Acta Mechanica

, Volume 229, Issue 8, pp 3355–3373 | Cite as

Analytical modeling of the coupled nonlinear free vibration response of a rotating blade in a gas turbine engine

  • P. A. Roy
  • S. A. Meguid
Original Paper


In this paper, we investigate the free vibration response of a rotating blade in a gas turbine engine. The blade is modeled as a tapered Timoshenko beam with nonlinear variations in its cross-section properties. The governing equations of motions are derived using Lagrangian mechanics and Rayleigh–Ritz method. These equations take into account centrifugal stiffening, axial and lateral coupling due to Coriolis effect, shear deformation, and rotary inertia. We examine the effect of the beam geometry upon its axial and lateral free vibration response. The effects of rotational speed, taper ratio, chord ratio, hub radius, and slenderness ratio on the natural frequencies are analyzed. The results of our analysis indicate that the taper ratio, slenderness ratio, and rotational speed of the beam govern its free lateral vibration response. The axial vibration of the beam is significantly affected by the slenderness ratio, but it is found to be independent of the hub radius.

List of symbols


Cross-section area of the beam


Coefficients for polynomial of the cross-section area


Cross-section area of the beam at the hub


Coriolis damping matrix


Young’s modulus


Centrifugal force


Shear modulus


Area moment of inertia


Coefficients for polynomial of the moment of inertia


Moment of inertia of the beam at the hub


Total stiffness matrix


Elastic property-dependent stiffness matrix

\(\mathbf{K}^{{\varvec{\Omega }} }\)

Rotational speed-dependent stiffness matrix


Length of the beam


Mass matrix


Force vector


Radius of the tip of the beam


Kinetic energy of the beam


Transformation matrix


Total potential energy


Work done by applied forces

\(U_{\gamma }\)

Potential energy due to shear strain

\(U_{\varepsilon }\)

Potential energy due to axial strain

\(W, U, \Phi \)

Displacement component of shape functions


Time-dependent generalized coordinates


Chord length


Chord length at the tip


Chord length at the hub


Chord ratio

\(\left( {\hat{{e}}_z ,\hat{{e}}_s ,\hat{{e}}_c } \right) \)

Rotating coordinate system


Displacement field vector


Thickness of the beam


Thickness of the beam at the tip


Thickness of the beam at the hub


Taper ratio


Shear coefficient


Mass of the beam


Index of order of polynomial for area


Index of order of polynomial for moment of inertia


Radius of the hub of the beam


Position vector of a typical point on the beam in stationary coordinate system


Position vector of a typical point on the beam in rotating coordinate system


Non-dimensional hub radius


Span of the beam


Axial displacement


Velocity vector


Lateral displacement


Distance of a typical fiber of the beam on a given cross-section area along lateral direction

\(\Lambda \)


\(\varOmega \)

Rotational speed

\(\bar{{\varOmega }}\)

Non-dimensional rotational speed

\(\beta \)

Stagger angle of the beam

\(\gamma _{sz}\)

Shear strain

\({\varvec{\upvarepsilon }}\)

Linear strain tensor

\(\varepsilon _\mathrm{s}\)

Axial strain

\(\theta \)

Rotational angle \(\varOmega t\)

{\(\xi \)}

Generalized coordinate

\(\rho \)


\(\varphi \)

cross-section rotation

\(\omega _{n}\)

Natural frequency (rad/s)

\(\bar{{\omega }}_n\)

Non-dimensional natural frequency


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

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