Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses

In this paper, nonlinear equations of three-dimensional motion are established for straight fluid-conveying pipes with general boundary conditions. Firstly ten springs at the two ends of the pipes are used to represent the general boundary conditions, and the displacements are expressed as the superposition of a Fourier cosine series and four supplementary functions to satisfy the boundary conditions. Then in the Lagrangian framework, three coupled equations of motion in compact matrix form are derived by using the extended Hamilton's principle, and the effect of the extra linear springs and lumped masses at arbitrary positions along the pipe are also considered in these equations. The natural frequencies of pipes with several different boundary conditions are computed by the linearized dynamic equations in order to validate the method through comparisons with results from other reliable sources. Nonlinear dynamic behavior of the pipes with different boundary conditions is analyzed by solving the nonlinear equations using the Runge-Kutta scheme at last. It is found that the method presented in the paper could conveniently and effectively solve the nonlinear vibration problem of fluid-conveying pipes with general boundary conditions and additional springs and masses.