Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

2, 40, 2002

REGULAR AND CHAOTIC VIBRATIONS OF

VAN DER POL-MATHIEU OSCILLATOR WITH

NON-IDEAL ENERGY SOURCE

Jerzy Warmiński

Department of Applied Mechanics, Technical University of Lublin

email: jwar@archimedes.pol.lublin.pl

Vibrations of a parametrically and self-excited system with a non-ideal
source of energy have been analysed in this paper. Themodel has consi-
sted of the van der Pol-Mathieu oscillator and non-ideal energy source,
which characteristic has been assumed as a straight line. Regular mo-
tion of the complete system, energy source – vibrating model, near the
main parametric resonance, has been solved analytically by the apply-
ingKrilov-Bogolubov-Mitropolskimethod. Possible types ofmotion and
transition fromregular to chaoticmotionhavebeen investigatedbyusing
the Lyapunov’s exponent criterion and attractor topological structure
analysis.

Key words: non-ideal systems, chaos, parametric and self-excited vibra-
tions, synchronisation

1. Introduction

Interaction between parametric and self-excited systems leads to many
interesting phenomena.A special effect, which is called a synchronisation phe-
nomenon, near the main and the second order parametric resonances was
presented by Tondl (1978), Yano (1989). Regular vibrations of the van der
Pol-Mathieu oscillator with different types of non-linearities were considered
there. Parametric vibrations pulled in self-excited vibrations near parametric
resonances. Outside those regions the system vibrated quasi-periodically with
amodulation amplitude.Analysis of regular vibrations of aparametrically and
self-excited system with the Rayleigh self-excitation model was presented by
Szabelski andWarmiński (1995). The internal loop inside themain parametric



416 J.Warmiński

resonance, as a result of an external force influence, for one degree of freedom
systemwas there discovered.Warmiński (2001) examined regular and chaotic
vibrations of the van derPol-Mathieu oscillator as well. It was presented there
that an additional external force can cause important quantitative changes in
synchronisation and chaotic regions.

The consideredmodels weremainly assumed as ideal systems. In a certain
class ofmodels, called non-ideal, the excitation of a system can not be expres-
sed as a pure function of time, but as an equation that relates the source of
energy to the system. Hence, non-ideal systems always have one additional
degree of freedom in comparison with similar ideal systems. The fundamental
analysis of non-ideal system vibrations was presented by Kononenko (1969),
where the dynamics of a non-ideal energy source and vibrating linear or non-
linear models was presented.

The influence of limited power supply on the stick-slip phenomena for a
non-ideal self-excited vibratingmodelwas investigated byPontes et al. (2000).
Regular vibrations of linear and nonlinear systems with self, parametric and
external excitation and a non-ideal energy source were deeply investigated by
Alifov and Frolov (1985). A dry friction model of self-excitation and a linear
parametric excitationwere considered in thatpaper.Two stable solutionswith
similar amplitudes in the resonance area were obtained for the ideal, as well
for non-ideal source of energy. Regular vibrations near the main parametric
resonance for the Rayleigh-Mathieu model with a non-ideal source of energy
were analysed byWarmiński et al. (2001). Analytical solutions for thatmodel
were obtained by the using analytical Krilov-Bogolubov-Mitropolski method.

The aim of this paper is to analyse regular vibrations near themain para-
metric resonance of the van der Pol-Mathieu oscillator forced by a non-ideal
energy source and to present possible transition of the system from regular
to chaotic motion. The obtained results will be compared with the equivalent
ideal system.

2. Model of vibrating system

Let us consider the parametric and self-excited model, which includes a
direct current (DC) motor with limited power, operating on a structure. The
excitation of the system is limited by the characteristic of the energy source.
Vibration of the systemdepends onmotion of themotor and the energy source
motion depends on vibration of the system. Then, coupling of the vibrating



Regular and chaotic vibrations of ... 417

oscillator and the energy sourcemotions takes place. Hence, it is important to
analyse what happens to the motor, as the response of the system changes.

Let us take into account one degree of freedom model with a non-ideal
source of energy presented in Fig.1.

Fig. 1. Model of a non-ideal parametrically and self-excited system

The model consists of a nonlinear cubic spring with periodically chan-
ging stiffness, which represents parametric excitation of the system, and self-
excitation described by the function fd(x,ẋ). The presence of the unbalanced
mass m0 in the non-idealmodel alters the potential V and kinetic T energies
of the ideal and linear problem, then

V =
1

2
kx2+m0g(x+r sinφ)

(2.1)

T =
1

2
(I0+m0r

2)φ̇2+
1

2
m0ẋ

2+ ẋφ̇rm0cosφ+
1

2
mẋ2

where
I0 – moment of inertia of the rotor
r,m0 – radius andmass of the rotating element.

Using of the second kind of Lagrange’s equations, we derive differential
equations of motion of the linear model

(m+m0)ẍ+kx = m0rφ̇
2 sinφ−m0rφ̈cosφ

(2.2)

(I0+m0r
2)φ̈+H(φ̇)=L(φ̇)−m0rẍcosφ+m0grcosφ

where
H(φ̇) – resistance torque

L(φ̇) – engine torque.



418 J.Warmiński

Taking into account that m0grcosφ is a small term and introducing non-
liner terms into equations (2.2), according to the paper byWarmiński (2001),
we express differential equations ofmotion of the complete non-idealmodel in
the dimensionless form

Ẍ +Fd(X,Ẋ)+(1−µcos2φ)(1+γX
2)X = q1(φ̈cosφ− φ̇

2 sinφ)
(2.3)

φ̈ = q2ẍcosφ+Γ(φ̇) φ̇ = ω

where
τ – dimensionless time, τ = pt
p – natural frequency of the system, p =

√
k/m1

and

m1 = m+m0 Ẋ =
dX

dτ
Ẍ =

d2X

dτ2

φ̇ =
dφ

dτ
Γ(φ̇)=

L(φ̇)−G(φ̇)

m0r
2+ I0

q1 =
m0r

M +m0
q2 =

m0r

I0+m0r2

The nonlinear damping function, which represents the self-excitation is as-
sumed as a van der Pol function: Fd(X,Ẋ) = (−α +βX

2)Ẋ, the nonlinear
stiffness is expressed by the parameter γ, and the parametric excitation am-
plitude by µ. The parameters α, β, γ, µ, q1, q2 are small and positive. The
system is forced by a non-ideal engine (DC motor), which characteristic is
assumed as a linear function of the angular velocity (Balthazar et al., 1997)

Γ(φ̇)= u1−u2φ̇ (2.4)

where
u1 – control parameter depending on the voltage
u2 – parameter depends on a type of the energy source.

3. Asymptotic analysis

Let us consider vibrations of the system near the main parametric reso-
nance. Then the system vibrates with the frequency ω close to the natural
frequency p (p =1), while the parametric excitation frequency is 2ω.
Introducing a small parameter ε, we can write

p−ω = ε∆ (3.1)



Regular and chaotic vibrations of ... 419

Let us also assume that motion of the system takes place around a ste-
ady state. It means that the second derivative φ̈ is not large. Expressing the
parameters by a small parameter

α = εα̃ β = εβ̃ γ = εγ̃

µ = εµ̃ q1 = εq̃1 q2 = εq̃2
Γ(φ̇)= εΓ̃(φ̇)

the differential equations of motion take the form

Ẍ +p2X = ε
[
(α̃− β̃Ẋ2)Ẋ + µ̃X cos2φ− γ̃X3+

+µ̃γ̃X3cos2φ− q̃1φ̈cosφ+ q̃1φ̇
2 sinφ

(3.2)

φ̈ = ε
[
Γ̃(φ̇)− q̃2Ẍ cosφ

]

Next, following Kononenko (1969), equations (3.2)1 are transformed into new
coordinates A, φ, ψ, where

X = Acos(φ+ψ) Ẋ =−Apsin(φ+ψ) φ̇ = ω (3.3)

and the second derivative of X has the form

Ẍ =−
dA

dt
psin(φ+ψ)−

(
ω+

dψ

dt

)
Apcos(φ+ψ) (3.4)

Taking into account (3.2)2 and (3.3) we obtain

dA

dt
cos(φ+ψ)−

dψ

dt
Asin(φ+ψ)= (ω−p)Asin(φ+ψ) (3.5)

Substitution of equations (3.3)-(3.5) into differential equations ofmotion (3.2)
yields

dω

dt
= ε
[
Γ̃(ω)+Apq̃2ωcosφcos(φ+ψ)

]

dA

dt
= ε
{
−
q̃1ω
2

p
sinφ+A

(
α̃−
1

4
A2β̃
)
sin(φ+ψ)+

3

4

A3γ̃

p
cos(φ+ψ)−

−
Aµ̃

2p

(
1+
3

4
A2γ̃
)
cos(φ−ψ)−

1

4
A3β̃ sin3(φ+ψ)+

+
1

4

A3γ̃

p
cos3(φ+ψ)−

Aµ̃

2p

(
1+
3

4
A2γ̃
)
cos(3φ+ψ)− (3.6)



420 J.Warmiński

−
1

8

A3γ̃µ̃

p
cos(φ+3ψ)−

1

8

A3γ̃µ̃

p
cos(5φ+3ψ)

}
sin(φ+ψ)+ε2...

dψ

dt
= ε
{
∆−

q̃1ω
2

Ap
sinφ+

(
α̃−
1

4
A2β̃
)
sin(φ+ψ)−

1

4
A2β̃ sin3(φ+ψ)+

+
3

4

A2γ̃

p
cos(φ+ψ)+

1

4

A2γ̃

p
cos3(φ+ψ)−

1

2

µ̃

p
cos(φ−ψ)−

−
1

2

µ̃

p
cos(3φ+ψ)−

1

8

A2γ̃µ̃

p
cos(5φ+3ψ)

}
cos(φ+ψ)+ε2...

To solve equations (3.6) the classical Krilov-Bogolubov-Mitropolski method is
applied. According to this method, in the first approximation we can write

ω = Ω +εU1(φ,Ω,a,ξ)

A = a+εU2(φ,Ω,a,ξ) (3.7)

ψ = ξ+εU3(φ,Ω,a,ξ)

where Ui(φ,Ω,a,ξ), i =1,2,3 are slowly changing periodic functions of time.
To find solutions for Ω, a, ξ in the first approximation the right sides of
equations (3.6) are averaged

dΩ

dt
=

ε

2π

2π∫

0

fΩ dϕ
da

dt
=

ε

2π

2π∫

0

fa dϕ
dξ

dt
=

ε

2π

2π∫

0

fξ dϕ (3.8)

and after integration we obtain

dΩ

dt
= ε
[
Γ̃(Ω)−

1

2
apq̃2Ωcosξ

]

da

dt
= ε
[1
2
α̃a−

1

8
a3β̃ −

1

4
a
µ̃

p
sin2ξ −

1

8
a3
γ̃µ̃

p
sin2ξ −

1

2
q̃1
Ω2

p
cosξ
]

(3.9)

dξ

dt
= ε
[
∆+
3

8
a2
γ̃

p
−
1

4

µ̃

p
cos2ξ −

1

4

γ̃µ̃

p
cos2ξ+

1

2
q̃1
Ω2

ap
sinξ
]

dφ

dt
= Ω

For the steady state, equations (3.9) have the form

Γ̃(Ω)−
1

2
apq̃2Ωcosξ =0



Regular and chaotic vibrations of ... 421

1

2
α̃a−

1

8
a3β̃ −

1

4
a
µ̃

p
sin2ξ −

1

8
a3
γ̃µ̃

p
sin2ξ −

1

2
q̃1
Ω2

p
cosξ =0 (3.10)

∆+
3

8
a2
γ̃

p
−
1

4

µ̃

p
cos2ξ−

1

4

γ̃µ̃

p
cos2ξ −

1

2
q̃
Ω2

ap
sinξ =0

and then the angular velocity, amplitude and phase of the vibrating system
for the steady state can be obtained from the above equations.

4. Stability analysis

The stability analysis of the periodic solutions is performed by using ap-
proximate differential equations of the first order (3.9), which can be written
in a shortened form

dΩ

dt
= F1(a,Ω,ξ)

da

dt
= F2(a,Ω,ξ)

dξ

dt
= F3(a,Ω,ξ) (4.1)

In the steady state, equations (4.1) are equal to zero

F1(a,Ω,ξ)= 0 F2(a,Ω,ξ)= 0 F3(a,Ω,ξ) =0

Taking into account a difference between perturbed and not perturbed equ-
ations, the differential equations in variations are written as follows

dδΩ
dt
=
(∂F1
∂Ω

)

0
δΩ +

(∂F1
∂a

)

0
δa+

(∂F1
∂ξ

)
δξ

dδa
dt
=
(∂F2
∂Ω

)

0
δΩ +

(∂F2
∂a

)

0
δa+

(∂F2
∂ξ

)
δξ (4.2)

dδξ
dt
=
(∂F3
∂Ω

)

0
δΩ +

(∂F3
∂a

)

0
δa+

(∂F3
∂ξ

)
δξ

Characteristic determinant of equation (4.2) has the form

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(∂F1
∂Ω

)

0
δΩ −ρ

(∂F1
∂a

)

0
δa

(∂F1
∂ξ

)
δξ

(∂F2
∂Ω

)

0
δΩ

(∂F2
∂a

)

0
δa−ρ

(∂F2
∂ξ

)
δξ

(∂F3
∂Ω

)

0
δΩ

(∂F3
∂a

)

0
δa

(∂F3
∂ξ

)
δξ −ρ

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=0 (4.3)



422 J.Warmiński

Index ”0” denotes partial derivatives of F functions in the equilibrium
point. Stability of approximate solutions (3.9) depends on the roots of cha-
racteristic equation (4.3). The solutions are stable if the eigenvalues of the
characteristic equation have negative real parts.

5. Regular vibrations

Fig. 2. Vibrations near the main parametric resonance versus control parameter u1,
(a) amplitude, (b) angular velocity

Numerical results were carried out by using Matlab-Simulink R© package.
The calculations were done for the system parameters taken from the most
interesting intervals of physical systems, seeWarmiński (2001) and Balthazar
et al. (1997). Values of parameters are following

α =0.1 β =0.05 γ =0.1

µ =0.2 q1 =0.2 q2 =0.3
(5.1)

and the characteristic of the energy source was assumed as a straight line
(3.4). In our calculations the control parameter is assumed as u1 ∈ (0.5,8.0)
and u2 = 1.5. Taking into account approximate equations (3.10) and finding
numerically the roots of characteristic equation (4.3)we plotted the amplitude
andangular velocity versus the control parameter (Fig.2). Stable andunstable
analytical solutions (AR) are marked by solid and dashed lines respectively,



Regular and chaotic vibrations of ... 423

small squares denotes numerical results obtained byRunge-Kutta-Gillmethod
(RKG)with a variable integration step length. Increasing the control parame-
ter u1 in interval u1 ∈ (0.5,1.3), we observe quasi-periodic motion, which is
obtained by mutual interaction between the self-excitation, parametric exci-
tation and energy source (DC motor). This type of motion is shown in Fig.2
by two dots, corresponding to the extreme deflections of the system. In the
region u1 ∈ (1.3,6.75), kind of a synchronisation effect is observed. It is pre-
dominated by the parametric and external excitation. An interesting effect in
the synchronisation region is visible. There are two maxima, resulting from
strong interaction between the vibrating system and energy source (limited
power supply). At the beginning, the amplitude increases, then decreases and
jumps to the second part of the stable resonance curve and increases once
again. Going out from the synchronisation area, the system jumps from pe-
riodic to quasi-periodic motion and finally vibrates quasi-periodically. When
the parameter u1 is decreasing, the jumpphenomenon from the quasi-periodic
to periodic motion takes place around u1 =∼ 3.75. It means that in the in-
terval u1 ∈ (3.75,6.75), the periodic and quasi-periodic motion can appear,
depending on initial conditions.

Fig. 3. Amplitude versus angular velocity near the main parametric resonance

Similar effects connected with the angular velocity are shown in Fig.2b.
It is necessary to underline that, inside the synchronisation region, we also
observe a local minimum of the angular velocity Ω. The amplitude charac-
teristic versus angular velocity near the main parametric resonance obtained
fromasymptotic analysis is presented inFig.3. This resonance curve has not a



424 J.Warmiński

typical shape. Inside the synchronisation region a internal loop appears. It can
be explainedwhile observing the synchronisation region inFig.2a andFig.2b.
Comparing the amplitude and angular velocity characteristics, the local ma-
ximum for both curves is visible. It affects directly the shape of the amplitude
versus angular velocity curve (Fig.3).Theappearing internal loopwas also ob-
served for the idealmodel (Warmiński, 2001).Nevertheless, ideal andnon-ideal
loops have different features. For the ideal model the internal loop appears on
the right branch of the resonance curve and only its upper part is stable. For
the non-ideal van der Pol oscillator the loop appears on the left branch and
there are stable solutions, however a piece of the lower part is unstable. This
effect is caused by the interaction between the self and parametric excitations
and due to the influence of the non-ideal source of energy. This is a different
result than that obtained for the non-ideal Rayleigh oscillator where the loop
was completely stable, seeWarmiński et al. (2001).

Fig. 4. Transition through resonance with a slowly increasing value of the control
parameter u1

Additionally, approximate equations of motion (3.9) were modelled in the
Matlab-Simulink package and were solved by the RKG method. The control
parameter was slowly increased in time (Fig.4). The time axis has two scales:
the time scale and corresponding u1 scale. The simulation of the averaged
equations shows evidently quasi-periodic motion out of the resonance region
and periodic (synchronised)motion inside themain parametric resonance. Lo-
cal decreasing of the amplitude and angular velocity is also visible. Behaviour
of the system for decreasing parameter u1 presents Fig.5. Coming into the
synchronisation area takes place near t = 1250 (u1 = 3.75). The dynamic
hysteresis effect is then visible.



Regular and chaotic vibrations of ... 425

Fig. 5. Transition through resonance with a slowly increasing value of the control
parameter u1

Fig. 6. Transition through resonance with a slowly increasing value of the control
parameter u1, numerical simulation

Toconfirmthe analytical investigation, equations (3.2)weredirectly solved
numerically.Fig.6 andFig.7presentnumerical resultsobtained for thepassage
through resonance with slowly increasing and decreasing value of the control
parameter u1. In these figures one can notice that inside the resonance region,
t ∈ (200,820) – for increasing and t ∈ (280,850) – for decreasing values of
the control parameter u1, two maxima of the amplitude are visible. These
results are in agreement with the previous analytical results shown in Fig.2
andFig.4. Thedifference is in the synchronisationwidth.The synchronisation
area obtained from the direct simulation is smaller than that obtained from
the analytical approach. Nevertheless, the main behaviour is the same.



426 J.Warmiński

Fig. 7. Transition through resonance with a slowly decreasing value of the control
parameter u1, numerical simulation

6. Chaotic vibrations

The system of differential equations (2.3) was solved numerically by using
the Runge-Kutta forth order method and numerical results were carried out
by the Dynamics package (Nusse and York, 1994). All simulations were do-
ne on the DecAlpha 200 workstation. To classify different types of motion,
the maximal Lyapunov exponent criterion was applied. At the beginning the
Lyapunov exponent diagram versus control parameter u1 was calculated for
assumed parameters (5.1). Results are presented in Fig.8.

Fig. 8. Lyapunov exponents diagram versus control parameter u1



Regular and chaotic vibrations of ... 427

In the whole considered region u1 ∈ (0,6), the maximal Lyapunov expo-
nent is not greater than zero (has not positive sign). Itmeans that the system
vibrates regularly. Phase diagrams (Fig.9) demonstrates possible different ty-
pes of regular motion.

Fig. 9. Phase diagrams for chosen parameters u1, (a) u1 =1.25, (b) u1 =1.5,
(c) u1 =3.2, (d) u1 =3.5

The phase diagram in Fig.9a presents vibration of the non-ideal system
before the main parametric resonance. The obtained attractor represented
by a black area (”torus”) on the phase plane corresponds to quasi-periodic
motion of the system, out of the synchronisation region. For larger value of
u1 parameter, the parametric excitation frequency synchronises with the self-
excitation andwith rotationof themotor.Thenasynchronisationphenomenon
takes place. In the phase diagram (Fig.9b) periodic motion is observed. This
kind of vibration corresponds to the region u1 ∈ (∼ 1.3,∼ 1.95) in Fig.8.
Having gone out from this region the system vibrates quasi-periodically once
again. The second synchronisation effect occurs in a smaller interval around
u1 ∈ (∼ 3.15,∼ 3.45). Nevertheless, in that case of synchronisation, periodic
motionhas two times larger period than that in theprevious case.For u1 =3.5
the system is out of the second synchronisation region and vibrates quasi-
periodically (Fig.9d). Values of the Lyapunov exponents for the considered
phase diagrams are presented in Table 1.



428 J.Warmiński

Table 1. Lyapunov exponents for chosen u1 parameters

u1 attractor type λ1 λ2 λ3 λ4

1.25 quasi-periodic 0 0 −0.062 −1.465

1.5 periodic 0 −0.118 −0.824 −0.824

3.2 periodic 0 −0.027 −0.027 −1.640

3.5 quasi-periodic 0 0 −0.084 −1.588

Periodic motion of the system takes place if the maximal value of the
Lyapunov exponent is equal to zero. For quasi-periodicmotion two Lyapunov
exponents are equal to zero. This analysis shows that the system motion is
evidently regular.

Basing on the results obtained for the ideal system (Warmiński, 2001), we
canmention that the increase of the parametric excitation amplitude µ leads
to chaos. The transition from regular to chaotic motion of the ideal system
was possible for µ > 1.

Let us assume that in the considered non-idealmodel, the parametric exci-
tation slightly increases to value µ =0.4 and the rest of parameters save their
values. A diagramwith Lyapunov exponents for this case is plotted in Fig.10.

Fig. 10. Lyapunov exponents versus control parameter u1, µ =0.4

In Fig.10 we can recognise three regions where the maximal Lyapunov
exponent has small but positive value (marked by no. 1, 2, 3). It means,
that in these three intervals of u1 parameter, the system vibrates chaotically.
The next diagram (Fig.11), plotted for µ = 1.0, shows that those chaotic
regions increase significantly. Themaximal Lyapunov exponent achievesmuch
greater positive values and chaotic areas are wider. The first chaotic region



Regular and chaotic vibrations of ... 429

Fig. 11. Lyapunov exponents diagram control parameter u1, µ =1.0

appears before the synchronisation, for small values of the control parameter
u1 ∈ (0.6,0.95). Two additional chaotic regions occur for u1 ∈ (2.4,2.7) and
u1 ∈ (3.6,4.0). TheLyapunovexponent in the secondand third regions achieve
considerable bigger values than in the first one.
Each of the chaotic regions possesses small windows of regular motion.

Transition from regular to chaotic motion is presented in Poincare’s maps
(Fig.12). The regular attractor visible in Fig.12a obtained for u1 = 0.8,
transforms itself into a strange chaotic attractor with a characteristic frac-
tal structure for u1 = 0.82. In the first chaotic region regular motion can
also appear (Fig.12c). In Fig.12d a synchronisation effect takes place, only
periodic motion occurs there, mapped by a simple single attractor. From this
region, the system tends to the second chaotic region (Fig.12e). For u1 =2.8
the fractal structure gradually disappears (Fig.12f) and vibrations go to regu-
lar form (Fig.12g). The strange chaotic attractor in the third region presents
Fig.12h. After exceeding the limit value u1 = 4.0 the system makes an exit
from chaos to regular motion. Values of the Lyapunov exponents for the ana-
lysed Poincare’s maps are presented in Table 2.
For the parameters u1 = 0.82, u1 = 2.6, u1 = 2.8, u1 = 3.6 the maximal

Lyapunov exponent is positive. It confirms the results visible on Poincare’s
sections where the strange chaotic attractors with a fractal structure are plot-
ted.



430 J.Warmiński

Fig. 12. Poincare’s maps for different u1 parameter (a) u1 =0.8, (b) u1 =0.82,
(c) u1 =0.84, (d) u1 =2.2, (e) u1 =2.6, (f) u1 =2.8, (g) u1 =3.5, (h) u1 =3.6,

(i) u1 =4.0, (j) u1 =4.2



Regular and chaotic vibrations of ... 431

Table 2. Lyapunov exponents for chosen u1 parameters, µ =1.0

u1 attractor’s type λ1 λ2 λ3 λ4

0.8 periodic 0.0 −0.055 −0.442 −1.184

0.82 chaotic 0.04 0.0 −0.265 −1.425

0.84 periodic 0.0 −0.0025 −0.201 −1.472

2.2 periodic 0.0 −0.5823 −0.5823 −0.9533

2.6 chaotic 0.16 0.0 −0.657 −1.3632

2.8 chaotic 0.026 0.0 −0.603 −1.37

3.5 periodic 0.0 −0.211 −0.884 −1.347

3.6 chaotic 0.17 0.0 −0.882 −1.593

4.0 periodic 0.0 −0.012 −0.923 −1.419

4.2 periodic 0.0 −0.508 −0.508 −1.397

7. Conclusions and remarks

This paper completes an analysis of parametric and self-excited systems
taking into consideration regular and chaotic motions of a non-ideal system.
Applying the analytical Krilov-Bogolubov-Mitropolski method, it is presented
that near the main parametric resonance kind of a synchronisation effect ta-
kes place. Nevertheless, transition through the resonance region has different
features to the equivalent ideal system. In the synchronisation region, there
appear two maxima and one local minimum as well for the amplitude of the
vibrating system, as for the angular velocity of the motor. This effect causes
that the amplitude curve versus excitation frequency has an additional inter-
nal loop. The loop appears on the left branch of the curve and its lower part
is partially unstable. Outside the synchronisation region the system vibrates
quasi-periodically. This result is quite different in comparison with that ob-
tained for the ideal van der Pol-Mathieu oscillator where the loop appeared
on the right branch and its lower part was completely unstable. For a small
value of the parametric excitation the system can vibrate regularly. Then, it
is possible to get two types of regular motions: periodic and quasi-periodic. If
the parametric excitation increases, the response of the system complicates.
A small increase in the parametric excitation value, from µ =0.2 to µ =0.4,
causes occurrence of three chaotic regions. Further increase of the parametric
excitation leads to widening of the chaotic region, larger positive values of
the Lyapunov exponents and, consequently, to more complicated behaviour



432 J.Warmiński

of the system. Transitions from the quasi-periodic motion to synchronisation
areas and from regular to chaotic vibrations were presented by observing the
attractors topology in the phase plane andPoincare’s section aswell as by the
Lyapunov exponent criterion. For the ideal system transition to chaos was po-
ssible only for big values of the parametric excitation (µ > 1). The considered
non-ideal system demonstrates transition from regular to chaotic motion for
smaller values of the parametric excitation µ =0.4.

References

1. Alifov A.A., Frolov K.W., 1985, Interaction of Nonlinear Oscillatory Sys-
tems with Energy Sources (in Russian), Nauka,Moscow

2. BalthazarJ.M.,RenteM.L.,DaviV.M., etal., 1997,Someobservations
onnumerical simulations of a nonlinear non-ideal problem,nonlinear dynamics,
inChaos, Control, and Their Applications To Engineering Sciences, Edit. Bal-
thazar J.M.,MookD.T., Rosario J.M., Vol 1,AmericanAcademyofMechanics
and Associacao Brasileira de Ciencias Mecanicas, Brazil, ISBN 85-900351-1-5,
97-104

3. Kononenko V.O., 1969,Vibrating Systems with Limited Power Supply, Illife

4. NusseH.E., Yorke J.A., 1994,Dynamics: Numerical Explorations, Springer
Verlag, NewYork

5. Pontes B.R., Oliveira V.A., Balthazar J.M., 2000, On friction-driven
vibrations in a mass block-belt-motor driven with a limited power supply, Jo-
urnal of Sound and Vibration, 234, 4, 713-723

6. Szabelski K., Warmiński J., 1995a, Self-excited system vibration with pa-
rametric and external excitations, Journal of Sound and Vibration, 187, 4,
595-607

7. Szabelski K., Warmiński J., 1995b, The parametric self excited non-linear
system vibrations analysis with the inertial excitation, Journal of Non-Linear
Mechanics, 30, 2, 179-189

8. TondlA., 1978,On the interactionbetween self-excited andparametric vibra-
tions,Monographs andMemoranda, 25, NationalResearch Institute forMachi-
ne Design, Prague

9. Warmiński J., 2001, Synchronisation effects and chaos in vanderPol-Mathieu
oscillator, Journal of Theoretical and Applied Mechanics, 39, 4, 861-884

10. Warmiński J., Balthazar J.M., Brasil R.M.L.R.F., 2001, Vibrations of
non-ideal parametrically and self-excited model, Journal of Sound and Vibra-
tion, 245, 2, 363-374



Regular and chaotic vibrations of ... 433

11. Yano S., 1989, Considerations on self-and parametrically excited vibrational
systems, Ingenieur-Archiv, 59, 285-295

Drgania regularne i chaotyczne oscylatora van der Pola-Mathieu

z nieidealnym źródłem energii

Streszczenie

W pracy przeprowadzono analizę drgań układu parametryczno-samowzbudnego
z nieidealnym źródłem energii. Model składa się z oscylatora van der Pola-Mathieu
oraz nieidealnego źródła energii, którego charakterystykę założono w postaci linii
prostej. Drgania regularne całego systemu, źródło drgań – model drgający, zostały
określone w otoczeniu głównego rezonansu parametrycznego na drodze badań anali-
tycznych z zastosowaniemmetodyKrilova-Bogolubova-Mitropolskiego.Możliwe typy
ruchóworaz przejście układu do ruchu chaotycznegoprześledzono poprzez zastosowa-
nie kryteriumwykładnikaLapunowaoraz analizę struktury topologicznej atraktorów.

Manuscript received January 8, 2001; accepted for print November 14, 2001