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Modal analysis of structures in periodic stateA. Lazarus

Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, Francearnaud.lazarus@upmc.fr

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This presentation focuses on the numerical computation of linear vibrational modes, or Floquet Forms, ofmechanical systems in periodic states such as rotating machineries with imperfections or any structures that arein compressive or tensile periodic states. To make our point, we present an original spectral method through thefundamental example of the oscillations of a 2D bi-articulated bar submitted to a periodic compressive load at itsend. We show that Floquet Forms generalize the concept of classic modal analysis for structures in equilibriumstates. Because of the complexity of the frequency spectrum of Floquet Forms as compared to the classic harmonicmodes of vibration, the type of instabilities encountered by periodically-varying structural systems is naturallymuch richer than systems in equilibrium.

1 Context and needModal analysis is a key concept in structural mechanics

and is today commonly used by engineers in various fieldssuch as Civil Engineering, automotive, aerospace or rotor-dynamics industries. It especially allows to reveal intrinsicvibrational properties of structures in equilibrium states,as well as their local stability, and it is therefore often anecessary step in the design of structures.

Thanks to Floquet theory [1], it should be possibleto extend this modal approach to structures in periodicstates, i.e. structures with mechanical or geometricalproperties that vary periodically with time. Modal analysisof structures in periodic states could be of practical interestfor many problems in mechanical engineering includingthe vibrational behavior of rotating machineries withimperfections, the design of structures submitted to periodiccompression or tension loadings or the stability analysis ofstructures undergoing large oscillations [2]. Surprisingly,to the best knowledge of the authors, this generalizationof modal analysis has never been completely and clearlyimplemented. Due to the conceptual complexity of thevibrational analysis of structures in periodic states, thedeveloped numerical methods mostly focused on dynamicstability [3, 4], neglecting the modal informations that areeither ignored, inaccurate or even inaccessible from thecomputations. As a consequence, the natural link betweenmodal analysis of equilibrium and periodic states yetsuggested by Floquet remains largely unknown by thestructural engineer and researcher community.

2 Task and FindingsHere, through the archetypal example of a 2D bi-

articulated bar submitted to a periodic compressive load atits end as illustrated in Fig.1.a, we present an original andaccessible spectral numerical framework that generalizesthe modal analysis of structures in equilibrium to structuresin periodic states. Although apparently simple, thisfundamental system encounters all the classic instabilities of2-dimensional dynamical systems that can be easily sorteddepending on the fundamental frequency of the state Ω andthe parameter η as illustrated in Fig.1.b-e.

Thanks to a single algorithm based on the sorting of theeigenvalue spectrum of Hill’s matrix [5], our method allowsto compute Floquet forms (FFs) of periodically conservativeor non conservative systems. FFs are almost periodic modalentities reducing to classic harmonic vibrational modesfor the particular case of structures in equilibrium states(for a system such as the one illustrated in Fig.1.a, FFs aresimply the 2 modes of vibration of the originally straightbi-articulated bar). Like for classic modal analysis, FFs

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Figure 1 – (a) The system under study is a bi-articulated barsubmitted to a compressive load with a period T = 2π/Ω atits end. The load is either periodically conservative (η = 0,

i.e. horizontal force) or non conservative (η = 1, i.e.following force). (b) Steady-state bifurcation on a

T -periodic orbit (Ω , 0 and η = 0). (c) Neimark-Sackerbifurcation on a almost-periodic orbit (Ω , 0 and η = 1). (d)

Static bifurcation on a new equilibrium state (Ω→ 0 andη = 0). (e) Hopf bifurcation on a T -periodic stationary state

with T , 2π/Ω (Ω→ 0 and η = 1).

carry the intrinsic vibrational signature of the structureand their spectrum allows to assess the linear stability ofthe periodic state as illustrated in Fig.2.b-e. In the generalcase when Ω , 0, the almost-periodicity of FFs as wellas the dependency of their frequency spectrum on Ω cancause the periodic state to lose stability and consequentlylead to steady-state, period doubling or Neimark-Sackerbifurcations as illustrated in Fig.1.b,c and Fig.2.b,c. In theparticular cases when Ω → 0, FFs reduce to harmoniceigenmodes and the equilibrium state can eventually becomestatically or dynamically unstable depending on the loadingparameter A as shown in Fig.1.d,e and Fig.2.d,e. Like in

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classic modal analysis, since FFs are first order perturbationsof stationary states, they carry the kinematic shape andtemporal signature of the bifurcated nonlinear vibrationalresponses such as illustrated in Fig.1.b-e. Their computationand analysis is therefore crucial for the design of structuresin periodic states.

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Figure 2 – Argand representation of the spectrum s and s̄ ofthe 2 Floquet Forms of the system shown in Fig.1.b-e at theonset of bifurcation. (b1) Steady state bifurcation. (b2) Flip

or period-doubling bifurcation. (c) Secondary Hopf, orNeimark-Sacker bifurcation. (d) Steady-state bifurcation orbuckling. (e) Dynamic instability or flutter leading to a Hopf

bifurcation.

3 Conclusions and perspectivesThe algorithm presented in this article is a unified

method to perform modal analysis of conservative or nonconservative systems in periodic states, including equilibria.This unique framework emphasizes the natural link betweenvibrational behavior of standstill and periodically-varyingsystems that are common in mechanical engineering. Likefor classic modal analysis, this generalized modal approachshould allow to design a broader range of structures bycomputing and analyzing FFs which give better physicalinsight and understanding on the linear stability of nonlineardynamical systems. This first study on a simple archetypal 2dof system will pave the way to the modal analysis of realstructures in periodic state with a large number of degreesof freedom. Due to the linear nature of those entities, theinteresting properties of classic vibrational modes shouldremain with FFs which could be promising candidates formodal reduction for structures in periodic states or predictorsin nonlinear algorithm such as Newton-Raphson method tocompute nonlinear periodic orbits of vibrating structures.

Références[1] G. Floquet. Sur la théorie des équations différentielles.

Annales scientifiques de l’École Normale Supérieure,8 : 3-132, 1879.

[2] V. V. Bolotin. The dynamic stability of elastic systems.Holden Day, 1964.

[3] R. Castelli and J.-P. Lessard. Rigorous numericsin Floquet theory : computing stable and unstablebundles of periodic orbits. SIAM Journal on AppliedDynamical Systems, 12 : 204-245, 2013.

[4] J. Zhou, T. Hagiwara and M. Araki. Spectralcharacteristics and eigenvalues computation of theharmonic state operators in continuous-time periodicsystems, Systems & Control Letters, 53 : 141-155,2004.

[5] A. Lazarus and O. Thomas. A harmonic-based methodfor computing the stability of periodic solutions ofdynamical systems. Comptes Rendus Mécanique, 338 :510-517, 2010.

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