Nonlinear dynamic analysis of cable-stayed arches under primary resonance of cables

Abstract

In this paper, the one-to-one interaction of a cable-stayed arch structure under the cable’s primary resonance is investigated. Based on the coupling condition at the arch tip, the partial differential equations governing the planar motion of the system are derived using the extended Hamiltonian principle, while with the application of the Galerkin method, these equations are transformed into a set of ordinary equations. Applying the method of multiple scales to these ordinary equations, the first approximated solutions and solvability condition are obtained. The one-to-one interaction between the cable and the arch is investigated under simultaneous internal and external resonances for an actual cable-stayed arch structure. Based on the shooting method and the pseudo-arclength algorithm, the dynamic solutions of the system are obtained, and a period-doubling route to chaos is analyzed. The effects of the cable’s initial tension, inclination angle, the arch’s rise-to-span ratio and intersection angle between the cable and the arch are explored, and the results show that the interaction response mainly depends on specific parameters.

Appendix

$$\begin{aligned} a_1^{ni}= & {} \int _0^{l_c } {m_c \varphi _n (x)\varphi _i (x)\hbox {d}x}\\ a_2^{ni}= & {} \int _0^{l_c } {c_c \varphi _n (x)\varphi _i (x)\hbox {d}x}\\ a_3^{mi}= & {} \int _0^{l_c } {E_c A_c } {y}''_0 \frac{\varphi _m (l_c )}{l_c \tan (\theta _g +\theta _c )}\varphi _i (x)\hbox {d}x-\int _0^{l_c } {H\varphi _i (x)} {\varphi }''_m (x)\hbox {d}x \\&-\int _0^{l_c } {E_c A_c } {y}''_0 \frac{1}{l_c }\int _0^{l_c } {{y}'_0 {\varphi }'_m (x)} \hbox {d}x\varphi _i (x)\hbox {d}x \\ a_4^{mi}= & {} -\int _0^{l_c } {E_c A_c } {y}''_0 \frac{\phi _m (l_g )}{l_c \sin (\theta _g +\theta _c )}\varphi _i (x)\hbox {d}x\\ a_5^{mpi}= & {} -\int _0^{l_c } {E_c A_c {\varphi }''_m (x)} \frac{\phi _p (l_g )}{l_c \sin (\theta _g +\theta _c )}\varphi _i (x)\hbox {d}x\\ a_6^{mpi}= & {} \int _0^{l_c } {E_c A_c {\varphi }''_m (x)} \frac{\varphi _p (l_c )}{l_c \sin (\theta _g +\theta _c )}\varphi _i (x)\hbox {d}x-\int _0^{l_c } {E_c A_c } {y}''_0 \frac{1}{2l_c }\int _0^{l_c } {{\varphi }'_m (x){\varphi }'_p (x)} \hbox {d}x\varphi _i (x)\hbox {d}x \\&-\int _0^{l_c } {E_c A_c {\varphi }''_m (x)} \frac{1}{l_c }\int _0^{l_c } {{y}'_0 {\varphi }'_p (x)} \hbox {d}x\varphi _i (x)\hbox {d}x \\ a_7^{mpqi}= & {} -\int _0^{l_c } {E_c A_c {\varphi }''_m (x)} \frac{1}{2l_c }\int _0^{l_c } {{\varphi }'_p (x){\varphi }'_q (x)} \hbox {d}x\varphi _i (x)\hbox {d}x\\ F_n= & {} \int _0^{l_c } {F(x)} \varphi _i (x)\hbox {d}x\\ b_1^{ni}= & {} \int _0^{l_g } {m_g \phi _i (\bar{{x}})\phi _n(\bar{{x}})} \hbox {d}\bar{{x}}\\ b_2^{ni}= & {} \int _0^{l_g } {c_g \phi _i (\bar{{x}})\phi _n(\bar{{x}})} \hbox {d}\bar{{x}}\\ b_3^{mi}= & {} \int _0^{l_g } {E_g I_g \phi _n^{(4)} (\bar{{x}})\phi _i (\bar{{x}})\hbox {d}\bar{{x}}} -\int _0^{l_g } {N\phi _n^{{\prime }{\prime }} (\bar{{x}})\phi _i (\bar{{x}})\hbox {d}\bar{{x}}} +\int _0^{l_g } {E_g A_g } {\bar{{y}}}''_0 \left( {\frac{\phi _p (l_g )}{l_g \tan (\theta _g +\theta _c )}} \right) \phi _i (\bar{{x}})\hbox {d}\bar{{x}} \\&-\int _0^{l_g } {E_g A_g } {\bar{{y}}}''_0 \frac{1}{l_g }\int _0^{l_g } {{y}'_0 {\phi }'_p (\bar{{x}})\hbox {d}\bar{{x}}} \phi _i (\bar{{x}})\hbox {d}\bar{{x}} \\ b_4^{mi}= & {} -\int _0^{l_g } {E_g A_g } {\bar{{y}}}''_0 \frac{\varphi _m (l_c )}{l_g \sin (\theta _g +\theta _c )}\phi _i (\bar{{x}})\hbox {d}\bar{{x}}\\ b_5^{mpi}= & {} -\int _0^{l_g } {E_g A_g } {\phi }''_m (\bar{{x}})\frac{\varphi _p (l_c )}{l_g \sin (\theta _g +\theta _c )}\phi _i (\bar{{x}})\hbox {d}\bar{{x}}\\ b_6^{mpi}= & {} \int _0^{l_g } {E_g A_g } {\phi }''_m (\bar{{x}})\left( {\frac{\phi _p (l_g )}{l_g \tan (\theta _g +\theta _c )}} \right) \phi _i (\bar{{x}})\hbox {d}\bar{{x}}-\int _0^{l_g } {E_g A_g } {\bar{{y}}}''_0 \frac{1}{2l_g }\int _0^{l_g } {{\phi }'_m (\bar{{x}}){\phi }'_p (\bar{{x}})\hbox {d}\bar{{x}}} \phi _i (\bar{{x}})\hbox {d}\bar{{x}} \\&-\int _0^{l_g } {E_g A_g } {\phi }''_m (\bar{{x}})\frac{1}{l_g }\int _0^{l_g } {{y}'_0 {\phi }'_p (\bar{{x}})\hbox {d}\bar{{x}}} \phi _i (\bar{{x}})\hbox {d}\bar{{x}} \\ b_7^{{mpqi}}= & {} -\int _0^{l_g } {E_g A_g } {\phi }''_m (\bar{{x}})\frac{1}{2l_g }\int _0^{l_g } {{\phi }'_p (\bar{{x}}){\phi }'_q (\bar{{x}})\hbox {d}\bar{{x}}} \phi _i (\bar{{x}})\hbox {d}\bar{{x}}. \end{aligned}$$