Abstract
The response of single and two-degree-of-freedom mechanical systems with elements exhibiting a hysteretic restoring force is studied under harmonic imposed motion to characterize the nonlinear dynamic properties of the system. The hysteretic behavior of the element is described by the Bouc–Wen model, which is simple but able to represent different hysteretic behaviors. Since for a given restoring force the nonlinear response is affected by the oscillation amplitude, frequency-response curves for various excitation levels are constructed. The curve of periodic solutions for increasing excitation amplitude has been investigated analyzing the evolution of the resonance frequencies. Furthermore, in addition to known behaviors, to some extent already observed for single-degree-of-freedom hysteretic oscillators, a richer class of solutions and bifurcations is found for a 2-DOF system. New branches of stable periodic oscillations are bifurcated where conditions of internal resonance are encountered by varying the excitation amplitude; these are characterized by the onset of novel modes with an oscillation shape significantly different than that of modes on the fundamental branch. Finally, comparison between systems close to and far from internal resonance allowed to enlighten the role of the nonlinear coupling in the vibration mitigation.
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Fotiu, P., Irschik, H., Ziegler, F.: Forced vibrations of an elasto-plastic and deteriorating beam. Acta Mech. 69(1–4), 193–203 (1987)
Vestroni, F., Noori, M.: Hysteresis in mechanical systems: modeling and dynamic response. Int. J. Nonlinear Mech. 37(8), 1261–1262 (2002)
Zhang, H., Wang, E., Min, F., Zhang, N., Su, C., Rakheja, S.: Nonlinear dynamics analysis of the semiactive suspension system with magneto-rheological damper. Shock Vib. 2015. Article ID 971731 (2015)
Ortin, J., Delaeyb, L.: Hysteresis in shape-memory alloys. Int. J. Nonlinear Mech. 37, 1275–1281 (2002)
Bernardini, D., Vestroni, F.: Non-isothermal oscillations of pseudoelastic devices. Int. J. Nonlinear Mech. 38(9), 1297–1313 (2003)
Laudani, A., Riganti, Fulginei F., Salvini, A.: Comparative analysis of Bouc–Wen and Jiles–Atherton models under symmetric excitations. Phys. B 435, 134–137 (2014)
Foliente, G.: Hysteresis modeling of wood joints and structural systems. J. Struct. Eng. 121(6), 1013–1022 (1995)
Bursi, O., Ceravolo, R., Reicher, S., Fragonara, L.: Identification of the hysteretic behaviour of a partial strength steel-concrete moment-resisting frame structure subject to pseudodynamic tests. Earthq. Eng. Struct. Dyn. 41, 1883–1903 (2012)
Yan, C., Yang, D., Ma, Z.J., Jia, J.: Hysteretic model of SRUHSC column and SRC beam joints considering damage effects. Mater. Struct. 50, 88 (2017)
Manzoori, A., Toopchi-Nezhad, H.: Application of an extended Bouc–Wen model in seismic response prediction of unbonded fiber-reinforced isolators. J. Earthq. Eng. 2016.1138166 (2016). https://doi.org/10.1080/13632469
Lamarque, C.-H., Turi Savadkoohi, A.T.: Dynamical behavior of a Bouc–Wen type oscillator coupled to a nonlinear energy sink. Meccanica 49, 1917–1928 (2014)
Laxalde, D., Thouverez, F., Sinou, J.-J.: Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber. Int. J. Nonlinear Mech. 41, 969–978 (2006)
Silva, L.L., Savi, M.A., Monteiro, P.C., Netto, T.A.: Effect of the piezoelectric hysteretic behavior on the vibration-based energy harvesting. J. Intell. Mater. Syst. Struct. 24(10), 1278–1285 (2013)
Domaneschi, M.: Simulation of controlled hysteresis by the semi-active Bouc–Wen model. Comput. Struct. 106(107), 245–257 (2012)
Carpineto, N., Lacarbonara, W., Vestroni, F.: Hysteretic tuned mass dampers for structural vibration mitigation. J. Sound Vib. 333, 1302–1318 (2014)
Carboni, B., Lacarbonara, W.: Nonlinear dynamic characterization of a new hysteretic device: experiments and computations. Nonlinear Dyn. 83, 23–39 (2016)
Ceravolo, R., Erlicher, S., Fragonara, L.Z.: Comparison of restoring force models for the identification of structures with hysteresis and degradation. J. Sound Vib. 332, 6982–6999 (2013)
Hassani, V., Tjahjowidodo, T., Do, T.N.: A survey on hysteresis modeling, identification and control. Mech. Syst. Signal Process. 49, 209–233 (2014)
Ma, F., Zhang, H., Bockstedte, A., Foliente, G.C., Paevere, P.: Parameter analysis of the differential model of hysteresis. J. Appl. Mech. 71, 342–349 (2004)
Awrejcewicz, J.: Hysteresis modelling and chaos prediction in one and two-DOF hysteretic models. Arch. Appl. Mech. 77, 261–279 (2007)
Awrejcewicz, J., Dzyubak, L., Lamarque, C.H.: Modelling of hysteresis using Masing–Bouc–Wen’s framework and search of conditions for the chaotic responses. Commun. Nonlinear Sci. Numer. Simul. 13, 939–958 (2008)
Ismail, M., Ikhouane, F., Rodellar, J.: The hysteresis Bouc–Wen model, a survey. Arch. Comput. Methods Eng. 16, 161–188 (2009)
Bouc, R.: Forced vibrations of mechanical systems with hysteresis. In: Proceedings of the 4th Conference on Nonlinear Oscillations, Prague (1967)
Wen, Y.K.: Method of random vibration of hysteretic systems. ASCE J. Eng. Mech. 102(2), 249–263 (1976)
Baber, T., Wen, Y.: Random vibration hysteretic, degrading systems. J. Eng. Mech. 107(6), 1069–1087 (1981)
Baber, T., Noori, M.: Random vibration of degrading, pinching systems. J. Eng. Mech. 111(8), 1010–1026 (1985)
Wong, C.W., Ni, Y.Q., Lau, S.L.: Steady-state oscillation of hysteretic differential model. I: response analysis. ASCE J. Eng. Mech. 120, 2271–2298 (1994)
Ikhouane, F., Mañosa, V., Rodellar, J.: Dynamic properties of the hysteretic Bouc–Wen model. Syst. Control Lett. 56, 197–205 (2007)
Erlicher, S., Bursi, O.S.: Bouc–Wen type models with stiffness degradation: thermodynamic analysis and applications. ASCE J. Eng. Mech. 134(10), 843–855 (2008)
Capecchi, D., Vestroni, F.: Periodic response of a class of hysteretic oscillators. Int. J. Nonlinear Mech. 25(2), 309–317 (1990)
Lacarbonara, W., Vestroni, F.: Nonclassical responses of oscillators with hysteresis. Nonlinear Dyn. 32(3), 235–258 (2003)
Li, H., Meng, G.: Nonlinear dynamics of a SDOF oscillator with Bouc–Wen hysteresis. Chaos Solitons Fractals 34, 337–343 (2007)
Capecchi, D., Masiani, R.: Reduced phase space analysis for hysteretic oscillators of Masing type. Chaos Solitons Fractals 7(10), 1583–1600 (1996)
Casini, P., Vestroni, F.: Characterization of bifurcating nonlinear normal modes in piecewise linear mechanical systems. Int. J. Nonlinear Mech. 46, 142–150 (2011)
Giannini, O., Casini, P., Vestroni, F.: Experimental evidence of bifurcating NNMs in piecewise linear systems. Nonlinear Dyn. 63, 655–666 (2011)
Casini, P., Giannini, O., Vestroni, F.: Persistent and ghost nonlinear normal modes in the forced response of non-smooth systems. Physica D Nonlinear Phenom. 241, 2058–2067 (2012)
Capecchi, D., Vestroni, F.: Asymptotic response of a two DOF elastoplastic system under harmonic excitation. Internal resonance case. Nonlinear Dyn. 7, 317–333 (1995)
Masiani, R., Capecchi, D., Vestroni, F.: Resonant and coupled response of hysteretic two-degree-of-freedom systems using harmonic balance method. Int. J. Nonlinear Mech. 37, 1421–1434 (2002)
Zhang, Y., Iwan, W.D.: Some observations on two piecewise-linear dynamic systems with induced hysteretic damping. Int. J. Nonlinear Mech. 38(5), 753–765 (2003)
Vakakis, A.F.: Non-linear normal modes (NNMs) and their applications in vibration theory: an overview. Mech. Syst. Signal Process. 11(1), 3–22 (1997)
Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, London (2008)
Vestroni, F., Luongo, A., Paolone, A.: A perturbation method for evaluating nonlinear normal modes of a piecewise linear two-degrees-of-freedom system. Nonlinear Dyn. 54(4), 379–393 (2008)
Pak, C.H.: On the coupling of non-linear normal modes. Int. J. Nonlinear Mech. 41, 716–725 (2006)
Casini, P., Giannini, O., Vestroni, F.: Effect of damping on the nonlinear modal characteristics of a piecewice-smooth system through harmonic forced response. Mech. Syst. Signal Process. 36, 540–548 (2013)
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This work has been partially supported by the MIUR (Ministry of Education, University and Research) under the Project PRIN 2015-2018, “Identification and Monitoring of Complex Structural Systems”.
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This paper is dedicated to the memory of Franz Ziegler.
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Casini, P., Vestroni, F. Nonlinear resonances of hysteretic oscillators. Acta Mech 229, 939–952 (2018). https://doi.org/10.1007/s00707-017-2039-5
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DOI: https://doi.org/10.1007/s00707-017-2039-5