Abstract
The dynamics of a simple model for an ocean wave energy converter is discussed. The model for the converter is a hybrid system consisting of a pair of harmonically excited mass–spring–dashpot systems and a set of four state-dependent switching rules. Of particular interest is the response of the model to a wide spectrum of harmonic excitations. Partially because of the piecewise-smooth dynamics of the system, the response is far more interesting than the linear components of the model would suggest. As expected with hybrid systems of this type, it is difficult to establish analytical results, and hence, with the assistance of an extensive series of numerical integrations, an atlas of qualitative results on the limit cycles and other forms of bounded oscillations exhibited by the system is presented. In addition, the presence of unstable limit cycles, the stabilization of the unforced system using low-frequency excitation, the peculiar nature of the response of the system to high-frequency excitation, and the implications of these results on the energy harvesting capabilities of the wave energy converter are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Olivar Tost, G., Piiroinen, P.T.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50(4), 629–701 (2008). doi:10.1137/050625060
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, London (2008)
Flieller, D., Riedinger, P., Louis, J.P.: Computation and stability of limit cycles in hybrid systems. Nonlinear Anal., Theory Methods Appl. 64(2), 352–367 (2006). doi:10.1016/j.na.2005.06.054
Goebel, R., Sanfelice, R., Teel, A.: Hybrid dynamical systems. IEEE Control Syst. Mag. 29(2), 28–93 (2009). doi:10.1109/MCS.2008.931718
Gonçalves, J.M.: Regions of stability for limit cycle oscillations in piecewise linear systems. IEEE Trans. Autom. Control 50(11), 1877–1882 (2005). doi:10.1109/TAC.2005.858674
Halse, C.K., Wilson, R.E., di Bernardo, M., Homer, M.E.: Coexisting solutions and bifurcations in mechanical oscillators with backlash. J. Sound Vib. 305(4–5), 854–885 (2007). doi:10.1016/j.jsv.2007.05.010
Hiskens, I.A.: Stability of hybrid system limit cycles: application to the compass gait biped robot. In: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 1, pp. 774–779 (2001). doi:10.1109/.2001.980200
Hiskens, I.A., Reddy, P.B.: Switching-induced stable limit cycles. Nonlinear Dyn. 50(3), 575–585 (2007). doi:10.1007/s11071-006-9175-0
Kessler, P., O’Reilly, O.M.: The ringing of Euler’s disk. Regul. Chaotic Dyn. 7(1), 49–60 (2002). doi:10.1070/RD2002v007n01ABEH000195
Kinkaid, N.M., O’Reilly, O.M., Papadopoulos, P.: On the transient dynamics of a multi-degree-of-freedom friction oscillator: A new mechanism for disc brake noise. J. Sound Vib. 287(4–5), 901–917 (2005). doi:10.1016/j.jsv.2004.12.005
Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. In: Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer, Berlin (2004)
Liberzon, D.: Switching in Systems and Control. Birkhäuser, Boston (2003)
Liberzon, D., Morse, S.A.: Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19, 59–70 (1999). doi:10.1109/37.793443
Orazov, B., O’Reilly, O.M., Savaş, O.: On the dynamics of a novel ocean wave energy converter. J. Sound Vib. 329(24), 5058–5069 (2010). doi:10.1016/j.jsv.2010.07.007
Rubensson, M., Lennartson, B.: Stability of limit cycles in hybrid systems using discrete-time Lyapunov techniques. In: Proceedings of the 39th IEEE Conference on Decision and Control, vol. 2, pp. 1397–1402 (2000). doi:10.1109/CDC.2000.912053
Rubensson, M., Lennartson, B.: Global convergence analysis for piecewise linear systems applied to limit cycles in a DC/AC converter. In: Proceedings of the American Control Conference, vol. 2, pp. 1272–1277 (2002). doi:10.1109/ACC.2002.1023195
van der Schaft, A.J., Schumacher, J.M.: An Introduction to Hybrid Dynamical Systems. In: Lecture Notes in Control and Information Sciences, vol. 251. Springer, London (2000). doi:10.1007/BFb0109998
Thota, P., Dankowicz, H.: TC-HAT \((\widehat{\mathrm{TC}})\): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems. SIAM J. Appl. Dyn. Syst. 7(4), 1283–1322 (2008). doi:10.1137/070703028
Wirkus, S., Rand, R., Ruina, A.: How to pump a swing. Coll. Math. J. 29(4), 266–275 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Orazov, B., O’Reilly, O.M. & Zhou, X. On forced oscillations of a simple model for a novel wave energy converter. Nonlinear Dyn 67, 1135–1146 (2012). https://doi.org/10.1007/s11071-011-0058-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0058-7