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Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft

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Abstract

The nonlinear axial-lateral-torsional free vibration of the rotating shaft is analyzed by employing the Rayleigh beam theory. The effects of lateral, axial and torsional deformations, gyroscopic forces and rotary inertia are taken into account, but the shear deformations are neglected. In the new developed dynamic model, the nonlinearities are originated from the stretching of beam centerline, nonlinear curvature and twist and inertial terms which leads to the coupling between the axial, lateral and torsional deformations. The deformed configuration of the cross section of the beam is represented by the axial and lateral deformations, also the geometry of the beam in the deformed configuration is represented by Euler angles. A system of coupled nonlinear differential equations is obtained which is examined by the method of multiple scales and the nonlinear natural frequencies are determined. The accuracy of the solutions is inspected by comparing the free vibration response of the system with the numerical integration of the governing equations. The effect of the spin speed and radius-to-length ratio of the rotating shaft on the free vibrational behavior of the system is inspected. The study demonstrates the effect of axial-lateral-torsional coupling on the nonlinear free vibrations of the rotating shaft.

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Abbreviations

A :

Cross-sectional area

E :

Young’s modulus of elasticity

I :

The second moment of area of the beam cross section

T :

Kinetic energy

L :

Length of the rotor

\(u_i ,i=1,2,3\) :

The components of displacement field along the x, y, z directions, respectively

N :

Number of mode shapes

R :

Radius of the shaft

e :

Extensional strain

t :

Time

x, y, z :

Variable coordinates

u :

Axial displacement of the shaft element

v, w :

Transverse deflections of the shaft element

\(\psi ,\theta ,\varphi \) :

Euler angles

\(\tau \) :

Nondimensional time

\(\theta _i ,i=1,2,3\) :

Rotation angles

\(\omega _i , i=1, 2, 3\) :

Angular velocity components

\(\Pi \) :

Potential energy

\(\xi \) :

Nondimensional coordinate

\(\alpha , \lambda , \beta \) :

Nondimensional parameters

\(\delta \) :

Variational operator

\(\varepsilon \) :

Booking device (small parameter)

\(\varepsilon _{ij} \) :

Strain components

\(\rho _i \) :

Curvature components

\(\omega \) :

Frequency of the motion in the first scale

\(\omega _n \) :

Natural frequency

\(\Omega \) :

Rotating speed of the shaft

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Authors and Affiliations

  1. Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

    S. H. Mirtalaie & M. A. Hajabasi

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  1. S. H. Mirtalaie
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  2. M. A. Hajabasi
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Correspondence to S. H. Mirtalaie.

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Mirtalaie, S.H., Hajabasi, M.A. Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft. Arch Appl Mech 87, 1465–1494 (2017). https://doi.org/10.1007/s00419-017-1265-6

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  • DOI: https://doi.org/10.1007/s00419-017-1265-6

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