Abstract
The problem of optimal energy harvesting for a piezoelectric element driven by mechanical vibrations is stated in terms of an ODE system with hysteresis under the time derivative coupling a mechanical oscillator with an electric circuit with or without inductance. In the piezoelectric constitutive law, both the self-similar piezoelectric butterfly character of the hysteresis curves and feedback effects are taken into account in a thermodynamically consistent way. The physical parameters of the harvester are chosen to be the control variable, and the goal is to maximize the harvested energy for a given mechanical load and a given time interval. If hysteresis is modeled by the Preisach operator, the system is shown to be well-posed with continuous data dependence. For the special case of the play operator, we derive first-order necessary optimality conditions and an explicit form of the gradient of the total harvested energy functional in terms of solutions to the adjoint system.
Similar content being viewed by others
References
Abdelkefi, A., Nayfeh, A.H., Haj, M.R.: Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation. Nonlinear Dyn. 67(2), 1221–1232 (2012)
Al Janaideh, M., Krejčí, P.: Inverse rate-dependent Prandtl–Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator. IEEE-ASME Trans. Mechatron. 18, 1498–1507 (2013)
Al Janaideh, M., Rakheja, S., Su, C.-Y.: Experimental characterization and modeling of rate-dependent hysteresis of a piezoceramic actuator. Mechatronics 17, 656–670 (2009)
Brokate, M.: Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ. Verlag Peter D. Lang, Frankfurt am Main (1987)
Brokate, M.: ODE control problems including the Preisach hysteresis operator: necessary optimality conditions. In: Feichtinger, G. (ed.) Dynamic Economic Models and Optimal Control (Vienna, 1991). North-Holland, Amsterdam, pp. 51–68 (1991)
Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. DCDS B 18, 331–348 (2013)
Casas, E., Tröltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31, 695–712 (2002)
Dai, X., Wen, Y., Li, P., Yang, J., Jiang, X.: A vibration energy harvester using magnetostrictive/piezoelectric composite transducer. In: Proceedings of Sensors Conference. IEEE, pp. 1447–1450 (2009)
Davino, D., Giustiniani, A., Visone, C.: Magnetoelastic energy harvesting: modeling and experiments. In: Berselli, G., Vertechy, R., Vassura, G. (eds.) Smart Actuation and Sensing Systems—Recent Advances and Future Challenges, pp. 487–512. InTech, Rijeka (2012)
Davino, D., Krejčí, P., Visone, C.: Fully coupled modeling of magnetomechanical hysteresis through ‘thermodynamic’ compatibility. Smart Mater. Struct. 22, 095009 (2013)
Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, Hoboken (2011)
Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multi- dimensional play operator. SIAM J. Control Opt. 49, 788–807 (2011)
Kamlah, M.: Ferroelectric and ferroelastic piezoceramics modeling of electromechanical hysteresis phenomena. Contin. Mech. Thermodyn. 13, 219–268 (2001)
Kaltenbacher, B., Krejčí, P.: A thermodynamically consistent phenomenological model for ferroelectric and ferroelastic hysteresis. ZAMM Z. Angew. Math. Mech. 96, 874–891 (2016)
Krejčí, P.: Hysteresis, Convexity, and Dissipation in Hyperbolic Equations. Gakkōtosho, Tokyo (1996)
Krejčí, P., Monteiro, G.A.: Oscillations of a temperature-dependent piezoelectric rod. Nonlinear Anal. Real World Appl. (to appear)
Krejčí, P., Timoshin, S.A.: Coupled ODEs control system with unbounded hysteresis region. SIAM J. Control Optim. 54, 1934–1949 (2016)
Kuhnen, K.: Modeling, identification and compensation of complex hysteretic nonlinearities—a modified Prandtl–Ishlinskii approach. Eur. J. Control 9, 407–418 (2003)
Kuhnen, K., Krejčí, P.: Compensation of complex hysteresis and creep effects in piezoelectrically actuated systems: a new Preisach modeling approach. IEEE Trans. Autom. Control 54, 537–550 (2009)
Nayfeh, A.H.: Perturbation Methods. Wiley, Hoboken (2004)
Peigney, M., Siegert, D.: Piezoelectric energy harvesting from traffic-induced bridge vibrations. Smart Mater. Struct. 22, 095019 (2013)
Renno, J.M., Daqaq, M.F., Inman, D.J.: On the optimal energy harvesting from a vibration source. J. Sound Vib. 329, 386–405 (2009)
Rakotondrabe, M., Clevy, C., Lutz, P.: Hysteresis and vibration compensation in a nonlinear unimorph piezocantilever. In: Proc. IEEE Int. Conf. Intell. Robots Syst., Nice, France, pp. 558–563 (2008)
Roundy, S.: On the effectiveness of vibration-based energy harvesting. J. Intell. Mater. Syst. Struct. 16, 809–823 (2005)
Sprekels, J., Tiba, D.: Optimization of differential systems with hysteresis, In: Barbu, V., Lasiecka, I., Tiba, D., Varsan, C. (eds.) Analysis and Optimization of Differential Systems: IFIP TC7 / WG7.2 International Working Conference on Analysis and Optimization of Differential Systems, September 10–14, 2002, Constanta, Romania. Springer, Boston, MA, pp. 387–398 (2003)
Stanton, S., Erturk, A., Mann, B., Inman, D.: Nonlinear piezoelectricity in electroelastic energy harvesters: modeling and experimental identification. J. Appl. Phys. 108, 074903 (2010)
Szarka, G.D., Stark, B.H., Burrow, S.G.: Review of power conditioning for kinetic energy harvesting systems. IEEE Trans. Power Electron. 27, 803–815 (2012)
Stephen, N.G.: On energy harvesting from ambient vibration. J. Sound Vib. 293, 409–425 (2006)
Triplett, A., Quinn, D.: The effect of non-linear piezoelectric coupling on vibration-based energy harvesting. J. Intell. Mater. Syst. Struct. 20, 1959–1967 (2009)
Wang, L., Yuan, F.G.: Vibration energy harvesting by magnetostrictive material. Smart Mater. Struct. 17, 045009 (2008)
Xiang, H.J., Wang, J.J., Shi, Z.F., Zhang, Z.W.: Theoretical analysis of piezoelectric energy harvesting from traffic induced deformation of pavements. Smart Mater. Struct. 22, 095024 (2013)
Acknowledgements
The authors thank both reviewers for their valuable comments. Moreover, financial support by the GAČR Grant GA15-12227S, RVO: 67985840, and FWF Grant P30054, is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by the GAČR Grant GA15-12227S, RVO: 67985840, and FWF Grant P30054.
Rights and permissions
About this article
Cite this article
Kaltenbacher, B., Krejčí, P. Analysis of an optimization problem for a piezoelectric energy harvester. Arch Appl Mech 89, 1103–1122 (2019). https://doi.org/10.1007/s00419-018-1459-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-018-1459-6