Principal resonance bifurcation of bending–torsion coupling of aero-engine compressor blade with assembled clearance under lateral displacement excitation of rotor shaft

Gang Han; Yushu Chen

doi:10.1007/s11071-014-1547-2

Nonlinear Dynamics

November 2014, Volume 78, Issue 3, pp 2049–2058 | Cite as

Principal resonance bifurcation of bending–torsion coupling of aero-engine compressor blade with assembled clearance under lateral displacement excitation of rotor shaft

Authors
Authors and affiliations

Gang HanEmail author
Yushu Chen

Original Paper

First Online: 23 July 2014

Received: 17 November 2013

Accepted: 15 June 2014

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1 Citations

Abstract

The principal resonance of bending–torsion coupling of a compressor blade with an assembled clearance and a cubic structural nonlinearity, subjected to the lateral displacement excitation of rotor shaft and aerodynamic loads, is analyzed to explore the topology transformation of amplitude–frequency response along with the changes of the physical parameters. The bifurcation equation of the first-order principal resonance response is derived from using the averaging method. The transition set and the bifurcation figures of the response solution are obtained by the singularity theory. The effects of the main physical parameters of the system on the topology transformation of amplitude–frequency response are discussed.

Keywords

Compressor blade Bending–torsion coupling Displacement excitation Transition set Assembled clearance

List of symbols

$A_i $: Harmonic amplitude
$\theta _i $: Harmonic phase angle
$D$: Displacement amplitude of rotor lateral vibration
$\bar{{D}}$: Dimensionless displacement amplitude of rotor lateral vibration
$a_h $: Dimensionless location of elastic axis from mid-chord, positive toward trailing edge of airfoil
$r_\alpha $: Dimensionless radius of gyration about the elastic axis
$h$: Bending deflection relative to rotor shaft
$h_1 $: Dimensionless bending deflection relative to rotor shaft
$b$: Semi-chord
$m$: Mass per unit span of blade
$t$: Time
$\tau $: Dimensionless time
$S_\alpha $: Static moment per unit span about elastic axis
$I_\alpha $: Moment of inertia about elastic axis
$V$: Freestream velocity relative to blades
$V^{*}$: Non-dimensional freestream velocity
$\delta $: Semi-clearance distance
$\delta _1 $: Dimensionless semi-clearance distance
$\{^{.}\}$: Differentiation with respect to $t$
$\{^{\prime }\}$: Differentiation with respect to $\tau $
$F(h)$: Piecewise linear displacement function in bending
$F_1 (h_1 )$: Dimensionless piecewise linear displacement function in bending
$M(\alpha )$: Nonlinear function in torsion
$Q_h (t)$: Aerodynamic force
$Q_\alpha (t)$: Aerodynamic moment
$S(h_1 )$: Nonlinear displacement function in bending
$x_\alpha $: Dimensionless location of mass center from the elastic axis, positive toward trailing edge of the airfoil
$K_h $: Bending stiffness of blade
$K_\alpha $: Torsional rigidity of blade
$\alpha $: Torsional deflection, positive clockwise
$\beta $: Nonlinear torsional rigidity coefficient
$\omega _h $: Uncoupled bending natural frequencies
$\omega _\alpha $: Uncoupled torsional natural frequencies
$\omega $: Dimensionless frequency of rotor lateral vibration
$\Omega $: Frequency of rotor lateral vibration
$\mu $: Mass ratio of blade
$\rho $: Air density
$\bar{{\omega }}$: Frequency ratio $\omega _h /\omega _\alpha $
$\lambda _k $: Modal frequency
$\zeta _h $: Damping coefficients in bending
$\zeta _\alpha $: Damping coefficients in torsion
$B_1 $: Bifurcation set
$H_1 $: Hysteresis set
$D_1 $: Double limit set
$B_2 $: Supplemental constraint bifurcation set
$H_2 $: Supplemental constraint hysteresis set
$D_2 $: Supplemental constraint double limit set

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Notes

Acknowledgments

The authors acknowledge the financial support of the National Science Foundation of China under the Grant 10632040.

Appendix

The coefficient expressions of Eqs. (5) and (6) are

$$\begin{aligned}&a_{11} =\frac{\bar{{\omega }}^{2}r_\alpha ^2 }{r_\alpha ^2 -x_a^2 }, \quad a_{12} =\frac{\left[ {2V^{*{2}}}r_\alpha ^2 +x_a V^{^{{*2}}(1+2a_h )-\mu x_a r_\alpha ^2 } \right] }{\mu \left( {r_\alpha ^2 -x_a ^{2}} \right) }, \\&a_{13} =\frac{2\zeta _\lambda r_\alpha ^2 \bar{{\omega }}}{r_\alpha ^2 -x_a^2 }, \quad a_{{14}} -\frac{2\zeta _\alpha x_a r_\alpha ^2 }{r_\alpha ^2 -x_a^2 }, \quad a_{15} = -\frac{x_a r_\alpha ^2 \beta }{r_\alpha ^2 -x_a^2 }, \\&a_{16} =\frac{r_\alpha ^2 D_1 \omega ^{2}\sin \gamma }{r_\alpha ^2 -x_a^2 }, a_{21} = -\frac{x_a }{r_\alpha ^2 }a_{11} ,\\&a_{22} =\frac{\left( {2V^{*{2}}x_a +V^{*{2}}(1+2a_h ) -\mu r_\alpha ^2 } \right) }{\mu \left( {x_a^2 -r_\alpha ^2 } \right) },\\&a_{23} =-\frac{x_a }{r_\alpha ^2 }a_{13} , \quad a_{24} =-\frac{1}{x_a }a_{{14}} ,\\&a_{25} =-\frac{1}{x_a }a_{15} , \quad a_{26} =-\frac{x_a }{r_\alpha ^2 }a_{16} . \end{aligned}$$

The matrix ${{\varvec{A}}}$ and vector $F(X,\tau )$ of Eq. (7) are

$$\begin{aligned}&\mathbf{A}=\left[ {{\begin{array}{cccc} 0&{} 1&{} 0&{} 0 \\ {-a_{11} }&{} 0&{} {-a_{12} }&{} 0 \\ 0&{} 0&{} 0&{} 1 \\ {-a_{21} }&{} 0&{} {-a_{22} }&{} 0 \\ \end{array} }} \right] , \quad F(X,\tau )=\left\{ {{\begin{array}{l} 0 \\ {F_1 } \\ 0 \\ {F_2 } \\ \end{array} }} \right\} , \end{aligned}$$

$$\begin{aligned} \left\{ {\begin{array}{l} \displaystyle F_1 \!=\!-a_{13} x_2 -a_{{14}} x_4 -\;a_{11} S(x_1 )-a_{15} x_3^3 +a_{16} \cos \omega \tau \\ \displaystyle F_2 \!=\!-a_{23} x_2 -\;a_{24} x_4 -a_{21} S(x_1 )-a_{25} x_3^3 +a_{26} \cos \omega \tau \\ \end{array}} \right. . \end{aligned}$$

The variables of Eq. (9) are

$$\begin{aligned} \overline{F_1 } =0, \quad \overline{F_2 } =F_1 , \quad \overline{F_3 } =0, \quad \overline{F_4 } =F_2 . \end{aligned}$$

The modal frequencies of derivation equation $\dot{X}=AX$ of Eq. (7) are

$$\begin{aligned}&\lambda _k =\sqrt{\frac{1}{2}\left[ {\left( {a_{22} +a_{11} } \right) \mp \sqrt{\left( {a_{22} +a_{11} } \right) ^{2}-4\left( {a_{22} a_{11} -a_{21} a_{12} } \right) }} \right] }\\&\quad (k=1,2), \end{aligned}$$

where the conditions of two different positive $\lambda _k $ are

$$\begin{aligned} \left\{ {\begin{array}{l} a_{22} +a_{11} >0 \\ \left( {a_{22} +a_{11} } \right) ^{2}-4\left( {a_{22} a_{11} -a_{21} a_{12} } \right) >0 \\ \end{array}} \right. . \end{aligned}$$

The fundamental solution sets $\phi _{sk} (\theta _k )$ and $\varphi _{sk}^*(\theta _k )$ of the derivation equation $\dot{X}=AX$ of Eq. (7) are

$$\begin{aligned} \left. {\begin{array}{ll} \varphi _{1k} =\cos \lambda _k \tau ,\\ \varphi _{1k}^*=\sin \lambda _k \tau \\ \varphi _{2k} =-\lambda _k \sin \lambda _k \tau ,\\ \varphi _{2k}^*=\lambda _k \cos \lambda _k \tau \\ \varphi _{3k} =\frac{1}{a_{12} }\left( {\lambda _k^2 -a_{11} } \right) \cos \lambda _k \tau ,\\ \varphi _{3k}^*=\frac{1}{a_{12} }\left( {\lambda _k^2 -a_{11} } \right) \sin \lambda _k \tau \\ \varphi _{4k} =-\frac{\lambda _k }{a_{12} }\left( {\lambda _k^2 -a_{11} } \right) \sin \lambda _k \tau ,\\ \varphi _{4k}^*=\frac{\lambda _k }{a_{12} }\left( {\lambda _k^2 -a_{11} } \right) \cos \lambda _k \tau \\ \end{array}} \right\} \quad (k=1,2). \end{aligned}$$

The fundamental solution sets $\phi _{sj} (\theta _j )$ and $\phi _{sj}^*(\theta _j )$ of the adjoint equation of the derivation equation $\dot{X}={{\varvec{A}}}\cdot X$of Eq. (7) are

$$\begin{aligned} \left. {\begin{array}{l} \phi _{1k} =\cos \lambda _k \tau ,\\ \phi _{1k}^*=\sin \lambda _k \tau \\ \phi _{2k} =-\frac{1}{\lambda _k }\sin \lambda _k \tau ,\\ \phi _{2k}^*=\frac{1}{\lambda _k }\cos \lambda _k \tau \\ \phi _{3k} =\frac{1}{a_{21} }\left( {\lambda _k ^{2}-a_{11} } \right) \cos \lambda _k \tau ,\\ \phi _{3k}^*=\frac{1}{a_{21} }\left( {\lambda _k ^{2}-a_{11} } \right) \sin \lambda _k \tau \\ \phi _{4k} =-\frac{1}{a_{21} \lambda _k }\left( {\lambda _k ^{2}-a_{11} } \right) \sin \lambda _k \tau ,\\ \phi _{4k}^*=\frac{1}{a_{21} \lambda _k }\left( {\lambda _k ^{2}-a_{11} } \right) \cos \lambda _k \tau \\ \end{array}} \right\} \quad (k=1,2). \end{aligned}$$

The expression of $\Delta _k^{\prime } $ in formula (10) is

$$\begin{aligned} \Delta _k^{\prime } =1+\frac{1}{a_{12} a_{21} }\left( {\lambda _k ^{2}-a_{11} } \right) ^{2}. \end{aligned}$$

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Authors and Affiliations

Gang Han
- 1
Email author
Yushu Chen
- 1

1.School of AstronauticsHarbin Institute of TechnologyHarbinChina

Principal resonance bifurcation of bending–torsion coupling of aero-engine compressor blade with assembled clearance under lateral displacement excitation of rotor shaft

Abstract

Keywords

List of symbols

Notes

Acknowledgments

References

Copyright information

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