WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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1190 B.S. Lazarov, J.S. Jensen / International Journal of Non-Linear Mechanics 42 (2007) 1186 –1193 ω/κ 1.7 1.6 1.5 1.4 1.3 1.3 1.2 1.2 0 0.01 1.1 1.1 0.05 0.1 1 1 0.2 0.5 0.9 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω/κ Re(μ u ) Im(μ u ) Fig. 5. Dispersion relation for = 0.1, = 0.01 and different values of the non-linear parameter A1,j Ā1,j . A1,j+1 of the attached oscillator can be calculated by solving the following equation: B1,j+1=A1,j+1 −1+ 2 2 +3 j+12 2i A1,j+1Ā1,j+1+ 2 (15) which is obtained from (10) and (13) by equating the coefficients of e it . Eq. (15) resembles the equation for the amplitude of the stationary response of externally excited Duffing oscillator. By using polar representation for the amplitudes A1,j+1 = RA(cos() + i sin()) and RB =|B1,j+1|, (15) can be written as 9 2 r 4 R 6 A + 6 j+1r 2 (r 2 − 1)R 4 A + ((r2 − 1) 2 + 4 2 r 2 )R 2 A − R2 B = 0, (16) where r = /. Eq. (16) can be represented as a cubic equation with respect to R2 A . It has either one, or three real roots for R2 A . In the second case, two of the solutions are stable and one of them is unstable, which is well known property of the Duffing oscillator [14]. There are two values of u corresponding to the two stable solutions of (16), and a unique value for B1,j+2 cannot be obtained. The two solutions for u,j+1 can be ordered by the magnitude of their real part. The one with larger absolute real value, l u,j+1 , produces an amplitude with a smaller absolute value, and the other one, u u,j+1 , produces an amplitude with a larger absolute value. Continuing in this way, an estimate for the upper and the lower bounds of the absolute value of the wave amplitude can be obtained: |B u 1,j+k |= e Re(u u,j+n ) |B u 1,j |, |B l 1,j+k |= n=1...k n=1...k e Re(l u,j+n ) |B l 1,j |. (17) In the case where only a single real solution for R 2 A exists, l u,j+k is equal to u u,j+k and |Bu 1,j+k |=|Bl 1,j+k |. 1.7 1.6 1.5 1.4 4. Numerical simulations The theoretical analysis derived in the previous section is validated by using numerical simulations for finite spring–mass chain, as shown in Fig. 1. Absorbing boundary conditions are applied at each end of the chain by adding dash-pot with a damping constant c = 1. The damping constant corresponds to the exact absorbing boundary condition for a continuous rod with distributed mass A=1 and stiffness EA=1, where is the mass density, A is the section area and E is the elastic modulus. A wave travelling in the right direction is generated by applying a harmonic force with frequency and unit amplitude at the left end of the chain. The first and the last 10 masses in the chain are without attached oscillators. The wave travels undisturbed, until it reaches the part with the attached oscillators, where a part of it is reflected back, and a part of it propagates until it reaches the right end of the chain. The system is integrated numerically, until steady state is reached. The root-mean-squared (RMS) value of the response amplitude [Bj ]= 1 1 t+T √ u 2 T t 2 j (t) dt (18) is calculated at the right end of the chain and is compared with the results obtained by using the analytical prediction derived in the previous section. The RMS value is calculated for a finite time interval T = 20, where = 2/ and is the frequency of the generated wave. The contribution of the higher order harmonics to the RMS value of the response amplitude is assumed to be small. 4.1. Influence of the non-linearities The transmission properties of a chain with 2000 attached oscillators calculated by using numerical simulations, and the procedure in Section 3.3, are shown in Fig. 6. The non-linear coefficients j = are taken to be the same for all attached oscillators. The results are obtained for several values of . The linear normalised frequency of the attached oscillators is 0.05. For small values of the non-linear coefficient the estimated
[B 2000 ] [B 2000 ] [B 2000 ] 0.25 0.2 0.15 0.1 0.05 0.25 B.S. Lazarov, J.S. Jensen / International Journal of Non-Linear Mechanics 42 (2007) 1186–1193 1191 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.2 0.15 0.1 0.05 0.25 ω/κ 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.2 0.15 0.1 0.05 ω/κ 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 ω/κ [B 2000 ] [B 2000 ] [B 2000 ] 0.25 0.2 0.15 0.1 0.05 0.25 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.2 0.15 0.1 0.05 ω/κ 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 ω/κ 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Fig. 6. Comparison of the theoretical prediction (dotted curves) of the transmitted amplitude with the one obtained by numerical integration for different values of = 0.0, 0.1, 0.25, 0.5, 1.0, 1.5, = 0.1, = 0.01. The RMS value of the amplitude of the first mass is 0.25. upper and lower values of the absolute value of the amplitude bound well the estimated RMS value of the amplitude from the numerical simulations. For large values of the non-linearity the results start to deviate. The frequency shift in the band gap can be clearly observed in all plots for = 0. In addition, the shape of the band gap changes, and the decaying rate decreases with increasing the non-linear coefficient. 4.2. Influence of the chain length The results obtained by both numerical simulations and analytical calculations for chains with different number of the attached oscillators are shown in Fig. 7. The non-linear parameter =0.3 is kept at a fixed value for all different spring–mass 0.25 0.2 0.15 0.1 0.05 ω/κ chains, and is taken to be the same for all attached oscillators. As can be seen, for small number of the attached oscillators the frequency shift of the band gap is significant. The reduction of the transmitted amplitude is relatively small compared with the one obtained for larger number of the attached oscillators. As the wave propagates along the chain, the amplitude is decreasing leading to frequency shift of the band gap back towards the linear natural frequency. The effect can be clearly observed by studying the plots. 4.3. Chains with variable non-linear coefficients If the wave frequency is located in the band gap, the amplitude is decaying fast, and the band gap is shifted back to
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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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4.4 Ridge waveguides 39 w ε =1 ?
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Relations to recent work 41 wave in
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Chapter 5 Advanced optimization pro
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5.1 Padé approximants 45 h 2h A f
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5.3 Space-time topology optimizatio
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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 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 and 92: 1068 3.3. Example 2: a 2-D waveguid
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Page 127 and 128: 1014 . ct . _ a) Q^ s^c "3
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Page 134 and 135: Z. Kristallogr. 220 (2005) 895-905
Page 136 and 137: Inverse design of phononic crystals
Page 144: Inverse design of phononic crystals
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
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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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