Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column

Abstract

We present a spectral method to compute the transverse vibrational modes, or Floquet Forms (FFs), of a 2D bi-articulated bar in periodic elastic state due to an end harmonic compressive force. By changing the directional nature of the applied load, the trivial straight Ziegler column exhibits the classic instabilities of stationary states of dynamical system. We use this simple structure as a numerical benchmark to compare the various spectral methods that consist in computing the FFs from the spectrum of a truncated Hill matrix. We show the necessity of sorting this spectrum and the benefit of computing the fundamental FFs that converge faster. Those FFs are almost-periodic entities that generalize the concept of harmonic modal analysis of structures in equilibria to structures in periodic states. Like their particular harmonic relatives, FFs allow to get physical insights in the bifurcations of periodic stationary states. Notably, the local loss of stability is due to the frequency lock-in of the FFs for certain modulation parameters. The presented results could apply to many structural problems in mechanics, from the vibrations of rotating machineries with shape imperfections to the stability of periodic limit cycles or of any slender structures with tensile or compressive periodic elastic stresses.

Nonlinear polarization coupling in freestanding nanowire/nanotube resonators

“…Moreover, it includes stability analysis based on computing the Floquet exponents in the frequency domain with a Hill eigenvalue problem. 23,24 The analytical treatment is used to clarify the physical origins of this transition and gives simple formulae for the transition depending on the geometric parameters of the NNs. The validity of these formulae is confirmed by comparison with simulations.…”

A Taylor series-based continuation method for solutions of dynamical systems

“…The subscript f stands for full and the subscript a stands for auxiliary. Once this recast is given, the auxiliary variables U a are expanded in Taylor series (2) in the same manner as for U. The expansions are then introduced in the quadratic equations (3) and the terms of same order in a are collected.…”

“…The starting point is computed automatically using the eigenvectors and the eigenvalues of the Jacobian matrix at the bifurcation point [3]. The stability of the periodic orbit is given by the Floquet exponents of the systems, computed with Hill's method as explained in [31] and [2]. With this system two Neimark-Sacker bifurcations are detected, one on each branch, for λ ≃ 0.03 on the branch arising from the second Hopf bifurcation and for λ ≃ 0.35 on the branch arising from the first Hopf bifurcation.…”

Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances

“…Amongst the 2n(2H + 1) solutions of (13), only 2n actually correspond to the Floquet exponents of the system while the rest are a numerical artifice due to the multiplicity of harmonics, and provide redundant information. To extract the pertinent solutions, one can either sort eigenvalues or eigenvectors, the latter being seemingly a more robust approach [27]. A point (X, ω) on the response curve corresponds to a stable periodic motion if the real parts of all its Floquet exponents is negative, and to an unstable one otherwise.…”