Reduced-order models for the analysis of a vertical rod under parametric excitation

Abstract

This paper focuses on the analysis of the parametric excitation of a vertical and immersed flexible rod, showing the influence of the choice of the shape function used in the Galerkin’s method. For this, three different reduced-order models (ROMs) are obtained from the continuous equation of transverse motion employing different shape functions. The first model (ROM(i)) uses an approximation of the actual vibration mode of the rod, written as a “Bessel-like” function. The second model (ROM(ii)) is based on a single trigonometric function as the shape function. Finally, a multi-modal ROM (ROM(iii)) is obtained using three trigonometric functions as a set of shape functions. Simulations are carried out aiming at verifying the capability of each model to properly represent the dynamics of the rod under parametric excitation. The quality of the numerical results obtained from the integration of the aforementioned ROMs is assessed by means of a comparison with a solution based on the finite element method (FEM). In addition to the numerical analysis, an analytical solution for the steady-state amplitude of a generic Duffing-Mathieu-Morrison oscillator is obtained using the method of multiple scales for the one degree of freedom ROMs. A case study is developed using the data of a vertical riser as an example of an engineering application. Maps of the steady-state amplitude as a function of the excitation amplitude and frequency are plotted using both the numerical simulations and the multiple scales solution. The results show that ROM(i) and ROM(iii) are in good agreement with the finite element solution. ROM(i) has the advantage of having only one degree of freedom and, consequently, can be studied using the analytical solution aforementioned. The use of a ROM with one degree of freedom using “Bessel-like” functions in the Galerkin’s scheme is concluded to have clear advantages from the practical point of view. The analytical solution allows this kind of ROM to give a post-critical amplitude map with low computational effort and that is in good agreement with the maps obtained with the simulation of the ROMs.