Dynamics of an articulated chain of rigid pipes discharging fluid under concomitant support excitation: A numerical anal
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(2020) 42:581
TECHNICAL PAPER
Dynamics of an articulated chain of rigid pipes discharging fluid under concomitant support excitation: A numerical analysis Igor Mancilla Lourenço1 · Renato Maia Matarazzo Orsino1 · Guilherme Rosa Franzini1 Received: 18 February 2020 / Accepted: 2 October 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract This work proposes a numerical investigation on the occurrence of parametric excitation in the classical Benjamin problem, when the support of the articulated chain of pipes is subjected to a prescribed oscillatory motion. Equations of motion for an articulated chain composed of two rigid pipes ejecting fluid are derived. Floquet theory is applied to the linearized mathematical model, and curves of instability as functions of the parametric excitation (amplitude and frequency) are obtained for selected internal flow velocities. In addition to the linear analysis, maps (showing the post-critical response as a function of some control parameters) are computed using numerical integrations of the nonlinear model. Among other findings, this paper reveals that the presence of the internal flow significantly affects the stability of the trivial solution. Furthermore, the concomitant parametric excitation and internal flow effects lead to maps of post-critical response with a marked erosion, i.e., small variations in the parameters of the mathematical model have significant effects in the observed response. Keywords Pipe discharging fluid · Parametric excitation · Floquet theory · Nonlinear dynamics · Post-critical analysis List of symbols A Amplitude of the prescribed base motion 𝐂 Dimensional nonlinear damping matrix ̄ Dimensional linearized damping matrix 𝐂 𝐂∗ Dimensionless nonlinear damping matrix ̄ ∗ Dimensionless linearized damping matrix 𝐂 𝐅 Dimensional force vector 𝐅∗ Dimensionless force vector g Numerical value for gravitational acceleration Ki , i = 1, 2 Dimensionless ith torsional stiffness 𝐊 Dimensional nonlinear stiffness matrix ̃ Dimensional linearized stiffness matrix, 𝐊 calculated without internal flow velocity and parametric stability ̄ Dimensional linearized stiffness matrix 𝐊 𝐊∗ Dimensionless nonlinear stiffness matrix ̄ ∗ Dimensionless linearized stiffness matrix 𝐊 ki , i = 1, 2 Values of the torsional springs L Total length of chain of pipes Technical Editor: Marcelo Areias Trindade. * Igor Mancilla Lourenço [email protected] 1
Offshore Mechanics Laboratory, Escola Politécnica, University of São Paulo, São Paulo, Brazil
Li , i = 1, 2 Length of the rigid pipes M Mass per unit length of the internal fluid 𝐌 Dimensional nonlinear mass matrix ̃ Dimensional linearized mass matrix, cal𝐌 culated without internal flow velocity and parametric stability ̄ Dimensional linearized mass matrix 𝐌 𝐌∗ Dimensionless nonlinear mass matrix ̄ ∗ Dimensionless linearized mass matrix 𝐌 m Mass per unit length of the pipes 𝐧x Unit vector of axes x 𝐧y Unit vector of axes y Qi Terms of Euler–Lagrange equation for generalized coordi
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