Nonlinear Dynamics

, Volume 88, Issue 4, pp 2915–2932 | Cite as

Nonlinear responses of spherical pendulum vibration absorbers in towerlike 2DOF structures

  • Takashi Ikeda
  • Yuji Harata
  • Akira Takeeda
Original Paper


This paper investigates the vibration control of a flexible towerlike structure using a single spherical pendulum vibration absorber (SPVA). The SPVA is attached to the structure, which is modeled as a two-degree-of-freedom system, subjected to horizontal, harmonic excitation. In the theoretical analysis, the nonlinearity of the restoring moment of the SPVA is considered, and new coordinates are implemented to avoid the non-integrability of the equations of motion due to a singularity. Van der Pol’s method is used to determine the frequency response curves for the structure and SPVA. When the excitation direction \(\alpha \) is \(0^{\circ }\), the structure is only excited in the x-direction; however, the structure vibrates in both the x- and y-directions due to nonlinear coupling between the structure and SPVA, known as autoparametric resonance. Clockwise and counterclockwise whirling motions appear in the structure and SPVA depending on the excitation frequency. On part of the branches of the response curves, Hopf bifurcations occur followed by amplitude-modulated motions, and combination resonances are also observed in the numerical simulations. When \(\alpha =45^\circ \), the structure is equally excited in the x- and y-directions; however, the vibrations in these directions appear at different amplitudes in the frequency response curves. Such phenomena also occur in the SPVA. The influence of the spring constant of the structure in the y-direction on the frequency response curves is also investigated. Experiments were conducted, and their results confirmed the validity of the theoretical analysis.


Nonlinear vibration Spherical pendulum vibration absorber Frequency response curve Hopf bifurcation Amplitude-modulated motion Combination resonance 

List of symbols


Amplitude of excitation

\(c_{x}, c_{y}\)

Damping coefficients of structure in the x- and y-directions, respectively

\(c_{{\theta x}}, c_{\theta y}\)

Damping coefficients of pendulum in the \(\theta _{x}\)- and \(\theta _{y}\)-directions, respectively


Acceleration of the gravity

\(k_{x}, k_{y}\)

Spring constants of structure in the x- and y-directions, respectively


Length of pendulum


Total mass of structure and pendulum \((=m_{0}+m)\)


Mass of structure


Mass of pendulum

\(p_{xn}, p_{yn}\)

Natural frequencies of the subsystems (\(n=1,2\)) defined by Eq. (9)


Natural frequency of the independent pendulum




Stationary Cartesian coordinate system (see Fig. 1)

\((x_{0}, y_{0}, z_{0})\)

Cartesian coordinate system fixed to the structure (see Fig. 1)

\(\alpha \)

Excitation direction

\(\theta \)

Inclination angle of pendulum

\(\theta _{x}\)

\(\theta \cos \varphi \)

\(\theta _{y}\)

\(\theta \sin \varphi \)

\(\mu \)

m / M

\(\varphi \)

Angular position of pendulum measured from the x-axis

\(\omega \)

Excitation frequency


  1. 1.
    Shieh, A.H.: Optimum performance of pendulum-type torsional vibration absorber. J. Aeronaut. Sci. 9(9), 337–340 (1942)CrossRefGoogle Scholar
  2. 2.
    Mustafa, G., Ertas, A.: Dynamics and bifurcations of a coupled column-pendulum oscillator. J. Sound Vib. 182(3), 393–413 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cuvalci, O., Ertas, A.: Pendulum as vibration absorber for flexible structures: experiments and theory. ASME J. Vib. Acoust. 118(4), 558–566 (1995)CrossRefGoogle Scholar
  4. 4.
    Lee, C.T., Shaw, S.W.: The non-linear dynamic response of paired centrifugal pendulum vibration absorbers. J. Sound Vib. 203(5), 731–743 (1997)Google Scholar
  5. 5.
    Haddow, A.G., Shaw, S.W.: Centrifugal pendulum vibration absorbers, an experimental and theoretical investigation. Nonlinear Dyn. 34(3–4), 293–307 (2003)CrossRefMATHGoogle Scholar
  6. 6.
    Alsuwaiyan, A.S., Shaw, S.W.: Steady-state responses in systems of nearly-identical torsional vibration absorbers. ASME J. Vib. Acoust. 125(1), 80–87 (2003)CrossRefGoogle Scholar
  7. 7.
    Ikeda, T.: Nonlinear responses of dual-pendulum dynamic absorbers. ASME J. Comput. Nonlinear Dyn. 6(1), 011012 (2011)CrossRefGoogle Scholar
  8. 8.
    Miles, J.W.: Stability of forced oscillations of a spherical pendulum. Quart. Appl. Math. 20(1), 21–32 (1962)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Miles, J.W.: Resonant motion of a spherical pendulum. Phys. D 11(3), 309–323 (1984)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bryant, P.J.: Breakdown to chaotic motion of a forced, damped, spherical pendulum. Phys. D 64(1–3), 324–339 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kana, D.D., Fox, D.J.: Distinguishing the transition to chaos in a spherical pendulum. Chaos Interdiscip. J. Nonlinear Sci. 5(1), 298–310 (1995)CrossRefGoogle Scholar
  12. 12.
    Aston, P.J.: Bifurcation of the horizontally forced spherical pendulum. Comput. Methods Appl. Mech. Eng. 170(3–4), 343–353 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Tritton, D.J., Groves, M.: Lyapunov exponents for the Miles’ spherical pendulum equations. Phys. D 126(1–2), 83–98 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Leung, A.Y.T., Kuang, J.L.: On the chaotic dynamics of a spherical pendulum with a harmonically vibrating suspension. Nonlinear Dyn. 43(3), 213–238 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cartwright, J.H.E., Tritton, D.J.: Chaotic dynamics and reversal statistics of the forced spherical pendulum: comparing the Miles equations with experiment. Dyn. Syst. Int. J. 25(1), 1–16 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Náprstek, J., Fischer, C.: Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper. Comput. Struct. 87(19–20), 1204–1215 (2009)CrossRefGoogle Scholar
  17. 17.
    Náprstek, J., Fischer, C.: Types and stability of quasi-periodic response of a spherical pendulum. Comput. Struct. 124(8), 74–87 (2013)CrossRefGoogle Scholar
  18. 18.
    Pospisil, S., Fischer, C., Náprstek, J.: Experimental analysis of the influence of damping on the resonance behavior of a spherical pendulum. Nonlinear Dyn. 78(1), 371–390 (2014)CrossRefGoogle Scholar
  19. 19.
    Murthy, V.R., Hammond, C.E.: Vibration analysis of rotor blades with pendulum absorber. J. Aircr. 18(1), 23–29 (1981)CrossRefGoogle Scholar
  20. 20.
    Stoker, J.J.: Nonlinear Vibrations. Wiley, New York (1950)MATHGoogle Scholar
  21. 21.
    Brent, R.P.: Algorithms for Minimization Without Derivatives, Chapt. 4. Prentice-Hall, Upper Saddle River (1973)Google Scholar
  22. 22.
    Yamamoto, T., Yasuda, K., Nagoh, T.: Super-summed-and-differential-harmonic oscillations in a nonlinear multidegree-of-freedom system. Bull. Jpn. Soc. Mech. Eng. 18(128), 1082–1089 (1975)CrossRefGoogle Scholar
  23. 23.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16(3), 285–317 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mechanical Systems EngineeringHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Yamaha Motor Co., Ltd.IwataJapan

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