Asymptotic Theory of Generalised Rayleigh Beams and the Dynamic Coupling

Abstract

We consider the effective dynamic response of an infinite asymmetric structure formed from a beam attached to a periodic array of resonators. The array couples the axial and flexural motions of the beam. We develop a point-wise description of the response of the system and demonstrate that when the separation of the resonators is small, the structure is approximated by the generalised Rayleigh beam: a beam attached to an elastic resonant layer that engages flexural-longitudinal wave coupling. The passage to the effective model utilises a dynamic homogenisation based on the method of meso-scale approximations, whose efficiency is not limited to the typical low-frequency regime of usual homogenisation approaches. The dispersion properties of this effective system are completely characterised and we illustrate the multi-coupling of waves through the analysis of Green’s matrix of this flexural system, which is constructed in the explicit form. The analytical results are also accompanied by illustrative numerical computations.

Appendix A: Resonance Modes

Here we discuss the response of the generalised Rayleigh beam at the resonant frequencies (11.25).

Resonant Response at $\omega _L$

Here $\omega _L$ corresponds to the longitudinal resonance of the medium atop the lateral beam. When $\omega _L$ is distinct from $\omega ^{\pm }_F$, the system (11.1) and (11.2) implies $V(x)=0$ and the longitudinal displacement obeys

$$\begin{aligned}(\rho A +\varPsi _{\text {eff}}(\omega _L) )\omega _L^2U(x)+EA U^{\prime \prime }(x)=0\;,\end{aligned}$$

for $-\infty<x<\infty$. Its solution is standard and given as

$$\begin{aligned}U(x)={A}\sin \left( \sqrt{\frac{\rho A +\varPsi _{\text {eff}}(\omega _L) }{EA}}\omega _L x\right) +B\cos \left( \sqrt{\frac{\rho A +\varPsi _{\text {eff}}(\omega _L) }{EA}}\omega _Lx\right) \;,\end{aligned}$$

where A and B are arbitrary constants. Hence in this resonance regime, flexural deformations of the system are not supported, whereas longitudinal waves can propagate through the medium.

Resonant Response at $\omega _F^{\pm }$

The frequency $\omega _F^{\pm }$ represents the flexural resonant response of the layer atop the beam. If this is distinct from $\omega _L$, one has at this frequency that U(x) and V(x) satisfy

$$\begin{aligned}\varPsi ^*_{\text {eff}}(\omega ^*) U({x})+\varPi ^*_{\text {eff}}(\omega ^*) V^\prime ({x})=0\end{aligned}$$

$$\begin{aligned}{-}\varPi ^*_{\text {eff}}(\omega ^*)U^{\prime }({x}) +\varUpsilon ^*_{\text {eff}}(\omega ^*) V^{\prime \prime }({x})=0\end{aligned}$$

where $\omega =\omega ^{\pm }_F$ and the notation $Q^*(\omega ^*)$ represents

$$\begin{aligned}Q^*(\omega ^*)=\lim _{\omega \rightarrow \omega ^*}(\omega -\omega ^*)Q(\omega )\;.\end{aligned}$$

This previous system admits the solution:

$$\begin{aligned}U(x)=-\frac{A\varPi ^*_{\text {eff}}(\omega ^*)}{\varPsi ^*_{\text {eff}}(\omega ^*)}\;, \qquad V(x)=A x +B\end{aligned}$$

where A and B are arbitrary constants, provided

$$\begin{aligned}\varPsi ^*_{\text {eff}}(\omega ^*)\varUpsilon ^*_{\text {eff}}(\omega ^*)+(\varPi ^*_{\text {eff}}(\omega ^*) )^2\ne 0\;.\end{aligned}$$

The result suggests at this resonance frequency the flexural displacement is linear and this is accompanied by a constant axial deformation.