Nonlinear Vibratory Energy Exchanges between a Two-Degree-of-Freedom Pendulum and a Nonlinear Absorber
Publication: Journal of Engineering Mechanics
Volume 145, Issue 8
Abstract
The multi-time-scale responses of a two-degree-of-freedom pendulum coupled with a nonlinear absorber were studied. The absorber was positioned in an arbitrary direction with respect to those of the pendulum oscillations. The phase-dependent slow invariant manifold of the system and its stable zones were traced at a fast time scale while system responses were studied at a slow time scale around the slow, invariant manifold, leading to detection of equilibrium and singular points. Moreover, the amplitude–frequency curves of the system were detected, showing the possibility of the existence of isolated branches, which could correspond to high energy levels for pendulum oscillations. All analytic developments were confronted with numerical results collected from direct numerical integration of system equations. Depending on the characteristics of external excitations, the system can face periodic or modulated regimes.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors would like to thank La Region Auvergne-Rhone-Alpes for supporting this work in the frame of the CALIPSO project.
References
Den Hartog, J. P. 1956. Mechanical vibrations. New York: McGraw-Hill.
Gendelman, O., L. I. Manevitch, A. F. Vakakis, and R. M’Closkey. 2001. “Energy pumping in nonlinear mechanical oscillators: Part I—Dynamics of the underlying Hamiltonian systems.” J. Appl. Mech. 68 (1): 34. https://doi.org/10.1115/1.1345524.
Gendelman, O. V. 2008. “Targeted energy transfer in systems with non-polynomial nonlinearity.” J. Sound Vib. 315 (3): 732–745. https://doi.org/10.1016/j.jsv.2007.12.024.
Gendelman, O. V., Y. Starosvetsky, and M. Feldman. 2008. “Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes.” Nonlinear Dyn. 51 (1–2): 31–46. https://doi.org/10.1007/s11071-006-9167-0.
Housner, G. W., L. A. Bergman, T. K. Caughey, A. G. Chassiakos, R. O. Claus, S. F. Masri, R. E. Skelton, T. T. Soong, B. F. Spencer, and J. T. P. Yao. 1997. “Structural control: Past, present, and future.” J. Eng. Mech. 123 (9): 897–971. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:9(897).
Ibrahim, R. 2008. “Recent advances in nonlinear passive vibration isolators.” J. Sound Vib. 314 (3–5): 371–452. https://doi.org/10.1016/j.jsv.2008.01.014.
Lamarque, C.-H., O. V. Gendelman, A. Ture Savadkoohi, and E. Etcheverria. 2011. “Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink.” Acta Mech. 221 (1–2): 175–200. https://doi.org/10.1007/s00707-011-0492-0.
Manevitch, L. I. 2001. “The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables.” Nonlinear Dyn. 25 (1): 95–109. https://doi.org/10.1023/A:1012994430793.
Matsuhisa, H., R. Gu, Y. Wang, O. Nishihara, and S. Sato. 1995. “Vibration control of a ropeway carrier by passive dynamic vibration absorbers.” JSME Int. J., Ser. C 38 (4): 657–662. https://doi.org/10.1299/jsmec1993.38.657.
Nayfeh, A. H., and D. T. Mook. 1995. Nonlinear oscillations. New York: Wiley.
Pham, T. T., C.-H. Lamarque, and A. Ture Savadkoohi. 2012. “Multi-resonance capturing in a two-degree-of-freedom system under two different harmonic excitations.” J. Vib. Control 18 (3): 451–466. https://doi.org/10.1177/1077546311404268.
Roberson, R. E. 1952. “Synthesis of a nonlinear dynamic vibration absorber.” J. Franklin Inst. 254 (3): 205–220. https://doi.org/10.1016/0016-0032(52)90457-2.
Starosvetsky, Y., and O. V. Gendelman. 2008a. “Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system.” J. Sound Vib. 312 (1): 234–256. https://doi.org/10.1016/j.jsv.2007.10.035.
Starosvetsky, Y., and O. V. Gendelman. 2008b. “Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry.” Physica D 237 (13): 1719–1733. https://doi.org/10.1016/j.physd.2008.01.019.
Ture Savadkoohi, A., C.-H. Lamarque, and Z. Dimitrijevic. 2012. “Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity.” Nonlinear Dyn. 70 (2): 1473–1483. https://doi.org/10.1007/s11071-012-0548-2.
Ture Savadkoohi, A., C.-H. Lamarque, M. Weiss, B. Vaurigaud, and S. Charlemagne. 2016. “Analysis of the 1:1 resonant energy exchanges between coupled oscillators with rheologies.” Nonlinear Dyn. 86 (4): 2145–2159. https://doi.org/10.1007/s11071-016-2792-3.
Vakakis, A. F. 2008a. “Nonlinear targeted energy transfer in mechanical and structural systems I.” In Solid mechanics and its applications. Dordrecht, Netherlands: Springer.
Vakakis, A. F. 2008b. “Nonlinear targeted energy transfer in mechanical and structural systems II.” In Solid mechanics and its applications. Dordrecht, Netherlands: Springer.
Vakakis, A. F., and O. Gendelman. 2001. “Energy pumping in nonlinear mechanical oscillators: Part II. Resonance capture.” J. Appl. Mech. 68 (1): 42. https://doi.org/10.1115/1.1345525.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 145 • Issue 8 • August 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Aug 22, 2018
Accepted: Dec 5, 2018
Published online: May 31, 2019
Published in print: Aug 1, 2019
Discussion open until: Oct 31, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.