Wave propagation in a sandwich structure

The propagation of elastic waves in a sandwich structure with two thin stiff face-plates and a thick compliant core is considered in this paper. A complete description of the dispersion relation with no restrictions on frequency and wavelength is provided. This is accomplished by transforming the wave equation to a Hamiltonian system and then using a transfer matrix approach for solving the Hamiltonian system. To provide insight, particular regimes of the frequency–wavelength plane are then considered. First, an explicit formula is derived for all natural frequencies at the long wavelength limit. It is shown that all waves with finite limiting frequency have zero group velocity, while those with vanishing limiting frequency correspond to longitudinal, shear and flexural waves. The displacement of the flexural waves are reminiscent of Mindlin plates, and an asymptotic procedure to find the shear correction factor is presented. Second, the lowest branch of the dispersion relation is studied in detail and mode shapes are used to motivate explicit but accurate description of this lowest branch. This approximate model is anticipated to be useful in simulations of large structures with sandwich structures.

[1]  S. Timoshenko,et al.  LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars , 1921 .

[2]  Adnan H. Nayfeh,et al.  Wave propagation in layered anisotropic media : with applications to composites , 1995 .

[3]  T. C. T. Ting,et al.  Anisotropic Elasticity: Theory and Applications , 1996 .

[4]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[5]  R. D. Mindlin.  Waves and Vibrations in Isotropic, Elastic Plates , 1989 .

[6]  N. A. Haskell.  The Dispersion of Surface Waves on Multilayered Media , 1953 .

[7]  E. Reissner,et al.  On bending of elastic plates , 1947 .

[8]  C. Bert,et al.  The behavior of structures composed of composite materials , 1986 .

[9]  S. Timoshenko,et al.  X. On the transverse vibrations of bars of uniform cross-section , 1922 .

[10]  S. BRODETSKY,et al.  Theory of Plates and Shells , 1941, Nature.

[11]  N. Stephen.  MINDLIN PLATE THEORY: BEST SHEAR COEFFICIENT AND HIGHER SPECTRA VALIDITY , 1997 .

[12]  J. C. Burkill,et al.  The theory of ordinary differential equations , 1956 .

[13]  L. Knopoff.  A matrix method for elastic wave problems , 1964 .

[14]  F. V. Atkinson,et al.  Discrete and Continuous Boundary Problems , 1964 .

[15]  E. Reissner.  The effect of transverse shear deformation on the bending of elastic plates , 1945 .

[16]  Anders Nilsson,et al.  Modelling the vibration of sandwich beams using frequency-dependent parameters , 2007 .

[17]  R. S. Khurmi.pdf,et al.  Strength of Materials , 1908, Nature.

[18]  S. Sorokin.  Analysis of propagation of waves of purely shear deformaton in a sandwich plate , 2006 .

[19]  Pizhong Qiao,et al.  Higher-order impact modeling of sandwich structures with flexible core , 2005 .

[20]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[21]  H. Lamb.  On waves in an elastic plate , 1917 .

[22]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[23]  Anders Nilsson,et al.  Wave propagation in and sound transmission through sandwich plates , 1990 .

[24]  Paolo Bonfiglio,et al.  Aspects of computational vibration transmission for sandwich panels , 2007 .

[25]  Sergey Sorokin,et al.  Analysis of Wave Propagation in Sandwich Plates with and without Heavy Fluid Loading , 2004 .

[26]  Rakesh K. Kapania,et al.  FREE VIBRATION ANALYSIS OF LAMINATED PLATES USING A LAYER-WISE THEORY , 1993 .

[27]  H. G. Allen.  Analysis and design of structural sandwich panels , 1969 .

[28]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[29]  Michel Castaings,et al.  Delta operator technique to improve the Thomson–Haskell‐method stability for propagation in multilayered anisotropic absorbing plates , 1994 .

[30]  R. Christensen,et al.  A HIGH-ORDER THEORY OF PLATE DEFORMATION, PART 1: HOMOGENEOUS PLATES , 1977 .

[31]  D. J. Mead,et al.  The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions , 1969 .

[32]  R. Christensen,et al.  A High-Order Theory of Plate Deformation—Part 2: Laminated Plates , 1977 .

[33]  A. Nilsson,et al.  Modeling Flexural Vibration of a Sandwich Beam Using Modified Fourth-order Theory , 2006 .

[34]  Yeoshua Frostig,et al.  Free Vibrations Of Sandwich Beams With A Transversely Flexible Core: A High Order Approach , 1994 .

[35]  W. Thomson,et al.  Transmission of Elastic Waves through a Stratified Solid Medium , 1950 .

[36]  Lord Rayleigh,et al.  On the Free Vibrations of an Infinite Plate of Homogeneous Isotropic Elastic Matter , 1888 .

[37]  Rakesh K. Kapania,et al.  Free vibration analysis of laminated plates using a layerwise theory , 1993 .

[38]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[39]  Ole Thybo Thomsen,et al.  Sandwich Structures 7: Advancing with Sandwich Structures and Materials. Proceedings of the 7th International Conference on Sandwich structures, Aalborg University, Aalborg, Denmark, 29-31 August 2005 , 2005 .