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Statements on nonlinear dynamics behavior of a pendulum, excited by a crank-shaft-slider mechanism

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Abstract

The nonlinear dynamics behavior analyzed, in this paper, consists in a pendulum vertically excited on the support by a crank-shaft-slider mechanism. The novelty is the obtainment and analysis of a mathematical model for the pendulum dynamics, under an excitation of a crank-slider, which is based on an extension of the mathematical model of the classical parametric pendulums. Through the modeling, it was verified that the nonlinear dynamics of the pendulum, excited by the crank-shaft-slider mechanism approaches to that of harmonic excitation, when one considered the length of the shaft is sufficient larger than the radius of the crank. The nonlinear dynamic analyses focused on observation of different kinds of motion for different values of dimensionless parameters of the adopted mathematical model. These parameters, includes the frequency of excitation, the amplitude and the geometry of the crank-shaft-slider mechanism. The adopted method of analyses used tools, such as, Lyapunov exponents, parameter space plots, basins of attractions, bifurcation diagrams, phase portraits, time histories and Poincaré sections. The kinds of motion include results on fixed point, oscillations, rotations, oscillations–rotations and chaotic motions.

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Abbreviations

a :

Length of the crank rod

b :

Length of the shaft

ε:

a over b length ratio

m :

Mass of pendulum

t :

Time

l :

Length of the pendulum

φ:

Angle between b bar and vertical axis

θ:

Angle of the crank rod

α:

Pendulum rotation angle

τ:

Dimensionless time

ω:

Ratio between excitation and natural frequency

ω0 :

Natural frequency of the pendulum

λ:

Maximum Lyapunov exponent

x p :

Position of pendulum in x coordinate

y p :

Position of pendulum in y coordinate

F :

Dimensionless parameter relating the angles θ and φ

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Acknowledgments

The authors would like to acknowledge the Brazilian agencies: CNPq, FAPESP and CAPES, for the financial support.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of São Paulo, Av. Trabalhador São-Carlense, nº 400, São Carlos, SP, 13566-590, Brazil

    Rafael Henrique Avanço & Hélio Aparecido Navarro

  2. Federal University of ABC, Santa Adélia Street, nº 166, Santo André, SP, Brazil

    Reyolando M. L. R. F. Brasil

  3. Department of Mechanical Engineering at Technological Institute of Aeronautics, Pça. Mal. Eduardo Gomes, nº 50, São José Dos Campos, SP, 12228-900, Brazil

    José Manoel Balthazar

  4. UNESP: Sorocaba, Control and Automation Engineering, Av. Três de Março, Sorocaba, SP, 18087-180, Brazil

    Átila Madureira Bueno

  5. Department of Mathematics, Federal University of Technology – Paraná, Ponta Grossa, PR, Brazil

    Angelo Marcelo Tusset

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  1. Rafael Henrique Avanço
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  2. Hélio Aparecido Navarro
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  6. Angelo Marcelo Tusset
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Correspondence to Rafael Henrique Avanço.

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Avanço, R.H., Navarro, H.A., Brasil, R.M.L.R.F. et al. Statements on nonlinear dynamics behavior of a pendulum, excited by a crank-shaft-slider mechanism. Meccanica 51, 1301–1320 (2016). https://doi.org/10.1007/s11012-015-0310-1

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