Nonlinear normal modes in an intrinsic theory of anisotropic beams

Abstract

Nonlinear normal modes in nonlinear oscillations of beams are derived from intrinsic equations, that is, using velocities and strains as primary degrees of freedom. Displacements and rotations are thus not system states but are instead obtained using the propagation of the local beam material reference frames, as in rigid-body dynamics. It is shown that the intrinsic variables suffice to describe the free vibrations of the beam. The approach does not need assumptions in the material properties, i.e., it is valid for general anisotropic behavior, or the beam kinematics, i.e., it is based on Cosserat's exact geometrical description of the deformable curve. Furthermore, the nonlinear modal equations in intrinsic coordinates are obtained from integrals involving only products of the mode shapes and known coefficients. Using this description, the nonlinear normal modes are sought through an asymptotic approximation to the invariant manifolds that define them in the space of intrinsic modal coordinates. Particular cases of homogeneous isotropic and composite cantilever beams are finally used to exemplify the approach. *

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

“…It is worth noticing that the VK model is naturally quadratic, whereas the Inxt model is cubic so that numerous additional variables (R p k , Q p k ) must be added, as in [42] for piezoelectric laminated beams. Other works [13,56] use elegant formulations of the equations of motion , first proposed in [31], using both the displacement and the velocity as variables. By expansion onto a suitable basis, a naturally quadratic formulation is obtained, which is exact without any restriction on the displacement/rotation magnitude.…”

An Intrinsic Description of the Nonlinear Aeroelasticity of Very Flexible Wings

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Robust Aeroelastic Control of Very Flexible Wings using Intrinsic Models

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“…The paper thus describes the development of a H ∞ control system on wings described by a intrinsic modal formulation, with the aim of providing gust alleviation and disturbance rejection using conventional flap actuation. The paper will first review the geometrically exact, intrinsic formulation of beam structures, expanding the description of Palacios 17 to a nonlinear cantilever beam on a moving base. Then, the 2-D aerodynamic model and the projection of the resulting nonlinear aeroelastic system on the modal basis will be discussed.…”