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On nonlinear dynamics behavior of an electro-mechanical pendulum excited by a nonideal motor and a chaos control taking into account parametric errors

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Abstract

This study analyzes a nonlinear system with a crank–shaft–slider mechanism linked to a pendulum. The crank is powered by a DC motor that moves horizontally the pendulum pivot. The speed of the motor is influenced by the pendulum dynamics, reason for calling the system as nonideal. In literature, the vibration of the pendulum support by a crank–shaft–slider is generally considered as harmonic what neglects the real complexity of the mechanism. It is also common, for the crank speed or the excitation frequency to be considered constant, so a unique degree-of-freedom can take place. The novelty in this research is the analysis of the feedback effect of the pendulum over the crank speed and the complexity of crank–slider mechanism without approaching to a harmonic motion. Different types of motion occur as oscillations, rotations, and chaos. Results with different initial conditions are worked out and basins of attractions plotted. Chaos was obtained when small variations of parameters are made. The methods of analysis include phase portraits, time histories, bifurcations diagrams, and basins of attraction. The verification of chaotic phenomena is performed through the 0–1 test. By the end, the feedback control is applied using the method of Tereshko by altering the energy of the chaotic system, leading the pendulum to a limit cycle.

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Abbreviations

a :

Length of the crank rod

b :

Length of the shaft

\(\epsilon\) :

a over b length ratio

m :

Mass of the pendulum

t :

Time

l :

Length of the pendulum

\(\varphi\) :

Angle between b bar and horizontal axes

\(\theta\) :

Angle of the crank rod

\(\alpha\) :

Angle of the pendulum

\(\tau\) :

Dimensionless time

\(\omega\) :

Dimensionless parameter for crank speed

\(\omega_{0}\) :

Natural frequency of the pendulum

x:

Horizontal position of the bob in the extreme of the pendulum

y:

Vertical position of the bob in the extreme of the pendulum

F:

Dimensionless parameter relating the angle \(\theta\) and \(\varphi\)

T:

Kinetic energy of the system

\(V_{p}\) :

Potential energy of the pendulum

\(L\) :

Lagrangian function

\(Q^{\text{NC}}\) :

Nonconservative generalized force of a coordinate

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Authors and Affiliations

  1. Mechanical Aeronautics Division, Aeronautics Technological Institute, Praça Marechal Eduardo Gomes, 50 - Vila das Acacias, São José dos Campos, SP, 12228-900, Brazil

    Rafael Henrique Avanço, José Manoel Balthazar & Airton Nabarrete

  2. Department of Mathematics, Federal University of Technology, Ponta Grossa, Paraná, Brazil

    Angelo Marcelo Tusset

  3. Department of Mechanical Engineering, University of São Paulo, Avenida Trabalhador São-carlense, nº400, São Carlos, SP, 13566-590, Brazil

    Helio Aparecido Navarro

Authors
  1. Rafael Henrique Avanço
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  2. Angelo Marcelo Tusset
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  3. José Manoel Balthazar
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  4. Airton Nabarrete
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  5. Helio Aparecido Navarro
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Correspondence to Rafael Henrique Avanço.

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Technical Editor: Aline Souza de Paula.

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Avanço, R.H., Tusset, A.M., Balthazar, J.M. et al. On nonlinear dynamics behavior of an electro-mechanical pendulum excited by a nonideal motor and a chaos control taking into account parametric errors. J Braz. Soc. Mech. Sci. Eng. 40, 23 (2018). https://doi.org/10.1007/s40430-017-0955-x

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