The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems

Abstract

Because nowadays structural engineers are willing to use or at least understand nonlinearities instead of simply avoiding them, there is a need for numerical tools performing analysis of nonlinear large-scale structures. Among these techniques, the harmonic balance (HB) method is certainly one of the most commonly used to study finite element models with reasonably complex nonlinearities. However, in its classical formulation the HB method is limited to the approximation of periodic solutions. For this reason, the present paper proposes to extend the method to the detection and tracking of codimension-1 bifurcations in the system parameters space. As an application, the frequency response of a spacecraft is studied, together with two nonlinear phenomena, namely quasiperiodic oscillations and detached resonance curves. This example illustrates how bifurcation tracking using the HB method can be employed as a promising design tool for detecting and eliminating such undesired behaviors.

Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber

“…The frequency response of system (2) for the parameters listed in Table 1 was computed using the algorithm proposed in reference [32]. Codimension-1 numerical continuation was carried out using the multi-harmonic balance method, which approximates periodic solutions using Fourier series.…”

“…Since the creation of the quasiperiodic solutions and of the DRC occurs through NS and fold bifurcations, respectively, these bifurcations were tracked in the three-dimensional space (x 1 , ω, F ) using the multi-harmonic balance method [32]. To facilitate the interpretation of the results, the bifurcation loci were projected onto the two-dimensional plane (x 1 , F ).…”

Frequency-Domain Sensitivity Analysis of Stability of Nonlinear Vibrations for High-Fidelity Models of Jointed Structures

“…is a matrix of massnormalised mode shapes of a linear structure calculated for a structure with the absence of the contact interactions; m is the number of mode shapes used for the model reduction. In order to calculate Φ the following auxiliary eigenproblem is solved 2 = KΦ MΦΩ (15) The solution of this eigenproblem provides a matrix of eigenvectors Φ and a matrix of natural frequencies corresponding to the eigenvectors:…”

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

“…The latter could be done by directly computing the truncated spectrum of Hill's matrix for a given periodic state. But a confusion subsists as for the sorting method one should use for the spurious computed spectrum, between eigenvalue [9,27,45] or eigenvector sorting [2,22,26,41]. Furthermore, only Floquet exponents are usually considered and modal informations from FFs are usually neglected.…”