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http://dx.doi.org/10.12989/sem.2017.61.4.497

Auto-parametric resonance of framed structures under periodic excitations  

Li, Yuchun (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University)
Gou, Hongliang (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University)
Zhang, Long (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University)
Chang, Chenyu (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University)
Publication Information
Structural Engineering and Mechanics / v.61, no.4, 2017 , pp. 497-510 More about this Journal
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 ${\Gamma}$-shaped frame is conducted for verifying the theoretical predictions and present calculation method.
Keywords
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;
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