Abstract
In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.
Highlights
Cable-supported structures have been widely used in civil engineering for its light weight, flexibleness, and low fundamental damping [1, 2]
Because of their large amplitude vibrations under environmental loads, these structures are susceptible to loss of serviceability or even failure
It is quite crucial to understand the dynamic characteristics of cablesupported structures under environmental loads [3, 4]
Summary
Cable-supported structures have been widely used in civil engineering for its light weight, flexibleness, and low fundamental damping [1, 2]. Ere are only a few studies that considered geometrical nonlinearity: Lv et al [27] established the governing equations of cable-stayed arch structures, and the possible internal resonance of cablestayed arch structures was investigated. Following the methodologies in [30], the discretization approach is used to obtain the modulation equations governing the nonlinear dynamic behavior of cable-stayed shallow arches with 1 : 1 internal resonances. Equations (6) and (7) are substituted into equations (2) and (3) and integrated where required, and the Galerkin method is used to obtain a nonlinear model with the following system of differential equations for the cable and shallow arch motions: θc + θg. Mass Extensional rigidity Initial tension Length Horizontal angle Damping coefficient
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