Archives of Acoustics, 47, 4, pp. 555-564, 2022
10.24425/aoa.2022.142891

### Asymmetrical PZT Applied to Active Reduction of Asymmetrically Vibrating Beam – Semi-Analytical Solution

Rzeszow University of Technology
Poland

Romuald KURAS
Rzeszow University of Technology
Poland

The article extended the idea of active vibration reduction of beams with symmetric modes to beams with asymmetric modes. In the case of symmetric modes, the symmetric PZT (s-PZT) was used, and the optimization of the problem led to the location of the s-PZT centre at the point with the greatest beam curvature. In the latter case, the asymmetric modes that occur due to the addition of the point mass cause an asymmetric distribution of the bending moment and transversal displacement of a beam. In this case, the optimal approach to the active vibration reduction requires both new asymmetric PZT (a-PZT) and its new particular distribution on the beam. It has been mathematically determined that the a-PZT asymmetry point (a-point), ought to be placed at the point of maximum beam bending moment. The a-PZT asymmetry was found mathematically by minimizing the amplitude of the vibrations. As a result, it was possible to formulate the criterion of the maximum bending moment of the beam. The numerical calculations confirmed theoretical considerations. So, it was shown that in the case of asymmetric vibrations, the a-PZTs reduced vibrations more efficiently than the s-PZT.
Keywords: beam; actuator (PZT); active vibration reduction; vibration reduction coefficient; effectiveness coefficient.
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#### References

Augustyn E., Kozien M.S., Pracik M. (2014), FEM analysis of active reduction of torsional vibrations of clamped-free beam by piezoelectric elements for separated mode, Archives of Acoustics, 39(4): 639–644, doi: 10.2478/aoa-2014-0069.

Barboni R., Mannini A., Fantini E., Gaudenzi P. (2000), Optimal placement of PZT actuators for the control of beam dynamics, Smart Materials and Structures, 9: 110–120, doi: 10.1088/0964-1726/9/1/312.

Brand Z., Cole M.O.T. (2020), Controllability and actuator placement optimization for active damping of a thin rotating ring with piezo-patch transducers, Journal of Sound and Vibration, 472: 115172, doi: 10.1016/j.jsv.2020.115172.

Branski A. (2011), An optimal distribution of actuators in active beam vibration – some aspects, theoretical considerations, [in:] Acoustic Waves – From Microdevices to Helioseismology, Beghi M.G. [Ed.], pp. 397–418, IntechOpen, Rijeka.

Branski A. (2012), Modes orthogonality of mechanical system simple supported beam-actuators-concentrated masses, Acta Physica Polonica A, 121(1A): 126–131.

Branski A. (2013), Effectiveness analysis of the beam modes active vibration protection with different number of actuators, Acta Physica Polonica A, 123(6): 1123–1127, doi: 10.12693/APhysPolA.123.1123.

Branski A., Lipinski G. (2011), Analytical determination of the PZT’s distribution in active beam vibration protection problem, Acta Physica Polonica A, 119(6A): 936–941.

Dhuri K.D., Seshu P. (2006), Piezo actuator placement and sizing for good control effectiveness and minimal change in original system dynamics, Smart Materials and Structures, 15(6): 1661–1672, doi: 10.1088/0964-1726/15/6/019.

Fawade A.S., Fawade S.S. (2016), Modeling and analysis of vibration controlled cantilever beam bounded by PZT Patch, International Journal of Engineering Inventions, 5(7): 7–14.

Fichtenholtz G.M. (1999), Differential and Integral Calculus, PWN, Warsaw.

Filipek R., Wiciak J. (2008), Active and passive structural acoustic control of the smart beam, The European Physical Journal Special Topics, 154: 57–63, doi: 10.1140/epjst/e2008-00517-2.

Fuller C.R., Elliot S.J., Nielsen P.A. (1997), Active Control of Vibration, Academic Press, London.

Gosiewski Z., Koszewnik A. (2007), The influence of the piezoelements placement on the active vibration damping system, [in:] Proceedings of the 8th Conference on Active Noise and Vibration Control Methods, pp. 69–79, Kraków, Poland.

Gupta V., Sharma M., Thakur N. (2011), Mathematical modeling of actively controlled piezo smart structures: a review, Smart Structures and Systems, 8(3): 275–302, doi: 10.12989/sss.2011.8.3.275.

Hansen C.H., Snyder S.D. (1997), Active Control of Noise and Vibration, E&FN SPON, London.

Hu K.M., Li H. (2018), Multi-parameter optimization of piezoelectric actuators for multi-mode active vibration control of cylindrical shells, Journal of Sound and Vibration, 426: 166–185, doi: 10.1016/j.jsv.2018.04.021.

Kaliski S. (1986), Vibrations and Waves [in Polish], PWN, Warszawa.

Kasprzyk S., Wiciak M. (2007), Differential equation of transverse vibrations of a beam with a local stroke change of stiffness, Opuscula Mathematica, 27(2): 245–252.

Khasawneh F.A., Segalman D. (2019), Exact and numerically stable expressions for Euler-Bernoulli and Timoshenko beam modes, Applied Acoustics, 151: 215–228, doi: 10.1016/j.apacoust.2019.03.015.

Kozien M.S. (2013), Analytical solutions of excited vibrations of a beam with application of distribution, Acta Physica Polonica A, 123(6): 1029–1033, doi: 10.12693/APhysPolA.123.1029.

Le S. (2009), Active vibration control of a flexible beam, Master’s Thesis, Mechanical and Aerospace Engineering, San Jose State University, doi: 10.31979/etd.r8xgwaar.

Parameswaran A.P., Ananthakrishnan B., Gangadharan K.V. (2015), Design and development of a model free robust controller for active control of dominant flexural modes of vibrations in a smart system, Journal of Sound and Vibration, 355: 1–18, doi: 10.1016/j.jsv.2015.05.006.

Przybyłowicz P.M. (2002), Piezoelectric Vibration Control of Rotating Structures, Scientific works of the Warsaw University of Technology. Mechanics, Vol. 197, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa.

Quarteroni A. (2009), Numerical Models for Differential Problems, Springer-Verlag, Milan.

Yassin B., Lahcen A., Zeriab E. (2018), Hybrid optimization procedure applied to optimal location finding for piezoelectric actuators and sensors for active vibration control, Applied Mathematical Modelling, 62: 701–716, doi: 10.1016/j.apm.2018.06.017.

Zhang S.Q., Schmidt R. (2012), LQR control for vibration suppression of piezoelectric integrated smart structures, Proceedings in Applied Mathematics and Mechanics, 12(1): 695–696, doi: 10.1002/pamm.201210336.

Zhang X., Takezawa A., Kang Z. (2018), Topology optimization of piezoelectric smart structures for minimum energy consumption under active control, Structural and Multidisciplinary Optimization, 58: 185–199, doi: 10.1007/s00158-017-1886-y.

Zoric N.D., Tomovic A.M., Obradovic A.M., Radulovic R.D., Petrovic G.R. (2019), Active vibration control of smart composite plates using optimized self-tuning fuzzy logic controller with optimization of placement, sizing and orientation of PFRC actuators, Journal of Sound and Vibration, 456: 173–198, doi: 10.1016/j.jsv.2019.05.035.

DOI: 10.24425/aoa.2022.142891