WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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3. Finite lattice structures ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 1063 Fig. 5. (a) The 6 6 mass–spring unit cell (denoted type 2) modelling a heavy stiff resonator (center 2 2 masses and springs) in soft suspension (surrounding springs) connected to a surrounding matrix, and (b) the corresponding band structure for wave propagation in the infinite periodic lattice. The inset shows a magnification of the band structure from o ¼ 0–2 kHz with the band gap shown hatched. Wave propagation in infinite, undamped, periodic lattices was analyzed in Section 2. In this section the behavior of finite periodic structures subjected to periodic loading is considered in order to study the effects of boundaries, damping, and imperfections in the periodic structure. The finite structures are treated by two examples: A 1-D-periodic structure that acts as a filter and a 2-D-periodic structure that can be used to guide waves.
1064 3.1. Model equations A finite number of 1-D or 2-D unit cells are considered. The displacement vector uðtÞ ¼ fu1u2yuNtot gT is introduced where Ntot is the total number of degrees of freedom in the model. In 1-D the vector takes the form: uðtÞ ¼fu1yuNuNþ1yuMNg T ; where M is the number of unit cells. In 2-D the vector is given as uðtÞ ¼fu1;1v1;1yu1;MyNv1;MyNyuMxN;MyNvMxN;MyNg T ; where Mx and My is the number of unit cells in the x and y directions, respectively. The model equations that govern the small-amplitude displacement of the masses can now be taken directly from Eq. (1) or (11)–(12) and written as M.u þ C’u þ Ku ¼ fe iOt ; ð29Þ where M and K is the assembled mass- and stiffness-matrices, respectively, C is an added viscous damping matrix, and f is a vector of forces of frequency O: A diagonal damping matrix is used to model the viscous damping with the components ci expressed in terms of the actual to critical damping ratio z i from the relation ci zi ¼ ffiffiffiffiffiffiffiffiffi 2 mi %ki p ; ð30Þ where mi is the mass corresponding to the ith dof and %ki is an equivalent stiffness defined as mio 2 i ; cf. Eq. (4) in 1-D and Eqs. (15) and (16) in 2-D. With the time dependency of the displacement vector uðtÞ ¼ae iOt inserted into Eq. (29), the linear set of equations to be solved for the amplitudes a is given as 3.2. Example 1: a 1-D filter ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 ð31Þ ð O 2 M þ iOC þ KÞa ¼ f: ð32Þ An application for band gaps in a 1-D lattice is as pass- or stop-band filters. In theory, for an infinite, and perfectly periodic lattice without damping, perfect filtering properties exist with alternating pass bands and complete stop bands. The properties of finite lattice structures subjected to periodic loading is here considered by analyzing the effect of the number of unit cells in the structure, viscous damping, and imperfections in the periodic structure. The unit cell considered in Section 2.1 with four masses and springs is used to describe the periodic structure. As previously stated, this can be seen as a discrete model of a 0:15 m unit cell consisting of 50% aluminum and 50% PMMA. In Section 2.1 it was shown that for the infinite lattice this unit cell displays a band gap between approximately 5.2–12 kHz: Another band gap appears above 13:5 kHz but the focus is here put on the response in the first gap. Fig. 6a shows the model of the structure with M unit cells. The structure is subjected to a periodic loading f cos Ot at the left end. Fig. 6b shows the corresponding computational mass– spring model with 10 unit cells ðM ¼ 10Þ and the four mass unit cell ðN ¼ 4Þ: In the following sections, the filtering properties of this structure are analyzed for different choice of system parameters. The response of the structure is typically given for the last mass in the lattice and presented as frequency response functions (FRF) showing the acceleration of the
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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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Page 37 and 38: 3.5 Plate structures 27 a) b) Figur
Page 39 and 40: 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
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Page 81 and 82: 1058 fundamental eigenfrequency o
Page 83 and 84: 1060 ðo 2 y;j;k ARTICLE IN PRESS J
Page 85: 1062 local resonator and splits up
Page 89 and 90: 1066 FRF (dB) 150 100 50 0 -50 -100
Page 91 and 92: 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)
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Page 109 and 110: ω/κ ω/κ 1.5 1.4 1.3 1.2 1.1 1 0
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Page 113 and 114: [B 400 ] 0.25 0.2 0.15 0.1 0.05 B.S
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Page 127 and 128: 1014 . ct . _ a) Q^ s^c "3
Page 129 and 130: 1016 O. Sigmund and J. S. Jensen (a
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Page 134 and 135: Z. Kristallogr. 220 (2005) 895-905
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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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