Resonance analysis between deck and cables in cable-stayed bridges with coupling effect of adjacent cables considered

Abstract

The nonlinear dynamic model of a shallow arch with multiple cables is developed to model a long-span cable-stayed bridge. Based on the veering phenomenon of cable-stayed bridges, the in-plane modal internal resonance between the first mode of the shallow arch and the first mode of the cable is investigated under both primary resonance and subharmonic resonance. Modulation equations of the dynamic system are obtained by Galerkin discretization and the multiple scales method, in which the equilibrium solution of modulation equations is obtained by the Newton–Raphson method. Meanwhile, the Runge–Kutta method is applied to directly solve the ordinary differential equations to verify the accuracy of the perturbation analysis. Numerical analysis shows that the internal resonance occurs in adjacent cables; the energy transfer mechanism and the dynamic behavior of system become more complex.

Appendices

Appendix A

$$ \begin{aligned} & m = \int_{0}^{1} {\phi^{2} } \left( s \right){\text{d}}s,\quad b_{1} = \int_{0}^{1} {\phi \left( s \right)y^{\prime \prime } \left( s \right)} {\text{ d}}s,\quad b_{2} = \int_{0}^{1} \phi \left( s \right)\phi^{\prime \prime } \left( s \right){\text{d}}s, \\ & b_{3} = \int_{0}^{1} {\phi^{\prime } \left( s \right)y^{\prime } \left( s \right){\text{d}}s} ,\quad b_{4} = \int_{0}^{1} {\phi^{\prime \prime \prime \prime } \left( s \right)} \phi \left( s \right){\text{d}}s,\quad b_{5} = \int_{0}^{1} {\phi^{\prime } \left( s \right)\phi^{\prime } \left( s \right)} {\text{ d}}s \\ & \Gamma_{nn} = \frac{1}{{\beta_{n}^{2} \int_{0}^{1} {\varphi_{n} \left( {x_{n} } \right)\varphi_{n} \left( {x_{n} } \right)} {\text{ d}}x_{n} }},\quad g_{nn} = \int_{0}^{1} {x_{n} \varphi_{n} \left( {x_{n} } \right)} {\text{ d}}x_{n} ,\quad f_{nn} = \int_{0}^{1} {y_{n}^{\prime \prime } } \left( {x_{n} } \right)\varphi_{n} \left( {x_{n} } \right){\text{d}}x_{n} , \\ & m_{nn} = \int_{0}^{1} {\varphi_{n}^{\prime } \left( {x_{n} } \right)\varphi_{n}^{\prime } \left( {x_{n} } \right) \, } {\text{d}}x_{n} ,\quad c_{nn} = \int_{0}^{1} {\varphi_{n}^{\prime } \left( {x_{n} } \right)y_{n}^{\prime } \left( {x_{n} } \right)} {\text{ d}}x_{n} ,\quad c_{0n} = \int_{0}^{1} {\varphi_{n}^{\prime } } \left( {x_{n} } \right){\text{d}}x_{n} , \\ & c_{n0} = \int_{0}^{1} {y_{n}^{\prime } } \left( {x_{n} } \right){\text{d}}x_{n} ,\quad h_{nn} = \int_{0}^{1} {\varphi_{n}^{\prime \prime } } \left( {x_{n} } \right)\varphi_{n} \left( {x_{n} } \right){\text{d}}x_{n} . \\ \end{aligned} $$

Appendix B

$$ \begin{aligned} d_{1} & = - \frac{{B\eta b_{1} }}{{m\beta^{4} }},\quad d_{2} = - \frac{{B\eta b_{2} }}{{m\beta^{4} }},\quad d_{5} = - \frac{{\eta b_{5} b_{2} }}{{2m\beta^{4} }}, \\ d_{31} & = \frac{{K_{1} \gamma_{1} \phi \left( {s_{1} } \right)\sin^{2} \theta_{1} }}{m} - \frac{{K_{1} \phi \left( {s_{1} } \right)c_{10} \sin 2\theta_{1} }}{2m},\quad d_{32} = \frac{{K_{2} \gamma_{2} \phi \left( {s_{2} } \right)\sin^{2} \theta_{2} }}{m} - \frac{{K_{2} \phi \left( {s_{2} } \right)c_{20} \sin 2\theta_{2} }}{2m},\quad d_{33} = \frac{{K_{3} \gamma_{3} \phi (s_{3} )\sin^{2} \theta_{3} }}{m} - \frac{{K_{3} \phi (s_{3} )c_{30} \sin 2\theta_{3} }}{2m}, \\ d_{40} & = - \frac{{\eta b_{5} b_{1} }}{{2m\beta^{4} }} - \frac{{b_{2} b_{3} }}{{m\beta^{4} }},\quad d_{41} = - \frac{{K_{1} \phi^{3} \left( {s_{1} } \right)\sin 2\theta_{1} \cos \theta_{1} }}{4m},\quad d_{42} = - \frac{{K_{2} \phi^{3} \left( {s_{2} } \right)\cos \theta_{2} \sin 2\theta_{2} }}{4m},\quad d_{43} = - \frac{{K_{3} \phi^{3} \left( {s_{3} } \right)\sin 2\theta_{3} \cos \theta_{3} }}{4m}, \\ d_{61} & = - \frac{{K_{1} \phi^{2} \left( {s_{1} } \right)c_{01} \sin 2\theta_{1} }}{2m},\quad d_{62} = - \frac{{K_{2} \phi^{2} \left( {s_{2} } \right)c_{02} \sin 2\theta_{2} }}{2m},\quad d_{63} = - \frac{{K_{3} \phi^{2} \left( {s_{3} } \right)c_{03} \sin 2\theta_{3} }}{2m}, \\ d_{71} & = - \frac{{K_{1} \phi \left( {s_{1} } \right)c_{11} \sin \theta_{1} }}{m},\quad d_{72} = - \frac{{K_{2} \phi \left( {s_{2} } \right)c_{22} \sin \theta_{2} }}{m},\quad d_{73} = - \frac{{K_{3} \phi \left( {s_{3} } \right)c_{33} \sin \theta_{3} }}{m}, \\ d_{81} & = - \frac{{K_{1} \phi \left( {s_{1} } \right)m_{11} \sin \theta_{1} }}{m},\quad d_{82} = - \frac{{K_{2} \phi \left( {s_{2} } \right)m_{22} \sin \theta_{2} }}{m},\quad d_{83} = - \frac{{K_{3} \phi \left( {s_{3} } \right)m_{33} \sin \theta_{3} }}{m}, \\ \end{aligned} $$

$$ \begin{aligned} r_{{11}} & = \left( { - \lambda _{1} \phi \left( {s_{1} } \right)c_{{10}} f_{{11}} \cos \theta _{1} + \gamma _{1} \lambda _{1} \phi \left( {s_{1} } \right)f_{{11}} \sin \theta _{1} } \right)\Gamma _{{11}} ,\quad r_{{12}} = \left( { - \lambda _{2} \phi \left( {s_{2} } \right)c_{{20}} f_{{22}} \cos \theta _{2} + \gamma _{2} \lambda _{2} \phi \left( {s_{2} } \right)f_{{22}} \sin \theta _{2} } \right)\Gamma _{{22}} ,\quad r_{{13}} = \left( { - \lambda _{3} \phi \left( {s_{3} } \right)c_{{30}} f_{{33}} \cos \theta _{3} + \gamma _{3} \lambda _{3} \phi \left( {s_{3} } \right)f_{{33}} \sin \theta _{3} } \right)\Gamma _{{33}} , \\ r_{{21}} & = - \frac{1}{2}\Gamma _{{11}} \lambda _{1} \phi ^{2} \left( {s_{1} } \right)f_{{11}} \cos ^{2} \theta _{1} ,\quad r_{{22}} = - \frac{1}{2}\Gamma _{{22}} \lambda _{2} \phi ^{2} \left( {s_{2} } \right)f_{{22}} \cos ^{2} \theta _{2} ,\quad r_{{23}} = - \frac{1}{2}\Gamma _{{33}} \lambda _{3} \phi ^{2} \left( {s_{3} } \right)f_{{33}} \cos ^{2} \theta _{3} \\ r_{{31}} & = - \left( {\lambda _{1} c_{{11}} f_{{11}} + h_{{11}} } \right)\Gamma _{{11}} ,\quad r_{{32}} = - \left( {\lambda _{2} c_{{22}} f_{{22}} + h_{{22}} } \right)\Gamma _{{22}} ,\quad r_{{33}} = - \left( {\lambda _{3} c_{{33}} f_{{33}} + h_{{33}} } \right)\Gamma _{{33}} \\ r_{{41}} & = - \frac{1}{2}\left( {\lambda _{1} m_{{11}} f_{{11}} + 2\lambda _{1} c_{{11}} } \right)\Gamma _{{11}} ,\quad r_{{42}} = - \frac{1}{2}\left( {\lambda _{2} m_{{22}} f_{{22}} + 2\lambda _{2} c_{{22}} } \right)\Gamma _{{22}} ,\quad r_{{43}} = - \frac{1}{2}\left( {\lambda _{3} m_{{33}} f_{{33}} + 2\lambda _{3} c_{{33}} } \right)\Gamma _{{33}} \\ r_{{51}} & = - \frac{1}{2}\lambda _{1} m_{{11}} h_{{11}} \Gamma _{{11}} ,\quad r_{{52}} = - \frac{1}{2}\lambda _{2} m_{{22}} h_{{22}} \Gamma _{{22}} ,\quad r_{{53}} = - \frac{1}{2}\lambda _{3} m_{{33}} h_{{33}} \Gamma _{{33}} , \\ \end{aligned} $$

$$ \begin{aligned} r_{61} & = - \left( {\lambda_{1} \phi \left( {s_{1} } \right)c_{10} f_{11} \cos \theta_{1} + \lambda_{1} \phi \left( {s_{1} } \right)c_{10} h_{11} \cos \theta_{1} - \gamma_{1} \phi \left( {s_{1} } \right)h_{11} \sin \theta_{1} } \right)\Gamma_{11} ,\quad r_{62} = - \left( {\lambda_{2} \phi \left( {s_{2} } \right)c_{20} f_{22} \cos \theta_{2} + \lambda_{2} \phi \left( {s_{2} } \right)c_{20} h_{22} \cos \theta_{2} - \gamma_{2} \phi \left( {s_{2} } \right)h_{22} \sin \theta_{2} } \right)\Gamma_{22} ,\quad r_{63} = - \left( {\lambda_{3} \phi \left( {s_{3} } \right)c_{30} f_{33} \cos \theta_{3} + \lambda_{3} \phi \left( {s_{3} } \right)c_{30} h_{33} \cos \theta_{3} - \gamma_{3} \phi \left( {s_{3} } \right)h_{33} \sin \theta_{3} } \right)\Gamma_{33} , \\ r_{71} & = - \frac{{\lambda_{1} \phi^{2} \left( {s_{1} } \right)h_{11} \Gamma_{11} \cos^{2} \theta_{1} }}{2},\quad r_{72} = - \frac{{\lambda_{2} \phi^{2} \left( {s_{2} } \right)h_{22} \Gamma_{22} \cos^{2} \theta_{2} }}{2},\quad r_{73} = - \frac{{\lambda_{3} \phi^{2} \left( {s_{3} } \right)h_{33} \Gamma_{33} \cos^{2} \theta_{3} }}{2}, \\ r_{81} & = - \lambda_{1} \phi \left( {s_{1} } \right)c_{01} h_{11} \Gamma_{11} \cos \theta_{1} ,\quad r_{82} = - \lambda_{2} \phi \left( {s_{2} } \right)c_{02} h_{22} \Gamma_{22} \cos \theta_{2} ,\quad r_{83} = - \lambda_{3} \phi \left( {s_{3} } \right)c_{03} h_{33} \Gamma_{33} \cos \theta_{3} \\ r_{91} & = \mu_{1} \phi \left( {s_{1} } \right)g_{11} \Gamma_{11} \beta_{1}^{2} \cos \theta_{1} ,\quad r_{92} = \mu_{2} \phi \left( {s_{2} } \right)s_{22} \Gamma_{22} \beta_{2}^{2} \cos \theta_{2} ,\quad r_{93} = \mu_{3} \phi \left( {s_{3} } \right)g_{33} \Gamma_{33} \beta_{3}^{2} \cos \theta_{3} , \\ r_{101} & = \phi \left( {s_{1} } \right)g_{11} \Gamma_{11} \beta_{1}^{2} \cos \theta_{1} ,\quad r_{102} = \phi \left( {s_{2} } \right)g_{22} \Gamma_{22} \beta_{2}^{2} \cos \theta_{2} ,\quad r_{103} = \phi \left( {s_{3} } \right)\Gamma_{33} \beta_{3}^{2} g_{33} \cos \theta_{3} . \\ \end{aligned} $$