Abstract
A control strategy for continuous, autonomous, linear mechanical systems, controlled via piezoelectric devices, and suffering Hopf bifurcations, triggered by positional nonconservative forces, is discussed. The strategy is based on the ‘principle of similarity’, proved in the literature to be successful in controlling externally excited systems. A continuous metamodel of Piezo-Electro-Mechanical system, loaded by position-dependent forces, is derived via the Extended Hamilton Principle. The similarity principle is introduced in the model, demanding for certain relations among mechanical and piezoelectric properties to be satisfied. A stability analysis is carried out via perturbation methods, also accounting for small deviations from similarity. It is shown that the similar control has always a detrimental effect in preventing the loss of stability of the mechanical systems and that this effect is robust under slight imperfections. As an example, a Generalized Beck beam is studied, internally and externally damped, under the simultaneous action of a follower and a dead load, for which the previous asymptotic results are corroborated by an exact analysis.
Similar content being viewed by others
References
Frahm, H.: Device for damping vibrations of bodies. US Patent 989958 (1911)
Den Hartog J.: Mechanical Vibrations. McGraw-Hill, New York (1956)
Viguié, R.: Tuning methodology of nonlinear vibration absorbers coupled to nonlinear mechanical systems. Ph.D. thesis, University of Liege (2010)
Yamaguchi H., Harnpornchai N.: Fundamental characteristics of multiple tuned mass dampers for suppressing harmonically forced oscillations. Earthq. Eng. Struct. 22(1), 51–62 (1993)
Abe M., Fujino Y.: Dynamic characterization of multiple tuned mass dampers and some design formulas. Earthq. Eng. Struct. 23(8), 813–835 (1994)
Kareem A., Kline S.: Performance of multiple mass dampers under random loading. J. Struct. Eng. 121(2), 348–361 (1995)
Rana R., Soong T.: Parametric study and simplified design of tuned mass dampers. Eng. Struct. 20(3), 193–204 (1998)
Ubertini F.: Prevention of suspension bridge flutter using multiple tuned mass dampers. Wind Struct. 13(3), 235 (2010)
Casalotti A., Arena A., Lacarbonara W.: Mitigation of post-flutter oscillations in suspension bridges by hysteretic tuned mass dampers. Eng. Struct. 69, 62–71 (2014)
Gendelman O., Gourdon E., Lamarque C.-H.: Quasiperiodic energy pumping in coupled oscillators under periodic forcing. J. Sound Vib. 294(4), 651–662 (2006)
Gourdon E., Alexander N., Taylor C., Lamarque C.-H., Pernot S.: Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. J. Sound Vib. 300(3), 522–551 (2007)
Vakakis A., Bergman L., Gendelman O., Gladwell G., Kerschen G., Lee Y., McFarland D.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, vol. 156. Springer, Netherlands (2009)
Luongo A., Zulli D.: Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm. Nonlinear Dyn. 70(3), 2049–2061 (2012)
Luongo, A., Zulli, D.: Aeroelastic instability analysis of NES-controlled systems via a mixed Multiple Scale/Harmonic Balance method. J. Vib. Control (2013). doi:10.1177/1077546313480542
Viguié R., Kerschen G.: Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology. J. Sound Vib. 326(3), 780–793 (2009)
Habib, G., Detroux, T., Viguié, R., Kerschen, G.: Nonlinear generalization of Den Hartog’s equal-peak method. Mech. Syst. Sign. Pr. 52–53, 17–28 (2015)
Alessandroni S., dell’Isola F., Porfiri M.: A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators. Int. J. Solids Struct. 39(20), 5295–5324 (2002)
Alessandroni S., Andreaus U., dell’Isola F., Porfiri M.: Piezo-electromechanical (PEM) Kirchhoff–Love plates. Eur. J. Mech. A-Solid. 23, 689–702 (2004)
Alessandroni S., Andreaus U., dell’Isola F., Porfiri M.: A passive electric controller for multimodal vibrations of thin plates. Comput. Struct. 83(15), 1236–1250 (2005)
Andreaus U., dell’Isola F., Porfiri M.: Piezoelectric passive distributed controllers for beam flexural vibrations. J. Vib. Control 10(5), 625–659 (2004)
dell’Isola F., Porfiri M., Vidoli S.: Piezo-electromechanical (PEM) structures: passive vibration control using distributed piezoelectric transducers. C.R. Mécanique 331, 69–76 (2003)
dell’Isola F., Santini E., Vigilante D.: Purely electrical damping of vibrations in arbitrary PEM plates: a mixed non-conforming FEM-Runge–Kutta time evolution analysis. Arch. Appl. Mech. 73(1–2), 26–48 (2003)
dell’Isola F., Maurini C., Porfiri M.: Passive damping of beam vibrations through distributed electric networks and piezoelectric transducers: prototype design and experimental validation. Smart Mater. Struct. 13(2), 299 (2004)
Maurini C., dell’Isola F., Del Vescovo D.: Comparison of piezoelectronic networks acting as distributed vibration absorbers. Mech. Syst. Sign. Pr. 18(5), 1243–1271 (2004)
Porfiri M., dell’Isola F., Frattale Mascioli F.: Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers. Int. J. Circ. Theor. Appl. 32(4), 167–198 (2004)
Rosi G., Pouget J., dell’Isola F.: Control of sound radiation and transmission by a piezoelectric plate with an optimized resistive electrode. Eur. J. Mech. A-Solid. 29(5), 859–870 (2010)
Abdel-Rohman M.: Design of tuned mass dampers for suppression of galloping in tall prismatic structures. J. Sound Vib. 171(3), 289–299 (1994)
Gattulli V., Di Fabio F., Luongo A.: One to one resonant double Hopf bifurcation in aeroelastic oscillators with tuned mass damper. J. Sound Vib. 262(2), 201–217 (2003)
Gattulli V., Di Fabio F., Luongo A.: Simple and double hopf bifurcations in aeroelastic oscillators with tuned mass dampers. J. Franklin Inst. 338(2-3), 187–201 (2001)
Pourzeynali S., Datta T.: Control of flutter of suspension bridge deck using TMD. Wind Struct. 5(5), 407–422 (2002)
Gattulli V., Di Fabio F., Luongo A.: Nonlinear tuned mass damper for self-excited oscillations. Wind Struct. 7(4), 251–264 (2004)
Abdel-Rohman M., Spencer B.: Control of wind-induced nonlinear oscillations in suspended cables. Nonlinear Dyn. 37(4), 341–355 (2004)
Bolotin V.: Nonconservative Problems of the Theory of Elastic Stability. Macmillan, New York (1963)
Wang Q., Quek S.: Enhancing flutter and buckling capacity of column by piezoelectric layers. Int. J. Solids Struct. 39(16), 4167–4180 (2002)
Wang Y., Wang Z., Zu L.: Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force. Arch. Appl. Mech. 83(4), 495–507 (2012)
dell’Isola F., Vidoli S.: Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping. Arch. Appl. Mech. 68(1), 1–19 (1998)
dell’Isola F., Henneke E., Porfiri M.: Synthesis of electrical networks interconnecting PZT actuators to damp mechanical vibrations. Int. J. Appl. Electromagn. 14(1), 417–424 (2001)
Andreaus, U., dell’Isola, F., Porfiri, M.: Multimodal vibration control by using piezoelectric transducers and passive circuits. In: Symposium on Electro-Magneto-Mechanics, pp. 307–317 (2003)
Germain P.: The method of virtual powers in continuum mechanics. Part two Microstruct. J. Appl. Math. 25(3), 556–575 (1973)
Luongo A., Zulli D.: Mathematical Models of Beams and Cables. Wiley, Hoboken (2013)
Luongo, A., Zulli, D.: A non-linear one-dimensional model of cross-deformable tubular beam. Int. J. Nonlinear Mech. (2014). doi:10.1016/j.ijnonlinmec.2014.03.008
Neff, P., Ghiba, I.-D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Therm. 1–43 (2013)
dell’Isola F., Madeo A., Placidi L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. Z. Angew. Math. Mech. 92(1), 52–71 (2011)
Carcaterra A., Akay A.: Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems. J. Acoust. Soc. Am. 121(4), 1971–1982 (2007)
Crandall S.: Dynamics of Mechanical and Electromechanical Systems. Mc Graw Hill Book Company, New York (1968)
IEEE: IEEE Standard on Piezoelectricity—IEEE Std 176–1987. Institute of Electrical and Electronic Engineers (1987)
Lanczos C.: Linear Differential Operators. Van Nostrand Reinhold, London (1961)
Luongo, A., D’Annibale, F.: Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces. Int. J. Mod. Phys. B 26(25), 1246015-1–1246015-6 (2012)
Luongo, A., D’Annibale, F.: Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads. Int. J. Nonlinear Mech. 55, 128–139 (2013)
Kirillov O., Seyranian A.: The effect of small internal and external damping on the stability of distributed non-conservative systems. J. Appl. Math. Mech. 69(4), 529–552 (2005)
Beck M.: Die knicklast des einseitig eingespannten, tangential gedrückten stabes. Z. Angew. Math. Phys. 3(3), 225–228 (1952)
Seyranian A., Mailybaev A.: Multiparameter stability theory with mechanical applications, vol. 13. World Scientific, Singapore (2003)
Luongo, A., D’Annibale, F.: On the destabilizing effect of damping on discrete and continuous circulatory systems. J. Sound Vib. 333(24), 6723–6741 (2014)
Luongo, A., D’Annibale, F.: A paradigmatic minimal system to explain the Ziegler paradox. Continuum Mech. Therm. (2014). doi:10.1007/s00161-014-0363-8
D’Annibale, F., Rosi, G., Luongo, A.: Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces. Meccanica (2014). doi:10.1007/s11012-014-0037-4
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
D’Annibale, F., Rosi, G. & Luongo, A. On the failure of the ‘Similar Piezoelectric Control’ in preventing loss of stability by nonconservative positional forces. Z. Angew. Math. Phys. 66, 1949–1968 (2015). https://doi.org/10.1007/s00033-014-0477-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0477-7