Nonlinear vibration and instability of embedded viscose-fluid-conveying pipes using DQM

In this paper, nonlinear vibration and instability response of an embedded pipe conveying viscose fluid is investigated. The pipe is considered as a Timoshenko beam embedded on an elastic foundation which is simulated by spring constant of the Winkler-model and the shear constant of the Pasternak-model. The external flow force, acting on the beam in the direction of the flexural displacement is described by the well-known Navier-Stokes equation. The corresponding governing equations are obtained using Hamilton's principle considering nonlinear strains and first shear deformation theory. In order to obtain the nonlinear frequency and critical fluid velocity for clamped supported mechanical boundary condition at two ends of the pipe, Differential quadrature method (DQM) is used in conjunction with a program being written in MATLAB. The effect of dimensionless parameters such as aspect ratios of length to radius of the pipe, Winkler and Pasternak modules, fluid velocity and viscosity as well as the material type of the pipe on the frequencies and instability of pipe are investigated. Results indicate that the internal moving fluid plays an important role in the instability of the pipe. Furthermore, the nonlinear frequency and instability increases as the values of the elastic medium constants  and viscosity of fluid increases.