Abstract
Harvesting energy from mechanical vibrations using piezoelectric materials presents itself as an interesting alternative energy source, particularly for embedded and integrated designs considering the high electric charge density that can be stored in these materials. To amplify the amount of energy available at narrow predefined frequency ranges, resonant cantilever devices are usually considered. Nevertheless, energy output is still small and highly sensitive to device parameters, mounting and operating conditions. Thus, these devices must be designed using optimization techniques, to ensure maximum extraction of energy available, and accounting for uncertainties in parameters, mounting and operating conditions. This work presents two methodologies to design cantilever piezoelectric energy harvesters using deterministic and robust optimization and accounting for the presence of uncertain parameters. The proposed methodology employs an electromechanical coupled finite element model to estimate mean and variance of harvestable power for given base excitation and parametric uncertainties. The electromechanical model is then used in two design methodologies, a robust design based on Taguchi’s method and a multiobjective deterministic Compromise Programming method. Both methods are shown to be capable of providing design solutions that allow maximization of nominal or mean harvesting performance and minimization of variability (increased robustness). As general design guidelines, it is shown that devices with larger mass lead to better mean performance but also to higher variability, thus a compromise solution is advisable. Also, a reduction of the effective harvesting circuit resistance, from nominally optimal value, may improve robustness without substantial decrease in mean performance.
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References
Ahmadian, H., Mottershead, J.E., Friswell, M.I.: Boundary condition identification by solving characteristic equations. J. Sound Vib. 247(5), 755–763 (2001)
Ali, S.F., Friswell, M.I., Adhikari, S.: Piezoelectric energy harvesting with parametric uncertainty. Smart Mat. Struct. 19(10), 105010 (2010)
Aloui, R., Larbi, W., Chouchane, M.: Global sensitivity analysis of piezoelectric energy harvesters. Compos. Struct. 228, 111317 (2019)
Ang, A.H.S., Tang, W.H.: Probability Concepts in Engineering Planning and Design, vol. 1. Wiley, Basic Principles (1975)
Beck, A.T., Gomes, W.J., Bazán, F.A.: On the robustness of structural risk optimization with respect to epistemic uncertainties. Int. J. Uncertain. Quantif. 2(1), 1–19 (2012)
Benasciutti, D., Moro, L., Zelenika, S., Brusa, E.: Vibration energy scavenging via piezoelectric bimorphs of optimized shapes. Microsyst. Technol. 16, 657–668 (2010)
Benjamin, J.R., Cornell, C.A.: Probability, Statistics, and Decision for Civil Engineers. Courier Corporation (2014)
Beyer, H.G., Sendhoff, B.: Robust optimization-a comprehensive survey. Comput. Methods Appl. Mech. Eng. 196(33–34), 3190–3218 (2007)
Carneiro, G.N., António, C.C.: Robustness and reliability of composite structures: effects of different sources of uncertainty. Int. J. Mech. Mater. Des. 15, 93–107 (2019)
Chattopadhyay, A., Seeley, C.E.: A simulated annealing technique for multiobjective optimization of intelligent structures. Smart Mater. Struct. 3, 98–106 (1994)
Chen, W., Wiecek, M.M., Zhang, J.: Quality utility-a compromise programming approach to robust design. J. Mech. Des. 121(2), 179–187 (1999)
Datta, R., Jain, A., Bhattacharya, B.: A piezoelectric model based multi-objective optimization of robot gripper design. Struct. Multidiscip. Optim. 53, 453–470 (2016)
Ducarne, J., Thomas, O., Deü, J.-F.: Placement and dimension optimization of shunted piezoelectric patches for vibration reduction. J. Sound Vib. 331, 3286–3303 (2012)
Dutoit, N.E., Wardle, B.L., Kim, S.G.: Design considerations for mems-scale piezoelectric mechanical vibration energy harvesters. Integr. Ferroelectr. 71(1), 121–160 (2005)
Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley (2011)
Franco, V.R., Varoto, P.S.: Parameter uncertainties in the design and optimization of cantilever piezoelectric energy harvesters. Mech. Syst. Sig. Process. 93, 593–609 (2017)
Franco Correia, V.M., Aguillar Madeira, J.F., Araújo, A.L., Mota Soares, C.M.: Multiobjective design optimization of laminated composite plates with piezoelectric layers. Compos. Struct. 169, 10–20 (2017)
Frecker, M.I.: Recent advances in optimization of smart structures and actuators. J. Intell. Mater. Syst. Struct. 14, 207–216 (2003)
Godoy, T.C., Trindade, M.A.: Effect of parametric uncertainties on the performance of a piezoelectric energy harvesting device. J. Braz. Soc. Mech. Sci. Eng. 34, 552–560 (2012)
Godoy, TC., Trindade, MA., Deü, J-F.: Topological optimization of piezoelectric energy harvesting devices for improved electromechanical efficiency and frequency range. In: Proceedings of 10th World Congress on Computational Mechanics (WCCM 2012), São Paulo, pp 4003–4016 (2014)
Hermansen, M.B., Thomsen, J.J.: Vibration-based estimation of beam boundary parameters. J. Sound Vib. 429(1), 287–304 (2018)
Hosseinloo, A.H., Turitsyn, K.: Design of vibratory energy harvesters under stochastic parametric uncertainty: a new optimization philosophy. Smart Mater. Struct. 25(5), 055023 (2016)
Joo, K.H., Min, D., Kim, J.G., Kang, Y.J.: New approach for identifying boundary characteristics using transmissibility. J. Sound Vib. 394(1), 109–129 (2017)
Kim, J., Lee, T.H., Song, Y., Sung, T.H.: Robust design optimization of fixed-fixed beam piezoelectric energy harvester considering manufacturing uncertainties. Sens. Actuat.: Phys. 260, 236–246 (2017)
Kim, M., Dugundji, J., Wardle, B.L.: Efficiency of piezoelectric mechanical vibration energy harvesting. Smart Mater. Struct. 24(5), 055006 (2015)
Lee, K.H., Park, G.J.: Robust optimization considering tolerances of design variables. Comput. Struct. 79(1), 77–86 (2001)
Leo, D.J.: Engineering Analysis of Smart Material Systems. Wiley (2007)
Lesieutre, G.A., Ottman, G.K., Hofmann, H.F.: Damping as a result of piezoelectric energy harvesting. J. Sound Vib. 269(3), 991–1001 (2004)
Liseli, J.L., Agnus, J., Lutz, P., Rakotondrabe, M.: Optimal design of piezoelectric cantilevered actuators for charge-based self-sensing applications. Sensors 19, 2582 (2019)
Lobato, FS., Steffen, Jr V.: Multi-Objective Optimization Problems: concepts and Self-Adaptive Parameters with Mathematical and Engineering Applications. Springer (2017)
Lopes, M.V., Eckert, J.J., Martins, T.S., dos Santos, A.A.: Multi-objective optimization of piezoelectric vibrational energy harvester orthogonal spirals for ore freight cars. J. Braz. Soc. Mech. Sci. Eng. 43, 295 (2021)
Lü, H., Yang, K., Huang, X., Yin, H.: Design optimization of hybrid uncertain structures with fuzzy-boundary interval variables. Int. J. Mech. Mater. Des. 17, 201–224 (2021)
Mann, B.P., Barton, D.A.W., Owens, B.A.M.: Uncertainty in performance for linear and nonlinear energy harvesting strategies. J. Intell. Mater. Syst. Struct. 23(13), 1451–1460 (2002)
Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26(6), 369–395 (2004)
McConnell, K.G., Varoto, P.S.: Vibration Testing: Theory and Practice, 2nd edn. Wiley (2008)
Mide Technology Material Properties of Piezoelectric Materials. https://support.piezo.com/article/62-material-properties. Accessed on 02 Dec 2020 (1989)
Mitcheson, P.D., Yeatman, E.M., Rao, G.K., Holmes, A.S., Green, T.C.: Energy harvesting from human and machine motion for wireless electronic devices. Proc. IEEE 96(9), 1457–1486 (2008)
Moreira, F.R.: Robust optimization multiobjective for engineering system design (in portuguese). PhD Thesis, Federal University of Uberlândia (2015)
Nabavi, S., Zhang, L.: Frequency tuning and efficiency improvement of piezoelectric MEMS vibration energy harvesters. J. Microelectromech. Syst. 28(1), 77–87 (2019)
Narita, F., Fox, M.: A review on piezoelectric, magnetostrictive, and magnetoelectric materials and device technologies for energy harvesting applications. Adv. Eng. Mater. 20(5), 1700743 (2018)
Pabst, U., Hagedorn, P.: Identification of boundary condition as part of model correction. J. Sound Vib. 182(4), 565–575 (1995)
Paiva, R.M.M., António, C.A.C., da Silva, L.F.M.: Multiobjective optimization of mechanical properties based on the composition of adhesives. Int. J. Mech. Mater. Des. 13, 1–24 (2017)
Park, G.J., Lee, T.H., Lee, K.H., Hwang, K.H.: Robust design: an overview. AIAA J. 44(1), 181–191 (2006)
Phadke, M.S.: Quality Engineering Using Robust Design. Prentice Hall PTR (1995)
Rafique, S.: Piezoelectric Vibration Energy Harvesting. Springer (2018)
Rao, S.S.: Engineering Optimization: Theory and Practice. Wiley (2009)
Ritto, T.G., Sampaio, R., Aguiar, R.R.: Boundary condition Bayesian identification from experimental data: a case study on a cantilever beam. Mech. Syst. Sig. Process. 68(69), 176–188 (2016)
Ritto, T.G., Sampaio, R., Cataldo, E.: Timoshenko beam with uncertainty on the boundary conditions. J. Braz. Soc. Mech. Sci. Eng. 30(4), 295–303 (2008)
Rupp, C.J., Evgrafov, A., Maute, K., Dunn, M.L.: Design of piezoelectric energy harvesting systems: a topology optimization approach based on multilayer plates and shells. J. Intell. Mater. Syst. Struct. 20, 1923–1939 (2009)
Salas, R.A., Ramírez, F.J., Montealegre-Rubio, W., Silva, E.C.N., Reddy, J.N.: A topology optimization formulation for transient design of multi-entry laminated piezocomposite energy harvesting devices coupled with electrical circuit. Int. J. Numer. Methods Eng. 113, 1370–1410 (2018)
Santos, H.F.L., Trindade, M.A.: Structural vibration control using extension and shear active-passive piezoelectric networks including sensitivity to electrical uncertainties. J. Braz. Soc. Mech. Sci. Eng. 33(3), 287–301 (2011)
Schuëller, G.I., Jensen, H.A.: Computational methods in optimization considering uncertainties-an overview. Comput. Methods Appl.Mech. Eng. 198(1), 2–13 (2008)
Seong, S., Hu, C., Lee, S.: Design under uncertainty for reliable power generation of piezoelectric energy harvester. J. Intell. Mater. Syst. Struct. 28(17), 2437–2449 (2017)
Sodano, H.A., Inman, D.J., Park, G.: A review of power harvesting from vibration using piezoelectric materials. Shock Vib. Dig. 36(3), 197–206 (2004)
Sudret, B., Marelli, S., Wiart, J.: Surrogate models for uncertainty quantification: An overview. In: 2017 11th European Conference on Antennas and Propagation (EUCAP), IEEE, pp. 793–797 (2017)
Tikani, R., Torfenezhad, L., Mousavi, M., Ziaei-Rad, S.: Optimization of spiral-shaped piezoelectric energy harvester using Taguchi method. J. Vib. Control 24(19), 4484–4491 (2018)
Trindade, M.A.: Optimization of active-passive damping treatments using piezoelectric and viscoelastic materials. Smart Mater. Struct. 16(6), 2159–2168 (2007)
Trindade, M.A., Benjeddou, A.: Refined sandwich model for the vibration of beams with embedded shear piezoelectric actuators and sensors. Comput. Struct. 86(5), 859–869 (2008)
Trindade, M.A., Benjeddou, A.: Effective electromechanical coupling coefficients of piezoelectric adaptive structures: critical evaluation and optimization. Mech. Adv. Mater. Struct. 16(3), 210–223 (2009)
Tsui, K.L.: An overview of Taguchi method and newly developed statistical methods for robust design. IIE Trans. 24(5), 44–57 (1992)
Zang, C., Friswell, M.I., Mottershead, J.: A review of robust optimal design and its application in dynamics. Comput. Struct. 83(4), 315–326 (2005)
Acknowledgements
This research was supported by CNPq, through research Grants 309193/2014-1, 134508/2015-7 and 309001/2018-8, and MCT/CNPq/FAPEMIG National Institute of Science and Technology on Smart Structures in Engineering, Grant 574001/2008-5, which the authors gratefully acknowledge. The first author also acknowledges CAPES for a doctoral scholarship.
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Martins, P.H., Trindade, M.A. & Varoto, P.S. Simplified robust and multiobjective optimization of piezoelectric energy harvesters with uncertain parameters. Int J Mech Mater Des 18, 63–85 (2022). https://doi.org/10.1007/s10999-021-09586-2
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DOI: https://doi.org/10.1007/s10999-021-09586-2