Classical Mechanics

Awrejcewicz, Jan; Koruba, Zbigniew

doi:10.1007/978-1-4614-3978-3

Cited by 33 publications

(35 citation statements)

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“…The study of electromechanical devices has focused almost entirely on linear systems, although nowadays it is widely recognized that an analysis based on linear models can be inadequate for devices with strong nonlinearities and for sufficiently large excursions from the equilibrium point [1][2][3]. On the other hand, the bifurcation theory has not been widely used in the analysis of the multidimensional controlled electromechanical systems.…”

Section: Introductionmentioning

confidence: 99%

“…In section 4, the stability of the normal form of the system is studied from the Lyapunov's direct method by means of a candidate Lyapunov function [1]. Consequently, the proportional constant and the reset time of a PID controller can be tuned to find out the stability of a predefined set point or 3 to predict a self-oscillating behavior. The previous procedure does not provide the derivative control action of the PID, which has been determined through the harmonic balance method (describing function) [1], [21][22].…”

Section: Introductionmentioning

confidence: 99%

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Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Pérez-Molina

Pérez-Polo

2014

Communications in Nonlinear Science and Numerical Simulation

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Please cite this article as: Pérez-Molina, M., F.Pérez-Polo, M., Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/ 10. 1016/j.cnsns.2014.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThis paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The BogdanovTakens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the AndronovPoincaré-Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Pérez-Molina

Pérez-Polo

2014

Communications in Nonlinear Science and Numerical Simulation

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“…The other control signal is an external force applied to the mobile piece, which is constrained by a linear damper and a nonlinear spring [1][2][3]. The control force has a similar structure to a proportional plus derivative (PD) controller.…”

Section: Introductionmentioning

confidence: 99%

“…These investigations have progressed due to the integrated formulation of mechanical, electric and magnetic circuits [2][3][4][5][6][7], as well as the combined use of chaos, bifurcation and control theory [8][9][10][11][12][13]. On the other hand, the appearance of micro electromechanical systems and the developing of new recording systems offer the possibility to examine new interesting dynamical behaviors [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Steady-state self-oscillations and chaotic behavior of a controlled electromechanical device by using the first Lyapunov value and the Melnikov theory

Pérez-Polo

Pérez-Molina

2014

Journal of Sound and Vibration

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In this paper regular and chaotic oscillations in a controlled electromechanical transducer are investigated. The nonlinear control laws are defined by an electric tension excitation and an external force applied to the mobile piece of the transducer.The paper shows that an Andronov-Poincaré-Hopf bifurcation appears as long as adequate parameters are chosen for the nonlinear control laws. The stability of the weak focuses associated to such bifurcation is examined according to the sign of the first Lyapunov value, showing that chaotic behavior can arise when the first Lyapunov value is varied harmonically. The appearance of a homoclinic orbit is investigated assuming an approximated model for the device. On the basis of the parametric equations of the homoclinic orbit and the presence of harmonic disturbances on the platform, it is demonstrated that chaotic oscillations can also appear, and they have been examined by means of the Melnikov theory. Chaotic behavior is corroborated by means of the sensitive dependence, Lyapunov exponents and power spectral density, and it is applied to drive the transducer mobile piece to a predefined set point assuming that noise due to the measurement process can appear in the control signals. The steady state error associated to such random noise is eliminated by adding a PI linear controller to the control force. Numerical simulations are used to corroborate the analytical results.

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“…The voice coil is a key unit of the speakers that performs energy conversion across the electrical, mechanical and acoustic components [4]. The energy conversion efficiency of the voice coils is under 10%, due to thermal and mechanical losses of over 90%, as the input power supplied to the voice coil is transformed into audio sound signals [5]. When operating the speakers for a very long time with excessive input power, the voice coils could experience lower sound quality and harmonic distortion, due to a high temperature.…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation on the Temperature Characteristics of the Voice Coil for a Woofer Using Thermal Equivalent Heat Conduction Models

Lee

Kim

2014

Entropy

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Abstract:The objective of this study is to numerically investigate the temperature and heat transfer characteristics of the voice coil for a woofer with and without bobbins using the thermal equivalent heat conduction models. The temperature and heat transfer characteristics of the main components of the woofer were analyzed with input powers ranging from 5 W to 60 W. The numerical results of the voice coil showed good agreement within ±1% of the data by Odenbach (2003). The temperatures of the voice coil and its units for the woofer without the bobbin were 6.1% and 5.0% on average, respectively; lower than those of the woofer with the bobbin. However, at an input power of 30 W for the voice coil, the temperatures of the main components of the woofer without the bobbin were 40.0% higher on average than those of the woofer obtained by Lee et al. (2013).

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Classical Mechanics

Cited by 33 publications

References 0 publications

Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Steady-state self-oscillations and chaotic behavior of a controlled electromechanical device by using the first Lyapunov value and the Melnikov theory

Numerical Investigation on the Temperature Characteristics of the Voice Coil for a Woofer Using Thermal Equivalent Heat Conduction Models

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