title: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. Journal of Sound and Vibration, 2004, 273 (1-2), pp.77-101. &#x27E8;10.1016/j.jsv.2003.04.005&#x27E9;. &#x27E8;hal-00830693&#x27E9;
text: Cubic coefficients The following coefﬁcients, which correspond to the trivially resonant terms, are equal to zero: 8p ¼ 1; y; N: upppp ¼ rpppp ¼ mpppp ¼ npppp ¼ 0; 8j > p; y; N: rppjj ¼ uppjj ¼ upjpj ¼ 0; mppjj ¼ nppjj ¼ npjpj ¼ 0; 8iop: rpiip ¼ upiip ¼ uppii ¼ 0; mpiip ¼ npiip ¼ nppii ¼ 0: The non-zero coefﬁcients are now given, 8p ¼ 1; y; N: 8i ¼ 1; y; N; iap: rpiii ¼ 1 h ð7o2i ð1Þ Dip i o2p Þðhpiii þ Apiii Þ þ 2o4i Bpiii ; ðA:2Þ 1 h upiii ¼ 6hpiii þ 6Apiii þ ð3o2i ð1Þ Dip i o2p ÞBpiii ; mpiii ¼ upiii ; 1 h ð9o2i ð1Þ Dip npiii ¼ 2 where Dð1Þ ip ¼ ðop o2i Þðo2p i 3o2p Þðhpiii þ Apiii Þ þ 2o2p o2i Bpiii ; 9o2i Þ: 8i ¼ 1; y; N rpijj ¼ o2i ðo2i 2o2j o2i ðo2i p uijj ¼ 2o2j 8o2j 6o2i þ o2p ÞDð1Þ ijp 2o2p Þ þ ðo2p ðo2i 2o2p o2p ÞDð1Þ ijp npjij ¼ 8o2j 2o2p Dð1Þ ijp þ mpijj ¼ 8o4j o2i ÞDð1Þ ijp 2 4oj Þ p Bijj ð1Þ o2i ÞDijp 2o2i 2o2j o2p 8o4j þ 2o2j o2i Dð1Þ ijp ðo2p 6o2p þ 2o2i ðo2p 8o2j o2i ÞDð1Þ ijp 2o2i Dð1Þ ijp o4p Bpjij ; i o2p þ o2i ÞBpjij ; o4i ½Apijj þ Apjij þ hpijj o2j ð o2p þ 4o2j o2i Þ p Bjij ; Dð1Þ ijp ½Apijj þ Apjij þ hpijj 6o2i o2j þ 2o2j o2p þ 2o2p o2i þ 2o2j Þ p Bijj 4o2j Bpijj þ ð4o2j ðo2p 2o4j ð3o2p þ o2i ðo2p 4o2j Þðo2p ½Apijj þ Apjij þ hijj 6o2i o2j þ 2o2j o2p þ 2o2p o2i þ i 2o2j Þh p Aijj þ Apjij þ hpijj o2p ÞDð1Þ ijp ðo2i 1 h 4ðApjij þ Apijj þ hpijj Þ ð1Þ Dijp upjij ¼ 4o2j Þðo2p 4o2j þ 3o2i Þ p 2o2i o2j p B þ Bjij ; ijj o2p ÞDð1Þ Dð1Þ ijp ijp ðo2i npijj ¼ 2o2p Þ þ ðo2p ðo2i 2o4j ðo2p 1; iap; 8j > i; y; N: Bpijj þ 8o4j o2p o2i þ 4o2j o2i Dð1Þ ijp o4p o2i ÞDð1Þ ijp ½Apijj þ Apjij þ hpijj þ o4i o4i Bpjij ; Bpijj 4o2j o2i Dð1Þ ijp o2p Bpjij ; where Dð1Þ ijp ¼ ðop þ oi 2oj Þðop þ oi þ 2oj Þð op þ oi þ 2oj Þð op þ oi 8i ¼ 1; y; N 2o2i ð4o2i rpiij ¼ 3o2p o2p ðo2p o2j ÞDð2Þ ijp upiij ¼ npjii ¼ 3o2j Þ 2o2i ð4o2i þ 1 Dð2Þ ijp 8o2i npiij ¼ 2o2p o2j ÞDð2Þ ijp j Apjii ½4ðApjii þ Apiij þ hpiij Þ 8o6i 2o2p o2i 8o2i 2o2j o4i ðo2j o2p ÞDð2Þ ijp 2o2p 2o2j h Dð2Þ ijp 2o2i o2j 2o2j o2p h o2i ðo2j 6o2p h o2j Bpjii Bpiij ; 4o2i ÞBpiij ; 4o2i þ o2p Þ i Dð2Þ ijp i þ 8o4i o2p ÞDð2Þ ijp 4o2i o2j o4p Dð2Þ ijp o4j o2p 2; 8j > i; y; N Bpiij ; Bpiij Bpjii ; oj Þðop þ 2oi þ oj Þð op þ 2oi þ oj Þð op þ 2oi 8i ¼ 1; y; N i Bpiij ; o4j þ 4o2i o2j þ o2j o2p Bpjii þ hpiij Apjii þ Apiij þ hpiij Dð2Þ ijp Apiij þ Apjii þ hpiij ðo2j 2o2j Dð2Þ ijp 6o2i o2j o2p ÞDð2Þ ijp 2 2 6oi oj þ 2o2j o2p þ 2o2i o2p where Dð2Þ ijp ¼ ðop þ 2oi i ðo2p Bpjii þ 8o4i þ 2o2i o2p 2o2j hpiij 4o2i Bpjii Apjii þ Apiij þ hpiij Dð2Þ ijp 8o2i p miij ¼ ðo2j Apiij o2p ÞDð2Þ ijp 6o2p o4i Apjii oj Þ2 ðop þ oj Þ2 o2j Þ þ ðop o2j ÞDð2Þ ijp ðo2p Apiij Bpiij ; Dð2Þ ijp ðo2p 6o2 h ðo2j þ 2o2i o2j Bpjii 3o2p 8o4i þ o4j þ o4p þ oj Þ2 ðop þ oj Þ2 h o2j ÞDð2Þ ijp 2o4i ð4o2i upjii ¼ 1; 8j > i; y; N; jap: o2j Þ þ ðop ðo2p 2oj Þ: 1; 8k > j; y; N: oj Þ: i upijk ¼ 1 Dð3Þ ijkp ½ð2o4j 6o4i 4o2k o2p þ 4o2i o2p þ 4o2i o2k þ 2o4k þ 4o2i o2j þ 2o4p 4o2j o2p Þðhpijk þ Apijk þ Apkij þ Apjik Þ 4o2k o2j þ ð o2k o4j o2i o4k 2o2j o2i o2p þ 3o2j o4p o4k o2j 2o2i o2k o2p þ 3o2i o4p o4j o2i o4i o2k þ 3o2k o4p þ o6i o6p 3o4k o2p þ o6k 2o2k o2j o2p o2j o4i ÞBpijk þ 10o2i o2k o2j þ o6j 3o4i o2p þ ð 2o4j o2i þ 4o4i o2p þ 4o2j o4i þ 6o2i o4k 4o4i o2k þ 4o2j o2i o2p 4o2i o2k o2j ÞBpjik 2o2i o4p 2o6i 4o2i o2k o2p þ ð 2o2i o4p þ 6o4j o2i þ 4o2i o2k o2p 4o2j o4i upjik ¼ 1 Dð3Þ ijkp 2o4i 2o4k þ 4o2i o2k 4o2j o2i o2p 2o6i þ 4o4i o2k ÞBpkij ; 2o2i o4k þ 4o4i o2p ½ð 2o4p þ 6o4j 4o2i o2k o2j 3o4j o2p 4o2k o2j þ 4o2k o2p 4o2i o2j þ 4o2i o2p 4o2j o2p Þðhpijk þ Apijk þ Apkij þ Apjik Þ þ ð2o6j þ 4o2k o2j o2p þ 4o2i o2k o2j þ 2o2j o4i þ 4o2k o4j 4o2j o2i o2p þ 2o2j o4p þ ð3o4j o2p þ o2k o4j 4o4j o2i 3o2j o4p o6i þ 3o4k o2p 10o2i o2k o2j þ ð4o2i o2k o2j þ 4o4j o2i þ 2o2j o4p þ 2o6j upkij ¼ Dð3Þ ijkp ½ð2o4p þ 2o4j þ 4o2i o2k þ 4o2k o2j o6j o6k þ 2o2j o2i o2p þ o6p 3o2i o4p þ o2j o4i ÞBpjik 6o2j o4i þ 2o4k o2j 4o4j o2p 4o2k o2j o2p þ 4o2j o2i o2p ÞBpkij ; 4o2k o4j 1 6o4k o2j ÞBpijk 3o2k o4p þ o2i o4k þ o4k o2j þ 3o4i o2p þ 2o2i o2k o2p þ o4j o2i þ o4i o2k þ 2o2k o2j o2p 4o4j o2p 4o2i o2p 4o2j o2p þ 2o4i Þðhpijk þ Apijk þ Apkij þ Apjik Þ 4o2i o2j þ ð 4o2i o2k o2j 6o4k þ 4o2k o2p 2o6k 2o4i o2k þ 6o2k o4j 2o2k o4p þ 4o2i o4k þ ð 2o6k 4o2i o4k þ 4o2k o2j o2p 4o2i o2k o2j þ 4o4k o2j 4o2k o2j o2p þ 4o2i o2k o2p 4o4k o2j þ 4o4k o2p ÞBpijk 4o2i o2k o2p 2o2k o4p þ 6o4i o2k 2o2k o4j þ 4o4k o2p ÞBpjik þ ð o2k o4j o2i o4k 2o2j o2i o2p þ 3o2j o4p o4k o2j 2o2i o2k o2p þ 3o2i o4p o4j o2i o4i o2k þ 3o2k o4p þ o6i o6p 3o4k o2p þ o6k 3o4j o2p þ 10o2i o2k o2j þ o6j 2o2k o2j o2p 3o4i o2p o2j o4i ÞBpkij ; rpijk ¼ 1 Dð3Þ ijkp ½ð o2k o4j o2i o4k 2o2j o2i o2p þ 3o2j o4p 2o2i o2k o2p þ 3o2i o4p o4j o2i 3o4j o2p þ 10o2i o2k o2j þ o6j þ ð4o2i o2k o4j o4k o2j o4i o2k þ 3o2k o4p þ o6i 2o2k o2j o2p 4o2k o4j o2p þ 4o2i o4k o2j o6p 3o4k o2p þ o6k o2j o4i Þðhpijk þ Apijk þ Apkij þ Apjik Þ 3o4i o2p 4o2j o4k o2p 6o4i o2k o2j þ 2o2k o2j o4p 4o4j o4k þ 2o2k o6j þ 2o2j o6k ÞBpijk þ 4o2i o2k o2j o2p þ ð4o2i o4k o2j þ 4o2i o2k o2j o2p þ 2o2i o2k o4p þ 2o2i o6k þ 2o6i o2k 6o2i o2k o4j 4o4i o2k o2p þ 4o4i o2k o2j ÞBpjik 4o2i o4k o2p þ ð 4o4i o4j þ 2o2i o2j o4p 6o2i o4k o2j þ 2o2i o6j þ 2o6i o2j 1 Dð3Þ ijkp ½ð 4o2i o2j þ 4o2i o2p þ 12o2i o2k þ 10o4j 6o4i 6o4k þ ð10o2i o2k o2j þ 3o6j 12o2j o2p Þðhpijk þ Apijk þ Apkij þ Apjik Þ 2o2k o2j o2p þ 2o2i o2k o2p 6o2j o2i o2p þ 9o2k o4j þ 3o2i o4p þ o6i o6p þ 5o2j o4p þ ð 18o2i o2k o2j 3o2j o4p þ 5o4k o2j þ 5o2j o4i þ 3o2i o4k þ o6p o6j þ 6o2j o2i o2p 5o2k o4p 3o6k 7o4j o2p þ o2i o4p 1 Dð3Þ ijkp o6i o6p þ 9o4j o2i 11o2j o4i 7o4j o2p o2k o4j þ 5o2i o2k o4p þ o6i o2j 6o2k o2j o2p 3o2i o4k 3o4k o2p þ o4i o2p þ 3o6j 7o4i o2p 11o4j o2i þ 2o2k o2j o2p 5o4i o2k þ 10o2i o2k o2j 6o2i o2k o2p þ 3o2k o4p 3o4k o2j Þðhpijk þ Apijk þ Apkij þ Apjik Þ 6o4i o2k o2p þ 18o4i o2k o2j þ 3o2i o6k o4j o2i þ 7o4i o2p 5o2k o4j ÞBpkij ; o6j þ 3o2k o4j þ o6k þ 9o2j o4i þ 3o6i þ 5o2i o4p þ o2j o4p þ ð3o4i o4p þ o8i 3o4i o2k ÞBpijk 5o4j o2i þ o2j o4i 2o2j o2i o2p þ 10o2i o2k o2j þ 3o2k o4p o6p þ o2i o4k 3o4i o2p 5o2i o4p ÞBpjik 6o2i o2k o2p ½ð 2o2j o2i o2p þ o4j o2p 3o4k o2p o6k 3o6i þ 3o4j o2p þ 7o4k o2p þ 3o4i o2k þ 6o2k o2j o2p þ 2o2i o2k o2p þ 5o2j o4p þ 3o4i o2k npkij ¼ 4o2k o2j þ 4o2k o2p þ 2o4p 11o4k o2j þ o2k o4p þ 3o2i o4k þ o4k o2p þ ðo4k o2j þ o6k 4o2i o4j o2p 4o4i o2j o2p þ 4o4i o2k o2j ÞBpkij ; þ 4o2i o2k o2j o2p þ 4o2i o2k o4j mpijk ¼ 4o4i o4k 5o4i o4k þ 6o2i o2k o2j o2p þ 5o2i o2j o4p 3o2i o2k o4j 5o4i o4j 7o2i o4k o2p 3o6i o2p 7o2i o4j o2p 6o4i o2j o2p þ o6i o2k o2i o6p þ 3o2i o6j ÞBpjik 3o2i o4k o2j þ ð3o2k o6j þ 6o2i o4j o2p þ 5o4i o4j þ 7o4i o2j o2p 3o4p o4j þ 3o6j o2p þ 11o2i o4k o2j þ 2o2i o2k o2j o2p þ o2j o6k þ ð 4o4i o2k o2j 1 Dð3Þ ijkp 5o2i o2j o4p 10o2i o2k o4j 4o4i o4j þ 3o4k o2j þ o2i o4p þ 2o2k o2j o2p þ o2k o6j o2k o2j o4p o8j ÞBpijk o6p 3o4j o2p þ 3o2j o4p þ 2o2j o2i o2p þ o6j o6k þ 2o2i o2k o2j 10o2i o2k o2p 3o4j o2i o6i þ o2k o4p Þðhpijk þ Apijk þ Apkij þ Apjik Þ 3o2k o4j þ ð 3o4i o4k 2o2k o4j o2p o2i o2k o4p 2o2j o4k o2p 2o2i o4k o2j 2o2i o4k o2p o4i o2k o2j þ 3o2i o6k o4i o2k o2p þ o6i o2k o2k o2j o4p 3o4p o4k þ 3o6k o2p 3o4j o4k o2i o2k o4j þ o2k o6p ÞBpijk þ ð 4o2i o2k o2j o2p þ 4o2i o4k o2j þ 4o4i o2k o2j 2o2i o6k þ 6o2i o2k o4p o8k þ 10o2i o2k o2j o2p þ 3o2j o6k 2o6i o2k þ 4o4i o4k 2o2i o2k o4j þ ð 2o4i o2j o2p þ 3o6i o2k o8i npijk ¼ 1 Dð3Þ ijkp ½ð 11o2k o4j þ 2o2j o2i o2p 2o4i o2k o2j o2i o2k o4j 4o4i o2k o2p 3o4i o4p o2i o2j o4p 2o4i o2k o2p o2i o2k o4p þ 3o6i o2j þ 10o2i o2k o2j o2p 3o4i o4j 3o4i o2p þ 9o4k o2j 3o2j o4i þ 3o6k o2i o4j o2p o2i o4k o2j ÞBpkij ; o6p þ o4j o2p 7o4k o2p o6j þ o4i o2k 6o2i o2k o2p þ o2j o4p þ 3o2i o4p þ 5o2k o4p þ o6i þ 3o4j o2i 2o2k o2j o2p þ 10o2i o2k o2j Þðhpijk þ Apijk þ Apkij þ Apjik Þ þ ð12o2i o2k o4j 6o2k o6j þ 2o2k o2j o4p 4o4j o4k 6o4i o2k o2j þ 10o2j o6k ÞBpijk 12o2j o4k o2p 6o2i o4k o2p 3o2i o2k o4j o2k o6p 3o4i o2k o2j 5o4j o4k þ 3o6i o2k þ o2i o6k þ 5o2i o2k o4p 7o4i o2k o2p 5o4i o4k 6o2j o4k o2p 7o2k o4j o2p ÞBpjik þ o8k þ 5o2k o2j o4p 3o6k o2p þ 3o4p o4k þ ð o4i o2j o2p 5o2k o2j o4p þ 7o2j o4k o2p þ 5o4j o4k þ 3o6j o2p o8j 5o2i o4k 4o2i o4k o2j þ 4o2i o2k o2j o2p þ 4o2k o4j o2p þ ð3o2k o6j þ 6o2i o2k o2j o2p þ o2j o6k þ 18o2i o4k o2j o2k o4j o2p 4o2i o4k o2p ÞBpjik 3o4i o4k þ 3o6i o2p o2i o4k o2p þ o2i o6j þ o2i o6k þ o2i o6p o2j o4k o2p 12o4i o2j o2p þ 2o2i o2j o4p þ 4o2i o4j o2p ½ðo2i o4k þ o4i o2p þ 3o2j o4i þ o4k o2p þ o4i o2k o2i o6j þ o6p o2j 6o2i o6j þ 4o2i o2k o2j o2p ÞBpkij ; þ 12o2i o2k o4j þ 10o6i o2j npjik ¼ 3o6i o2j 3o4j o4k 6o2i o4k o2j 9o4i o2k o2j o2k o6j 3o4p o4j þ o6i o2j þ 2o2i o2k o2j o2p þ 11o4i o2k o2j þ o6p o2j 10o2i o2k o4j 3o4i o4j 2o2i o4j o2p ÞBpkij ; 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