Non-linear reduced-order model for parametric excitation analysis of an immersed vertical slender rod

• Published in: International journal of non-linear mechanics 2016-04, Vol.80, p.29-39
• Main Authors: Franzini, G.R., Mazzilli, C.E.N.
• Format: Article
• Language: English
• Subjects: Amplitudes; Galerkin methods; Immersed slender rod; Instability; Mathematical analysis; Mathematical models; Nonlinearity; Parametric excitation; Reduced-order model; Riser; Stability; Vibration mode
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2015.09.019

A unidirectional three-mode reduced-order model (ROM) for the lateral motion of a slender and immersed rod subjected to harmonic and axial top motion was derived from the continuum equation of motion. Simple trigonometric functions were employed as approximations for the vibration modes and projection functions in Galerkin׳s method. The non-linear character of the ROM comes from the extensibility of the rod axis and the quadratic hydrodynamic damping. The focus of this investigation is the principal Mathieu׳s instability with respect to the first vibration mode, i.e., the condition in which the top-motion frequency is twice the structural first natural frequency. Time histories of modal amplitude, as well as maps of post-critical steady-state vibration amplitudes were obtained and discussed. It is seen that, within the principal parametric instability region of the first mode, the time history corresponding to the second classic (sinusoidal) mode oscillates with dominant frequency of the first classic (sinusoidal) mode. Another finding is that, besides the principal Mathieu׳s instability region, there are also other regions of instability, but with considerably smaller amplitudes. This aspect is due to the non-linear character of the coupled system of equations that defines the ROM. •Reduced-order model.•Parametric excitation of an immersed vertical slender rod.•Modal-amplitude time-histories.•Post-critical amplitude maps.