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.

Full text

Noname manuscript No.
(will be inserted by the editor)
Modal and stability analysis of structures in periodic elastic
states: application to the Ziegler column
Barend Bentvelsen ·Arnaud Lazarus
Received: date / Accepted: date
Abstract We present a spectral method to compute
the transverse vibrational modes, or Floquet Forms (FFs),
of a 2Dbi-articulated bar in periodic elastic state due to
an end harmonic compressive force. By changing the di-
rectional nature of the applied load, the trivial straight
Ziegler column exhibits the classic instabilities of sta-
tionary states of dynamical system. We use this sim-
ple 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 ben-
efit of computing the fundamental FFs that converge
faster. Those FFs are almost-periodic entities that gen-
eralize the concept of harmonic modal analysis of struc-
tures in equilibria to structures in periodic states. Like
their particular harmonic relatives, FFs allow to get
physical insights in the bifurcations of periodic station-
ary states. Notably, the local loss of stability is due to
the frequency lock-in of the FFs for certain modula-
tion parameters. The presented results could apply to
many structural problems in mechanics, from the vibra-
tions 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.
Keywords Floquet theory ·Modal analysis ·Stability
analysis ·Structural mechanics ·Bifurcation analysis
B. Bentvelsen ·A. Lazarus
Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, UMR
7190, Institut Jean Le Rond d’Alembert, F-75005, Paris,
France
E-mail: arnaud.lazarus@upmc.fr
1 Introduction
Modal analysis is a key concept in the study of me-
chanical vibrations that is today commonly used by
scientists and engineers in various fields from molec-
ular chemistry [36] to aerospace industries [10]. It is
a spectral numerical method consisting in decompos-
ing the first order perturbation around an equilibrium
state of a dynamical system in a linear superposition
of harmonic eigenmodes [13]. In practice, those modes
allow to reveal intrinsic vibrational properties [28] or
the local stability behavior of structures in equilibria
[21,31]. They can also be used as a projection basis to
reduce the dimensionality of linear or nonlinear vibra-
tional structural models [23, 25].
Thanks to Floquet theory [11, 44], it should be pos-
sible to generalize this modal approach to dynamical
systems in periodic stationary states, i.e. states whose
properties vary periodically with time [4]. In structural
dynamics, this includes rotating machineries with im-
perfections [12] or structures submitted to periodic com-
pression or tension axial loadings such as slender beams
or plates undergoing large vibrations [29]. Mathemati-
cally, a perturbation around a periodic state can be de-
composed in a linear surperposition of almost periodic
eigenfunctions, called Floquet forms (FFs). Like vibra-
tional modes for an equilibrium, FFs could reveal the
intrinsic vibrational properties of a structure in periodic
state and determine its local stability. Floquet theory is
numerically applied in periodically time-varying linear
systems [4] or nonlinear dynamical systems [14,30] but
the spectral computation and physical meaning of FFs
have been overlooked.
In the time domain, two main families of numeri-
cal methods exist that derive from Floquet theory. One
possibility is to compute the Monodromy or state tran-

2 Barend Bentvelsen, Arnaud Lazarus
sition matrix and its eigenvalues (Floquet multipliers)
that assess the local stability of the perturbed periodic
stationary state [33,43]. Although it may be possible
to recover FFs from the eigenvectors of this matrix,
they are usually ignored. A second technique consists
in using the Lyapunov-Floquet transformation to re-
cast a linear time-periodic system in a time-invariant
one [37,38]. This transformation could allow to com-
pute and analyze FFs but it has mostly been used as
a step, which coupled with center manifold reduction
techniques and normal form theories, enables the study
of nonlinear time-periodic systems undergoing bifurca-
tions [5,32,39].
In the frequency domain, the calculation of the spec-
trum of Hill’s matrix [17] associated with the perturbed
periodic state should theoretically give direct access to
FFs, but the computational reality is quite different as
upon numerical truncation, the convergence of Hill’s
matrix is subjected to controversy [6,34, 35]. As a con-
sequence, two main approaches have been developed
to apply Floquet theory. The first numerical approach
consists in imposing particular fundamental frequen-
cies of FFs, or Floquet exponents, to maintain the self-
adjointness of the original perturbed Hill problem and
discard any spurious spectrum. Whether it is marginal
boundary Floquet exponents associated with harmonic
and subharmonic FFs [1,19] or intermediate ones var-
ied within the reciprocal primitive lattice of a Bloch
wave analysis [7,8,18], those methods compute the pa-
rameters of the periodic state associated with a given
Floquet mode. This is different from modal analysis
that would supposedly seek for the FFs associated with
a given perturbed stationary state. 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]. Further-
more, only Floquet exponents are usually considered
and modal informations from FFs are usually neglected.
Floquet modes have been computed for the linear vibra-
tion analysis of non axisymmetric rotating machineries
such as cracked or geometrically imperfect rotors but
the computations are based on Hill’s truncated matrix
without sorting algorithm [15,16,20,24], a method that
is known to give erroneous results in some cases, espe-
cially when dealing with resonant nonlinear dynamical
systems.
Here, we show how to compute FFs in the spectral
domain and highlight their potential for modal analysis
of structures in periodic states, through a simple bench-
mark model: the Ziegler column [3, 42]. The discrete
dynamical system we consider is a classic 2Dmodel
consisting of two articulated rigid bars, connected by
elastic hinges, that are submitted to an end periodic
compressive load so that the elastic state of the Ziegler
column is periodically modulated [32]. We focus on the
transverse oscillatory modes of the structure around its
trivial configuration that is the undeformed straight col-
umn in space, with periodic elasticity in time.
In Section 2, we describe the nonlinear equations of
motion of our model as well as the time-varying linear
ordinary differential equations describing the transverse
linear oscillations about the trivial state. By considering
a non-conservative following or conservative horizon-
tal compressive loading, we show that our benchmark
model captures the classic bifurcations and local insta-
bilities of fixed points and periodic states of discrete
dynamical systems. We then recall the Floquet-Hill fre-
quency method and review the various spectral sorting
techniques that exist to compute FFs. In Section 3, we
perform the modal and stability analysis of the peri-
odically conservative case. In Section 4, we investigate
the influence of a non-conservative positional loading on
the computation of FFs and their associated stability.
In both periodically conservative and non-conservative
cases, we compare the efficiency of the spectral sorting
methods for computing FFs. All our stability results are
validated through the Monodromy matrix algorithm.
Like for classic modal analysis, the free transverse
vibration of a structure in periodic elastic state can be
decomposed in a linear combination of its FFs. As clas-
sic modes are constant eigenfunctions, harmonically vi-
brating; FFs are periodic eigenfunctions, harmonically
modulated. Computation of FFs is crucial since the loss
of local stability is due to frequency lock-in of FFs in
the parameter space of the periodic state. In the conser-
vative case, the Floquet modes are uncoupled. Period
doubling and stationary bifurcations are explained by
the frequency lock-in of a Floquet mode and its conju-
gate in the state space. This mechanism is a general-
ization of the buckling of an equilibrium configuration
seen through its vibrational harmonic modes. In the
non-conservative case, the computed FFs are coupled.
Secondary Hopf bifurcations are explained by the fre-
quency lock-in between two physical FFs in the param-
eter space of the periodic state. Similarly, Hopf bifur-
cation is explained by a frequency lock-in between two
classic modes. Finally, we show that not sorting the
spectrum of Hill’s truncated matrix leads to erroneous
stability results. We also highlight the fact that sorting
the eigenvectors instead of the eigenvalues of the Hill’s
matrix converge faster in the non-conservative case and
for slow modulation in the conservative case.
Our results give new physical insights on the natural
relation between classic harmonic modes of vibrations

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 3
and Floquet modes for structures that are in periodic
elastic states. This paper also clarifies the debate about
the Floquet-Hill frequency method to assess the stabil-
ity of periodic states by clearly showing the necessity of
sorting the spectral outcomes of the Hill matrix. Those
results could pave the way for a modal analysis of struc-
tures in periodic states and the use of Floquet forms for
stability analysis, structural design, or possible candi-
dates for modal reduction techniques.
Fig. 1 The 2Dstructure under study is a bi-articulated bar
submitted to a compressive periodic load at its end either
conservative (η= 0, i.e. horizontal force) or non conservative
(η= 1, i.e. following force).
2 Nonlinear and linearized equations of motion
of the Ziegler column in periodic elastic state
2.1 Nonlinear equation of motion
To illustrate the concept of Floquet forms, we consider
the archetypal example of the planar bi-articulated bar
illustrated in Fig. 1, also known as the Ziegler column
[32,40,42]. The rigid and inextensible bars of length
2lhave a mass m. The two bars are allowed to ro-
tate at points Oand Bthanks to elastic hinges char-
acterized by a rotational stiffness k. At rest, the bi-
articulated structure is lying in the horizontal direction
(O, x). The structure is possibly moving in the plane
(O, x, y) under the action of a T-periodic end compres-
sive force P(t) = P(t+T) = Acos(Ωt) with funda-
mental frequency Ω= 2π/T . This force is either pe-
riodically conservative (η= 0, i.e. horizontal force) or
non-conservative (η= 1, i.e. following force). The mo-
tion of this structure is completely parameterized by
the two angles θ1(t) and θ2(t) between the horizontal
axis (O, x) and the first and second bar, respectively. In
this paper, we are interested in the linear vibrations of
this two degrees-of-freedom structure about the trivial
straight state θ1(t) = θ2(t) = 0. Such a model is qual-
itatively representative of a tremendous amount of ap-
plications in structural mechanics as it exhibits most of
the classic bifurcations of dynamical systems, although
the perturbed stationary state is spatially trivial. Be-
cause the applied end load is possibly periodic, classic
modal analysis fails to analyze such a simple system
whose elasticity may vary periodically with time and
Floquet modes will be needed.
Balancing the quantity of acceleration of each bar
of the bi-articulated elastic system with the applied ex-
ternal moments (the expression of those quantities are
given in Appendix 1), the nonlinear equation of mo-
tion of the Ziegler column, reads, in the physical space
(θ1(t), θ2(t)):
0 = 16
3ml2¨
θ1+ 2ml2¨
θ2cos(θ1−θ2)
+ 2ml2˙
θ2
2sin(θ1−θ2)+2kθ1−kθ2
+ 2lA cos(Ωt) [cos(θ1) sin(ηθ2)−sin(θ1) cos(ηθ2)]
0 = 4
3ml2¨
θ2+ 2ml2¨
θ1cos(θ1−θ2)
−2ml2˙
θ2
1sin(θ1−θ2) + kθ2−kθ1
+ 2lA cos(Ωt) [cos(θ2) sin(ηθ2)−sin(θ2) cos(ηθ2)] (1)
By introducing the dimensionless time variable τ=ωnt
with a reference natural frequency ωn=pk/(ml2)
and multiplying the first and second line of Eq.(1) by
3/(16k) and 3/(4k), respectively, one can rewrite Eq.(1)
in the dimensionless form
0 = ¨
θ1+3
8¨
θ2cos(θ1−θ2)
+3
8˙
θ2
2sin(θ1−θ2) + 3
8θ1−3
16θ2
+λcos(βτ ) [cos(θ1) sin(ηθ2)−sin(θ1) cos(ηθ2)]
0 = ¨
θ2+3
2¨
θ1cos(θ1−θ2)
−3
2˙
θ2
1sin(θ1−θ2) + 3
4θ2−3
4θ1
+ 4λcos(βτ ) [cos(θ2) sin(ηθ2)−sin(θ2) cos(ηθ2)] (2)
where ˙
( ) denotes differentiation with respect to τ,β=
Ω/ωnis the ratio between the excitation and the refer-
ence natural frequency and λ=A/(8k/3L) represents
the dimensionless amplitude of the harmonic compres-
sive load. The dimensionless period of the applied com-
pressive force now reads ¯
T= 2π/β.
Eq.(2) is an implicit nonlinear system of Ordinary
Differential Equations (ODEs) with periodic coefficients
in the form fx(t),˙
x(t), t, =0, where the 4-dimensional
state vector reads x(t) = {θ1, θ2,˙
θ1,˙
θ2}T. By changing
the dimensionless loading parameters λand βin the

4 Barend Bentvelsen, Arnaud Lazarus
Dim-less time, τ
0 200 400 600
Angular displacement,
θ
(
τ
)
-14
-12
-10
-8
-6
-4
-2
0
θ1(τ)
θ2(τ)
Dim-less time, τ
0 200 400 600
Angular displacement,
θ
(
τ
)
-200
-100
0
100
200
θ1(τ)
θ2(τ)
Dim-less time, τ
0 200 400 600
Angular displacement, θ(τ)
-100
-50
0
50
100
θ1(τ)
θ2(τ)
Dim-less time, τ
0 200 400 600
Angular displacement, θ(τ)
-100
-50
0
50
100
θ1(τ)
θ2(τ)
(a)
(b)
(c)
(d)
Fig. 2 Nonlinear dynamical response for various loading pa-
rameters and initial conditions θ1(0) = θ2(0) = −1◦and
˙
θ1(0) = ˙
θ2(0) = 0. A little amount of viscous damping has
been introduced to help the simulations. (a) Bifurcation to
an equilibrium state for η= 0, β= 0 and λ= 0.072. (b) Flip
bifurcation to a dynamic state with a 2 ¯
T-period for η= 0,
β= 0.584 and λ= 0.036. (c) Hopf bifurcation to a periodic
stationary state for η= 1, β= 0 and λ= 0.4. (d) Secondary
Hopf or Neimark-Sacker bifurcation on a quasi-periodic state
for η= 1, β= 0.1 and λ= 0.6. Insets show the bifurcated
stationary states in the state space θ(τ),˙
θ(τ).
conservative case η= 0 or non-conservative one η= 1
and for a given set of initial conditions x(0), this simple
systems exhibits most of the qualitative vibrational be-
havior of stationary states of nonlinear dynamical sys-
tems as shown in Fig. 2.
2.2 Modal analysis of the trivial periodic elastic state
θ0
1(τ) = θ0
2(τ) = 0
To get a deeper physical understanding of the rich qual-
itative behavior exhibited by the Ziegler column in Fig.
2, we study the linear vibrations around the trivial spa-
tial state θ0
1(τ) = θ0
2(τ) = 0 that verifies the dimension-
less equation of motion given in Eq.(2).
Replacing θ1(τ) and θ2(τ) by their first order per-
turbed expressions θ1(τ) = θ0
1(τ) + εθ∗
1(τ) and θ2(τ) =
θ0
2(τ)+εθ∗
2(τ) in the nonlinear equation of motion Eq.(2)
and equating the first power of the small parameter ε,
one obtains the linearized equation of motion in the
vicinity of the considered state θ0
1(τ) = θ0
2(τ) = 0,
M¨
u(τ) + K(τ)u(τ) = 0(3)
where
u(τ) = θ∗
1(τ)
θ∗
2(τ),M=13
8
3
21and
K(τ) = 3
8−3
16
−3
4
3
4+λcos(βτ )−1η
0 4η−4
are the vector of physical degrees of freedom, the mass
and stiffness matrix, respectively.
For β= 0, Eq.(3) is the one of a linearized Ziegler
column under constant compressive loading and quan-
tities such as natural vibrational frequencies, critical
buckling and critical flutter loads can be classically de-
termined. By looking at the eigenvalues of the stiffness
matrix for η= 0, it is possible to assess the criti-
cal buckling load of the fundamental elastic state. The
condition det(K(λ)) = 0 gives a critical buckling load
λb= (9−3√5)/32. By analyzing the dynamical problem
given in Eq.(3) for no compressive loading, λ= 0, one
can compute the dimensionless natural frequencies of
the straight Ziegler column. The condition det(K(λ=
0) −ω2M) = 0 gives us the natural frequencies of
the unloaded system ω1= ((27 −3√74)/14)1/2and
ω2= ((27 + 3√74)/14)1/2. Finally, by finding the min-
imal λfor which the condition det(K(λ)−ω2M) = 0
leads to <(ω)>0 for β= 0 and η= 1, we obtain
λf= ((135/8) −2p1575/256)/25, the critical flutter
instability threshold above which the Ziegler column in
constant elastic state undergoes a Hopf bifurcation [42].
Working with the 4-dimensional vector of state vari-
ables y(τ) = {θ∗
1, θ∗
2,˙
θ∗
1,˙
θ∗
2}Tinstead of u(τ), the linear

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 5
system of ODEs given in Eq.(3) can be recast in the
generalized form
B˙
y(τ)−A(τ)y(τ) = 0(4)
with B=02M
M 02and A(τ) = −K(τ)02
02M
and where 0nis the n×nnull matrix. Inverting the
constant matrix B, the structural vibrational problem
of Eq.(3) can be transformed from the physical space
to the dynamical state space,
˙
y(τ) = J(τ)y(τ) (5)
with
J(τ) = B−1A(τ) = 1
2J0
c+J1
ccos(βτ )
=J0+J1eiβτ +J−1e−iβτ ,
J0=1
2J0
c=



0 0 1 0
0 0 0 1
−3
2
15
14 0 0
3−33
14 0 0




and J1=J−1=1
2J1
c=λ



0 0 0 0
0 0 0 0
8
7(4
7η−12
7) 0 0
−12
7(−20
7η+32
7)00




.
One sees from the linear ODE of Eq.(5) that the time-
varying Jacobian J(τ) of the nonlinear equation of mo-
tion Eq.(2) evaluated at θ0
1(τ) = θ0
2(τ) = 0 is ¯
T-periodic
with minimal period ¯
T= 2π/β in our case. Eq.(5) mod-
els the linear vibrations about the straight Ziegler col-
umn in a periodic elastic state. To study this linear
ODE with periodically time-varying coefficients, we can
use Floquet theory [11, 44].
According to Floquet theory, the N-dimensional lin-
ear system Eq.(5) has N= 4 linearly independent solu-
tions yn(τ), so that any solution y(τ) can be written:
y(τ) =
N
X
n=1
cnyn(τ) (6)
where cnare Nconstants that depend upon the initial
conditions y(0) and yn(τ) are called fundamental so-
lutions. According to the same theory, it is possible to
express the fundamental solutions yn(τ) in the so-called
Floquet form
yn(τ) = pn(τ)esnτ(7)
where pn(τ) = pn(τ+¯
T) is a N-dimensional com-
plex vector function of period ¯
Tand snis a complex
number called a Floquet exponent. Developing the un-
known periodic function pn(τ) in a complex Fourier
series pn(τ) = Pph
neihβτ , the FFs in Eq.(7) can be
rewritten in term of the fundamental frequency β=
2π/ ¯
T:
yn(τ) =
+∞
X
h=−∞
ph
ne(ihβ+sn)τ(8)
The FF yn(τ) is an almost periodic oscillation whose
spectrum depends on the periodicity ¯
Tof the spatially
trivial elastic state of the Ziegler column. In the par-
ticular case when β= 0, the eigenfunction p(τ) is con-
stant and the normal form in Eq.(7)-(8) is a harmonic
mode of vibration as it is defined in classic structural
dynamics. The FFs for structures in periodic state can
therefore be seen as a generalization of a vibrational
mode of a structure in equilibrium.
In this work, we would like to compute the FFs
of a Ziegler column for a given compressive end pe-
riodic load. One direct way to do it is to expand the
¯
T-periodically time-varying Jacobian of Eq.(5) in the
Fourier series
J(τ) =
+∞
X
k=−∞
Jkeikβτ .(9)
In the particular case of a harmonic end compressive
load with the Jacobian given in Eq.(5), Eq.(9) is simpli-
fied in closed form as the contributions of J(τ) are lim-
ited to the fundamental and first harmonic, i.e. Jk=0N
for |k|>1.
Replacing the perturbed solution y(τ) by its ex-
panded Floquet normal form Eq.(8) and the jacobian
J(τ) by its Fourier expansion Eq.(9) in the periodically
time-varying linear ODE Eq.(5), and balancing every
harmonic, we obtain an extended eigenvalue problem
in the spectral domain
HH−s1qH=0(10)
where, for example,
H2=





J0+i2β14J−1J−2J−3J−4
J1J0+iβ14J−1J−2J−3
J2J1J0J−1J−2
J3J2J1J0−iβ14J−1
J4J3J2J1J0−i2β14






.
is the complex Hill matrix truncated to the order H=
2, 1Nis the identity matrix of size N= 4, 1is the iden-
tity matrix of size N×(2H+ 1) and 0is a null vector
of same dimension. More details about the derivation
of the complex and real Hill matrix from the harmonic
balance method can be found in Appendix 2 and Ap-
pendix 3, respectively.
Computing the spectrum of the square Hill matrix
HHwith size N×(2H+ 1), we get N×(2H+ 1)

6 Barend Bentvelsen, Arnaud Lazarus
eigenvalues sland complex eigenvectors qH
lof size N×
(2H+ 1). For H= 2, qH=2
lis in the form qH=2
l=
p−2
lp−1
lp0
lp1
lp2
lTwhere ph
lis the harmonic contri-
bution of the lth Floquet form yl(τ) given in Eq.(8).
Replacing the computed eigenvalues sland eigenvectors
qH
lin the expression Eq.(8), we get N×(2H+ 1) nu-
merically approximated Floquet forms yl(τ). Since we
were theoretically waiting for NFloquet forms in the
state space, some information is redundant and some
interpretation is needed.
When H→ ∞, the computed eigenvalues and eigen-
vectors are not all independent. Actually, there are N
independent families of solutions that verify the rela-
tions sn+k=sn+ikβ and ph+k
n+k=ph
nfor n= 1 . . . N ,
−∞ < k < +∞and −∞ <h<+∞. By replacing
the Nfamilies of infinite eigenvalues and eigenvectors
in Eq.(8), only NFloquet forms yn(τ) are obtained,
the rest of the information being redundant. In prac-
tice, when truncating Hto a finite value, only some of
the N×(2H+ 1) computed eigenvalues sland eigen-
vectors qH
leventually converge, as His increased, to
the aforementioned relations; some of the spectrum is
spurious, whatever the chosen truncation order. The
presence of this spurious spectrum lies in the fact that
the infinite Hill operator H∞is self-adjoint when the
truncated one HHis not (a short explanation is given
at the end of Appendix 3). As a consequence, there is
a necessity to sort the computed spectrum to obtain
the Nconverged Floquet forms. Based on the partic-
ular relations between eigenvalues or eigenvectors that
exist for H→ ∞, two different sorting strategies can
be used:
1. Eigenvalue sorting: For the converged part of the
computed spectrum, we have the particular rela-
tions sn+k=sn+ikβ for n= 1 . . . N and −H≤
k≤+H. By taking the Neigenvalues whose imag-
inary parts are contained in the spectral primitive
cell −β/2≤ =(sl)< β/2, the latter will eventually
converge to Floquet exponents as His increased.
Replacing those Nconverged eigenvalues slwith
their associated eigenvectors qH
lin Eq.(8), we can
reconstruct the NFloquet forms yn(τ). The conver-
gence of the sorted eigenvalues has been rigorously
proved [27,45].
2. Eigenvector sorting: Since the converged eigenvec-
tors verify ph+k
n+k=ph
nfor −H≤k≤+Hand
−H≤h≤+H, the Nfundamental eigenvectors
qH
nassociated with the fundamental Floquet expo-
nents sn+k=sn+ikβ for k= 0, are the most
symmetric ones as compared to p0
n[22]. To com-
pute those Nfundamental eigenvectors in practice,
we compute the N×(2H+1) weighted means wl=
Phh|ph
l|/Ph|ph
l|. In this dual space, the converged
... ...
... ...
... ...
(c) (d)
(e)
(a) (b)
Fig. 3 Bifurcation and local stability analysis of a periodic
stationary state by studying the spectrum of the destabilizing
Floquet form in the Argand plane. (a) Static instability lead-
ing to a steady-state bifurcation. (b) Dynamical instability
responsible for the Hopf bifurcation. (c) Steady bifurcation
of a T-periodic state. (d) Flip or period doubling bifurcation
of a periodic state. (e) Secondary Hopf or Neimark-Sacker
bifurcation of a periodic state.
spectrum verifies wn+k=wn+k. The Neigenvec-
tors and associated eigenvalues that lead to the N
fundamental Floquet forms yn(τ) through Eq.(8),
are the ones inside the primitive cell −1/2≤wl<
1/2. Currently, there is no mathematical proof on
the convergence of this sorting method but we will
show that computing the Nfundamental FFs asso-
ciated with k= 0 is often more efficient than the
eigenvalue sorting method, especially for small fun-
damental frequency of the periodic state β.
Like a vibrational mode for a perturbed equilibrium,
the Ncomputed FFs yn(τ), with complex spectrum
Ph(sn+ihβ), allow to determine the local stability of
a perturbed periodic stationary state. Notably, if it ex-
ists a subscript gfor which <(sg)>0, the perturbed
stationary state increases exponentially in the direc-
tion of the gth mode yg(τ) and the stationary state
is said to be locally unstable. By analyzing how the
spectrum Ph(sg+ihβ) and its complex conjugate in
the state space Ph(¯sg+ihβ) cross the imaginary axis

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 7
in the Argand plane, as shown in Fig. 3, it is possible to
characterize the bifurcation that will undergo the per-
turbed stationary state. Note that for both algorithms,
it is important to exclude one of the limit of the ranges
−β/2≤ =(sl)< β/2 or −1/2≤wl<1/2 in order to
keep Neigenvalues or eigenvectors even if the conjugate
spectra lock in i(β/2±mβ) as shown in Fig. 3(d).
In the following, we compare and challenge the afore-
mentioned algorithms by computing the FFs and local
stability of the Ziegler column in the straight trivial
spatial state and periodic elastic state as shown in Fig.
1. In practice, this is done by computing and sorting the
spectrum of the truncated Hill matrix HHof Eq.(10)
derived from the Jacobian J(τ) given in Eq.(5). Section
3 concerns the case of a conservative compressive end
force with η= 0 where only the bifurcations of Figs.
3(a),(c) and (d) can be observed. Section 4 deals with a
nonconservative following end compressive load, i.e. for
η= 1, where Hopf bifurcations as illustrated in Figs.
3(b) and (e) can happen as well.
3 Periodically conservative case (η= 0)
In this section, we analyze the influence of the mod-
ulation loading parameters βand λon the transverse
vibrational modes and stability of the Ziegler column
under an end compressive horizontal load (η= 0). We
start with the classic case of a constant compressive
force, i.e. for β= 0 and study the influence of period-
icity when β6= 0 on the harmonic modes and stability.
We finish with some remarks on the particular limit
β→0.
3.1 Constant elastic state (β= 0)
When β= 0, the applied compressive dimensionless
load ¯
P(τ) = λcos(βτ ) is constant in time and the Hill
matrix of Eq.(10) truncated to the order H= 2 be-
comes
H2=





J004040404
04J0040404
0404J00404
040404J004
04040404J0






(11)
with
J0=



0 0 1 0
0 0 0 1
−3
2
15
14 0 0
3−33
14 0 0




+λ



0 0 0 0
0 0 0 0
8
7−12
70 0
−12
7
32
70 0




.
(a)
(b)
Fig. 4 Evolution of the spectrum of the two Floquet forms
(or classic harmonic modes) as a function of loading param-
eter λfor η= 0 and β= 0. (a) Evolution of the natural
frequencies of the bi-articulated bar in compression. (b) Evo-
lution of the growth rate of the two modes. The grey regions
in (a) and (b) indicate that the trivial state θ0
1(τ) = θ0
2(τ)=0
is locally unstable. Insets: Top and bottom respectively show,
for λ= 0, the angles θ∗
1(τ) and θ∗
2(τ) of the in-phase and out-
of-phase classic modes with natural frequency ω1and ω2, re-
spectively. The dashed-dotted lines represent the moduli of
the complex eigenfunctions |p0
n|and −|p0
n|that envelope the
motion.
From Eq.(11) we see that the lines and rows of the trun-
cated Hill matrix of Eq.(10) are independent, whatever
the truncation order H. If H= 0, no spectral sorting is
needed as the Hill matrix reduces to its central block J0
whose spectrum gives N= 4 eigenvalues and eigenvec-
tors that correspond to the Nclassic harmonic modes
yn(τ) = p0
nesnτwhen replaced in Eq.(8). If H > 0,
the Hill matrix leads to (2H+ 1) identical families of
Nindependent eigenvalues and eigenvectors of J0. Ap-
plying the eigenvector sorting method would give the N
correct harmonic modes but the eigenvalue sorting algo-
rithm will return an empty spectrum as the computed
eigenvalues will never be in the spectral primitive cell
−β/2≤ =(sl)< β/2 since β= 0. Thus, only the eigen-
vector sorting algorithm convey the notion that classic
normal modes are particular cases of fundamental FFs.

8 Barend Bentvelsen, Arnaud Lazarus
Fig. 4 is the classical dynamical vision of buckling.
It shows the evolution of the computed spectrum of the
N= 4 harmonic FFs as a function of dimensionless
compressive load λ/λbwhere λbis the already men-
tioned buckling load. At λ= 0, the bi-articulated beam
has two classic harmonic vibrational modes in the phys-
ical space: one with natural frequency ω1where the lin-
earized angles θ∗
1(τ) and θ∗
2(τ) vibrates in phase; one
with natural frequency ω2where θ∗
1(τ) and θ∗
2(τ) are
out-of-phase. The time evolutions of the linearized an-
gles θ∗
1(τ) and θ∗
2(τ) of those two modes are shown in
the insets of Fig. 4(b). The dash-dotted lines represent
the constant amplitude of the eigenfunction modulus
|p0
n|whose value is undefined, unless normalized. The
free vibratory response of the bi-articulated beam, solu-
tion of Eq.(5) with the Jacobian J(τ) given in Eq.(11),
can be written as a linear superposition of these two
normal modes. As the compressive load λis increased,
the frequencies of the two fundamental FFs decrease
until the smallest one eventually goes to zero at λ=λb.
The lock-in of the conjugate spectrum s1and ¯s1on
the real axis induces a positive growth rate <(s1)>0
and therefore a bifurcation of the trivial spatial straight
state of the Ziegler column along the in-phase static
mode. This local instability is the one shown in Fig.
3(a) and is responsible for the nonlinear response shown
in Fig. 2(a). Note that in this case, the two FFs are un-
coupled in the physical space as highlighted in Fig. 4
by the fact that the two color lines never combine.
3.2 Periodic elastic state (β6= 0)
When β6= 0, the straight bi-articulated bar is in a pe-
riodic elastic state as the Jacobian J(τ) of Eq.(5) is ¯
T-
periodic with ¯
T= 2π/β. In the particular conservative
case where η= 0, the Hill matrix of Eq.(10), truncated
to the order H= 2, becomes
H2=





J0+i2β14J1040404
J1J0+iβ14J10404
04J1J0J104
0404J1J0−iβ14J1
040404J1J0−i2β14






(12)
with
J0=



0 0 1 0
0 0 0 1
−3
2
15
14 0 0
3−33
14 0 0




and J1=λ



0 0 0 0
0 0 0 0
8
7−12
70 0
−12
7
32
70 0




.
Unlike in Eq.(11), the lines and rows of the truncated
HHof Eq.(12) are now fully coupled via the harmonic
(c)
(a)
(b)
θ1(τ)
θ(τ)
2
θ1(τ)
θ(τ)
2
θ1(τ)
θ(τ)
2
Fig. 5 Vibratory response for η= 0, β= 1.55ω1,λ= 0.75λb
and H= 3. (a) Time evolution of the angles θ∗
1(τ) and θ∗
2(τ)
of the first fundamental Floquet form y1(τ) = p1(τ)es1τover
the first two periods 2 ¯
Twhere ¯
T= 2π/β. The dash-dotted
lines show the moduli of the periodic eigenfunctions |pn(τ)|
and −|pn(τ)|that envelope the almost periodic motions. (b)
Same as (a) but for the second fundamental Floquet form.
(c) Time evolution of the angles θ∗
1(τ) and θ∗
2(τ) of the free
vibratory response y(τ) of Eq.(5) for the initial conditions
θ∗
1(0) = 1 and θ∗
2(0) = ˙
θ∗
1(0) = ˙
θ∗
2(0) = 0. The response
has been computed either with a classic direct iterative ODE
solver (dashed line) or by recombining the FFs (full line).
contribution of the Jacobian J1and the sorting of the
spectrum of Hill’s matrix given in Eq.(12) is a necessity
to compute the N= 4 FFs in the state space.
Figs. 5(a),(b) show the two fundamental FFs, yn(τ) =
pn(τ)esnτ, computed with the eigenvector sorting algo-
rithm, about the trivial spatial state θ0
1(τ) = θ0
2(τ)=0
for η= 0, β= 1.55ω1,λ= 0.75λband H= 3. Those
two typical physical FFs are the periodically modulated
generalization of the classic harmonic modes shown in
the insets of Fig. 4(b). Notably, the first FF of Fig. 5(a)
almost-periodically vibrates with a θ∗
1(τ) and θ∗
2(τ) in
phase and a fundamental frequency close to ω1when
the second FF in Fig. 5(b) vibrates out-of-phase with a
fundamental frequency close to ω2. Unlike classic har-

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 9
(a)
(b)
Fig. 6 Evolution of the spectrum of the N= 4 fundamental
FFs as a function of β/ω1for η= 0, λ= 0.75λband H=
25. (a) Evolution of the frequency spectrum location of the
FFs, =(sn) + Phihβ. (b) Evolution of the growth rate of the
FFs, <(sn). The grey regions in (a) and (b) indicate that the
straight bi-articulated bar is unstable.
monic modes, the modulus of the eigenfunction p(τ),
whose value is undefined unless normalized, is not con-
stant but ¯
T-periodic as illustrated by the dashed-dotted
lines in Figs. 5(a),(b). Like for β= 0, the moduli |p(τ)|
and −|p(τ)|envelope the almost-periodic motion. The
superposition property of FFs given in Eq.(6) is high-
lighted in Fig. 5(c) where we show the free vibratory
response of the straight bi-articulated bar computed ei-
ther through FFs or classic ODE time integrator. The
perturbation y(τ), solution of Eq.(5), can be decom-
posed in the normal basis of its FFs which notably
means it would have been possible to find a set of initial
conditions for which only one FF contributes to the so-
lution y(τ) (e.g. in Fig. 5 for θ∗
1(0) = 0.646, θ∗
2(0) = 1
and ˙
θ∗
1(0) = ˙
θ∗
2(0) = 0, only the first FF would con-
tribute to the linear response and Fig. 5(c) would have
been identical to Fig. 5(a)).
Fig. 6 shows the evolution of the N= 4 spectra
sn+Phihβ of the computed fundamental FFs as a
function of β/ω1for λ= 0.75λband H= 25 (to ensure
convergence). Like in the classic case illustrated in Fig.
4 for β= 0, the fundamental frequencies =(sn), and
moreover the frequency spectra =(sn) + Phihβ, vary
with the modulation parameters of the elastic state λ
and β. Also, since the system is conservative, the two
FFs are uncoupled in the physical space as highlighted
in Fig. 6 by the independence between the red and blue
modal colors. Furthermore, like the classic harmonic
modes of Fig. 4, it exists some modulation parameters
for which the straight bi-articulated bar becomes unsta-
ble, i.e. the spectrum of the in-phase FF =(sn)+Phihβ
lock in the state space with its conjugate leading to a
positive associated growth rate <(sn) (we do not see
here the lock-in of the out-of-phase FF in Fig. 6 because
the latter happens at higher modulation frequency than
the displayed one). As illustrated in Figs. 3(c),(d) and
displayed in Fig. 6(a), FFs have a poly-harmonic spec-
trum =(sn) + Phihβ and can lock either in i(β±mβ)
or in i(β/2±mβ). These two phenomena respectively
correspond to a ¯
Tand 2 ¯
T-periodic FF that diverges
exponentially and is responsible for the steady and flip
bifurcation shown for example in Fig. 2(b). The prin-
cipal region of instability, i.e. the one with the largest
(a)
(b)
Fig. 7 Stability chart of the conservative system in the (λ, β)
space for η= 0, β6= 0 and a variable converged truncation
order Hgiven in Fig. 14(a). (a) T(red) and 2T-instability (or-
ange) regions computed through the eigenvector or eigenvalue
sorting algorithms. (b) Same stability chart but by analyzing
all the spectrum of the Hill matrix, i.e. with no sorting of the
eigenvectors or eigenvalues. The black color show supplemen-
tary instability regions which are spurious numerical results
due to the truncation of the Hill matrix that would happen
whatever H.

10 Barend Bentvelsen, Arnaud Lazarus
βrange, corresponds to a lock-in of the fundamental
frequency of a FF when secondary regions are related
to lock-in of subharmonics.
Fig. 7(a) displays the stability chart of the straight
bi-articulated bar with an end compressive horizontal
force in the modulation parameters space (λ, β). For
each parameters, we chose a truncation order Hso that
the maximum difference δbetween the N= 4 Flo-
quet multipliers computed with the eigenvector sort-
ing method and the ones obtained from a converged
monodromy matrix algorithm [30] is δ= 1 ×10−6.
The resulting minimal truncation order map, H, as a
function of (λ, β) to ensure convergence is displayed in
Fig. 14(a) of Appendix 4. Colored regions, or Math-
(c)
(a)
(b)
θ1(τ)
θ(τ)
2
θ1(τ)
θ(τ)
2
Fig. 8 Fundamental FFs for η= 0, β= 100ω1,λ= 0.75λb
and H= 1. (a) Time evolution of the angles θ∗
1(τ) and
θ∗
2(τ) of FF1 over the first two natural periods 2 ¯
T0where
¯
T0= 2π/ω1. The dash-dotted lines show the moduli of the pe-
riodic eigenfunctions |pn(τ)|and −|pn(τ)|that envelope the
almost periodic motion. (b) Same as (a) but for FF2. (c) (left)
Raw spectrum of the truncated Hill matrix. The N= 4 fun-
damental Floquet exponents computed by eigenvector sorting
are shown with red circles. The region −β/2≤ =(sl)< β/2
for eigenvalue sorting is shown in grey. (right) FFT of the two
quasi-harmonic FFs.
ieu tongues, that theoretically originate at 2β/mω1for
λ→0 with ma positive integer, correspond to param-
eters for which at least one of the growth rate <(sn) of
the 4 FFs is positive. Red regions are T-instability do-
mains associated with lock of the frequency spectrum
of the in-phase FF in β±mβ when orange regions show
2T-instability zones related to a β/2±mβ lock-in of the
in-phase FF as shown in Fig. 6. In the particular conser-
vative case η= 0 and for the same truncation order map
of Hgiven in Fig. 14(a), the use of the eigenvalue sort-
ing algorithm lead to the exact same stability chart dis-
played in Fig. 7(a). Indeed, we observe the eigenvector
sorting method gives the N= 4 Floquet exponents sn
that are in the primitive spectral cell −β/2≤sn< β/2
when in ¯
Tor ¯
2T-periodic instability regions. Fig. 7(b)
shows the stability chart but by using no sorting algo-
rithms, i.e. by analyzing all the eigenvalues slof the
Hill matrix to see whether <(sl)>0. The differences
between both stability charts are highlighted in black.
Whatever the truncation order H, the truncated Hill
matrix will always give some spurious eigenvalues that
are inherent to the harmonic balance method. Those
spurious eigenvalues are more visible for high λand
close to the instability regions.
3.3 Asymptotic cases (β→+∞) and (β→0)
In the asymptotic cases where β→+∞or β→0,
i.e. in the situations where the modulation time scale
is far from the natural time scale of the system given
here by ω1and ω2, several qualitative and quantitative
comments can be made about the FFs.
Fig. 8 illustrates the β→+∞scenario by showing
the two fundamental FFs of the straight bi-articulated
bar in periodic elastic state for η= 0, λ= 0.75λband
β= 100ω1. The time evolution of the two angles θ∗
1(τ)
and θ∗
2(τ) of the two FFs are shown in Figs. 8(a) and
(b). For high modulation frequencies, the harmonic con-
tribution of the compressive force is averaged out and
the bi-articulated elastic bar behaves like a classic ef-
fective oscillator. The two FFs asymptotically tend to
the two classic in-phase and out-of-phase modes of the
system with natural frequencies ω1and ω2as shown
in the insets of Fig. 4(b). Notably, the ¯
T-periodic en-
velopes of the almost-periodic FFs, |p(τ)|and −|p(τ)|,
appear constant over the natural period ¯
T0since the
small oscillations over ¯
Tare negligible. Fig. 8(c) dis-
plays the raw spectrum of the Hill matrix (left) as well
as the reconstructed spectrum of the FFs (right). In
the β→+∞case, the FFs tend to classic harmonic
modes with a spectrum composed of a single oscilla-
tion frequency. This spectrum is easily recovered with

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 11
(a)
(b)
(c)
Fig. 9 Fundamental FFs for η= 0, β= 0.05ω1,λ= 0.75λb
and H= 15. (a) Time evolution of the angle θ∗
1(τ) of FF1 over
the first two periods 2 ¯
Twhere ¯
T= 2π/β with their envelopes
|pn(τ)|and −|pn(τ)|. (b) Same as (a) but for FF2. (c) (left)
Raw spectrum of the truncated Hill matrix. The N= 4 fun-
damental Floquet exponents computed by eigenvector sorting
are shown with red circles. The region −β/2≤ =(sl)< β/2
for eigenvalue sorting is shown in grey. (right) FFT of the two
FFs.
both eigenvector and eigenvalue sorting algorithms. Ac-
tually, both methods lead to the same eigenvalue out-
put as the N= 4 fundamental Floquet exponents ob-
tained with the eigenvector sorting algorithm and rep-
resented by red circle in Fig. 8(c), are the one inside
the −β/2≤ =(sl)< β/2.
Fig. 9 illustrates the β→0 situation by showing
the two in-phase and out-of-phase FFs of Fig. 8, but
for a very small modulation frequency β= 0.05ω1. The
time evolution of the angle θ∗
1(τ) of the two FFs are
shown in Figs. 9(a) and (b) (θ∗
2(τ) is not displayed
for a sake of clarity). For slow modulation frequency,
the end compressive load quasi-statically takes all the
amplitudes between the minimum λ=−0.75λband
maximum +0.75λb. As a consequence, the elasticity of
our system is quasi-statically modulated and the trans-
verse natural frequencies of the straight bi-articulated
bar appear to almost continuously vary (with a small
step β) around ω1and ω2. We can see on Figs. 9(a)
and (b) that the motion is modulated in frequency and
amplitude. The FFTs of the two FFs are displayed on
the right side of Fig. 9(c) and show two broad spec-
trum in the vicinity of ω1and ω2whose almost con-
tinuous width depend on the modulation amplitude λ.
For small β, many frequencies are contained in the FFs
and a high truncation order His needed for the spec-
trum of Hill’s matrix to converge. The raw spectrum
of Hill matrix is shown on the left of Fig. 9(c) for
H= 15. It is separated in four distinct packets, cen-
tered around ω1,ω2,−ω1and −ω2, that correspond to
N= 4 families of eigenvalues sn+k=sn+ikβ where
−H≤k≤+H(apart from the edges of the packets
where the computed eigenvalues are spurious due to
truncation errors). The eigenvector sorting algorithm
finds the N= 4 fundamental Floquet exponents, high-
lighted by red circles, that correspond to the most con-
verged eigenvalues (center of the packets). Because of
the small modulation frequency β, the eigenvalue sort-
ing algorithm is far from being converged for H= 15 as
only two, yet not converged, eigenvalues relative to the
first FFs are located in the −β/2≤ =(sl)< β/2 region.
For the two eigenvalues associated with the spectrum
packet around ω2and −ω2to enter that region, more
than H= 100 would be needed. In the limit situation
β→0, the eigenvalue sorting algorithm is not adapted
to efficiently compute the FFs and their spectrum, on
the contrary to the eigenvector sorting method that
computes the most converged fundamental Floquet ex-
ponents and FFs whatever β. In the non-conservative
case studied in next section, the same kind of conver-
gence issue will happen in the determination of instabil-
ity regions associated with Neimark-Sacker bifurcation
when using the eigenvalue sorting algorithm.
4 Non-Conservative case (η= 1)
In this section, we analyze the influence of the mod-
ulation loading parameters βand λon the transverse
vibrational modes of the Ziegler column under a non-
conservative end positional loading (η= 1). We start
with the classic case of a constant compressive force,
i.e. for β= 0, and study the influence of periodicity
when β6= 0 on the modes and stability.
4.1 Constant elastic state (β= 0)
In the particular case β= 0, the applied compressive
dimensionless load ¯
P(τ) = λcos(βτ ) and therefore the

12 Barend Bentvelsen, Arnaud Lazarus
periodically time-varying Jacobian J(τ) of Eq.(5) are
independent of time. The Hill matrix of Eq.(10) trun-
cated to the order Hreduces to a block diagonal matrix
with only the fundamental part of the Jacobian J0as
shown in Eq.(11). In this section, since η= 1, the Ja-
cobian J0reads,
J0=



0 0 1 0
0 0 0 1
−3
2
15
14 0 0
3−33
14 0 0




+λ



0 0 0 0
0 0 0 0
8
7−5
70 0
−12
7
12
70 0




.(13)
We can infer from the shape of the Jacobian in Eq.(13)
that the problem is now non-conservative as the bottom
left 2 ×2 block matrix of J0depending on λis non
symmetric. Whether we compute the whole spectrum
of J0, i.e. the truncated Hill matrix for H= 0, or we use
the eigenvector sorting algorithm for H > 0 (like in the
conservative case, the eigenvalue sorting algorithm can
not be used for β= 0), one numerically approximates
N= 4 FFs for a given modulation parameter λ.
(a)
(b)
Fig. 10 Evolution of the spectrum of the two Floquet forms
(or classic harmonic modes) as a function of loading param-
eter λfor η= 1 and β= 0. (a) Evolution of the natural
frequencies of the bi-articulated bar in compression. (b) Evo-
lution of the growth rate of the two modes. The grey regions
in (a) and (b) indicate that the trivial state θ0
1(τ) = θ0
2(τ)=0
is locally unstable.
Fig. 10 shows the plot of the evolution of the com-
puted spectrum of the N= 4 fundamental FFs as a
function of dimensionless compressive load λ/λfwhere
λfis the already mentioned flutter load. At λ= 0, the
unloaded straight bi-articulated bar has two classic in-
phase and out-of-phase vibrational modes with natural
frequency ω1and ω2, respectively. Unlike the conserva-
tive case, the two harmonic FFs are coupled for η= 1
and influence each other so that their spectra eventually
lock-in. As the compressive load λgrows, the frequency
of the in-phase mode increases when the one of the out-
of-phase mode decreases. At λ=λf, the spectra of the
two physical FFs lock in a finite =(sn)>0, inducing a
positive growth rate <(sn) of the resulting locked mode
(c)
(a)
(b)
θ1(τ)
θ(τ)
2
θ1(τ)
θ(τ)
2
Fig. 11 Vibratory response for η= 1, β= 1.55ω1,λ=
0.75λfand H= 5. (a) Time evolution of the angles θ∗
1(τ) and
θ∗
2(τ) of the first fundamental FF y1(τ) = p1(τ)es1τover the
first two periods 2 ¯
Twhere ¯
T= 2π/β. The dash-dotted lines
show the moduli of the periodic eigenfunctions |pn(τ)|and
−|pn(τ)|that envelope the almost periodic motions. (b) Same
as (a) but for the second fundamental FF (only θ∗
2(τ) is shown
for a sake of clarity). (c) Time evolution of the angles θ∗
1(τ)
and θ∗
2(τ) of the free vibratory response y(τ) of Eq.(5) for the
initial conditions θ∗
1(0) = 1 and θ∗
2(0) = ˙
θ∗
1(0) = ˙
θ∗
2(0) = 0.
The response has been computed either with a classic direct
iterative ODE solver or by recombining the FFs.

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 13
(for λ>λf, only one in-phase mode subsists in the
physical space which separates in an unstable and sta-
ble one in the state space). This destabilization mech-
anism corresponds to the flutter instability case shown
in Fig. 3(b) that is responsible for the Hopf bifurcation
illustrated in Fig. 2(c). Increasing λfurther, the spectra
of the remaining FF lock again, this time in the state
space and on =(sn) = 0. This second situation is qual-
itatively similar to the instability by divergence shown
in Fig. 4 and Fig. 3(a) for the conservative case. The
flutter instability only happens on a range of loading
parameter λand evolves toward a divergence instabil-
ity for sufficiently high compressive loads.
4.2 Periodic elastic state (β6= 0)
When β6= 0, the straight Ziegler column is in a pe-
riodic elastic state and its perturbed motion is gov-
erned by Eq.(5) with a ¯
T-periodic Jacobian J(τ) where
¯
T= 2π/β. The truncated matrix has therefore the
shape of a tridiagonal matrix by block as shown in
Eq.(12) but since η= 1 in this section, the fundamental
and first harmonic contributions of J(τ) now reads
J0=



0 0 1 0
0 0 0 1
−3
2
15
14 0 0
3−33
14 0 0




and J1=λ



0 0 0 0
0 0 0 0
8
7−5
70 0
−12
7
12
70 0




.
(14)
We use the eigenvector sorting algorithm on the spec-
trum of the truncated Hill matrix with J0and J1given
in Eq.(14) to compute the N= 4 FFs of the straight
Ziegler column under a given end following compressive
load ¯
P(τ) = λcos(βτ ).
Figs. 11(a),(b) display the perturbed angles θ∗
1(τ)
and θ∗
2(τ) of two fundamental FFs, in the form yn(τ) =
pn(τ)esnτ, about the trivial spatial state θ0
1(τ) = θ0
2(τ) =
0 for η= 1, β= 1.55ω1,λ= 0.75λfand H= 5. Like
in the conservative case η= 0 illustrated in Fig. 5, the
FFs are the almost-periodic generalization of the clas-
sic harmonic modes shown in the insets of Fig. 10(b).
For β6= 0, the eigenfunction pn(τ) are periodic and the
moduli |pn(τ)|and −|pn(τ)|envelope the oscillation. In
the non conservative case with η= 1, the following end
compressive force modulates both FFs as illustrated in
Fig. 11(a),(b) where the amplitude modulation of the
periodic |p1(τ)|and |p2(τ)|are of similar intensity. The
superposition property of the normal forms still holds
in the non conservative case and is highlighted in Fig.
11(c). For a given set of initial condition, the pertur-
bation y(τ) solution of Eq.(5) computed with a classic
(a)
(b)
Fig. 12 Evolution of the spectrum of the N= 4 fundamen-
tal FFs as a function of β/ω1for η= 1 and λ= 0.75λf.
(a) Evolution of the frequency spectrum location of the FFs,
=(sn) + Phihβ. (b) Evolution of the growth rate of the
FFs, <(sn). The grey regions in (a) and (b) indicate that
the straight Ziegler column is locally unstable.
direct time ODE solver is in perfect agreement with the
solution recombined from FFs with Eq.(6).
Fig. 12 shows the evolution of the N= 4 spectra
sn+Phihβ of the computed FFs as a function of β/ω1
for λ= 0.75λfand H= 25. Similarly to Figs. 4, 6 and
10, the frequency spectra =(sn) + Phihβ varies with
the modulation parameters λand βand can eventually
lock-in. Because the FFs have a poly-harmonic spec-
trum and η= 1, it exists several ranges of frequency
modulation βwhere the spectra of two different FFs
are locked in frequencies that are not positive integer
multiple of β/2 (those are the regions where the red
and blue colors in Fig. 12 mix and where every growth
rates <(sn) are different from zero in Fig. 12(b)). Those
situations are qualitatively illustrated in Fig. 3(e) and
are responsible for the Neimark-Sacker bifurcation il-
lustrated in Fig. 2(d). The wider region of instability
responsible for those secondary-hopf bifurcations corre-
sponds to a lock-in of the fundamental harmonic of the
two different FFs when the other ones are due to lock-
in of sub-harmonics. It is interesting to note that in the
main instability region (far right of Fig. 12), the spec-
tra of the two FFs ultimately lock in i(β/2±mβ) when

14 Barend Bentvelsen, Arnaud Lazarus
(a)
(c)
(b)
Fig. 13 Stability chart of the non conservative system in the
(λ, β) space for η= 1, β6= 0 and a variable converged trun-
cation order Hgiven in Fig. 14(b). (a) Neimark-Sacker insta-
bility (green) and 2T-instability (orange) regions computed
through the eigenvector sorting algorithm. (b) Same stabil-
ity chart but by analyzing the raw spectrum of the Hill ma-
trix. The black color shows supplementary instability regions,
a spurious numerical artifact inherent to the truncated Hill
matrix. (c) Same stability chart but by using the eigenvalue
sorting algorithm. The black color indicates supplementary
instability regions where the eigenvalue sorting algorithm is
not yet converged for the truncation order of Fig. 14(b).
increasing β, i.e. the system undergoes a 2 ¯
T-instability.
This main instability was already observed in Fig. 6.
Fig. 13(a) displays the stability chart of the straight
Ziegler column with an end compressive positional fol-
lowing force in the modulation parameters space (λ, β).
Like in Fig. 7, the displayed stability chart obtained
with the eigenvector sorting algorithm has been vali-
dated trough the computation of the monodromy ma-
trix and its Floquet multipliers in the time domain [30],
leading to a map of truncation order Hin the (λ, β)
space shown in Fig. 14(b) of Appendix 4. The system
exhibits instability tongues that correspond to param-
eters for which at least one of the growth rate <(sn) of
the N= 4 fundamental FFs is positive. Here, green re-
gions are associated with secondary Hopf bifurcations
and correspond to modulation ranges where the two
physical FFs are locked in frequencies that are not in-
teger multiple of β/2 when the orange region shows the
2T-instability situation related to a β/2±mβ lock-in
of the fundamental of the resulting in-phase FF in the
state space as illustrated in Fig. 12. Like in the conser-
vative case in Fig. 7, the analysis of the full spectrum
of Hill’s matrix (no sorting) leads to a wrong stabil-
ity diagram displayed in Fig. 13(b). Moreover, unlike
for ¯
Tand ¯
2T-instability regions, the eigenvalue sort-
ing algorithm, represented by the stability chart of Fig.
13(c), is not giving the correct Neimark-Sacker insta-
bility regions for the optimal truncation order of Fig.
14(b). The reason is that for some modulation param-
eters and at this optimal truncation order, the sorted
Floquet exponents snthat are in the primitive spectral
cell −β/2≤sn< β/2 are not yet converged because
they are not the fundamental ones responsible for the
Neimark-Sacker lock-in instability. On the contrary, the
FFs computed with the eigenvector sorting algorithm
are the fundamental ones, a property that is crucial
when dealing with secondary Hopf bifurcations or sta-
ble FFs for small βas explained in Section 3.3.
5 Conclusions
The presented study has shown how to practically im-
plement and compute the vibrational modes, or Flo-
quet Forms (FFs), of a structure in periodic elastic
state through the archetypal example of a Ziegler col-
umn subjected to an end harmonic compressive force.
The latter has been used to compare the classic spec-
tral methods that exist to compute the FFs through
Hill’s matrix. Our results highlighted the absolute ne-
cessity to sort the spectrum of Hill’s matrix and the
benefit of the eigenvector sorting algorithm that selects
the fundamental FFs, which are the most converged
ones. We also showed the similarities and differences
between the classic harmonic modes about equilibria
and the almost-periodic FFs about periodic stationary
states. Mathematically and physically, FFs can be seen
as a poly-harmonic generalization of harmonic modes.
At high frequency modulation, FFs tend to effective

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 15
classic vibrational modes as the modulated state is av-
eraged to its fundamental part. On the contrary, at low
frequency modulation, a spectral leakage occurs and
FFs contain an almost continuous frequency spectrum,
with a finite bandwidth. In the particular case of zero
frequency modulation, FFs are equivalent to classic har-
monic modes where the natural frequencies is specified
by the phase of the periodic elastic state.
The spectral calculation of FFs could be used in
many engineering systems, from rotating machineries
with geometric imperfections to the transverse vibra-
tions of slender structures with compressive or tensile
periodic stresses. For practical purposes in computa-
tional mechanics though, further numerical developments
would be needed for a computation of the converged
spectrum of large Hill matrices in reasonable CPU time.
Notably, the current eigenvector sorting algorithm ap-
plies on the full raw spectrum, an impractical situation
for high dimensional systems. It would therefore be cru-
cial to find a way to include this sorting algorithm at the
level of the eigenvectors calculation in order to reduce
the spectrum computation to the first few physical fun-
damental FFs. Another possibility would be to take ad-
vantage of the nature of the eigenvalue problem and use
domain decomposition methods as well as paralleliza-
tion techniques to improve computational times. Along
the development of the modal analysis of structures in
periodic states, FFs should be tested as a reduction ba-
sis for linear and nonlinear problems in structural vibra-
tion or as preferential directors in predictor-corrector
algorithms. Another important application lies in the
insight given by FFs on dynamic instabilities as shown
numerically in this manuscript. Notably, it could be in-
teresting to use the frequency lock-in of FFs as an ex-
perimental instability precursor for structures in period
states. An experimental Ziegler column [3] would be a
relevant test bench to assess the feasibility of instability
detections through the spectral signature of FFs.
Appendix 1: Equation of motion of the Ziegler
column
With the help of Newton’s second law applied on the
two rotating rigid bars parameterized by θ1(t) and θ2(t),
it is possible to establish the nonlinear equation of mo-
tion of the Ziegler column described in Fig. 1. By equat-
ing the quantity of acceleration Abar1and Abar2on one
side, to the sum of the external moments Mbar1and
Mbar2, on the bar 1 and 2, respectively, on the other
side, we obtain a system of two nonlinear Ordinary Dif-
ferential Equations:
Abar1=d
dt(∂T
∂˙
θ1
)−∂T
∂θ1
=Mk
1+Mc
1+MP
1
Abar2=d
dt(∂T
∂˙
θ2
)−∂T
∂θ2
=Mk
2+Mc
2+MP
2(15)
In Eq.(15), T(θ1, θ2,˙
θ1,˙
θ2) is the kinetic energy of the
system of two rotating rigid bars reading
T=8
3ml2˙
θ2
1+2
3ml2˙
θ2
2+ 2ml2˙
θ1˙
θ2cos (θ1−θ2).(16)
The moments Mk
1and Mk
1represent the restoring elas-
tic moments due to the rotational springs and read, in
bar 1 and 2, respectively:
Mk
1=kθ1+k(θ1−θ2) and Mk
2=k(θ2−θ1).(17)
The moments MP
1and MP
2are due to the end external
harmonic force F(t) and read, in bar 1 and 2, respec-
tively:
MP
1= 2lA cos(Ωt) [−cos(θ1) sin(ηθ2) + sin(θ1) cos(ηθ2)]
MP
2= 2lA cos(Ωt) [−cos(θ2) sin(ηθ2) + sin(θ2) cos(ηθ2)]
(18)
If η= 0 (in the case of a force remaining horizontal
upon deformation of the structure), the moments MP
1
and MP
2can be derived from the gradient of a potential
energy, and the system is said to be conservative, or pe-
riodically conservative as the value of the moments are
periodically varying with time. If η= 1 (in the case of
moments depending on the position of the structure in
space), the previous property is not true and the sys-
tem is non-conservative. Replacing Eqs.(16)-(18) into
Eq.(15), one obtains the nonlinear system of equation
of motions of the two-degrees-of-freedom as expressed
in Eq.(1).
Appendix 2: Derivation of the complex Hill ma-
trix
In this section, we explain how to derive the Hill ma-
trix of the time-periodic linearized equation of motion
Eq.(5), ˙
y(τ) = J(τ)y(τ), in the complex domain. The
first step consists in rewriting the ¯
T-periodic Jacobian
of size N= 4, J(τ) = J(τ+2π/β), in a complex Fourier
series. In our particular case of a Ziegler column sub-
mitted to a harmonic end compressive load, J(τ) can
be expanded in the closed-form J(τ) = J0+J−1e−iβτ +
J1eiβτ where J0,J1and J−1are given in Eq.(5).
According to Floquet theory, Floquet forms (FFs)
are the N= 4 particular fundamental solutions y(τ)
in the form y(τ) = p(τ)esτ where p(τ) = p(τ+¯
T)

16 Barend Bentvelsen, Arnaud Lazarus
and sis a complex number. Since the eigenfunction
p(τ) is ¯
T-periodic, the FF can also be expanded in the
complex Fourier series y(τ) = P+∞
h=−∞ phe(ihβ+s)τand
the associated velocity ˙
y(τ) reads ˙
y(τ) = P∞
h=−∞(s+
ihβ)phe(s+ihβ )τ.
Replacing the expanded expressions of y(τ), ˙
y(τ)
and J(τ) in Eq.(5), one can recast the equation of mo-
tion from the time domain to the frequency domain
such that
0=J(τ)y(τ)−˙
y(τ)
0=
∞
X
h=−∞ hJ0phe(s+ihβ)τ+J−1phe(s+i(h−1)β)τ
+J1phe(s+i(h+1)β)τ−(s+ihβ)phe(s+ihβ )τi(19)
From Eq.(19), by taking into account that the esτ can
be factorized and removed, it is now straightforward
to apply the harmonic balance method. The latter is
based on the property that in order for the sum of all
harmonics eihβτ for h=−∞. . . +∞to be balanced to
zero, the sum of the contributions in front of each har-
monics eihβτ has to be balanced to zero. For practical
purposes, this remark leads to several vectorial alge-
braic equations of dimension N= 4. For the first five
harmonics −2, −1, 0, 1 and 2, those equations read:
e−2iβτ :J−1p−1+(J0−(s−2iβ))p−2+J1p−3= 0
e−iβτ :J−1p0+(J0−(s−iβ))p−1+J1p−2= 0
e0iβτ :J−1p1+(J0−s)p0+J1p−1= 0
eiβτ :J−1p2+(J0−(s+iβ))p1+J1p0= 0
e2iβτ :J−1p3+(J0−(s+ 2iβ))p2+J1p1= 0
(20)
Eq.(20) is an eigenvalue problem, truncated to the
order H= 2, that can be transform in the matrix
form of Eq.(10), H2−s1q2=0where H2is the
Hill matrix truncated to the order H= 2, given in
Eq.(12), sis the complex eigenvalue of H2and q2=
p−2p−1p0p1p2Tis the associated eigenvector. Note
that the construction of the complex Hill matrix for
higher truncated order Hfollows the exact same rea-
soning shown for H= 2. The problem we encounter
when the eigenvalue problem Eq.(20) is truncated, e.g.
to the order H= 2 in the form of the Hill matrix H2,
is that we have to drop the contributions p−3and p3.
The consequence is that Eq.(20) are only approximated
because some equations are not correct anymore. This
is the reason why a sorting algorithm is necessary to
compute the Floquet forms of the Hill matrix.
In the general case of a more complex periodic Ja-
cobian than the harmonic one we dealt with in this
article, the latter cannot be expanded in a closed-form
Fourier series but rather in the general form of Eq.(9),
J(τ) = P+∞
k=−∞ Jkeikβτ . In this scenario, the aforemen-
tioned spectral expansions of y(τ) and ˙
y(τ) remain the
same but the product J(τ)y(τ) changes, so that Eq.(19)
now becomes
0=
+∞
X
h=−∞
∞
X
k=−∞
(Jkph−k−(s+ihβ)ph)eihβ τ .(21)
Eq.(21) is an infinite value problem that can be recast
in the matrix form (H∞−s1)q∞=0where H∞is
the general complex infinite Hill matrix whose form is
given in Eq.(10) for a truncation order H= 2.
Appendix 3: Derivation of the real Hill matrix
It exists some situations where the presence of the pure
imaginary number “i” in the complex Hill matrix is
a problem. An alternative is to deal with a real Hill
matrix, although the formalism is more complicated.
The difference with Appendix 2 is simply to expand
the time-periodic equation of motion Eq.(5), ˙
y(τ) =
J(τ)y(τ), in real Fourier series.
In our particular case of a harmonic modulation, the
¯
T-periodic Jacobian simply reads
J(τ) = 1
2J0
c+J1
ccos(βτ ) (22)
where the expressions of J0
cand J1
care given in Eq.(5).
The N= 4 Floquet forms y(τ) given in the complex
domain in Eq.(8) can be expressed by the real expansion
y(τ) = 1
2a0+
∞
X
h=1 ahcos(hβτ ) + bhsin(hβτ)esτ
(23)
and the associated time derivative reads
˙
y(τ) = 1
2a0s+
∞
X
h=1 sah+hβbhcos(hβτ)
+sbh−hβahsin(hβτ)esτ (24)
Replacing the expanded expressions of y(τ), ˙
y(τ) and
J(τ) of Eq.(22)-(24) in Eq.(5), we can recast the equa-
tion of motion from the time domain to the spectral

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column 17
domain
0=J(τ)y(τ)−˙
y(τ)
0=
∞
X
h=1 1
2J0
cah−sah−hβbhcos(hβτ)
+
∞
X
h=1 1
2J0
cbh−sbh+hβahsin(hβτ)
1
2J0
c−sa0
2+J1
c a0
2cos(βτ )
+
∞
X
h=1 ah
2cos (1 + h)βτ + cos (1 −h)β τ 
+bh
2sin (1 + h)βτ −sin (1 −h)βτ!(25)
Like for Eq.(19) and (20), the harmonic balance
method allows us to recast the problem in 2H+ 1
algebraic equations of dimension Nby independently
equating to zero all the constant terms in cos(0βτ ), the
first harmonics cos(βτ ) and sin(βτ ), the second har-
monics cos(2βτ ) and sin(2βτ ) and so on. By putting
those equations in a matrix form, we obtain an eigen-
value problem HH−s1qH=0where HHis the
N×(2H+ 1)-dimensional square real Hill matrix trun-
cated at order H,sand qHare the N×(2H+ 1)
complex eigenvalues and eigenvectors of HH, respec-
tively. If we order the eigenvector in the form qH=
1
2a0a1. . . aHb1. . . bHT, the real Hill matrix reads,
e.g. for H3and q3=1
2a0a1a2a3b1b2b3T:
H3=










1
2J0
c1
2J1
c0404040404
J1
c1
2J0
c1
2J1
c04−β140404
041
2J1
c1
2J0
c1
2J1
c04−2β1404
04041
2J1
c1
2J0
c0404−3β14
04β1404041
2J0
c1
2J1
c04
04042β14041
2J1
c1
2J0
c1
2J1
c
0404043β14041
2J1
c1
2J0
c









(26)
When replacing the eigenvalues sand associated eigen-
vectors qof the real Hill matrix in the real Fourier ex-
pansion of the fundamental solutions given in Eq.(23),
one gets a numerical approximation of the FFs of the
systems. But like in the complex domain, because of
truncation errors, the N= 4 families of (2H+ 1) com-
puted solutions y(τ) do not all converge to the FFs and
a sorting algorithm is needed. The computed eigenval-
ues are the same whether it comes from the real or
complex Hill matrix. Consequently, the eigenvalue al-
gorithm still holds with the real Hill matrix: by keep-
ing the Neigenvalues inside the spectral primitive cell
−β/2≤ =(sl)< β/2, the latter (with their associated
eigenvectors) will eventually lead to Nconverged FFs
as His increased. The eigenvector sorting algorithm
that allows to compute the Nfundamental FFs also
holds with the real Hill matrix. It still consists in com-
puting the weighted means wl=Ph|ph
l|/Ph|ph
l|for
−H≤h≤Hand keeping the spectrum associated with
the Nweighted means that belong to the primitive cell
−1/2≤wl<1/2. The only supplementary step to
compute wlwith the real Hill matrix is that we need
the complex/real transformation formulas p0=a0/2
for h= 0 and p−h= (ah+ibh)/2 or ph= (ah−ibh)/2
for h > 0.
In the general case of a periodic modulation, the
¯
T-periodic Jacobian would read
J(τ) = 1
2J0
c+
∞
X
h=1 Jh
ccos(hβτ ) + Jh
ssin(hβτ ).(27)
In this situation, the linear equation of motion ˙
y(τ)−
J(τ)y(τ) = 0, expanded in the real spectral domain,
becomes much more complicated. However, the har-
monic balance method can still be applied, leading to
an eigenvalue problem HH−s1qH=0. The expres-
sion of the real Hill matrix HHis yet more compli-
cated than the one of Eq.(26) in the harmonic mod-
ulation case. If decomposed in a sum of block matri-
ces and by ordering the eigenvector in the form qH=
1
2a0a1. . . aHb1. . . bHT,HHreads:
HH=

1
2J0
c1
2Jc1
2Js
JT
cKc+TcKs−Ts
JT
sKs+TsTc−Kc
(28)
where
Jc=J1
cJ2
c. . . JH
cand Js=J1
sJ2
s. . . JH
s
are (N×HN)-dimensional block vectors (JT
cand JT
s
are the transpose of the above block vectors, not of the
full matrices Jcand Jsmeaning one has to be careful
to not transpose the matrices Jh
cand Jh
sin the process
but simply ordering them in a column block vector),
Kc,s =1
2







J2
c,s J3
c,s J4
c,s . . . JH+1
c,s
J3
c,s J4
c,s J5
c,s . . . JH+2
c,s
J4
c,s J5
c,s J6
c,s . . . JH+3
c,s
.
.
..
.
..
.
.....
.
.
JH+1
c,s JH+2
c,s JH+3
c,s . . . J2H
c,s







are (NH ×NH)-dimensional block matrices with har-
monic contributions of the Jacobian either on cosine or

18 Barend Bentvelsen, Arnaud Lazarus
sine, and where
Tc=1
2







J0
cJ1
cJ2
c. . . JH−1
c
J1
cJ0
cJ1
c. . . JH−2
c
J2
cJ1
cJ0
c. . . JH−3
c
.
.
..
.
..
.
.....
.
.
JH−1
cJH−2
cJH−3
c. . . J0
c







and
Ts=1
2







2β14−J1
s−J2
s. . . −JH−1
s
J1
s4β14−J1
s. . . −JH−2
s
J2
sJ1
s6β14. . . −JH−3
s
.
.
..
.
..
.
.....
.
.
JH−1
sJH−2
sJH−3
s. . . 2Hβ14







are (NH ×NH)-dimensional block matrices. Although
seemingly complicated if compared to the general com-
plex Hill matrix given in Eq.(10) that is the sum of a
complex block diagonal matrix and a real Toeplitz block
matrix, the general real Hill matrix is relatively easy to
numerically implement. Indeed, it is composed of Kc
and Kswhich are Hankel block matrices, Tcthat is a
Toeplitz matrix and Tsthat is the sum of a real block
diagonal matrix and a Toeplitz matrix. Applying the
(a)
(b)
Fig. 14 Optimal spectral truncation order map Hconv in
the β, λ space to ensure convergence. (a) Conservative case
η= 0. (b) Non conservative scenario η= 1.
eigenvector sorting algorithm on the real Hill matrix
HHof Eq.(28) allows to compute the Nfundamental
FFs of a system in a general periodic state.
Appendix 4: Spectral convergence of the stabil-
ity charts
The stability charts of Figs. 7 and 13 have been com-
puted and validated with a classic monodromy matrix
algorithm in the time domain [30]. For each parame-
ter (β, λ), the N= 4 Floquet multipliers of the Mon-
odromy matrix were computed with a sufficiently small
time step to ensure convergence and served as a refer-
ence solution. The Hill matrix was then constructed for
various increasing truncation order H. For each H, the
Floquet multipliers ρ0
nwhere obtained from the N= 4
fundamental Floquet exponents sncomputed with the
eigenvector sorting algorithm explained in Section 2.2,
thanks to the relation ρ0
n=esn¯
Twhere ¯
T= 2π/β is
the dimensionless period of the considered perturbed
elastic state. We defined a converged spectral trunca-
tion order Hconv as the minimal Hfor which the N= 4
differences |ρ0
n−ρn|were not exceeding 1 ×10−6. The
map of the converged truncation order Hconv in the
(β, λ) space is given in Figs.14(a) and (b) for η= 0 and
η= 1, respectively.
The number of required harmonics Hconv is gener-
ally larger as the modulation amplitude λis enhanced
and the frequency modulation βis decreased. Also more
harmonics are required in the instability regions than
in the stable ones. Those converged truncation order
maps are the optimal ones when using the eigenvector
sorting algorithm. If no sorting, the stability map would
not converge and if using the eigenvalue sorting algo-
rithm, one could need higher truncation order Hconv,
especially for small βor in the non conservative case.
Note finally that it appears from Fig. 14 that more har-
monics are needed in the non conservative case than in
the conservative one. This trend is however exagerated
as most of the numerical data converge for less than
H= 25 and only a very thin region, located around
β/ω1≈0.75 and corresponding to a 2 ¯
Tinstability of
the second FF, needs H= 50.
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