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Nonlinear Dynamics

, Volume 83, Issue 3, pp 1705–1726 | Cite as

Localization phenomena in torsional rotating shaft systems with multiple centrifugal pendulum vibration absorbers

  • Keisuke Nishimura
  • Takashi Ikeda
  • Yuji Harata
Original Paper

Abstract

The behavior of multiple centrifugal pendulum vibration absorbers (CPVAs) is investigated when they are installed on a rotor in order to suppress torsional vibrations of an elastic, rotating shaft. Linear torsion springs are installed at the supporting points of the CPVAs in order to observe localization phenomena of torsional vibrations in rotating shaft systems. Natural frequency diagrams reveal the existence of a resonance point which implies that localization phenomena may occur in the CPVAs. Van der Pol’s method is employed to determine frequency response curves considering the nonlinearities of the CPVAs. Theoretical results show that localization phenomena occur in the CPVAs at two driving shaft speed ranges near the resonance point where the fluctuation frequency of the driving shaft speed equals the mean rotational speed. Localization phenomena with constant and modulated amplitudes, including chaotic vibrations, appear depending on the stiffnesses of the torsion springs and the radial distances of the CPVAs. Bifurcation sets are calculated to show the rotational speed range where localization phenomena occur. The influence of the fluctuation amplitude of the driving shaft speed on response curves is also examined for both high and low mean rotational speeds. Furthermore, frequency response curves are calculated when the fluctuation frequency of the driving shaft speed equals twice the mean rotational speed.

Keywords

Nonlinear torsional vibration  Centrifugal pendulum vibration absorber Frequency response curve Localization phenomenon Hopf bifurcation Amplitude-modulated motion 

List of symbols

a

Amplitude of fluctuating phase angle of driving shaft

C

Damping coefficient of driven shaft

\(c_{i}\)

Damping coefficient of pendulum i

K

Torsional stiffness of driven shaft

\(k_{i}\)

Stiffness of torsion spring installed at supporting point of pendulum i

\(l_{i}\)

Length of pendulum i

M

Total mass of rotor and pendula \((=m_{0}+\Sigma _{ }m_{i})\)

\(m_{0}\)

Mass of rotor

\(m_{i}\)

Mass of pendulum i

p

Natural frequency

\(r_{0}\)

Radius of gyration of rotor

\(r_{i}\)

Radial distance between supporting point of pendulum i and rotor center

t

Time

(xyz)

Stationary Cartesian coordinate system (see Fig. 1)

(\(x^{\prime }\), \(y^{\prime }\), \(z^{\prime }\))

Cartesian coordinate system rotating with \(\varPsi \) (see Fig. 1)

\(\alpha _{i}\)

Angular position of pendulum i

\(\theta _{i}\)

Inclination angle of pendulum i

\(\mu _{0}\)

\(=m_{0}/M\)

\(\mu _{ i}\)

\(=m_{i}\) /M

\(\varPsi \)

Phase angle of driven shaft and rotor

\(\varPsi _{0}\)

Phase angle of driving shaft

\(\psi \)

\(=\varPsi -\omega t\)

\(\varOmega \)

Fluctuation frequency of driving shaft speed

\(\omega \)

Mean rotational speed of driving shaft, driven shaft and rotor

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mechanical Systems Engineering, Institute of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

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