On the complex mode shapes and natural frequencies of clamped-clamped fluid-conveying pipe

In the present study, the closed-form expression for complex mode shapes of a fluid-conveying pipe using Timoshenko beam theory is developed for the first time. By applying the method of separation of variables, the complex mode shapes and eigenvalues are obtained. To the best knowledge of the authors, there is no study carried out on the complex eigenvalue problem for vibration analysis of the Timoshenko fluid-conveying pipes. Given this oversight, in this study the effects of fluid velocity on the imaginary and real parts of the mode shapes and natural frequencies of a clamped-clamped fluid-conveying pipe are investigated. The results show that for very low fluid velocities, the first and the second mode shapes are similar to that of an equivalent beam. For high fluid velocities, the first mode shape experiences some deviation so that it looks like the second mode shape of a pipe having very low fluid velocity. The results show that fluid flow in the pipeline effectively reduces its bending moment as well as the energy of vibration. In addition, the nodal points corresponding to the second mode are replaced with quasi-node points.