Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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5 Complete Bifurcation Behaviors of a Henon Map 47
of the stable and unstable periodic solutions with respect to the positive and negative
mapping structures were analyzed. A complete picture of the periodic solutions of
positive and negative mappings is given; the positive and negative mappings are
in a pair. The Poincare mapping sections of the Neimark bifurcation of periodic
solutions were illustrated, and the chaotic layers for the discrete system with the
Henon map were observed.
References
1. Henon M (1976) A two-dimensional mapping with a strange attractor. Commun Math Phys
50:69–77
2. Marotto FR (1979) Chaotic behavior in the Henon mapping. Commun Math Phys 68:187–194
3. Curry JH (1979) On the Henon transformation. Commun Math Phys 68:129–140
4. Cvitanovic P, Gunaratne GH, Procaccia I (1988) Topological and metric properties of Henon-
type strange attractors. Phys Rev A 38(3):1503–1520
5. Gallas JAC (1993) Structure of the parameter space of the Henon map. Phys Rev Lett
70(18):2714–2717
6. Zhusubaliyev ZT, Rudakov VN, Soukhoterin EA, Mosekilde E (2000) Bifurcation analysis of
the Henon map. Discrete Dyn Nat Soc 5:203–221
7. Gonchenko SV, Meiss JD, Ovsyannikov II (2006) Chaotic dynamics of three-dimensional
Henon maps that originate from a homoclinic bifurcation. Regul Chaotic Dyn 11(2):191–212
8. Hruska SL (2006) Rigorous numerical models for the dynamics of complex Henon mappings
on their chain recurrent sets. Discrete Continuous Dyn Syst 15(2):529–558
9. Gonchenko SV, Gonchenko VS, Tatjer JC (2007) Bifurcations of three-dimensional diffeomor-
phisms with non-simple quadratic homoclinic tangencies and generalized Henon maps. Regul
Chaotic Dyn 12(3):233–266
10. Lorenz EN (2008) Compound windows of the Henon-map. Physica D 237:1689–1704
11. Luo ACJ, Han RPS (1992) Period doubling and multifractals in 1-D iterative maps. Chaos
Solitons Fractals 2(3):335–348
12. Luo ACJ (2005) The mapping dynamics of periodic motions for a three-piecewise linear sys-
tem under a periodic excitation. J Sound Vib 283:723–748
Chapter 6
Study on the Performance
of a Two-Degree-of-Freedom Chaotic
Vibration Isolation System
Jing-Jun Lou, Ying-Chun Wang, and Shi-Jian Zhu
Abstract The nonlinear dynamics and the vibration isolation effectiveness of a
two-degree-of-freedom nonlinear vibration isolation system are numerically stud-
ied. The complex nonlinear behavior in the force-frequency plane of the system
is analyzed. Cascades of bifurcation of the system with different excitation ampli-
tude are also obtained. The power flow transmissibility is analyzed to validate the
performance of the system in vibration isolation. The numerical results show that
the reduction of the line spectra when the system is chaotic is much greater than
that when the system is nonchaotic, and that the overall effectiveness of vibration
isolation at chaos is better than that at nonchaos.
6.1 Introduction
More than three decades of intense studies of nonlinear dynamics have shown that
chaos occurs widely in engineering and natural systems. Historically, it has usually
been regarded as a nuisance and designed out if possible. It has been noted only
as irregular or unpredictable behavior and often attributed to random external influ-
ences. Further studies showed that chaotic phenomena are completely deterministic
and characteristic for typical nonlinear systems.
These studies posed questions about the practical application of chaos. One of
the possible answers is to control chaotic behavior in such a way as to make it pre-
dictable. Recently, there have been examples of the potential usefulness of chaotic
behavior, and this has caused engineers and applied scientists to become more
interested in chaos [1].
It was proposed by the authors the method using the chaotic regime of the
nonlinear vibration isolation system for reduction of line spectrum of radiated noise
J.-J. Lou (
)
College of Naval Architecture and Power, Naval University of Engineering, Wuhan 430033,
People’s Republic of China
e-mail: jingjun
lou@hotmail.com
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
6,
c
Springer Science+Business Media, LLC 2010
49
50 J.-J. Lou et al.
from a marine vessel in [2]. Experimental chaos in nonlinear vibration isolation
system was observed, and the possible practical application of the chaos method
in line spectra reduction was confirmed in [3]. In this work, the method of chaotic
vibration isolation for reduction of line spectrum is numerically studied.
As is well known, noise spectra of the radiated waterborne noise of surface and
under-surface ships are generally in two categories. One is the broad-band noise hav-
ing a continuous spectrum. The other is the line spectrum which contains lines at
discrete frequencies. The nature of the radiated noise spectra changes as the naviga-
tion speed changes. At high speed, the signature is dominated by broad-band noise,
while at low speed the signature is dominated by line spectrum, and the machinery
is the leading noisemaker.
Insertion of resilient isolators between the machinery and the base is one of the
most common methods for controlling unwanted vibration. Isolators in service are
usually assumed to be linear and the performance characteristics of isolators under
the assumption of linearity have been widely reported [4, 5]. The linear vibration
isolation system has vibration attenuation within a rather wide frequency range. But
its ability in line spectrum reduction is limited.
Thus it becomes important to include nonlinearity presented in practical isola-
tors. Because of the limitations of the linear vibration isolators, nonlinear isolators
were studied in some literatures. However, the investigation was constrained to the
periodic vibration [6,7]. In spite of the achievement in the application of the chaotic
vibration mechanics to chaotic vibratory rollers [8], it is neglected that vibration
excitation and vibration isolation are two sides in the field of vibration engineering.
And no efforts are made to make use of chaos in vibration isolation. The authors
tried to utilize the characteristics of chaos to vibration isolation, and a method of
chaotic vibration isolation was advanced for machinery vibration control and line
spectra reduction [2]. When chaos takes place in a nonlinear vibration isolation
system under a single frequency excitation, the line spectrum of response at the
excitation frequency grows into a broad-band one. Therefore, the frequency config-
uration of the radiated noise is altered. What is more important, the concentrated
energy spreads from the excitation frequency to a broad-band frequency range. To
validate the effectiveness of the method of chaotic vibration isolation, the dynamics
and the power flow transmissibility of the two-degree-of-freedom nonlinear vibra-
tion isolation system is studied in the present work.
6.2 Model of the Two-Degree-of-Freedom Nonlinear
Vibration Isolation System
A majority of stationary equipments are installed by means of vibration isolators
attached to rigid supporting structures. There have been numerous publications that
used a single-degree-of-freedom (SDOF) system to model both active and passive
vibration isolation systems [9, 10]. However, some stationary equipments requiring
vibration isolation are installed on nonrigid structures such as higher floors of
6 Two-Degree-of-Freedom Chaotic Vibration Isolation System 51
Fig. 6.1 Two-degree-of-
freedom vibration isolation
model
F
0
cosWT
M
1
M
2
X
1
X
2
C
1
C
2
K
1
,K
3
K
2
multistory buildings. This problem also becomes critically important for objects
installed in vehicles, such as car engines, machinery on surface and under-surface
ships, etc. In these cases, the mass of the isolated object is substantially greater than
the “effective mass” of the supporting structure [11].
As we all know, the equipment on a flexible base has six degrees of freedom.
If the translational vibration in the vertical direction is considered only the isolation
system with flexible base can be simplified as a two-degree-of-freedom system as
showninFig.6.1. In this work, the base is simplified as a single-degree-of-freedom
lumped system, namely, a dynamical model of a combination of one equivalent
mass, one equivalent stiffness, and one equivalent damper. In Fig. 6.1, M
1
is the
mass of the isolated equipment, M
2
is the mass of the base, X
1
is the displacement of
the isolated equipment, and X
2
is the displacement of the base. The elastic support
of the base mass is assumed to be linear, and the stiffness and damping coefficient
are K
2
and C
2
respectively. The isolator is of linear damping C
1
and cubic hardening
behavior with the first order term K
1
and the third order term K
3
. The excitation
force F
0
cos ˝T is a cosine function of the time T ,and˝ is a single, constant, and
known input frequency.
From Newtonian mechanics, the differential equations of motion of the system
in Fig. 6.1 are
(
M
1
R
X
1
CC
1
.
P
X
1
P
X
2
/ C K
1
.X
1
X
2
/CK
3
.X
1
X
2
/
3
D F
0
cos T CM
1
g
M
2
R
X
2
C
1
.
P
X
1
P
X
2
/ K
1
.X
1
X
2
/K
3
.X
1
X
2
/
3
C C
2
P
X
2
CK
2
X
2
D 0
(6.1)
For K
1
;K
3
¤ 0, by introducing the dimensionless quantities x
1
DX
1
p
K
3
=K
1
,
x
2
D X
2
p
K
3
=K
1
;!
1n
D
p
K
1
=M
1
,andt D T!
1n
, we put the differential
equations of motion in a dimensionless form:
dX
1
dT
D !
1n
p
K
1
=K
3
dx
1
dt
;
dX
2
dT
D !
1n
p
K
1
=K
3
dx
2
dt
(6.2)
52 J.-J. Lou et al.
d
2
X
1
dT
2
D !
2
1n
p
K
1
=K
3
d
2
x
1
dt
2
;
d
2
X
2
dT
2
D !
2
1n
p
K
1
=K
3
d
2
x
2
dt
2
(6.3)
Then, (6.1) grows into a dimensionless form
8
<
:
Rx
1
C
1
. Px
1
Px
2
/ C .x
1
x
2
/ C .x
1
x
2
/
3
D f cos !t C G
Rx
2
u
1
. Px
1
Px
2
/ u.x
1
x
2
/ u.x
1
x
2
/
3
C
2
Px
2
C u
K
2
K
1
x
2
D 0
(6.4)
where
1
D C
1
.
p
M
1
K
1
;fD
s
K
3
K
3
1
F
0
;!D
!
1n
;GD
s
K
3
K
3
1
M
1
g;
u D
M
1
M
2
;
2
D u
C
2
C
1
1
(6.5)
6.3 Dynamical Analysis
In order to discuss the influence of the excitation amplitude f and excitation fre-
quency ! on the dynamics of system (6.4), let
1
D 0:02;
2
D 0:2; u D 2; K
2
=K
1
D 100
As mentioned previously, for cases on high floors or on surface and under-surface
ships, the mass of the isolated object is substantially greater that the “effective mass”
of the supporting structure, and hence u D 2 is selected.
In order to analyze the influence of the excitation frequency ! on the dynamics of
system (6.4), the global bifurcation with regard to the change of the excitation fre-
quency with the excitation amplitude fixed is analyzed, using the Poincar´e section
method. The degree of periodicity of the response has been declared mainly on the
basis of the Poincar´e map, but quantitative dynamic measures – such as the fre-
quency response spectrum, the Lyapunov exponents, and the fractal dimensions –
have been calculated in many specific situations, too.
The global bifurcation diagram with f D 4:0 is shown in Fig. 6.2. Decreasing
the excitation frequency from 5 to 0.8 with the step-size equal to 0.01, two cascades
of period-doubling bifurcation occur which originates from a period 1 (P-1) motion.
However, the excitation amplitude is so small that chaos does not appear.
The global bifurcation diagram with f D 8:8 is shown in Fig. 6.3. Besides two
cascades of period-doubling bifurcation, two sequences of chaotic motion occur.
The global bifurcation process is as follows:
(1) The system remains P-1 motion when ! 4:46. The phase plot of the P-1
motion when ! D 4:80 and f D 8:8 are shown in Fig. 6.4.
6 Two-Degree-of-Freedom Chaotic Vibration Isolation System 53
Fig. 6.2 Global bifurcation diagram with regard to excitation frequency .f D 4:0/
Fig. 6.3 Global bifurcation diagram with regard to excitation frequency .f D 8:8/
(2) When ! D 4:45, the first period-doubling bifurcation occurs and a period 2
(P-2) motion is obtained. The phase plot of the P-2 motion when ! D 4:10 and
f D 8:8 are shown in Fig. 6.5.
(3) When ! is decreased to 3.96, a response of period 4 (P-4) is observed after twice
period-doubling bifurcation. The phase plot of the P-4 motion when ! D 3:74
and f D 8:8 are shown in Fig. 6.6.
(4) Further decrease of the excitation frequency leads to deeper bifurcation cas-
cades, and responses of period 8 (P-8) and chaos at ! D 3:71 and ! D 3:70,
respectively. The phase plot of the P-8 motion when ! D 3:71 and f D 8:8 are
showninFig.6.7. The phase plot and Poincar´e map of the chaotic motion when
! D 3:70 and f D 8:8 are shown in Fig. 6.8.
54 J.-J. Lou et al.
Fig. 6.4 Phase plot of P-1 motion .f D 8:8; ! D 4:80/
Fig. 6.5 Phase plot of P-2 motion .f D 8:8; ! D 4:10/
(5) When 2:12 ! 3:60, the P-1 motion presents itself again. Then, another
sequence of period-doubling bifurcation occurs, and the system remains P-2
motion when 1:96 ! 2:11.
(6) However, this period-doubling bifurcation does not accelerate as the excita-
tion frequency decreases, and instead it is broken by the P-1 motion when
1:18 ! 1:95.
6 Two-Degree-of-Freedom Chaotic Vibration Isolation System 55
Fig. 6.6 Phase plot of P-4 motion .f D 8:8; ! D 3:74/
Fig. 6.7 Phase plot of P-8 motion .f D 8:8; ! D 3:71/
(7) When ! D 1:17, the third sequence of period-doubling bifurcation comes forth
and the P-2 motion is obtained.
(8) The P-4 and P-8 motions are observed when ! D 1:10 and ! D 1:09 respec-
tively, and then chaos occurs.
(9) Finally, the system rests on P-1 motion when ! 1:05.
56 J.-J. Lou et al.
Fig. 6.8 Phase plot and Poincar´e map of chaotic motion .f D 8:8; ! D 3:70/
6.4 Analysis of the Power Flow
In the third section, the dynamical evolution of the two-degree-of-freedom nonlin-
ear vibration isolation system (6.4) in the control parameter space is analyzed. Two
cascades of deep period-doubling bifurcation and two sequences of chaotic motion
are observed when the excitation frequency is decreased from 5 to 0.8, while the ex-
citation amplitude remains 8.8. However, the isolation characteristics of the system
(6.4), rather than the dynamical evolution, is of much more concern.
In the nonlinear theory of vibration isolation, one encounters the problem of
defining a suitable performance index for the isolation. This is because a harmonic
response with the same frequency as that of the excitation is not guaranteed. As
mentioned previously, the response may contain subharmonics and superharmonics,
and sometimes the response may even be nonperiodic, namely, chaotic. Usually, if
other harmonics are present, a working index of isolation effectiveness may be de-
fined as the ratio of the r.m.s. values of the response and the excitation. In the case
of a chaotic response, the ratio of the power spectral densities of the response and
the excitation may be used [7]. But the indices to be used for various types of ex-
citation, unlike in a linear system, cannot be related through simple expressions.
Furthermore, in the design of a vibration isolation system, what we care most is the
attenuation of the vibratory energy. The power flow, hence, is a good candidate.
As we all know, the force transmissibility is defined as the ratio of the force
transferred to the base vs. the excitation force. Correspondingly, we can define the
power flow transmissibility as [12]
T
P
D
P
out
P
in
(6.6)
where P
out
is the power flow transferred to the base through the isolator, and P
in
is
the power flow into the whole system.
6 Two-Degree-of-Freedom Chaotic Vibration Isolation System 57
Express the reciprocal of T
P
with decibel, and (6.6) becomes
L
P
D 10 lg
1
T
P
D 10 lg
P
in
P
out
(6.7)
where L
P
is called the power flow attenuation rate (PFAR).
However, this working index L
P
is devoid of any information about the response
frequency content and cannot indicate the weakening of the line spectrum. One of
the most advantages of the chaotic vibration isolation lies in isolating the line spec-
trum. Another index should be used.
Following the definition of the vibration level difference, let
D
P
D 10 lg
P
LS
in
P
LS
out
(6.8)
as the power flow line spectrum drop (PFLSD), where P
LS
in
is the input power flow
at the frequency containing the line spectrum, and P
LS
out
is output power flow at the
same frequency.
Thus, the isolation effectiveness can be analyzed throughout the bifurcation pro-
cess from the point of view of energy. The evolution of the PFAR and the PFLSD
at the excitation frequency of system (6.4) with f D 8:8 is shown in Fig. 6.9.
The representative phase plots of P-1 and chaotic motions when ! D 4:80 and
3.70 throughout the two cascades of deep period-doubling bifurcation are shown in
Figs. 6.4 and 6.8 respectively. The corresponding characteristics of the power flow
when ! D 4:80 and3.70isshowninFigs.6.10 and 6.11, and the corresponding
PFLSD is 35.11 and 40.31 dB respectively. Namely, the reduction of the line spectra
when the system is chaotic is 5 dB greater than that when the system is periodic. The
values of the PFAR and the PFLSD at different excitation frequencies are listed in
Table 6.1.
As shown in Figs. 6.10 and6.11andTable6.1, the nonlinear vibration isolation
system is good for the reduction of the line spectra. For some parameters, the PFAR
in chaotic state is 17 dB higher than that in P-1 state at best, and the PFLSD in
chaotic state is 5–20 dB higher than that in P-1 state. Data in rows of No. 2–3 vs.
that of No. 4–6 and data in rows of No. 10 vs. that of No. 11–12, furthermore, shows
that the PFLSD in 1/2 (or 1/4, 1/8) subharmonicstates is close to that in chaotic state.
That is, the isolation effectiveness of the line spectrum is widely improved after a
sequence of period-doubling bifurcations.
6.5 Conclusion
The complex dynamic behaviour of the two-degree-of-freedom nonlinear vibration
isolation system is studied numerically, and its ability in line spectrum reduction is
also analyzed.