Karhunen–Loève decomposition of coupled axial/bending vibrations of beams subject to impacts

This work presents a study of the oscillations of a vertical slender beam, clamped in its upper extreme, pinned in its lower one and constrained inside an outer cylinder in its lower portion. The beam is subject to distributed axial loads, due to its own weight, leading to geometric softening of its lower portion and thus yielding a large number of vibroimpacts with the outer cylinder. This is due to the axial–bending coupling, often called geometric stiffening and largely discussed in the last two decades. Here, it is accounted for by using a non-linear finite element model proposed in a previous work, in which non-linear strain–displacement relations are considered. To help understand this non-linear coupled vibro-impact problem, the Karhunen–Loève decomposition, also known as the proper orthogonal decomposition, is applied to its simulated dynamics. The results show that the micro-impacts, accompanying the beam–hole impacts and mainly due to the beam compressive softening, and the reaction forces at the top and bottom positions, are well represented only when using a non-linear axial–bending coupling. It is also shown that 15 proper orthogonal modes are sufficient to reconstruct the dynamics of the impacting beam under a 3% error margin.