinear Normal Modes of Linear Oscillator With Strongly Nonlinear Damped Attachment,” Nonlinear Dyn., 37 (2), pp. 115–128.
8
Shaw, S. W., and Pierre, C., 1993, “Normal Modes for Nonlinear Vibratory Systems,” J. Sound Vib.  [CrossRef], 164 , pp. 85–124.
9
Vakakis, A. F., Manevitch, L. I., Mikhlin, Yu. V., Pilipchuk, V. N., and Zevin, A. A., 1996, "Normal Modes and Localization in Nonlinear Systems", Wiley, New York.
10
Jang, X., McFarland, M., Bergman, L. A., and Vakakis, A. F., 2003, “Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results,” Nonlinear Dyn., 33 , pp. 7–102.
11
Gendelman, O. V., Gourdon, E., and Lamarque, C. H., 2005, “Quasi-periodic Energy Pumping in Coupled Oscillators Under Periodic Forcing,” J. Sound Vib., (in press).
12
Shaw, J., Shaw, S. W., and Haddow, A. G., 1989, “On the Response of the Nonlinear Vibration Absorber,” Int. J. Non-Linear Mech.  [CrossRef], 24 , pp. 281–293.
13
Natsiavas, S., 1992, “Steady State Oscillations and Stability of Non-Linear Dynamic Vibration Absorbers,” J. Sound Vib., 156 (2), pp. 227–245.
14
Rice, H. J., and McCraith, J. R., 1987, “Practical Non-Linear Vibration Absorber Design,” J. Sound Vib., 116 (3), pp. 545–559.
15
Jackson, E. A., 1994, "Perspectives of Nonlinear Dynamics", Vol. 1 , Cambridge University Press, Cambridge, UK.

Figures

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.

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.

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