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.

为了更好地理解斜拉桥的动力特性，本研究使用解析方法研究了斜拉浅拱在两次外部谐波激励下的动力特性。首先，应用汉密尔顿原理获得系统的无量纲平面振动方程，并利用伽勒金离散化方法获得拱和两条索的三个常微分方程。其次，采用多尺度法推导了涉及系统动态响应幅度和相位的调制方程。第三，考虑三个同时发生共振的情况。最后，通过频率响应，幅度响应，相平面和分叉图来说明参数研究结果。混沌现象也会被检测到并呈现出来。为了验证已开发的解析解决方案，通过应用Runge–Kutta方法整合原始的常微分方程，进行了数值模拟。结果表明，在三种模拟情况下，从分析溶液和Runge-Kutta方法获得的结果均达到可接受的一致性。获得的结果表明，在三种模拟情况下，系统的动态响应在频率和幅度响应上都表现出相似性，而相平面肖像（例如，期间1，期间2，期间3解）中存在一些质的差异。他们的分叉图。数值模拟是通过应用Runge-Kutta方法整合原始的常微分方程来进行的。结果表明，在三种模拟情况下，从分析溶液和Runge-Kutta方法获得的结果均达到可接受的一致性。获得的结果表明，在三种模拟情况下，系统的动态响应在频率和幅度响应上表现出相似性，而相平面肖像（例如，期间1，期间2，期间3解）中存在一些定性差异。他们的分叉图。数值模拟是通过应用Runge-Kutta方法整合原始的常微分方程来进行的。结果表明，在三种模拟情况下，从分析溶液和Runge-Kutta方法获得的结果均达到可接受的一致性。获得的结果表明，在三种模拟情况下，系统的动态响应在频率和幅度响应上都表现出相似性，而相平面肖像（例如，期间1，期间2，期间3解）中存在一些质的差异。他们的分叉图。