Revisited mode-expansion method for elastic strips

The mode-expansion method is widely applied in waveguide analysis. Particular applications usually include only a few modes to obtain explicit analytical results for simple waveguides. Here, we apply the method to the evaluation of the dynamic properties of elastic strips; to obtain useful results for thin strips, tens of plate modes must be involved in the expansion, representing the vibration shape of the strip. Although this increases the numerical complexity of the analysis, it is still easy for both numerical implementation and for physical interpretation of results. The final result we seek in this paper is the matrix of harmonic impedance of strips that provides dependence of displacements on both upper and lower sides of a strip of rectangular cross-section on arbitrary traction applied to these sides, where both the traction and displacements are expressed in the spatial Fourier series (other sides of strips are assumed to be stress-free). Detailed discussion of numerical issues is provided for isotropic strips and example applications are presented for strips interacting with surface waves or with other strips arranged in a "woodpile" super-lattice.