Super-Harmonic and Internal Resonances of a Suspended Cable with Nearly Commensurable Natural Frequencies

In this paper, super-harmonic and internal resonance characteristics ofa viscously damped cable with nearly commensurable natural frequenciesare investigated by use of a novel method. The proposed frequency-domainsolution method is based on the combined use of a three-dimensionalnonlinear finite element approach and the incremental harmonic balancetechnique. It is an accurate algorithm in the sense that it accommodatesmulti-harmonic components and no mode-based model reduction is utilizedin the solution process. The alternating frequency/amplitude-controlledalgorithm enables complete solution to the frequency-response curvesincluding unstable branches, sub- and super-harmonic resonance andinternal resonance. A suspended cable paradigm under internal resonancecondition is studied using the proposed method. Nonlinear response andmodal interaction characteristics of the cable at different frequencyregions are identified from analysis of response profiles and harmoniccomponent features. The super-harmonic and internal resonance responsesare respectively characterized based on the harmonic distribution. Underan in-plane harmonic excitation, the two-to-one internal resonancebetween the in-plane and out-of-plane modes and the super-harmonicresonance around the second symmetric in-plane mode are revealed. Strongnonlinear interaction among different modes in the parameter spaceranging from primary resonance to super-harmonic resonance is observed.