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Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances

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Abstract

An asymmetrical rotating shaft with unequal mass moments of inertia and flexural rigidities in the direction of principal axes is considered. In this system, there are two excitation sources, including a harmonic excitation due to the dynamic imbalances and a parametric excitation due to shaft asymmetry. Nonlinearities are due to the in-extensionality of the shaft and large amplitude. In this study, harmonic and parametric resonances due to the mentioned effects are considered. The influences of inequality of mass moments of inertia and flexural rigidities in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation of steady state response of the rotating asymmetrical shaft are investigated. In addition, the characteristic of stable stationary points and loci of bifurcation points as function of damping coefficient are determined. In order to analyze the resonances of the system the multiple scales method is applied to the complex form of partial differential equations of motion. The achieved results show a good agreement with those of numerical computation.

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Abbreviations

A :

Cross section area

A 11 :

Longitudinal stiffness

c :

External damping coefficient

D 11,D 22,D 33 :

Torsional and flexural stiffness respectively

α :

Strain along the neutral axis of the shaft

E :

Elasticity modulus

e y ,e z :

Eccentricity distributions with respect to axes y and z

G :

Shear modulus

I 1,I 2,I 3 :

Polar and diametrical mass moments of inertia, respectively

l :

Length of the rotating shaft

m :

Mass per unit length of the shaft

u :

Longitudinal displacement

v,w :

Transverse displacements

ψ,θ,β :

Euler angles

k i (i=1,…,3):

Shaft curvatures

Ω :

Rotational speed

Δ I :

Difference between the two mass moments of inertia

Δ D :

Difference between the two flexural rigidities

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Authors and Affiliations

  1. Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-177, Tehran, Iran

    Majid Shahgholi & Siamak Esmaeilzadeh Khadem

Authors
  1. Majid Shahgholi
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  2. Siamak Esmaeilzadeh Khadem
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Correspondence to Siamak Esmaeilzadeh Khadem.

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Shahgholi, M., Khadem, S.E. Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances. Nonlinear Dyn 70, 1311–1325 (2012). https://doi.org/10.1007/s11071-012-0535-7

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  • DOI: https://doi.org/10.1007/s11071-012-0535-7

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