A note about the appearance of non-hyperbolic solutions in a mechanical pendulum system

The investigation of the behavior of a nonlinear system consists in the analysis of different stages of its motion, where the complexity varies with the proximity of a resonance region. Near this region the stability domain of the system undergoes sudden changes due basically to competition and interaction between periodic and saddle solutions inside the phase portrait, leading to the occurrence of the most different phenomena. Depending of the domain of the chosen control parameter, these events can reveal interesting geometric features of the system so that the phase portrait is not capable to express all them, since the projection of these solutions on the two-dimensional surface can hide some aspects of these events. In this work we will investigate the numerical solutions of a particular pendulum system close to a secondary resonance region, where we vary the control parameter in a restrict domain in order to draw a preliminary identification about what happens with this system. This domain includes the appearance of non-hyperbolic solutions where the basin of attraction in the center of the phase portrait diminishes considerably, almost disappearing, and afterwards its size increases with the direction of motion inverted. This phenomenon delimits a boundary between low and high frequency of the external excitation.344173230931

Escape in a nonideal electro-mechanical system

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 vice-versa, 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.335340Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Chaotic Vibrations Of A Nonideal Electro-mechanical System

Nonideal systems are those in which one takes account of the influence of the oscillatory system on the energy supply with a limited power. In this paper, a particular nonideal system is investigated, consisting of a pendulum whose support point is vibrated along a horizontal guide by a two bar linkage driven by a DC motor, considered to be a limited power supply. Under these conditions, the oscillations of the pendulum are analyzed through the variation of a control parameter. The voltage supply of the motor is considered to be a reliable control parameter. Each simulation starts from zero speed and reaches a steady-state condition when the motor oscillates around a medium speed. Near the fundamental resonance region, the system presents some interesting nonlinear phenomena, including multi-periodic, quasiperiodic, and chaotic motion. The loss of stability of the system occurs through a saddle-node bifurcation, where there is a collision of a stable orbit with an unstable one, which is approximately located close to the value of the pendulum's angular displacement given by αC = π/2. The aims of this study are to better understand nonideal systems using numerical simulation, to identify the bifurcations that occur in the system, and to report the existence of a chaotic attractor near the fundamental resonance.38Out/1316991706Belato, D., (1998) Não Linearidades Do Eletro-pêndulo, p. 128. , Master Dissertation, Faculdade de Engenharia Mecânica, UNICAMP, Campinas, São Paulo, BrazilKononenko, V.O., (1969) Vibrating Systems with a Limited Power Supply, , Iliffe, LondonKranospol'skaya, T.S., Shvets, A.Yu., Chaotic interactions in a pendulum-energy source system (1990) Prikladnaya Mekhanika, 26 (5), pp. 90-96Kranospol'skaya, T.S., Shvets, A.Yu., Chaos in vibrating systems with a limited power-supply (1993) Chaos, 3, pp. 387-395McRobie, F.A., Thompson, J.M.T., Global integrity in engineering dynamics - Methods and applications (1992) Applied Chaos, pp. 31-49. , Kim, J.H., Stringer, J. (Eds.), Wiley, New YorkNayfeh, A.H., Mook, D.T., (1979) Nonlinear Oscillations, , Wiley, New YorkSuherman, S., Plaut, R.H., Use of a flexible internal support to suppress vibrations of a rotating shaft passing through a critical speed (1997) Journal of Vibration and Control, 3, pp. 213-23

Using transient and steady state considerations to investigate the mechanism of loss of stability of a dynamical system

This work presents the complete set of features for solutions of a particular non-ideal mechanical system near the fundamental and near to a secondary resonance region. The system comprises a pendulum with a horizontally moving suspension point. Its motion is the result of a non-ideal rotating power source (limited power supply), acting oil the Suspension point through a crank mechanism. Main emphasis is given to the loss of stability, which occurs by a sequence of events, including intermittence and crisis, when the system reaches a chaotic attractor. The system also undergoes a boundary-crisis, which presents a different aspect in the bifurcation diagram due to the non-ideal supposition. (c) 2004 Published by Elsevier B.V

A note about the appearance of non-hyperbolic solutions in a mechanical pendulum system

The investigation of the behavior of a nonlinear system consists in the analysis of different stages of its motion, where the complexity varies with the proximity of a resonance region. Near this region the stability domain of the system undergoes sudden changes due basically to competition and interaction between periodic and saddle solutions inside the phase portrait, leading to the occurrence of the most different phenomena. Depending of the domain of the chosen control parameter, these events can reveal interesting geometric features of the system so that the phase portrait is not capable to express all them, since the projection of these solutions on the two-dimensional surface can hide some aspects of these events. In this work we will investigate the numerical solutions of a particular pendulum system close to a secondary resonance region, where we vary the control parameter in a restrict domain in order to draw a preliminary identification about what happens with this system. This domain includes the appearance of non-hyperbolic solutions where the basin of attraction in the center of the phase portrait diminishes considerably, almost disappearing, and afterwards its size increases with the direction of motion inverted. This phenomenon delimits a boundary between low and high frequency of the external excitation

Chaotic vibrations of a nonideal electro-mechanical system

Nonideal systems are those in which one takes account of the influence of the oscillatory system on the energy supply with a limited power (Kononenko, 1969). In this paper, a particular nonideal system is investigated, consisting of a pendulum whose support point is vibrated along a horizontal guide by a two bar linkage driven by a DC motor, considered to be a limited power supply. Under these conditions, the oscillations of the pendulum are analyzed through the variation of a control parameter. The voltage supply of the motor is considered to be a reliable control parameter. Each simulation starts from zero speed and reaches a steady-state condition when the motor oscillates around a medium speed. Near the fundamental resonance region, the system presents some interesting nonlinear phenomena, including multi-periodic, quasiperiodic, and chaotic motion. The loss of stability of the system occurs through a saddle-node bifurcation, where there is a collision of a stable orbit with an unstable one, which is approximately located close to the value of the pendulum's angular displacement given by alpha (C)= pi /2. The aims of this study are to better understand nonideal systems using numerical simulation, to identify the bifurcations that occur in the system, and to report the existence of a chaotic attractor near the fundamental resonance. (C) 2001 Elsevier B.V. Ltd. All rights reserved

An Overview On Non-ideal Vibrations

We analyze the dynamical coupling between energy sources and structural response that must not be ignored in real engineering problems, since real motors have limited output power.386613621Sommerfeld, A., Beiträge zum dynamischen ausbau der festigkeitslehe (1902) Physikal Zeitschr, 3Kononenko, V.O., (1969) Vibrating Systems with a Limited Power Supply, , (in Russian: 1959), English translation, Illife BooksBleckman, I.I., Self-synchronization of certain vibratory devices (1953) Eng. Trans., 16Evan-Iwanowski, R.M., (1976) Resonance Oscillators in Mechanical Systems, , ElsevierDimentberg, M.F., (1988) Statistical Dynamics of Nonlinear and Time Varying Systems, , John Wiley and SonsDimentberg, M.F., McGovern, L., Norton, R.L., Chapdelaine, J., Harrison, R., Dynamics of an unbalanced shaft interacting with a limited power supply (1997) Nonlinear Dyn., 13, pp. 171-187Nayfeh, A.H., Mook, D.T., (1979) Nonlinear Oscillations, , John Wiley and SonsBalthazar, J.M., Mook, D.T., Weber, H.I., Fenili, A., Belato, D., De Mattos, M.C., Wieczorek, S., On vibrating systems with a limited power supply and their applications to engineering sciences 49th Brazilian Seminar of Mathematical Analysis, State University of Campinas, Campinas, SP, Brazil, Short Course, March 5-9, 1999, pp. 137-277. , in: Honig, C.S. (ed)Balthazar, J.M., Mook, D.T., Brazil, R.M.L.R.F., Weber, H.I., Fenili, A., Belato, D., Felix, J.L.P., Recent results on vibrating problems with limited power supply Sixth Conference on Dynamical Systems Theory and Applications, Lodz, Poland, December, 10-12, 2001, pp. 27-50. , in: Awrejcewicz, J., Brabski, J. and Nowakowski, J. (eds)Yamanaka, H., Murakami, S., Optimum designs of operating curves for rotating shaft systems with limited power supplier (1989) Current Topics in Structural Mechanics, PVP, 179, pp. 181-185. , in: Chung, H. (ed.)ASME NYBalthazar, J.M., Cheshankov, B.I., Rushev, D.T., Barbanti, L., Weber, H.I., Remarks on the passage through resonance of a vibrating system, with two degree of freedom (2001) J. Sound Vib., 239 (5), pp. 1075-1085Christ, H., Stationärer und Instatioärer Betrieb Eines Federnd Gelagerten, Unwuchtigen Motors (1966), Dissertation Universität KarlsruheWauer, J., Bürle, P., Dynamics of a flexible slider-crank mechanism driven by a non-ideal source of energy (1997) Nonlinear Dyn., 13, pp. 221-242Wauer, J., Suherman, S., Vibration suppression of rotating shafts passing through resonances by switching shaft stiffness (1997) J. Vib. Acoust., 120, pp. 170-180Suherman, S., Transient analysis and vibrating suppression of a cracked rotating shaft with a ideal and a non-ideal motor passing through a critical speed (1996), PhD Thesis, Virginia Polytechnic Institute and State UniversityIwatsubo, T., Kanki, H., Kawai, R., Vibration of asymmetric rotor through critical speed with limited power supply (1972) J. Mech. Eng. Sci., 14 (3), pp. 184-194Suzuki, S.H., Dynamic behavior of a beam subject to a force of time-dependent frequency (1978) J. Sound Vib., 57, pp. 59-64Suzuki, S.H., Dynamic behavior of a beam subject to a force of time-dependent frequency (continued) (1978) J. Sound Vib., 60 (3), pp. 417-422Balthazar, J.M., Rente, M.L., Mook, D.T., Weber, H.I., Some observations on numerical simulations of a non-ideal dynamical system (1997) Nonlinear Dynamics, Chaos, Control and Their Applications to Engineering Sciences, 1, pp. 97-104. , in: Balthazar, J.M., Mook, D.T. and Rosario, J.M. (eds)Balthazar, J.M., Mook, D.T., Weber, H.I., Mattos, M.C., Some remarks on the behaviour of non-ideal dynamical systems (1997) Nonlinear Dynamics, Chaos, Control and Their Applications to Engineering Sciences, 1, pp. 88-96. , in: Balthazar, J.M., Mook, D.T. and Rosario, J.M. (eds)De Mattos, M.C., Balthazar, J.M., Wieczork, S., Mook, D.T., An experimental study of vibrations of non-ideal systems Proceedings of DETC'97, ASME Design Engineering Technical Conference, September 14-17, Sacramento, California, USA, CD-ROM, 1997, p. 10De Mattos, M.C., Balthazar, J.M., On the dynamics of an armature controlled dc motor mounted on an elastically table Proceedings of 15th Brazilian Congress of Mechanical Engineering, November 22-16, Águas de Lindóia, São Paulo, Brazil, 1999, p. 10Brasil, R.M.F.L., Balthazar, J.M., Nonlinear oscillations of a portal frame structure excited by a non-ideal motor (2000) Control of Oscillations and Chaos, 2, pp. 275-278. , in: Chernousko, A.L. and Fradkov, A.I. (eds)Palacios, J.F., Balthazar, J.M., Brasil, R.M.L.R.F., On non-ideal dynamics of nonlinear portal frame analysis using averaging method Proceedings of the Ninth International Symposium on Dynamic Problems of Mechanics, Florianópolis, SC, Brazil, 5-9 March 2001, pp. 341-346. , in: Espindola, J.J., Lopes, E.O.M. and Bazan, F.S.V. (eds)Fenili, A., On slewing structure: Modeling and dynamical analysis (2000), PhD Thesis, State University of Campinas, SP, Brazil(in Portuguese)Fenili, A., Balthazar, J.M., Weber, H.I., Mook, D.T., Nonlinear analysis of the motion of a flexible, rotating, cantilever beam (2002), (submitted)Fenili, A., Balthazar, J.M., Weber, H.I., Mook, D.T., On the comparison between two mathematical models for flexible slewing structures-linear and nonlinear curvature (2001) Nonlinear Dynamics, Chaos and Their Applications to Engineering Sciences, 4: Recent Developments in Nonlinear Phenomena, pp. 372-382. , in: Balthazar, J.M., Gonçalves, P.B., Brasil, R.M.F.L.R.F., Caldas I.L. and Rizatto, F.B. (eds)Fenili, A., Balthazar, J.M., Mook, D.T., Weber, H.I., Application of the center manifold reduction to the slewing flexible non-ideal model (2002) J. Brazil. Soc. Mech. Sci., , (in press)Fenili, A., Balthazar, J.M., Mook, D.T., Some remarks about the experimental analysis of slewing flexible structures and mathematical modeling Proceedings of the Ninth International Symposium on Dynamic Problems of Mechanics, Florianópolis, SC, Brazil, 5-9 March 2001, pp. 341-346. , in: Espindola, J.J., Lopes, E.O.M. and Bazan, F.S.V. (eds)Fenili, A., Balthazar, J.M., Mook, D.T., A brief note on experimental identification of dc-motor parameters (2001) Sci. Eng. J., 10 (1), pp. 105-108Pontes, B.R., De Oliveira, V.A., Balthazar, J.M., On friction-driven vibrations in a mass block-belt-motor with limited power supply (2000) J. Sound Vib., 234 (4), pp. 713-723Pontes, B.R., De Oliveira, V.A., Balthazar, J.M., On the dynamic response of a mechanical system with dry friction and limited power supply (2001) Nonlinear Dynamics, Chaos and Their Applications to Engineering Sciences, 4: Recent Developments in Nonlinear Phenomena, pp. 355-371. , in: Balthazar, J.M., Gonçves P.B., Brasil, R.M.F.L.R.F., Caldas I.L. and Rizatto, F.B. (eds)Alifov, A., Frolov, K.V., Investigation of self-excited oscillations with friction, under conditions of parametric excitation and limited power of energy source (1977) Mekhanika Tvedogo Tela, 15 (4), pp. 25-33Warminski, J., Balthazar, J.M., Brasil, R.M.L.R.F., Vibrations of non-ideal parametrically and self-excited model (2001) J. Sound Vib., 234 (4), pp. 713-723De Souza, S.L.T., Caldas, I.L., Balthazar, J.M., Brasil, R.M.L.R.F., Analysis of regular and irregular dynamics of a non-ideal gear rattling problem (2002) J. Brazil. Soc. Mech. Sci., pp. 111-114Krasnopol'skaya, T.S., Shevts, A.Y., Chaotic interactions in a pendulum energy source system (1990) Prikladnaya Mekhanika, 26 (5), pp. 90-96Belato, D., Nonlinear analysis of non-ideals holonomic dynamical systems (2002), PhD Thesis, Faculdade de Engenharia Mecânica, UNICAMP, Campinas, São Paulo, Brazil(in Portuguese)Belato, D., Weber, H.I., Balthazar, J.M., Mook, D.T., Chaotic vibrations of a non-ideal electromechanical system (2001) Int. J. Solids Struct., 38, pp. 669-1706Palacios, J.F., Balthazar, J.M., Brasil, R.M.L.R.F., Some comments on a control technique by using internal resonance and saturation phenomenon: Applications to a simple machine foundation VII Pan American Congress of Applied Mechanics, Chile, Telmuco, 2002, pp. 141-144Brasil, R.M.L.R.F., Garzeri, F.J., Balthazar, J.M., An experimental study of the nonlinear dynamics of a portal frame foundation for a non-ideal motor Proceedings of DETC'01 ASME 2001 Design Engineering Technical Conference and Computers and Information in Engineering Conference, Pittsburgh, Pennsylvania, September 9-12, CD ROM, 200