Sliding beams, Part II: time integration

In this paper we obtain solutions for the discretized incremental system equations, as obtained in Part I, under certain initial and boundary conditions and/or speciÞed applied loads, using the variable domain beam element. As a check on the validity of implementation, we Þrst limit ourselves to linear analysis and obtain results for the axially inextensible sliding beams which we compare with the results reported in the literature. Second we set the axial velocity to zero and solve some special cases when the length of the beam is constant. In this case, we check the formulation and its implementation for non-linearities in the system due to large displacements. Finally, we solve the sliding beam problem for small amplitude oscillations, with a non-linear solver and compare the results with those obtained by the linear solver used for inextensible sliding beams. With these preliminary tests completed, we obtain the transient response of the free and forced large amplitude vibrations of the ßexible sliding beam and demonstrate the need for using a nonlinear analysis for this complex system. Finally, we consider the stability of the motion of periodically time varying ßexible sliding beams and show that in the case of parametric resonance, the unstable regions obtained in the linear analysis, which imply unbounded amplitudes, are indeed stable and bounded when non-linear terms are taken into account