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Structural Engineering and Mechanics Volume 61, Number 4, February25 2017 , pages 497-510 DOI: https://doi.org/10.12989/sem.2017.61.4.497 |
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Auto-parametric resonance of framed structures under periodic excitations |
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Yuchun Li, Hongliang Gou, Long Zhang and Chenyu Chang
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| Abstract | ||
| A framed structure may be composed of two sub-structures, which are linked by a hinged joint. One sub-structure is the primary system and the other is the secondary system. The primary system, which is subjected to the periodic external load, can give rise to an auto-parametric resonance of the second system. Considering the geometric-stiffness effect produced by the axially internal force, the element equation of motion is derived by the extended Hamilton\'s principle. The element equations are then assembled into the global non-homogeneous Mathieu-Hill equations. The Newmark\'s method is introduced to solve the time-history responses of the non-homogeneous Mathieu-Hill equations. The energy-growth exponent/coefficient (EGE/EGC) and a finite-time Lyapunov exponent (FLE) are proposed for determining the auto-parametric instability boundaries of the structural system. The auto-parametric instabilities are numerically analyzed for the two frames. The influence of relative stiffness between the primary and secondary systems on the auto-parametric instability boundaries is investigated. A phenomenon of the \"auto-parametric internal resonance\" (the auto-parametric resonance of the second system induced by a normal resonance of the primary system) is predicted through the two numerical examples. The risk of auto-parametric internal resonance is emphasized. An auto-parametric resonance experiment of a | ||
| Key Words | ||
| auto-parametric resonance; framed structures; finite element modeling; non-homogeneous Mathieu-Hill equation; energy-growth exponent/coefficient (EGE/EGC);finite-time Lyapunov exponent (FLE); experiment | ||
| Address | ||
| Yuchun Li, Hongliang Gou, Long Zhang and Chenyu Chang : Department of Hydraulic Engineering, College of Civil Engineering, Tongji University, Shanghai, China | ||