Finite element analysis for free vibration of pipes conveying fluids-physical significance of complex mode shapes

A finite element formulation is presented for the natural vibration analysis of pipes conveying fluids. The solution of the resulting quadratic eigenvalue problem generally yields complex eigenvalues and eigenvectors. The present study then develops a robust mathematical procedure that combines the real and imaginary components of the eigenvectors to form physically attainable (i.e., real) mode shapes. The procedure yields a family of solutions that is more general than previously known solutions. The well-known classical mode shape is shown to be recoverable as a special case from the present solution. The study provides new insights on the effects of viscous damping, axial compressive force, and the flexibility of intermediate pipe supports on the response. Additionally, the study develops a novel algorithm based on Hermitian angles between eigenvectors to automate the tracing of mode evolution in the frequency-velocity plots.