In this paper, the novel model of fluid-conveying imperfect pipe supported at both ends is established by considering the geometric imperfection and the geometric nonlinearity induced by mid-plane stretching. The integral-partial differential equation is discretized by the Galerkin method and solved by a fourth-order Runge–Kutta integration algorithm. Compared with the supercritical pitchfork bifurcation of the perfect pipe conveying fluid, the results show that the cusp bifurcation occurs in the imperfect pipe when increasing the flow velocity. Excellent agreement is observed between the numerical results and the analytical results. The two stable asymmetry bifurcation branches bring interesting phenomena in the post-buckling state. The global nonlinear dynamic behaviors of the imperfect pipe are studied by establishing the bifurcation diagrams. The influence of the geometric imperfection amplitude on the nonlinear response is leading to cusp bifurcation comparing with pitchfork bifurcation of the perfect pipe. When pulsation frequency is set as the bifurcation parameter, there are clear nonresonance ranges, low energy resonance ranges and high energy resonance ranges. In the high energy resonance ranges, the first mode vibration coexisting with the sub-harmonic resonance and combination resonance occurs. As the mean flow velocity and pulsation amplitude are set as bifurcation parameters, the vibration of the imperfect pipe becomes more and more complicated. The vibration exhibits far richer dynamic behaviors including periodic, multi-periodic, quasi-periodic, and chaotic motions. The viscoelastic damping can effectively suppress the vibration response and transfer the high energy resonance to the low energy resonance state. The improved model and corresponding results provide useful information for further studying the dynamic behaviors of fluid-conveying pipe with geometric imperfections.

中文翻译：

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几何不完美管道输送脉动流体的非线性参数振动

在本文中，考虑几何缺陷和中平面拉伸引起的几何非线性，建立了两端支撑的流体输送缺陷管道的新模型。积分偏微分方程通过 Galerkin 方法离散化，并通过四阶 Runge-Kutta 积分算法求解。与完美管道输送流体的超临界干草叉分叉相比，结果表明，当流速增加时，不完美管道会出现尖头分叉。在数值结果和分析结果之间观察到了极好的一致性。两个稳定的不对称分岔分支在后屈曲状态下带来了有趣的现象。通过建立分岔图研究了不完美管道的全局非线性动力学行为。与完美管道的干草叉分叉相比，几何缺陷幅度对非线性响应的影响导致尖头分叉。以脉动频率为分岔参数时，有明显的非共振区、低能共振区和高能共振区。在高能共振范围内，出现与次谐波共振和组合共振并存的第一模态振动。由于将平均流速和脉动幅度设置为分岔参数，不完美管道的振动变得越来越复杂。振动表现出更丰富的动态行为，包括周期、多周期、准周期和混沌运动。粘弹性阻尼能有效抑制振动响应，将高能共振转变为低能共振状态。改进的模型和相应的结果为进一步研究具有几何缺陷的流体输送管道的动力学行为提供了有用的信息。