Abstract
This paper deals with the cyclic-periodic structure with piezoelectric shunt circuits or a parallel piezoelectric network. The objective is to obtain the vibration suppression effect of them on the cylic-periodic structure. The background of the research is about vibration reduction of bladed disks in aero-engines, and the system is simulated by a lumped parameter model. The tuned and mistuned dynamic equations of the system are derived. The method of equivalent blisk model (EBM) is used to identify the lumped parameters of a finite element model, which is related to the experimental model. Then the Modified Modal Assurance Criterion (MMAC) is used to evaluate the vibration suppression ability of shunt circuits and the network. The numerical results show that both of these two systems can control vibration well, even the mechanical and electrical mistuning level are up to 10 %, and there is an optimal resistance which can be used to design the system to get a good performance of vibration suppression. An experiment is performed to validate the analytical results. The experimental results demonstrate that the ability of vibration suppression for both two systems are almost the same, but the piezo-network has an advantage in vibration delocalization of the mistuned system.
Nomenclature
-
-
damping of a sector
-
-
stiffness between blade and disk
-
-
coupling stiffness between sectors
-
-
stiffness between disk and ground
-
-
short-circuit stiffness of a piezoelectric material
-
-
blade mass of a sector
-
-
disk mass of a sector
-
-
non-dimensional electrical charge
-
-
Time
-
-
displacement of blade
-
-
displacement of disk
-
-
non-dimensional displacement of blade
-
-
non-dimensional displacement of disk
-
-
capacitance in circuit
-
-
intrinsic capacitance of a piezoelectric material
-
-
electrical charge
-
-
exciting force on blade
-
-
exciting force on disk
-
-
electrical current
-
-
mistuned variables
-
-
tuned variable
-
-
total number of sectors
-
-
non-dimensional excitation on blade
-
-
non-dimensional excitation on disk
-
-
resistance in circuit
-
-
voltage
-
-
non-dimensional coupling stiffness
-
-
non-dimensional disk stiffness
-
-
non-dimensional capacitance in circuit
-
-
non-dimensional intrinsic capacitance of a piezoelectric material
-
-
non-dimensional short-circuit stiffness of a piezoelectric material
-
-
stationary response
-
-
non-dimensional resistance in circuit
-
-
electromechanical coupling factor
-
-
non-dimensional frequency of excitation
-
-
mass ratio of disk to blade
-
-
mechanical damping ratio
-
-
modal damping of mistuned electromechanical system
-
-
modal damping of mistuned mechanical system
-
-
standard deviation
-
-
non-dimensional time
-
-
ratio of relative generalized force
-
-
ratio of relative modal damping
-
-
ratio of relative modal stiffness
-
-
ratio of relative infinity norm
-
-
frequency of excitation
-
-
reference frequency
-
-
natural frequency of mistuned electromechanical system
-
-
natural frequency of mistuned mechanical system
-
-
mode shape of mistuned electromechanical system
-
-
mode shape of mistuned mechanical system
References
1. Castanier MP, Pierre C Modeling and analysis of mistuned bladed disk vibration: status and emerging directions. J Propul Power. 2006, 22, 384–96.10.2514/1.16345Search in Google Scholar
2. Griffin JH A review of friction damping of turbine blade vibration, Int J Turbo Jet-Engines, 1990, 7, 297–308.10.1515/TJJ.1990.7.3-4.297Search in Google Scholar
3. Kosmatka JB, Mehmed O. Experimental spin testing of integrally damped composite blades. In: 39th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and material conferences. Newport (RI). Reston (VA): American Institute of Aeronautics and Astronautics, Inc, 1998.Search in Google Scholar
4. Hagood NW, Flotow AV. Damping of structural vibrations with piezoelectric materials and passive electrical networks. J Sound Vib. 1991;146:243–68.10.1016/0022-460X(91)90762-9Search in Google Scholar
5. Wu SY. Piezoelectric shunts with a parallel RL circuit for structural damping and vibration control. In: Proceedings of the SPIE 2720, smart structures and materials 1996: passive damping and isolation. San Diego, USA. California, 1996.10.1117/12.239093Search in Google Scholar
6. Hollkamp JJ. Multimodal passive vibration suppressing with piezoelectric materials and resonant shunts. J Intell Mater Syst Struct. 1994;5:49–57.10.1177/1045389X9400500106Search in Google Scholar
7. Wu SY. Method for multiple mode shunt damping of structural vibration using a single piezoelectric transducer. In: Proceedings of the SPIE 3327, smart structures and materials 1998: passive damping and isolation. San Diego, USA. California, 1998.10.1117/12.310680Search in Google Scholar
8. Wu SY. Piezoelectric shunts for simultaneous vibration reduction and damping of multiple vibration modes. United States patent No. 5783898, 1998.Search in Google Scholar
9. Wu SY, Bicos AS. Structural vibration damping experiments using improved piezoelectric shunts. In: Proceedings of the SPIE 3045, smart structures and materials 1997: passive damping and isolation. San Diego, USA. California, 1997.10.1117/12.274217Search in Google Scholar
10. Behrens S, Moheimani SOR. Current flowing multiple-mode piezoelectric shunt dampener. In: Proceedings of the SPIE 4697, smart structures and materials 2002: damping and isolation, San Diego, USA. California, 2002.10.1117/12.472658Search in Google Scholar
11. Kim SJ, Yun CY, Paek BJ. Optimal design of a piezoelectric passive damper for vibration plates. In: Proceedings of the SPIE 3989, smart structures and materials 2000: damping and isolation. San Diego, USA. California, 2000.10.1117/12.384590Search in Google Scholar
12. Moheimani SOR. A survey of recent innovations in vibration damping and control using shunted piezoelectric transducers. IEEE Trans Control Syst Technol. 2003;11:482–94.10.1109/TCST.2003.813371Search in Google Scholar
13. Zhang S, Schmidt R, Qin X. Active vibration control of piezoelectric bonded smart structures using PID algorithm. Chin J Aeronautics. 2015;28:305–13.10.1016/j.cja.2014.12.005Search in Google Scholar
14. Livet S, Berthillier M, Collet M, Cote JM. Numerical and experimental optimized shunted piezoelectric circuit for turbomachinery blades. In: 12th IFToMM World Congress. Besançon, France, 2007.Search in Google Scholar
15. Livet S, Collet M, Berthillier M, Jean P, Cote JM. Turbomachinery blades damping thanks to optimized shunted piezoelectric circuits. In: The 15th International Symposium on: Smart Structures and Materials & Nondestructive Evaluation and Health Monitoring. International Society for Optics and Photonics. 2008: 692812–692812–12.10.1117/12.776898Search in Google Scholar
16. Min JB, Duffy KP, Provenza AJ. Shunted piezoelectric vibration damping analysis including centrifugal loading effects. In: Proc., 51st AIAA/ASME/ASCE /AHS/ASC Structures, Structural Dynamics and Materials Conf.Orlando, Florida, 2010.10.2514/6.2010-2716Search in Google Scholar
17. Min JB, Duffy KP, Choi BB, Provenza AJ, Kray N. Numerical modeling methodology and experimental study for piezoelectric vibration damping control of rotating composite fan blades. Comput Struct. 2013;128:230–42.10.1016/j.compstruc.2013.06.001Search in Google Scholar
18. Kauffman JL, Lesieutre GA. Vibration reduction of turbomachinery bladed disks with changing dynamics using piezoelectric materials. AIAA Paper. 2011: 2011–2003.10.2514/6.2011-2003Search in Google Scholar
19. Zhou B, Thouverez F, Lenoir D Essentially nonlinear piezoelectric shunt circuits applied to mistuned bladed disks. J Sound Vib. 2014;333:2520–42.10.1016/j.jsv.2013.12.019Search in Google Scholar
20. Erturk A, Inman DJ. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. J Vibr Acoust Trans ASME. 2008;130:041002–041002.10.1115/1.2890402Search in Google Scholar
21. Jordi BP, Manel PV. Electromechanical model of a multi-layer piezoelectric cantilever. In: 7th International Conference on thermal, mechanical and multiphysics simulation and experiments in micro-electronics and micro-systems. Como, Italy, 2006.Search in Google Scholar
22. Anderson TA, Sexton DW. A vibration energy harvesting sensor platform for increased industrial efficiency. In: Proceedings of the SPIE 6174, smart structures and materials 2006: sensors and smart structures technologies for civil, mechanical, and aerospace systems. San Diego, USA. California, 2006.10.1117/12.659586Search in Google Scholar
23. Mokrani B, Bastaits R, Horodinca M, Romanescu I, Burda I, Viguié R, et al. Parallel piezoelectric shunt damping of rotationally periodic structures. Adv Mater Sci Eng. 2015. DOI:10.1155/2015/162782.Search in Google Scholar
24. Tang J, Wang KW. Vibration control of rotationally periodic structures using passive piezoelectric shunt networks and active compensation. J Vibr Acoust Trans ASME. 1999;121:379–90.10.1115/1.2893991Search in Google Scholar
25. Tang J, Wang KW. Active-passive hybrid piezoelectric networks for vibration control: comparison and improvement. Smart Mater Struct. 2001;10:794–806.10.1088/0964-1726/10/4/325Search in Google Scholar
26. Tang J, Wang KW. Vibration delocalization of nearly periodic structures using coupled piezoelectric networks. J Vibr Acoust Trans ASME. 2003;125:95–108.10.1115/IMECE2001/AD-23704Search in Google Scholar
27. Hongbiao Y, Wang KW, Jianhua Z. Piezoelectric networking with enhanced electromechanical coupling for vibration delocalization of mistuned periodic structures-theory and experiment. J Sound Vib. 2006;295:246–65.10.1016/j.jsv.2006.01.006Search in Google Scholar
28. Hongbiao Y, Wang KW. Piezoelectric networks for vibration suppression of mistuned bladed disk. J Vibr Acoust Trans ASME. 2007;129:559–66.10.1115/1.2775511Search in Google Scholar
29. Hongbiao Y, Wang KW. Vibration suppression of mistuned coupled-blade-disk systems using piezoelectric circuitry network. J Vibr Acoust Trans ASME. 2009;31:021008–021008–12.Search in Google Scholar
30. Li L, Fan Y. Vibration dissipation characteristics of symmetrical piezoelectric networks with passive branches. In: Proceedings of ASME Turbo Expo 2012: turbine technical conference and exposition, Copenhagen, Denmark, 2012.10.1115/GT2012-69208Search in Google Scholar
31. Li L, Deng P, Fan Y. Dynamic characteristics of a cyclic-periodic structure with a piezoelectric network. Chin J Aeronautics. 2015;28:1426–37.10.1016/j.cja.2015.08.014Search in Google Scholar
32. Nipkau J. Analysis of mistuned blisk vibrations using a surrogate lumped mass model with aerodynamic influences PhD-Thesis, 2011. https://opus4.kobv.de/opus4-btu/frontdoor/index/index/docId/2133.Search in Google Scholar
33. Freipp ML, Atalla MJ. Review of modal sensing and actuation techniques. Shock Vibration Digest. 2001;33:3–14.10.1177/058310240103300101Search in Google Scholar
34. Yang JS, Zhou HG, Hu YT, Jiang Q. Performance of a piezoelectric harvester in thickness-stretch mode of a plate. IEEE Trans Ultrason Ferroelectr Freq Control. 2005;52:1872–6.10.1109/TUFFC.2005.1561644Search in Google Scholar PubMed
35. Wang PY, Li L. Vibration of bi-periodic bladed disk In: ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, American Society of Mechanical Engineers, Düsseldorf, North Rhine-Westphalia, Germany, 2014: V07BT33A021-V07BT33A021–9.Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
