This article proposes a new, simple, and efficient strategy, allowing one to optimize the diffusion operator between a passive beam coupled to a like beam equipped with a periodically distributed network of shunted piezoelectric patch actuators. A multimodal wave dispersion model is used to compute the diffusion operator and analyze the stability properties of the combined system. Based on this mathematical tool, specific optimization procedures are introduced to allow maximization or minimization of the wave transmissibility between the passive and the active distributed beam. A specific example is used to demonstrate the capability of the shunted piezoelectric system to induce total reflection through the total absorption of incoming propagating flexural waves while guaranteeing the stability, robustness, and realizability of such a system.
Banks, H.T., Wang, Y. and Smith, R.C.1996. Smart Material Structures Modeling Estimation and Control , Wiley, Chichester.
2.
Banerjee, J.R. and Sobey, A.J.2005. ``Dynamic Stiffness Formulation and Free Vibration Analysis of a Three Layered Sandwich Beam,'' International Journal of Solids and Structures, 42:2181-2197.
3.
Benjeddou, A.2000. ``Advances in Piezoelectric Finite Element Modeling of Adaptive Structural Elements: A Survey,'' Computers and Structures , 76(1-3):347-363.
4.
Brillouin, L.1946. Wave Propagation in Periodic Structures, Dover Publication Inc., Dover, New York.
5.
Collet, M., Walter, V. and Delobelle, P.2003. ``Active Damping of a Micro-cantilever Piezocomposite Beam,'' Journal of Sound and Vibration, 260(3):453-476.
6.
Collet, M., Delobelle, P. and Walter, V.2004. ``Active Damping with Piezoelectric MEMS Devices,'' In: Smart Structures and Material Conference (Active Damping) , SPIE, pp. 301-309.
7.
Collet, M., David, P. and Berthillier, M.2005. ``Active Acoustical Impedance using Distributed Electrodynamical Shaker,'' In: 16th International Conf on Adaptive Structures and Technologies, ICAST, pp. 263-275, Paris.
8.
Collet, M., David, P. and Berthillier, M.2006. ``Active Acoustical Impedance using Distributed Electrodynamical Transducers,'' In: 13th International Symposium on Smart Structures and Materials, NDE, IEEE, pp. 6166-60, San Diego.
9.
D'Andrea, R. and Recht, B.2004. ``Distributed Control of Systems over Discrete Groups,''IEEE Transactions on Automatic Control, 49(9):1421-1603.
10.
Denke, P.H., Eide, G.R. and Pickard, J.1975. ``Matrix Difference Equation Analysis of Vibrating Periodic Structures,'' AIAA Journal, 13(2):160-166.
11.
Duhamel, D., Mace, B.R. and Brennanl, M.J.2006. ``Finite Element Analysis of the Vibrations of Waveguides and Periodic Structures,'' Journal of Sound and Vibration , 294:205-220.
12.
Elmore, W. and Heald, M.1985. Physics of Waves, Dover, New York.
13.
Fuller, C.R. and Fahy, F.J.1982. ``Characteristics of Wave Propagation and Energy Distributions in Cylindrical Elastic Shells Filled with Fluid,'' Journal of Sound and Vibration, 81(4):501-518.
14.
Graff, K.G.1991. Wave Motion in Elastic Solids, Oxford University Press, London.
15.
Hagood, N.W. and von Flotow, A.H.1991. ``Damping of Structural Vibrations with Piezoelectric Materials and Passive Electrical Networks, '' Journal of Sound and Vibration, 146(2):243-268.
16.
Houillon, L., Ichchou, M.N. and Jezequel, L.2005. ``Wave Motion in Thin-walled Structures, '' Journal of Sound and Vibration, 281(3-5):483-507.
17.
Hughes, T.J.R.1995. ``Multiscale Phenomena: Green's Functions, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods,'' Computer Methods in Applied Mechanics and Engineering , 127(1-4):387-401.
18.
Ichchou, M.N., Akrout, S. and Mencik, J.-M.2007. ``Guided Waves Group and Energy Velocities via Finite Elements,'' Journal of Sound and Vibration, 305:931-944.
19.
Jeffreys, H. and Jeffreys, B.S.1956. Methods of Mathematical Physics, Cambridge University Press, Cambridge.
20.
Lenczner, M. and Montseny, G.2005a. ``Diffusive Realization of Operators Solutions of Partial Differential Operational Equations,''CRAS Séries I, 341(12):737-740.
21.
Lenczner, M., Montseny, G. and Yakouby, Y.2005b. ``Approximations of Distributed Control Operators on Quasi-local Architectures,'' In: 17ème Congrès Mondial IMACS Calcul Scientifique, Mathématiques Appliquées et Simulation, LAAS, pp. 235-252.
22.
Mahapatra, R.D., Singhal, A. and Gopalakrishnan, S.2006. ``A Higher-order Finite Waveguide Model for Spectral Analysis of Composite Structures, '' Computer Methods in Applied Mechanics and Engineering, 195:1116-1135.
23.
Mead, D.J.1973. ``A General Theory of Harmonic Wave Propagation in Linear Periodic Systems with Multiple Coupling,'' Journal of Sound and Vibration, 27(2):235-260.
24.
Mencik, J.-M. and Ichchou, M.N.2005. ``Multi-mode Propagation and Diffusion in Structures through Finite Elements,'' European Journal of Mechanis. A/Solids , 24:877-898.
25.
Mencik, J.-M. and Ichchou, M.N.2008. ``A Substructuring Technique for Finite Element Wave Propagation in Multi-layered Systems,'' Computer Methods in Applied Mechanics and Engineering, 197(6-8):505-523.
26.
Meyer, Y., Verdot, T. and Collet, M.2007. ``Active Isolation of Electronic Micro-components with Piezoelectrically Transduced Silicon Mems Devices,'' Journal of Smart Materials and Structures, 16(1):128-134.
27.
Monnier, P. and Collet, M.2005. ``Definition of the Mechanical Design Parameters to Optimize Efficiency of Integraol Force Feedback Active Damping Strategy, '' Journal Structural Control and Health Monitoring, 12(1):65-89.
28.
Nelson, P.A. and Elliott, S.J.1992. Active Control of Sound, Academic Press , San Diego, London.
29.
Ohayon, R. and Soize, C.1998. Structural Acoustics and Vibration, Academic Press, San Diego.
30.
Preumont, A.1997. Vibration Control of Structures: An Introduction, Kluwer AP, Dordecht, Boston , London.
31.
Ryzhik, L., Papanicolaou, G. and Keller, J.B.1996. ``Transport Equation for Elastic and other Waves in Random Media,'' Wave Motion, 24:327-370.
32.
Sinoploi, B., Schenato, L. and Shankar, S.2003. ``Distributed Control Applications within Sensors Networks,''Proceedings of the IEEE, 91-8:1235-1247.
33.
Varadan, V.K., Awadelkarim, O.O. and Gardner, J.W.2004. Microsensors, Mems, and Smart Devices, Wiley, Chichester.
34.
von Flotow, A.H.1986. ``Disturbance Propagation in Structural Networks,''Journal of Sound and Vibration, 106(3):433-450.
35.
Zhong, W. and Williams, F.1995. ``On the Direct Solution of Wave Propagation for Repetitive Structures,'' Journal of Sound and Vibration, 181(3):485-501.
