# The Effect of Infinitesimal Damping  on  the Dynamic Instability Mechanism of  Conservative Systems

Authors: Dimitris S. Sophianopoulos; George T. Michaltsos; Anthony N. Kounadis
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

The local instability of 2 degrees of freedom (DOF) weakly damped systems is thoroughly discussed using the Liénard-Chipart stability criterion. The individual and coupling effect of mass and stiffness distribution on the dynamic asymptotic stability due to mainly infinitesimal damping is examined. These systems may be as follows: (a) unloaded (free motion) and (b) subjected to a suddenly applied load of constant magnitude and direction with infinite duration (forced motion). The aforementioned parameters combined with the algebraic structure of the damping matrix (being either positive semidefinite or indefinite) may have under certain conditions a tremendous effect on the Jacobian eigenvalues and then on the local stability of these autonomous systems. It was found that such systems when unloaded may exhibit periodic motions or a divergent motion, while when subjected to the above step load may experience either a degenerate Hopf bifurcation or periodic attractors due to a generic Hopf bifurcation. Conditions for the existence of purely imaginary eigenvalues leading to global asymptotic stability are fully assessed. The validity of the theoretical findings presented herein is verified via a nonlinear dynamic analysis.


---

## Body

## 1. Introduction

In previous studies of the 3rd author, based on 2-DOF and 3-DOF cantilevered models [1] under partial follower loading
(nonconservative systems), it was shown that in a small region of divergence
instability,
flutter (dynamic instability) may occur before divergence (static instability),
if very small damping is included [2, 3]. Bolotin et al. [4] using an aeroelastic
model presented a similar result. Païdoussis et al. [5] and Païdoussis [6] have
shown that flutter may occur in an inherently conservative system but for large
damping. However, the effect of damping, being of paramount importance in
nonconservative autonomous systems, was in general ignored when these systems
are subjected to a step conservative (potential) loading. This is so because it
was widely accepted that dynamic stability in nondissipative conservative
systems, which are stable, does not
change by the inclusion of damping [7].The local dynamic stability of discrete systems under
step conservative loading when small dissipative forces are included is
governed by the matrix-vector differential equation [8–11]: (1.1)
M
q
¨
+
C
q
˙
+
V
q
=
0
, where the dot denotes a
derivative with respect to time t; q
(
t
) is an n-dimensional state vector with coordinates q
i
(
t
)
(
i
=
1
,
…
,
n
)
;
M
,
C
and V are n
×
n real symmetric matrices.
More specifically, matrix M associated with the total kinetic energy of the system is a function of the
concentrated masses m
i
(
i
=
1
,
…
,
n
)
, being always positive definite; matrix   C  the elements of which are the damping coefficients
c
i
j
(
i
,
j
=
1
,
…
,
n
) may be positive definite, positive semidefinite as in the case of pervasive
damping [12, 13], or indefinite [14–16]; V is a generalized stiffness matrix with coefficients k
i
j
(
i
,
j
=
1
,
…
,
n
) whose elements V
i
j are also linear functions of a suddenly applied external load λ with
constant direction and infinite duration [17], that is, V
i
j
=
V
i
j
(
λ
;
k
i
j
).
Apparently, due to this type of loading, the system under discussion is autonomous. When the external loading λ
is applied statically, one can obtain
the static (divergence) instability or buckling loads λ
j
c
(
j
=
1
,
…
,
n
) by vanishing of the determinant of the stiffness
matrix V
(
λ
;
k
i
j
),
that is, (1.2)
|
V
(
λ
;
k
i
j
)
|
=
0. Clearly, (1.2) yields an nth degree algebraic
equation in λ. Assuming distinct critical
states, the matrix V
(
λ
;
k
i
j
) is positive
definite for λ
<
λ
1
c
,
positive
semidefinite for λ
=
λ
1
c
, and indefinite for λ
>
λ
1
c
.Kounadis in two very recent publications [10, 11] has
established the conditions under which the above autonomous dissipative systems
under step (conservative) loading may exhibit dynamic bifurcational modes of
instability before divergence (static)
instability, that is, for λ
<
λ
1
c, when infinitesimal damping is included. These
bifurcational modes may occur through either a degenerate Hopf bifurcation (leading to periodic motion around
centers) or a generic Hopf
bifurcation (leading to periodic attractors or to flutter). These unexpected
findings (implying failure of Ziegler’s kinetic criterion and other singularity
phenomena) may occur for a certain combination of values of the mass
(primarily) and stiffness distribution of the system in connection with a positive semidefinite or an indefinite damping matrix.The question which now arises is whether there are
combinations of values of the above-mentioned parameters (mass and stiffness
distribution) which in connection with the algebraic structure of damping matrices
may lead to dynamic bifurcational modes of instability when the system under
discussion isunloaded. Such local
(due to unforced motion) dynamic instability will be sought through the set of
asymptotic stability criteria of Liénard-Chipart [8, 18] which are elegant and more
readily employed than the well-known Routh-Hurwitz stability
criteria.As another main objective of this work, some new dynamic
bifurcations related to the algebraic structure of the damping matrix when the
systems are loaded by the above type of step conservative load will be also
discussed, using the Liénard-Chipart criterion
by analyzing 2-DOF models for which a lot of numerical results are available. Finally,
the conditions of a double purely imaginary root leading to a new dynamic bifurcation,
whose response is similar to that of a generic Hopf bifurcation, are properly
established.


## 2. Basic Equations

Solution of (1.1) can be sought in
the form (2.1)
q
=
r
e
ρ
t
, where ρ is in general a complex number and r is a complex vector independent of
time t.Introducingq from
(2.1) into  (1.1), we
get (2.2)
(
ρ
2
M
+
ρ
C
+
V
)
r
=
0.For given stiffness coefficientsk
i
j
(
i
,
j
=
1
,
…
,
n
), the
generalized stiffness matrix V is a linear function of λ. Thus, if matrices M, C, V are given, solutions of  (2.2) are intimately related to the algebraic
properties of the matrix-valued function L
(
ρ
)
=
ρ
2
M
+
ρ
C
+
V, and more specifically to the Jacobian
eigenvalues [
ρ
=
ρ
(
λ
)
] obtained through the vanishing of the determinant: (2.3)
|
ρ
2
M
+
ρ
C
+
V
|
=
0
, whose expansion gives the characteristic (secular) equation
for an N-DOF system: (2.4)
ρ
2
n
+
α
1
ρ
2
n
−
1
+
⋯
+
α
2
n
−
1
ρ
+
α
2
n
=
0
, where the real
coefficients α
i
(
i
=
1,
…
,
2
n
) are determined by means of Bôcher formula [19].
The eigenvalues (roots) of  (2.4) ρ
j
(
j
=
1,
…
,
2
n
) are, in general, complex conjugate pairs ρ
j
=
ν
j
±
μ
j
i (where ν
j
and μ
j are real numbers and   i
=
−
1) with corresponding complex conjugate eigenvectors r
j and r
¯
j
(
j
=
1
,
…
,
n
). Since ρ
j
=
ρ
j
(
λ
),
clearly ν
j
=
ν
j
(
λ
)
,
μ
j
=
μ
j
(
λ
)
,
r
j
=
r
j
(
λ
)
,  and r
¯
j
=
r
¯
j
(
λ
).
Thus, the solutions of  (1.1) are of the
form (2.5)
A
e
ν
j
t
cos
μ
j
t
,


B
e
ν
j
t
⁢
sin
⁡
μ
j
t
, where A and B are constants which are determined from the initial conditions. Solutions in (2.5) are bounded,
tending to zero as t
→
∞,
if all eigenvalues of  (2.4) have negative real parts, that is, when ν
j
<
0 for all j. In this case, the algebraic
polynomial (2.4) is called a Hurwitz
polynomial (since all its roots have negative real parts) and the origin
(
q
=
q
˙
=
0
) of the system is asymptotically stable.


### 2.1. Criteria for Asymptotic Stability

Thenecessary and sufficient conditions which assure that all roots of  (2.4) have negative real parts (i.e., v
j
<
0 for all j) which means that the
corresponding polynomial |
L
(
ρ
)
| is a Hurwitz
polynomial are of great practical importance.Consider the more general case of a polynomial inz
with real coefficients α
i
(
i
=
1
,
…
,
n
): (2.6)
f
(
z
)
=
α
0
z
n
+
α
1
z
n
−
1
+
⋯
+
α
n
−
1
z
+
α
n
=
0

(
α
0
>
0
) for which we will seek the necessary and sufficient conditions so that all its  roots have negative real parts.Denoting
byz
κ
(
κ
=
1
,
…
,
m
) the real roots and by r
j
±
i
s
j
(
j
=
1
,
…
,
(
n
−
m
)
/
2
)
;
i
=
−
1
) the complex roots of  (2.6), we may assure that all these roots in the
complex plane lie to the left of the
imaginary axis, that is, (2.7)
z
κ
<
0
,

r
j
<
0

(
κ
=
1
,
…
,
m
;
j
=
1
,
…
,
n
−
m
2
)
. Then one can write (2.8)
f
(
z
)
=
α
0
∏
κ
=
1
m
(
z
−
z
κ
)
∏
j
=
1
n
−
m
(
z
2
−
2
r
j
z
+
r
j
2
+
s
j
2
)
.Since due to inequality (2.7), each term in the last
part of (2.8) has positive coefficients, it is deduced that all coefficients
of (2.6) are also positive. However,
this (i.e., α
i
>
0
for all i with α
0
>
0)
is a necessary but by no means sufficient condition for all roots of (2.6) to lie in the left half-plane (i.e., Re
(
z
)
<
0).According toRouth-Hurwitz criterion [18]
of asymptotic stability for all roots of (2.6) to
have negative real parts, the necessary and sufficient conditions are (2.9)
Δ
1
>
0
,
Δ
2
>
0
,
…
,
Δ
n
>
0 where (2.10)
Δ
1
=
α
1
,
Δ
2
=
|
α
1
α
3
α
0
α
2
|
,
Δ
3
=
|
α
1
α
3
0
α
0
α
2
α
4
0
α
1
α
3
|
,
…
,
Δ
n
=
|
α
1
α
3
α
5
⋯
⋯
⋯
⋯
α
0
α
2
α
4
⋯
⋯
⋯
⋯
0
α
1
α
3
⋯
⋯
⋯
⋯
0
α
0
α
2
α
4
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
α
i
|   (with α
κ
=
0 for κ
>
n). Note the last equality Δ
n
=
α
n
Δ
n
−
1
.It should be noted that when the above necessary
conditionsα
i
>
0  (for all i) hold, inequalities (2.9) are not independent. For instance, for n
=
4, the Routh-Hurwitz
conditions reduce to the single inequality  Δ
3
>
0; for n
=
5, they reduce to Δ
2
>
0  and Δ
4
>
0; while for n
=
6, they reduce also to two inequalities, Δ
3
>
0
,
Δ
5
>
0.
This case was discussed by Liénard and Chipart who established the following
elegant criterion for asymptotic stability [8].The Liénard-Chipart Stability Criterion
For a polynomial with real
coefficientsf
(
z
)
=
α
0
z
n
+
α
1
z
n
−
1
+
…
+
α
n
−
1
z
+
α
n
=
0
(
α
0
>
0
) to have all roots with negative real parts, it is necessary
and sufficient that
(1)
all coefficients off(z) be positive, that is,
(2.11a)
α
i
>
0

(
i
=
1
,
…
,
n
)
;
(2)
the determinant inequalities be also
positive, that is,(2.11b)
Δ
n
−
1
>
0
,
Δ
n
−
3
>
0
,
…
,
where Δ
κ denotes as before the Hurwitz
determinant of κth order.
It can be shown that if the Hurwitz determinants ofodd order are positive, then
those of even order are also positive, and vice versa. This holds
even when only part of the
coefficients α
i of f(z) (with α
0
>
0
) are positive. According to this, the Liénard-Chipart criterion
is defined as follows.
Necessary and sufficient conditions for all roots of the real polynomial f
(
z
)
=
α
0
z
n
+
α
1
z
n
−
1
+
…
+
α
n
−
1
z
+
α
n
=
0
,
(
α
0
>
0
) to have negative
real parts can be given in any one of the following forms [18]:
(1)
(2.12a)
α
n
>
0
,
α
n
−
2
>
0
,
…
;
w
i
t
h
⁢
{
either
Δ
1
>
0
,
Δ
3
>
0
,
…
,
or
Δ
2
>
0
,
Δ
4
>
0
,
…
,
(2)
(2.12b)
α
n
>
0
,
α
n
−
1
>
0
,
α
n
−
3
>
0
,
…
;
w
i
t
h
{
either
Δ
1
>
0
,
Δ
3
>
0
,
…
,
or
Δ
2
>
0
,
Δ
4
>
0
,
…
.
This stability criterion was rediscovered by Fuller [20].
For instance, for a 2-DOF cantilevered
model, the characteristic (secular) (2.4) is (2.13)
ρ
4
+
α
1
ρ
3
+
α
2
ρ
2
+
α
3
ρ
+
α
4
=
0

(
α
0
=
1
)
.
According to the last criterion,
all roots of (2.13) have negative real parts provided that α
4
>
0
,
α
2
>
0
,
Δ
1
=
α
1
>
0
and Δ
3
=
α
3
(
α
1
α
2
−
α
3
)
−
α
1
2
α
4
>
0. Clearly, from the last inequality, it follows that α
3
>
0.
Hence, the positivity of α
1 and α
3 was assured via the above conditions (i.e., α
4
>
0
,
α
2
>
0
,
Δ
1
>
0
,
Δ
3
>
0).


## 2.1. Criteria for Asymptotic Stability

Thenecessary and sufficient conditions which assure that all roots of  (2.4) have negative real parts (i.e., v
j
<
0 for all j) which means that the
corresponding polynomial |
L
(
ρ
)
| is a Hurwitz
polynomial are of great practical importance.Consider the more general case of a polynomial inz
with real coefficients α
i
(
i
=
1
,
…
,
n
): (2.6)
f
(
z
)
=
α
0
z
n
+
α
1
z
n
−
1
+
⋯
+
α
n
−
1
z
+
α
n
=
0

(
α
0
>
0
) for which we will seek the necessary and sufficient conditions so that all its  roots have negative real parts.Denoting
byz
κ
(
κ
=
1
,
…
,
m
) the real roots and by r
j
±
i
s
j
(
j
=
1
,
…
,
(
n
−
m
)
/
2
)
;
i
=
−
1
) the complex roots of  (2.6), we may assure that all these roots in the
complex plane lie to the left of the
imaginary axis, that is, (2.7)
z
κ
<
0
,

r
j
<
0

(
κ
=
1
,
…
,
m
;
j
=
1
,
…
,
n
−
m
2
)
. Then one can write (2.8)
f
(
z
)
=
α
0
∏
κ
=
1
m
(
z
−
z
κ
)
∏
j
=
1
n
−
m
(
z
2
−
2
r
j
z
+
r
j
2
+
s
j
2
)
.Since due to inequality (2.7), each term in the last
part of (2.8) has positive coefficients, it is deduced that all coefficients
of (2.6) are also positive. However,
this (i.e., α
i
>
0
for all i with α
0
>
0)
is a necessary but by no means sufficient condition for all roots of (2.6) to lie in the left half-plane (i.e., Re
(
z
)
<
0).According toRouth-Hurwitz criterion [18]
of asymptotic stability for all roots of (2.6) to
have negative real parts, the necessary and sufficient conditions are (2.9)
Δ
1
>
0
,
Δ
2
>
0
,
…
,
Δ
n
>
0 where (2.10)
Δ
1
=
α
1
,
Δ
2
=
|
α
1
α
3
α
0
α
2
|
,
Δ
3
=
|
α
1
α
3
0
α
0
α
2
α
4
0
α
1
α
3
|
,
…
,
Δ
n
=
|
α
1
α
3
α
5
⋯
⋯
⋯
⋯
α
0
α
2
α
4
⋯
⋯
⋯
⋯
0
α
1
α
3
⋯
⋯
⋯
⋯
0
α
0
α
2
α
4
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
α
i
|   (with α
κ
=
0 for κ
>
n). Note the last equality Δ
n
=
α
n
Δ
n
−
1
.It should be noted that when the above necessary
conditionsα
i
>
0  (for all i) hold, inequalities (2.9) are not independent. For instance, for n
=
4, the Routh-Hurwitz
conditions reduce to the single inequality  Δ
3
>
0; for n
=
5, they reduce to Δ
2
>
0  and Δ
4
>
0; while for n
=
6, they reduce also to two inequalities, Δ
3
>
0
,
Δ
5
>
0.
This case was discussed by Liénard and Chipart who established the following
elegant criterion for asymptotic stability [8].The Liénard-Chipart Stability Criterion
For a polynomial with real
coefficientsf
(
z
)
=
α
0
z
n
+
α
1
z
n
−
1
+
…
+
α
n
−
1
z
+
α
n
=
0
(
α
0
>
0
) to have all roots with negative real parts, it is necessary
and sufficient that
(1)
all coefficients off(z) be positive, that is,
(2.11a)
α
i
>
0

(
i
=
1
,
…
,
n
)
;
(2)
the determinant inequalities be also
positive, that is,(2.11b)
Δ
n
−
1
>
0
,
Δ
n
−
3
>
0
,
…
,
where Δ
κ denotes as before the Hurwitz
determinant of κth order.
It can be shown that if the Hurwitz determinants ofodd order are positive, then
those of even order are also positive, and vice versa. This holds
even when only part of the
coefficients α
i of f(z) (with α
0
>
0
) are positive. According to this, the Liénard-Chipart criterion
is defined as follows.
Necessary and sufficient conditions for all roots of the real polynomial f
(
z
)
=
α
0
z
n
+
α
1
z
n
−
1
+
…
+
α
n
−
1
z
+
α
n
=
0
,
(
α
0
>
0
) to have negative
real parts can be given in any one of the following forms [18]:
(1)
(2.12a)
α
n
>
0
,
α
n
−
2
>
0
,
…
;
w
i
t
h
⁢
{
either
Δ
1
>
0
,
Δ
3
>
0
,
…
,
or
Δ
2
>
0
,
Δ
4
>
0
,
…
,
(2)
(2.12b)
α
n
>
0
,
α
n
−
1
>
0
,
α
n
−
3
>
0
,
…
;
w
i
t
h
{
either
Δ
1
>
0
,
Δ
3
>
0
,
…
,
or
Δ
2
>
0
,
Δ
4
>
0
,
…
.
This stability criterion was rediscovered by Fuller [20].
For instance, for a 2-DOF cantilevered
model, the characteristic (secular) (2.4) is (2.13)
ρ
4
+
α
1
ρ
3
+
α
2
ρ
2
+
α
3
ρ
+
α
4
=
0

(
α
0
=
1
)
.
According to the last criterion,
all roots of (2.13) have negative real parts provided that α
4
>
0
,
α
2
>
0
,
Δ
1
=
α
1
>
0
and Δ
3
=
α
3
(
α
1
α
2
−
α
3
)
−
α
1
2
α
4
>
0. Clearly, from the last inequality, it follows that α
3
>
0.
Hence, the positivity of α
1 and α
3 was assured via the above conditions (i.e., α
4
>
0
,
α
2
>
0
,
Δ
1
>
0
,
Δ
3
>
0).


## 3. Mathematical Analysis

Subsequently, using the spring
cantilevered dynamical model of 2-DOF shown in Figure1, we will examine in
detail the effect of violation of one
or more of the conditions of  Liénard-Chipart criterion
on its asymptotic stability. The response of this dynamic model carrying two
concentrated masses is studied when it is either unloaded or loaded by a
suddenly applied load of constant magnitude and direction with infinite
duration. Such autonomous dissipative systems with infinitesimal damping (including
the case of zero loading) are properly discussed. If at least one root of the
secular equation (2.13) has a positive real part, the corresponding solution (2.5)
will contain an exponentially increasing function and the system will become
unstable.Figure 1
2-DOF autonomous system under conservative loading.The seeking of animaginary root of the secular equation (2.13)
which represents a border line between dynamic stability and instability is a
first but important step in our discussion. Clearly, an imaginary root gives
rise to an oscillatory motion of the form e
i
μ
t
,
(
i
=
−
1
,
μ
r
e
a
l
n
u
m
b
e
r
) around the trivial state. However, the existence
of at least one multiple imaginary root of
the κth order of multiplicity leads to a solution containing functions of
the form e
i
μ
t
,
t
e
i
μ
t
,
…
,
t
κ
−
1
e
i
μ
t which increases with time. Hence, the multiple imaginary root on the imaginary
axis of the complex plane denotes local dynamic
instability. The discussion of such a situation is also another objective
of this study.The nonlinear equations of motion for the 2-DOF model
of Figure1 with rigid links of equal length ℓ are given by [11] (3.1)
(
1
+
m
)
θ
¨
1
+
θ
¨
2
cos
⁡
(
θ
1
−
θ
2
)
+
θ
˙
2
2
sin
⁡
(
θ
1
−
θ
2
)
+
c
11
θ
˙
1
+
c
12
θ
˙
2
+
k
θ
1
−
θ
2
+
θ
1
−
λ
sin
⁡
θ
1
=
0
,
θ
¨
2
+
θ
¨
1
cos
⁡
(
θ
1
−
θ
2
)
−
θ
˙
1
2
sin
⁡
(
θ
1
−
θ
2
)
+
c
22
θ
˙
2
+
c
12
θ
˙
1
−
θ
1
+
θ
2
−
λ
sin
⁡
θ
2
=
0
, where m
=
m
1
/
m
2
,
k
=
k
1
/
k
2
,
λ
=
P
ℓ
/
k
2.Linearization of (3.1)
after setting (3.2)
Θ
=
[
θ
1
θ
2
]
=
e
ρ
t
[
φ
1
φ
2
]
=
e
ρ
t
φ   gives (3.3)
(
ρ
2
M
+
ρ
C
+
V
)
φ
=
0
, where (3.4)
M
=
[
m
11
m
12
m
12
m
22
]
=
[
1
+
m
1
1
1
]
,


C
=
[
c
11
c
12
c
12
c
22
]
,


V
=
[
V
11
V
12
V
12
V
22
]
=
[
k
+
1
−
λ
−
1
−
1
1
−
λ
]
.Note that in case of a Rayleigh’s dissipative
function the damping coefficients are,c
11
=
c
1
+
c
2
,
c
12
=
−
c
2
,
c
21
=
−
c
2, and c
22
=
c
2,
where  c
i
(
i
=
1
,
2
) is dimensionless coefficient for the ith rigid link. This case (for which det
C
=
|
c
|
=
c
1
c
2) is a specific situation of the damping matrix C which is not discussed herein.Thestatic buckling (divergence) equation is given by (3.5)
λ
2
−
(
2
+
k
)
λ
+
k
=
0
, whose lowest root is the first buckling load λ
1
c equal to (3.6)
λ
1
c
=
0.5
(
k
+
2
−
k
2
+
4
)
. Clearly, for the entire
interval of values of k
>
0,  (3.6)
yields    0
≤
λ
1
c
<
1.The characteristic (secular) equation is(3.7)
ρ
4
+
α
1
ρ
3
+
α
2
ρ
2
+
α
3
ρ
+
α
4
=
0
, where (3.8)
α
1
=
1
m
(
m
11
c
22
+
m
22
c
11
−
2
m
12
c
12
)
,
α
2
=
1
m
(
m
11
V
22
+
m
22
V
11
−
2
V
12
m
12
+
|
c
|
)
,
α
3
=
1
m
(
c
11
V
22
+
c
22
V
11
−
2
V
12
c
12
)
,
α
4
=
1
m
(
V
11
V
22
−
V
12
2
)
.Let us first examine the effect
of violation of Liénard-Chipartcriterion on the system stabilityin
the case ofzero loading (i.e., λ
=
0). Then expressions in (3.8) due to relations (3.4) are written as
follows:
(3.9)
α
1
=
1
m
[
(
1
+
m
)
c
22
+
c
11
−
2
c
12
]
,
α
2
=
1
m
[
k
+
m
+
4
+
c
11
c
22
−
c
12
2
]
,
α
3
=
1
m
[
c
11
+
c
22
(
k
+
1
)
+
2
c
12
]
,
α
4
=
k
m
. According to Liénard-Chipart criterion,
inequalities (2.12a) imply (3.10)
α
4
>
0
,


α
2
>
0
,


Δ
1
,
Δ
3
>
0
, where Δ
1
=
α
1
>
0 and Δ
3
=
α
3
(
α
1
α
2
−
α
3
)
−
α
1
2
α
4
>
0.
Clearly, from the last inequality, it follows that α
3
>
0.Forc
i
i
>
0
(
i
=
1
,
2
)
,
k
>
0, and m
>
0 (implying α
4
>
0), it is deduced that this criterion is violated if either one of α
1
o
r
α
2 is zero or Δ
3 is zero. These three cases will be discussed
separately in connection with the algebraic structure of the damping matrix C
=
[
c
i
j
]
.Case 1 (α
1
=
0
w
i
t
h
α
2
>
0).
Ifα
1
=
0 (yielding Δ
3
=
−
α
3
2
<
0), then (3.11)
(
1
+
m
)
c
22
+
c
11
−
2
c
12
=
0.                                                                                      Equation (3.11),
being independent of λ and
k, is satisfied only when the damping matrix C is indefinite, that is, (3.12)
c
11
c
22
−
c
12
2
<
0

(
c
i
i
>
0
⁢
f
o
r
⁢
i
,
j
=
1
,
2
)
. Indeed, the last inequality due to relation
(3.11) implies (3.13)
(
1
+
m
)
2
c
22
2
+
2
(
m
−
1
)
c
11
c
22
+
c
11
2
>
0 which is always satisfied,
regardless of the value of c22/c11, since for m
>
0, the
discriminant of (3.13) (equal to −
16
c
11
2
m) is always
negative.
Thus,
we have explored theunexpected finding that an unloaded (stable) system
becomes dynamically unstable at any
small disturbance in case of an indefinite damping matrix even when infinitesimal damping is included.
Since
all coefficients of (3.7) are positive from the theory of algebraic equations
it follows that this equation cannot have positive root. Also the case of existence of a pair of pure imaginary
roots associated with Δ
3
=
0 is ruled out, since Δ
3
<
0 (due to α
1
=
0). Hence, (3.7) has either two negative roots combined with a pair of
complex conjugate roots with positive real part or two pairs of complex
conjugate roots with opposite real
parts. Both cases imply local dynamic instability.Case 2 (α
2
=
0
w
i
t
h
α
1
>
0).
Ifα
2
=
0 (implying also Δ
3
<
0), then (3.14)
|
c
|
=
−
k
−
m
−
4
<
0

(
k
,
m
>
0
)
. Namely, the damping matrix [
c
i
j
]
is indefinite but with large negative
determinant (rather unrealistic case). Since the Liénard-Chipart criterion
is violated, the model is again locally dynamically unstable.
Since all coefficients of (3.7) are positive, from
the theory of algebraic equations, it is deduced that this equation
cannot have positive root. Also the
case of existence of a pair of pure imaginary roots associated with Δ
3
=
0 is ruled out, since Δ
3
<
0 (due to α
2
=
0). Hence, (3.7) has either two negative roots combined with a pair of complex conjugate roots
with positive real part or two pairs
of complex conjugate roots with opposite real parts. Both cases imply local
dynamic instability.Case 3 (Δ
3
=
0).
In
this case, stability conditions in (3.10) are satisfied except for the last one, since Δ
3
=
0 which yields [11]
(3.15)
Δ
3
=
α
3
(
α
1
α
2
−
α
3
)
−
α
1
2
α
4
=
0. Note that λ
<
λ
1
c implies α
4
>
0 (i.e., det
V
>
0).
This is a necessary condition for
the secular (3.7) to have one pair of pure
imaginary roots ±
μ
i
,
(
i
=
−
1
).
Indeed, this can be readily established by inserting  ρ
=
±
μ
i into (3.7) and then equating to zero real and
imaginary parts.
Consider
now the more general case ofnonzero
loading (i.e., λ
≠
0). Using
relations in (3.8),
(3.15) can be written as follows: (3.16)
A
λ
2
+
B
λ
+
Γ
=
0
, where
(3.17a)
A
=
(
c
11
+
c
22
)
(
m
+
2
)
α
¯
1
−
m
(
c
11
+
c
22
)
2
−
[
(
1
+
m
)
c
22
+
c
11
−
2
c
12
]
2
,
(3.17b)
B
=
−
α
¯
1
{
(
m
+
2
)
[
c
11
+
(
k
+
1
)
c
22
+
2
c
12
]
+
(
c
11
+
c
22
)
[
k
+
m
+
4
+
|
c
|
]
}

+
2
m
[
c
11
+
(
k
+
1
)
c
22
+
2
c
12
]
(
c
11
+
c
22
)
+
(
k
+
2
)
[
(
1
+
m
)
c
22
+
c
11
−
2
c
12
]
2
,
(3.17c)
Γ
=
(
k
+
m
+
4
+
|
c
|
)
[
c
11
+
(
k
+
1
)
c
22
+
2
c
12
]
α
¯
1
−
m
[
c
11
+
(
k
+
1
)
c
22
+
2
c
12
]
2

−
k
[
(
1
+
m
)
c
22
+
c
11
−
2
c
12
]
2
, where
(3.17d)
c
11
=
c
12
2
+
|
c
|
c
22
,


α
¯
1
=
m
α
1
.
For real λ,
the discriminant D of
(3.16) must be greater than
or equal to zero, that is,  (3.18)
D
=
B
2
−
4
A
Γ
≥
0. Subsequently, attention is focused
on the following: (a)
matrix C is positive semidefinite (i.e., |
c
|
=
0 with c
i
i
>
0
,
i
=
1
,
2)
and (b) matrix C is indefinite (|
c
|
<
0 with c
i
i
>
0
,
i
=
1
,
2).
Using the symbolic manipulation ofMathematica [21], one can find that
(3.19)
D
=
|
c
|
f
(
|
c
|
)
, where f
(
|
c
|
) is an algebraic polynomial of 5th degree  in |
c
|
.Case 4 (|
c
|
=
0
,
f
≠
0).
For|
c
|
=
0, (3.16) implying D
=
0
admits a double root, which due to (3.17a), (3.17b), (3.17d) is given
by (3.20)
λ
1
H
=
2
c
12
2
+
c
12
c
22
(
k
−
m
)
−
c
22
2
(
k
+
m
+
2
)
c
12
2
−
c
22
2
−
c
12
c
22
m
. Using the Reduce command
embedded in Mathematica, one can find
the conditions under which 0
<
λ
1
H
<
λ
1
c,
given in the appendix, relation (A.1).Case 5 (f
=
0
,
|
c
|
≠
0).
Moreover, it was found symbolically that the 5th degree polynomialf
(
|
c
|
) possesses three
real roots (one double and one single), and two pure imaginary ones. Discussing their nature, one can find that the
double root of f
(
|
c
|
),
being equal to |
c
|
=
−
(
c
12
−
c
22
)
2
−
c
22
2
m
<
0, yields (3.21)
c
12
=
1
2
[
c
11
+
(
m
+
1
)
c
22
]
. Then, the double root of (3.16) becomes (3.22)
λ
2
H
=
2
c
11
+
c
22
(
k
+
m
+
2
)
c
11
+
c
22 which is always greater than λ
1
c and hence
of minor importance for the present analysis.
The third real root off
(
|
c
|
),
if substituted in (3.16), yields again a double root in λ,
always less than zero, which is rejected. Thus, only the case of a positive semidefinite damping matrix may
lead to an acceptable value of the corresponding load (i.e., 0
<
λ
H
<
λ
1
c) associated with a degenerate Hopf bifurcation, as theoretically was shown by Kounadis
[10, 11].Case 6 (λ
=
0).
Ifλ
=
0,
(3.16) implies Γ
=
0,
which after symbolic manipulation of (3.17c) can be written in the following
form: (3.23)
Γ
=
1
c
22
2
(
|
c
|
3
+
A
2
|
c
|
2
+
A
1
|
c
|
+
A
0
)
=
0
, where A
i
(
i
=
0
,
1
,
2
) are given in the appendix, relations (A.2).
It is evident that A
2
>
0 and A
0
≥
0, a fact implying that (3.23)
can be satisfied only for |
c
|
≤
0 if also A
1
≥
0; otherwise (i.e.,
if A
1
<
0) the system may be locally stable or unstable. For |
c
|
=
0,
one can find the corresponding values of c
i
i
(
i
=
1
,
2
), given in (A.3)
and (A.4) in the appendix, which are always positive. This special case, for
which the trivial (unloaded) state is associated with a pair of pure imaginary eigenvalues (necessary condition
for a Hopf bifurcation), implies a local dynamic instability.Conditions for A Double Imaginary Root
For adouble imaginary root, the first
derivative of the secular equation (3.7) must be also zero, which yields (3.24)
4
ρ
3
+
3
α
1
ρ
2
+
2
α
2
ρ
+
α
3
=
0. Inserting into (3.24) ρ
=
μ
i
, we obtain μ
2
=
0.5
α
2
=
α
3
/
3
α
1 and thus α
3
=
3
α
1
α
2
/
2. Introducing this expression of α
3 into (3.15), it follows that α
1
=
0,
which also implies that α
3
=
0 and hence (3.24) becomes ρ
2
=
−
0.5
α
2.
If ρ
2
=
−
0.5
α
2 is inserted into the secular equation ρ
4
+
α
2
ρ
2
+
α
4
=
0,
for a double imaginary root, it
follows that α
2
2
−
4
α
4
=
0 which due to relations (3.8) yields
(3.25)
(
m
2
+
4
)
λ
2
−
2
λ
(
m
2
+
(
2
−
k
)
m
+
2
k
+
8
+
(
m
+
2
)
|
c
|
)
|
+
|
c
|
2
+
2
(
k
+
m
+
4
)
|
c
|
+
(
k
+
m
+
4
)
2
−
4
m
k
=
0.
For real λ,
the discriminant D of (3.25) must satisfy the inequality  (3.26)
D
=
|
c
|
2
+
(
4
−
m
k
)
|
c
|
−
(
m
+
k
)
2
≥
0 which yields (3.27)
either

|
c
|
<
0.5
(
m
k
−
4
−
m
2
(
k
2
+
4
)
+
4
k
2
+
16
)
,
o
r

|
c
|
>
0.5
(
m
k
−
4
+
m
2
(
k
2
+
4
)
+
4
k
2
+
16
)
. Using the conditions found above (3.28)
α
1
=
α
3
=
0
, relations (3.9) yield (3.29)
(
1
+
m
)
c
22
+
c
11
−
2
c
12
=
0
,
c
11
(
1
−
λ
)
+
c
22
(
k
+
1
−
λ
)
+
2
c
12
=
0.   Adding the last two equations, we
obtain (3.30)
(
2
−
λ
)
c
11
+
(
k
+
m
+
2
−
λ
)
c
22
=
0. Since k
,
m
>
0
, and λ
<
λ
1
c
<
1
, it follows that both coefficients of c11 and c22 in (3.30) are positive. Hence, c11 and c22 are of opposite sign (i.e., c
11
/
c
22
<
0) and consequently |
c
|
=
c
11
c
22
−
c
12
2
<
0;
thus the 2nd of inequalities (3.27) is excluded.Solving simultaneously the system
of equationsα
2
2
−
4
α
4
=
0, α
1
=
α
3
=
0 in k, m, λ, two
ternaries of values for k, m, and λ are obtained, given in the appendix, expressions (A.5). For all these values to be greater than zero, the Reduce command embedded in Mathematica
[21] yields two sets of conditions, given also in the appendix, relations
(A.6). Further symbolic computations are needed for establishing the
conditions for a double pure imaginary root for loading values less than λ
1
c.
Nevertheless, suitable combinations of values of c
i
j, k, and m may
be found. This will be demonstrated in Section 4. The corresponding dynamic
response, since the system is associated with a codimension-2 local bifurcation, is anticipated to
be related to isolated periodic orbits
which will be established via a straightforward complete nonlinear dynamic
analysis.


## 4. Numerical Results

In
this section, numerical results corresponding to all the above cases of
violation of the Liénard-Chipart stability
criterion are given below in
tabular and graphical forms.Case 1 (α
1
=
0).
(a)λ
=
0. For an unloaded system with k
=
m
=
1, choosing c
11
=
0.015 and c
22
=
0.002,
(3.11) yields  c
12
=
0.0095 and as expected the damping matrix is indefinite with  determinant |
c
|
=
−
0.0006025. The two pairs of corresponding eigenvalues are ρ
1
,
2
=
−
0.00332577
±
0.41421
i and ρ
3
,
4 = 0.00335877
±
0.41421
i
(
i
=
−
1
),
implying local dynamic instability.
Solving numerically the system of nonlinear equations (3.1), we find that the
dynamic response of the system is associated with a divergent motion, as
depicted in Figure 2, with the aid of the time series
[
τ
,
θ
1
(
τ
)
], time velocities [
τ
,
θ
˙
1
(
τ
)
] and
phase-plane portraits
[
θ
1
(
τ
)
,
θ
˙
1
(
τ
)
].
(b)λ
≠
0. For a system with k
=
5
,
m
=
4, and λ
=
0.5
<
λ
1
c
=
0.807418 and for c
11
=
0.01
,
c
22
=
0.002,
(2.13) yields c
12
=
−
0.0175 implying |
c
|
=
−
2.5625
×
10
−
4
. The trivial state is locally dynamically unstable,
since ρ
1
,
2
=
−
0.00397748
±
0.4351
i and ρ
3
,
4
=
0.00397748
±
0.4351
i
. The corresponding dynamic response is again
related to a divergent (unbounded) motion,
as shown in the phase-plane portraits of Figures 3(a) and 3(b).(a) Time series[
τ
,
θ
1
(
τ
)
], (b) time velocities [
τ
,
θ
˙
1
(
τ
)
], and (c) phase-plane portraits [
θ
1
(
τ
)
,
θ
˙
1
(
τ
)
],
for the system of Case 1(a), exhibiting a divergent motion.
(a)
(b)
(c)Phase-plane portraits[
θ
i
(
τ
)
,
θ
˙
i
(
τ
)
] (i
=
1
,
2) for the system of Case 1(b),
associated with a divergent motion.
(a)
(b)Case 2 (α
2
=
0).
(a)λ
=
0: If
k
=
m
=
1 relation (3.14) is satisfied, for example, for the damping coefficients
c
11
=
0.50
,
c
22
=
2.00, yielding c
12
=
7 and |
c
|
=
−
6.
For this, rather unrealistic, subcase, the corresponding eigenvalues are equal
to ρ
1
=
−
1.86617
,
ρ
2
=
−
0.102227
, and ρ
3
,
4
=
1.37995
±
1.8269
i (local instability). Hence, the response of
the system is also related to a divergent (unbounded) motion, presented graphically in the phase-plane curves of
Figures 4(a) and 4(b).
(b)λ
≠
0.
Similarly, for a system with k
=
0.10
,
m
=
0.20 (for which λ
1
c
=
0.0487508) in order that α
2
=
0
, we must choose an indefinite damping matrix
with  |
c
|
=
−
4.25.
Setting, for example, c
11
=
2.375
,
c
12
=
3.00
,
c
22
=
2.00, and λ
=
0.227273
<
λ
1
c
,  the trivial state is locally dynamically unstable
with ρ
1
=
−
2.46657
,
ρ
2
=
−
0.00503929
, and ρ
3
,
4
=
4.2983
±
1.6612
i
. The system exhibits a divergent (unbounded) motion, as shown in Figures 5(a) and 5(b).Phase-plane portraits[
θ
i
(
τ
)
,
θ
˙
i
(
τ
)
]
(
i
=
1
,
2
) for the system of Case 2(a),
associated with a divergent motion.
(a)
(b)Phase-plane portraits[
θ
i
(
τ
)
,
θ
˙
i
(
τ
)
]
(
i
=
1
,
2
) for the system of Case 2(b),
associated with a divergent motion.
(a)
(b)Case 3 (Δ
3
=
0
w
i
t
h
α
1
α
2
≠
0).
(a)Positive
semidefinite damping matrix
(
|
c
|
=
0
). Choosing c
11
=
0.01
,
c
22
=
0.0004
(and thus c
12
=
0.002), the 1st requirement of the 2nd set of conditions given in the appendix, relation (A.1), is satisfied (i.e., c
12
>
c
22).
The 2nd requirement, that is, m
>
2
(
c
12
−
c
22
)
/
c
22
, yields m
>
8, and hence one can choose m
=
10. The 3rd requirement in (A.1) implies that 0.8333
<
k
<
3 and thus
one can take k
=
1. Solving numerically (3.15) with respect to λ, we
obtain the value of λ
H
=
0.307692256
<
λ
1
c
=
0.381966, associated with a pair of
pure imaginary eigenvalues, while the other pair has negative real parts. The
evolution of both pairs of eigenvalues in the complex plane as λ
varies is presented in Figure 6 for λ
<
λ
1
c.
For λ
=
λ
H, a degenerate Hopf bifurcation
occurs and the system exhibits a periodic motion, whose amplitude depends
on the initial conditions. Relevant results in graphical form can be found in
recent publications of the 3rd author [10, 11].
(b)Indefinite damping matrix
(
|
c
|
<
0
).
It has been proven by Kounadis [10, 11] that in this subcase (for λ
<
λ
1
c) all the necessary and sufficient conditions
for a generic Hopf bifurcation   are fulfilled and hence the
system experiences a periodic attractor response (stable limit cycles) with
constant final amplitudes regardless of the initial conditions. Numerical
results are given in the aforementioned papers by Kounadis.
(c)Δ
3
=
0
and
λ
=
0. If at the same time |
c
|
=
0, one can find the values of
c
i
i
(
i
=
1
,
2
) through (A.3) and (A.4) in the appendix, which are
always positive. A further investigation of this case as well as of the case
|
c
|
<
0 for the global stability of the system can be performed through a
nonlinear dynamic analysis.
(d)Double pure imaginary eigenvalues. For this
special case, three combinations of damping matrix coefficients c
i
j are examined. These, along with the corresponding critical values of k, λ and
m, satisfying relations (A.5) of the appendix, are given in Table 1. Note
that Cases 3(d)1 and 3(d)2 are the outcome of the 1st set from relations
(A.5), while Case 3(d)3 is the outcome from the 2nd set. Clearly, in all cases, λ
<
λ
1
c.Table 1
Values of damping coefficients (c
i
j) and critical system parameters (k, λ
c
r, m) for three subcases with double pure imaginary eigenvalues.
Case no.
c11
c12
c22
k
λ
c
r
m
λ
1
c
|
c
|
3(d)1
3.45
1.00
−1.40
2.65201108
0.164480292
0.035714286
0.665195
−5.83
3(d)2
3.60
1.00
−1.20
3.178585026
0.244040821
0.333333333
0.711568
−5.32
3(d)3
3.40
1.00
−1.20
2.346316274
0.629282032
0.166666666
0.631633
−5.08Evolution of both pairs of eigenvalues in the complex plane for the system of
Case3(a), associated with a degenerate Hopf bifurcation.
(a)
(b)In the three above subcases, the evolution of both pairs ofλ-dependent eigenvalues
in the complex plane is depicted in Figures 7, 8, 9(a), and  9(b), from which it
is evident that for all λ
<
λ
1
c,
except for λ
<
λ
c
r (where a codimension-2 bifurcation
occurs), the pairs of eigenvalues remain always in opposite planes of the Im axis, but symmetric with respect to the Re
axis. This symmetry is always present for the pair with negative real parts,
while for the other pair (with positive real parts), this feature remains until
their imaginary part vanishes simultaneously at a certain value of the loading λ
less than λ
1
c.Figure 7
Evolution of both pairs of eigenvalues in the complex plane for the system of Case3(d)1.Figure 8
Evolution of both pairs of eigenvalues in the complex plane for the system of Case3(d)2.Evolution of both pairs of eigenvalues in the complex plane (a), and
corresponding detail (b), for the system of Case3(d)3.
(a)
(b)The
dynamic response of the system for all these subcases is associated with
isolated periodic orbits (whose final amplitude is constant and independent of
the initial conditions), as shown in the phase-plane
trajectories of Figures10, 11, and 12.Phase-plane portraits[
θ
i
(
τ
)
,
θ
˙
i
(
τ
)
]
(
i
=
1
,
2
) for the system of Case 3(d)1, associated
with an isolated periodic orbit.
(a)
(b)Phase-plane portraits[
θ
i
(
τ
)
,
θ
˙
i
(
τ
)
]
(
i
=
1
,
2
) for the system of Case 4c2, associated
with an isolated periodic orbit.
(a)
(b)Phase-plane portraits[
θ
i
(
τ
)
,
θ
˙
i
(
τ
)
]
(
i
=
1
,
2
) for the system of Case 3(a), associated
with an isolated periodic orbit.
(a)
(b)The corresponding dynamic bifurcations related to the abovedouble pure imaginary eigenvalues behave like a generic Hopf bifurcation, whose basic
feature is the intersection of the λ-dependent path of
one eigenvalue with the imaginary axis. On the other hand, in all the above
subcases, the branches of two consecutive λ-dependent
eigenvalues meet the imaginary axis at the same point with λ
=
λ
c
r.
Namely, the transversality condition
is satisfied through two intersected lines at the same point of the imaginary
axis, but whose branches in the left (negative) and right (positive) half planes
belong to the 1st and 2nd pairs of eigenvalues, respectively.Finally, Figure13 verifies
the unexpected phenomenon (Kounadis [11]) of discontinuity in the dynamic loading λ
H associated with either a degenerate or a generic Hopf bifurcation.Plotsm versus λ
H for
a system with k
=
1 exhibiting a discontinuity for the case of (a) degenerate and
(b) generic Hopf bifurcation.
(a)
(b)


## 5. Concluding Remarks

This
study discusses in detail the coupling effect of (infinitesimal mainly) damping
with the mass and stiffness distribution in a 2-DOF cantilevered model under
step potential loading. Such an autonomous system may be associated either with
a positive semidefinite or indefinite damping matrix (with positive or negative
diagonal elements). Attention is focused on the violation of the Liénard-Chipart stability criterion
when this system is eitherunloaded or loaded by a suddenly applied load
of constant magnitude and direction with infinite duration (step potential
loading). The most important findings of this study are the following.(1) Usage of Liénard-Chipart, simple
and readily employed, stability criterion compared to that of Routh-Hurwitz brought
into light a variety of new types of dynamic bifurcations reported below.(2) The mass and stiffness distribution combined
either with a positive semidefinite or an indefinite damping matrix may have a
considerable effect on theasymptotic
stability, prior to divergence
instability, even in case of infinitesimal damping.(3) The cantilevered model whenunloaded (being statically stable),
strangely enough, under certain conditions becomes dynamically unstable to any
small disturbance leading to a divergent (unbounded) motion.(4) The above model whenloaded under analogous of the previous conditions exhibits also a divergent motion at a certain value of
the external load.It is worth noting that both the above cases of divergent motion may occur fornegligibly small negative determinant of
the damping (indefinite) matrix when α
1
=
0, while for
α
2
=
0
(regardless of whether
λ
=
0 or λ
≠
0), the determinant of the damping matrix is
negative but finite.(5) WhenΔ
3
=
0 and α
1
α
2
≠
0
, four distinct responses may occur as follows.(a)
If|
c
|
=
0 (positive semidefinite damping
matrix), the system exhibits a periodic motion associated with a degenerate Hopf bifurcation. Then, the
final amplitude of motion depends on the initial conditions.
(b)
When the damping matrix isindefinite (even with infinitesimal but
negative determinant, |
c
|
<
0), the system may exhibit a periodic attractor
response associated with a generic Hopf bifurcation. It is of paramount practical importance the case where such
an unexpected dynamic instability occurs at a load λ
H less than the 1st buckling load.
In both the above cases,
it was confirmed the recently reported unexpected finding [11] of discontinuity
of the dynamic loading λ
H (associated with either a
degenerate or a generic Hopf bifurcation) occurring at a certain value of the mass distribution.
(c)
When at the same timeλ
=
0, we
have a local dynamic instability for
|
c
|
≤
0, whose global stability can be established through a nonlinear dynamic analysis.
(d)
The case of adouble pure imaginary eigenvalue may occur for an indefinite damping matrix with finite determinant and negative ratio of the corresponding
diagonal elements. In this special case, there are two pairs of eigenvalues in the complex plane which touch the
imaginary axis at the same point for a certain value λ
=
λ
c
r. This
situation yields local instability leading to a motion with final constant
amplitude regardless of the initial conditions. Namely, such a dynamic
bifurcation behaves in a way similar to that of a generic Hopf bifurcation. This new type of dynamic bifurcation was
also verified via a nonlinear dynamic analysis.



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*Source: 471080-2008-05-26.xml*