DRAFT: A HARMONIC-BASED METHOD FOR COMPUTING THE STABILITY OF PERIODIC OSCILLATIONS OF NON-LINEAR STRUCTURAL SYSTEMS

In this paper, we present a validation on a practical example of a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Base d on Floquet theory and Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eige nvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing t he Harmonic Balance Method and Asymptotic Numerical Method continuation technique with the developed Hill method, we o btain a purely-frequency based continuation tool able to com pute the stability of the continued periodic solutions in a reduc ed omputation time. This procedure is validated by considering a externally forced string and computing the complete bifurcat ion diagram with the stability of the periodic solutions. The part icular coupled regimes are exhibited and found in excellent agreem ent with results of the literature, allowing a method validatio n. Introduction Determining the local stability of a dynamical system periodic solution is of primary interest in an engineering con text since only stable solutions are experimentally encoun tered. Moreover, a change in the stability can lead to significant qu alitative, and possibly dramatic, changes in the system respon se. A Address all correspondence to this author. commonly-used method consists in computing the monodromy matrix of the linear periodic system governing the evolutio n of the perturbed solution and compare its eigenvalues (Floque t m ltipliers) moduli to one. Two well known numerical strategie s are classically used. The first technique consists in a time-int gration of the Jacobi matrix of the initial dynamical system over one period [1]. In the second technique, the monodromy matrix is si mply a by-product of a shooting continuation method [2]. The efficiency of those methods has been proved for years, but, inherently, the eigenvalues accuracy depends on the ch osen time-step size, leading to possible long time computations . Furthermore, those time-domain approaches are not necessaril y best suited for nonlinear periodic solution computed, for insta ce, with a harmonic balance method. In the latter, eigenvalues o f the linear operator can naturally be computed with Hill’s me thod also based on Fourier series expansion. This harmonic-base d method deals with linear periodic systems and has been intro duced by Hill one century ago for the determination of the lun ar perigee [3]. Since then, it has been applied to a wide range of physical problems like the computation of energy eigenvalu es of the Schr̈ odinger equation [4,5], the stability of limit cycles in electrical circuits [6] or the vibratory behavior of flexibl e rotating machines [7,8]. However, although Hill’s method often prov ides satisfactory results, the meaning of the computed eigenval ues is generally misunderstood and may lead to wrong results. Actu ally, this method is not of a trivial use since it requires to a pprox1 Copyright c © 2010 by ASME imate the spectra of infinite-dimensional operators, i.e. to sort the most converged eigenvalues among all the numerical ones . A recent work of the authors proposes a simple numerical strategy to sort these eigenvalues and therefore proper ly d terminate the stability of periodic solutions of dynamical systems [9]. A simple criterion, based on the energy distributi on of the computed eigenvectors, is proposed and enables to extra ct the most converged eigenvalues (Floquet exponents) of the trun cated problem, and therefore determinate the local stability of t he solution. By mixing the standard harmonic balance method with th e so-called asymptotic numerical method continuation techn ique based on the quadratic recast of the dynamical system [10], i t becomes a straightforward and efficient procedure to continue the periodic solutions as well as their associated Floquet expo nents. Whereas the case of a forced Duffing oscillator is proposed in [ 9] as a validation example, this article focuses on another one . The main features of the developed harmonic-based continuation tool are first exposed. Then, the case of an externally excited string is considered. This problem is interesting sin ce it extends the results obtained on the Duffing oscillator in two wa ys. Firstly, more than one mechanical degree of freedom is inclu ded. Secondly, since both transverse polarizations are conside red in the model, one-to-one internal resonances are present, all owing a particularly complicated bifurcation diagram and showin g several stable and unstable solutions as well as additional pit chfork bifurcations. Finally, this problem is qualitatively simi lar to the one of a perfect circular plate, where one to one resonances a re observed between companion modes due to the symmetry of revolution. In the case of the string, out of plane motion are observed, which are equivalent to the travelling waves observ ed for the circular plate [11,12]. STABILITY OF PERIODIC SOLUTIONS General formulation We consider continuous-time dynamical systems governed by the general equation ẋ (t) = f (