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Theoretical analysis of dynamic behaviors of cable-stayed bridges excited by two harmonic forces

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Abstract

To better understand the dynamic behaviors of cable-stayed bridges, this study investigates the dynamic behaviors of a cable-stayed shallow arch subjected to two external harmonic excitations using the analytical approach. First, dimensionless planar vibration equations of the system are obtained by applying the Hamilton principle, and three ordinary differential equations of the arch and the two cables are obtained by using the Galerkin discretization method. Second, modulation equations involving the amplitude and phase of the dynamic response of the system are derived by applying the method of multiple scales. Third, three simultaneous resonance cases are considered. Finally, parametric study results are illustrated through frequency responses, amplitude responses, phase plane and bifurcation diagrams. Chaos phenomenon is also detected and presented. To validate the developed analytical solutions, numerical simulations are conducted by applying the Runge–Kutta method to integrate the original ordinary differential equations. The results demonstrate that acceptable consistency is reached in the results obtained from the analytical solutions and the Runge–Kutta method in the three simulated cases. The obtained results show that the system’s dynamic responses in the three simulated cases exhibit similarities in their frequency and amplitude responses, while some qualitative differences exist in the phase plane portraits (e.g., period-1, period-2, period-3 solutions) and their bifurcation diagrams.

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Acknowledgements

This study is financially supported by National Science Foundation of China under Grant Nos. 11972151, 11572117, 11502076 and 11872176, and China Scholarship Council.

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Authors and Affiliations

  1. College of Civil Engineering, Hunan University, Changsha, Hunan Province, China

    Houjun Kang & Yunyue Cong

  2. Department of Civil, Architectural and Environmental Engineering, Missouri University of Science and Technology, 1401 N. Pine St., Rolla, MO, 65409, USA

    Yunyue Cong & Guirong Yan

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  1. Houjun Kang
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  2. Yunyue Cong
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Correspondence to Guirong Yan.

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Appendix

Appendix

In order to simplify the expression of Galerkin integral coefficients in Eqs. (9) to (10), the following integrals are introduced.

\( d_{mm} = \int_{0}^{1} {y^{\prime}_{m} \varphi^{\prime}_{m} {\rm d}x_{m} } \), \( d_{33} = \int_{0}^{1} {y^{\prime}_{\rm a} (x)\phi^{\prime}_{\rm a} (x){\rm d}x} \), \( d_{0m} = \int_{0}^{1} {\varphi^{\prime}_{m} (x_{m} ){\rm d}x_{m} } \), \( d_{m0} = \int_{0}^{1} {y^{\prime}_{m} {\rm d}x_{m} } \), \( f_{mm} = \int_{0}^{1} {y^{\prime\prime}_{m} \varphi_{m} {\rm d}x_{m} } \), \( h_{mm} = \int_{0}^{1} {\varphi_{m} \varphi^{\prime\prime}_{m} {\rm d}x_{m} } \), \( l_{mm} = \int_{0}^{1} {\varphi^{\prime}_{m} \varphi^{\prime}_{m} {\rm d}x_{m} } \), \( s_{mm} = \int_{0}^{1} {x_{m} \varphi_{m} {\rm d}x_{m} } \), \( r_{33} = \int_{0}^{1} {\phi_{\rm a}^{(4)} \phi_{\rm a} {\rm d}x} \), \( \Gamma_{mm} = 1/\int_{0}^{1} {\varphi_{m} \varphi_{m} {\rm d}x_{m} } \quad (m = 0,1,2,3) \)where m = 0 denotes that the term does not exist, \( \varphi_{m} \) and \( y_{m} \) represent the mode shapes and initial configurations of cables and shallow-arch, respectively, and \( \varphi_{3} = \phi_{\rm a} \), \( y_{3} = y_{\rm a 0} \).

$$ b_{11} = - \Gamma_{33} P_{1} f_{03} ,\quad b_{12} = - \Gamma_{33} P_{2} f_{03} ,\quad b_{15} = - \frac{1}{2}\Gamma_{33} \sum\limits_{j = 1}^{2} {\kappa_{j} \cos^{4} \vartheta_{j} \phi_{\rm a}^{4} (s_{j} )} - \frac{1}{2}\Gamma_{33} \eta l_{33} h_{33} /\beta_{\rm a}^{4} , $$
$$ b_{13} = \omega_{\rm a}^{2} = - \Gamma_{33} \sum\limits_{j = 1}^{2} {\kappa_{j} \cos \vartheta_{j} \sin \vartheta_{j} \phi_{\rm a}^{2} (s_{j} )d_{j0} } + \Gamma_{33} \sum\limits_{j = 1}^{2} {\kappa_{j} \gamma_{\rm cj} \phi_{\rm a}^{2} (s_{j} )\sin^{2} \vartheta_{j} } - (\Gamma_{33} \eta d_{33} f_{33} - \Gamma_{33} r_{33} )/\beta^{4} , $$
$$ b_{14} = - \frac{1}{2}\Gamma_{33} \sum\limits_{j = 1}^{2} {\kappa_{j} \phi_{\rm a}^{3} (s_{j} )\cos^{2} \vartheta_{j} \sin \vartheta_{j} } - \Gamma_{33} \sum\limits_{j = 1}^{2} {\kappa_{j} \cos^{3} \vartheta_{j} \phi_{\rm a}^{3} (s_{j} )d_{j0} } + \Gamma_{33} \sum\limits_{j = 1}^{2} {\kappa_{j} \gamma_{j} \cos^{2} \vartheta_{j} \sin \vartheta_{j} \phi_{\rm a}^{3} (s_{j} )} - \frac{1}{2}\Gamma_{33} \eta l_{33} f_{33} /\beta_{\rm a}^{4} - \Gamma_{33} \eta d_{33} h_{33} /\beta_{\rm a}^{4} , $$
$$ b_{16} = - \Gamma_{33} \kappa_{1} d_{11} \sin \vartheta_{1} \phi_{\rm a} (s_{1} ),\quad b_{17} = \left( { - \frac{1}{2}d_{01} \sin 2\vartheta_{1} - \cos^{2} \vartheta_{1} d_{11} - \cos^{2} \vartheta_{2} d_{10} \varphi^{\prime}_{1} (1) + \gamma_{\rm c1} \cos \vartheta_{1} \sin \vartheta_{1} \varphi^{\prime}_{1} (1)} \right)\Gamma_{33} \kappa_{1} \phi_{\rm a}^{2} (s_{1} ), $$
$$ b_{18} = \left( { - \frac{1}{2}l_{11} \sin \vartheta_{1} - \cos \vartheta_{1} d_{11} \varphi^{\prime}_{1} (1)} \right)\Gamma_{33} \kappa_{1} \phi_{\rm a} (s_{1} ),\quad b_{19} = - \Gamma_{33} \kappa_{2} d_{22} \sin \vartheta_{2} \phi_{\rm a} (s_{2} ), $$
$$ b_{110} = - \left( {\frac{1}{2}\sin 2\vartheta_{2} d_{02} - \cos \vartheta_{2}^{2} d_{22} - \cos \vartheta_{2}^{2} d_{02} \varphi^{\prime}_{2} (1) + \gamma_{\rm c2} \cos \vartheta_{2} \sin \vartheta_{2} \varphi^{\prime}_{2} (1)} \right)\Gamma_{33} \kappa_{2} \phi_{\rm a}^{2} (s_{2} ), $$
$$ b_{111} = - \left( {\frac{1}{2}l_{22} \sin \vartheta_{2} + d_{22} \cos \vartheta_{2} \varphi^{\prime}_{2} (1)} \right)\Gamma_{33} \kappa_{2} \phi_{\rm a} (s_{2} ),\quad b_{112} = - (d_{01} + \frac{1}{2}\varphi^{\prime}_{1} (1))\Gamma_{33} \kappa_{1} \cos^{3} \vartheta_{1} \phi_{\rm a}^{3} (s_{2} ), $$
$$ b_{113} = - \left( {\frac{1}{2}l_{11} + d_{01} \varphi^{\prime}_{1} (1)} \right)\Gamma_{33} \kappa_{1} \cos^{2} \vartheta_{1} \phi_{\rm a}^{2} (s_{1} ),\quad b_{114} = - \frac{1}{2}\Gamma_{33} \kappa_{2} \cos^{3} \vartheta_{2} \phi_{\rm a}^{3} (s_{2} )(d_{10} + \varphi^{\prime}_{2} (1)), $$
$$ b_{115} = - \frac{1}{2}\Gamma_{33} \kappa_{2} \cos^{2} \vartheta_{2} \phi_{\rm a}^{2} (s_{2} )(l_{22} - d_{02} \varphi^{\prime}_{2} (1)),\quad b_{116} = - \frac{1}{2}\Gamma_{33} \kappa_{1} \cos \vartheta_{1} \phi_{\rm a} (s_{1} )\varphi^{\prime}_{1} (1)l_{11} ,\quad b_{210} = - \frac{1}{2}\lambda_{1} \Gamma_{11} l_{11} h_{11} /\beta_{1}^{2} $$
$$ b_{117} = - \frac{1}{2}\Gamma_{33} \kappa_{2} \cos \vartheta_{2} \phi_{\rm a} (s_{2} )\varphi^{\prime}_{2} (1)l_{22} ,\quad b_{21} = \mu_{1} \Gamma_{11} \cos \vartheta_{1} \phi_{\rm a} (s_{1} )S_{11} ,\quad b_{22} = \Gamma_{11} \cos \vartheta_{1} \phi_{\rm a} (s_{1} )S_{11} , $$
$$ b_{23} = (\gamma_{\rm c1} \sin \vartheta_{1} - \lambda_{1} \cos \vartheta_{1} d_{10} )\Gamma_{11} \phi_{\rm a} (s_{1} )f_{11} /\beta_{1}^{2} ,\quad b_{24} = - \frac{1}{2}\Gamma_{11} \lambda_{1} f_{11} \phi_{\rm a}^{2} (s_{1} )\cos^{2} \vartheta_{1} /\beta_{1}^{2} , $$
$$ b_{25} = - \frac{1}{2}\lambda_{1} \Gamma_{11} d_{11} f_{11} /\beta_{1}^{2} - \Gamma_{11} h_{11} /\beta_{1}^{2} ,\quad b_{26} = - \left( {\frac{1}{2}l_{11} f_{11} + d_{11} h_{11} } \right)\lambda_{1} \Gamma_{11} /\beta_{1}^{2} ,\quad b_{29} = - \frac{1}{2}\lambda_{1} \Gamma_{11} \cos^{2} \vartheta_{1} \phi_{\rm a}^{2} (s_{1} )h_{11} /\beta_{1}^{2} , $$
$$ b_{27} = - \lambda_{1} \Gamma_{11} \cos \vartheta_{1} \phi_{\rm a} (s_{1} )(d_{01} f_{11} + d_{10} h_{11} )/\beta_{1}^{2} + \gamma_{\rm c1} \lambda_{1} \Gamma_{11} \sin \vartheta_{1} \phi_{\rm a} (s_{1} )h_{11} /\beta_{1}^{2} ,\quad b_{28} = - \lambda_{1} \Gamma_{11} \cos \vartheta_{1} \phi_{\rm a} (s_{1} )d_{01} h_{11} /\beta_{1}^{2} , $$
$$ b_{31} = \mu_{2} \cos \vartheta_{2} \phi_{\rm a} (s_{2} )\Gamma_{22} S_{22} ,\quad b_{32} = \Gamma_{22} \cos \vartheta_{2} \phi_{\rm a} (s_{2} )S_{22} ,\quad b_{33} = \lambda_{2} \Gamma_{22} \phi_{\rm a} (s_{2} )f_{22} (\sin \vartheta_{2} - \cos \vartheta_{2} d_{20} )/\beta_{2}^{2} , $$
$$ b_{34} = - \frac{1}{2}\lambda_{2} \Gamma_{22} \cos^{2} \vartheta_{2} \phi_{\rm a}^{2} (s_{2} )f_{22} /\beta_{2}^{2} ,\quad b_{35} = - \lambda_{2} \Gamma_{22} d_{11} f_{22} /\beta_{2}^{2} - \Gamma_{22} h_{22} /\beta_{2}^{2} ,\quad b_{36} = \lambda_{2} \Gamma_{22} (d_{22} h_{22} - \frac{1}{2}l_{22} f_{22} )/\beta_{2}^{2} , $$
$$ b_{37} = - \lambda_{2} \Gamma_{22} \cos \vartheta_{2} \phi_{\rm a} (s_{2} )(d_{02} f_{22} + d_{20} h_{22} )/\beta_{2}^{2} + \gamma_{\rm c2} \lambda_{2} \Gamma_{22} \sin \vartheta_{2} \phi_{\rm a} (s_{2} )h_{22} /\beta_{2}^{2} ,\quad b_{38} = - \lambda_{2} \Gamma_{22} \cos \vartheta_{2} \phi_{\rm a} (s_{2} )d_{02} h_{22} /\beta_{2}^{2} , $$
$$ b_{39} = - \frac{1}{2}\lambda_{2} \varGamma_{22} \cos^{2} \vartheta_{2} \phi_{\rm a}^{2} (s_{2} )h_{22} /\beta_{2}^{2} ,\quad b_{310} = - \frac{1}{2}\lambda_{2} \varGamma_{22} l_{22} h_{22} /\beta_{2}^{2} . $$

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Kang, H., Cong, Y. & Yan, G. Theoretical analysis of dynamic behaviors of cable-stayed bridges excited by two harmonic forces. Nonlinear Dyn 102, 965–992 (2020). https://doi.org/10.1007/s11071-020-05763-8

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