, Volume 51, Issue 6, pp 1301–1320 | Cite as

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

  • Rafael Henrique Avanço
  • Hélio Aparecido Navarro
  • Reyolando M. L. R. F. Brasil
  • José Manoel Balthazar
  • Átila Madureira Bueno
  • Angelo Marcelo Tusset


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.


Pendulum Parametric Crank-shaft-slider Chaos 

List of symbols


Length of the crank rod


Length of the shaft


a over b length ratio


Mass of pendulum




Length of the pendulum


Angle between b bar and vertical axis


Angle of the crank rod


Pendulum rotation angle


Dimensionless time


Ratio between excitation and natural frequency


Natural frequency of the pendulum


Maximum Lyapunov exponent


Position of pendulum in x coordinate


Position of pendulum in y coordinate


Dimensionless parameter relating the angles θ and φ



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


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Rafael Henrique Avanço
    • 1
  • Hélio Aparecido Navarro
    • 1
  • Reyolando M. L. R. F. Brasil
    • 2
  • José Manoel Balthazar
    • 3
  • Átila Madureira Bueno
    • 4
  • Angelo Marcelo Tusset
    • 5
  1. 1.Department of Mechanical EngineeringUniversity of São PauloSão CarlosBrazil
  2. 2.Federal University of ABCSanto AndréBrazil
  3. 3.Department of Mechanical Engineering at Technological Institute of AeronauticsSão José Dos CamposBrazil
  4. 4.UNESP: Sorocaba, Control and Automation EngineeringSorocabaBrazil
  5. 5.Department of MathematicsFederal University of Technology – ParanáPonta GrossaBrazil

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