Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 3, pp. 693-714, Warsaw 2008

NONLINEAR NORMAL MODES OF COUPLED

SELF-EXCITED OSCILLATORS IN REGULAR AND

CHAOTIC VIBRATION REGIMES

Jerzy Warmiński

Lublin University of Technology, Department of Applied Mechanics, Lublin, Poland

e-mail: j.warminski@pollub.pl

Vibration analysis of coupled self-excited nonlinear oscillators have be-
en studied in the paper. Possible regularmotion generated by nonlinear
damping has been determined by extracting Nonlinear Normal Modes
(NNM) from the model. Influence of the nonlinear terms and intensity
of self-excitation on the system response and vibrationmodes have been
presented. Parameters leading to chaotic motion have been found and
NonlinearNormalModes, whichmay appear nearby the chaotic respon-
se have been presented as well. The considered two degree of freedom
example shows that the autonomous system (without time dependent
excitation)may transit to chaotic vibrations if the systemposseses a po-
tential function with ”potential wells”. However, NNMs separated for a
very sensitive region close to chaotic vibrations, does not converge with
motion of the original system in the third approximation order.

Keywords: self-excitation,nonlinearvibrations,nonlinearnormalmodes,
chaotic motion

1. Introduction

Self-excited oscillations belong to a special class of vibrationswhichmay occur
without any external or internal periodic forcing. They appear due to specific
internal properties of a system. As a classical example of a self-excited system
we canmention: vibration of airplanewings (flutter), unwanted vibrations du-
ring machining processes (chatter), vibrations of a vehicle wheel (shimmy),
etc. Two main types of self-excitation are usually distinguished: (a) soft self-
excitation represented in the phase space by a stable limit cycle and, (b) hard



694 J. Warmiński

self-excitation represented by an unstable limit cycle. In the second type, de-
pending on initial conditions, the trajectory tends to an equilibrium point or
to infinity.Therefore, this type self-excitation is sometimes called catastrophic
(Warmiński, 2001). If the self-excitation interacts with other vibration types,
such as an external periodic force or parametric vibrations, many new une-
xpected phenomena can be observed (Szabelski and Warmiński, 1995, 1997;
Warmiński, 2003).

Themechanism of self-excitation can bemodelled by differential equations
with time delay terms or by introduction of nonlinear functions of the sta-
te space coordinates which can model this physical phenomenon. The second
approach is discussed in the present paper. The mathematical model of the
self-excited system does not include terms directly depending on time. The
dynamics is governed by autonomous, ordinary differential equations which
include nonlinear termsmodelling the self-excitation. Van der Pol’s or Rayle-
igh’smodels of self-excitation are incommonuse in literature.Theyare treated
as equivalent, what is not quite true if we consider other nonlinear terms inc-
luded in the model (Warmiński, 2001). Rayleigh’s model of self-excitation, as
a more suitable for mechanical systems, is considered in the paper.

Another importantproblemconnectedwith stronglynonlinear andcoupled
multi-degree-of-freedom systems concerns a proper extraction of vibrationmo-
des. If the system is linear, then its motion can be represented by a superpo-
sition of every single vibration mode.When motion of one particle is known,
thenmotion of the rest particles is also determined by a linear functional de-
pendence of the space coordinates. The linearmode separation can be applied
to linear and weakly viscously damped systems. The method of NNMs pre-
sented by Szemplińska-Stupnicka (1997, 1973) allows for construction of NNM
around resonances for externally forced systems.

However, if damping is large or is represented by nonlinear functions, then
the response of the systemmay depend not only on the displacement but also
on velocity (Rosenberg, 1960; Vakakis, 1997). In the case of a nonlinear auto-
nomous coupled self-excited system, the method of nonlinear normal modes
proposed by Shaw and Pierre (1993) seems to be very promising to its analy-
sis. Thismethod is suitable for strongly nonlinear and also velocity-dependent
models. To the author’s knowledge, there are no papers in the literature de-
voted to this topic. Some preliminary results on nonlinear normal modes of
self-excited systems were presented by Warmiński (2004, 2006) during two
conferences related to NNMs.

The purpose of this paper is to study motion of two coupled, nonlinear,
self-excited oscillators in a different parameter configuration by application of



Nonlinear normal modes of coupled self-excited oscillators... 695

nonlinear normal modes, and next to find their most essential properties in
the regular and chaotic regimes.

2. Model of the oscillatory system

The considered model is composed of two nonlinear oscillators coupled by a
linear spring f12(x1,x2). Each oscillator includes nonlinear stiffness, repre-
sented by nonlinear functions f1(x1) and f2(x2), respectively, and nonlinear
damping given by fd1(ẋ1), fd2(ẋ2).

Fig. 1. Physical model of the systemwith one (a) and two (b) nonlinear dampers

Assuming nonlinear stiffness of Duffing’s type and Rayleigh’s type of self-
excitation, the differential equations of motion take form

m1ẍ1+(−α1+β1ẋ
2
1)ẋ1+ δ1x1+γ1x

3
1+ δ12(x1−x2)= 0

(2.1)

m2ẍ2+(−α2+β2ẋ
2
2)ẋ2+ δ2x2+γ2x

3
2− δ12(x1−x2)= 0

where, fd1(ẋ1) = (−α1 +β1ẋ
2
1)ẋ1, fd2(ẋ2) = (−α2 +β2ẋ

2
2)ẋ2 are nonlinear

Rayleigh’s functions, and f1(x1)= δ1x1+γ1x
3
1, f2(x2)= δ2x2+γ2x

3
2 nonlinear

stiffness of Duffing’s type. Both oscillators are coupled by a linear springwith
the stiffness δ12. Set (2.1) consists of two nonlinear autonomous differential
equations without time in an explicit form. Assuming that α1 = 0, β1 = 0,
α2 = 0, β2 = 0, γ1 = 0, γ2 = 0, we get a linear conservative system with
natural frequencies determined by the formula

ω01,2 =

{
1

2
(δ1+ δ12)+M(δ12+ δ2)∓

(2.2)

∓
√
[(δ1+ δ12)+M(δ12+ δ2)]2−4M(δ1δ12+ δ1δ2+δ12δ2)

}1
2

where M =m1/m2.



696 J. Warmiński

Nevertheless, due to existing nonlinear terms which either depend on the
displacement or velocity, the decoupling of the system should take into acco-
unt both aspects: nonlinear stiffness and nonlinear damping. Therefore, the
nonlinear normal modes will be constructed for a strongly nonlinear system.

3. Non-linear normal modes

For the nonlinear normalmodes formulation the set of equations (2.1) is rew-
ritten in the form

ẋ1 = y1 ẏ1 = f1(x1,y1,x2,y2)

ẋ2 = y2 ẏ2 = f2(x1,y1,x2,y2)
(3.1)

where

f1 ≡ f1(x1,y1,x2,y2)=
1

m1
{−δ1x1− δ12(x1−x2)+ε[(α̃1− β̃1y

2
1)y1− γ̃1x

3
1]}

f2 ≡ f2(x1,y1,x2,y2)=
1

m2
{−δ2x2+ δ12(x1−x2)+ε[(α̃2− β̃2y

2
2)y2− γ̃2x

3
2]}

The coefficients of nonlinear terms are expressed as

α1 = εα̃1 β1 = εβ̃1

α2 = εα̃2 β2 = εβ̃2

where ε > 0 denotes a formal small parameter which allows for grouping of
all nonlinear terms.
The coordinates x1 and y1 of the first oscillator are chosen as master

coordinates, and they are denoted as

x1 =u y1 = v (3.2)

According to Shaw and Pierre (1993), the coordinates of the second oscillator
(slave coordinates) are expressed as functions of the master coordinates

x2 =X2(u,v) y2 =Y2(u,v) (3.3)

We assume that the functional dependencies X2(u,v) and Y2(u,v) exist, thus
the displacement and velocity of the second oscillator can be expressed by
the displacement and velocity of the first oscillator. The functions X2(u,v),



Nonlinear normal modes of coupled self-excited oscillators... 697

Y2(u,v) are constraint equations and they represent, the so called, modal sur-
faces. Their time derivatives can be found as

Ẋ2(u,v) =
∂X2
∂u
u̇+
∂X2
∂v
v̇ Ẏ2(u,v) =

∂Y2
∂u
u̇+
∂Y2
∂v
v̇ (3.4)

Expressing differential equations of motion (3.1) by the master-slave coordi-
nate notation yields

v= u̇ v̇= f1(u,v,X2(u,v),Y2(u,v))

ẋ2 ≡Y2(u,v) ẏ2 ≡ Ẏ2(u,v) = f2(u,v,X2(u,v),Y2(u,v))
(3.5)

and next, substituting the above equations into (3.4), we get

Y2(u,v)=
∂X2(u,v)

∂u
v+
∂X2(u,v)

∂v
f1(u,v,X2(u,v),Y2(u,v))

(3.6)

f2(u,v,X2(u,v),Y2(u,v)) =
∂Y2(u,v)

∂u
v+
∂Y2(u,v)

∂v
f1(u,v,X2(u,v),Y2(u,v))

Equations (3.6) determine themodal surfaces. However, bearing inmind that
the consideredmodel is nonlinear, receiving analytical solutions in the general
case can produce difficulties or can be even impossible. Therefore, assuming
that the oscillations take place around the zero equilibrium position, we can
expand the constraint functions in power series

X2(u,v) = a1u+a2v+a3u
2+a4uv+a5v

2+a6u
3+a7u

2v+

+a8uv
2+a9v

3+ . . .
(3.7)

Y2(u,v) = b1u+ b2v+ b3u
2+ b4uv+b5v

2+ b6u
3+ b7u

2v+

+b8uv
2+ b9v

3+ . . .

The expansion takes into account terms up to the cubic order, which is in
agreement with nonlinear functions included in Eq. (2.1). Time derivatives of
Eqs. (3.7) take forms

ẋ2 = Ẋ2(u,v) = a1u̇+a2v̇+2a3uu̇+a4uv̇+a4u̇v+2a5vv̇+3a6u
2u̇+

+2a7uu̇v+a7u
2v̇+a8u̇v

2+2a8uvv̇+3a9v
2v̇+ . . .

(3.8)

ẏ2 = Ẏ2(u,v)= b1u̇+ b2v̇+2b3uu̇+ b4uv̇+ b4u̇v+2b5vv̇+3b6u
2u̇

+2b7uu̇v+ b7u
2v̇+ b8u̇v

2+2b8uvv̇+3b9v
2v̇+ . . .



698 J. Warmiński

Considering the above expressions, the last two equations of (3.1) and going
back to the original notation, we get

y2 = a1y1+a2f1+2a3x1y1+a4x1f1+a4y
2
1 +2a5y1f1+3a6x

2
1y1+2a7x1y

2
1

+a7x
2
1f1+a8y

3
1 +2a8x1y1f1+3a9y

2
1f1+ . . .

(3.9)

f2 = b1y1+b2f1+2b3x1y1+ b4x1f1+ b4y
2
1 +2b5y1f1+3b6x

2
1y1+

+2b7x1y
2
1 + b7x

2
1f1+ b8y

3
1 +2b8x1y1f1+3b9y

2
1f1+ . . .

Next we substitute (3.7) and (3.8) into functions f1(x1,y1,x2,y2) =
= f1(u,v,X2(u,v),Y2(u,v)), f2(x1,y1,x2,y2) = f2(u,v,X2(u,v),Y2(u,v)) in
Eqs. (3.9).
From the last two equations of (3.1), it results that

y2− ẋ2 =0 f2(x1,y1,x2,y2)− ẏ2 =0 (3.10)

Thus, grouping the terms of (3.10) in a proper order with respect to the
master coordinates, we receive a set of two equations composed of the terms:
u, v, u2, uv, v2, u3, u2v, uv2, v3, . . . , v9. Equation (3.10) is satisfied for
non-trivial u and v, only if the coefficients near the mentioned terms are
equal to zero. It allows one to determine the unknown parameters a1,a2, . . .,
b1,b2, . . . of expansion (3.7). Taking into account that the highest order terms
are negligible, it has been decided to solve the problem up to the third order
of the coordinates u and v. Terms of higher order are truncated from the
expansion. Eventually, we get a set of eighteenth algebraic nonlinear equations
(nine for each of Eqs. (3.10)) with eighteen unknown parameters a1, . . . ,a9,
b1, . . . ,b9.
The parameters of modal surfaces are determined from equations:

— term u

b1+
a2
m1
(δ1+ δ12+a1δ12)= 0

(3.11)

b2
m1
(δ1+ δ12−a1δ12)+

1

m2
[δ12−a1(δ12− δ2)+εb1α̃2] = 0

— term v

−a1+ b2−
a2
m1
(a2δ12+εα̃1)= 0

(3.12)

−b1+
1

m1
(−a2b2δ12− b2εα̃1)+

1

m2
[a2(−δ12− δ2)+b2εα̃2] = 0



Nonlinear normal modes of coupled self-excited oscillators... 699

—term u2

b3+
1

m1
[−a2a3δ12−a1a4δ12+a4(δ1+ δ12)] = 0

(3.13)

1

m1
[−a3b2δ12−a1b4δ12+ b4(δ1+ δ12)]+

1

m2
[a3(−δ12− δ2)+ b3εα̃2] = 0

— term uv

−2a3+ b4+
1

m1
[−2a2a4δ12−2a1a5δ12+a5(2δ1+2δ12)−a4εα̃1] = 0

(3.14)

−2b3+
1

m1
[−a4b2δ12−a2b4δ12−2a1b5δ12+b5(2δ1+2δ12)− b4εα̃1]+

+
1

m2
[a4(−δ12− δ2)+ b4εα̃2] = 0

— term v2

−a4+ b5+
1

m1
(−3a2a5δ12−2a5εα̃1)= 0

(3.15)

−b4+
1

m1
(−a5b2δ12−2a2b5δ12−2b5εα̃1)+

1

m2
[a5(−δ12− δ2)+ b5εα̃2] = 0

— term u3

b6+
1

m1
[−a3a4δ12−a1a7δ12+a7(δ1+ δ12)+a2(−a6δ12+εγ̃1)] = 0

(3.16)

1

m1
[−a6b2δ12−a3b4δ12−a1b7δ12+ b7(δ1+ δ12)+ b2εγ̃1]+

+
1

m2
[a6(−δ12− δ2)+ b6εα̃2− b

3
1εβ̃2−a

3
1εγ̃2] = 0

— term u2v

−3a6+ b7+
1

m1
[−a24δ12−2a3a5δ12−2a2a7δ12−2a1a8δ12+

+a8(2δ1+2δ12)−a7εα̃1] = 0
(3.17)

−3b6+
1

m1
[−a7b2δ12−a4b4δ12−2a3b5δ12−a2b7δ12−2a1b8δ12+

+b8(2δ1+2δ12)− b7εα̃1]+

+
1

m2
[a7(−δ12− δ2)+ b7εα̃2−3b

2
1b2εβ̃2−3a

2
1a2εγ̃2] = 0



700 J. Warmiński

—term uv2

−2a7+ b8+

+
1

m1
[−3a4a5δ12−3a2a8δ12−3a1a9δ12+a9(3δ1+3δ12)−2a8εα1] = 0

(3.18)

−2b7+
1

m1
[−a8b2δ12−a5b4δ12−2a4b5δ12−2a2b8δ12−3a1b9δ12+

+b9(3δ1+3δ12)−2b8εα̃1]+

+
1

m2
[a8(−δ12− δ2)+ b8εα̃2−3b1b

2
2εβ̃2−3a1a

2
2εγ̃2] = 0

— term v3

−a8+b9+
1

m1
[−2a25δ12−3a9εα̃1+a2(−4a9δ12+εβ̃1)] = 0

(3.19)

−b8+
1

m1
[−a9b2δ12−2a5b5δ12−3a2b9δ12−3b9εα̃1+ b2εβ̃1]+

+
1

m2
[a9(−δ12− δ2)+ b9εα̃2−b

3
2εβ̃2−a

3
2εγ̃2] = 0

The parameters a1, a2, b1, b2 can be found independently of the first four
equations, while a3, a4, a5, b3, b4, b5 are equal to zero because of lack of
quadratic nonlinear terms in the original dynamic equations, the rest, i.e. a6,
a7, a8, a9, b6, b7, b8, b9 are found for the assumed numerical data.

As can be noticed in the next Section, in the numerical example, the set
of equations (3.11)-(3.19) gives two different real solutions which represent
the first and second nonlinear normal vibration modes. Substituting these
solutions into the first two equations (3.1), we obtain uncoupled nonlinear
differential equations for the first and secondmode, respectively

vi = u̇i v̇i = f1i(ui,vi,X2i(ui,vi),Y2i(ui,vi)) i=1,2 (3.20)

Taking into account that v̇i = üi, equations (3.20) take form:

— for the first mode

ü1+ω
2
01u1+(−αI +βIu̇

2)u̇1+(ηIu1+ρIu̇1)u1u̇1+γIu
3
1 =0 (3.21)

— for the secondmode

ü2+ω
2
02u2+(−αII +βIIu̇

2
2)u̇2+(ηIIu2+ρIIu̇2)u2u̇2+γIIu

3
2 =0 (3.22)



Nonlinear normal modes of coupled self-excited oscillators... 701

The coefficients ω01, ω02 represent natural frequencies of the linear system
and they are in full accordance with linear eigenvalues (2.2). The coefficients
αI, αII, βI, βII are called modal coefficients of Rayleigh’s self-excitation, γI,
γII, ηI, ηII, ρI, ρII –modal nonlinear terms of the first and second vibration
modes, respectively.

4. Numerical example of regular motion

Exemplary calculations have been done for following data (Warmiński, 2001):
— variant I – one self-excited damper

α1 =0.01 α2 =0 β1 =0.05 β2 =0

γ1 =0.1 γ2 =0.1 δ1 = δ2 =1 δ12 =0.3

m1 =1 m2 =2

(4.1)

— variant II – two self-excited dampers

α1 =α2 =0.01 β1 =β2 =0.05

δ1 = δ2 =1 δ12 =0.3

γ1 = γ2 =0.1 m1 =1 m2 =2

(4.2)

Natural frequencies calculated from (2.2) take values: ω01 = 0.766,
ω02 = 1.168, while the coefficients necessary for separation of the nonline-
ar modes a1, . . . ,a9, b1, . . . ,b9 defined in the previous section take values
(Table 1).
After applicationof thedecouplingprocedure,weget adifferential equation

for the two non-linear oscillators:
— variant I – one self-excited damper
(a) mode I

ü+0.5869u+(−0.000813+0.0163045u̇2)u̇+
(4.3)

+(−0.02387u+0.25684u̇)uu̇+0.2174u3 =0

(b) mode II

ü+1.3631u+(−0.01+0.04669u̇2)u̇+
(4.4)

−(0.00414u+0.00384u̇)uu̇+0.09418u3 =0



702 J. Warmiński

Table 1.Coefficients of nonlinear normal modes

Variant I

Mode I
a1 =2.377 a2 =−0.03062 a3 =0 a4 =0 a5 =0
a6 =−0.3926 a7 =0.0796 a8 =−0.8559 a9 =0.112318
b1 =0.017973 b2 =2.37693 b3 =0 b4 =0 b5 =0
b6 =−0.04004 b7 =−0.1697 b8 =−0.03216 b9 =−0.855365

Mode II
a1 =−0.2103 a2 =−0.00271 a3 =0 a4 =0 a5 =0
a6 =0.0194 a7 =0.0138 a8 =0.013 a9 =0.011
b1 =0.00369 b2 =−0.21033 b3 =0 b4 =0 b5 =0
b6 =−0.01854 b7 =0.0234 b8 =−0.0174 b9 =0.0132

Variant II

Mode I
a1 =2.377 a2 =−0.01531 a3 =0 a4 =0 a5 =0
a6 =−0.3926 a7 =−0.1376 a8 =−0.8559 a9 =−0.1843
b1 =0.00899 b2 =2.37693 b3 =0 b4 =0 b5 =0
b6 =0.0841 b7 =−0.1734 b8 =0.044 b9 =−0.8572

Mode II
a1 =−0.2103 a2 =−0.001354 a3 =0 a4 =0 a5 =0
a6 =0.0196 a7 =0.0133 a8 =0.013 a9 =0.011
b1 =0.00185 b2 =−0.21035 b3 =0 b4 =0 b5 =0
b6 =−0.018 b7 =0.0236 b8 =−0.0169 b9 =0.0134

—variant II – two self-excited dampers
(a) mode I

ü+0.5869u+(−0.00541+0.10531u̇2)u̇+
(4.5)

+(0.04129u+0.25676u̇)uu̇+0.2178u3 =0

(b) mode II

ü+1.3631u+(−0.01+0.0468u̇2)u̇−
(4.6)

−(0.00398u+0.00394u̇)uu̇+0.094113u3 =0

Equations (4.3), (4.4) and (4.5), (4.6) represent uncoupledmotion of two non-
linear oscillators for each nonlinearmode and for the first and second variant,
respectively. Note that the coefficients of the linear components u obtained
after transformation are equal to the square of the natural frequency determi-
ned in a classical way, (2.2), which confirms correctness of the transformation.



Nonlinear normal modes of coupled self-excited oscillators... 703

Comparing both numerical variants, we see that the equations have similar
nature (posses the same nonlinear terms), only values of the coefficients are
different.

To verify the analytical results, equations (4.3), (4.4) and adequately (4.5),
(4.6) are solved numerically and the results are comparedwith a direct nume-
rical simulation of original equations (3.1).

Because the system is self-excited, there is no special leading excitation
frequency related with natural frequencies of the system. Numerical analysis
shows that depending on initial conditions, the first or the secondmode of the
system can be activated. On the phase plane x1y1 (Fig.2a) we get two limit
cycleswhich represent the first and the secondnonlinearmode.Corresponding
basins of attraction (BA) of two possible solutions are presented in the phase
plane x2y2 in Fig.2b.

Fig. 2. Phase plane x1y1 with limit cycles (a) and basins of attraction in the phase
plane x2y2 (b) for the first and second vibrationmodes; model with one

self-excitation term (variant I, (4.1))

The time histories in Fig.3 represent motion of the system in physical
and normal coordinates. The normal coordinates have been obtained by nu-
merical integration and the inverse transformation of the physical coordinates
(x1,y1,x2,y2) in the coordinates (u1,v1,u2,v2). In Fig.3a, the first vibration
mode is activated – evidently synchronousmotion of bothmasses takes place.
After transformation to nonlinear normal modes, by using (4.3) and (4.4), we
get motion expressed only by the first normal mode u1, while the secondmo-
de u2 is close to zero (Fig.3b). The secondmode visible in Fig.3c corresponds
to anti-symmetric motion. After transformation to the normal coordinates



704 J. Warmiński

(Fig.3d), the motion is represented only by the second normal mode u2. The
first mode u1 is not activated.

Fig. 3. Time histories in physical and normal coordinates obtained for the first (a)
and the second (b) vibrationmode; model with one self-excitation term (variant I,

(4.1))

After application of the decoupling procedure, the vibrationmodes are re-
presented by two independent oscillators described by (4.3) and (4.4). Those
equations includenonlinear dampingandnonlinear stiffnesswhichplay impor-
tant role in the systemdynamics. This is themost essential difference between
the classical linearmode transformation and that presented in this paper.Mo-
tion of each nonlinear oscillator is represented by a limit cycle, which means
that NNMs fit well qualitatively to the original self-excited model.

In the secondvariant, (4.2), two self-exciteddampersare included(Fig.1b).
We can expect that, in the general case, two modes of the system can be
activated. Of course, for a specific set of parameters, also one-mode response
is possible, or when the system satisfies some symmetry conditions, so called
similar normal modes can be obtained. Then motion of the nonlinear system
is manifested in modes similar to its linear counterpart. Analysis of these
specific features are out of scope of this paper. In general, the systemwith two
self-excitations is composed of two nonlinear normal modes. Time histories
of physical coordinates obtained for such a case (data (4.2)) are presented in
Fig.4a and 4b.



Nonlinear normal modes of coupled self-excited oscillators... 705

Fig. 4. Time histories in physical coordinates (a), (b) and normal coordinates
obtained for the first (c) and the second (d) vibrationmode; model composed of two

self-excitation terms (variant II, (4.2))

After nonlinear transformation, by using the second set of coefficients (Ta-
ble 1) we get a modal response of the system. Two extracted normal modes
u1 and u2 are presented in Fig.4c and 4d. Themotion is very well separated.

The slave coordinates x2, y2 are related with the master ones u, v by
themodal functions X2(u,v), Y2(u,v). Thesemodal surfaces for the first and
secondmodes, respectively, are presented in Fig.5.

An exemplary trajectory which tends to a stable limit cycle and stays on
the modal surfaces is also plotted in this figure. Decoupling of motion on the
two self-excited oscillators is clearly visible. As results from the above figures,
the nonlinear modal surfaces strongly depend both on the displacement and
velocity. It strictly comes from the self-excitation terms which are nonlinear
functions of velocity. In the classical linear approach, the influence of velocity
is neglected.

To activate only one mode, we start from the initial conditions ut=0 = 1
and vt=0 = 0 which, after functional transformation, leads to the following
initial conditions of the original system: x10 = 1, y10 = 0, x20 = 1.98437,
y20 = 0.0931 for mode I, and x10 = 1, y10 = 0, x20 = −0.190714,
y20 =−0.01611 for mode II.

A trial of activation only one-mode vibrations by putting a proper ini-
tial condition does not lead to a single mode response of the system. At the



706 J. Warmiński

Fig. 5. Nonlinear modal surfaces; model with two self-excitations terms – variant II;
(a), (b) mode I, (c), (d) mode II

beginning, only one mode dominates (Fig.6), but due to a small numerical
perturbation, the second mode, in long time period, is also activated (Fig.7).
This is caused by the soft type of self-excitation, which means that the equ-
ilibrium point is unstable and the trajectories tend to a stable quasi-periodic
limit cycle represented by a torus in the phase space.

”Superposition” of the solutions obtained from both separated oscillators
should correspond to the result obtained from direct numerical simulation of
the original system.

In Fig.8a, we see a projection of a quasi-periodic torus on the phase plane
of the master coordinates, obtained from the direct numerical simulation of



Nonlinear normal modes of coupled self-excited oscillators... 707

Fig. 6. Time histories in normal coordinates obtained for the first (a) and the
second (b) vibrationmode; model with two self-excitation terms (variant II, (4.2)),

short time after activation

Fig. 7. Time histories in normal coordinates obtained for the first (a) and the
second (b) vibrationmodes, model with two self-excitation terms (variant II, (4.2)),

long time after activation

Fig. 8. Quasi periodic tori projected on the physical master coordinates x1, y1
obtained from the original system (a), and after superposition of motion of modal

oscillators (b); model with two self-excitation terms (variant II, (4.2))



708 J. Warmiński

original equations (3.1), and inFig.8b superposition ofmodal oscillators (4.5),
(4.6). Both results are qualitatively in a good agreement, but a quantitative
difference is visible. The difference comes from the approximation of nonli-
near functions of equations (3.9) by truncation terms higher than the third
order. For a strongly nonlinearmodel and large values of the displacement and
velocity, it leads to a quantitative difference.
The proposed technique by application of NNM allows for separation of

motion. It takes into account the influence of nonlinear terms with respect to
the displacement and velocity. However, the basic assumption of this approach
is that there exists a functional relation between the master and slave coor-
dinates. This condition can not be always satisfied. In the next Section, we
present more complicated dynamics of the considered system.

5. Chaotic response of the model

On the basis of the literature study, we can expect that a two degree of fre-
edom system, even without time-dependent excitation, may transit to chaotic
motion. In Warmiński (2001), the transition from regular to chaotic motion
was found for a system possessing a four-well potential function.
Let us consider a simpler variant of the model, i.e. variant I with one self-

excited damper. To get the required potential function, we assume a strongly
nonlinear systemcomposedof twononlinearoscillators havinganegative linear
partof the stiffness and largenonlinear terms.Numerical analysis of the system
is performed for the following data

α1 =0.01 α2 =0 β1 =0.05 β2 =0

γ1 =3 γ2 =3 δ1 = δ2 =−1 δ12 =0.3

m1 =1 m2 =2

(5.1)

Note that δ1, δ2 have negative values and γ1, γ2 are large. The potential
surface is determined by a function

V =
1

2
δ1(x

2
1+x

2
2)+
1

2
δ12(x1−x2)

2+
1

4
(γ1x

4
1+γ2x

4
2) (5.2)

The potential V (x1,x2) for the assumed data is plotted in Fig.9. This func-
tion has four local minima (”wells”) and one maximum at the origin of the
coordinate system. A cross-section of the surface for x2 = 0 is presented in
Fig.9b.



Nonlinear normal modes of coupled self-excited oscillators... 709

Fig. 9. Potential surface versus system physical coordinates (a) and the cross-section
for x2 =0 (b)

Coupling of both oscillators depends on the connecting spring stiffness,
which is denoted by the parameter δ12. Therefore, the potential surface cross-
sections are plotted for a few different coupling parameters (dashed lines in
Fig.9). The solid line denotes the function plotted for the main value of this
parameter δ12 =0.3. We see that for δ12 =1 the potential wells vanish.

To check the influence of this stiffness and its possible behaviour, the
Lyapunov exponents versus the bifurcation parameter δ12 are computed
(Fig.10). In the range of the bifurcation parameter δ12 ∈ (∼ 0.12,∼ 0.5),
themaximal Lyapunov exponent is positive, whichmeans that the response of
the system is chaotic. Poincarémaps for theparameter δ12 =0.3 are presented
in Fig.11.

Fig. 10. Lyapunov exponents versus bifurcation parameter δ12



710 J. Warmiński

Fig. 11. Strange chaotic attractor in x1y1 (a), and x2y2 (b) Poincaré sections;
model with one self-excitation term (variant (5.1))

Motion of the four-well potential system for assumed data (5.1) becomes
irregular, and its nature is represented by a strange chaotic attractor (Fig.11).
If we apply the decoupling strategy by introducing NNM, we may expect
to get two nonlinear modal oscillators which should give a chaotic response
of the system. These modal oscillators would be one-degree-of-freedom and
autonomous subsystems. Because uncoupled modal oscillators belong to 2D
phase space, therefore these systems can not produce chaotic motion. At least
a 3D phase spacemodel is required to perform chaotic motion. It allows us to
conclude that it is not possible to receive a chaotic response as a composition
(”superposition”) of two 2D nonlinear subsystems.
However, the methodology presented in Section 4 can be formally applied

for parameters assumed in (5.1). It means that out of the chaotic region,
motion of the slave coordinates x2, y2 can be related with the master ones
x1, y1.
Repeating the procedure presented in details in Section 4, we receive for

δ12 =0.6 (which corresponds to regular motion, see Fig.10), twomodal oscil-
lators:
—mode I

ü−0.7359u+(−0.00615+0.00866u̇2)u̇+(0.0284u+0.8661u̇)uu̇+2.23714u3 =0
(5.3)

—mode II

ü−0.1359u+(−0.00385−0.0459u̇2)u̇−(0.01915u+3.395u̇)uu̇+2.04518u3 =0
(5.4)



Nonlinear normal modes of coupled self-excited oscillators... 711

Both oscillators posses a negative linear part of the stiffness, which comes
from the assumed parameters. Direct numerical simulations shows that for
δ12 = 0.6 regular motion takes place and, depending on initial conditions,
three different attractors are obtained. These attractors, numbered by 1, 2, 3,
are presented in Fig.12 on Poincaré sections with respect to x1, y1 and x2,
y2 coordinates. The attractors show possiblemotion around a single potential
well (attractors 2 and 3) and they are shifted with respect to the equilibrium
point, or motion around two potential wells (attractor 1) with full symmetry
about the equilibrium position. It is worth to add that the trajectory reaches
the attractor after a long time of irregular motion, which represents a kind
of temporary chaos. This means that the system is very sensitive to initial
conditions and small perturbations of parameters.

Fig. 12. Attractors of the regular response in x1y1 (a) and x2y2 (b) Poincaré
sections; model with one self-excitation term, δ12 =0.6

The motion for the first and the second NNM, received after system de-
composition, is represented by equations (5.3) and (5.4). Solutions to these
equations in the phase plane are plotted in Fig.13.

As results from Fig.13a, the trajectory tends to the equilibrium position
located in the left potential well, while the second mode (Fig.13b) goes to
infinity. Thus, we can conclude that the third order approximation does not
represent the modal response of the system properly. The nonlinear terms,
whichhavebeen truncated in (3.7) over the thirdorder,mayplayan important
role in the modal decomposition of the considered, very sensitive, nonlinear
systemwith four potential wells.



712 J. Warmiński

Fig. 13. Modal response of the system for δ12 =0.6; (a) numerical solution of
Eq. (5.3) and (b) Eq. (5.4)

6. Conclusions

The analysis presented in the paper concerns dynamics of a nonlinear self-
excited system.TheNNMsareapplied todecouplemotionof the system.Using
a transformation which takes into account nonlinear stiffness and nonlinear
damping terms, the modal surfaces strongly dependent on the displacement
and velocity, are constructed. The proposed technique allows for decoupling
of the self-excitation generated by both oscillators. Two separate limit cycles
have been obtained. The results in the third order approximation are in a go-
od agreement if the system has a positively defined linear part of the stiffness
matrix.However, for the systemwith four potential wells, which is determined
by parameters with a negative linear part of the stiffness and strong nonline-
ar terms, the response can be regular or chaotic. Generally, in such a case,
the system is very sensitive to initial conditions and parameter perturbations.
The method of NNMs fails such a structure. If the response is regular, the
motion can be decoupled into twomodal oscillators. However, the third order
approximation does not satisfy the convergence of the modal solutions. Con-
struction of NNMs in a higher order approximation is theoretically possible
but from the practical point of view rather laborious. Separation of chaotic
motion on twomodal oscillators can not give a proper solution because any of
the separated subsystemsmay not produce a chaotic response due to too low
(2D) dimension of the oscillator.



Nonlinear normal modes of coupled self-excited oscillators... 713

Acknowledgments

This work has been partially supported by the EUgrantMTKD-CT-2004-014058

”Modern Composite Materials Applied in Aerospace, Civil and Mechanical Engine-

ering: Theoretical Modelling and Experimental Verification”.

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Nieliniowe postacie drgań sprzęgniętych oscylatorów samowzbudnych

w obszarach ruchu regularnego i chaotycznego

Streszczenie

W pracy przedstawiono analizę drgań sprzęgniętych nieliniowych oscylatorów sa-
mowzbudnych. Ruch regularny układu, generowany przez nieliniowe tłumienie, okre-
ślono poprzez zastosowanie nieliniowych postaci drgań. Zbadanowpływ członów nie-
liniowych i intensywność samowzbudzenia na odpowiedź układu oraz postacie drgań.
Określono parametry układu prowadzące do ruchu chaotycznego oraz nieliniowe po-
staciedrgańwystępującewpobliżu tegoobszaru.Stwierdzono, żeukładautonomiczny
(bez wymuszeń jawnie zależnych od czasu) o dwóch stopniach swobody,może przejść
do ruchu chaotycznego jeśli posiada funkcję potencjału z tzw. ”dołkami”. Jednak,
nieliniowe postacie drgańwyznaczone dla tego czułego regionu, w pobliżu chaosu, nie
są zgodne z wynikami otrzymanymi z bezpośredniej symulacji numerycznej.

Manuscript received January 9, 2008; accepted for print January 30, 2008