WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 1069 Fig. 9. (a) The structure with Mx My unit cells with a periodic loading f cos Ot acting centrally at the top boundary and simple supports at the bottom corners, and (b) the corresponding computational model for Mx ¼ My ¼ 7 and type 1 unit cells. the supports. This corresponds well to the motion predicted by Eq. (28), where the inclusions and the matrix material are predicted to move essentially rigidly in anti-phase with nearly identical amplitudes.
1070 ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 Fig. 10. The structural response with Mx My type 1 unit cells included in the structure, (a) shows the response in point A and (b) in point B. 3.3.3. Viscous damping Fig. 14a shows the effect of adding damping to the structure with Mx ¼ My ¼ 21 and type 1 unit cells. The undamped FRF curve is shown for comparison together with curves for damping ratios of z ¼ 0:1% and 1.0%. Fig. 14b shows the corresponding curves with type 2 unit cells. Clearly, with type 1 unit cells the band gap is still noticeable with damping present, even when the damping is so strong that all resonance peaks have nearly disappeared from the response. However, with type 2 unit cells in the structure the band gap practically disappears for strong
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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
Page 41 and 42: Chapter 4 Optimization of photonic
Page 43 and 44: 4.1 Waveguide bends and junctions 3
Page 45 and 46: 4.2 Photonic crystal building block
Page 47 and 48: 4.2 Photonic crystal building block
Page 49 and 50: 4.4 Ridge waveguides 39 w ε =1 ?
Page 51 and 52: Relations to recent work 41 wave in
Page 53 and 54: Chapter 5 Advanced optimization pro
Page 55 and 56: 5.1 Padé approximants 45 h 2h A f
Page 57 and 58: 5.3 Space-time topology optimizatio
Page 59 and 60: 5.3 Space-time topology optimizatio
Page 61 and 62: Relations to recent work 51 Figure
Page 63 and 64: Chapter 6 Conclusions In the first
Page 65 and 66: References K. Asakawa, Y. Sugimoto,
Page 67 and 68: References 57 W. R. Frei, D. A. Tor
Page 69 and 70: References 59 F. Maestre, A. Münch
Page 71 and 72: References 61 O. Sigmund, J. S. Jen
Page 73 and 74: References 63 X. M. Zhou and G. K.
Page 75 and 76: Dansk resumé 65 i strukturen. En r
Page 77 and 78: 1054 ARTICLE IN PRESS J.S. Jensen /
Page 81 and 82: 1058 fundamental eigenfrequency o
Page 83 and 84: 1060 ðo 2 y;j;k ARTICLE IN PRESS J
Page 85 and 86: 1062 local resonator and splits up
Page 87 and 88: 1064 3.1. Model equations A finite
Page 89 and 90: 1066 FRF (dB) 150 100 50 0 -50 -100
Page 91: 1068 3.3. Example 2: a 2-D waveguid
Page 103 and 104: wave-guiding properties show promis
Page 105 and 106: (a) (b) Acceleration response (dB)
Page 107 and 108: B.S. Lazarov, J.S. Jensen / Interna
Page 109 and 110: ω/κ ω/κ 1.5 1.4 1.3 1.2 1.1 1 0
Page 111 and 112: [B 2000 ] [B 2000 ] [B 2000 ] 0.25
Page 113 and 114: [B 400 ] 0.25 0.2 0.15 0.1 0.05 B.S
Page 115 and 116: 1002 (a) (c) (e) 0. Sigmund and J.
Page 117 and 118: 1004 O. Sigmund and J. S. Jensen wh
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Page 121 and 122: 1008 O. Sigmund and J. S. Jensen we
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Page 125 and 126: 1012 (a) (b) O. Sigmund and J. S. J
Page 127 and 128: 1014 . ct . _ a) Q^ s^c "3
Page 129 and 130: 1016 O. Sigmund and J. S. Jensen (a
Page 131 and 132: 1018 O. Sigmund and J. S. Jensen 4.
Page 134 and 135: Z. Kristallogr. 220 (2005) 895-905
Page 136 and 137: Inverse design of phononic crystals
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Inverse design of phononic crystals
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Inverse design of phononic crystals
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320 Optimization of the dissipation
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322 The reflection of the wave is f
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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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WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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