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Periodic Oscillation and Bifurcation Analysis of Pendulum with Spinning Support Using a Modified Continuous Piecewise Linearization Method

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Abstract

The large-amplitude oscillation of a pendulum with spinning support was investigated using a modified continuous piecewise linearization method (CPLM). In contrast to previous studies, the present study investigated the response of the spinning support pendulum when the non-dimensional rotation parameter (\( {\Lambda} \)) is greater than one. The analysis showed that the natural frequency increased monotonically with Λ, while the oscillation history produced a distinct qualitative change as \( {\Lambda} \) increases from \( {\Lambda} < 1 \) to \( {\Lambda} > 1 \), confirming the presence of a bifurcation at \( {\Lambda} = 1 \). It was also observed that the response exhibits a bi-stable equilibrium and a double-well potential when \( {\Lambda} > 1 \). Finally, the modified CPLM solution was shown to produce a maximum error of less than 0.30% for \( {\text{A}} \le 179^\circ \) and \( {\Lambda} \le 1 \), which is better than other published results. This shows the potential of the modified CPLM to obtain accurate periodic solutions of complex nonlinear systems.

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Abbreviations

\( \varphi \,\left( {\text{rad}} \right) \) :

Angular displacement of pendulum

\( \varphi_{r} \,\left( {\text{rad}} \right) \) :

Displacement at the beginning of a discretization

\( \varphi_{s} \,\left( {\text{rad}} \right) \) :

Displacement at the end of a discretization

\( \dot{\varphi }_{r} \,\left( {\text{rad/s}} \right) \) :

Velocity at the beginning of a discretization

\( \dot{\varphi }_{s} \,\left( {\text{rad/s}} \right) \) :

Velocity at the end of a discretization

\( \omega_{rs} \,\left( {\text{rad/s}} \right) \) :

CPLM constant representing circular frequency of CPLM solution

\( {\varphi}_{rs} \,\left( {\text{rad}} \right) \) :

CPLM constant representing phase angle of CPLM solution when \( K_{rs} > 0 \)

Δt(s):

Time interval covered by a discretization

\( {\Lambda}\left( - \right) \) :

Non-dimensional rotation parameter or dimensionless rotational speed

\( A\,\left( {\text{rad}} \right) \) :

Amplitude of the pendulum

\( A_{rs} \,\left( {\text{rad}} \right);\;B_{rs} \,\left( {\text{rad}} \right) \) :

Integration constants for CPLM solution when Krs < 0

\( C_{rs} \,\left( {\text{rad}} \right) \) :

CPLM constant representing steady-state response of CPLM solution

\( F_{rs} \,\left( {{\text{N}}/{\text{kg}}} \right) \) :

Linearized restoring force for the discretization bounded by points r and s

\( G_{rs} \,\left( {\text{rad/s}} \right);H_{rs} \,\left( {\text{rad}} \right) \) :

Integration constants for CPLM solution when Krs = 0

\( K_{rs} \,\left( {{\text{N}}/{\text{kg}}\,{\text{rad}}} \right) \) :

Linearized stiffness for the discretization bounded by points r and s

\( R_{rs} \,\left( {\text{rad}} \right) \) :

CPLM constant representing amplitude of CPLM solution when Krs > 0

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Acknowledgements

The authors are grateful to Dr. E.C. Ebieto of the Department of Mechanical Engineering, University of Port Harcourt, for proof-reading the draft manuscript and making useful suggestions. The reviewers of the manuscript are also acknowledged for their comments which helped to improve the quality of the paper.

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Authors and Affiliations

  1. Applied Mechanics and Design (AMD) Research Group, Department of Mechanical Engineering, Faculty of Engineering, University of Port Harcourt, Port Harcourt, Nigeria

    Akuro Big-Alabo & Chinwuba Victor Ossia

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  1. Akuro Big-Alabo
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  2. Chinwuba Victor Ossia
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Correspondence to Akuro Big-Alabo.

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Appendix

Appendix

Pseudocode Algorithm for Modified CPLM Solution for Pendulum with Spinning Support

figure a

The pseudocode algorithm above is for the negative velocity oscillation stage (i.e. \( \dot{\varphi } < 0 \)) when the pendulum swings from \( + \,A \) to \( - \,A \). This stage constitutes the first half-cycle of the oscillation. For the remaining half-cycle when the pendulum swings from \( - \,A \) back to \( + \,A \) a similar algorithm is applicable and the necessary changes can be made by referring to “Modified Continuous Piecewise Linearization Method” section.

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Big-Alabo, A., Ossia, C.V. Periodic Oscillation and Bifurcation Analysis of Pendulum with Spinning Support Using a Modified Continuous Piecewise Linearization Method. Int. J. Appl. Comput. Math 5, 114 (2019). https://doi.org/10.1007/s40819-019-0697-9

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