![]() ![]() ![]() In this chapter, some recent results on the metamaterial properties of one- and two-dimensional magnonic crystals are presented. Roberto Zivieri, in Solid State Physics, 2012 Abstract In all cases, the distinction of extraneous modes may be facilitated by simultaneous examinination of their associated damping ratios. Stabilization diagrams may indeed be useful, although the expected pattern is often distorted by phenomena such as frequency splitting and stabilization of extraneous (noise) frequencies. Their basis for structural mode distinction lies with the expectation that structural frequencies will tend to stabilize (remain invariant) as the order increases, whereas extraneous frequencies will change ‘randomly’ within the considered frequency range. Note that Δ ijl ≜ E ijl | / E ij, with E ij representing the energy of the ith vibration response due to the jth excitation, and E ijl that part of E ij that is associated with mode l.įrequency stabilization diagrams represent the evolution of estimated natural frequencies with increasing model order. ∥ matrix norm, and ɛ a selected threshold. ![]() With Δ ijl representing the dispersion of mode l in the ijth transfer function, ∥ In a discrete scenario, we assume that the surfaces S k are represented by a collection of vertices □ For instance, for every point p ∈ S, the L-B eigensystem gives rise to a m-dimensional feature vector s m ( p ) : = ( λ j − 1 / 2 ϕ j ( p ) ) j = 1 m, where the number m is user supplied. The L-B approach is especially effective when registering shapes with different poses. SR strategies have been investigated to match L-B eigensystems as good as possible for surfaces undergoing isometric deformations see, e.g., Rustamov (2007). Two isometric surfaces have the same L-B eigensystems ( Shi et al., 2008). ![]() This L-B eigensystem can be used to define a shape signature for a surface up to isometry. Īs the L-B operator Δ S is self-adjoint and elliptic, it has a system of eigenvalue and corresponding eigenfunctions ( λ j, ϕ j) with −Δ S ϕ j = λ j ϕ j and λ j ≤ λ j+1 for all j ∈ N. ![]()
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