Spatial nonlinear vibrations of a pipeline section supported at the ends are studied. A pipe bent under both its own weight and constant pressure of the fluid contained therein is subjected to hydraulic shock. The model of bending and rotational motions of the pipeline is used. The gravity forces and the Coriolis (inertia) forces as well as the mutual effect of internal pressure and changes in the curvature of the axial line of the pipe are taken into account. Oscillatory movements of the pipeline are described by a system of two nonlinear differential equations. The first form of oscillation is considered. For an approximate analysis of the dynamics of the pipeline deformation, the inertial and inertial-elastic stages are introduced. At the first stage, only pressure in the fluid and inertial forces are taken into account. The second stage of the bending and rotational motions of the pipeline is a continuation of its inertial stage. At the end of the first stage, the action of the shock load ceases. The Cauchy problem with zero initial conditions is also solved by using the numerical Runge—Kutta method. A comparison of the results of approximate analytical and numerical solutions is given. Changes in the bending of the midpoint of span and the angle of rotation of the steel pipeline are presented as a function of time for different amplitudes of dynamic pressure.

中文翻译：

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内冲击压力作用下管道的空间非线性振荡

研究了端部支撑的管道截面的空间非线性振动。在自身重量和恒定压力下弯曲的管子承受液压冲击。使用管道的弯曲和旋转运动模型。考虑了重力和科里奥利（惯性）力以及内部压力和管道轴线曲率变化的相互影响。管道的振荡运动由两个非线性微分方程组描述。考虑振荡的第一种形式。为了对管道变形的动力学进行近似分析，引入了惯性和惯性弹性阶段。在第一阶段，仅考虑流体压力和惯性力。管道弯曲和旋转运动的第二阶段是其惯性阶段的延续。在第一阶段结束时，冲击载荷的作用停止。使用数值Runge-Kutta方法也可以解决初始条件为零的柯西问题。给出了近似解析解和数值解的结果的比较。对于不同的动压幅度，跨度中点的弯曲和钢管旋转角度的变化是时间的函数。给出了近似解析解和数值解的结果的比较。对于不同的动压幅度，跨度中点的弯曲和钢管旋转角度的变化是时间的函数。给出了近似解析解和数值解的结果的比较。对于不同的动压幅度，跨度中点的弯曲和钢管旋转角度的变化是时间的函数。