[Paper Review] Dispersion and localization in structured Rayleigh beams

A. Piccolroaz, A. B. Movchan|arXiv (Cornell University)|Aug 21, 2014

Thermoelastic and Magnetoelastic Phenomena2 references3 citations

TL;DR

This paper investigates dispersive wave behavior and localization in structured Rayleigh beams with rotational inertia, prestress, and Winkler foundation support, using a mathematical analogy to couple-stress materials. It derives closed-form dispersion equations and shows that rotational inertia induces wide low-frequency band gaps and strong localization, enabling wave filtering and potential applications in elastic wave shielding.

ABSTRACT

This paper brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, our main focus is on the analysis of dispersive waves, band-gaps and localized waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler--Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations.

Motivation & Objective

• To analyze dispersive wave propagation and band-gap formation in Rayleigh beams with rotational inertia and pre-stress.
• To establish a mathematical analogy between Rayleigh beams and couple-stress elastic materials for shared wave dynamics.
• To investigate how rotational inertia and elastic foundations influence wave localization and group velocity in periodic beam systems.
• To derive and solve quasi-periodic Green’s functions and dispersion equations for structured Rayleigh beams.
• To explore the potential of such systems for elastic wave filtering and dynamic shielding applications.

Proposed method

• Formulates the governing equation for a prestressed Rayleigh beam on a Winkler foundation, incorporating translational and rotational inertia: $EI hinspace v^{ ext{iv}} - P hinspace v'' + ho A hinspace ar{v} - ho I hinspace ar{v}'' + eta v = 0$.
• Establishes formal mathematical equivalence between the Rayleigh beam equation and the anti-plane shear wave equation in couple-stress materials, highlighting shared higher-order dispersion terms.
• Derives closed-form dispersion equations for periodic multi-mass systems in Rayleigh beams, using dimensionless parameters for mass $\overline{M}$ and moment of inertia $\overline{I}_M$.
• Constructs quasi-periodic Green’s functions for both translational and rotational inertia effects, enabling analysis of defect modes and localized waveforms.
• Solves the dispersion equation numerically to generate dispersion diagrams and analyze band-gap structure and group velocity behavior.
• Uses the rational function form of the dispersion equation in $Z = \cos(Kd)$ to enable accurate and efficient solution of band structures.

Experimental results

Research questions

• RQ1 How does rotational inertia in Rayleigh beams affect the formation and width of low-frequency band gaps?
• RQ2 What is the role of prestress and Winkler foundation stiffness in shaping the dispersion characteristics of structured Rayleigh beams?
• RQ3 How do the combined effects of translational and rotational inertia influence group velocity and wave localization in periodic beam systems?
• RQ4 In what way does the mathematical analogy between Rayleigh beams and couple-stress materials inform the analysis of wave dispersion and localization?
• RQ5 Can quasi-periodic Green’s functions be constructed in closed form for Rayleigh beams with rotational inertia, and how do they enable defect mode analysis?

Key findings

• Rotational inertia in Rayleigh beams leads to significant flattening of dispersion curves, especially in the high-frequency range, indicating reduced group velocity.
• For the second band, increasing rotational inertia causes a sign reversal in group velocity and curvature, indicating complex wave dynamics not present in Euler–Bernoulli beams.
• Wide low-frequency band gaps emerge when rotational inertia is dominant, as seen in the case where mass is negligible but rotational inertia is non-zero.
• The dispersion equation is a rational function of $\cos(Kd)$, enabling accurate and efficient solution of band structures for periodic systems.
• Defect modes and localized waveforms are observed in stop bands, with localization strength increasing with rotational inertia.
• The system exhibits non-dispersive behavior for a specific prestress value $P = EA$, analogous to non-dispersion in couple-stress materials at $J = 2\rho\ell^2$.