ABSTRACT In this work, the dynamic behavior of self-synchronization and synchronization through mechanical interactions between the nonlinear self-excited oscillating system and two non-ideal sources are examined by numerical simulations. The physical model of the system vibrating consists of a non-linear spring of Duffing type and a nonlinear damping described by Rayleigh&apos;s term. This system is additional forced by two unbalanced identical direct current motors with limited power (non-ideal excitations). The present work mathematically implements the parametric excitation described by two periodically changing stiffness of Mathieu type that are switched on/off

J L Palacios

J M Balthazar

J Warminski

CiteSeerX

DowProceedings of IDETC/CIE 2005: 
ASME 2005 Design Engineering Technical Conferences and 
Computers and Information in Engineering Conference 
Long Beach, California, USA, September 24-25, 2005 
DETC2005-84756 
ON NUMERICAL SIMULATIONS OF A NONLINEAR SELF-EXCITED SYSTEM WITH 
TWO NON-IDEAL SOURCES 
 
 
J.L Palacios 
Universidade Regional Integrada do Alto Uruguai e 
das Missões, Departamento de Ciências Exatas e da 
Terra-URI, CEP 98802-470, Santo Ângelo, RS Tel: 
(0xx55) 33137984, fax: (0xx55) 33137902. 
jorgelpfelix@yahoo.com.br  
J.M. Balthazar 
Universidade Estadual Paulista, Instituto de 
Geociências e Ciências Exatas, Departamento de 
Estatística, Matemática Aplicada e Computação-
UNESP, CP 178, CEP 13500-230, Rio Claro, SP. Tel: 
(0xx19) 35340220R233. Prof. visitante do DPM -FEM 
–UNICAMP.  jmbaltha@rc.unesp.br 
 
R.M.L.R.F. Brasil 
Universidade de São Paulo, Departamento 
de Estrutura e Fundações – USP, CEP 
05508-900, São Paulo, SP. e-mail: 
rmlrdfbr@usp.br 
 
 
J. Warminski 
Department of Applied Mechanics, Technical 
University of Lubin. Nadbystrzycka 36, 20-618 
Lubin, Poland. Tel.: +48-81-5381197, fax: +48-
81-5250808. e-mail: 
jwar@archimedes.pol.lublin.pl 
 
 
 
 
Proceedings of IDETC/CIE 2005
ASME 2005 International D sign Engineering Tech ical Co ferenc s 
& Computers and Inform tion in Engineering Conference 
September 24-28, 2005, Long Beach, California USA 
 
DETC2005-84756 
 
ABSTRACT 
In this work, the dynamic behavior of self-synchronization 
and synchronization through mechanical interactions between 
the nonlinear self-excited oscillating system and two non-ideal 
sources are examined by numerical simulations. The physical 
model of the system vibrating consists of a non-linear spring of 
Duffing type and a nonlinear damping described by Rayleigh’s 
term. This system is additional forced by two unbalanced 
identical direct current motors with limited power (non-ideal 
excitations). The present work mathematically implements the 
parametric excitation described by two periodically changing 
stiffness of Mathieu type that are switched on/off. 
 
INTRODUCTION 
In engineering practice, we can distinguish systems where 
oscillations are caused by different reasons. The well-known 
oscillations are: a) Self-excited systems in which, roughly 
speaking, a constant input produces periodic output; b) 
Parametrically excited systems characterized by periodically 
changing in time parameters and c) Systems excited by an 
external force. 
All these systems were comprehensively analyzed in 
literature separately. However, we can find some papers that are  
nloaded From: https://proceedings.asmedigitalcollection.asme.org on 07/01/2019 Terms of Udevoted to interaction between different kinds of vibrations, for 
example: self and external excited vibrations, self and 
parametric excited vibrations, as well as between self-
parametric and external excited vibrations [15, 16]. 
We notice that when a forcing function is independent of 
the system it acts on, then the problem is called ideal. In such 
case, the excitation may be formally expressed as a pure 
function of time.  If in a certain model its ideal source is 
replaced by a non-ideal source, the excitation can be put in the 
form, where it is a function, which depends on the response of 
the system. Therefore, non-ideal source cannot be expressed as 
a pure function of time but rather as an equation that relates the 
source to the system of equations that describes the model. 
Hence, non-ideal models always have one additional degree of 
freedom when compared with similar ideal models [1, 2, 3, 4, 
5, 10]. 
In current literature, the name “self-synchronization” is 
used for synchronous rotation in the absence of any “direct” 
Kinematic coupling between rotating components. The 
behavior of the phenomenon of self-synchronization has been 
studied by a number of authors. Among them we mention the 
works of  [6, 7, 9]. Recently [11, 12, 13] studied self-
synchronization of the two identical DC motors, with a limited 1 Copyright © 2005 by ASME 
se: http://www.asme.org/about-asme/terms-of-use
Dowpower supply and with masses attached eccentrically to their 
rotating shafts supported by a structural frame. 
When synchronization is achieved by proper 
interconnections in the systems, i. e. without any artificially 
introduced external action, then the system is referred to as 
self-synchronized. A classical example of self-synchronization 
is the pair of pendulum clocks hanging from a light weight 
beam that was reported by Huygens. He observed that both 
pendulums oscillated with the same frequency [6]. 
In this paper, we are interested in analyzing the influence 
of the response of the nonlinear and self-excited systems on the 
two unbalanced identical direct current motors, which is being 
an extension of the work of [16]. 
 
MODEL 
 
Let us consider a nonlinear and self-excited model, which 
includes two identical direct current (DC) motors with limited 
power, operating on a structure (Fig. 1).  
The excitation of the system is limited by the 
characteristics of the energy source. Vibration of the system 
depends on the motion of the motors, and the energy sources 
motion depends on vibration of the system, as well. Then, 
coupling of the vibrating oscillator and the two DC motors 
takes place. 
Hence, it is important to analyses what will happen to the 
motors, as the response of the system changes. 
 
 
 
Figure 1. Non-ideal nonlinear and self-excited system 
model 
 
Taking into account the differential equation of the 
complete electro-mechanical system presented in [16] and Fig. 
1, we can write for ( 1, 2)s = : 
 
( ) ( ) cos coss s s s s s s s s s s sJ L H m r x m grϕ ϕ ϕ ϕ ϕ= − + −  
2 3
1 1 1
2
2
1
ˆ( ) ( )
                              ( sin cos )j j j j j j
j
mx c c x x kx k x
m r ϕ ϕ ϕ ϕ
=
+ − + + +
= −∑
 
(1)  
nloaded From: https://proceedings.asmedigitalcollection.asme.org on 07/01/2019 Terms of where x  is oscillatory coordinate of vibrating body; sϕ  is 
rotational coordinate of each DC motor; sϕ  is rotational speed 
of rotors; sJ  is a moment of inertia of each motor; sm  is 
unbalanced mass of each motor; sr  is eccentricity of each 
unbalanced mass; ( )s sL ϕ  is a controlled torque of each DC 
motor; ( )s sH ϕ  is a resistance torque of each DC motor. 
In order, to obtain the governing equations of the system in 
their dimensionless form, we define the non-dimensional time 
tτ ω=  and the non-dimensional displacement 
1 1
MX x
m r
= , 
where k
M
ω =  is natural frequency of the system and 
1 2M m m m= + + . 
Hence, we will obtain: 
 
ˆ coss s s s s s sâ b R Xϕ ϕ β ϕ′′ ′ ′′= − +  
2 3
2
2
1
( ) ( )
                        ( sin cos )j j j j j
j
X qX X X pX
R
µ
ϕ ϕ ϕ ϕ
=
′′ ′ ′+ − + + +
′ ′′= −∑
               (2) 
 
The dimensionless parameters in Eq. (2) are defined as: 
2 2
1 1
ˆ ss
s
aa
m r Iω
= , 2
1 1
ˆ s
s
s
bb
m r Iω
= , 
1 1
s s
s
m rR
m r
= , 1s
s
m
MI
β = , 
1c
M
µ
ω
= , 
2
1 1 1
3
ĉ m rq
M
ω
= , 
2
1 1 1
2
k m rp
M
= , 2
1 1
s
s
JI
m r
= . 
 
From the mathematical point of view two different 
parametric excitations were placed in the equation of motion in 
their dimensionless form. From second equation of Eq. (2) we 
will obtain: 
2
2 2
1
2
2
1
( ) (1 cos 2 )(1 )
                          ( sin cos )
j j
j
j j j j j
j
X qX X pX X
R
µ α ϕ
ϕ ϕ ϕ ϕ
=
=
′′ ′ ′+ − + + − +
′ ′′= −
∑
∑
  
                                                                                     (3) 
 
NUMERICAL RESULT 
 
Next, we carried out, a number of numerical simulations, 
in order to observe the interaction between the two identical 
non-ideal DC motors and nonlinear and self-excited structural 
system (as regular and irregular motions). Furthermore, we 
observe the self-synchronization and synchronization 
phenomena in pre-resonance, resonance and post-resonance 
regions. 
Equation 2 was simulated by using the block Diagrams in 
SIMULINK that models the non-ideal nonlinear 2 Copyright © 2005 by ASME 
Use: http://www.asme.org/about-asme/terms-of-use
Down(parametrically) self-excited system. To obtain different 
regimes in the system we varied the torques of each DC motor 
and   the initial rotation of second motor. 
The first set of numerical results, shown in Fig. 2, 
illustrates the development of self-synchronization by intervals 
when the torques of each DC motors are approximately equal 
1â  = 1 and 2â  = 0.9. 
 
 
 
 
Figure 2. a) Velocity of rotors, b) phase plane for 
1â  = 1 and 2â  = 0.9: self-synchronization by intervals. 
 
Figure 2(a), shows that the rotors rotate in the same 
direction and arrive at some average angular velocity in steady 
state motion where the angular velocities are in phase and out 
of phase by intervals (the rotors synchronize phase and anti-
phase). The frequencies of the rotors are below resonance (pre-
resonance). Figures 2(b) shows the phase plane of the 
dynamical behavior of system, in this case, beat phenomenon. 
The second   set of numerical results, shown in Fig. 3, we 
illustrates the development of self-synchronization when the 
torques of each DC motor are equal 1â  = 1.0 and 2â  = 0.93. 
Figure 3(a), shows that the rotors turn in the same direction 
and arrive it at some synchronous velocity in phase and in 
steady state motion, the velocities of rotors are below of 
resonance region (pre-resonance). Fig. 3(b) shows the 
dynamical behavior of system on phase plane of periodic 
motion. 
In the fourth set of numerical results, shown in Fig. 4, we 
illustrate the absence of self-synchronization when the torques 
of each DC motor are different 1â  = 3.0 and 2â  = 2.2. 
Figure 4(a), shows that the rotors turn in the same direction 
and arrive it at some average angular velocity in steady state 
motion, the velocities of rotors are in the post resonance region  
loaded From: https://proceedings.asmedigitalcollection.asme.org on 07/01/2019 Termsand synchronization anti-phase. Figures 4(b) shows the 
dynamical behavior of system on the phase plane of motion p-
periodic. 
 
 
 
 
Figure 3. a) Velocity of rotors and b) phase plane for  
1â  = 1.0 and 2â  = 0.93: self-synchronization. 
 
 
 
 
Figure 4. a) Velocity of rotors and b) phase plane for  
1â  = 3.0 and 2â  = 2.2: absence of self-synchronization. 
 
Figure 5, shows different regimes in the phase plane for 
the case of a non-ideal parametrically and self excited system 
when was considered the Eq. (3) in the motion equation and we 3 Copyright © 2005 by ASME 
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Dowvaried the parameter 2α  of the parametric excitation of the 
second DC motor. Other parameters are fixed. For 2α =1, we 
see a development chaotic, 2α =1.5 we see a development 
periodic with p-period, 2α =2 we see that is tending to a limit 
cycle with 1-period, 2α =3 we see a development periodic with 
1-period. 
 
 
 
 
 
 
Figure 5. For values different of the parameter 2α  of the 
parametric excitation of second DC motor:  
(a) 2α  = 1, (b) 2α  = 1.5, (c) 2α  = 2 and (d) 2α  = 3.  
nloaded From: https://proceedings.asmedigitalcollection.asme.org on 07/01/2019 TermFinally, Figure 6 shows the results of the influence self-
excited coefficient q  on the cubic nonlinear of the system. In 
this case, the interaction is examined when the two rotors arrive 
it at some synchronous velocity, with the cubic nonlinear 
coefficient fixed p  and for values different self-excited 
coefficient q . 
 
 
 
 
Figure 6. Phase plane for 1â  = 2.0 and 2â =2.01:  
(a) q =0.05, p =16 and (b) q =4.0, p =16. 
 
CONCLUSIONS 
 
In this paper, we analyzed a type of non-linear 
phenomenon of self-synchronization and synchronization in 
non-ideal self-excited mechanical systems, i. e. we observe the 
behavior of the self-synchronization when is considered only 
mechanical interactions between the self-excited oscillating 
system and two unbalanced energy source with limited power.  
The numerical simulation shows a bifurcation sequences 
on phase plane for slow transition through resonance. 
We remarked that with control parameters  (the constants 
torques) we might control the presence and absence of self-
synchronization as synchronization in phase and anti-phase; in 
this case, we observe motions of regular (periodic, quasi-
periodic) and irregular (chaotic).   
From a mathematical point of view not yet applicable in 
the mechanical model, two different parametric excitations 
were placed in the equation of motion. We observe a 
bifurcation sequences from development chaotic until periodic 
for each variation of the parametric excitation. 4 Copyright © 2005 by ASME 
s of Use: http://www.asme.org/about-asme/terms-of-use
  
  
  
    
DownACKNOWLEDGMENTS 
The first author acknowledges support by Universidade 
Regional Integrada URI. The second and third authors 
acknowledge financial support by FAPESP - Fundação de 
Amparo à Pesquisa do Estado de São Paulo and CNPq - 
Conselho Nacional de Pesquisas, both Brazilian Research 
Funding Agencies. 
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DETC2005-84756 ON NUMERICAL SIMULATIONS OF A NONLINEAR SELF-EXCITED SYSTEM WITH TWO NON-IDEAL SOURCES

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DETC2005-84756 ON NUMERICAL SIMULATIONS OF A NONLINEAR SELF-EXCITED SYSTEM WITH TWO NON-IDEAL SOURCES

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