Cable’s mode interactions under vertical support motions: boundary resonant modulation
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Cable’s triad and two-to-one mode interactions excited by support motions are modeled and analyzed in a unified boundary modulation formulation. Based upon proper scaling and a boundary resonance concept, the small support motion is modeled as a nonzero boundary modulation term for cable’s reduced (slow) dynamics through attacking cable’s continuous dynamic equations directly by the multiple scale method. Boundary resonance coefficients, characterizing the boundary modulation effect, are derived analytically for both cable’s triad and two-to-one mode resonant dynamics. It is found that the boundary resonance coefficients depend on both cable’s boundary modal information and cable’s initial deformation/sag. Frequency response diagrams based on cable’s reduced models (modulation equations) are obtained, with stability and bifurcation determined. Finally, these approximate analytical results are verified by the numerical results through applying the finite-difference method directly to cable’s original partial differential equations.
KeywordsCable–support interaction Boundary resonant modulation Mode interaction Multiple scale method Boundary resonance coefficient
This study is funded by Program for Supporting Young Investigator, Hunan University. And it is also supported by National Science Foundation of China under Grant Nos. 11502076 and 11572117. Interesting comments and criticism by the reviewers are also gratefully acknowledged.
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