Three-dimensional nonlinear extreme vibrations of cantilevers based on a geometrically exact model

Finite element computation of nonlinear modes and frequency response of geometrically exact beam structures

“…Beginning with the first mode of vibration, shown in Fig. 4(a-c), it can be seen that it manifests a slightly hardening behavior, meaning that the resonant frequency Ω increases as the vibration amplitude increases, a characteristic well documented in the literature, see [79,12,37]. Point A represents a point of moderate amplitude of displacement along the backbone curve and point C a point of high amplitude.…”

“…Additionally, the cantilever beam is the simplest example that can be selected in order to validate the numerical strategy proposed in this paper. Many works have been published in the past on nonlinear vibrations of cantilever beams and the interested reader can refer to [12] for an overview, as well as the recent numerical contributions [36,35,49,37].…”

“…11, it is noted that the convergence of the auxiliary variables closely mirrors the convergence of θ , which makes sense as the critical auxiliary variables are those that govern the rotation, c and s, which are based on θ as shown in Eq. (37). U x and U y converge "faster" than θ , meaning fewer harmonics are needed.…”

“…Highly flexible structures, however, can be subjected to deflections of extreme amplitude that necessitate richer beam models in order to be simulated appropriately. One solution is to turn to classical three dimensional finite element simulations [34], available in most commercial FE codes, which incorporate geometrical nonlinearities and do not impose restrictions on the maximum deflection that can be simulated accurately [1,[35][36][37]. But, in this case, an enormous number of elements (and, consequently, of degrees of freedom, leading to sometimes prohibitive computation times) is required for accurate simulation of the slender structures considered in this article, largely due to the small thickness and their flexibility in bending.…”

“…This method leads to a simple formulation in the form of partial differential equations with transverse degrees of freedom only, which are discretized by an expansion onto a normal mode basis. The strategy produces highly accurate results when compared with reference numerical FE simulations or experiments [49,37,50,51], although it is restricted to the case of a single cantilever beam.…”