WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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4. Conclusions In order to use phononic band gap materials in mechanical structures that utilize the selective wave blocking abilities of these materials, the effects associated with the boundaries, damping, and imperfections need to be taken into account. In this work the vibrational response of 1-D and 2-D mass–spring structures subjected to periodic loading has been investigated. The focus has been put on analyzing structures with a periodic micro-structure (unit cells) with gaps in the band structure for the corresponding infinite periodic lattice. Two examples were used to demonstrate the effect on the response of boundaries, viscous damping, and imperfections—a 1-D structure acting as a wave filter, and wave guiding in 2-D structures. It was shown how the response in the band gap frequency range depends on the number of unit cells in the structure. In 1-D and for one of the two types of unit cells analyzed in 2-D, it was seen how the response in the band gap is insensitive to moderate amounts of viscous damping, whereas for the other 2-D unit cell analyzed, the band gap effect almost disappears with strong damping added. In the 1-D example it was shown also that the response in the band gap is insensitive to small imperfections in the periodic structure. It was demonstrated in both the 1-D and 2-D example, that the free boundaries cause local resonances that affect the response in the band gap. Two 2-D wave guides were analyzed, created by replacing the periodic structure with a homogeneous structure in a straight and a 901 bent path. It was shown how the vibrational response is confined to these paths in the band gap frequency ranges. Further work deals with the formulation and solution of optimization problems that can be used to design of applications of phononic band gap materials and structures. Acknowledgements This work was supported by the Danish Research Council through the project ‘‘Phononic bandgap materials: Analysis and optimization of wavetransmission in periodic materials’’. The author would like to thank Jon Juel Thomsen, Ole Sigmund, and Martin P. Bends^e for valuable comments and suggestions. References ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 1077 [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals, Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [2] E. Yablonovitch, Photonic crystals: semiconductors of light, Scientific American 285 (2001) 34–41. [3] Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure, Philosophical Magazine 24 (1887) 145–159. [4] L. Brillouin, Wave Propagation in Periodic Structures, Dover Publications Inc, New York, 1953. [5] C. Elachi, Waves in active and passive periodic structures: a review, Proceedings of the IEEE 64 (1976) 1666–1698. [6] D.J. Mead, Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995, Journal of Sound and Vibration 190 (1996) 495–524. [7] S. Parmley, T. Zobrist, T. Clough, A. Perez-miller, M. Makela, R. Yu, Phononic band structure in a mass chain, Applied Physics Letters 67 (1995) 777–779.
1078 ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 [8] A. Diez, G. Kakarantzas, T.A. Birks, P.St.J. Russel, Acoustic stop-bands in periodically microtapered optical fibers, Applied Physics Letters 76 (2000) 3481–3483. [9] D.J. Griffiths, C.A. Steinke, Waves in locally periodic media, American Journal of Physics 69 (2001) 137–154. [10] M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure, Journal of Sound and Vibration 158 (1992) 377–382. [11] T. Suzuki, P.K.L. Yu, Complex elastic wave band structures in three-dimensional periodic elastic media, Journal of the Mechanics of Physics and Solids 46 (1998) 115–138. [12] P. Langlet, A.-C. Hladky-hennion, J.-N. Decarpigny, Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method, Journal of the Acoustical Society of America 98 (1995) 2792–2800. [13] W. Axmann, P. Kuchment, An efficient finite element method for computing spectra of photonic and acoustic band-gap materials, Journal of Computational Physics 150 (1999) 468–481. [14] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials, Science 289 (2000) 1734–1736. [15] J.O. Vasseur, P.A. Deymier, G. Frantziskonis, G. Hong, B. Djafari-rouhani, L. Dobrzynski, Experimental evidence for the existence of absolute acoustic band gaps in two-dimensional periodic composite media, Journal of Physics: Condensed Matter 10 (1998) 6051–6064. [16] C.G. Poulton, A.B. Movchan, R.C. Mcphedran, N.A. Nicorovici, Y.A. Antipov, Eigenvalue problems for doubly periodic elastic structures and phononic band gaps, Proceedings of the Royal Society of London A 456 (2000) 2543–2559. [17] M.S. Kushwaha, Classical band structure of periodic elastic composites, International Journal of Modern Physics 10 (1996) 977–1094. [18] M. Torres, F.R. Montero de Espinosa, D. Garc!ıa-Pablos, N. Garc!ıa, Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects, Physical Review Letters 82 (1999) 3054– 3057. [19] I.E. Psarobas, N. Stefanou, A. Modinos, Phononic crystals with planar defects, Physical Review B 62 (2000) 5536– 5540. [20] O. Sigmund, Microstructural design of elastic band gap structures, in: G.D. Cheng, Y. Gu, S. Liu, Y. Wang (Eds.), Proceedings of the Second World Congress of Structural and Multidisciplinary Optimization, CD-rom, Liaoning Electronic Press, Dalian, China, 2001. [21] O. Sigmund, J.S. Jensen, Topology optimization of phononic band gap materials and structures, Fifth World Congress on Computational Mechanics, Vienna, Austria, 2002, available at http://wccm.tuwien.ac.at. [22] P.G. Martinsson, A.B. Movchan, Vibrations of lattice structures and phononic band gaps 56 (2003) 45–64.
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WAVES AND VIBRATIONS IN INHOMOGENEO
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Denne afhandling er af Danmarks Tek
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List of Thesis Papers [1] J. S. Jen
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Contents Preface i List of Thesis P
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Chapter 1 Introduction This thesis
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eplacemen Phononic and photonic ban
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Optimal material distribution - top
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Chapter 2 The bandgap phenomenon In
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2.1 Band diagram and forced vibrati
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2.3 Experimental demonstration 11 F
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y 2.5 Nonlinearities 13 Response in
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Relations to recent work 15 Amplitu
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Chapter 3 Bandgap structures as opt
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3.1 Vibration-quenching structures
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3.2 Maximizing wave reflection 21 d
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3.4 Maximizing wave dissipation 23
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3.4 Maximizing wave dissipation 25
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3.5 Plate structures 27 a) b) Figur
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3.6 Acoustic design 29 design domai
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Chapter 4 Optimization of photonic
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4.1 Waveguide bends and junctions 3
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4.2 Photonic crystal building block
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4.2 Photonic crystal building block
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Page 51 and 52: Relations to recent work 41 wave in
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Page 61 and 62: Relations to recent work 51 Figure
Page 63 and 64: Chapter 6 Conclusions In the first
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Page 81 and 82: 1058 fundamental eigenfrequency o
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Page 85 and 86: 1062 local resonator and splits up
Page 87 and 88: 1064 3.1. Model equations A finite
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Page 136 and 137: Inverse design of phononic crystals
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324 which averaged over a wave peri
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326 where the modified stress compo
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328 optimization algorithm is enhan
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330 reflectance are also seen (Fig.
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332 It should be emphasized that ot
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334 b Dissipation, D c Dissipation,
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336 7.2. Three-phase design ARTICLE
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338 ARTICLE IN PRESS J.S. Jensen /
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340 ARTICLE IN PRESS J.S. Jensen /
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968 ARTICLE IN PRESS J.S. Jensen, N
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970 To solve the wave equation with
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972 In the following section, we sh
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974 Design variable, t Design varia
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976 Eigenvalue ratio, ω n+1 2 /ωn
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978 to a design parameter te are gi
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980 ARTICLE IN PRESS J.S. Jensen, N
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982 respect to the higher bound C1
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984 instead vary the value of b. Th
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986 References ARTICLE IN PRESS J.S
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a thick solid was considered. A few
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The above expressions provide insta
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In all the examples shown a density
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domain. To the very left the passiv
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Jensen JS, Sigmund O (2005) Topolog
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paper extends on this work. For an
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R 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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objective 1 0.9 0.8 0.7 0.6 0.5 0.4
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The optimization algorithm employs
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Appl. Phys. Lett., Vol. 84, No. 12,
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J. S. Jensen and O. Sigmund Vol. 22
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Topology optimization and fabricati
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Normalized transmission 1.0 0.8 0.6
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Fig. 4. Scanning electron micrograp
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Broadband photonic crystal waveguid
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2. Design and fabrication Silicon-o
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Fig. 3. Steady-state magnetic field
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Topology optimised broadband photon
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1202 IEEE PHOTONICS TECHNOLOGY LETT
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1204 IEEE PHOTONICS TECHNOLOGY LETT
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11. P. Debackere, S. Scheerlinck, P
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nanoimprinted patterns are transfer
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emoved and the spectra changed dras
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Y-splitter W1 waveguide 60-degree b
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5. Photonic Wire Waveguides Figure
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Figure 5: Optimized designs and cor
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Figure 8: Optimized designs and cor
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[13] Borel P I, Frandsen L H, Harp
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1606 J. S. JENSEN To compute a freq
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1608 J. S. JENSEN Cramer’s rule [
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1610 J. S. JENSEN The following pro
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1612 J. S. JENSEN Table I. Coeffici
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1614 J. S. JENSEN h 2h Α fcos Ωt
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1616 J. S. JENSEN The derivative of
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1618 J. S. JENSEN which is solved f
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1620 J. S. JENSEN Based on the expr
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1622 J. S. JENSEN Response, ln(u ti
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1624 J. S. JENSEN h 2h fcos Ωt Fi
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1626 J. S. JENSEN details and have
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1628 J. S. JENSEN Equation (A10) co
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1630 J. S. JENSEN 7. Coyette J-P, L
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The number of masses in the chain t
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5 Results The optimization algorith
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Figure 9. Response for optimized de
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Space-time topology optimization fo
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good for low to moderate stiffness
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V ¼ 0:3 at a given time instant fo
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The parameters used in this example
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Normalized amplitude 1 0.9 0.8 0.7
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at the Department of Mechanical Eng
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600 J.S. JENSEN techniques for obta
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602 J.S. JENSEN 1 and 2, where the
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604 J.S. JENSEN 4. Numerical analys
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606 J.S. JENSEN Energy, E a) Energy
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608 J.S. JENSEN relaxed somewhat in
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610 J.S. JENSEN when the contrast i
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612 J.S. JENSEN 6. Summary and conc
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614 J.S. JENSEN Sigmund, O. and Jen
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