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Research Papers

Phononic Band Gap Systems in Structural Mechanics: Finite Slender Elastic Structures and Infinite Periodic Waveguides

[+] Author and Article Information
Michele Brun

Assistant Professor
Dipartimento di Ingegneria Meccanica,
Chimica e dei Materiali,
Università di Cagliari,
Cagliari I-09123, Italy;
Department of Mathematical Sciences,
University of Liverpool,
Liverpool L69 3BX, UK
e-mail: mbrun@unica.it

Alexander B. Movchan

Professor
Department of Mathematical Sciences,
University of Liverpool,
Liverpool L69 3BX, UK
e-mail: abm@liv.ac.uk

Ian S. Jones

Professor
School of Engineering,
John Moores University,
Liverpool L3 3AF, UK
e-mail: I.S.Jones@ljmu.ac.uk

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 19, 2012; final manuscript received November 9, 2012; published online June 6, 2013. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 135(4), 041013 (Jun 06, 2013) (9 pages) Paper No: VIB-12-1111; doi: 10.1115/1.4023819 History: Received April 19, 2012; Revised November 09, 2012

The paper presents a novel spectral approach, accompanied by an asymptotic model and numerical simulations for slender elastic systems such as long bridges or tall buildings. The focus is on asymptotic approximations of solutions by Bloch waves, which may propagate in a infinite periodic waveguide. Although the notion of passive mass dampers is conventional in the engineering literature, it is not obvious that an infinite waveguide problem is adequate for analysis of long but finite slender elastic systems. The formal mathematical treatment of a Bloch wave would reduce to a spectral analysis of equations of motion on an elementary cell of a periodic structure, with Bloch–Floquet quasi-periodicity conditions imposed on the boundary of the cell. Frequencies of some classes of standing waves can be estimated analytically. One of the applications discussed in the paper is the “dancing bridge” across the river Volga in Volgograd.

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Figures

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Fig. 1

Volga bridge, ((a), (b) HTB Volgograd News). Flexural deformation of the main upper body of the bridge: (b) real structure, (c) simplified numerical model, and (d) modified 3D structured wave guide with lightweight resonators.

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Fig. 2

(a) Dispersion diagram for a periodic phononic band gap structure representing a bridge on periodically placed elastic pillars. (b) Frequency comparison between analytical computations of the pass bands of the periodic structure corresponding to flexural vibrations and eigenfrequencies f = 1.921 Hz and f = 1.894 Hz of the two span bridge computed with finite elements. Two sets of curves corresponding to flexural vibrations in vertical and transverse directions are shown.

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Fig. 3

The flexural eigenmodes of the bridge used for numerical comparison in Fig. 2.

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Fig. 4

Two-dimensional elementary cell with the system of resonators. (a) The first eigenmode. (b) Truss model used for the analytical estimates of the eigenfrequencies: each mass Mi undergoes the horizontal and vertical displacements u1(i) and u2(i), respectively, with i = 1,2. Parameter values are: d = 4 m; the thickness s = 0.2 m; the radii of the disks are 0.1 m and 0.075 m; h1 = 2 m, h2 = 1 m, β = π/6; the longitudinal stiffness coefficients are γ = 0.14 GPa, γ1 = 0.018 GPa; the main plate has mass density ρ = 7850 kg/m3 and shear modulus μ = 80 GPa; the disks and the elastic links have mass density ρM = 7850 kg/m3 and ργ = 200 kg/m3, respectively. The added mass is approximately 7% of the total mass.

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Fig. 5

Dispersion diagrams and corresponding eigenmodes for the 2D structured wave guide (finite element computations). (a) The original beam structure and the eigenmode to suppress. FA and FB correspond to the analytical values given in Eq. (10). (b) The modified structure and the first several eigenmodes with corresponding normalized frequencies. F¯A and F¯B are finite element results for the analytical frequencies FA and FB. Vibrating modes are shown in Ref. [34].

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Fig. 6

First four standing wave modes of the resonant structure. The upper deck of the bridge is fixed. The first mode, at normalized frequency F=fd/v=0.0181, is designed to eliminate the flexural vibration of the deck of the bridge. In the computation d = 4 m and v=μ/ρ=3194 m/s is the shear wave speed in the upper deck.

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Fig. 7

Modified bridge: the flexural vibration of the main body has been suppressed. The built-in resonators have taken on the vibrational motion. The added mass is 1.17% of the total mass. The scale bar indicates the normalized displacement magnitude. The vibrating mode is shown in Ref. [34].

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Fig. 8

Eigenmodes of the multistory buildings. The horizontal foundation of the structure is fixed, whereas all the remaining boundary is traction free. Panel 1: first four lower frequency eigenmodes of the multistructure, each showing energy absorption by the resonators. The scale bar indicates the normalized displacement magnitude. Normalized eigenfrequencies F=fd/v: (a) 0.0225; (b) 0.2254; (c) 0.3936; (d) 0.4446. Panel 2: two lower frequency eigenmodes of the frame structure. Normalized eigenfrequencies: (e) 0.0144; (f) 0.0260. (g) The dispersion diagram showing normalized frequency F versus k2d. (h) Square elementary cell with dimension d = 1 m. The external structure has thickness s = 0.1 m, the resonator is composed of a rectangular solid with dimensions 0.3 m × 0.14 m and three thin ligaments with dimensions 0.2 m × 0.01 m. Material parameters are: shear modulus μ = 80 GPa, Poisson ratio ν = 0.28, mass density ρ = 7850 Kg/m3. Vibrating modes are shown in Ref. [34].

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