Abstract
In this work a particular system is investigated consisting of a pendulum whose point of support is vibrated along a horizontal guide by a two bar linkage driven from a DC motor, considered as a limited power source. This system is nonideal since the oscillatory motion of the pendulum influences the speed of the motor and viceversa, reflecting in a more complicated dynamical process. This work comprises the investigation of the phenomena that appear when the frequency of the pendulum draws near a secondary resonance region, due to the existing nonlinear interactions in the system. Also in this domain due to the power limitation of the motor, the frequency of the pendulum can be captured at resonance modifying completely the final response of the system. This behavior is known as Sommerfeld effect and it will be studied here for a nonlinear system.
Nonideal system; pendulum; escape; sommerfeld effect; bifurcation
Escape in a nonideal electromechanical system
D. Belato^{I}; H. I. Weber^{II}; J. M. Balthazar^{III}
^{I}DPM  Faculdade de Engenharia Mecânica, UNICAMP, C. P. 6122, 13083970 Campinas, SP. Brazil, belato@fem.unicamp.br
^{II}DEM  Pontíficia Universidade Católica, PUC/RJ, 22453900 Rio de Janeiro, RJ. Brazil, hans@mec.pucrio.br
^{III}Instituto de Geociências e Ciências Exatas, UNESP/Rio Claro, C. P. 178, 13500230 Rio Claro, SP. Brazil, jmbaltha@rc.unesp.br
ABSTRACT
In this work a particular system is investigated consisting of a pendulum whose point of support is vibrated along a horizontal guide by a two bar linkage driven from a DC motor, considered as a limited power source. This system is nonideal since the oscillatory motion of the pendulum influences the speed of the motor and viceversa, reflecting in a more complicated dynamical process. This work comprises the investigation of the phenomena that appear when the frequency of the pendulum draws near a secondary resonance region, due to the existing nonlinear interactions in the system. Also in this domain due to the power limitation of the motor, the frequency of the pendulum can be captured at resonance modifying completely the final response of the system. This behavior is known as Sommerfeld effect and it will be studied here for a nonlinear system.
Keywords: Nonideal system, pendulum, escape, sommerfeld effect, bifurcation
Introduction
In the configuration of a mechanical system several times some of its dynamical proprieties are neglected due to the imposed suppositions for its correct working inside a determined frequency band. Nevertheless, out of this band these suppositions can be broken, revealing new conditions of operation where more elaborated tools of investigation must be used in order to avoid undesirable regimes of motion. These regimes appear mainly during the passage through resonance, which for a nonlinear system reflects in multiple critical domains of the control parameter, where the existing bifurcational processes alter completely the preliminary dynamical characteristics of the system. Generally, critical regimes of motion are reached with the variation of the control parameter value (Belato et al (2001)), where due to the approximation of the frequency of motion to the resonance region motion can become critical. When this happens during a slow and continuous variation of the external frequency, we say that the frequency is "captured at resonance" and this event is related to the disappearance of a separatrix of motion inside the phase portrait, where the existing manifold becomes resonant.
These dynamical proprieties occur in any mechanical system but when the power supplied by the energy source is unlimited, the passage through resonance is fast during the runup or shutdown of the mechanism and it does not affect the motion of the system. However, problems will appear when a slow runup or a slow shutdown happens, because the frequency of excitation can become critical by its capture at resonance. The slow passage through the resonance can be observed when the energy source power is limited (Nayfeh and Mook (1979)), because the motor does not get sufficient power to provide a perfect operation of the mechanism. Although this latter sounds impracticable, there are interesting topics of research in this field (Dimentberg et al (1997) and Belato (2001)), where the application of this supposition is done in order to understand and to avoid undesirable characteristics of motion during any stage of operation in a mechanism.
In a complete description of these systems: a dynamical system with a limited energy source is called nonideal when the motion of the oscillatory system interferes in motion the driving force, because there is a dynamic interaction between the motor and the pendulum and viceversa. As the action of the motor depends on the motion of the pendulum, it cannot be described by a determined function on time but its motion must be represented by a differential equation increasing the degrees of freedom of the system. In this case, the frequency of excitation is not constant but it is described by a function with given amplitude, which increases near a resonance region. It is the variation of this amplitude of the frequency that allows the capture in a slow and continuous passage through resonance, and that makes the vibrations between the mechanism and its foundation greater and greater. This will lead to a possible destruction of the mechanism. This process is known as Sommerfeld effect and it is characterized by the "flow of energy" from the energy source to the oscillation system and viceversa, and it can lead to the synchronization of all motions of the mechanism (Belato (2001)).
Generally, the escape from the potential well in the forced nonlinear oscillator can be associated with homoclinic tangles and fractals basin, which is often followed by chaotic motions (Thompson et al (1987)). Thompson (1989) studied the escape of the asymmetric potential and observed that the stability loss of the system occurs through a fold bifurcation. When an attractor set undergoes a discontinuous change, disappearing from the phase space, the bifurcation is catastrophic and it is known as bluesly catastrophe or boundary crisis. A catastrophe in a dissipative dynamical system that causes an attractor to completely lose stability will result in a transient trajectory making a rapid jump in the phase space to some other attractor (Stewart and Ueda (1991)). This event occurs due to a fractal basin boundary, and in the presence of even infinitesimal noise we cannot predict to which of the remote attractors the system will jump (Thompson and Soliman (1991)). Stewart et al (1995) studied the optimal escape of periodically forced oscillations from a potential well. All the phenomena mentioned above, are related with softening systems near the fundamental resonance region.
In this work, we will investigate the numerical results of a particular nonideal system that consists of a simple pendulum connected to a DC motor through a shaftcrank mechanism (See "Fig. 1"). The point of support of the pendulum undergoes a horizontal displacement defined by s_{A} = a cosq + b where q defines the angular displacement of the motor. In this case, the nonideal condition is obtained for a determinate choice of the parameters of the DC motor, which becomes it of limited power. The analysis will concentrate near a secondary resonance region where the excitation frequency is close to q' » 2.5 for the chosen parameters. In this region a global bifurcation known as bluesky catastrophe occurs, which is characterized by the disappearance of a limited attractor of the phase space and the system jumps to a remote attractor that can be limited or not. This event characterizes the escape phenomenon from a potential well, leading to the rotational solutions of the pendulum (Belato (1998)). Although, this reflects in a nonlinear event we will study it on the nonideal concept in order to understands its characteristics.
Other features of this system were analyzed in previous works, near the main resonance region (q' » 1), including: periodic, multiperiodic, quasiperiodic and chaotic motion (Belato et al (2001) and Belato (2001)). Belato et al (2000) studied the evolution of the phase portrait of a similar ideal Hamiltonian system concerning the variation of a control parameter, where the studied dynamical system is composed by a shaftcrank mechanism and the simple pendulum. It summarizes the nonlinear behavior of our system without considering the nonideal conditions, i.e., the differential equations of motion of the mechanism do not include the equations of the motor.
This paper is organized as follows. The Section 2 contains the complete differential equations that govern the motion of the electromotorpendulum system. Section 3 gives the results of numerical simulation and its interpretation and Section 4 contains the conclusions.
Nomenclature
a = length of the connecting rod, m
b = length of the crank mechanism, m
c = parameters of the differential equations, dimensionless
c_{A }= vicous damping coefficient at the pin, Ns/m
c_{m} = internal loss coefficient in the motor, mNs/rad
g = gravity coefficient, m/s^{2}
i = current, A
I = current, dimensionless
i_{0} = medium current, dimensionless
J = moment of inertia of the load and motor, kg/m^{2}
K_{E} = voltage constant, Vs/rad
K_{T} = torque constant, Nm/A
L = pendulum length, m
L = inductance, H
m = pendulum mass, kg
p = constant, dimensionless
q = constant, dimensionless
R = electrical resistance, W
t = time, s
t* = time, dimensionless
T_{f} = constant friction torque in the motor, Nm
V = motor voltage,V
Greek Symbols
a = angular displacement of the pendulum, rad.
a' = angular speed of the pendulum, dimensionless.
m_{l} = viscous damping coefficient at the pendulum, mNs/rad
q = angular displacement of the motor, rad.
= mean angular speed of the motor, dimensionless.
q' = angular speed of the motor, dimensionless
w_{0} = natural frequency of the pendulum, rad/s
Î_{1} = ratio between a and b, dimensionless
Î_{2} = ratio between a and l, dimensionless
Mathematical Model
Defining the dimensionless variables: t* = w_{0}t and I = i/i_{0}, where w_{0} = (g/l)^{1/2} is the natural frequency of the pendulum and i_{0} is the mean current in the armature, we obtain the dimensionless complete system of differential equations that governs the motion of the electromotor pendulum mechanism (See details in Belato (2001)) given by:
where , , , , c_{3} = ma^{2}, c_{4} = mal, , , , , , defines the parameters of the equations and the primes denote derivatives with respect to t*.
The chosen control parameter is the motor voltage, represented by the parameter c_{8}. In previous works, this parameter was considered as a constant for the analysis of the steady state behavior of the system. Here, we will adopt this control parameter as a continuous function in the time in order to analyze the transient behavior of the electromotor pendulum system simulating the runup process of the mechanism. For this, we choose the following function:
where p and q are constants values. Therefore, this work consists in the description and identification of the pendulum behavior when the q parameter is varied.
Numerical Results
We carried out a large number of numerical simulations with different initial conditions and numerical values of the physical parameters of the problem. SimulinkÔ Toolbox of the MATLABÔ is used to simulate the system of differential equations and the numerical integrator used is the Runge Kutta fifth order (RK45) with variable steplength. The used parameters for the numerical simulation of the system of differential equations (1) are: Î_{1}=Î_{2}= 0.233, c_{1} = c_{2} = c_{6} = 0, c_{3} = 0.00098, c_{4} = 0.0042, c_{5} = 0.01055, c_{7} = 0.01, c_{9} = 33.83, c_{10} = 78.40 and c_{8} = ptanh(qt*) is the parameter used to control the diversity of nonlinear behavior of this system, with p = 290 and q is varied. The adopted initial conditions are: q'(0) = q(0) = 0, a'(0) = a(0) = 0 and I(0) = 0.
When the value of the control parameter is constant, results for different motor voltages will not consider the transient response, and it is possible to detect different steadystate behaviors of the pendulum when the motor speed increases. There are three main resonance regions for this system, namely: the fundamental resonance (q' » 1.0), where a chaotic attractor appears, and two secondary resonance regions. Near the first one (q' » 2.5 for the choice parameters), it is verified that the inevitable escape from the potential well leads to divergence to infinity and in this case the pendulum starts a rotational motion, i.e., the pendulum escapes from the potential well, its motion is captured by the motor motion and both enter in a synchronized state, Belato (2001).
"Figure 2a" shows the behavior of the pendulum motion as the motor speed increases, for a constant value of the control parameter c_{8}. The point (a',a) = (0,0) determines the minimum of the potential well for q' < 2, while it determines the local maximum between the potential twowell for q' > 5 and the separatrix (heteroclinic orbits), calculated using the undamped unforced pendulum equation, separates bounded and unbounded solutions. When the motor is introduced in the mechanism (forced term), the initial configuration of phase portrait changes due the presence of the bifurcations that alter the behavior of the system in the main resonance regions. In one of them, called the first secondary resonance region observed close to q' » 2.5, occurs a global bifurcation known as bluesky catastrophe. A periodic or quasiperiodic attractor loses stability disappearing from the phase space and the pendulum jumps to one of two remote unlimited attractors, characterize by a rotational motion. In reality, in this frequency band occurs the appearance of a homoclinic connection in the system. Due to this connection, the point (a',a) = (0,0) becomes unstable when the motor speed travels through the domain 2 < q' < 3, where for the values q' > 5.0 the pendulum motion converges to one of two bound periodic solutions oscillating around the points a » ±(p/2), Belato (1998). The stability loss in this region occurs due to the existence of a saddlenode bifurcation and the graphic below represents a schematic of all the bifurcational process of the electromotor pendulum, when a constant control parameter is varied.
Now, considering the control parameter as defined in "Eq. 2" we will investigate the pendulum behavior near the first secondary resonance region, determined by q' » 2.5 for the chosen parameters, in order to simulate small variations on the transient response during the operation of the mechanism. In this case, we can determine if the passage through resonance will or not affect its motion, since for this system the capture at resonance incites the escape of the pendulum from the potential onewell, diverging totally from the original attractor. To illustrate it, we adopt the value p = 290 to obtain the numerical results presented in "Fig. 3" and "Fig. 4".
"Fig. 3" simulates the motion condition of the mechanism when q = 0.05. For this value, the system behavior summarizes in: (a) the motor speed is not constant due the nonideal condition, but it converges for a function with small amplitude when the system reaches the steadystate solution, "Fig. 3a"; (b) during the steadystate stage, the phase space of the motor presents a almost periodic motion, but there are small dispersions of points mainly near the positions q = ± (p/2), "Fig. 3c"; (c) the transient response of the pendulum with greater amplitude converges for a limited attractor, "Fig 3b and d". Thus, for the chosen parameters, the behavior of the pendulum is periodic and it does not belong to a resonance region.
Now, considering smaller curvature of the function in (2) with q = 0.04, we can simulate a slower passage through the critical speed of this system, "Fig. 4 and 5". When the frequency of the pendulum passes by a resonant domain (i.e. q' » 2.5), the motor speed is captured at resonance and in this stage, the pendulum escapes from the potential well reaching an unlimited remote attractor, characterized by rotational motion in clockwise (or anticlockwise) direction, "Fig. 4". "Fig. 4a ", we can observe an increase in the amplitude of the motor speed, but its steadystate behavior looks regular and it is synchronized with the pendulum motion, "Fig. 4b and c". No irregular or chaotic motion in the system is detected in this frequency band. In reality, the escape of the pendulum in this frequency band is a nonlinear event related with the existence of a homoclinic connection, which is responsible for the destruction of limited solutions in the center of the phase portrait. Therefore, the proximity of the control parameter value c_{8} from the bifurcation point reveals a lack of precision to predict the final state of this system, because in this domain occurs a decrease of the basin attraction that leads to its destruction, Belato (2001).
"Fig. 5" represents the initial variation of the function c_{8} for q = 0.04 and q = 0.05.
Conclusions
In this work, a particular nonideal nonlinear electromechanical system was analyzed. In a secondary resonance region, the system undergoes a global bifurcation (bluesky catastrophe), which turns the solutions in this domain of the control parameter unbounded. When we consider a small variation on the transient behavior, we cannot predict when the system will converge to the limited attractor or when it will be captured by the resonant motor speed inciting the escape of the pendulum and reaching an unlimited remote attractor. Although, the system does not reveal any irregular or chaotic behavior, its final analysis becomes unpredictable since characteristics of the transient response can alter the motion conditions of the system in this frequency band.
Acknowledgments
The authors thank to the FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) by the financial support for the development of this work, through the doctoral scholarship of Débora Belato.
Presented at COBEM 99 15th Brazilian Congress of Mechanical Engineering, 2226 November 1999, São Paulo, SP. Brazil.
Technical Editor: José Roberto F. Arruda.
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Publication Dates

Publication in this collection
08 Sept 2003 
Date of issue
Nov 2002