Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

2, 40, 2002

NON-LINEAR DYNAMIC ANALYSIS OF AN AXIALY

MOVING VISCOELASTIC BEAM

Krzysztof Marynowski

Department of Machines Dynamics, Technical University of Łódź

email: kmarynow@ck-sg.p.lodz.pl

Stability and oscillation characteristics of an axially moving beam ha-
ve been investigated. Two different models of the beam material, i.e.,
Kelvin-Voigt and Maxwell have been considered. The numerical solu-
tions of full nonlinear and linearized equations have been compared.The
effects of axially travelling speed and the internal damping ondynamical
stability of the axially moving beam have been studied in details. Nu-
merical studies of the Kelvin-Voigt andMaxwell models show that both
models give similar results only for small values of the internal dam-
ping (dimensionless internal damping coefficient smaller than 5 ·10−6).
Formaterialswith larger damping coefficient the consideredmodels give
different results.

Key words: moving beam, internal damping, dynamic stability

Notation

A – cross section area of the beam
b – width of the beam
c – axial transport speed
cf – wave velocity
d – thickness of the beam
E – Young’s modulus of the beammaterial
J – cross section moment of inertia
k1,k2,k3 – dimensionless coefficients
l – length of the beam
M – bendingmoment



466 K.Marynowski

N – axial stress
P – tension force
r1,r2,r3 – dimensionless coefficients
s – dimensionless axial transport speed
t – time
w – transverse displacement of the beam
x,y – Cartesian co-ordinates
z – dimensionless transverse displacement of the beam
β – dimensionless internal damping coefficient
ε – strain component in the x direction
γ – internal damping coefficient
ε1,η,κ – dimensionless coefficients
ρ – mass density of the beam.

1. Introduction

Elastic continua translating at high speed such as band sawblades,magne-
tic types, paper webs, plastic sheets, films, transmission cables are present in
various industrial applications. Excessive vibrations of moving systems incre-
ase defects and can lead to failure of the translatingmaterials. The analysis of
vibration and dynamic stability of such systems are very important for design
of manufacturing devices.

In modeling axially moving materials one can use one-dimensional beam
theory (e.g.Wickert, 1992) or two-dimensional plate theory (e.g. Marynowski
and Kołakowski, 1999). Although the plate theory gives the most accurate
description of physical phenomena that occur in the web, it is very complica-
ted mathematically and requires time-consuming calculations. The previous
studies show that for a large class of practically important webs with a small
flexural stiffness the beam theory gives equally accurate results as the plate
theory (Marynowski, 1999).

The other important problem one can meet while considering axially mo-
ving webs is how to model the web material. A lot of earlier works in this
field focused on dynamic investigations of string-like and beam-like axially
moving isotropic systems (e.g.Wickert andMote, 1990; Wickert, 1993; Moon
andWickert, 1997). In all these works, the webmaterial was taken to be line-
arly elastic. However, paper webs, new plastics and compositematerials webs,
which are used in the industry need more realistic rheologic models. Many
investigators studied linear viscoelastic models. Kovalenko (1959) considered



Non-linear dynamic analysis of ... 467

the problem of a column of a constant stiffness with the internal damping li-
nearly proportional to the strain rate. Stevens (1966) considered the stability
of an initially straight, simply supported column subjected to an axial load
on the assumption that simple spring-dashpot models might adequately re-
present the column material. Fung et al. (1998) studied transverse vibrations
of an axially moving string subjected to an initial stress. The string material
was considered as the Kelvin-Voigt element in series with a spring.

In this paper twodifferent rheologicmodels of thematerial, namelyKelvin-
Voigt andMaxwell ones are used to describe the dynamics of the axially mo-
ving beam. Additionally, the results obtained from the analysis of the line-
arized equations are compared with the results of the integration of the full
non-linear equations.

2. Mathematical model of the moving beam

A viscoelastic axially moving beam of the length l is considered. The be-
am moves with the axial velocity c. The geometry of the system and the
introduced co-ordinates are shown in Fig.1.

Fig. 1. Axially moving beam

Theproblemof transverse oscillations of axiallymoving continua in a state
of a uniform initial stress has already been investigated. The results of earlier
studies of the axially moving band (Marynowski and Kołakowski, 1999) give
the following equation of the beammotion in the y direction

ρA(−w,tt−2cw,xt− c2w,xx)+Mx,xx+(Nxw,x),x =0 (2.1)

The uniform initial tension force P0 provides the required initial stress for the
material of the model. The nonlinear strain component in the x direction is



468 K.Marynowski

related to the displacement w by

ε(x,t)=
1

2
w2,x(x,t) (2.2)

Theone-dimensional constitutive equation of amaterial of thedifferential type
obeys the relation

Γσ=Ξε (2.3)

where Γ and Ξ are differential operators defined as

Γ =
R
∑

j=0

aj
dj

dtj
Ξ =

Q
∑

j=0

bj
dj

dtj
(2.4)

where aj, bj are constant coefficients.

2.1. Kelvin-Voigt model of the material

The Kelvin-Voigt rheologic model is shown in Fig.2a. In this case the
differential constitutive equation can be written as

a0σ= b1ε,t+b0ε (2.5)

where

a0 =1 b0 =E b1 = γ (2.6)

To obtain mathematical description of the viscoelastic beam model one
should multiply Eq. (2.1) with the operator Γ . The bending moment M is
given

M =−EJzw,xx−Jzγw,xxt (2.7)

Using Eqs (2.2), (2.5) and (2.6) one obtains

w,tt+2cw,xt+ c
2w,xx+

EJ

ρA
w,xxxx+

Jγ

ρA
w,xxxxt−

P0

ρA
w,xx−

3E

2ρ
w2,xw,xx−

(2.8)

−2
γ

ρ
(w,xw,xtw,xx+ cw,xw

2
,xx)−

γ

ρ
(w2,xw,xxt+ cw

2
,xw,xxx)= 0

The boundary conditions

w(0, t) =w(l,t) = 0 w,xx(0, t) =w,xx(l,t)= 0 (2.9)



Non-linear dynamic analysis of ... 469

Let the dimensionless parameters be

z=
w

d
ξ=
x

l
s=
c

cf
= c

√

Aρ

P0

τ = t
cf

l
=
t

l

√

P0

Aρ
cf =

√

P0

Aρ

(2.10)

Substitution of Eqs (2.10) into Eq. (2.8) gives a dimensionless nonlinear
equation of motion of the Kelvin-Voigt viscoelastic beam

z,ττ +2sz,ξτ +(s
2−1)z,ξξ +sz,ξ+ε1z,ξξξξ+βz,ξξξξτ −

3

2
κz2,ξz,ξξ−

(2.11)

−ηs(2z2,ξξz,ξ +z
2
,ξz,ξξξ)−η(2z,ξz,ξτz,ξξ +z

2
,ξz,ξξτ)= 0

where

β=
Jγ

l3
√
P0ρA

ε1 =
EJ

P0l
2

κ=
Ed2A

P0l
2

η=
γd2A

l3
√
P0ρA
(2.12)

Fig. 2. (a) Kelvin-Voigt and (b)Maxwell rheologic models

2.2. Maxwell rheologic model of the material

TheMaxwell rheologic model is shown in Fig.2b. In this case the differen-
tial constitutive equation can be written as

a0σ+a1σ,t = b1ε,t (2.13)

where

a0 =E a1 = γ b1 =Eγ (2.14)



470 K.Marynowski

To obtain mathematical description of the viscoelastic beam model one
should multiply Eq. (2.1) with the operator Γ and using Eqs (2.2) and (2.3)
one obtains

w,ttt+3cw,xtt+(3c
2− c2f)w,xxt+ c(c

2− c2f)w,xxx+r1w,tt+

+2r1cw,xt+r1(c
2− c2f)w,xx+r2w,xxxx = (2.15)

= r3(2w,xw,xtw,xx+2cw,xw
2
,xx+w

2
,xw,xxt+cw

2
,xw,xxx)

where

cf =

√

P0

Aρ
r1 =

E

γ
r2 =

E2J

Aργ
r3 =

E

ρ
(2.16)

The boundary conditions and the dimensionless parameters are given in
Eqs (2.9) and (2.10), respectively. Substitution of Eqs (2.10) into Eq. (2.15)
gives a dimensionless nonlinear equation ofmotion of theMaxwell viscoelastic
beam

z,τττ +k1z,ττ +3sz,ξττ +(3s
2−1)z,ξξτ +2sk1z,ξτ +

+k1(s
2−1)z,ξξ +s(s2−1)z,ξξξ +k2z,ξξξξ = (2.17)

= k3(2z,ξz,ξτz,ξξ +2sz
2
,ξξz,ξ +z

2
,ξz,ξτ +sz

2
,ξz,ξξξ)

where

k1 =
El

γcf
k2 =

E2J

P0lγcf
k3 =

d3AE

P0l
2

(2.18)

2.3. Solution to the problem

Theproblems represented byEq. (2.11) for theKelvin-Voigt rheologicmo-
del of the beam material and Eq. (2.17) for the Maxwell one together with
the boundary conditions given in Eqs (2.9) have been solved using the Ga-
lerkin method. The following finite series representation of the dimensionless
transverse displacement has been assumed

z(ξ,τ) =
n
∑

i=1

sin(iπξ)qi(τ) (2.19)

where qi(τ) is the generalized displacement.
Substituting Eq. (2.19) into Eqs (2.11), (2.17) and using the orthogonality

condition one determines a set of ordinary differential equations. Sets of the
ordinary equations are shown in Appendix for n=3.



Non-linear dynamic analysis of ... 471

3. Numerical results and discussion

Numerical investigations have been carried out for the beam model of
the steel web. Parameters data: length l = 1m, width b = 0.2m, thick-
ness d = 0.0015m, mass density ρ = 7800kg/m3, Young’s modulus: E =
0.2 · 1012N/m2, initial stress N0 = 2500N/m, n = 3. Initial conditions:
q1 = 1, q1,t = 0, ... q3,ttt = 0. The Runge-Kutta method was used to in-
tegrate the ordinary differential equations and analyse the dynamic behaviour
of the system.

3.1. Kelvin-Voigt model of material

Fig. 3. Phase portrait and time history of the solution of the linearized system
(A.1); (a) – s=0, β=10−4; (b) – s=1.4, β=10−4; (c) – s=1.41, β=10−5;

(d) – s=1.45, β=10−5

Atfirst, the linearized damped systemwas investigated. To show thedyna-
micbehaviourof the systemnaturaldampedoscillations of thefirstgeneralized
coordinate q1 for different values of the axial speed s of the beammodel were



472 K.Marynowski

investigated. In the subcritical region of transport speeds (s < scr) one can
observe free flexural damped vibrations around the trivial equilibrium posi-
tion (Fig.3a). In supercritical transport speeds (s > scr) for a small internal
damping the web experiences divergent instability (Fig.3b) and next flutter
instability (Fig.3d). Between these two instability regions there is a second
stability domain. The existence of the second stable region is dependent on
the internal damping of the web material. When the internal damping in-
creases the width of the second stable region decreases. The time history of
the first generalized coordinate q1 in the second stable region is shown in
Fig.3c. The location of the instability regions of the linearized system with
the Kelvin-Voigt model of an axially moving material is shown in Fig.4.

Fig. 4. Instability regions of the linearized systemwith the Kelvin-Voigt rheologic
model of the axially movingmaterial

Subsequently, the non-linear systemwith theKelvin-Voigt model ofmate-
rial was investigated. A bifurcation diagram of the non-linear system for the
internal damping coefficient β = 10−5 is shown in Fig.5. The dimensionless
transport speed shasbeenusedas thebifurcationparameter.One canobserve
supercritical bifurcation at the transport speed s = scr = 1.12. For s < scr
only one attractor exists (q1 =0) and for s> scr this critical point becomes
repeller and one can observe two attractors (non-zero critical points). The
phase portraits and time histories of the solutions of the non-linear system are
shown in Fig.6.
It is worth to note that the analysis of the non-linear systemdoes not indi-

cat the existance of various formsof instability regions of the linearized system.



Non-linear dynamic analysis of ... 473

Fig. 5. Bifurcation diagram of the nonlinear systemwith the Kelvin-Voigt rheologic
model of material (β=10−5)

Fig. 6. Phase portrait and time history of the solution of the nonlinear system (A.1);
(a) – s=1.3, β=10−4; (b) – s=1.54, β=0; (c) – s=1.54, β=10−5



474 K.Marynowski

Though the analysis of the linearized system predicts exponentially growing
oscillations in the divergence instability region of transport speeds, non-linear
damped oscillations which tend to the stable critical point occur (Fig.6a). At
transport speeds above the first divergence instability region of the linearized
system the undamped non-linear system experiences global motion between
two critical points (Fig.6b). For different values of the internal damping and
initial conditions the system may reach various equilibrium positions in the
supercritical transport speeds region (Fig.6c).

3.2. Maxwell model of material

Fig. 7. Instability regions of the linearized systemwith theMaxwell rheologic model
of the axially movingmaterial

The stability and instability regions calculated for the linearized system
(A.2) with the Maxwell rheologic model of the beam material are shown in
Fig.7. The results corresponding to the Maxwell model indicate that in the
range of supercritical transport speeds only for smaller values of internal dam-
ping coefficient (β < 5 · 10−6) the critical speed of the beam is the same
as the one obtained with the Kelvin-Voigt model (compare with Fig.4). The
Maxwell model does not confirm the existence of the second stability region
located between the divergence and the flutter instability regions. A phase
portrait and time history of oscillations in this region of transport speed are
shown in Fig.8a.



Non-linear dynamic analysis of ... 475

Fig. 8. Phase portrait and time history of the solution of the linearized system
(A.2); (a) – s=1.41, β=10−5; (b) – s=1.0, β=10−5; (c) – s=0, β=10−4;

(d) – s=1.4, β=10−4

For larger values of the internal damping the system loses its stability
due to the flutter instability. This is the significant difference between both
considered linearized models, as the Kelvin-Voigt model does not allow the
identification of this instability region.Thecritical valueof the transport speed
decreases with the increase of the damping coefficient β. The phase portraits
and time histories of the system response in both flutter instability regions are
shown in Fig.8b,c,d.

Next, thenon-linear systemwith theMaxwell rheologicmodel of theaxially
moving material was investigated. A bifurcation diagram of the non-linear
system for the internal damping coefficient β = 10−5 is shown in Fig.9. For
s< scr =0.5 only one attractor exists (q1 =0).For s> scr one can observe at
first the region of transport speeds where unbounded solutions occur. Above
this region the non-linear oscillations occur which are characterized by one
large limit cycle (region I in Fig.9). In the second divergence instability region



476 K.Marynowski

Fig. 9. Bifurcation diagram of the nonlinear systemwith theMaxwell rheologic
model of the axially movingmaterial (β=10−5, I – large limit cycle region,

II – small limit cycle region)

of the linearized system one can observe the non-linear oscillations which are
characterized by two small limit cycles (region II in Fig.9).

The phase portraits and time histories of the system response (the first
generalized co-ordinate q1) of the non-linear system for small values of the
internal damping co-efficient (β ¬ 10−5) are shown in Fig.10. At the critical
speed the non-linear system exhibits the flatter instability. The characteristic
system response in this region is shown in Fig.10a. Numerical studies of the
non-linear system show that in the supercritical range of the transport speed
the system exhibits the divergent instability (Fig.10b).With further increase
of the transport speed the large limit cycle oscillations around two equilibria
are developed (Fig.10c). This type of oscillations is similar to the one observed
in the Kelvin-Voigt model (Fig.6b). Phase portraits and time histories of the
system response showing the small stable limit cycle around two different
equilibria are shown in Fig.10d,e.

The phase portraits and time histories of the system response (the first
generalized co-ordinate q1) of the non-linear system (A.2) for larger values of
the internal damping co-efficient (β > 3·10−5) are shown inFig.11.Numerical
studies of the nonlinear system (A.2) show that in the subcritical range of the
transport speed s one observes damped natural oscillations (Fig.11a). At the
critical transport speed the system loses its stability by divergence. Fig.11b
shows thenon-linear systemresponse for s≈ scr.The increase of the transport
speed and the transition to the divergence instability region of the linearized
system (A.2) do not indicate a qualitative change in the response character in
comparison with the previous case with small damping.



Non-linear dynamic analysis of ... 477

Fig. 10. Phase portrait and time history of the solution of the nonlinear system
(A.2); (a) – s=1, β=8.9 ·10−6; (b) – s=1.12, β=10−5; (c) – s=1.65,
β=10−5; (d) – s=2, β=10−5, q1(0)=−1; (e) – s=2, β=10−5, q1(0)= 1



478 K.Marynowski

Fig. 11. Phase portrait and time history of the solution of the nonlinear system
(A.2); (a) – s=0.5, β=3.6 ·10−5; (b) – s=1.11, β=3.6 ·10−5

4. Conclusions

Dynamic investigations of an axially moving beam subject to a constant
axial stress are carried out in this paper. As the beam material models the
Kelvin-Voigt and Maxwell ones are considered. General forms of differential
equations of transverse oscillations of the systems are derived together with
the differential constitutive law for their rheologic models.
Numerical investigations have been carried out for the beam model of

a steel web. The analysis of the linearized equations with the Kelvin-Voigt
material model shows that in the subcritical range of the transport speed an
increase in this speed causes a decrease in the frequency of natural oscillations.
At the critical speed the system exhibits divergent instability. The analysis of
the linearizedMaxwell material model shows the system loses its stability due
to flutter instability. This is the significant difference between both considered
models, as the Kelvin-Voigt model does not allow the identification of this
instability region.
For supercritical transport speeds and small internal dampingboth lineari-

zedmodels showthat the systemexperiencesdivergent andflutter instabilities.
TheKelvin-Voigtmodel reveals that between these two instability regions the-
re is a second stability area. The width of this region depends on the internal
damping of the webmaterial.When the internal damping increases the width
of the second stable region decreases more and more and finally disappears.
The Maxwell model does not confirm the existence of the second stability
region of transport speed.



Non-linear dynamic analysis of ... 479

The dynamic analysis of the non-linear damped system undergoing con-
stant axial stress shows that in the supercritical transport speed region non-
trivial equilibrium positions bifurcate from the straight configuration of the
web, and global motion between the co-existing equilibrium positions occurs.
At the same transport speed, for different values of the internal damping and
initial conditions, the system may reach various equilibrium positions. Insi-
de the instability regions one can observe different dynamical behaviour de-
pending on the considered model. The nonlinear Kelvin-Voigt model shows
existence of a stable limit cycle in both regions of the divergence instability
while thenonlinearMaxwellmodel indicates suchbehaviour only in the second
region.
TheKelvin-Voigt andMaxwellmodelsgivedynamically similar results only

for small values of the internal damping (β < 5 ·10−5). As the experimental
estimation of the internal damping in the steel web indicates larger values of β
(Osiński, 1997), the description of the dynamical behaviour of such a system
requires experimental verification of both considered models.

Acknowledgement

This paper was supported by the State Committee for Scientific Research (KBN)

under grant No. 7T08E03417.

A. Appendix

The set of ordinary differential equations of the viscoelastic beam model
with the Kelvin-Voigt model of the material (n=3)

q̈1 = (s
2−1)π2q1−ε1π4q1+

16

3
sq̇2−βπ4q̇1−

− a1
(3

8
q31 +3q1q

2
2 +
27

4
q1q
2
3 +
9

8
q21q3+

9

2
q22q3

)

+

+ a2s
(848

21
q1q2q3+

2992

35
q2q
2
3 +
112

15
q21q2+

1408

105
q32

)

−

− a2
(3

4
q21q̇1+

3

4
q21q̇3+

3

2
q1q̇1q3+3q

2
2q̇3+

+ 6q2q̇2q3+4q1q2q̇2+2q̇1q
2
2 +
9

2
q̇1q
2
3 +9q1q3q̇3

)

q̈2 = 4(s
2−1)π2q2−16ε1π4q2−

16

3
sq̇1+

48

5
sq̇3−16βπ4q̇2−



480 K.Marynowski

− a1
(

3q21q2+6q
3
2 +27q2q

2
3 +9q1q2q3

)

+

+ a2s
( 8

15
q31 +
44712

385
q33 +
1952

105
q1q
2
2 +
1016

35
q1q
2
3 +
936

35
q21q3+

1568

15
q22q3

)

−
(A.1)

− a2
(

6q1q2q̇3+6q1q̇2q3+12q
2
2q̇2+6q̇1q2q3+

+ 4q1q̇1q2+36q2q3q̇3−
27

2
q1q3q̇3+2q

2
1q̇2+18q

2
3q̇2

)

q̈3 = 9(s
2−1)π2q3−81εzπ4q3−

48

5
sq̇2−81βπ4q̇3−

− a1
(3

8
q31 +
243

8
q33 +
9

2
q1q
2
2 +
27

4
q21q3+27q

2
2q3

)

+

+ a2s
(

−
10656

105
q1q2q3+

78192

385
q2q
2
3 +
144

35
q21q2−

128

15
q32

)

−

− a2
(3

4
q21q̇1+

9

2
q21q̇3+4q1q̇1q2+

243

4
q23q̇3+

+ 9q1q̇1q3+2q1q2q̇2+3q̇1q
2
2 +18q

2
2q̇3+36q2q̇2q3

)

where

β=
Jγ

l
√
P0ρA

ε1 =
EJ

P0l
2

a1 =
Ed2Aπ4

P0l
2

a2 =
γd2Azπ

4

l3
√
P0ρA

The set of ordinary differential equations of the viscoelastic beam model
with theMaxwell model of the material (n=3)

...
q
1 = −k1q̈1+8sq̈2−π2(1−3s2)q̇1+

16

3
k1sq̇2−π2[k1(1−s2)+π2k2]q1+

+
32

3
π2s(1−s2)q2−2k3π4

(3

8
q21q̇1+

3

4
q1q3q̇1+3q2q3q̇2+

+
3

8
q21q̇3+

3

2
q22q̇3+2q1q2q̇2+

9

2
q1q3q̇3+q

2
2q̇1+

9

4
q23q̇1

)

+

+ 2k3sπ
5
(

2.134q32 +1.189q
2
1q2+6.543q1q2q3+40.745q2q

2
3

)

...
q
2 = −k1q̈2−8sq̈1−4π2(1−3s2)q̇2−

16

3
k1sq̇1−4π2[k1(1−s2)+

+ 4π2k2]q2+
72

5
sq̈3+

48

5
k1sq̇3−

8

3
π2s(1−s2)q1+

216

5
π2s(1−s2)q3−



Non-linear dynamic analysis of ... 481

− 2k3π2
(

6q22q̇2+2q1q2q̇1+3q2q3q̇1+9q1q3q̇2+ q
2
1q̇2+9q

2
3q̇2+3q1q2q̇3+

(A.2)

+ 18q2q3q̇3
)

+2k3sπ
5
(

0.084q31 +18.492q
3
3 +2.985q1q

2
2 +4.636q1q

2
3 +

+ 4.257q21q3+16.643q
2
2q3

)

...
q
3 = −k1q̈3−

72

5
sq̈2−9π2(1−3s2)q̇3−

48

5
k1sq̇2−

96

5
π2s(1−s2)q2−

− 9π2[k1(1−s2)+9π2k2]q3−2k3π4
(3

8
q21q̇1+

9

2
q1q3q̇1+3q1q2q̇2+

+
3

2
q22q̇1+

243

8
q23q̇3+18q2q3q̇2+9q

2
2q̇3+

9

4
q21q̇3

)

+

+ 2k3sπ
5
(

−1.357q32 +0.655q
2
1q2+16.151q1q2q3−62.315q2q

2
3

)

where

k1 =
El

γcf
k2 =

E2J

P0lγcf
k3 =

d2AE

P0l
2

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Nieliniowa analiza dynamiki przesuwającej się osiowo wiskoelastycznej

belki

Streszczenie

W pracy badano stateczność dynamiczną ruchu oraz drgania przesuwającej się
osiowo belki. Do opisu własności materiału belki zastosowano dwa modele reologicz-
ne: model Kelvina-Voigta oraz model Maxwella. Dla obu badanych modeli wypro-
wadzono nieliniowe równania różniczkowe o pochodnych cząstkowych opisujące ruch
poprzeczny belki. Przybliżone rozwiązanie równań ruchu otrzymano stosując metodę
Galerkina.Badanianumeryczneprzeprowadzonodlamodelubelkowegoprzesuwającej
się osiowo cienkiej wstęgi stalowej. Badanowpływprędkości przesuwu oraz tłumienia
wewnętrznego na stateczność dynamiczną układu. Wyniki badań wskazują, że tylko
przy małych wartościach bezwymiarowego współczynnika tłumienia wewnętrznego
β < 5 · 10−6 układy z obydwoma badanymi modelami reologicznymi charakteryzują
się podobnym zachowaniem dynamicznym. Przy wyższym tłumieniu wewnętrznym
otrzymano różniące się wyniki badań dynamiki zarówno układu zlinearyzowanego,
jak i układu nieliniowego.

Manuscript received May 28, 2001; accepted for print October 1, 2001