An approximate method for pipes conveying fluid with strong boundaries

An approximate method is proposed for the strong nonlinear and non-homogenous boundary value problem of a pipe conveying fluid for the first time. Usually, the boundary value is satisfied transcendentally in the truncation processing acting on the partial differential governing equation. However, the nonlinear and non-homogenous boundary disables it. To overcome this problem, the method of modal correction together with the modal projection is proposed. This method treats nonlinear and non-homogenous boundaries as generalized governing equations. Since then, nonlinear and non-homogenous terms in the boundary could be discussed fully based on the harmonic balance method (HBM). The discussion on natural frequencies suggests the standard of the convergence of the modal projection. The harmonic convergence can be judged by the solution with more harmonics. By treating those coefficients of harmonics as time-varying parameters, state equations will be produced. Based on them, both of the stability of the approximate solution and the type of bifurcation could be judged. Besides, coefficients of each order harmonic on different modal projections or spatial corrections could reveal the detailed information of the response, such as the harmonic caused by the nonlinearity or the power distribution on different modal projection. By comparing with the Dirac operator and the multiscale method, the advantage of the proposed method on dealing with strong boundaries is verified. Nonlinear and non-homogenous boundaries are not the trap for the gyroscopic system anymore. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

An approximate method is proposed for the strong nonlinear and non-homogenous boundary value problem of a pipe conveying fluid, using modal correction and projection to treat the boundaries as generalized governing equations. The discussion on natural frequencies and harmonic convergence helps in judging the stability of the solution and the type of bifurcation, while revealing detailed information of the response. The proposed method shows advantages in dealing with strong boundaries compared to other existing methods.