Thesis presented to the Chair of Mechanics of the Conservatoire National des Arts et Métiers (CNAM) in June/2000 as part of requirements to obtain the degree of : Doctor in Mechanics
Advisor: Prof. Roger Ohayon, D.Sc.

This research was supported by the Brazilian Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) through a D.Sc. scolarship and the French Army Procurement Agency (DGA) through Grant D.G.A./D.S.P./S.T.T.C./MA. no.97-2530.

Abstract

This thesis presents a numerical analysis of the structural vibrations damping obtained by passive, active and hybrid active-passive damping treatments. For that, a finite element model of a sandwich beam which layers can be elastic, piezoelectric or viscoelastic is presented. It is validated through comparisons with analytical, experimental and numerical results found in the literature and, then, extended to the case of laminated surface layers. The representation of the frequency-dependence of the viscoelastic materials properties is studied using the Anelastic Displacement Fields (ADF), Golla-Hughes-McTavish (GHM) and iterative Modal Strain Energy (MSE) models. A modal reduction of the resulting state-space system using a complex base is proposed and validated through comparisons with the results found in the literature. An iterative algorithm is proposed to evaluate the optimal control (Linear Quadratic Regulator, LQR) respecting the maximum electric field allowed by the piezoelectric patches. The comparison between the damping performances of the extension and shear piezoelectric actuation mechanisms shows that the latter are more effective for small amplitudes and high frequencies and for sandwich structures with rigid surface layers and soft core. Three hybrid damping treatments, obtained by modifying the relative position of the viscoelastic and piezoelectric layers, were analyzed and compared. It is shown that active constrained layer (ACL) treatments, consisting in replacing the elastic constraining layer by a piezoelectric actuator, are effective only for very thin viscoelastic layers. At the same time, treatments using passive constrained layer (PCL) associated with a piezoelectric actuator bonded directly on the opposite surface of the structure (AC/PCL) or between the viscoelastic layer and the structure (AC/PSOL) are, generally, more effective. However, the three treatments provide effective and robust control systems. The study of the vibration control of a sandwich beam with viscoelastic core, through two piezoelectric patches bonded symmetrically on its upper and lower surfaces, using three controllers, namely LQR, derivative and Linear Quadratic Gaussian (LQG), is carried out. It is shown that optimal controllers LQR and LQG are more effective than the derivative one, since they are less dependent on the colocalisation of the actuators and sensors. Finally, the influence of the operating temperature on the damping performance of an active constrained layer treatment is analyzed. Results show that it is possible to obtain uniform performances in an interval of temperatures.

Click here to download the PDF version [1.95 M]

References

[1] Agnes, G.S. and Napolitano, K. Active constrained layer viscoelastic damping. In 34 th AIAA/ASME/ASCE/AHS/ASC Struct., Structural Dyn., and Mater. Conf., AIAA, Reston (USA), pp. 3499-3506, 1993.

[2] Assaf, S. Mod´ elisation par la M´ ethode des ´ El´ ements Finis du Comportement Vibratoire des Poutres et Plaques Sandwiches : M´ etal Mat´ eriaux Visco´ elastiques. Th` ese de Doctorat, Universit´ e de Technologie de Compi` egne, Compi` egne, 1991.

[3] Azvine, B., Tomlinson, G.R., and Wynne, R. Use of active constrained-layer damping for controlling resonant vibration. Smart Mater. Struct., 4(1):1-6, 1995.

[4] Azvine, B., Tomlinson, G.R., Wynne, R., and Sensburg, O. Vibration suppression of flexible structures using active damping. In Breitbach, E.J., Wada, B.K., and Natori, M.C., eds., Proceedings of the 4 th Int. Conf. Adaptive Struct. Tech., Technomic Pub. Co., Lancaster (USA), pp. 340-356, 1993.

[5] Badre-Alam, A., Wang, K.W., and Gandhi, F. Optimization of enhanced active constrained layer (EACL) treatment on helicopter flexbeams for aeromechanical stability augmentation. Smart Mater. Struct., 8(2):182-196, 1999.

[6] Bagley, R.L. and Torvik, P.J. Fractional calculus - a different approach to analysis of viscoelastically damped structures. AIAA J., 21(5):741-748, 1983.

[7] Bagley, R.L. and Torvik, P.J. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J., 23(6):918-925, 1985.

[8] Baz, A. Boundary control of beams using active constrained layer damping. J. Vib. Acoust., 119(2):166-172, 1997.

[9] Baz, A. Dynamic boundary control of beams using active constrained layer damping. Mech. Syst. Signal Proc., 11(6):811-825, 1997.

[10] Baz, A. Optimization of energy dissipation characteristics of active constrained layer damping. Smart Mater. Struct., 6(3):360-368, 1997.

[11] Baz, A. Robust control of active constrained layer damping. J. Sound Vib., 211(3):467-480, 1998.

[12] Baz, A. and Ro, J. Finite element modeling and performance of active constrained layer damping. In Meirovitch, L., ed., 9 th VPI&SU Conf. on Dyn. & Control of Large Struct., Blacksburg (USA), pp. 345-357, 1993.

[13] Baz, A. and Ro, J. Partial treatment of flexible beams with active constrained layer damping. In Guran, A., ed., Recent Developments in Stability, Vibration and Control of Structural Systems, ASME, New York, vol. AMD-167, pp. 61-80, 1993.

[14] Baz, A. and Ro, J. Optimum design and control of active constrained layer damping. J. Vib. Acoust., 117(B):135-144, 1995.

[15] Benjeddou, A. Recent advances in hybrid active-passive vibration control (submitted). J. Vib. Control, 1999.

[16] Benjeddou, A., Trindade, M.A., and Ohayon, R. A unified beam finite element model for extension and shear piezoelectric actuation mechanisms. J. Intell. Mater. Syst. Struct., 8(12):1012-1025, 1997.

[17] Bent, A.A., Hagood, N.W., and Rodgers, J.P. Anisotropic actuation with piezoelectric fiber composites. J. Intell. Mater. Syst. Struct., 6(3):338-349, 1995.

[18] Biot, M.A. Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys. Rev., 97(6):1463-1469, 1955.

[19] Brackbill, C.R., Lesieutre, G.A., Smith, E.C., and Govindswamy, K. Thermomechanical modeling of elastomeric materials. Smart Mater. Struct., 5(5):529-539, 1996.

[20] Chen, T. and Baz, A. Performance characteristics of active constrained layer damping versus passive constrained layer damping with active control. In Varadan, V.V. and Chandra, J., eds., Smart Struct. & Mater. 1996: Mathematics and Control in Smart Structures, SPIE, Bellingham (USA), vol. 2715, pp. 256-268, 1996.

[21] Christensen, R.M. Theory of Viscoelasticity: An Introduction. Academic Press, Inc., New York, 2 nd edition, 1982.

[22] Crassidis, J., Baz, A., and Wereley, N. H ¥ control of active constrained layer damping. J. Vib. Control, 6(1):113-136, 2000.

[23] Crawley, E.F. and Anderson, E.H. Detailed models of piezoceramic actuation of beams. J. Intell. Mater. Syst. Struct., 1(1):4-25, 1990.

[24] Crawley, E.F. and deLuis, J. Use of piezoelectric actuators as elements of intelligent structures. AIAA J., 25(10):1373-1385, 1987.

[25] DiTaranto, R.A. Theory of vibratory bending for elastic and viscoelastic layered finite-length beams. J. Appl. Mech., 32:881-886, 1965.

[26] Dosch, J.J., Inman, D.J., and Garcia, E. A self-sensing piezoelectric actuator for collocated control. J. Intell. Mater. Syst. Struct., 3(1):167-185, 1992.

[27] Dovstam, K. Augmented Hooke’s law in frequency domain: a three dimensional, material damping formulation. Int. J. Solids Struct., 32(19):2835-2852, 1995.

[28] Enelund, M. and Lesieutre, G.A. Time domain modeling of damping using anelastic displacement fields and fractional calculus. Int. J. Solids Struct., 36(29):4447-4472, 1999.

[29] Franklin, G.F. and Powell, J.D. Digital Control of Dynamic Systems. Addison-Wesley, Reading (USA), 1980.

[30] Friot, E. and Bouc, R. Contrˆ ole optimal par r´ etroaction du rayonnement d’une plaque munie de capteurs et d’actionneurs pi´ ezo-´ electriques non colocalis´ es. In 2 eme Colloque GDR Vibroacoustique, LMA, Marseille, pp. 229-248, 1996.

[31] Friswell, M.I. and Inman, D.J. Hybrid damping treatments in thermal environments. In Tomlinson, G.R. and Bullough, W.A., eds., Smart Mater. Struct., IOP Publishing, Bristol (UK), pp. 667-674, 1998.

[32] Friswell, M.I., Inman, D.J., and Lam, M.J. On the realisation of GHM models in viscoelasticity. J. Intell. Mater. Syst. Struct., 8(11):986-993, 1997.

[33] Ganapathi, M., Patel, B.P., Polit, O., and Touratier, M. A C 1 finite element including transverse shear and torsion warping for rectangular sandwich beams. Int. J. Numer. Methods Eng., 45(1):47-75, 1999.

[34] Goh, C.J. and Caughey, T.K. On the stability problem caused by finite actuator dynamics in the collocated control of large space structures. Int. J. Control, 41(3):787-802, 1985.

[35] Golla, D.F. and Hughes, P.C. Dynamics of viscoelastic structures - a time-domain, finite element formulation. J. Appl. Mech., 52(4):897-906, 1985.

[36] Ha, S.K., Keilers, C., and Chang, F.-K. Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators. AIAA J., 30(3):772- 780, 1992.

[37] Hagood, N.W., Chung, W.H., and von Flotow, A. Modelling of piezoelectric actuator dynamics for active structural control. J. Intell. Mater. Syst. Struct., 1(7):327-354, 1990.

[38] Hagood, N.W., Kindel, R., Ghandi, K., and Gaudenzi, P. Improving transverse actuation of piezoceramics using interdigitated surface electrodes. In Hagood, N.W., ed., Smart Struct. and Mater. 1993: Smart Struct. and Intel. Syst., SPIE, Bellingham (USA), vol. 1917, pp. 341-352, 1993.

[39] Herman Shen, M.-H. Analysis of beams containing piezoelectric sensors and actuators. Smart Mater. Struct., 3(4):439-447, 1994.

[40] Huang, S.C., Inman, D.J., and Austin, E.M. Some design considerations for active and passive constrained layer damping treatments. Smart Mater. Struct., 5(3):301- 313, 1996.

[41] Im, S. and Atluri, S.N. Effects of a piezo-actuator on a finitely deformed beam subjected to general loading. AIAA J., 27(12):1801-1807, 1989.

[42] Inman, D.J. and Lam, M.J. Active constrained layer damping treatments. In Ferguson, N.S., Wolfe, H.F., and Mei, C., eds., 6 th Int. Conf. on Recent Advances in Struct. Dyn., Southampton (UK), vol. 1, pp. 1-20, 1997.

[43] Jemai, B., Ichchou, M.N., Jezequel, L., and Noe, M. Comparison between a large bandwidth LQG and modal control methods applied to complex flexible structure. In 2 nd Int. Conf. on Active Control in Mechanical Engineering, Lyon (France), October 1997.

[44] Johnson, A.R., Tessler, A., and Dambach, M. Dynamics of thick viscoelastic beams. J. Eng. Mater. Tech., 119(3):273-278, 1997.

[45] Johnson, C.D., Keinholz, D.A., and Rogers, L.C. Finite element prediction of damping in beams with constrained viscoelastic layers. Shock Vib. Bulletin, 50(1):71-81, 1981.

[46] Kapadia, R.K. and Kawiecki, G. Experimental evaluation of segmented active constrained layer damping treatments. J. Intell. Mater. Syst. Struct., 8(2):103-111, 1997.

[47] Kerwin, Jr., E.M. Damping of flexural waves by a constrained viscoelastic layer. J. Acoust. Soc. Am., 31(7):952-962, 1959.

[48] Kim, G., Jensen, T., DeGiorgi, V., Bender, B., Wu, C.C., Flipen, D., Lewis, D., Zhang, Q., Mueller, V., Kahn, M., Silberglett, R., and Len, L.K. Composite piezoelectric assemblies for torsional actuators. Technical Report NRL/MR/6380-97-7997, Naval Research Lab. and PSU, 1997.

[49] Kirk, D.E. Optimal Control Theory. Prentice-Hall, Englewood Cliffs, 1970.

[50] Krommer, M. and Irschik, H. On the influence of the electric field on free transverse vibrations of smart beams. Smart Mater. Struct., 8(3):401-410, 1999.

[51] Lam, M.J., Inman, D.J., and Saunders, W.R. Vibration control through passive constrained layer damping and active control. J. Intell. Mater. Syst. Struct., 8(8):663-677, 1997.

[52] Lam, M.J., Inman, D.J., and Saunders, W.R. Variations of hybrid damping. In Davis, L.P., ed., Smart Struct. & Mater. 1998: Passive Damping and Isolation, SPIE, Bellingham (USA), vol. 3327, pp. 32-43, 1998.

[53] Leibowitz, M.M. and Vinson, J.R. On active (piezoelectric) constrained layer damping in composite sandwich structures. In Breitbach, E.J., Wada, B.K., and Natori, M.C., eds., Proceedings of the 4 th Int. Conf. Adaptive Struct. Tech., Technomic Pub. Co., Lancaster (USA), pp. 530-541, 1993.

[54] Lesieutre, G.A. Finite elements for dynamic modeling of uniaxial rods with frequency-dependent material properties. Int. J. Solids Struct., 29(12):1567-1579, 1992.

[55] Lesieutre, G.A. and Bianchini, E. Time domain modeling of linear viscoelasticity using anelastic displacement fields. J. Vib. Acoust., 117(4):424-430, 1995.

[56] Lesieutre, G.A. and Govindswamy, K. Finite element modeling of frequency-dependent and temperature-dependent dynamic behavior of viscoelastic materials in simple shear. Int. J. Solids Struct., 33(3):419-432, 1996.

[57] Lesieutre, G.A. and Lee, U. A finite element for beams having segmented active constrained layers with frequency-dependent viscoelastics. Smart Mater. Struct., 5(5):615-627, 1996.

[58] Lesieutre, G.A. and Mingori, D.L. Finite element modeling of frequency-dependent material damping using augmenting thermodynamic fields. J. Guidance, 13(6):1040-1050, 1990.

[59] Lewis, F.L. Optimal Control. John Wiley & Sons, 1986.

[60] Liao, W.H. and Wang, K.W. A new active constrained layer configuration with enhanced boundary actions. Smart Mater. Struct., 5(5):638-648, 1996.

[61] Liao, W.H. and Wang, K.W. On the active-passive hybrid control of structures with active constrained layer treatments. J. Vib. Acoust., 119(4):563-572, 1997.

[62] Liao, W.H. and Wang, K.W. On the analysis of viscoelastic materials for active constrained layer damping treatments. J. Sound Vib., 207(3):319-334, 1997.

[63] Lin, M.W., Abatan, A.O., and Rogers, C.A. Application of commercial finite element codes for the analysis of induced strain-actuated structures. J. Intell. Mater. Syst. Struct., 5(6):869-875, 1994.

[64] Liu, Y. and Wang, K.W. Enhanced active constrained layer damping treatment with symmetrically and non-symmetrically distributed edge elements. In Davis, L.P., ed., Smart Struct. & Mater. 1998: Passive Damping and Isolation, SPIE, Bellingham (USA), vol. 3327, pp. 61-72, 1998.

[65] McTavish, D.J. and Hughes, P.C. Modeling of linear viscoelastic space structures. J. Vib. Acoust., 115:103-110, 1993.

[66] Mead, D.J. Passive Vibration Control. John Wiley & Sons, New York, 1999.

[67] Mead, D.J. and Markus, S. The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. J. Sound Vib., 10(2):163-175, 1969.

[68] Nashif, A., Jones, D., and Henderson, J. Vibration Damping. John Wiley & Sons, New York, 1985.

[69] Ohayon, R. and Soize, C. Structural Acoustics and Vibration : Mechanical Models, Variational Formulations and Discretization. Academic Press, London (UK), 1998.

[70] Park, C.H., Inman, D.J., and Lam, M.J. Model reduction of viscoelastic finite element models. J. Sound Vib., 219(4):619-637, 1999.

[71] Petitjean, B. and Legrain, I. Contrˆ ole actif large bande des vibrations d’une plaque en feedback: aspects th´ eoriques et pratiques. In 2 eme Colloque GDR Vibroacoustique, LMA, Marseille, pp. 215-228, 1996.

[72] Plouin, A.-S. and Balm` es, E. Pseudo-modal representations of large models with viscoelastic behavior. In Int. Modal Analysis Conf., SEM, Bethel (USA), pp. 1440- 1446, 1998.

[73] Plump, J.M. and Hubbard Jr., J.E. Modeling of an active constrained layer damper. In 12 th Int. Congress on Acoustics, Toronto (Canada), pp. #D4-1, 1986.

[74] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. The Mathematical Theory of Optimal Processes. Wiley-Interscience, New York, 1962.

[75] Preumont, A. Vibration Control of Active Structures: An Introduction. Kluwer Academic Publishers, Dordrecht (The Netherlands), 1997.

[76] Rahmoune, M., Benjeddou, A., Ohayon, R., and Osmont, D. New thin piezoelectric plate models. J. Intell. Mater. Syst. Struct., 9(12):1017-1029, 1998.

[77] Rahmoune, M., Osmont, D., Benjeddou, A., and Ohayon, R. Finite element modeling of a smart structure plate system. In Santini, P., Rogers, C.A., and Morotsu, Y., eds., Proceedings of the 7 th Int. Conf. Adaptive Struct. Tech., Technomic Pub. Co., Lancaster (USA), pp. 463-473, 1996.

[78] Rongong, J.A., Wright, J.R., Wynne, R.J., and Tomlinson, G.R. Modelling of a hybrid constrained layer/piezoceramic approach to active damping. J. Vib. Acoust., 119(1):120-130, 1997.

[79] Saravanos, D.A. and Heyliger, P.R. Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. J. Intell. Mater. Syst. Struct., 6(3):350-363, 1995.

[80] Shen, I.Y. Hybrid damping through intelligent constrained layer treatments. J. Vib. Acoust., 116(3):341-349, 1994.

[81] Shen, I.Y. Stability and controllability of Euler-Bernoulli beams with intelligent constrained layer treatments. J. Vib. Acoust., 118(1):70-77, 1996.

[82] Shen, I.Y. A variational formulation, a work-energy relation and damping mechanisms of active constrained layer treatments. J. Vib. Acoust., 119(2):192-199, 1997.

[83] Soong, T.T. and Hanson, R.D. Recent development in active and hybrid control research. In Housner, G.W. and Masri, S.F., eds., Int. Workshop on Structural Control, USC, Honolulu (Hawaii), vol. CE-9311, pp. 483-490, 1993.

[84] Sun, C.T. and Zhang, X.D. Use of thickness-shear mode in adaptive sandwich structures. Smart Mater. Struct., 4(3):202-206, 1995.

[85] The Institute of Electrical and Electronics Engineers, Inc. IEEE Standard on Piezoelectricity n.176-1987. 1987.

[86] Tsai, M.S. and Wang, K.W. Integrating active-passive hybrid piezoelectric networks with active constrained layer treatments for structural damping. In Clark, W.W., Xie, W.C., Allaei, D., Namachchivaya, N.S., and O’Reilly, O.M., eds., Active/Passive Vib. Contr. & Nonlinear Dyn. of Struct., ASME, New York, vol. DE-95/AMD-223, pp. 13-24, 1997.

[87] van Nostrand, W.C. and Inman, D.J. Finite element model for active constrained layer damping. In Anderson, G.L. and Lagoudas, D.C., eds., Active Mater. & Smart Struct. SPIE, vol. 2427, pp. 124-139, 1995.

[88] Varadan, V.V., Lim, Y.-H., and Varadan, V.K. Closed loop finite-element modeling of active/passive damping in structural vibration control. Smart Mater. Struct., 5(5):685-694, 1996.

[89] Veley, D.E. and Rao, S.S. A comparison of active, passive and hybrid damping in structural design. Smart Mater. Struct., 5(5):660-671, 1996.

[90] Veley, D.E. and Rao, S.S. A comparison of active, passive and hybrid damping treatments in structural design. In Johnson, C.D., ed., Smart Struct. & Mater. 1996: Passive Damping and Isolation, SPIE, Bellingham (USA), vol. 2720, pp. 184-195, 1996.

[91] Wang, G. and Wereley, N.M. Frequency response of beams with passively constrained damping layers and piezo-actuators. In Davis, L.P., ed., Smart Struct. & Mater. 1998: Passive Damping and Isolation, SPIE, Bellingham (USA), vol. 3327, pp. 44- 60, 1998.

[92] Wang, J., Yong, Y.-K., and Imai, T. Finite element analysis of the piezoelectric vibrations of quartz plate resonators with high-order plate theory. Int. J. Solids Struct., 36(15):2303-2319, 1999.

[93] Yang, S.M. and Lee, Y.J. Interaction of structure vibration and piezoelectric actuation. Smart Mater. Struct., 3(4):494-500, 1994.

[94] Yellin, J.M. and Shen, I.Y. A self-sensing active constrained layer damping treatment for a Euler-Bernoulli beam. Smart Mater. Struct., 5(5):628-637, 1996.

[95] Yellin, J.M. and Shen, I.Y. An analytical model for a passive stand-off layer damping treatment applied to an Euler-Bernoulli beam. In Davis, L.P., ed., Smart Struct. & Mater. 1998: Passive Damping and Isolation, SPIE, Bellingham (USA), vol. 3327, pp. 349-356, 1998.

[96] Yildirim, V., Sancaktar, E., and Kiral, E. Free vibration analysis of symmetric cross-ply laminated composite beams with the help of the transfer matrix approach. Commun. Num. Meth. Eng., 15(9):651-660, 1999.

[97] Yiu, Y.C. Finite element analysis of structures with classical viscoelastic materials. In 34 th AIAA/ASME/ASCE/AHS/ASC Struct., Structural Dyn. & Mater. Conf., AIAA, La Jolla, CA, pp. 2110-2119, 1993.

[98] Zhang, X.D. and Sun, C.T. Formulation of an adaptive sandwich beam. Smart Mater. Struct., 5(6):814-823, 1996. Back to main homepage