Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams

Abstract

This work addresses the large amplitude nonlinear vibratory behavior of a rotating cantilever beam, with applications to turbomachinery and turbopropeller blades. The aim of this work is twofold. Firstly, we investigate the effect of rotation speed on the beam nonlinear vibrations and especially on the hardening/softening behavior of its resonances and the appearance of jump phenomena at large amplitude. Secondly, we compare three models to simulate the vibrations. The first two are based on analytical models of the beam, one of them being original. Those two models are discretized on appropriate mode basis and solve by a numerical following path method. The last one is based on a finite-element discretization and integrated in time. The accuracy and the validity range of each model are exhibited and analyzed.

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

“…Von Kármán models can be viewed as an intermediate and consistent model between linear models and full large rotation models [53,54]. On the contrary, large rotation models have to be considered in the other cases (beams with ends free to move such as cantilever beams, see [55,56]), out of the scope of the article.…”

“…N . Depending on the formulation of the geometrical nonlinearities in the finite element discretization, this expansion can be exact (it is the case for 3D elements or for shell/plate/beam elements with a von Kármán strain/displacement nonlinear law [22,34]) or truncated, for instance in the case of large rotation elements (such as geometrically exact beam elements for which the internal force vector include sine and cosine functions of the rotation degrees of freedom of the cross section [56,58]). Notice that Eq.…”

Reduced Order Models for Nonlinear Dynamic Analysis With Application to a Fan Blade

“…In the framework of rotating structures, a comparative study between several models of a rotating cantilever beam in terms of accuracy and validity is presented in [10]. These models are mainly used in the study of helicopter and turbo-machinery blades [11], modelisations of slender beams or thin shells [12] and fluid-structure interactions [13].…”

A generic and efficient Taylor series–based continuation method using a quadratic recast of smooth nonlinear systems

“…'s are frequently introduced to described the rotation of the cross-section, and as a consequence, the functions sine and cosine also frequently appear in the formulations. Let us illustrate again with the rod model used in the work of Thomas et al 40 for computing nonlinear vibrations. With u, w, and denoting the axial, transversal, and rotational displacement of the rod cross-section, the generalized strain e, k, and for tension, bending, and transverse shear reads e = (1 + u ′ ) cos + w ′ sin k = ′ (64)…”