Abstract
This paper describes a nonlinear phenomenon in the dynamical behavior of a nonlinear system under two nonideal excitations: the selfsynchronization of unbalanced direct current motors. The considered model is taken as a Duffing system that is excited by two unbalanced direct current motors with limited power supplies. The results obtained by using numerical simulations are discussed in details
Nonlinear system; nonideal system; selfsynchronization
A short note on a nonlinear system vibrations under two nonideal excitations
J. L. Palacios^{I}; J. M. Balthazar^{II}; R. M. L. R. F. Brasil^{III}
^{I}Universidade Regional Integrada de Santo Ângelo Departamento de Ciências Exatas e da TerraURI C .P. 203 98802470 Santo Ângelo, RS. Brazil jfelix@urisan.tche.br
^{II}Universidade Estadual Paulista Instituto de Geociências e Ciências Exatas Departamento de Estatística, Matemática Aplicada e Computação UNESP C. P. 178 13500230 Rio Claro, SP. Brazil jmbaltha@rc.unesp.br
^{III}Universidade de São Paulo, Escola Politécnica Departamento de Estrutura e Fundações USP C. P. 61546 9524970 São Paulo, SP. Brazil rmlrdfbr@usp.br
ABSTRACT
This paper describes a nonlinear phenomenon in the dynamical behavior of a nonlinear system under two nonideal excitations: the selfsynchronization of unbalanced direct current motors. The considered model is taken as a Duffing system that is excited by two unbalanced direct current motors with limited power supplies. The results obtained by using numerical simulations are discussed in details
Keywords: Nonlinear system, nonideal system, selfsynchronization
Introduction
We remark that the study of nonideal vibrating systems, that is, those where the excitation is influenced by the response of the system, is still considered to be a major challenge in theoretical and practical engineering research.
When the excitation is not influenced by the response, it is said to be an ideal excitation or an ideal source of energy. On the other hand, when the excitation is influenced by the response of the system, it is said to be nonideal. Thus, depending on the excitation, one refers to vibrating systems as ideal or nonideal.
The behavior of ideal vibrating systems is well known in current literature, but there are few published results on nonideal ones. Generally, nonideal vibrating systems are those for which the power supply is limited. The behavior of the vibrating systems departs from the ideal case as power supply becomes more limited. For nonideal dynamical systems, one must add an equation that describes how the energy source supplies the energy to the equations that govern the corresponding ideal dynamical system. Thus, as a first characteristic, the nonideal vibrating system has one more degree of freedom than its ideal counterpart.
The first kind of nonideal problem arising in current literature is the socalled Sommerfeld effect, discovered in 1902 (see Sommerfeld, 1902), discussed in a book by (Kononenko, 1969), and entirely devoted to the subject. Recently, a review of different theories concerning this subject, was presented in (Balthazar et al, 2001), (Balthazar et al., 2002) and (Palacios, 2002).
Selfsynchronization of shafts is a wellknown nonlinear phenomenon, whereby two (or more) unbalanced shafts on a common movable structure may rotate synchronously due to interaction via structural vibrations only, even in the absence of any direct kinematics coupling. The phenomenon has been extensively studied by asymptotic methods to predict possible (multiple) steadystate rotational motions and to evaluate their stability, mostly with application to the design of vibrators with a reduced number of driving motors. Certain cases of undesirable shaft selfsynchronization in engineering have also been studied, but only steadystate motions were analyzed. Results of numerical simulation of transient selfsynchronization of rotating shafts, one potential application being gas turbine engines with multiple shafts, was studied by (Dimentberg, 2001).
In this paper, two unbalanced dc motors are used to demonstrate the selfsynchronization that may occur when the shafts rotation speeds become temporarily close to one another depending on the torque, considered as the control variable, and of a support with nonlinear stiffness.
This paper is an extension of the following previous works: (Palacios, 2002) that studied a portal frame with nonlinear characteristic of elasticity under one nonideal excitation; (Balthazar et al., 2001) and (Warminski and Balthazar, 2001) and (Warminsky and Balthazar, 2003) that studied the nonstationary regime of a DC motor with limited power supply; (Kang, 2002) that studied the nonlinear dynamic of nonlinear systems subjected to double excitations; (Dimentberg et al., 2001) that studied the selfsynchronization of rotors and Sommerfeld effect and (Blekhman, 1988) that studied the selfsynchronization of two unbalanced rotating machines mounted on a linearly elastic support. A first announcement of this work was done by (Palacios et al, 2003).
Nomenclature
â = control parameter, dimensionless
= parameter related to a type of motor, dimensionless
F_{d} = Nonlinear damping function
F_{s} = Nonlinear stiffening function
J_{1} = moment of inertia of rotor 1, kgm^{2}
J_{2}= inertia moment of rotor 2, Kgm^{2}
k_{c} = friction coefficient in the bearing
k_{M}= motor parameter
M_{1} =driving torque of the motor 1, Nm
M_{2} = driving torque of the motor 2, Nm
m_{0} =total mass of vibrating parts, kg
x = generalized coordinate, dimensionless
Greek Symbols
j_{1}= angular displacement of motor 1, rad.
j_{2}= angular displacement of motor 2, rad.
= natural frequency of the system, dimensionless
Subscripts
1 = relative to the horizontal displacement
2 = relative to the vertical displacement
Dynamical Model of the Nonlinear System
Consider a nonlinear mechanical system consisting of two unbalanced rotors driven by two dc motors with limited power supplies and mounted on an elastic support with nonlinear stiffening and damping. Figure 1 illustrates such a system.
The differential equations of motion may be written as follows (see, for instance, Nóbrega, 1994; Dimentberg et al., 2001; Warminski and Balthazar, 2001; Balthazar et al., 2001; Kang et al., 2002; Blekhman, 1988 and Palacios, 2002):
where
m_{0}= is the total mass of vibrating parts,
J_{1}, J_{2}= moment of inertia of motors 1 and 2,
F_{d} ()= Nonlinear damping function (for Van der Pol model, it is g(x^{2}1)),
F_{s}(x)= Nonlinear stiffening function (for Duffing model, it is k_{1}x+k_{2}x^{3}),
j_{1}, j_{2} = Rotation angles of the rotors measured from the lowest vertical position,
x = Vertical displacement of the supporting body from the equilibrium position,
k_{c} = Friction coefficient of bearings,
k_{M} = Motor parameter,
M_{1} , M_{2} = The torques of the motors,
q_{1},q_{2},a_{j} (j = 1...4) = Parameters depending on the eccentricities of rotors and physical parameters of the system.
The motor torque can be expressed (Balthazar et al., 2001), for nonstationary regime, by the following expressions:
and, if stationary regime is considered, the above expressions become:
where U_{1}, U_{2} are voltages applied across the armature of the motors.
Next, we will consider the nonideal Duffing model in dimensionless form:
where
t=wt Defined as dimensionless time,
â_{1}, â_{2} = Dimensionless torques applied by the dc motors, used as control parameters,
= identical dc motors model,
=dimensionless damping coefficient,
= dimensionless natural frequency of the Duffing oscillator,
= dimensionless cubic stiffness coefficient.
In the next section we will study the selfsynchronization of the nonideal system represented by differential equations (4) using the characteristic curve of the motors in the straightline approximation (3).
Numerical Simulation Results
To simulate the mathematical model of the system (4), MATLABSIMILUNKÒ Software, was used. Differential equations were solved by integration method of RungeKutta. Numerical, dimensionless data for the nonideal nonlinear system are shown in Table 1.
Results of computations are presented Fig. 24. Behavior of the system was observed during the selection of the torques as â_{1}»â_{2}, â_{1}¹â_{2}.
We analyze the selfsynchronization of the two unbalanced rotors in postresonance and resonance regions related to the difference between rotors velocities. Figures 2(a) and 4(a) have an expanded portion bellow.
First Set of Numerical Simulation Results:
Figure 2 illustrates the development of selfsynchronization in the nonideal and nonlinear mechanical system when the dimensionless torques are â_{1}=5.0, â_{2}=5.04. We observe that the rotors turn in the same direction and arrive to some average rotational velocity in steady state motion, see Fig. 2(a), where the rotational velocities are in antiphase (the rotors synchronize antiphase), see the zoom of Fig. 2(a) bellow. The velocities of the rotors are out of the resonance region. Note also that the rotor velocity (difference) approaches zero in the stationary regime, see Fig 2(c). At the same time, the vertical displacement of the supporting body decreases; see Fig. 2(b).
Second Set of Numerical Simulation Results:
Figure 3 illustrates the absence of selfsynchronization when the torques are different, â_{1}=5.0 and â_{2}=1.5. It is seen that the rotational velocities of the rotors have different values.
The rotational velocity of the second rotor is captured at the resonance region of the system and the rotational velocity of the first rotor is above of the resonance region (see Fig. 3(a)). Note that the vertical displacement of the supporting body does not decrease (see Fig. 3(c)), and note also that the rotor velocity difference (see. Fig. 3(b)) does not tend to zero; i.e. the selfsynchronization is absent.
Third Set of Numerical Simulation Results:
Figure 4 shows the synchronization for a certain time interval, where we may observe the reduction of the damping of the support. We also observe that the response follows the some synchronization of the rotation of the rotors. In this case the constant torques considered were â_{1}=1.5 and â_{2}=3.0.
On other hand, we observe chaotic behavior in the interaction between the Duffing oscillator and the two unbalanced rotors when damping of the support is reduced (see Figs 4(b) and (c)).
Conclusions
A practical problem of synchronization of a nonideal and nonlinear vibrating system was posed and investigated by means of numerical simulations. It has also been shown, that by making constant the variation of torques, we may control the selfsynchronization and synchronization (in the system).
This work has as its motivation the investigation of a class of vibrating machines: crushers, mills, screens, feeder, etc. We developed mathematical models and implemented numerical analysis that may also be used for control purposes.
Acknowledgements
The first author thanks to the department of Exacts Sciences of URISAN that provided computational material support and the second and third authors thank grants by FAPESP and CNPq, both Brazilian Research Funding Agencies
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Publication Dates

Publication in this collection
18 Mar 2004 
Date of issue
Dec 2003