Nonlinear Dynamics

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

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

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


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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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