WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

More documents

Recommendations

Info

ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 1061 Fig. 4. (a) The 5 5 mass–spring unit cell (denoted type 1) modelling a stiff inclusion (center 3 3 masses and springs) in a surrounding matrix, and (b) the corresponding band structure for wave propagation in the infinite periodic lattice structure. As appears from Fig. 4b a gap appears in the band structure for oE46:6–57:3 kHz between the third and fourth bands. In this frequency range waves cannot propagate in the infinite lattice regardless of the direction of propagation. The band gap calculated for this mass-spring unit cell model corresponds qualitatively to the band gap found for the corresponding continuum model, see e.g., Refs. [15,20]. 2.2.2. Band gaps for a heavy resonator: type 2 unit cell Alternatively, band gaps can be obtained in the lower frequency range by placing a heavy inclusion in soft suspension with a surrounding matrix material. The heavy inclusion acts as a
1062 local resonator and splits up the band structure. Recently, band gaps for such structures were demonstrated in Refs. [14,22], with the former pointing out a possible application in noise reduction due to the low band gap frequency range. Fig. 5a shows the unit cell and the corresponding band structure is shown in Fig. 5b. The masses and springs in the model have been chosen as mmat ¼ 1:27 10 2 kg; minc ¼ 9:93 10 2 kg; kmat;1 ¼ kmat;3 ¼ 2 kmat;2 ¼ 2 kmat;4 ¼ 4:10 10 9 kg=s 2 ; kinc;1 ¼ kinc;3 ¼ 2 kinc;2 ¼ 2 kinc;4 ¼ 118 10 9 kg=s 2 ; ksusp;1 ¼ ksusp;3 ¼ 2 ksusp;2 ¼ 2 ksusp;4 ¼ 4:00 10 6 kg=s 2 : ð23Þ With the parameters in Eq. (23) the model corresponds to a 0:02 m 0:02 m unit cell of an epoxy matrix with a copper inclusion suspended in a thin massless layer of silicone rubber. 3 The band gap frequency range can be accurately predicted. The local resonance of the heavy inclusion in the soft suspension splits up the band structure and determines the lower bound for the band gap rffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi Ksusp 7ksusp o1E ¼ E1:34 kHz: ð24Þ Minc 4minc The first possible propagation mode above the band gap is when the inclusion and the matrix material can move essentially rigidly in anti-phase. With the matrix motion denoted x and the motion of the inclusion y; the rigid motion is governed by the 2-d.o.f. system Mmat .x ¼ Ksuspðy xÞ; ð25Þ Minc .y ¼ Ksuspðx yÞ; ð26Þ that yields the upper bound frequency sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2E KsuspðMinc þ MmatÞ ¼ 7ksuspð4minc þ 32mmatÞ E1:88 kHz ð27Þ MincMmat ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 4minc32mmat as well as the ratio of amplitudes at the upper frequency bound y x ¼ 1 o2 Mmat 2 ¼ Ksusp Mmat ¼ Minc 32mmat E 1:02: ð28Þ 4minc It is noted that the frequency bounds in Eqs. (24) and (27) correspond well to the gap frequencies shown in the inset in Fig. 5b. 3 Material data: Eepo ¼ 4:1 GPa; r epo ¼ 1140 kg=m 3 ; Ecop ¼ 118 GPa; r cop ¼ 8940 kg=m 3 ; Erub ¼ 4 MPa:
Page 1 and 2:
WAVES AND VIBRATIONS IN INHOMOGENEO
Page 3:
Denne afhandling er af Danmarks Tek
Page 7 and 8:
List of Thesis Papers [1] J. S. Jen
Page 9 and 10:
Contents Preface i List of Thesis P
Page 11 and 12:
Chapter 1 Introduction This thesis
Page 13 and 14:
eplacemen Phononic and photonic ban
Page 15 and 16:
Optimal material distribution - top
Page 17 and 18:
Chapter 2 The bandgap phenomenon In
Page 19 and 20:
2.1 Band diagram and forced vibrati
Page 21 and 22:
2.3 Experimental demonstration 11 F
Page 23 and 24:
y 2.5 Nonlinearities 13 Response in
Page 25 and 26:
Relations to recent work 15 Amplitu
Page 27 and 28:
Chapter 3 Bandgap structures as opt
Page 29 and 30:
3.1 Vibration-quenching structures
Page 31 and 32:
3.2 Maximizing wave reflection 21 d
Page 33 and 34: 3.4 Maximizing wave dissipation 23
Page 35 and 36: 3.4 Maximizing wave dissipation 25
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
Page 77 and 78: 1054 ARTICLE IN PRESS J.S. Jensen /
Page 81 and 82: 1058 fundamental eigenfrequency o
Page 83: 1060 ðo 2 y;j;k ARTICLE IN PRESS J
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 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)
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
Page 119 and 120: 1006 (a) - - - - - - _ - - - - I I
Page 121 and 122: 1008 O. Sigmund and J. S. Jensen we
Page 123 and 124: 1010 O. Sigmund and J. S. Jensen Th
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
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144:
Page 147 and 148:
320 Optimization of the dissipation
Page 149 and 150:
322 The reflection of the wave is f
Page 151 and 152:
324 which averaged over a wave peri
Page 153 and 154:
326 where the modified stress compo
Page 155 and 156:
328 optimization algorithm is enhan
Page 157 and 158:
330 reflectance are also seen (Fig.
Page 159 and 160:
332 It should be emphasized that ot
Page 161 and 162:
334 b Dissipation, D c Dissipation,
Page 163 and 164:
336 7.2. Three-phase design ARTICLE
Page 165 and 166:
338 ARTICLE IN PRESS J.S. Jensen /
Page 167 and 168:
340 ARTICLE IN PRESS J.S. Jensen /
Page 169 and 170:
968 ARTICLE IN PRESS J.S. Jensen, N
Page 171 and 172:
970 To solve the wave equation with
Page 173 and 174:
972 In the following section, we sh
Page 175 and 176:
974 Design variable, t Design varia
Page 177 and 178:
976 Eigenvalue ratio, ω n+1 2 /ωn
Page 179 and 180:
978 to a design parameter te are gi
Page 181 and 182:
980 ARTICLE IN PRESS J.S. Jensen, N
Page 183 and 184:
982 respect to the higher bound C1
Page 185 and 186:
984 instead vary the value of b. Th
Page 187 and 188:
986 References ARTICLE IN PRESS J.S
Page 189 and 190:
a thick solid was considered. A few
Page 191 and 192:
The above expressions provide insta
Page 193 and 194:
In all the examples shown a density
Page 195 and 196:
domain. To the very left the passiv
Page 197 and 198:
Jensen JS, Sigmund O (2005) Topolog
Page 199 and 200:
paper extends on this work. For an
Page 201 and 202:
R 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 203 and 204:
objective 1 0.9 0.8 0.7 0.6 0.5 0.4
Page 205 and 206:
The optimization algorithm employs
Page 207 and 208:
Appl. Phys. Lett., Vol. 84, No. 12,
Page 210 and 211:
J. S. Jensen and O. Sigmund Vol. 22
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Topology optimization and fabricati
Page 220 and 221:
Normalized transmission 1.0 0.8 0.6
Page 222 and 223:
Fig. 4. Scanning electron micrograp
Page 224 and 225:
Broadband photonic crystal waveguid
Page 226 and 227:
2. Design and fabrication Silicon-o
Page 228 and 229:
Fig. 3. Steady-state magnetic field
Page 230 and 231:
Topology optimised broadband photon
Page 232 and 233:
1202 IEEE PHOTONICS TECHNOLOGY LETT
Page 234:
1204 IEEE PHOTONICS TECHNOLOGY LETT
Page 237 and 238:
11. P. Debackere, S. Scheerlinck, P
Page 239 and 240:
nanoimprinted patterns are transfer
Page 241 and 242:
emoved and the spectra changed dras
Page 243 and 244:
Y-splitter W1 waveguide 60-degree b
Page 245 and 246:
5. Photonic Wire Waveguides Figure
Page 247 and 248:
Figure 5: Optimized designs and cor
Page 249 and 250:
Figure 8: Optimized designs and cor
Page 251 and 252:
[13] Borel P I, Frandsen L H, Harp
Page 253 and 254:
1606 J. S. JENSEN To compute a freq
Page 255 and 256:
1608 J. S. JENSEN Cramer’s rule [
Page 257 and 258:
1610 J. S. JENSEN The following pro
Page 259 and 260:
1612 J. S. JENSEN Table I. Coeffici
Page 261 and 262:
1614 J. S. JENSEN h 2h Α fcos Ωt
Page 263 and 264:
1616 J. S. JENSEN The derivative of
Page 265 and 266:
1618 J. S. JENSEN which is solved f
Page 267 and 268:
1620 J. S. JENSEN Based on the expr
Page 269 and 270:
1622 J. S. JENSEN Response, ln(u ti
Page 271 and 272:
1624 J. S. JENSEN h 2h fcos Ωt Fi
Page 273 and 274:
1626 J. S. JENSEN details and have
Page 275 and 276:
1628 J. S. JENSEN Equation (A10) co
Page 277 and 278:
1630 J. S. JENSEN 7. Coyette J-P, L
Page 279 and 280:
The number of masses in the chain t
Page 281 and 282:
5 Results The optimization algorith
Page 283 and 284:
Figure 9. Response for optimized de
Page 286 and 287:
Space-time topology optimization fo
Page 288 and 289:
good for low to moderate stiffness
Page 290 and 291:
V ¼ 0:3 at a given time instant fo
Page 292 and 293:
The parameters used in this example
Page 294 and 295:
Normalized amplitude 1 0.9 0.8 0.7
Page 296:
at the Department of Mechanical Eng
Page 299 and 300:
600 J.S. JENSEN techniques for obta
Page 301 and 302:
602 J.S. JENSEN 1 and 2, where the
Page 303 and 304:
604 J.S. JENSEN 4. Numerical analys
Page 305 and 306:
606 J.S. JENSEN Energy, E a) Energy
Page 307 and 308:
608 J.S. JENSEN relaxed somewhat in
Page 309 and 310:
610 J.S. JENSEN when the contrast i
Page 311 and 312:
612 J.S. JENSEN 6. Summary and conc
Page 313:
614 J.S. JENSEN Sigmund, O. and Jen
show all

WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?