From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: (a) Model I: a sandwich beam with resonators in series and (b) model II: a sandwich beam with resonators in parallel

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Sketch of a unit cell (left) and its equivalent model (right) for (a) model I and (b) model II

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: A two degrees-of-freedom system

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Two spring–mass systems

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Attenuation factor β for the sandwich beam with resonators in series (model I) (a) m2/m1=0.5; (b) m2/m1=1.0; and (c) m2/m1=1.5 (a = 0.012 m, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Attenuation factor β for the sandwich beam with resonators in series (model I) (a) k2/k1=0.5; (b) k2/k1=1.0; and (c) k2/k1=1.5 (a = 0.012 m, m2/m1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Effect of m2/m1 for model I on edge frequencies of (a) the first bandgap and (b) the second bandgap (a = 0.012 m, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Effect of k2/k1 for model I on edge frequencies of (a) the first bandgap and (b) the second bandgap (a = 0.012 m, m2/m1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Effect of a for model I on edge frequencies of (a) the first bandgap and (b) the second bandgap (m2/m1=0.5, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Dispersion curves for model I, obtained by the continuum model and finite element (FE) model (a = 0.012 m, m2/m1=1.5, k2/k1=1)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Nondimensionalized wave number for model I as a function of dimensionless frequency ω¯, obtained by the continuum model with distributed masses and the one with effective mass (a = 0.012 m, m2/m1=1.5, k2/k1=1)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Dimensionless effective mass per unit length for model I with varying (a) m2/m1 and (b) k2/k1 (a = 0.012 m)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Attenuation factor β for the sandwich beam with resonators in parallel (model II) (a) m2/m1=0.5 and (b) m2/m1=1.5 (a = 0.012 m, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Attenuation factor β for the sandwich beam with resonators in parallel (model II) (a) k2/k1=0.5 and (b) k2/k1=1.5 (a = 0.012 m, m2/m1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Comparison of attenuation factor β for the sandwich beam having a two-resonator system with m2/m1=1.0 and k2/k1=1.0 (model II) and for the beam with one spring–mass system (a = 0.012 m)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Effect of m2/m1 for model II on edge frequencies of (a) the first bandgap and (b) the second bandgap (a = 0.012 m, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Effect of k2/k1 for model II on edge frequencies of (a) the first bandgap and (b) the second bandgap (a = 0.012 m, m2/m1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Effect of a for model II on edge frequencies of (a) the first bandgap and (b) the second bandgap (m2/m1=0.5, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Dispersion curves for model II, obtained by the continuum model and FE model (a = 0.012 m, m2/m1=1.5, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Nondimensionalized wave number for model II as a function of dimensionless frequency ω¯, obtained by the continuum model with distributed masses and the one with effective mass (a = 0.012 m, m2/m1=1.5, k2/k1=1.0)

From: Wave Propagation in Sandwich Structures With Multiresonators Date of download: 9/18/2016 Copyright © ASME. All rights reserved. From: Wave Propagation in Sandwich Structures With Multiresonators J. Vib. Acoust. 2016;138(4):041009-041009-10. doi:10.1115/1.4033197 Figure Legend: Dimensionless effective mass per unit length for model II with varying (a) m2/m1 and (b) k2/k1 (a = 0.012 m)