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

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Abstract

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

A

Cross-section area of the beam

\(A_{a}\)

Coefficients for polynomial of the cross-section area

\(A_{r}\)

Cross-section area of the beam at the hub

C

Coriolis damping matrix

E

Young’s modulus

\(F_\mathrm{cf}\)

Centrifugal force

G

Shear modulus

I

Area moment of inertia

\(I_{a}\)

Coefficients for polynomial of the moment of inertia

\(I_{r}\)

Moment of inertia of the beam at the hub

K

Total stiffness matrix

\(\mathbf{K}^{\mathbf{m}}\)

Elastic property-dependent stiffness matrix

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

Rotational speed-dependent stiffness matrix

L

Length of the beam

M

Mass matrix

P

Force vector

R

Radius of the tip of the beam

T

Kinetic energy of the beam

T

Transformation matrix

U

Total potential energy

\(U_{W}\)

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

XYZ

Time-dependent generalized coordinates

c

Chord length

\(c_{R}\)

Chord length at the tip

\(c_{r}\)

Chord length at the hub

\(\bar{{c}}\)

Chord ratio

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

Rotating coordinate system

d

Displacement field vector

h

Thickness of the beam

\(h_{R}\)

Thickness of the beam at the tip

\(h_{r}\)

Thickness of the beam at the hub

\(\bar{{h}}\)

Taper ratio

k

Shear coefficient

m

Mass of the beam

\(n_{A}\)

Index of order of polynomial for area

\(n_{I}\)

Index of order of polynomial for moment of inertia

r

Radius of the hub of the beam

r

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

\(\mathbf{r}_{{\mathbf{r}}}\)

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

\(\bar{{r}}\)

Non-dimensional hub radius

s

Span of the beam

u

Axial displacement

v

Velocity vector

w

Lateral displacement

z

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

\(\Lambda \)

Lagrangian

\(\varOmega \)

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

\(\rho \)

Density

\(\varphi \)

cross-section rotation

\(\omega _{n}\)

Natural frequency (rad/s)

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

Non-dimensional natural frequency

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Notes

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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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