Modeling, dynamics, and parametric studies of a multi-cable-stayed beam model

Abstract

To study the dynamic behavior of cable-stayed bridges, a linear multi-cable-stayed beam model is developed to investigate its in-plane and out-of-plane transverse vibrations. From the full-bridge perspective, an in-plane nonlinear single-mode discrete model is established. First, the in-plane and out-of-plane motion equations of the system and their boundary conditions are derived. Second, by employing the separation of variables method, the linear eigenvalue problems are solved. The influences of the mass ratio, stiffness ratio, and cable sag on the occurrence of global, local, and coupled vibration modes are studied. Third, frequency response, amplitude response, phase diagram, time history, and power spectrum are extracted to investigate the system’s nonlinear dynamic behaviors. The obtained results demonstrate that for the in-plane motion, the occurrence of global and local modes of the system depends on the mass and stiffness ratios between cable and beam significantly; for the out-of-plane motion, without the elastic support of the cable, the global modes occur, which can be suppressed by adjusting the mass and stiffness ratios between cable and beam but may in turn induce the cable’s local vibration modes; for the nonlinear analysis, the single-degree-of-freedom system behaves like a hardening spring. Its lower branch behaves more complicated than the higher one and has a double-periodic steady-state solution. The system with large damping ratio behaves shows weak hardening spring property.

Appendix

For compact expression, the following integrals are introduced to define the coefficients $c_{i}$ and the excitation amplitude $F_{i}$ in Eq. (28).

$$\begin{aligned} \Gamma _{c1}= & {} \int _0^1 {\phi _{c1}^{2} dx} ,\quad \Gamma _{c2} =\int _0^1 {\phi _{c2}^{2} dx} ,\quad \Gamma _{c3} =\int _0^1 {{\phi '}_{c1}^{2} dx} ,\quad \Gamma _{c4} =\int _0^1 {{\phi ' }_{c2}^{2} dx} , \nonumber \\ \Gamma _{c5}= & {} \int _0^1 {\phi _{c1} {\phi ''}_{c1} dx} ,\quad \Gamma _{c6} =\int _0^1 {\phi _{c2} {\phi '' }_{c2} dx} ,\quad \Gamma _{c7} =\int _0^1 {{{y}'}'_{c1} \phi _{c1} dx} ,\quad \Gamma _{c8} =\int _0^1 {{{y}'}'_{c2} \phi _{c2} dx}, \nonumber \\ \Gamma _{b1}= & {} \int _0^{s_{1} } {\phi _{b1}^{2} dx} +\int _{s_{1} }^{s_{2} } {\phi _{b2}^{2} dx} +\int _{s_{2} }^1 {\phi _{b3}^{2} dx}, \nonumber \\ \Gamma _{b2}= & {} \int _0^{s_{1} } {f_{1} \phi _{b1} dx} +\int _{s_{1} }^{s_{2} } {f_{1} \phi _{b2} dx} +\int _{s_{2} }^1 {f_{1} \phi _{b3} dx}, \nonumber \\ \Gamma _{b3}= & {} \int _0^{s_{1} } {f_{2} \phi _{b1} dx} +\int _{s_{1} }^{s_{2} } {f_{2} \phi _{b2} dx} +\int _{s_{2} }^1 {f_{2} \phi _{b3} dx}, \nonumber \\ m_{v}= & {} \beta _{c}^{2} \left( {\Gamma _{c1} +\Gamma _{c2} +\frac{\gamma ^{3}}{\rho }\Gamma _{b1} } \right) , \end{aligned}$$

(A.1)

where $f_{i}$ are the amplitudes of external harmonic excitation applied to the components of the beam. Therefore, the coefficients $c_{i}$ and $F_{i}$ are expressed as

$$\begin{aligned} c_{1}= & {} \frac{\beta _{c}^{2} }{m_{v} }\left( {\xi _{c} \left( {\Gamma _{c1} +\Gamma _{c2} } \right) +\xi _{b} \frac{\gamma ^{3}}{\rho }\Gamma _{b1} } \right) , \nonumber \\ c_{2}= & {} -\frac{\mu }{m_{v} }\left( {\frac{1}{2}\Gamma _{c3} \Gamma _{c7} +\frac{1}{2}\Gamma _{c4} \Gamma _{c8} +\bar{{\varepsilon }}_{1} \Gamma _{c5} +\bar{{\varepsilon }}_{2} \Gamma _{c6} } \right) , \nonumber \\ c_{3}= & {} -\frac{\mu }{2m_{v} }\left( {\Gamma _{c3} \Gamma _{c5} +\Gamma _{c4} \Gamma _{c6} } \right) , \nonumber \\ F_{1}= & {} \frac{\beta _{c}^{2} }{m_{v} }\frac{\gamma ^{3}}{\rho }\Gamma _{b2},\nonumber \\ F_{2}= & {} \frac{\beta _{c}^{2} }{m_{v} }\frac{\gamma ^{3}}{\rho }\Gamma _{b3}. \end{aligned}$$

(A.2)

The coefficients in Eq. (43) are given as

$$\begin{aligned} \Lambda _{1}= & {} -\frac{6c_{3} F_{2}^{2}}{(1-\Omega _{2}^{2})^{2}}+\frac{4c_{2}^{2} F_{2}^{2}}{(\Omega _{2}^{2} -1)^{2}}-\frac{8c_{2}^{2} F_{2}^{2}}{(\Omega _{2}^{2} -4)(\Omega _{2}^{2} -1)^{2}}, \nonumber \\ \Lambda _{2}= & {} -\frac{2c_{2}^{2} F_{2} }{3(1-\Omega _{2}^{2} )}-\frac{3c_{3} F_{2} }{1-\Omega _{2}^{2} }+\frac{4c_{2}^{2} F_{2} }{(\Omega _{2} -2)\Omega _{2} (\Omega _{2}^{2} -1)}, \nonumber \\ \Lambda _{3}= & {} \frac{10}{3}c_{2}^{2} -3c_{3}. \end{aligned}$$

(A.3)