Dynamics and stability of an extending beam attached to an axially moving base immersed in dense fluid

In the present study, we construct a theoretical model for investigating the dynamics and stability of a flexible slender cantilever which is attached to an axially moving base fully immersed in an incompressible fluid. Meanwhile, the cantilevered beam is subjected to a time dependent axial extension. The coordinate transformation is utilized to derive the governing equations with consideration of an axial added mass coefficient and realistic initial conditions. Based on the Galerkin approach and Runge-Kutta technique, the numerical results for the dynamical behavior of the system under conditions of steady rate of extension and speed of the moving base are displayed. It is demonstrated that there is a critical value of extension rate at which the beam loses stability in the case when the base is fixed. As the base moves beyond a certain speed, however, the beam returns to be stable even if the extension rate is above the critical value. Furthermore, the beam system can exhibit peak response as the base moving speed is much higher than the extension rate.