Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Lekomtsev S. V., Matveenko V. P., Senin A. N. Passive damping of vibrations of a cylindrical shell interacting with a flowing fluid. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 2, pp. 207-226. DOI: 10.18500/1816-9791-2023-23-2-207-226, EDN: XMXZRM

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 830)
Article type: 

Passive damping of vibrations of a cylindrical shell interacting with a flowing fluid

Lekomtsev Sergey Vladimirovich, Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences
Matveenko Valerii P., Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences
Senin Alexander Nikolaevich, Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences

The possibility of passive damping of vibrations of a thin-walled cylindrical shell interacting with a flowing fluid is evaluated. The mechanism is based on connecting the open piezoelectric ring fixed on the surface of the structure to an external shunt electric circuit consisting of series-connected resistance and inductance coil. Their optimal values were selected numerically using the developed finite-element algorithm. The proposed approach is based on solving a series of modal problems. It allows us to obtain higher damping ratios compared to those evaluated by the commonly used analytical expressions and leads to the smallest difference in the natural frequencies of the structure and the electric circuit. In modeling a spatial shell, its curvilinear surface is approximated by a set of flat segments. Each of them is supposed to comply with the relations of the theory of multilayer plates and the equations of linear theory of piezoelasticity written for the case of plane stress state. With this approach, in the vectors of electric field and electric induction it is possible to keep nonzero only such components that are normal to the electroded surface of the piezoelectric ring. The constitutive relations, describing the vortex-free dynamics of an ideal compressible fluid in the case of small perturbations, are formulated in terms of the perturbation velocity potential. The corresponding wave equation is written in the coordinate system associated with the structure and is transformed together with the impermeability condition and boundary conditions to a weak form using the Bubnov – Galerkin method. The paper analyzes the variation of the complex eigenvalues of an electromechanical system depending on the values of resistance and inductance of a series electric circuit. Different methods for calculating the optimal parameters of the system are compared. The frequency response curves demonstrating a decrease in the amplitude of forced harmonic vibrations at a given fluid flow velocity are obtained.

The study was supported by the grant of the Russian Scientific Foundation (project No. 18-71-10054).
  1. Hagood N. W., von Flotow A. H. Damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of Sound and Vibration, 1991, vol. 146, iss. 2, pp. 243–268. https://doi.org/10.1016/0022-460X(91)90762-9
  2. Park C. H., Inman D. J. Enhanced piezoelectric shunt design. Shock and Vibration, 2003, vol. 10, pp. 127–133. https://doi.org/10.1155/2003/863252
  3. Fleming A. J., Moheimani S. O. R. Control orientated synthesis of high-performance piezoelectric shunt impedances for structural vibration control. IEEE Transactions on Control Systems Technology, 2005, vol. 13, iss. 1, pp. 98–112. https://doi.org/10.1109/TCST.2004.838547
  4. Porfiri M., Maurini C., Pouget J.-P. Identification of electromechanical modal parameters of linear piezoelectric structures. Smart Materials and Structures, 2007, vol. 16, iss. 2, pp. 323–331. https://doi.org/10.1088/0964-1726/16/2/010
  5. Thomas O., Ducarne J., Deu J.-F. Performance of piezoelectric shunts for vibration reduction. Smart Materials and Structures, 2012, vol. 21, iss. 1, Art. 015008. https://doi.org/10.1088/0964-1726/21/1/015008
  6. Soltani P., Kerschen G., Tondreau G., Deraemaeker A. Piezoelectric vibration damping using resonant shunt circuits: An exact solution. Smart Materials and Structures, 2014, vol. 23, iss. 12, Art. 125014. https://doi.org/10.1088/0964-1726/23/12/125014
  7. Heuss O., Salloum R., Mayer D., Melz T. Tuning of a vibration absorber with shunted piezoelectric transducers. Archive of Applied Mechanics, 2016, vol. 86, pp. 1715–1732. https://doi.org/10.1007/s00419-014-0972-5
  8. Toftekær J. F., Benjeddou A., Høgsberg J. General numerical implementation of a new piezoelectric shunt tuning method based on the effective electromechanical coupling coefficient. Mechanics of Advanced Materials and Structures, 2020, vol. 27, iss. 22, pp. 1908–1922. https://doi.org/10.1080/15376494.2018.1549297
  9. Gripp J. A. B., Rade D. A. Vibration and noise control using shunted piezoelectric transducers: A review. Mechanical Systems and Signal Processing, 2018, vol. 112, pp. 359–383. https://doi.org/10.1016/j.ymssp.2018.04.041
  10. Presas A., Luo Y., Wang Z., Valentin D., Egusquiza M. A review of PZT patches applications in submerged systems. Sensors, 2018, vol. 18, iss. 7, 2251. https://doi.org/10.3390/s18072251
  11. Marakakis K., Tairidis G. K., Koutsianitis P., Stavroulakis G. E. Shunt piezoelectric systems for noise and vibration control: A review. Frontiers in Built Environment, 2019, vol. 5, 64. https://doi.org/10.3389/fbuil.2019.00064
  12. Moheimani S. O. R., Fleming A. J. Piezoelectric Transducers for Vibration Control and Damping. 1st ed. London, Springer, 2006. 287 p. https://doi.org/10.1007/1-84628-332-9
  13. Zhang J. M., Chang W., Varadan V. K., Varadan V. V. Passive underwater acoustic damping using shunted piezoelectric coatings. Smart Materials and Structures, 2001, vol. 10, iss. 2, pp. 414–420. https://doi.org/10.1088/0964-1726/10/2/404
  14. Larbi W., Deu J.-F., Ohayon R. Finite element formulation of smart piezoelectric composite plates coupled with acoustic fluid. Composite Structures, 2012, vol. 94, iss. 2, pp. 501–509. https://doi.org/10.1016/j.compstruct.2011.08.010
  15. Sun Y., Li Z., Huang A., Li Q. Semi-active control of piezoelectric coating’s underwater sound absorption by combining design of the shunt impedances. Journal of Sound and Vibration, 2015, vol. 355, pp. 19–38. https://doi.org/10.1016/j.jsv.2015.06.036
  16. Larbi W. Numerical modeling of sound and vibration reduction using viscoelastic materials and shunted piezoelectric patches. Computers & Structures, 2020, vol. 232, 105822. https://doi.org/10.1016/j.compstruc.2017.07.024
  17. Pernod L., Lossouarn B., Astolfi J.-A., Deu J.-F. Vibration damping of marine lifting surfaces with resonant piezoelectric shunts. Journal of Sound and Vibration, 2021, vol. 496, 115921. https://doi.org/10.1016/j.jsv.2020.115921
  18. Lekomtsev S. V., Oshmarin D. A., Sevodina N. V. An approach to the analysis of hydroelastic vibrations of electromechanical systems, based on the solution of the nonclassical eigenvalue problem. Mechanics of Advanced Materials and Structures, 2021, vol. 28, pp. 1957–1964. https://doi.org/10.1080/15376494.2020.1716120
  19. Iurlov M. A., Kamenskikh A. O., Lekomtsev S. V., Matveenko V. P. Passive suppression of resonance vibrations of a plate and parallel plates assembly, interacting with a fluid. International Journal of Structural Stability and Dynamics, 2022, vol. 22, iss. 9, 2250101. https://doi.org/10.1142/S0219455422501012
  20. Matveenko V. P., Iurlova N. A., Oshmarin D. A., Sevodina N. V., Iurlov M. A. An approach to determination of shunt circuits parameters for damping vibrations. International Journal of Smart and Nano Materials, 2018, vol. 9, iss. 2, pp. 135–149. https://doi.org/10.1080/19475411.2018.1461144
  21. Zienkiewicz O. C. Finite Element Method in Engineering Science. London, McGraw-Hill, 1971. 521 p.
  22. Reddy J. N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. 2nd ed. Boca Raton, CRC Press, 2004. 858 p. https://doi.org/10.1201/b12409
  23. ANSI/IEEE Std176-1987. IEEE Standard on Piezoelectricity. New York, IEEE, 1988. 66 p. https://doi.org/10.1109/IEEESTD.1988.79638
  24. Thomas O., Deu J.-F., Ducarne J. Vibrations of an elastic structure with shunted piezoelectric patches: Efficient finite element formulation and electromechanical coupling coefficients. International Journal for Numerical Methods in Engineering, 2009, vol. 80, iss. 2, pp. 235–268. https://doi.org/10.1002/nme.2632
  25. Moon S. H., Kim S. J. Active and passive suppressions of nonlinear panel flutter using finite element method. American Institute of Aeronautics and Astronautics Journal, 2001, vol. 39, iss. 11, pp. 2042–2050. https://doi.org/10.2514/2.1217
  26. Yao G., Li F.-M. The stability analysis and active control of a composite laminated open cylindrical shell in subsonic airflow. Journal of Intelligent Material Systems and Structures, 2014, vol. 25, iss. 3, pp. 259–270. https://doi.org/10.1177/1045389X13491020
  27. Benjeddou A., Deu J.-F., Letombe S. Free vibrations of simply-supported piezoelectric adaptive plates: An exact sandwich formulation. Thin-Walled Structures, 2002, vol. 40, iss. 7–8, pp. 573–593. https://doi.org/10.1016/S0263-8231(02)00013-7
  28. Sheng G. G., Wang X. Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells. Applied Mathematical Modelling, 2010, vol. 34, iss. 9, pp. 2630–2643. https://doi.org/10.1016/j.apm.2009.11.024
  29. Allik H., Hughes J. R. Finite element method for piezoelectric vibration. International Journal for Numerical Methods in Engineering, 1970, vol. 2, iss. 2, pp. 151–157. https://doi.org/10.1002/nme.1620020202
  30. Il’gamov M. A. Kolebaniya uprigikh obolochek, soderzhaschikh zhidkost’ i gaz [Vibrations of Elastic Shells Containing Liquid and Gas]. Moscow, Nauka, 1969. 182 p. (in Russian).
  31. Paıdoussis M. P. Fluid-structure Interactions: Slender Structures and Axial Flow. Vol. 2. 2nd ed. London, Elsevier Academic Press, 2016. 944 p. https://doi.org/10.1016/C2011-0-08058-4
  32. Bochkarev S. A., Lekomtsev S. V., Matveenko V. P. Hydroelastic stability of a rectangular plate interacting with a layer of ideal flowing fluid. Fluid Dynamics, 2016, vol. 51, iss. 6, pp. 821–833. https://doi.org/10.1134/S0015462816060132
  33. Zienkiewicz O. C., Taylor R. L. The Finite Element Method. Vol. 2. Solid Mechanics. 5th ed. Oxford, Boston, Butterworth-Heinemann, 2000. 459 p.
  34. Reddy J. N. An Introduction to Nonlinear Finite Element Analysis: With applications to heat transfer, fluid mechanics, and solid mechanics. 2nd ed. Oxford University Press, 2015. 768 p.
  35. Bochkarev S. A., Lekomtsev S. V., Matveenko V. P., Senin A. N. Hydroelastic stability of partially filled coaxial cylindrical shells. Acta Mechanica, 2019, vol. 230, iss. 11, pp. 3845–3860. https://doi.org/10.1007/s00707-019-02453-4
  36. Tisseur F., Meerbergen K. The quadratic eigenvalue problem. SIAM Review, 1988, vol. 43, iss. 2, pp. 235–286. https://doi.org/10.1137/S0036144500381988
  37. Lehoucq R. B., Sorensen D. C. Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM Journal on Matrix Analysis and Applications, 1996, vol. 17, iss. 4, pp. 789–821. https://doi.org/10.1137/S0895479895281484
  38. Larbi W., Deu J.-F., Ohayon R. Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid. International Journal for Numerical Methods in Engineering, 2007, vol. 71, iss. 12, pp. 1412–1435. https://doi.org/10.1002/nme.1987
  39. Lekomtsev S. V., Oshmarin D. A., Sevodina N. V. An approach to the analysis of hydroelastic vibrations of electromechanical systems, based on the solution of the nonclassical eigenvalue problem. Mechanics of Advanced Materials and Structures, 2021, vol. 28, iss. 19, pp. 1957–1964. https://doi.org/10.1080/15376494.2020.1716120
  40. Yurlova N. A., Sevodina N. V., Oshmarin D. A., Iurlov M. A. Algorithm for solving problems related to the natural vibrations of electro-viscoelastic structures with shunt circuits using ANSYS data. International Journal of Smart and Nano Materials, 2019, vol. 10, iss. 2. pp. 156–176. https://doi.org/10.1080/19475411.2018.1542356