Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

Abstract

This paper deals with the nonlinear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the nonlinear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to a third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the nonlinear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed.The model is constructed by the introduction of a reference configuration which is deduced from the nonlinear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.

Robust dynamic analysis of detuned-mistuned rotating bladed disks with geometric nonlinearities

“…corresponding to the rotational axis (0, 0, 1) in R. From now on, the convention of summation over repeated latin indices is used. The unknown displacement field in R is denoted as u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) and is solution of the following nonlinear boundary value problem [37].…”

“…In the present context, it is assumed that there is no rigid body motion of the rotor and that the bladed disk structure rotates around a fixed axis. Such assumptions then allows for obtaining all the above equations as can be shown in [38,37,39,40].…”

“…Let B NL (2πν; δ K ) be the random variable depending on δ K , defined by Eq. (37), and constructed using the NL-SROM1. For two values of δ K controlling the mistuning level, Figure 21 (δ K = 0.03) and Figure 22 (δ K = 0.1) display the confidence region of random variable B NL (2πν; δ K ), estimated with a probability level of 0.95, for configurations P 0 (tuned), and for P 6 , P 11 , and P 25 (detuned defined in Appendix A).…”

Mistuning analysis and uncertainty quantification of an industrial bladed disk with geometrical nonlinearity

“…Concerning the reduced damping operator, a modal damping model is added. Finally, the reduced external load is written as (12) in which x 0 is the current position in the natural configuration [24]. Concerning the choice of the projection basis, the one related to the linear eigenvalue problem of the rotating tuned conservative structure, for which the gyroscopic coupling effects are ignored, is chosen.…”

Experimental evaluation and model of a nonlinear absorber for vibration attenuation

“…The one-DOF nonlinear model is composed of a mass-spring-damper system with a nonlinear spring and a nonlinear damper, subjected to an excitation of its base (see the scheme displayed in Figure 1). We introduce a parameterized family of one-DOF nonlinear oscillators, for which the proposed algebraic model is inspired/coherent with the linear viscoelasticity theory in finite displacements (nonlinear model) without memory [38,39]. The motivation of this choice is the following: the nonlinear absorber that will be designed will not have material nonlinearities but only nonlinear geometrical effects and will have a viscoleastic type behavior related to the choice of the material.…”