Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system)
The study of the transfer of energy between subsystems coupled in hybrid system is very important for different applications. This paper presents an analytical analysis of the transfer of energy between linear and nonlinear oscillators for free vibrations when oscillators are statically, as well a...
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Інститут математики НАН України
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irk-123456789-1646832020-02-11T01:28:07Z Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) Hedrih (Stevanović), K. Статті The study of the transfer of energy between subsystems coupled in hybrid system is very important for different applications. This paper presents an analytical analysis of the transfer of energy between linear and nonlinear oscillators for free vibrations when oscillators are statically, as well as dynamically, connected into double-oscillator system, as the two new hybrid systems, every with two degrees of freedom. The analytical analysis showed that the elastic connection between oscillators caused the appearance of a like two-frequency regime of time function, and that the energy transfer between subsystems appears. Also, the dynamical linear constraint between oscillators, each with one degree of freedom, coupled in hybrid system changes dynamics from single frequency regimes into like twofrequency regimes. The dynamical constraint as a connection between subsystems is realized by rolling element with inertia properties. In this case, an analytical analysis of the transfer energy between linear and nonlinear oscillators for free vibrations is also performed. The two Lyapunov exponents corresponding to each of two eigen modes are expressed by using energy of the corresponding eigen time component. Вивчення переносу енергії між підсистемами, що поєднані у гібридну систему, є дуже важливим для різних застосувань. У даній статті проведено аналітичне дослідження переносу енергії між лінійним та нелінійним осциляторами при вільних коливаннях для випадків як статичного, так і динамічного поєднання осциляторів у подвійно-осциляторну систему у вигляді двох нових гібридних систем із двома ступенями вільності кожна. Аналітичне дослідження показало, що пружне поєднання осциляторів зумовлює встановлення двочастотно-подібного режиму функції часу і спричиняє перенос енергії між підсистемами. Динамічний лінійний зв'язок між осциляторами, що поєднані у гібридну систему і мають один ступінь вільності кожен, змінює динаміку з одночастотних режимів до двочастотно-подібних режимів. Динамічний зв'язок як поєднання підсистем реалізовано елементом, що котиться і має інерційні властивості. Також проведено аналітичне дослідження переносу енергії між лінійним і нелінійним осциляторами у цьому випадку. Для двох експонент Ляпунова, що відповідають кожній з двох власних мод, побудовано вирази з використанням енергії відповідних власних компонент часу. 2008 Article Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) / K. Hedrih (Stevanović) // Український математичний журнал. — 2008. — Т. 60, № 6. — С. 796–814. — Бібліогр.: 44 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164683 517.9 en Український математичний журнал Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Hedrih (Stevanović), K. Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) Український математичний журнал |
| description |
The study of the transfer of energy between subsystems coupled in hybrid system is very important for
different applications. This paper presents an analytical analysis of the transfer of energy between linear
and nonlinear oscillators for free vibrations when oscillators are statically, as well as dynamically,
connected into double-oscillator system, as the two new hybrid systems, every with two degrees of
freedom. The analytical analysis showed that the elastic connection between oscillators caused the
appearance of a like two-frequency regime of time function, and that the energy transfer between
subsystems appears. Also, the dynamical linear constraint between oscillators, each with one degree of
freedom, coupled in hybrid system changes dynamics from single frequency regimes into like twofrequency regimes. The dynamical constraint as a connection between subsystems is realized by rolling
element with inertia properties. In this case, an analytical analysis of the transfer energy between linear
and nonlinear oscillators for free vibrations is also performed.
The two Lyapunov exponents corresponding to each of two eigen modes are expressed by using
energy of the corresponding eigen time component. |
| format |
Article |
| author |
Hedrih (Stevanović), K. |
| author_facet |
Hedrih (Stevanović), K. |
| author_sort |
Hedrih (Stevanović), K. |
| title |
Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) |
| title_short |
Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) |
| title_full |
Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) |
| title_fullStr |
Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) |
| title_full_unstemmed |
Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) |
| title_sort |
energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/164683 |
| citation_txt |
Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system) / K. Hedrih (Stevanović) // Український математичний журнал. — 2008. — Т. 60, № 6. — С. 796–814. — Бібліогр.: 44 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT hedrihstevanovick energyinteractionbetweenlinearandnonlinearoscillatorsenergytransferthroughthesubsystemsinahybridsystem |
| first_indexed |
2025-07-14T17:17:23Z |
| last_indexed |
2025-07-14T17:17:23Z |
| _version_ |
1837643537460494336 |
| fulltext |
UDC 517.9
K. Hedrih (Stevanović) (Univ. Niš, Math. Inst. SANU)
ENERGY INTERACTION BETWEEN LINEAR
AND NONLINEAR OSCILLATORS
(ENERGY TRANSIENT THROUGH THE SUBSYSTEMS
IN THE HYBRID SYSTEM)*
ENERHETYÇNA VZA{MODIQ MIÛ LINIJNYMY
TA�NELINIJNYMY OSCYLQTORAMY
(PROCES PEREXODU ENERHI} ÇEREZ PIDSYSTEMY
U HIBRYDNIJ SYSTEMI)
The study of the transfer of energy between subsystems coupled in hybrid system is very important for
different applications. This paper presents an analytical analysis of the transfer of energy between linear
and nonlinear oscillators for free vibrations when oscillators are statically, as well as dynamically,
connected into double-oscillator system, as the two new hybrid systems, every with two degrees of
freedom. The analytical analysis showed that the elastic connection between oscillators caused the
appearance of a like two-frequency regime of time function, and that the energy transfer between
subsystems appears. Also, the dynamical linear constraint between oscillators, each with one degree of
freedom, coupled in hybrid system changes dynamics from single frequency regimes into like two-
frequency regimes. The dynamical constraint as a connection between subsystems is realized by rolling
element with inertia properties. In this case, an analytical analysis of the transfer energy between linear
and nonlinear oscillators for free vibrations is also performed.
The two Lyapunov exponents corresponding to each of two eigen modes are expressed by using
energy of the corresponding eigen time component.
Vyvçennq perenosu enerhi] miΩ pidsystemamy, wo po[dnani u hibrydnu systemu, [ duΩe vaΩlyvym
dlq riznyx zastosuvan\. U danij statti provedeno analityçne doslidΩennq perenosu enerhi] miΩ
linijnym ta nelinijnym oscylqtoramy pry vil\nyx kolyvannqx dlq vypadkiv qk statyçnoho, tak
i dynamiçnoho po[dnannq oscylqtoriv u podvijno-oscylqtornu systemu u vyhlqdi dvox novyx
hibrydnyx system iz dvoma stupenqmy vil\nosti koΩna. Analityçne doslidΩennq pokazalo, wo
pruΩne po[dnannq oscylqtoriv zumovlg[ vstanovlennq dvoçastotno-podibnoho reΩymu funkci]
çasu i spryçynq[ perenos enerhi] miΩ pidsystemamy. Dynamiçnyj linijnyj zv’qzok miΩ oscylq-
toramy, wo po[dnani u hibrydnu systemu i magt\ odyn stupin\ vil\nosti koΩen, zming[ dynamiku
z odnoçastotnyx reΩymiv do dvoçastotno-podibnyx reΩymiv. Dynamiçnyj zv’qzok qk po[dnannq
pidsystem realizovano elementom, wo kotyt\sq i ma[ inercijni vlastyvosti. TakoΩ provedeno
analityçne doslidΩennq perenosu enerhi] miΩ linijnym i nelinijnym oscylqtoramy u c\omu
vypadku.
Dlq dvox eksponent Lqpunova, wo vidpovidagt\ koΩnij z dvox vlasnyx mod, pobudovano vy-
razy z vykorystannqm enerhi] vidpovidnyx vlasnyx komponent çasu.
1. Introduction. 1.1. The study of the transfer of energy between subsystems
coupled in hybrid system (see [1 – 16]) is very important for different applications.
The new manuscripts by author (see [17 – 19, 41 – 44]) presents analytical analysis of
the transfer of energy between plates for free and forced transversal vibrations of an
elastically connected double-plate system. The analytical analysis showed that the
elastic connection between plates caused the appearance of a two-frequency regime of
time functions, which corresponds to one eigenamplitude function of one mode, and
also that time functions of different vibration modes are uncoupled, but energy transfer
between plates in one eigen mode appears. It was shown for each shape of vibrations.
Series of two Lyapunov exponents corresponding to one eigen amplitude mode are
expressed by using energy of the corresponding eigen amplitude time component.
Then it is very important to investigate transfer of energy between subsystems in the
hybrid system, and also between different subprocesses in the system with hybrid
process containing these subprocesses coupled with different type of connections. The
Parts of this research were supported by the Ministry of Sciences, Technologies and Development of
Republic Serbia trough Mathematical Institute SANU Belgrade Grant ON 144002 “Theoretical and
Applied Mechanics of Rigid and Solid Body. Mechanics of Materials” and Faculty of Mechanical
Engineering University of Niš.
© K. HEDRIH (STEVANOVIĆ), 2008
796 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 797
elements of connections between subsystems are possible to realize by static,
dynamical and rheological elements and also by hybrid connection containing elements
with static, dynamic and rheological properties. This is the reason to investigate basic
types of elementary connections between subsystem coupled in the hybrid system
beginning with subsystems, every of both, with one degree of freedom and compare
kinetic parameters, properties of the simple dynamical processes in the separate
decoupled subsystems into separate their integrity and compare processes in coupled
subsystem taking into account interaction and changes in the dynamics of the
components in system present in their subsystems. It is especially important to analyze
energy in the decoupled subsystem as a separate system with proper integrity of
dynamical processes.
1.2. When, at an international conference ICNO in Kiev in 1969, my professor of
mechanics and mathematics, D. Rašković (see [16, 20]) introduced me to academician
Yurii Alekseevich Mitropolskii (see [21 – 24]) and when I started really to understand
the differences between linear and nonlinear dynamics, I knew I was on the right path
of research which enchanted me ever more by understanding new phenomena and their
variety in nonlinear dynamics of realistic engineering and other dynamical systems.
(First my knowledge about properties of nonlinearity and the nonlinear function I
obtained in high school from my excellent professor of mathematics Draginja Nikolić
and during my research Matura work on the subject of elementary functions and their
graphics as a final high school examination. )
For beginning of this paper, it is necessary to present a survey of original results of
the author and of researchers from Faculty of Mechanical Engineering University of
Niš (see [1 – 3, 10, 15, 20, 25 – 32]), inspired and/or obtained by the Krylov – Bogo-
lyubov – Mitropolskii asymptotic method (see [21 – 24]), and as a direct influence of
professor Rašković [16] scientific instruction and published Mitropolskii’s papers and
monographs. These results have been published in scientific journals, and were
presented on the scientific conferences and in numerous degree works and theses
supervised by Mitropolskii (in 1972 – 1975), Rašković (in 1964 – 1974), and Hedrih
(in 1976 – 2006).
The original results obtained by Hedrih (Stevanović), Rašković, Kozić, Pavlović,
Mitić, Filipovski, Janevski, and Simonović contain asymptotic analysis of the nonlinear
oscillatory motions of elastic bodies: beams, plates, shells and shafts. The
multifrequency oscillatory motion of elastic bodies was studied. Corresponding
differential equations systems of first approximations for amplitudes and phases
multifrequency regime of elastic bodies nonlinear oscillations were composed. The
characteristic properties of nonlinear systems passing through coupled multifrequency
resonant state and mutual influences excited modes were discovered.
In the same cited papers amplitude-frequency and phase frequency curves for
stationary and nonstationary coupled multifrequency resonant states, based on the
numerical experiment on the differential equations systems are presented.
Using ideas of averaging and asymptotic methods of Krylov, Bogolyubov, and
Mitropolskii. K. Hedrih gives the asymptotic approximations of the solutions for one-,
two-, three- and four-frequency vibrations of elastic beams, shaft and thin elastic plates,
as well as of the thin elastic shells with positive constant Gauss’ curvatures and finite
deformations, and differential equations system for amplitude and phase corresponding
vibration regimes.
Some results of an investigation of multifrequency vibrations in single frequency
regime in nonlinear systems with many degrees of freedom and with slowchanging
parameters are presented by Stevanović and Rašković article from 1974 (see [20]).
Application of the Krylov – Bogolyubov – Mitropolskii asymptotic method for study of
elastic bodies nonlinear oscillations and energetic analysis of the elastic bodies
oscillatory motions give new results in theses by Stevanović in 1975. One-frequency
transversal oscillations of thin rectangular plate with nonlinear constitutive material
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
798 K. HEDRIH (STEVANOVIĆ)
stressstrain relations and nonlinear transversal vibrations of a plate with special
analysis of influence of weak nonlinear boundary conditions are contents of the articles
by Hedrih (1979, 1981) and than asymptotic solution of the nonlinear equations of the
thin elastic shell with positive Gauss’ curvature in two-frequency regime is pointed out
in the article by Hedrih (1983). Two-frequency oscillations of the thin elastic shells
with finite deformations and interactions between harmonics have been studied by
Hedrih and Mitić (see [27]) and multifrequency forced vibrations of thin elastic shells
with a positive Gauss’s curvature and finite displacements by Hedrih in 1984, and also
on the mutual influence between modes in nonlinear systems with small parameter
applied to the multifrequencies plate oscillations are studied by Hedrih, Kozić, Pav-
lović, and Mitić (see [28]), as well as Herdih and Simonović (see [42, 43]), and
Janevski (see [44]).
Multifrequency forced vibrations of thin elastic shells with a positive Gauss’
curvature and finite deformations and initial deformations influence of the shell middle
surface to the phase-frequency characteristics of the nonlinear stationary forced shell’s
vibrations and numerical analysis of the four-frequency vibrations of thin elastic shells
with Gauss’ positive curvature and finite deformations by Hedrih and Mitić [27] and
also initial displacement deformation influence of the thin elastic shell middle surface
to the resonant jumps appearance by Hedrih and Mitić (1987). By means of the
graphical presentations from the cited references, analysis was made and some
conclusions about nonlinear phenomenon in multifrequency vibrations regimes were
pointed out. Some of these conclusions we quote here: nonlinearities are the reason for
the appearance of interaction between modes in multifrequency regimes; in the coupled
resonant state one or several resonant jumps appear on the amplitude-frequency and
phase frequency curves; these resonant jumps are from smaller to greater amplitudes
and vice versa.
Unique trigger of singularities (see [13, 33 – 38]) with one unstable homoclinic
saddle type point, and with two singular stable knot type points appear in one
frequency stationary resonant state. It is visible on the phase-frequency as well as on
the amplitude-frequency for stationary resonant state.
In the case of the multifrequency coupled resonant state and in the appearance of
the more resonant coupled modes in resonant range of corresponding frequencies,
unique trigger, and multiplied triggers appear (see [33]). Maximum number of triggers
coupled is adequate to number of coupled modes and resonant frequencies of external
excitations. Multiplied triggers contain as many unstable saddle homoclinic points in
the mapped plane as the number of resonant frequencies of external excitations. For
example, if a four-frequency coupled resonant process in u-v plane is in question, four
homoclinic saddle type points appear. The appearance of these unstable homoclinic
points requires further study, since it induces instability elements in a stationary
nonlinear multifrequency process.
Using differential equations systems of the first approximation of multifrequency
regime of stationary and nonstationary resonant states, we analyzed the energy of
excited modes and transfer of energy from one to other modes (see [26, 28]). On the
basis of this analysis, the question of excitation of lower frequency modes by higher
frequency mode in the nonlinear multifrequency vibration regimes was opened.
1.3. In many engineering systems with nonlinearity, high frequency excitations
are sources of the appearance of multifrequency resonant regimes with high frequency
modes as well as low frequency modes. It is visible from many experimental research
results and also theoretical results (see [14]). The interaction between amplitudes and
phases of the different modes in the nonlinear systems with many degrees of the
freedom as well in the deformable body infinite numbers frequency vibration free and
forced regimes is observed theoretically by averaging asymptotic methods of Krylov,
Bogolyubov, and Mitropolskii. This knowledge has great practical importance.
Application of the Krylov – Bogolyubov – Mitropolskii asymptotic method as well
as energy approach given in monographs by Mitropolskii (see [21, 22]) for study of the
elastic bodies nonlinear oscillations and energetic analysis of the elastic bodies
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 799
oscillatory motions give new results in master and doctoral theses by Stevanović in
1972 and 1975.
In numerous papers (see [1, 2, 9, 15, 27, 28, 41]) author presented transfer of
energy between modes in nonlinear deformable body vibrations by using averaging and
asymptotic methods of Krylov, Bogolyubov, and Mitropolskii for obtaining system of
the differential equations of amplitudes and phases in first approximations and
expression for energy of the excited modes depending of amplitudes, phases and
frequencies of different nonlinear modes. By means of these obtained asymptotic
approximations of the solutions, the energy analysis of the interaction of the modes in
the cases of then multifrequency vibration regimes in the nonlinear elastic systems
(beams, plates and shells) excited by initial conditions for free and forced vibrations
were made and transfer energy between modes is identified. Also, for the case of the
forced frequency of the external excitation in the resonant frequency range near to one
of the natural eigen frequency of the basic linear system two or more resonant energy
jumps are present and it is possible to identified by the obtained system of the
differential equations of amplitudes and phases in first approximations. Trigger of
coupled singularities, as well as coupled triggers of the energy values is also present in
the nonlinear system multifrequency resonant stationary and nonstationary regimes
during to increasing and decreasing values of the external excitation frequencies though
corresponding mode resonant ranges.
In conclusion of tills part we can summarize the following: oscillatory processes in
dynamical systems depend on systems character. In such systems energy is also
transformed from one form to another and has different flows inside a dynamical
system. Transformation of kinetic energy into potential energy and vice versa occurs
in conservative systems, but when lineal systems are in question, the energy carried by
a considered harmonic (mode) of adequate frequency remains constant during a
dynamical process, as does the total systems energy. There is no mutual influence
between harmonics and the system may be presented by partial oscillators, the number
of which is equal to the number of oscillations freedom degrees, or to the number of
free vibrations own circular frequencies. During that the total energy of a single partial
oscillator remains constant and the transformation of kinetic energy into potential
occurs. In sash linear system transfer energy between modes no occurs (see [16]).
When nonlinear conservative systems are in question such conclusion for linear
systems would be incorrect. The theoretical and experimental studies reveal that the
interactions between widely separated nonlinear modes result in various bifurcations,
the coexistence of multiple attractors, and chaotic attractors. The theoretical results
show also that damping may be destabilizing. The different types of nonlinear
phenomena in single degree of freedom nonlinear system dynamics are investigated
between other researchers, also by Hedrih (see [39, 40]).
1.4. For introducing to the problem of the energy transfer or transient in the hybrid
nonlinear systems, it is useful to take, for simple analysis, into consideration the change
energy between parts of the energy carrying on the generalized coordinates φ and ρ
in the very known system, known under name spring pendulum system with two
degree of freedom. For the analysis of the energy in the spring pendulum we can write
the kinetic and potential energies in the forms
E m lk = + +[ ]1
2
2 2 2˙ ( ) ˙ρ ρ φ and E c mg lp = + + −1
2
12ρ ρ φ( )( cos ),
where m is mass of the pendulum, l length of pendulum string-neglected mass spring
in the static equilibrium state of the pendulum, and c spring axial rigidity and φ and
ρ are respectfully, angle and extension part of length of the string-spring of the
pendulum with comparison of the spring length in static equilibrium state of the
pendulum, take as the generalized coordinates of the system. For the linearized case
for kinetic energy, after neglecting small member — part of kinetic energy on the
generalized coordinate φ we can taking into account following expression:
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
800 K. HEDRIH (STEVANOVIĆ)
expression E m lk2
2 21
2
= +( ) ˙ρ φ changes into approximation E m lk2
21
2
≈ ( ˙)φ .
Only for small oscillations — perturbations from equilibrium position it is possible to
use approximation of the expression for kinetic and potential energy in the form
E m lk ≈ +[ ]1
2
2 2˙ ( ˙)ρ φ and E c mglp ≈ +1
2
1
2
2 2ρ φ
in for that linearized case the generalized coordinates are normal coordinates of the
small oscillations of the spring pendulum around equilibrium position ρ = 0, φ = 0
and coordinates are decoupled. In this linearized case of the spring pendulum model,
the energy carried on the these normal coordinates are uncoupled and transfer or
transient of the total energy don’t appeared between proper parts of the separate normal
coordinate and on the separate processes defined by normal coordinates are
conservative systems every with one degree of the freedom. In this case on the every
of the coordinate there are conversion of the energy from kinetic to potential, but some
of the both on one normal coordinates is constant:
E mkρ ρ≈ 1
2
2˙ and E cpρ ρ≈ 1
2
2 ,
E m lkφ φ≈ 1
2
2( ˙) and E mglpφ φ≈ 1
2
2 .
This is visible from system differential equations of the linearized system
m c˙̇ρ ρ+ = 0,
ml mgl2 0˙̇φ φ+ =
or in the form
˙̇ρ ω ρ+ =2
2 0 , ω2
2 = c
m
,
˙̇φ ω φ+ =1
2 0, ω1
2 = g
l
.
But for the nonlinear case the interaction between coordinates is present and then
energy transient appears:
E m l lk = + + +[ ]1
2
22 2 2 2 2 2˙ ˙ ˙ ˙ρ φ ρ φ ρ φ
and
E c mgl mgp = + − + −1
2
1 12ρ φ ρ φ( cos ) ( cos ).
We can separate the following parts:
I. Kinetic and potential energy carrying on the coordinate ρ are
E mkρ ρ= 1
2
2˙ and E c mgpρ ρ ρ= +1
2
2 .
By analyzing these previous expressions we can see that with these expressions for
decoupled oscillator with coordinate ρ, we have pure linear oscillator or harmonic
oscillator with coordinate ρ and frequency ω2
2 = c
m
, and separated process is
isochronous.
II. Kinetic and potential energy carrying on the coordinate φ are
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 801
E mlkφ φ= 1
2
2 2˙ and E mglpφ φ= −( cos )1 .
By analyzing these previous expressions we can see that with these expression for
decoupled oscillator with coordinate φ , we have pure nonlinear oscillator with
coordinate φ, and separated process is no isochronous. For linearized case this
oscillator have frequency ω1
2 =
g
l
.
III. Then we can conclude that formally we have coupled two oscillators, one pure
linear with one degree of freedom, and second nonlinear, also with one degree of
freedom. In the hybrid system these oscillators are coupled and energy of the coupling
containing two parts: one kinetic and second potential energy. Then, in the coupling,
hybrid connections with static and dynamic properties are introduced.
Kinetic and potential energies of the coordinate φ and ρ interaction in the
nonlinear hybrid model are:
E m lk( , )
˙
φ ρ ρ ρφ= [ + ]1
2
2 2 and E mgp( , ) cosφ ρ ρ φ= − . (1)
For nonlinear case differential equations are
m c mg˙̇ ( cos )ρ ρ φ+ + − =1 0 ,
ml d
dt
m l mgl2 2 0˙̇ ( ) ˙ sinφ ρ ρφ φ+ +[ ] + =
or in the form
˙̇ ( cos )ρ ω ρ φ+ + − =2
2 1 0g ,
l l l l2
1
2 22 2 0˙̇ ( ) ˙ ˙ ( ) ˙̇ sinφ ρ ρφ ρ ρφ ω φ+ + + + + = ,
or
˙̇ ( cos )ρ ω ρ φ+ = − −2
2 1g , (2)
˙̇ ( sin ) ˙ ˙ ( ) ( ) ˙̇φ ω φ ω φ φ ρφ ρ ρ ρ φ+ = − − + − +1
2
1
2
2 2
2 1 2
l
l
l
l (3)
or in nonlinear approximation forms for small oscillations around zero coordinates ρ =
= 0, φ = 0 or of the around stable equilibrium position of the spring pendulum
˙̇
! !
ρ ω ρ φ φ φ φ+ ≈ − − + − + …
2
2
2 4 6 8
2 24 6 8
g , (4)
˙̇
! !
˙ ˙ ( ) ( ) ˙̇φ ω φ ω φ φ φ ρφ ρ ρ ρ φ+ ≈ − − + − …
− + − +1
2
1
2
3 5 7
2 23 5 7
2 1 2
l
l
l
l . (5)
If we introduce phase coordinate, then we can write
v = ρ̇ ,
˙ ( cos )v g= − − −ω ρ φ2
2 1 ,
u = φ̇ ,
˙ ( sin ) ˙ ˙ ( ) ( ) ˙̇u
l
l
l
l= − + − − + − +ω φ ω φ φ ρφ ρ ρ ρ φ1
2
1
2
2 2
2 1 2
or in the approximation
v = ρ̇ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
802 K. HEDRIH (STEVANOVIĆ)
˙
! !
v g≈ − − − + − + …
ω ρ φ φ φ φ
2
2
2 4 6 8
2 24 6 8
,
u = φ̇ ,
˙
! !
˙ ˙ ( ) ( ) ˙̇u
l
l
l
l≈ − − − + − …
− + − +ω φ ω φ φ φ ρφ ρ ρ ρ φ1
2
1
2
3 5 7
2 23 5 7
2 1
2 .
From system equations (2), (3), as well from their approximations (4), (5) we can
see that their right-hand parts are nonlinear and are functions of generalized
coordinates, as well as of the generalized coordinates first and second derivatives. Also
we can see that generalized coordinates φ and ρ around their zero values, when ρ =
= 0, φ = 0 at the stable equilibrium position of the spring pendulum are also main
coordinates of the linearized model. It is reason that the asymptotic averaged method is
applicable for obtaining first asymptotic approximation of the solutions and it is
possible to use for energy analysis of the transfer energy between energies carried by
generalized coordinates φ and ρ in this nonlinear system with two degree of freedom,
but formally we can take into account that we have two oscillators, one non linear and
one linear every with one degree of freedom as two subsystems coupled in the hybrid
system with two degree of freedom, by hybrid connection realized by statical and
dynamical connections. This interconnection have two part of energy interaction
between subsystems expressed by kinetic and potential energy in the form (1).
Taking into consideration some conclusion from considered system of the spring
pendulum we can conclude also that it is important to consider more simple case of the
coupling between linear and nonlinear systems with one degree of freedom with
different types of the coupling realized by simple static or dynamic elements, for to
investigate hybrid phenomena in the coupled subsystems.
2. Energy analysis of the oscillatory processes and of the modes in nonlinear
system. When nonlinear conservative systems are in question such conclusion for
linear systems would be incorrect. The theoretical and experimental studies reveal that
the interactions between widely separated modes result in various bifurcations, the
coexistence of multiple attractors, and chaotic attractors. The theoretical results show
also that damping may be destabilizing.
The interaction between high-frequency and low-frequency modes observed
experimentally and demonstrated theoretically is of great practical importance. In
many engineering systems, high-frequency excitations can be caused by rotating
machinery and some debalances.
Kinetic energy and potential energy in first asymptotic approximation for nonlinear
conservative system nonlinear modes using normal coordinates of unperturbed
corresponding linear system are (see [1 – 3]):
Ek =
s
s n
ks
=
=
∑
1
E =
s
s n
s
=
=
∑ ( )
1
2ξ̇ +
+ ε ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξg s r n n s r n n1 2 1 1 2 1, , , , , , , , ˙ , ˙ , , ˙ , ˙ , , ˙ , ˙… … … …( )− − ,
Ep
s
s n
s s s r n nf= ( ) + … …( )
=
=
−∑
1
2 2
1 2 1ω ξ ξ ξ ξ ξ ξ ξ, , , , , , , ,
where
ξ θ ψs s s sa= +( )cos , s n= …1 2, , , ,
are first asymptotic approximations of normal coordinates, and as are amplitudes, and
θs + ψ s are phases as a functions of time and which are calculate from differential
equations first approximations (see [21, 22]).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 803
2.1. Nonlinear oscillator. Kinetic and potential energies and Rayleigh dissipative
function of nonlinear oscillator with one degree of freedom and generalized coordinate
x1 are:
E m xk( ) ˙1 1 1
21
2
= , E c x c xp( ) ˜1 1 1
2
1 1
41
2
1
4
= + and Φ( ) ˙1 1 1
21
2
= b x ,
where m1 is masses, c1 is the spring rigidity coefficient of the linear elasticity low,
and c̃1 the spring rigidity coefficient of the nonlinear elasticity low, b1 coefficient of
the system linear dumping force. For this nonlinear oscillator it is right: d
dt
Ek( )1( +
+ Ep( )1 ) = – 2 1Φ( ) and for the case of the free vibrations.
For this case differential equation is in the following form:
˙̇ ˙ ˜x x x xN1 1 2 1
2
1 1
2
1
32+ + = −δ ω ω ,
where
ω1
2 1
1
= c
m
, 2 1
1
1
δ = b
m
, ˜ ˜
ωN
c
m1
2 1
1
=
and characteristic equation of the basic liner equation corresponding to previous has the
following characteristic numbers: λ1 2, = – δ1 � i ω δ1
2
1
2− = – δ1 � ip1 for the
small damping coefficient δ1 < ω1, and solution for free vibrations is: x t1( ) =
= R e t
01
1−δ cos p t1 01+( )α . To obtain approximation by using averaged method, we
propose solution in the following form:
x t1( ) = R t e t
1
1( ) −δ cos ( )Φ1 t ,
where R t1( ) and Φ1( )t are unknown functions. Also we can write: Φ1( )t = p t1 +
+ φ1. After averaging with respect to the full phase Φ1( )t we obtain the following
system of the averaged equations:
˙ ( )R t1 = 0,
˙ ( )φ1 t = – 3
8 1
1
2
1
2 2 1
p
R t eN
t˜ ( )ω δ−
and after integration we obtain for amplitude and phase the following first
approximation:
R t1( ) = R01 = const,
φ1( )t = 3
16 1 1
1
2
01
2 2 1
δ
ω δ
p
R eN
t˜ − + α01
and for full phase:
Φ1( )t = p t1 + 3
16 1 1
1
2
01
2 2 1
δ
ω δ
p
R eN
t˜ − + α01
and solution in the first averaged approximation form is:
x t1( ) = R e tt
01 1
1−δ cos ( )Φ = R e t
01
1−δ cos ˜p t
p
R eN
t
1
1 1
1
2
01
2 2
01
3
16
1+ +
−
δ
ω αδ
we can see that amplitude of the solution in the first averaged approximation form is in
the form R e t
01
1−δ and that phase are also function of the time, and also frequency
˜ ( )p t1 = p1 + 3
8 1
1
2
01
2 2 1
p
R eN
tω̃ δ− is changeable with time in the first approximation
obtained by averaged method.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
804 K. HEDRIH (STEVANOVIĆ)
By using previous averaged solution we obtain Lyapunov exponent in the form
λ
ω
δ1 1
2
1
2 1
2
1
1
2
1 0= +
= − <
→∞
lim ln ( ) ˙ ( )
t t
x t x t
or in the form
a b
Fig. 1. Two hybrid systems containing coupled subsystems by (a) static constraint,
coupled by linear spring rigidity c and (b) dynamical constraint, coupled by
rolling element of the mass m — dynamic coupling: one nonlinear (left) and
second linear (right).
˜ lim ln ( ) ( ) ˙ ( ) lim lnλ ω
ω ω ω
δ1 1
2 1
2
1
2 1
4
1
2 1
2
1 1
2 1
1
2
1 1
2 2
0= + +
=
= − <
→∞ →∞t
N
tt
x t x t x t
t
E
m
sist .
In our research, we can investigate system with small nonlinearity and small
vibrations around periodic vibrations.
2.2. Linear oscillator. Kinetic and potential energies and Rayleigh dissipative
function of linear oscillator with one degree of freedom and generalized coordinate x2
are:
E m xk( ) ˙2 2 2
21
2
= , E c xp( )2 2 2
21
2
= and Φ2 2 2
21
2
= b ẋ ,
where m2 is mass, c2 is the spring rigidity coefficient of the linear elasticity low,
b2 coefficient of the system linear dumping force.
For this system it is possible to show that is d
dt
E Ek p( ) ( )2 2+( ) = –2 Φ2 . For this
case differential equation is in the following form: ˙̇x2 + 2 2 2δ ẋ + ω2
2
2x = 0, where
ω2
2 =
c
m
2
2
, 2 2δ =
b
m
2
2
, and with characteristic numbers λ1 2, = – δ2 � i ω δ2
2
2
2−
for the small damping coefficient δ2 < ω2. Solution for free vibrations is x t2( ) =
= R e t
0
2−δ cos p t2 2+( )α .
2.3. Hybrid system — coupled nonlinear and linear oscillators by statical
constraint. Kinetic and potential energies and Rayleigh dissipative function of the
hybrid system, containing two subsystems — one linear oscillator and one nonlinear
oscillator, with two degree of freedom expressed by generalized coordinates x1 and
x2 (see Fig. 1 (a)) are:
E m x m xk = +1
2
1
21 1
2
2 2
2˙ ˙ ,
E c x c x c x x c xp = + + − +1
2
1
4
1
2
1
21 1
2
1 1
4
1 2
2
2 2
2˜ ( ) ,
Φ = +1
2
1
21 1
2
2 2
2b x b x˙ ˙ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 805
where m1 and m2 are masses, c1, c and c2 are the spring rigidity coefficients of
the linear elasticity low, and c̃1 the spring rigidity coefficient of the nonlinear
elasticity low, b1 and b2 coefficient of the system linear dumping forces. For this
system it is possible to show that is d
dt
E Ek p( )+ = – 2 Φ.
Energy interaction in this hybrid system, containing two coupled subsystems by
statical constraint, is potential energy of the spring for coupling nonlinear and linear
subsystem and is expressed in the form
E c x xp( , ) ( )1 2 2 1
21
2
= − .
Coupled system of differential equations of the hybrid system containing two sub-
systems, one nonlinear and one linear are in the form
˙̇ ˙ ˜x x a x a x xN1 1 2 1
2
1
2
1 1
2
2 1
2
1
32+ + +( ) − = −δ ω ω ,
˙̇ ˙x x a x a x2 2 2 2
2
2
2
2 2
2
12 0+ + +( ) − =δ ω ,
where are
ωi
i
i
c
m
2 = , 2δi
i
i
b
m
= , a c
mi
i
2 = , ˜ ˜
ωN
i
c
m1
2 1= , i = 1, 2.
Taking into account that consideration of the homogeneous system does not lose
generality of the phenomena, next our considerations are applied to this homogeneous
hybrid system.
For the basic linear equations of the coupled system of the differential equations of
the hybrid system containing two subsystems, one linearized and one linear are in the
form:
˙̇ ˙x x a x a x1 1 1 1
2
1
2
1 1
2
22 0+ + +( ) − =δ ω ,
˙̇ ˙x x a x a x2 2 2 2
2
2
2
2 2
2
12 0+ + +( ) − =δ ω
and for case that linearized and linear systems are equal ω1
2 = ω2
2 and δ1 = δ2 and
a1
2 = a2
2 , we can define:
A
1
1
=
, B
2
2
=
δ
δ
, C =
+ −
− +
ω
ω
1
2
1
2
1
2
1
2
1
2
1
2
a a
a a
and frequency equation is in the form:
λ λ
λ δλ ω
λ δλ ω
2
2
1
2
1
2
1
2
1
2 2
1
2
1
2
2
2
0A B C+ + =
+ + + −
− + + +
=
a a
a a
with characteristic numbers:
λ δ1 2 1, = − ∓ i p , λ δ ω δ δ1 2 1 1
2
1
2
1 1, = − − = −∓ ∓i i p
for the small damping coefficient δ1 < ω1,
λ δ3 4 1, ˜= − ∓ i p , λ δ ω δ δ3 4 1 1
2
1
2
1
2
1 12, ˜= − + − = −∓ ∓i ia p ,
where
p1 1
2
1
2= −ω δ
for the small damping coefficient δ1 < ω1,
˜ ˜p p a2 1 1
2
1
2
1
22= = + −ω δ
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
806 K. HEDRIH (STEVANOVIĆ)
for the small damping coefficient δ1 < ω1, and solution of the linear coupled system
we can write in the following two-frequency form:
x t e R p t R p tt
1 01 1 01 02 2 02( ) cos cos ˜= +( ) + +( )[ ]−δ α α ,
x t e R p t R p tt
2 01 1 01 02 2 02( ) cos cos ˜= +( ) − +( )[ ]−δ α α ,
where amplitudes and phases R i0 and α0i are constants.
By using averaged method, a first approximation of the solution of the hybrid
system, containing coupled nonlinear and linear system, we propose solutions in the
following forms:
x t e R t t R t tt
1 1 1 21 2( ) ( )cos ( ) ( )cos ( )= +[ ]−δ Φ Φ ,
x t e R t t R t tt
2 1 1 21 2( ) ( )cos ( ) ( )cos ( )= −[ ]−δ Φ Φ ,
where R ti( ) and Φi t( ) are unknown functions. Also we can write: Φi t( ) = p t1 +
+ φi . Then after averaging with respect to the full phase Φ1( )t we obtain the
following system of the first asymptotic approximation of the system differential
equations for amplitude R ti( ) and phase Φi t( ):
˙ ( )R t1 0= ,
˙ ( ) ˜ ( ) ( )φ ω δ
1
1
1
2
1
2
2
2 23
16
2 1t
p
R t R t eN
t= − +[ ] − ,
˙ ( )R t2 0= ,
˙ ( )
˜
˜ ( ) ( )φ ω δ
2
2
1
2
2
2
1
2 23
16
2 1t
p
R t R t eN
t= − +[ ] −
and after integration we obtain
R t R1 01( ) = = const ,
φ
δ
ω αδ
1
1
1
2
01
2
02
2 2
01
3
32
2 1( ) ˜t
p
R R eN
t= +[ ] +− ,
R t R2 02( ) = = const ,
φ
δ
ω αδ
2
2
1
2
01
2
02
2 2
02
3
32
2 1( )
˜
˜t
p
R R eN
t= +[ ] +− .
Solutions in the first asymptotic approximation in averaged form of the hybrid system
are
x t1( ) = e R p t
p
R R et
N
t− −+ +[ ] +
δ δ
δ
ω α01 1
1
1
2
01
2
02
2 2
01
3
32
2cos ˜ +
+ R p t
p
R R eN
t
02 2
2
1
2
01
2
02
2 2
02
3
32
2cos ˜
˜
˜+ +[ ] +
−
δ
ω αδ ,
x t2( ) = e R p t
p
R R et
N
t− −+ +[ ] +
δ δ
δ
ω α01 1
1
1
2
01
2
02
2 2
01
3
32
2cos ˜ –
– R p t
p
R R eN
t
02 2
2
1
2
01
2
02
2 2
02
3
32
2cos ˜
˜
˜+ +[ ] +
−
δ
ω αδ
we can see that amplitudes of the solution in the first approximation is in the form
R ei
t
0
−δ and that phases are also functions of the time, and also frequencies
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 807
p t p
p
R R eN
t
1 1
1
1
2
01
2
02
2 23
16
2( )
˜
˜= + +[ ] −ω δ
and
˜ ( ) ˜
˜
˜p t p
p
R R eN
t
2 2
2
1
2
01
2
02
2 23
16
2= + +[ ] −ω δ
are changeable with time in the first approximation obtained by averaged method.
By using previous averaged solution we can obtain Lyapunov exponents in the
forms
λ
ω
δ1 1
2
1
2 1
21
2
1 0= +
= − <
→∞
lim ln ( ) ˙ ( )
t t
x t x t ,
λ
ω
δ2 2
2
2
2 2
21
2
1 0= +
= − <
→∞
lim ln ( ) ˙ ( )
t t
x t x t .
Also, taking into account that system is nonlinear
λ̃1 = lim ln ( )
˜
( ) ˙ ( )
t
N
t
x t x t x t
→∞
+ +
1
2
1
1
2 1
2
1
2 1
4
1
2 1
2ω
ω ω
=
= lim ln
t t
E
m→∞
1
2 2 1 1
2
subsist (1)
ω
= –δ < 0.
For the nonhomogeneous case we can define
A
1
1
=
, B
2
2
=
δ
δ
1
2
, C =
+ −
− +
ω
ω
1
2
1
2
1
2
2
2
2
2
2
2
a a
a a
and frequency equation is in the form:
λ λ
λ δ λ ω
λ δ λ ω
2
2
1 1
2
1
2
1
2
2
2 2
2 2
2
2
2
2
2
0A B C+ + =
+ + + −
− + + +
=
a a
a a
or in the form
λ δ λ ω λ δ λ ω2
1 1
2
1
2 2
2 2
2
2
2
1
2
2
22 2 0+ + +( ) + + +( ) − =a a a a
with four roots: λ1 2, = – δ̂1 � i p̂1 and λ3 4, = – δ̂2 � i p̂2 . Own amplitude numbers
we obtain from
A
a
A
a
C
s s
s s
s
1
1
2
2
2
1 1
2
1
22
( ) ( )
˜=
+ + +
=
λ δ λ ω
or
A
K
A
K
C
s
s
s
s s
1
21
2
22
( )
( )
( )
( )= =
and solution of the linear coupled system we can write in the following form:
x t K e R p t K e R p tt t
1 21
1
01 1 01 21
2
02 2 02
1 2( ) cos ˆ cos ˆ( ) ˆ ( ) ˆ
= +( ) + +( )− −δ δα α ,
x t K e R p t K e R p tt t
2 22
1
01 1 01 22
2
02 2 02
1 2( ) cos ˆ cos ˆ( ) ˆ ( ) ˆ
= +( ) + +( )− −δ δα α ,
where amplitudes and phases R i0 and α0i are constants.
By using asymptotic averaged method, a first asymptotic approximation of the
solution of the hybrid system, containing coupled nonlinear and linear system as
subsystems, we propose solutions in the following forms:
x t K e R t t K e R tt t
1 21
1
1 1 21
2
02 2
1 2( ) ( )cos ( ) cos ( )( ) ˆ ( ) ˆ
= +− −δ δΦ Φ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
808 K. HEDRIH (STEVANOVIĆ)
x t K e R t K e R tt t
2 22
1
01 1 22
2
02 2
1 2( ) cos ( ) cos ( )( ) ˆ ( ) ˆ
= +− −δ δΦ Φ ,
where R ti( ) and Φi t( ) are unknown functions. Also we can write: Φi t( ) = p̂ ti +
+ φi . And all next is similar as in previous considered part.
2.4. Hybrid system — coupled nonlinear and linear oscillators by dynamical
constraint. In Fig. 1(b) we can see hybrid system containing two subsystems, one
linear and one nonlinear coupled by dynamical constraint. Dynamical constraint
consists of the one disk with mass m and mass inertia axial moment JC with
possibility of rolling between two masses m1 and m2 of the subsystems. In our
research, we can investigate small nonlinearity in the subsystem, and also in the hybrid
system and also small vibrations around periodic.
Angular velocity of the disk rotation, with mass m and mass inertia axial moment
JC , between two masses m1 and m2 is: ωC =
˙ ˙x x
R
2 1
2
−
, and velocity of the mass
center is: νC =
˙ ˙x x1 2
2
+
. Kinetic energy of the coupling nonlinear and linear
subsystems is
E J Jk C C C Cm m
x x x x
R( , )
˙ ˙ ˙ ˙
1 2
2 2 1 2
2
2 1
21
2
1
2 2 2
= +[ ] = +
+ −
ν ω
or
Ek a x a x x x a( , ) ˆ ˙ ˆ ˙ ˙ ˙ ˆ1 2 11 1
2
22 2
2
1 2 12
1
2
2= + +( ),
where
â m
R
C
11 24 4
= + J
, â m
R
C
22 24 4
= + J
and â m
R
C
12 24 4
= − J
.
Then we have hybrid system with dynamic, but also linear, constraint between
subsystems as a dynamic coupled subsystems.
Kinetic and potential energies and Rayleigh dissipative function of the hybrid
system, containing two subsystems — one linear oscillator and one nonlinear oscillator,
with two degree of freedom expressed by generalized coordinates x1 and x2 (see
Fig. 1(a)) are
E Jk Cm x m x m
x x x x
R
= + + +
+ −
1
2
1
2
1
2 2 21 1
2
2 2
2 1 2
2
2 1
2
˙ ˙
˙ ˙ ˙ ˙
,
E c x c x c xp = + +1
2
1
4
1
21 1
2
1 1
4
2 2
2˜ ˙
and
Φ = +1
2
1
21 1
2
2 2
2b x b x˙ ˙ ,
where m1 and m2 are masses, c1, c and c2 are the spring rigidity coefficients of
the linear elasticity low, and c̃1 the spring rigidity coefficient of the nonlinear
elasticity low, b1 and b2 coefficient of the system linear dumping forces. For this
system it is possible to show that is d
dt
E Ek p( )+ = −2Φ .
Energy interaction in this system is potential energy of the spring for coupling
nonlinear and linear system and is expressed in the form:
E m m
R
x m m
R
x x x m
Rk
C C C= + +
+ + +
+ −
1
2 4 4
1
2 4 4
1
2
2
4 4
1 2 1
2
2 2 2
2
1 2 2
J J J˙ ˙ ˙ ˙
or in the form
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 809
E a x a x a x xk = + +( )1
2
211 1
2
22 2
2
12 1 2˜ ˙ ˜ ˙ ˜ ˙ ˙ ,
where
˜ ˆa m m
R
a aC
11 1 2 11 114 4
= + + = +J
,
˜ ˆa m m
R
a aC
22 2 2 22 224 4
= + + = +J
, ˜ ˆa m
R
aC
12 2 124 4
= − =J
.
Coefficient ã12 = m
4
–
JC
R4 2 is coefficient of the subsystems coupling and as the
constraint is dynamical then this coefficient is coefficient of inertia. When this
coefficient is equal to zero, then the system coordinate x1 and x2 are decoupled and
there are not energy of the coupling, but there are energy of the influence of the
dynamic constraint by additional members.
Kinetic energy of the first subsystem as a one part of the hybrid system is
E m m
R
xk
C
( ) ˙1 1 2 1
21
2 4 4
= + +
J
or E a xk = 1
2 11 1
2˜ ˙ .
Kinetic energy of the second subsystem as a one part of the hybrid system is
E m m
R
xk
C
( ) ˙2 2 2 2
21
2 4 4
= + +
J
or E a xk = 1
2 22 2
2˜ ˙ .
Kinetic energy of the coupling of the subsystems as a two parts of the hybrid system is
E x x m
Rk
C
( , ) ˙ ˙1 2 1 2 2
1
2
2
4 4
= −
J
or E a x xk = ˜ ˙ ˙12 1 2.
Additional part of the kinetic energy of the first subsystem — reduction of the
dynamic constraint to the first subsystem
E m
R
x a xk d
C
( ) ˙ ˆ ˙1 2 1
2
11 1
21
2 4 4
1
2
= +
=J
.
Additional part of the kinetic energy of the first subsystem — reduction of the
dynamic constraint to the second subsystem
E m
R
x a xk d
C
( ) ˙ ˆ ˙2 2 2
2
22 2
21
2 4 4
1
2
= +
=J
.
When the coefficient of subsystems coupling equals zero: ã12 = m
R
C
4 4 2−
J
= 0, then
subsystems do not have kinetic energy interaction, but have additional part of kinetic
energy of the first subsystem — reduction of the dynamic constraint to the first
subsystem and additional part of the kinetic energy of the second subsystem —
reduction of the dynamic constraint to the second subsystem.
System of differential equations is based on the kinetic and potential energy and
Rayleigh dissipative function
E a x a x a x xk = + +( )1
2
211 1
2
22 2
2
12 1 2˜ ˙ ˜ ˙ ˜ ˙ ˙ ,
E c x c x c xp = + +1
2
1
4
1
21 1
2
1 1
4
2 2
2˜ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
810 K. HEDRIH (STEVANOVIĆ)
Φ = +1
2
1
21 1
2
2 2
2b x b x˙ ˙ .
System of differential equations based on the kinetic and potential energy and Rayleigh
dissipative function are in the following form:
˜ ˙̇ ˜ ˙̇ ˜ ˙a x a x c x c x b x11 1 12 2 1 1 1 1
3
1 1 0+ + + + = ,
˜ ˙̇ ˜ ˙̇ ˙a x a x c x b x22 2 12 1 2 2 2 2 0+ + + = .
After multiplying first and second equation or the previous system by 1
11â
and 1
22â
respective and after introducing the following notations:
κ1
12
11
=
˜
˜
a
a
, κ2
12
22
=
˜
˜
a
a
, ˜
˜
ω1
2 1
11
= c
a
,
˜
˜
ω2
2 1
22
= c
a
, ˜̃
˜
˜
˜
˜
ω ωN N
c
a
m
a1
2 1
11
1
2 1
11
= = , 2˜
˜
δi
i
ii
b
a
= , i = 1, 2,
we can write the following system of differential equations of the hybrid system:
˙̇ ˙̇ ˜ ˙ ˜ ˙ ˜̃x x x x xN1 1 2 1
2
1 1 1 1
2
1
32+ + + = −κ ω δ ω ,
˙̇ ˙̇ ˜ ˜ ˙x x x x2 2 1 2
2
2 2 22 0+ + + =κ ω δ .
For the basic linear equations of the linear dynamically coupled system of the
differential equations of the hybrid system containing two subsystems, one linearized
and one linear are in the form
˙̇ ˙̇ ˜ ˜ ˙x x x x1 1 2 1
2
1 1 12 0+ + + =κ ω δ ,
˙̇ ˙̇ ˜ ˜ ˙x x x x2 2 1 2
2
2 2 22 0+ + + =κ ω δ
we can define the following matrices:
A
1
1
=
κ
κ
1
2
, B
2
2
=
δ
δ
1
2
, C =
˜
˜
ω
ω
1
2
2
2
0
0
and frequency equation is in the form
λ λ
λ δ λ ω κ λ
κ λ λ δ λ ω
2
2
1 1
2
1
2
2
2 2
2 2
2
2
2
0A B C+ + =
+ +
+ +
=
˜
˜
or in the form
λ δ λ ω λ δ λ ω κ κ λ2
1 1
2 2
2 2
2
1 2
42 2 0+ +( ) + +( ) − =˜ ˜
with four roots λ1 2, = – δ̂1 � i p̂1 and λ3 4, = – δ̂2 � i p̂2 . Own amplitude numbers
we obtain from
A A
C
s
s
s
s s
s
1
1
2
2
2
1 1
22
( ) ( )
˜
−
=
+ +
=
κ λ λ δ λ ω
or
A
K
A
K
C
s
s
s
s s
1
21
2
22
( )
( )
( )
( )= =
and the solution of the basic linear coupled system we can write in the following form:
x t K e R p t K e R p tt t
1 21
1
01 1 01 21
2
02 2 02
1 2( ) cos ˆ cos ˆ( ) ˆ ( ) ˆ
= +( ) + +( )− −δ δα α ,
x t K e R p t K e R p tt t
2 22
1
01 1 01 22
2
02 2 02
1 2( ) cos ˆ cos ˆ( ) ˆ ( ) ˆ
= +( ) + +( )− −δ δα α
where amplitudes and phases R i0 and α0i are constants, depending of initial
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 811
conditions.
By using asymptotic averaged method, a first asymptotic approximation of the
solution of the hybrid system, containing dynamical coupled nonlinear and linear
system, we propose solutions in the following forms:
x t K e R t t K e R tt t
1 21
1
1 1 21
2
02 2
1 2( ) ( ) cos ( ) cos ( )( ) ˆ ( ) ˆ
= +− −δ δΦ Φ ,
x t K e R t K e R tt t
2 22
1
01 1 22
2
02 2
1 2( ) cos ( ) cos ( )( ) ˆ ( ) ˆ
= +− −δ δΦ Φ ,
where R ti( ) and Φi t( ) are unknown functions. Also we can write Φi t( ) = p̂ t1 + φi .
After averaging with respect to the full phase Φ1( )t we obtain the following system of
the first asymptotic averaged approximation of the equations for amplitudes R ti( ) and
phases Φi t( ):
˙ ( )R t1 0= ,
˙ ( )φ1 t = 3
16 1 21
1
22
2
22
1
21
2 1
2
p K K K K
N( ) ( ) ( ) ( )
˜
−[ ]ω ×
× e K R t e K K R tt t− −( ) [ ] + [ ] [ ]{ }2
21
1 3
1
2 2
21
1
21
2 2
2
21 22
ˆ ( ) ˆ ( ) ( )( ) ( )δ δ ,
˙ ( )R t2 0= ,
˙ ( )φ2 t =
3
16 2 21
2
22
1
22
2
21
1 1
2
ˆ
˜
( ) ( ) ( ) ( )p K K K K
N−[ ]ω ×
× 2 2
21
1 3
1
2 2
21
1
21
2 2
2
21 2e K R t e K K R tt t− −( ) [ ] + [ ] [ ]{ }ˆ ( ) ˆ ( ) ( )( ) ( )δ δ .
After integrating the system of averaged equations we obtain first approximation of
the amplitudes and phases of the solution
˙ ( )R t R1 01= = const ,
φ1( )t = – 3
16 1 21
1
22
2
22
1
21
2 1
2
p K K K K
N( ) ( ) ( ) ( )
˜
−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
21 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
+ α01,
R t R2 02( ) = = const ,
φ2( )t = – 3
16 2 21
2
22
1
22
2
21
1 1
2
ˆ
˜
( ) ( ) ( ) ( )p K K K K
N−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
21 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
+ α02 .
Full phases Φi t( ) are
Φ1( )t = p̂ t1 – 3
16 1 21
1
22
2
22
1
21
2 1
2
p K K K K
N( ) ( ) ( ) ( )
˜
−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
21 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
+ α01,
Φ2( )t = ̂p t2 – 3
16 2 21
2
22
1
22
2
21
1 1
2
ˆ
˜
( ) ( ) ( ) ( )p K K K K
N−[ ]ω ×
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
812 K. HEDRIH (STEVANOVIĆ)
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
21 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
+ α02 .
Solution in the first averaged asymptotic approximation
x t1( ) = K e Rt
21
1
01
1( ) ˆ−δ cos ˆ ˜
( ) ( ) ( ) ( )p t
p K K K K
N1
1 21
1
22
2
22
1
21
2 1
23
16
−
−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
+
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
2
01
1 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
α +
+ K e Rt
21
2
02
2( ) ˆ−δ cos ˆ
ˆ
˜
( ) ( ) ( ) ( )p t
p K K K K
N2
2 21
2
22
1
22
2
21
1 1
23
16
−
−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
+
2
1
21
1 3
01
2
2
2
21
1
21
2 2
02
2
02
1 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
α ,
x t2( ) = K e Rt
22
1
01
1( ) ˆ−δ cos ˆ ˜
( ) ( ) ( ) ( )p t
p K K K K
N1
1 21
1
22
2
22
1
21
2 1
23
16
−
−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
+
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
2
01
1 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
α +
+ K e Rt
22
2
02
2( ) ˆ−δ cos ˆ
ˆ
˜
( ) ( ) ( ) ( )p t
p K K K K
N2
2 21
2
22
1
22
2
21
1 1
23
16
−
−[ ]ω ×
× e K R e K K R
t t− −
( ) [ ] + [ ] [ ]
+
2
1
21
1 3
01
2 2
2
21
1
21
2 2
02
2
02
1 2
2
ˆ
( )
ˆ
( ) ( )
ˆ ˆ
δ δ
δ δ
α .
By using previous averaged solution we can obtain Lyapunov exponents in the
forms
λ
ω
δ1 1
2
1
2 1
2
1
1
2
1 0= +
= − <
→∞
lim ln ( )
˜
˙ ( ) ˆ
t t
x t x t ,
λ
ω
δ2 2
2
2
2 2
2
2
1
2
1 0= +
= − <
→∞
lim ln ( )
˜
˙ ( ) ˆ
t t
x t x t .
Also, taking into account that system is nonlinear we can introduce first Lyapunov
exponent in the forms
λ̃1 = lim ln ( )
˜
˜
˙ ( )
˜
˙ ( )
t
N
t
x t x t x t
→∞
+ +
1
2
1
1
2 1
2
1
2 1
4
1
2 1
2ω
ω ω
=
= lim ln
˜t t
E
m→∞
1
2 1
2
subsist(1)
12 ω
= −δ̂1 < 0.
3. Concluding remarks. As the energy (kinetic, potential and the power of energy
dissipation caused by dissipative forces), “carried” by a harmonic of a corresponding
oscillations “stroll” frequency depends both on the amplitude square and on the square
of its time derivatives, or frequency, the harmonic amplitudes, phases or frequencies
change during the oscillatory process and regime itself as well as the interaction
between them causes the energy change. The appearance of energy transfer from one
harmonic onto other or others of higher or lower frequencies can also be noticed here.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ENERGY INTERACTION BETWEEN LINEAR AND NONLINEAR OSCILLATORS … 813
In nonlinear systems we can observe the idea of equivalent systems exchange by
the use of elementary linear simple oscillators which would be uncoupled and would
make an equivalent replacement for a linearized system. And after that we may, using
the asymptotic methods of nonlinear mechanics. for instance the method of Krylov –
Bogolyubov – Mitropolskii, compose a system of necessary approximation of the
differential equations for nonlinear oscillation harmonic amplitudes and phases that are
close to an unperturbed oscillations. From such a system of adequate approximation
differential equations for amplitudes and phases that are mutually coupled by nonlinear
members, we may, using either quantitative or qualitative analysis, derive certain
conclusions about the flows and the transfer of energy by following the phase and
harmonics trajectories through the phase space of dynamical system local states around
singularities.
A generalization of a analytical analysis of the transfer energy between linear and
nonlinear oscillators for free vibrations with different type constraints as a couple
between two subsystems every of them with one degree of freedom is also important
but it is new task.
1. Hedrih (Stevanović) K. Interpretation of the transfer of energy from high-frequency to low-
frequency modes by averaging asymptotic method Krylov – Bogolyubov – Mitropolsky //
Asymptotics in Mech.: Proc. Second Int. Conf. / Eds Ali H. Nayfeh, K. V. Rozdestvensky. –
St.Petersburg, 1997. – P. 97 – 104.
2. Hedrih (Stevanović) K. Energetic analysis of oscillatory processes and the modes in nonlinear
systems // Solid Mech., Serbian – Greek: Symp. 1997. Sci. Meetings Serb. Acad. Sci. and Arts,
SANU Belgrade. – 1997. – 77, book 3. – P. 137 – 146.
3. Hedrih (Stevanović) K. Multifrequency forced vibrations of thin elastic shell // CD Proc. Fifth
EUROMECH Nonlinear Dynamics Conf., Eindhoven Univ. Technology / Eds D. H. van Campen,
M. D. Lazurko, W. P. J. M. van den Over. – 2005. – P. 2417 – 2426.
4. Hedrih (Stevanović) K. The dissipation function of a nonconservative system of mass particles //
Tensor. – 2002. – 63, # 2. – P. 176 – 186.
5. Hedrih (Stevanović) K. Mathematical analogy and phenomenological mapping: vibrations of multi
plate and multi beam homogeneous systems // Sci. Bul. “Politekhnica” Univ. Timisoara, Romania
(Trans. Mech.) / Eds L. Bereteu, T. Cioara, M. Toth-Tascau, C. Vigaru. – 2005. – 50, Spec. Issue. –
P. 1224 – 6077.
6. Hedrih (Stevanović) K. Partial fractional order differential equations of transversal vibrations of
creep-connected double plate systems // Fract. Different. and Appl. / Eds Alain Le Mahaute,
J. A. Tenreiro Machado, Jean Claude Trigeassou, Jocelyn Sabatier. – 2005. – P. 289 – 302.
7. Hedrih (Stevanović) K. Frequency equations of small oscillations mixed systems of the coupled
discrete and continuous subsystems // Mekhanika tverdogo tela (Rigid Body Mechanics). – 2003. –
33. – P. 174 – 189.
8. Hedrih (Stevanović) K. Eigen amplitude vectors and functions extended orthogonality of small
oscillations mixed systems of the coupled discrete and continuous subsystems // Facta univ., ser.
mech., automat. contr. and rob. – 2005. – 4, # 17. – P. 225 – 243.
9. Hedrih (Stevanović) K. Transversal vibrations of the axially moving double belt system with creep
layer // Preprints, 2nd IFAC Workshop Fract. Different. and Appl. (19 – 21 July, 2006, Porto,
Portugal). – 2006. – P. 230 – 235. http:/www.iser.ipp.pl.
10. Hedrih (Stevanović) K. Transversal vibrations of creep connected multi plate homogeneous
systems // CD Proc. Fifth EUROMECH Nonlinear Dynamics Conf. Eindhoven Univ. Technology /
Eds D. H. van Campen, M. D. Lazurko, W. P. J. M. van den Over. – 2005. – P. 1445 – 1454.
11. Hedrih (Stevanović) K. Discrete continuum method // COMPUTATIONAL MECHANICS,
WCCM VI in Conjunction with APCOM’04 / Eds Z. H. Zao, M. W. Zuang, W. X. Zhong. – 2004. –
P. 1 – 11.
12. Hedrih (Stevanović) K. Creep vibrations of a fractional derivative order constitutive relation
deformable bodies // Appl. Mech. Amer. – 2004. – 10. – P. 548 – 551.
13. Hedrih (Stevanović) K. On rheonomic systems with equivalent holonomic conservative systems
applied to the nonlinear dynamics of the watt’s regulator // Eleventh World Congr. in Mechanism
and Machine Sci. (IFToMM): Proc. – Tianjin, China: China Machine press, 2004. – Vol. 2. –
P. 1475 – 1479.
14. Nayfeh Ali. H. Transfer of energy from high-frequency to low-frequency modes // Second Int.
Conf. “Asymptotics in Mech.”: St.Petersburg Marine Techn. Univ., Russia, 13 – 16 October. – 1996.
– P. 44.
15. Stevanović (Hedrih) K. Application of the asymptotic method for study of the elastic bodies
nonlinear oscillations — energy analysis of the elastic bodies oscillatory motions: Doctor. Dis. Niš,
1975. – 331 p. (in Serbian).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
814 K. HEDRIH (STEVANOVIĆ)
16. Rašković D. Teorija oscilacija (Theory of Oscillations). – Naučna knjiga, 1965. – 503 p.
(in Serbian).
17. Hedrih (Stevanović) K. Transversal vibrations of double-plate systems // Acta Mech. Sinica. –
2006. – 22. – P. 487 – 501.
18. Hedrih (Stevanović) K. Modes of the homogeneous chain dynamics // Signal Proc. – 2006. – 86. –
P. 2678 – 2702.
19. Hedrih (Stevanović) K. Integrity of dynamical systems // J. Nonlinear Anal. – 2005. – 63. – P. 854
– 871.
20. Stevanović (Hedrih) K., Rašković D. Investigation of multi-frequencies vibrations in single-
frequency regime in nonlinear systems with many degrees of the freedom and with slowchanging
parameters // J. Nonlinear Vibrations Problems. – 1974. – # 15. – P. 201 – 202.
21. Mitropolskii Yu. A. Nestatsionarnye protsessy v nelineinykh sistemakh. – Kiev: AN USSR, 1955
(in Russian).
22. Mitropolskii Yu. A. Problemy asimptoticheskoi teorii nestatsionarnykh kolebanii. – Moskva: Nauka,
1964 (in Russian).
23. Mitropolskii Yu. A. Some problems in the development in nonlinear mechanics theory and
applications // Facta univ., ser. mech., automat. contr. and rob. – 1995. – 1, # 5. – P. 539 – 560.
24. Mitropolskii Yu. A. On application of asymptotic methods of nonlinear mechanics for solving some
problems of oscillation theory // Ibid. – 1996. – 2, # 6. – P. 1 – 9.
25. Hedrih (Stevanović) K. One-frequency nonlinear forced vibrations of uniform beams // Theor. and
Appl. Mech. – 1978. – # 4. – P. 39 – 50.
26. Hedrih (Stevanović) K. One-frequency proper nonlinear vibration of thin plate // Ibid. – 1978. –
# 4. – P. 51 – 65.
27. Hedrih (Stevanović) K., Mitić Sl. Dvofrekventne oscilacije plitke ljuske sa konacnim
deformacijama i uzajamni uticaj harmonica // Nelinearni Problemi Dinamike, Arandjelovac. – 1983.
– P. 197 – 203.
28. Hedrih (Stevanović) K., Kozić P., Pavlović R. O uzajamnom uticaju harmonika u nelinearnim
sistemima s malim parametrom // Rec. trav. Inst. Math. Nouv. ser. – 1984. – 4. – P. 91 – 102.
29. Hedrih (Stevanović) K. Multifrequency forced vibrations of thin elastic shells with a positive
Gausse’s curvature and finite deformations // Theor. and Appl. Mech. – 1985. – # 11. – P. 59 – 72.
30. Hedrih (Stevanović), K., Pavlov B. Strange attractors of the phase portrait of motion of a heavy
material point along the circle with an oscillating centre and under the influence of two frequency
couple // Proc. 2nd Int. Conf. Nonlinear Mech., Beijing 1993: Abstrts. – 1993. – 514. – P. 938 – 944.
31. Stevanović (Hedrih) K. Two-frequency nonstationary forced vibrations of the beams // Math. Phys.
– 1972. – 12.
32. Hedrih (Stevanović) K. Selected chapters from theory of nonlinear vibrations (in Serbian). – First
Edition. – Niš, 1975. – 180 p.
33. Hedrih (Stevanović) K. A trigger of coupled singularities // Meccanica. – 2004. – 39, # 3. – P. 295
– 314.
34. Hedrih (Stevanović) K. Homoclinic orbits layering in the coupled rotor nonlinear dynamics and
chaotic clock models. – Springer, 2005. – 421 p.
35. Hedrih (Stevanović) K. Contribution to the coupled rotor nonlinear dynamics // Nonlinear Sci. –
Belgrade: Acad. Nonlinear Sci., 2004.
36. Hedrih (Stevanović) K. Phase portraits and homoclinic orbits visualization of nonlinear dynamics
of multiple step reductor/multiplier // Eleventh World Congr. in Mechanism and Machine Sci.
(IFToMM): Proc. – Tianjin, China: China Machine press, 2004. – Vol. 2. – P. 1508 – 1512.
37. Hedrih (Stevanović) K., Veljović Lj. Nonlinear dynamics of the heavy gyro-rotor with two rotating
axes // Facta Univ., Ser. Mech., Automat. Contr. and Rob. – 2004. – 4, # 16. – P. 55 – 68.
38. Hedrih (Stevanović) K., Simonović J. Nonlinear phenomena in dynamics of car model // Ibid. –
2003. – 3, # 14. – P. 865 – 879.
39. Hedrih (Stevanović) K. Nonlinear dynamics of a heavy material particle along circle which rotates
and optimal control // Solid. Mech. and Appl. / Ed G. M. L. Gladwell. – 2005. – 26. – 504 p.
40. Hedrih (Stevanović) K. Vector method of the heavy rotor kinetic parameter analysis and nonlinear
dynamics. – Niš, 2001.
41. Hedrih (Stevanović) K. Energy transfer in double plate system dynamics // Acta Mech. Sinica. –
2008. – 24, # 3. – P. 331 – 344.
42. Hedrih (Stevanović) K., Simonović J. Transversal vibrations of a double circular plate system with
visco-elastic layer excited by a random temperature field // Int. J. Nonlinear Sci. and Numer.
Simulation. – 2008. – 9, # 1. – P. 47 – 50.
43. Hedrih (Stevanović) K., Simonović J. Transversal vibrations of a non-conservative double circular
plate system // Facta univ., ser. mech., automat. contr. and rob. – 2007. – 6, # 1. – P. 1 – 64.
44. Janevski G. Two-frequency nonlinear vibration of the antis metric laminated and angle — play
plate // Ibid. – 2005. – 4, # 17. – P. 345 – 358.
Received 14.09.06,
after revision — 02.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
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