Nonlinear normal modes in an intrinsic theory of anisotropic beams

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 Cosseratʹs 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.