Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems |
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Authors: | W. Lacarbonara G. Rega |
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Affiliation: | a Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, via Eudossiana 18, 00184 Roma, Italy b Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, via A. Gramsci 53, 00197 Roma, Italy |
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Abstract: | The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented. |
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Keywords: | Internal resonances Non-linear orthogonality Non-linear normal mode Shallow arch Buckled beam Suspended cable |
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