On the existence of antisymmetric or symmetric Lamb waves at non-linear higher harmonics

Abstract

This article theoretically studies the symmetry characteristics of Rayleigh-Lamb guided waves in nonlinear, isotropic plates. It has been known that the nonlinearity driven double harmonic in Lamb waves does not support antisymmetric motion. However the proof of this has not been obvious. Moreover, little is known on nonlinearity driven Lamb harmonics higher than double. These gaps were here studied by the method of perturbation coupled with wavemode orthogonality and forced response. This reduced the nonlinear problem to a forced linear problem which was subsequently investigated to formulate an energy level constraint as the defining factor for the absence of antisymmetry at any order of higher harmonic. This constraint was then used to explain the reason behind the absence of antisymmetric Lamb waves at the double harmonic. Further, it was shown that antisymmetric motion is prohibited at all the higher-order even harmonics, whereas all the higher order odd harmonics allow both symmetric and antisymmetric motions. Finally, experimental results corroborating theoretical conclusions are presented.

Cumulative second harmonics in weakly nonlinear plates and shells

“…It has been shown that the secondary wave field is cumulative, that is the amplitude increases linearly with propagation distance, when the primary and secondary wave field are in internal resonance [8][9][10][11][12] . The internal resonance condition is met when: (i) the wave motion is synchronized (i.e., the phase velocities match) and (ii) there is nonzero power flux from the primary mode to the secondary mode 8 .…”

“…The secondary modes are plotted as double harmonics. It has been shown that only symmetric Lamb modes can be generated as second harmonics from a single source [9][10][11] . Thus, Figure 4a shows primary Lamb modes and secondary symmetric Lamb modes, and Figure 4b shows primary SH modes and secondary symmetric Lamb modes.…”

“…Full details can be found elsewhere [6][7][8][9][10][11][12][13][14][15] . A curvilinear coordinate system is used for the shell, and both kinematic and material nonlinearities are included in the formulation.…”

Interaction of torsional and longitudinal guided waves in weakly nonlinear circular cylinders

“…occurs to L(0,1), L(0,3), L(0,6), and L(0,8). L(0,5) and L(0, 9) correspond to symmetric Lamb modes in the plate, while L(0,1), L(0,3), L(0,6), and L(0,8) correspond to antisymmetric Lamb modes. In plates, antisymmetric secondary modes receive no power flux due to their symmetry properties.…”

“…Likewise, Cantrell 5 studied the harmonic generation in cyclically stressed wavy slip metals and correlated the nonlinearity parameter b with the percentage of total life expended. The higher harmonic generation of guided waves in plates has been studied by Deng,6,7 de Lima and Hamilton, 8 Srivastava and Lanza di Scalea, 9 and M€ uller et al 10 These studies formulated the general conditions for the generation of cumulative higher harmonics in plate, such as nonzero power flux from the fundamental to the secondary wave fields and a phase matching condition. M€ uller et al 10 also considered group velocity matching as a necessary condition for cumulative second harmonic generation.…”