A semianalytical approach to obtain the proper orthogonal modes (POMs) is described for the non-linear oscillation of a cantilevered pipe conveying fluid. Theoretically, while the spatial coherent structures arc the cigenfunctions of the time-averaged spatial autocorrelation functions, it emerges that once the Galerkin projection of the proper orthogonal modes is carried out using the uniform cantilever-beam modes, the spatial dependency of the integral eigenvalue problem can be eliminated by analytical manipulation which avoids any spatial discretization error. As the solution of the integral equation is obtained semi-analytically by linearly projecting the proper orthogonal modes on the cantilever-beam modes, any linear or non-linear operation can be carried out analytically on the proper orthogonal modes. Futhermore, the reduced-order eigenvalue problem minimizes the numerical pollution which leads to spurious eigenvectors, as may arise in the case of a large-scale eigenvalue problem (without the Galerkin projection of the eigenvectors on the cantilever-beam modes). This methodology can conveniently be used to study the convergence of the numerically calculated proper orthogonal modes obtained from the full-scale eigenvalue problem.

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Persistent URL dx.doi.org/10.1115/IMECE2002-32211
Conference 2002 ASME International Mechanical Engineering Congress and Exposition
Sarkar, A. (A.), & Paidoussis, M.P. (M. P.). (2002). A cantilever conveying fluid: Coherent modes versus beam modes. In American Society of Mechanical Engineers, Applied Mechanics Division, AMD (pp. 1119–1127). doi:10.1115/IMECE2002-32211