58 A. Kuttich et al. 0.05 0.1 0.15 0.2 0.25 0.3 4 100 200 300 K33 φ 0.1 0.15 8 14 (a) 0.05 0.1 0.15 0.2 0.25 0.3 2 2.3 2.6 K33 φ (b) Fig. 6.4 Minimal objective function value for different values of K33 nominal solution for nominal eigenfrequency (black solid line), nominal solution for worst-case eigenfrequency (red solid line), robust solution for worst-case eigenfrequency (blue solid line) 6.5 Conclusion We presented an optimization approach for the direct minimization of the H1-norm of the frequency transfer function of a single mass system with RL- and RLC-shunts. Furthermore, we introduced a robust optimization approach to deal with uncertainty in the structural eigenfrequency !sc. Our numerical results yield important information about the benefit of the robust optimization approach depending on the electromechanical coupling coefficient K33.Whereas K33 influences the benefit of the robust optimization approach for systems with RL-shunts, K33 has nearly no impact on the robust solution for systems with RLC-shunts. To sum up, the robust optimization is useful for systems with RL-shunts considering an K33 smaller than 0:15. For systems with RLC-shunts the effort of robust optimization seems to be not necessary and, hence, uncertainty in the structural eigenfrequency may be neglected. Acknowledgements The authors like to thank the German Research Foundation DFG for funding this research within the SFB 805. References 1. Beck, B.S.: Negative capacitance shunting of piezoelectric patches for vibration control of continuous systems. PhD thesis, Georgia Institute of Technology (2012) 2. Niederberger, D.: Smart damping materials using shunt control. PhD thesis, Swiss Federal Institute of Technology Zürich (2005) 3. Moheimani, S.O.R., Fleming, A.J.: Piezoelectric Transducers for Vibration Control and Damping. Springer, London (2006) 4. Hagood, N.W., von Flotow, A.H.: Damping of structural vibrations with piezoelectric materials and passive electrical networks. J. Sound Vib. 146(2), 243–268 (1991) 5. Neubauer, M., Oleskiewicz, R., Popp, K., Krzyzynski, T.: Optimization of damping and absorbing performance of shunted piezo elements utilizing negative capacitance. J. Sound Vib. 298, 84–107 (2006) 6. Soltani, P., Kerschen, G., Tondreau, G., Deraemaeker, A.: Piezoelectric vibration damping using resonant shunt circuits: an exact solution. Smart Mater. Struct. 23, 1–11 (2014) 7. Zambolini-Vicente, B.G.G.L., Silva, V.A.C., de Lima, A.M.G.: Robust design of shunt circuits for passive control of vibrations of composite structures. In: Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling (2014) 8. Mokrani, B., Burday, I., Tian, Z., Preumont, A.: Adaptive inductor for vibration damping in presence of uncertainty. In: Proceedings of SMART2015 7th ECCOMAS Thematic Conference on Smart Structures and Materials, Ponta Delgada, Azores, June 3–6 2015
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