Journal of Applied and Computational Mechanics
Improving the linear stability of the visco-elastic Beck's beam via piezoelectric controllers
Document Type : Special Issue Paper
Authors
1 Department of Civil, Construction-Architectural and Environmental Engineering, University of L’Aquila, 67100 L’Aquila, Italy
2 International Research Center on Mathematics and Mechanics of Complex Systems, University of L’Aquila, 67100, L’Aquila, Italy
Abstract
Control strategies for the visco-elastic Beck's beam, equipped with distributed piezoelectric devices and suffering from Hopf bifurcation triggered by a follower force, are proposed in this paper. The equations of motion of the Piezo-Electro-Mechanical (PEM) system are derived through the Extended Hamilton Principle, under the assumption that the piezoelectric patches are shunted to the so-called zero-order network and zero-order analog electrical circuit. An exact solution for the eigenvalue problem is worked out for the PEM system, while an asymptotic analysis is carried out to define three control strategies, recently developed for discrete PEM systems, that are here adapted to improve the linear stability of the visco-elastic Beck's beam. An extensive parametric study on the piezo-electrical quantities, based on an exact linear stability analysis of the PEM system, is then performed to investigate the effectiveness of the controllers.
Keywords
- Piezo-Electro-Mechanical systems
- Vibration Control
- Linear Stability Analysis
- Hopf bifurcation
- Beck’s beam
Main Subjects
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[28] 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 Materials and Structures, 2004, 13(2), 299.
[29] Porfiri, M., dell’Isola, F., Frattale Mascioli, F., Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers, International Journal of Circuit Theory and Applications, 2004, 32(4), 167–198.
[30] Rosi, G., Control of sound radiation and transmission by means of passive piezoelectric networks: modelling, optimization and experimental implementation, Ph.D. thesis, Sapienza University of Rome, University of Paris 6, 2010.
[31] Maurini, C., dell’Isola, F., Del Vescovo, D., Comparison of piezoelectronic networks acting as distributed vibration absorbers, Mechanical Systems and Signal Processing, 2004, 18(5), 1243–1271.
[32] Bolotin, V.V., Nonconservative problems of the theory of elastic stability, Macmillan, New York, 1963.
[33] Beck, M., Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes, Zeitschrift für angewandte Mathematik und Physik ZAMP, 1952, 3(3), 225–228.
[34] Ziegler, H., Die stabilitätskriterien der elastomechanik, Ingenieur Archiv, 1952, 20(1), 49–56.
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[36] Kirillov, O.N., Nonconservative stability problems of modern physics, Walter de Gruyter, Berlin/Boston, 2013.
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[38] Kirillov, O., Seyranian, A., The effect of small internal and external damping on the stability of distributed non-conservative systems, Journal of Applied Mathematics and Mechanics, 2005, 69(4), 529–552.
[39] Luongo, A., D’Annibale, F., A paradigmatic minimal system to explain the Ziegler paradox, Continuum Mechanics and Thermodynamics, 2015, 27(1-2), 211–222.
[40] Luongo, A., D’Annibale, F., On the destabilizing effect of damping on discrete and continuous circulatory systems, Journal of Sound and Vibration, 2014, 333(24), 6723–6741.
[41] D’Annibale, F., Ferretti, M., Luongo, A., Improving the linear stability of the beck’s beam by added dashpots, International Journal of Mechanical Sciences, 2016, 110, 151–159.
[42] Wang, Q., Quek, S.T., Enhancing flutter and buckling capacity of column by piezoelectric layers, International Journal of Solids and Structures, 2002, 39(16), 4167–4180.
[43] Wang, Y., Wang, Z., Zu, L., Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force, Archive of Applied Mechanics, 2012, 83(4), 495–507.
[44] D’Annibale, F., Rosi, G., Luongo, A., Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces, Meccanica, 2015, 50(3), 825–839.
[45] D’Annibale, F., Rosi, G., Luongo, A., Controlling the limit-cycle of the ziegler column via a tuned piezoelectric damper, Mathematical Problems in Engineering, 2015, 2015.
[46] D’Annibale, F., Rosi, G., Luongo, A., Piezoelectric control of hopf bifurcations: A non-linear discrete case study, International Journal of Non-Linear Mechanics, 2016, 80, 160–169.
[47] D’Annibale, F., Piezoelectric control of the hopf bifurcation of ziegler’s column with nonlinear damping, Nonlinear Dynamics, 2016, 86(4), 2179–2192.
[48] Luongo, A., D’Annibale, F., Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads, International Journal of Non-Linear Mechanics, 2013, 1–12.
[49] Luongo, A., D’Annibale, F., Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic beck’s beam, Mathematics and Mechanics of Solids, 2017, 22(6), 1347–1365.
[2] Den Hartog, J.P., Mechanical vibrations, Courier Corporation, 1985.
[3] Yamaguchi, H., Harnpornchai, N., Fundamental characteristics of multiple tuned mass dampers for suppressing harmonically forced oscillations, Earthquake engineering & structural dynamics, 1993, 22(1), 51–62.
[4] Abé, M., Fujino, Y., Dynamic characterization of multiple tuned mass dampers and some design formulas, Earthquake engineering & structural dynamics, 1994, 23(8), 813–835.
[5] Kareem, A., Kline, S., Performance of multiple mass dampers under random loading, Journal of structural engineering, 1995, 121(2), 348–361.
[6] Rana, R., Soong, T., Parametric study and simplified design of tuned mass dampers, Engineering structures, 1998, 20(3), 193–204.
[7] Gattulli, V., Di Fabio, F., Luongo, A., Simple and double hopf bifurcations in aeroelastic oscillators with tuned mass dampers, Journal of the Franklin Institute, 2001, 338(2-3), 187–201.
[8] Gattulli, V., Di Fabio, F., Luongo, A., One to one resonant double Hopf bifurcation in aeroelastic oscillators with tuned mass damper, Journal of Sound and Vibration, 2003, 262(2), 201–217.
[9] Gattulli, V., Di Fabio, F., Luongo, A., Nonlinear tuned mass damper for self-excited oscillations, Wind and Structures, 2004, 7(4), 251–264.
[10] Ubertini, F., Prevention of suspension bridge flutter using multiple tuned mass dampers, Wind and Structures, 2010, 13(3), 235–256.
[11] Viguié, R., Tuning methodology of nonlinear vibration absorbers coupled to nonlinear mechanical systems, Ph.D. thesis, PhD Thesis, 2010.
[12] Casalotti, A., Arena, A., Lacarbonara, W., Mitigation of post-flutter oscillations in suspension bridges by hysteretic tuned mass dampers, Engineering Structures, 2014, 69, 62–71.
[13] Gendelman, O.V., Gourdon, E., Lamarque, C.H., Quasiperiodic energy pumping in coupled oscillators under periodic forcing, Journal of Sound and Vibration, 2006, 294(4-5), 651–662.
[14] Gourdon, E., Alexander, N.A., Taylor, C.A., Lamarque, C.H., Pernot, S., Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results, Journal of sound and vibration, 2007, 300(3-5), 522–551.
[15] Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S., Nonlinear targeted energy transfer in mechanical and structural systems, vol. 156, Springer Science & Business Media, 2008.
[16] Luongo, A., Zulli, D., Dynamic analysis of externally excited nes-controlled systems via a mixed multiple scale/harmonic balance algorithm, Nonlinear Dynamics, 2012, 70(3), 2049–2061.
[17] Luongo, A., Zulli, D., Aeroelastic instability analysis of nes-controlled systems via a mixed multiple scale/harmonic balance method, Journal of Vibration and Control, 2014, 20(13), 1985–1998.
[18] D’Annibale, F., Rosi, G., Luongo, A., On the failure of the ’similar piezoelectric control’ in preventing loss of stability by nonconservative positional forces, Zeitschrift für angewandte Mathematik und Physik, 2015, 66(4), 1949–1968.
[19] Del Vescovo, D., Giorgio, I., Dynamic problems for metamaterials: review of existing models and ideas for further research, International Journal of Engineering Science, 2014, 80, 153–172.
[20] Giorgio, I., Galantucci, L., Della Corte, A., Del Vescovo, D., Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications, International Journal of Applied Electromagnetics and Mechanics, 2015, 47(4), 1051–1084.
[21] Pagnini, L.C., Piccardo, G., The three-hinged arch as an example of piezomechanic passive controlled structure, Continuum Mechanics and Thermodynamics, 2016, 28(5), 1247–1262.
[22] Alessandroni, S., Andreaus, U., dell’Isola, F., Porfiri, M., A passive electric controller for multimodal vibrations of thin plates, Computers and Structures, 2005, 83(15), 1236–1250.
[23] Andreaus, U., dell’Isola, F., Porfiri, M., Piezoelectric passive distributed controllers for beam flexural vibrations, Journal of Vibration and Control, 2004.
[24] Alessandroni, S., dell’Isola, F., Porfiri, M., A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators, International Journal of Solids and Structures, 2002, 39(5295-5324).
[25] Alessandroni, S., Andreaus, U., dell’Isola, F., Porfiri, M., Piezo-electromechanical (pem) kirchhoff–love plates, European Journal of Mechanics A/Solids, 2004, 23, 689–702.
[26] dell’Isola, F., Porfiri, M., Vidoli, S., Piezo-electromechanical (PEM) structures: passive vibration control using distributed piezoelectric transducers, Comptes Rendus de l’Academie des Sciences, Mécanique, 2003, 331, 69–76.
[27] 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, Archive of Applied Mechanics, 2003, 73(1-2), 26–48.
[28] 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 Materials and Structures, 2004, 13(2), 299.
[29] Porfiri, M., dell’Isola, F., Frattale Mascioli, F., Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers, International Journal of Circuit Theory and Applications, 2004, 32(4), 167–198.
[30] Rosi, G., Control of sound radiation and transmission by means of passive piezoelectric networks: modelling, optimization and experimental implementation, Ph.D. thesis, Sapienza University of Rome, University of Paris 6, 2010.
[31] Maurini, C., dell’Isola, F., Del Vescovo, D., Comparison of piezoelectronic networks acting as distributed vibration absorbers, Mechanical Systems and Signal Processing, 2004, 18(5), 1243–1271.
[32] Bolotin, V.V., Nonconservative problems of the theory of elastic stability, Macmillan, New York, 1963.
[33] Beck, M., Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes, Zeitschrift für angewandte Mathematik und Physik ZAMP, 1952, 3(3), 225–228.
[34] Ziegler, H., Die stabilitätskriterien der elastomechanik, Ingenieur Archiv, 1952, 20(1), 49–56.
[35] Seyranian, A., Mailybaev, A., Multiparameter stability theory with mechanical applications, vol. 13, World Scientific, Singapore, 2003.
[36] Kirillov, O.N., Nonconservative stability problems of modern physics, Walter de Gruyter, Berlin/Boston, 2013.
[37] Kirillov, O.N., A theory of the destabilization paradox in non-conservative systems, Acta Mechanica, 2005, 174(3-4), 145–166.
[38] Kirillov, O., Seyranian, A., The effect of small internal and external damping on the stability of distributed non-conservative systems, Journal of Applied Mathematics and Mechanics, 2005, 69(4), 529–552.
[39] Luongo, A., D’Annibale, F., A paradigmatic minimal system to explain the Ziegler paradox, Continuum Mechanics and Thermodynamics, 2015, 27(1-2), 211–222.
[40] Luongo, A., D’Annibale, F., On the destabilizing effect of damping on discrete and continuous circulatory systems, Journal of Sound and Vibration, 2014, 333(24), 6723–6741.
[41] D’Annibale, F., Ferretti, M., Luongo, A., Improving the linear stability of the beck’s beam by added dashpots, International Journal of Mechanical Sciences, 2016, 110, 151–159.
[42] Wang, Q., Quek, S.T., Enhancing flutter and buckling capacity of column by piezoelectric layers, International Journal of Solids and Structures, 2002, 39(16), 4167–4180.
[43] Wang, Y., Wang, Z., Zu, L., Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force, Archive of Applied Mechanics, 2012, 83(4), 495–507.
[44] D’Annibale, F., Rosi, G., Luongo, A., Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces, Meccanica, 2015, 50(3), 825–839.
[45] D’Annibale, F., Rosi, G., Luongo, A., Controlling the limit-cycle of the ziegler column via a tuned piezoelectric damper, Mathematical Problems in Engineering, 2015, 2015.
[46] D’Annibale, F., Rosi, G., Luongo, A., Piezoelectric control of hopf bifurcations: A non-linear discrete case study, International Journal of Non-Linear Mechanics, 2016, 80, 160–169.
[47] D’Annibale, F., Piezoelectric control of the hopf bifurcation of ziegler’s column with nonlinear damping, Nonlinear Dynamics, 2016, 86(4), 2179–2192.
[48] Luongo, A., D’Annibale, F., Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads, International Journal of Non-Linear Mechanics, 2013, 1–12.
[49] Luongo, A., D’Annibale, F., Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic beck’s beam, Mathematics and Mechanics of Solids, 2017, 22(6), 1347–1365.
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