passive non-linear targeted energy transfer; vibration absorbtion
Abstract :
[en] This review paper discusses recent efforts to passively move unwanted energy from a
primary structure to a local essentially non-linear attachment (termed a non-linear energy sink)
by utilizing targeted energy transfer (TET) (or non-linear energy pumping). First, fundamental
theoretical aspects of TET will be discussed, including the essentially non-linear governing
dynamical mechanisms for TET. Then, results of experimental studies that validate the TET
phenomenon will be presented. Finally, some current engineering applications of TET will be
discussed. The concept of TET may be regarded as contrary to current common engineering
practise, which generally views non-linearities in engineering systems as either unwanted or, at
most, as small perturbations of linear behaviour. Essentially non-linear stiffness elements are
intentionally introduced in the design that give rise to new dynamical phenomena that are very
beneficial to the design objectives and have no counterparts in linear theory. Care, of course, is
taken to avoid some of the unwanted dynamic effects that such elements may introduce, such as
chaotic responses or other responses that are contrary to the design objectives.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others Mechanical engineering
Author, co-author :
Lee, Y. S.; University of Illiois at Urbana-Champaign > Department of Aerospace Engineering,
Vakakis, Alexander F.; National Technical University of Athens > School of AppliedMathematical and Physical Sciences
Bergman, L. A.; University of Illiois at Urbana-Champaign > Department of Aerospace Engineering,
McFarland, D. M.; University of Illiois at Urbana-Champaign > Department of Aerospace Engineering,
Kerschen, Gaëtan ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux
Nucera, F.; Mediterranean University, Reggio Calabria, Italy > Department ofMechanics andMaterials,
Tsakirtzis, S.; National Technical University of Athens > School of AppliedMathematical and Physical Sciences,
Language :
English
Title :
Passive non linear TET and its application to vibration absorption: a review
Publication date :
2008
Journal title :
Proceedings of the Institution of Mechanical Engineers. Part K, Journal of Multi-body Dynamics
Frahm, H. Device for damping vibrations of bodies. US Pat. 989958, 1909.
Sun, J. Q., Jolly, M. R., and Norris, M. Passive, adaptive and active tuned vibration absorbers - a survey. Trans. ASME, J. Mech. Des., 1995, 117 234-242.
Housner, G., Bergman, L., Caughey, T., Chassiakos, A., Claus, R., Masri, S., Skelton, R., Soong, T., Spencer, B., and Yao, J. Structural control: past, present, and future. ASCE J. Eng. Mech., 1997, 123, 897-971.
Zuo, L. and Nayfeh, S. Minimax optimization of multi-degree-of-freedom tuned-mass dampers. J. Sound Vibr., 2004, 272(3-5), 893-908.
Krenk, S. Frequency analysis of the tuned mass damper. Trans. ASME, J. Appl. Mech., 2005, 72, 936-942.
El-Khatib, H., Mace, B., and Brennan, M. Suppresion of bending waves in a beam using a tuned vibration absorber. J. Sound Vibr., 2005, 288, 1157-1175.
Zuo, L. and Nayfeh, S. The two-degree-of-freedom tuned-mass damper for suppression of single-mode vibration under random and harmonic excitation. Trans. ASME J. Vibr. Acoust., 2006, 128, 56-65.
Roberson, R. Synthesis of a nonlinear dynamic vibration absorber. J. Franklin Inst., 1952, 254, 205-220.
Pak, C., Song, S., Shin, H., and Hong, S. A study on the behavior of nonlinear dynamic absorber (in Korean). Korean Soc. Noise Vibr. Eng, 1993, 3, 137-143.
Wagg, D. Multiple non-smooth events in multi-degree-of-freedom vibro-impact systems. Nonlinear Dyn., 2006, 43, 137-148.
Pun, D. and Liu, Y. B. On the design of the piecewise linear vibration absorber. Nonlinear Dyn., 2000, 22(4), 393-113.
Shaw, S. and Wiggins, S. Chaotic motions of a torsional vibration absorber. Trans. ASME, J. Appl. Mech., 1988, 55, 952-958.
Tondl, A., Ruijgrok, M., Verhulst, F., and Nabergoj, R. Autoparametric resonance in mechanical systems, 2003 (Cambridge University Press, New York).
Golnaraghi, M.Vibration suppression of flexible structures using internal resonance. Mech. Res. Commun., 1991, 18, 135-143.
Vilallonga, E. and Rabitz, H. Vibrational energy transfer at the gas-solid interface: the role of collective and of localized vibrational modes. J. Chem. Phys., 1986, 85, 2300-2314.
Vilallonga, E. and Rabitz, H. A hybrid model for vibrational energy transfer at the gas-solid interface: discrete surface atoms plus a continuous elastic bulk. J. Chem. Phys., 1990, 92, 3957-3976.
Sievers, A. and Takeno, S. Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett., 1988, 61, 970-973.
Shepelyansky, D. Derealization of quantum chaos by weak nonlinearity. Phys. Rev. Lett., 1993, 70, 1787-1790.
Sokoloff, J. Reduction of energy absorption by phonons and spin waves in a disordered solid due to localization. Phys. Rev. B, 2000, 61, 9380-9386.
Khusnutdinova, K. and Pelinovsky, D. On the exchange of energy in coupled Klein-Gordon equations. Wave Motion, 2003, 38, 1-10.
Morgante, A., Johansson, M., Aubry, S., and Kopidakis, G. Breather-phonon resonances in finite-size lattices: 'phantom breathers'? J. Phys. A., 2002, 35, 4999-5021.
Kopidakis, G., Aubry, S., and Tsironis, G. Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett., 2001, 87, 165501.1-4.
Aubry, S., Kopidakis, G., Morgante, A., and Tsironis, G. Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers. Phys. B, 2001, 296, 222-236.
Maniadis, P., Kopidakis, G., and Aubry, S. Classical and quantum targeted energy transfer between nonlinear oscillators. Phys. D, 2004, 188, 153-177.
Memboeuf, A. and Aubry, S. Targeted energy transfer between a rotor and a Morse oscillator: a model for selective chemical dissociation. Phys. D, 2005, 207, 1-23.
Hodges, C. Confinement of vibration by structural irregularity. J. Sound Vibr., 1982, 82, 411-124.
Pierre, C. and Dowell, E. Localization of vibrations by structural irregularity. J. Sound Vibr., 1987, 114, 549-564.
Bendiksen, O. Mode localization phenomena in large space structures. AIAA J., 1987, 25, 1241-1248.
Cai, C., Chan, H., and Cheung, Y. Localized modes in a two-degree-coupling periodic system with a nonlinear disordered subsystem. Chaos Solitons Fractals, 2000, 11, 1481-1492.
Anderson, P. Absence of diffusion in certain random lattices. Phys. Rev., 1958, 109, 1492-1505.
Hodges, C. and Woodhouse, J. Vibration isolation from irregularity in a nearly periodic structure: theory and measurements. J. Acoust Soc. Am., 1983, 74, 894-905.
Vakakis, A. Dynamics of a nonlinear periodic structure with cyclic symmetry. Acta Mech., 1992, 95, 197-226.
Vakakis, A. and Cetinkaya, C. Mode localization in a class of multi-degree-of-freedom nonlinear systems with cyclic symmetry. SIAM J. Appl. Math., 1993, 53, 265-282.
Vakakis, A., Raheb, M., and Cetinkaya, C. Free and forced dynamics of a class of periodic elastic systems. J. Sound Vibr., 1994, 172, 23-46.
King, M. and Vakakis, A. A very complicated structure of resonances in a nonlinear system with cyclic symmetry: nonlinear forced localization. Nonlinear Dyn., 1995, 7, 85-104.
Aubrecht, J. and Vakakis, A. Localized and non-localized nonlinear normal modes in a multi-span beam with geometric nonlinearities. Trans. ASME, J. Appl. Mech., 1996, 118, 533-542.
Salenger, G. and Vakakis, A. Discretenes effects in the forced dynamics of a string on a periodic array of non-linear supports. Int. J. Non-Linear Mech., 1998, 33, 659-673.
Fang, X., Tang, J., Jordan, E., and Murphy, K. Crack induced vibration localization in simplified bladed-disk structures. J. Sound Vibr., 2006, 291, 395-418.
Campbell, D., Flach, S., and Kivshar, Y. Localizing energy through nonlinearity and discreteness. Phys. Today, 2004, 57(1), 43-49.
Sato, M., Hubbard, B., and Sievers, A. Colloquium: nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev. Mod. Phys., 2006, 78, 137-157.
Gendelman, O., Manevitch, L., Vakakis, A., and M'Closkey, R. Energy pumping in coupled mechanical oscillators, part I: dynamics of the underlying Hamiltonian systems. Trans. ASME, J. Appl. Mech., 2001, 68, 34-41.
Nayfeh, A. and Pal, P. Non-linear non-planar parametric responses of an inextensional beam. Int. J. Non-Linear Mech., 1989, 24, 139-158.
Pai, P. and Nayfeh, A. Non-linear non-planar oscillations of a cantilever beam under lateralbase excitations. Int. J. Non-Linear Mech., 1990, 25, 455-474.
Nayfeh, A. and Mook, D. Energy transfer from high-frequency to low-frequency modes in structures. Trans. ASME, J. Appl. Mech., 1995, 117, 186-195.
Malatkar, P. and Nayfeh, A. On the transfer of energy between widely spaced modes in structures. Nonlinear Dyn., 2003, 31, 225-242.
Cusumano, J. Low-dimensional, chaotic, nonplanar motions of the elastica: experiment and theory. PhD Thesis, Cornell University, 1990.
Gendelman, O. and Vakakis, A. F. Transitions from localization to nonlocalization in strongly nonlinear damped oscillators. Chaos Solitons Fractals, 2000, 11, 1535-1542.
Vakakis, A. Inducing passive nonlinear energy sinks in vibrating systems. Trans. ASME, J. Vibr. Acoust., 2001, 123, 324-332.
Gendelman, O. Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dyn., 2001, 25, 237-253.
Vakakis, A. and Gendelman, O. Energy pumping in coupled mechanical oscillators, part ii: resonance capture. Trans. ASME J. Appl. Mech., 2001, 68, 42-48.
Panagopoulos, P., Gendelman, O., and Vakakis, A. Robustness of nonlinear targeted energy transfer in coupled oscillators to changes of initial conditions. Nonlinear Dyn., 2007, 47(4), 377-387.
Bohr, T., Bak, P., and Jensen, M. Transition to chaos by interaction of resonances in dissipative systems, ii. Josephson junctions, charge-density waves, and standard maps. Phys. Rev. A, 1984, 30, 1970-1981.
Itin, A., Neishtadt, A., and Vasiliev, A. Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave. Phys. D, 2000, 141, 281-296.
Vainchtein, D., Rovinsky, E., Zelenyi, L., and Neishtadt, A. Resonances and particle stochastization in nonhomogeneous electromagnetic fields. J. Nonlinear Sci., 2004, 14, 173-205.
Haberman, R. Energy bounds for the slow capture by a center in sustained resonance. SIAM J. Appl. Math., 1983, 43, 244-256.
Kath, W. Necessary conditions for sustained reentry roll resonance. SIAM J. Appl. Math., 1983, 43, 314-324.
Kath, W. Conditions for sustained resonance. II. SIAM J. Appl. Math., 1983, 43, 579-583.
Haberman, R., Rand, R., and Yuster, T. Resonant capture and separatrix crossing in dual-spin spacecraft. Nonlinear Dyn., 1999, 18, 159-184.
Belokonov, V. and Zabolotnov, M. Estimation of the probability of capture into a resonance mode of motion for a spacecraft during its descent in the atmosphere. Cosmic Res., 2002, 40, 467-478.
Arnold, V. Dynamical systems III (encyclopaedia of mathematical sciences), 1988 (Springer-Verlag, Berlin Heidelberg, Germany).
Burns, T. and Jones, C. A mechanisms for capture into resonance. Phys. D, 1993, 69, 85-106.
Neishtadt, A. Scattering by resonances. Celest Mech. Dyn. Astron., 1997, 65, 1-20.
Neishtadt, A. On adiabatic invariance in two-frequency systems. Hamiltonian systems with three or more degrees of freedom, NATO ASI Series C 533, 1999, pp. 193-212.
Quinn, D. Resonance capture in a three degree-of-freedom mechanical system. Nonlinear Dyn., 1997, 14, 309-333.
Quinn, D. Transition to escape in a system of coupled oscillators. Int. J. Non-Linear Mech., 1997, 32, 1193-1206.
Zniber, A. and Quinn, D. Frequency shining in nonlinear resonant systems with damping. In the ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, Illinois, 2003, DETC2003/VIB-48444.
Gendelman, O. and Lamarque, C. Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium. Chaos Solitons Fractals, 2005, 24(2), 501-509.
Musienko, A. and Manevitch, L. Comparison of passive and active energy pumping in mechanical nonlinear system. In the ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, California, 2005, DETC2005-84806.
Thorp, J., Seyler, C., and Phadke, A. Electromechanical wave propagation in large electric power systems. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 1998, 45(6), 614-22.
Vakakis, A. Analysis and identification of linear and nonlinear normal modes in vibrating systems. PhD Thesis, California Institute of Technology, 1991.
Gourdon, E. and Lamarque, C. H. Energy pumping with various nonlinear structures: numerical evidences. Nonlinear Dyn., 2005, 40(3), 281-307.
Pilipchuk, V., Vakakis, A., and Azeez, M. Study of a class of subharmonic motions using a non-smooth temporal transformation (NSTT). Phys. D, 1997, 100, 145-164.
Gendelman, O., Manevitch, L., Vakakis, A., and Bergman, L. A degenerate bifurcation structure in the dynamics of coupled oscillators with essential stiffness nonlinearities. Nonlinear Dyn., 33, 2003, 1-10.
Gendelman, O. V. Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn., 2004, 37(2), 115-128.
Gourdon, E. and Lamarque, C. H. Nonlinear energy sink with uncertain parameters. J. Comput Nonlinear Dyn., 2006, 1(3), 187-195.
Musienko, A. I., Lamarque, C. H., and Manevitch, L. I. Design of mechanical energy pumping devices. J. Vibr. Control, 2006, 12(4), 355-371.
Vakakis, A. Designing a linear structure with a local nonlinear attachment for enhanced energy pumping. Meccanica, 2003, 38, 677-686.
Vakakis, A., Manevitch, L., Gendelman, O., and Bergman, L. Dynamics of linear discrete systems connected to local, essentially non-linear attachments. J. Sound Vibr., 2003, 264, 559-577.
Vakakis, A., McFarland, D. M., Bergman, L., Manevitch, L., and Gendelman, O. Isolated resonance captures and resonance capture cascades leading to single- or multi-mode passive energy pumping in damped coupled oscillators. Trans. ASME, J. Vibr. Acoust., 2004, 126, 235-244.
Manevitch, L., Gendelman, O., Musinko,A.,Vakakis, A., and Bergman, L. Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment. Phys. D, 2003, 178, 1-18.
Vakakis, A., Manevitch, L., Musienko, A., Kerschen, G., and Bergman, L. Transient dynamics of a dispersive elastic wave guide weakly coupled to an essentially nonlinear end attachment. Wave Motion, 2005, 41, 109-132.
Gendelman, O., Gorlov, D., Manevitch, L., and Musienko, A. Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses. J. Sound Vibr., 286, 2005, 1-19.
Lee, Y., Kerschen, G., Vakakis, A., Panagopoulos, P., Bergman, L., and McFarland, D. M. Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Phys. D, 2005, 204(1-2), 41-69.
Kerschen, G., Lee, Y., Vakakis, A., McFarland, D. M., and Bergman, L. Irreversible passive energy transfer in coupled oscillators with essential nonlinearity. SIAM J. Appl. Math., 2006, 66(2), 648-679.
Kerschen, G., Gendelman, O., Vakakis, A. F., Bergman, L. A., and McFarland, D. M. Impulsive periodic and quasi-periodic orbits of coupled oscillators with essential stiffness nonlinearity. Commun. Nonlinear Sci. Numer. Simul., 2008, 13(5), 959-978.
Manevitch, L., Gourdon, E., and Lamarque, C. Parameters optimization for energy pumping in strongly non-homogeneous 2 DOF system. Chaos Solitons Fractals, 2007, 31(4), 900-911.
Kerschen, G., Kowtko, J., McFarland, D. M., Bergman, L., and Vakakis, A. Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators. Nonlinear Dyn., 2007, 47(1), 285-309.
Tsakirtzis, S., Panagopoulos, P., Kerschen, G., Gendelman, O., Vakakis, A., and Bergman, L. Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments. Nonlinear Dyn., 2007, 48(3), 285-318.
Panagopoulos, P. N., Vakakis, A. E, and Tsakirtzis, S. Transient resonant interactions of finite linear chains with essentially nonlinear end attachments leading to passive energy pumping. Int. J. Solids Struct., 2004, 41(22-23), 6505-6528.
Vakakis, A. and Rand, R. Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities. Int. J. Non-Linear Mech., 2004, 39, 1079-1091.
Gourdon, E. and Lamarque, C. Energy pumping for a larger span of energy. J. Sound Vibr., 2005, 285(3), 711-720.
Georgiades, F., Vakakis, A. F., and Kerschen, G. Broad-band passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment. Int. J. Non-Linear Mech., 2007, 42(5), 773-788.
Panagopoulos, P., Georgiades, F., Tsaklrtzls, S., Vakakis, A. R, and Bergman, L. A. Multi-scaled analysis of the damped dynamics of an elastic rod with an essentially nonlinear end attachment. Int. J. Solids Struct., 2007, 44(1), 6256-6278.
Tsakirtzis, S., Vakakis, A. F., and Panagopoulos, P. Broadband energy exchanges between a dissipative elastic rod and a multi-degree-of-freedom dissipative essentially non-linear attachment. Int. J. Non-Linear Mech., 2007, 42(1), 36-57.
Huang, N., Shen, Z., Long, S., Wu, M., Shih, H., Zheng, Q., Yen, N.-C., Tung, C., and Liu, H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A., Math. Phys. Sci., 1998, 454, 903-995.
McFarland, D. M., Bergman, L., and Vakakis, A. Experimental study of non-linear energy pumping occurring at a single fast frequency. Int. J. Non-Linear Mech., 2005, 40, 891-899.
Kerschen, G., Vakakis, A., Lee, Y., McFarland, D. M., Kowtko, J., and Bergman, L. Energy transfers in a system of two coupled oscillators with essential nonlinearity: 1:1 resonance manifold and transient bridging orbits. Nonlinear Dyn., 2005, 42(3), 283-303.
Lee, Y., Kerschen, G., McFarland, D. M., Hill, W., Nichkawde, C., Strganac, T., Bergman, L., and Vakakis, A. Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 2: experiments. AIAA J., 2007, 45(10), 2391-2400.
McFarland, D. M., Kerschen, G., Kowtko, J., Lee, Y., Bergman, L., and Vakakis, A. Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators. J. Acoust Soc. Am., 2005, 118, 791-799.
Kerschen, G., McFaland, D. M., Kowtko, J., Lee, Y., Bergman, L., and Vakakis, A. Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity. J. Sound Vibr., 2007, 299, 822-838.
Kerschen, G., Vakakis, A., Lee, Y., McFarland, D. M., and Bergman, L. Toward a fundamental understanding of the Hilbert-Huang transform. In the International Modal Analysis Conference XXIV, St. Louis, Missouri, 30 January-2 February 2006.
Kerschen, G., Vakakis, A., Lee, Y., McFarland, D. M., and Bergman, L. Toward a fundamental understanding of the Hilbert-Huang transform in nonlinear structural dynamics. J. Vibr. Control, 2008, 14(1-2), 77-105.
Vakakis, A. Shock isolation through the use of nonlinear energy sinks. J. Vibr. Control, 2003, 9, 79-93.
Jiang, X. and Vakakis, A. R Dual mode vibration isolation based on non-linear mode localization. Int. J. Non-Linear Mech., 2003, 38(6), 837-850.
Georgiadis, F., Vakakis, A., McFarland, D. M., and Bergman, L. Shock isolation through passive energy pumping caused by nonsmooth nonlinearities. Int. J. Bifurcation Chaos, 2005, 15(6), 1989-2001.
Jiang, X., McFarland, D. M., Bergman, L., and Vakakis, A. Steady-state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dyn., 2003, 33, 87-102.
Gendelman, O. V. and Starosvetsky, Y. Quasi-periodic response regimes of linear oscillator coupled to nonlinear energy sink under periodic forcing. Trans. ASME, J. Appl. Mech., 2007, 74(2), 325-331.
Gendelman, O., Gourdon, E., and Lamarque, C. Quasiperiodic energy pumping in coupled oscillators under periodic forcing. J. Sound Vibr., 2006, 294(4-5), 651-662.
Gendelman, O., Starosvetsky, Y., and Feldman, M. Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: description of response regimes. Nonlinear Dyn., 2008, 51(1-2), 31-46.
Starosvetsky, Y. and Gendelman, O. Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: optimization of anonlinear vibration absorber. Nonlinear Dyn., 2008, 51(1-2), 47-57.
Gourdon, E., Alexander, N., Taylor, C., Lamarque, C., and Pernot, S. Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. J. Sound Vibr., 2007, 300(3-5), 522-551.
Lee, Y., Vakakis, A., Bergman, L., and McFarland, D. M. Suppression of limit cycle oscillations in the van der Pol oscillator by means of passive nonlinear energy sinks (NESs). Struct. Control Health Monit., 2006, 13(1), 41-75.
Lee, Y., Vakakis, A., Bergman, L., McFarland, D. M., and Kerschen, G. Triggering mechanisms of limit cycle oscillations in a two-degree-of-freedom wing flutter model. J. Fluids Struct., 2005, 21(5-7), 485-529.
Lee, Y., Vakakis, A., Bergman, L., McFarland, D. M., and Kerschen, G. Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 1: theory. AIAA J., 2007, 45(3), 693-711.
Lee, Y., Vakakis, A., Bergman, L., McFarland, D. M., and Kerschen, G. Enhancing robustness of aeroelastic instability suppression using multi-degree-of-freedom nonlinear energy sinks. AIAA J., 2008 (in press).
Nucera, F., Vakakis, A., McFarland, D. M., Bergman, L., and Kerschen, G. Targeted energy transfers in vibro-impact oscillators for seismic mitigation. Nonlinear Dyn., 2007, 50(3), 651-677.
Nucera, F., McFarland, D. M., Bergman, L., and Vakakis, A. Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: computational results. J. Sound Vibr., 2008 (in press).
Nucera, F., Iacono, F. L., McFarland, D. M., Bergman, L., and Vakakis, A. Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: experimental results. J. Sound Vibr., 2008, 313(1-2), 57-76.
Viguié, R., Kerschen, G., Golinval, J.-C., McFarland, D. M., Bergman, L., Vakakis, A., and van de Wouw, N. Using targeted energy transfer to stabilize drill-string systems. In the International Modal Analysis Conference XXV, Orlando, Florida, 19-22 February 2007.
Bellizzi, S., Cochelin, B., Herzog, P., and Mattel, P.-O. An insight of energy pumping in acoustic. In the Second International Conference on Nonlinear Normal Modes and Localization in Vibrating Systems, Samos, Greece, 19-23 June 2006.
Sanders, J. and Verhulst, R Averaging methods in non-linear dynamical systems, 1985 (Springer-Verlag, New York).
Keener, J. On the validity of the two-timing method for large times. SIAM J. M. Anal., 1977, 8, 1067-1091.
Sanders, J. Asymptotic approximations and extension of time-scales. SIAM. Math. Anal., 1980, 11, 758-770.
Bosley, D. and Kevorkian, J. Adiabatic invariance and transient resonance in very slowly varying oscillatory Hamiltonian systems. SIAM J. Appl. Math., 1992, 52, 494-527.
Lochak, P. and Meunier, C. Multiphase averaging for classical systems: with applications to adiabatic theorems, 1988 (Springer-Verlag, New York, Berlin, Heidelberg).
Kevorkian, J. Passage through resonance for a one-dimensional oscillator with slowly varying frequency. SIAM J. Appl. Math., 1971, 20, 364-373.
Neishtadt, A. Passage through a separatrix in a resonance problem with a slowly-varying parameter. Prikl. Mat. Mekh, 1975, 39, 621-632.
Sanders, J. On the passage through resonance. SIAM J. Math. Anal., 1979, 10, 1220-1243.
Nayfeh, A. and Mook, D. Nonlinear oscillations, 1979 (John Wiley & Sons, New York).
Bakhtin, V. Averaging in multifrequency systems. Funktsional'nyi Analiz i Ego Prilozheniya, 1986, 20, 1-7 (translated).
Dodson, M., Rynne, B., and Vickers, J. Averaging in multifrequency systems. Nonlinearity, 1989, 2, 137-148.
Nayfeh, A. Introduction to perturbation techniques, 1980 (John Wiley & Sons, New York).
Bourland, F., Haberman, R., and Kath, W. Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture. SIAM J. Appl. Math., 1991, 51, 1150-1167.
Nandakumar, K. and Chatterjee, A. The simplest resonance capture problem, using harmonic balance based averaging. Nonlinear Dyn., 2004, 37, 271-284.
Manevitch, L. The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear Dyn., 2001, 25, 95-109.
Pilipchuk, V. Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformation. J. Sound Vibr., 1996, 192, 43-64.
Pilipchuk, V., Vakakis, A., and Azeez, M. Sensitive dependence on initial conditions of strongly nonlinear periodic orbits of the forced pendulum. Nonlinear Dyn., 1998, 16, 223-237.
Pilipchuk, V. Application of special nonsmooth temporal transformations to linear and nonlinear systems under discontinuous and impulsive excitation. Nonlinear Dyn., 1999, 18, 203-234.
Pilipchuk, V. Non-smooth time decomposition for non-linear models driven by random pulses. Chaos Solitons Fractals, 2002, 14, 129-143.
Pilipchuk, V. Temporal transformations and visualization diagrams for nonsmooth periodic motions. Int. J. Bifurcation Chaos, 2005, 15(6), 1879-1899.
Roberts, S. and Shipman, J. Two-point boundary value problems: shooting methods, 1972 (American Elsevier Publishing Company, Inc., New York).
Meirovitch, L. Methods of analytical dynamics, 1970 (McGraw-Hill, Inc., New York).
Guckenheimer, J. and Holmes, P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, 1983 (Springer-Verlag, NewYork).
Kuznetsov, Y. Elements of applied bifurcation theory, 1995 (Springer-Verlag, NewYork).
Doedel, E., Champneys,A., Fairgrieve, T., Kuznetsov,Y., Sandstede, B., and Wang, X. AUTO97: continuation and bifurcation software for ordinary differential equations, 1997.
Dhooge, A., Govaerts, W., and Kuznetsov, Y. MATCONT: a Matlab Package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw., 2003, 29(2), 141-164.
Boashash, B. Time frequency signal analysis and processing, a comprehensive reference, 2003 (Elsevier Ltd, Amsterdam, Boston).
Laskar, J. Introduction to frequency map analysis. Hamiltonian systems with three or more degrees of freedom, NATO ASI Series C 533, 1999, pp. 134-150.
Laskar, J. Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Phys. D, 1993, 67, 257-281.
Chandre, C., Wiggins, S., and Uzer, T. Time-frequency analysis of chaotic systems. Phys. D, 2003, 181, 171-196.
Vela-Arevalo, L. and Marsden, J. Time-frequency analysis of the restricted three-body problem: transport and resonance transitions. Class. Quantum Gravity, 2004, 21, S351-S375.
Lind, R., Snyder, K., and Brenner, M. Wavelet analysis to characterise non-linearities and predict limit cycles of an aeroelastic system. Mech. Syst. Signal Process., 2001, 15, 337-356.
Feldman, M. Non-linear free vibration identification via the Hilbert transform. J. Sound Vibr., 1997, 208, 475-189.
Kim, I. and Kim, Y. Damage size estimation by the continuous wavelet ridge analysis of dispersive bending waves in a beam. J. Sound Vibr., 2005, 287, 707-722.
Staszewski, W. Identification of non-linear systems using multi-scale ridges and skeletons of the wavelet transform. J. Sound Vibr., 1998, 214, 639-658.
Huang, N., Shen, Z., and Long, S. A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech., 1999, 31, 417-457.
Huang, N., Wu, M.-L., Long, S. R., Shen, S., Qu, W., Gloersen, P., and Fan, K. A confidence limit for the empirical mode decomposition and Hilbert spectral analysis. Proc. R. Soc. London, A. Math. Phys. Sci., 2003, 459, 2317-2345.
Cheng, J., Yu, D., and Yang, Y. Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform. Mech. Syst. Signal Process., 2007, 21(3), 1197-1211.
Cheng, J., Yu, D., and Yang, Y. Research on the intrinsic mode function (IMF) criterion in EMD method. Mech. Syst. Signal Process., 2006, 20, 817-824.
Zhang, R., King, R., Olson, L., and Xu, Y.-L. Dynamic response of the trinity river relief bridge to controlled pile damage: modeling and experimental data analysis comparing Fourier and Hilbert-Huang techniques. J. Sound Vibr., 2005, 285, 1049-1070.
Cheng, J., Yu, D., and Yang, Y. Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings. Mech. Syst. Signal Process., 2005, 19, 259-270.
Rilling, G., Flandrin, P., and Gono̧alvès, P. On empirical mode decomposition and its algorithms. In the IEEE-Eurasip Workshop on Nonlinear Signal and Image Processing, Grado, Italy, June 2003.
Pilipchuk, V. The calculation of strongly nonlinear systems close to vibration-impact systems. Prikl. Mat. Mekh., 1985, 49, 572-578.
Schaaf, R. Global solution branches of two point boundary value problems, lecture notes in mathematics, vol. 1458, 1990 (Springer-Verlag, Heidelberg, NewYork).
Vakakis, A., Manevitch, L., Mikhlin, Y., Pilipchuk, V., and Zevin, A. Normal modes and localization in nonlinear systems, 1996 (John Wiley & Sons, Inc., NewYork).
Verhulst, E Nonlinear differential equations and dynamical systems, 2nd edition, 1990 (Springer-Verlag Inc., NewYork)
Salemi, P., Golnaraghi, M., and Heppler, G. Active control of forced and unforced structural vibration. J. Sound Vibr., 1997, 208, 15-32.
Masri, S. and Caughey, T. A nonparametric identification technique for nonlinear dynamic systems. Trans. ASME, J. Appl. Mech., 1979, 46, 433-141.
Kowtko, J. Experiments with nonlinear energy sinks: a novel approach to vibrational energy dissipation. Master's Thesis, University of Illinois at Urbana-Champaign, 2005.
