dynamic response and localization in strongly damaged waveguides by g. carta, m. brun, and a. b....
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![Page 1: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/1.jpg)
Dynamic response and localization in strongly damaged waveguides
by G. Carta, M. Brun, and A. B. Movchan
Proceedings AVolume 470(2167):20140136
July 8, 2014
©2014 by The Royal Society
![Page 2: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/2.jpg)
Railway viaduct in Piacenza, Italy (image taken from http://www.tensacciai.it, accessed on 31 January 2014).
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 3: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/3.jpg)
(a) Example of an elongated damaged structure: rail bridge over Westmoreland Road, Bath, UK (image taken from Google Maps Street View, https://maps.google.com/, retrieved on 31 January
2014); (b) schematic of a finite strip with equispaced cracks; (c) eigen...
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 4: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/4.jpg)
(a) Finite elastic strip with equispaced transverse cracks, that consists of five repeating cells; (b) detail of a repetitive cell.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 5: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/5.jpg)
Eigenfrequencies of finite strips possessing different lengths.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 6: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/6.jpg)
Dispersion diagrams of the infinite periodic strip for ρϵ/h=1/5 (a) and ρϵ/h=1/100 (b), obtained from the finite-element software COMSOL Multiphysics.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 7: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/7.jpg)
First eight modes (at increasing eigenfrequencies) relative to the standing waves (figure 5b) at k=π/l ((a), (b), (e), (f)) and k=0 ((c), (d), (g), (h)), provided by the finite-element code.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 8: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/8.jpg)
Periodic cell of a beam with elastic connections.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 9: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/9.jpg)
Elastic strip subjected to symmetric rotations (a) and antisymmetric displacements (b) at the ends.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 10: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/10.jpg)
Physical plane representation of the propagation zones.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 11: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/11.jpg)
Dispersion curves for an intact beam (solid grey lines) and for a damaged beam (solid thick black lines) with ρϵ=h/5 (a) and ρϵ=h/100 (b).
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 12: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/12.jpg)
Non-dimensional frequency parameter ϕ versus normalized wavenumber kl for ρϵ/h=1/5 (a) and ρϵ/h=1/100 (b).
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 13: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/13.jpg)
Lowest two eigenmodes of simple beams with different boundary conditions.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society
![Page 14: Dynamic response and localization in strongly damaged waveguides by G. Carta, M. Brun, and A. B. Movchan Proceedings A Volume 470(2167):20140136 July 8,](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649e8f5503460f94b93efb/html5/thumbnails/14.jpg)
First normalized eigenfrequency, as a function of the corresponding normalized stiffness, of the beams shown in figure 12.
G. Carta et al. Proc. R. Soc. A 2014;470:20140136
©2014 by The Royal Society