Abstract

Blisks suffer from flutter, a self-sustained vibration caused by aerodynamic coupled forces. This instability could cause serious damage to the blades and the machine. Flutter stability is usually analyzed based on the eigenvalue method in the aspect of the linear structural dynamic system, which transforms a dynamics stability analysis into a point of equilibrium in an infinite time scale. However, in reality, most of the blisk vibrations arise on a finite time horizon. The transient vibration amplification may cause serious damage. This paper proposes a transient flutter stability analysis method in a finite time for structural mistuned blisk based on the energy growth method. First, two common blisk models coupled aerodynamic force with different complexity are built and are all expressed in the state space representation. A novel energy growth method is then employed to analyze the transient stability and to find the maximum energy growth of the models. The optimal initial condition which leads to the maximum energy growth is obtained. A new flutter stability criterion is developed to consider the transient stability based on the energy growth method and the infinite time stability based on the eigenvalue method. The new transient stability method is verified by two numerical studies. It is found that the structural mistuned blisk model which is traditionally predicted stable still has a transient instability in a finite time due to the non-normal property of the dynamic state matrix.

References

1.
Seeley
,
C. E.
,
Wakelam
,
C.
,
Zhang
,
X.
,
Hofer
,
D.
, and
Ren
,
W.-M.
,
2017
, “
Investigations of Flutter and Aerodynamic Damping of a Turbine Blade: Experimental Characterization
,”
ASME J. Turbomach.
,
139
(
8
), p.
081011
.10.1115/1.4035840
2.
Veers
,
P. S.
,
Ashwill
,
T. D.
,
Sutherland
,
H. J.
,
Laird
,
D. L.
,
Lobitz
,
D. W.
,
Griffin
,
D. A.
,
Mandell
,
J. F.
,
Musial
,
W. D.
,
Jackson
,
K.
,
Zuteck
,
M.
,
Miravete
,
A.
,
Tsai
,
S. W.
, and
Richmond
,
J. L.
,
2003
, “
Trends in the Design, Manufacture and Evaluation of Wind Turbine Blades
,”
Wind Energy
,
6
(
3
), pp.
245
259
.10.1002/we.90
3.
Tan
,
Y.
,
Zang
,
C.
, and
Petrov
,
E. P.
,
2019
, “
Mistuning Sensitivity and Optimization for Bladed Disks Using High-Fidelity Models
,”
Mech. Syst. Signal Process.
,
124
, pp.
502
523
.10.1016/j.ymssp.2019.02.002
4.
Castanier
,
M. P.
, and
Pierre
,
C.
,
2006
, “
Modeling and Analysis of Mistuned Bladed Disk Vibration: Current Status and Emerging Directions
,”
J. Propul. Power
,
22
(
2
), pp.
384
396
.10.2514/1.16345
5.
Figaschewsky
,
F.
, and
Kühhorn
,
A.
,
2015
, “
Analysis of Mistuned Blade Vibrations Based on Normally Distributed Blade Individual Natural Frequencies
,”
ASME
Paper No. GT2015-43121.10.1115/GT2015-43121
6.
Dugundji
,
J.
,
1979
, “
Flutter Analysis of a Tuned Rotor With Rigid and Flexible Disks
,” Gas Turbine and Plasma Dynamics Laboratory, Massachusetts Institute of Technology, Cambridge, UK, Report No.
146
.http://hdl.handle.net/1721.1/104419
7.
Besem
,
F. M.
,
Kielb
,
R. E.
, and
Key
,
N. L.
,
2016
, “
Forced Response Sensitivity of a Mistuned Rotor From an Embedded Compressor Stage
,”
ASME J. Turbomach.
,
138
(
3
), p.
031002
.10.1115/1.4031866
8.
Carta
,
F. O.
,
1967
, “
Coupled Blade-Disk-Shroud Flutter Instabilities in Turbojet Engine Rotors
,”
ASME J. Eng. Power
,
89
(
3
), pp.
419
426
.10.1115/1.3616708
9.
Hall
,
K. C.
, and
Lorence
,
C. B.
,
1993
, “
Calculation of Three-Dimensional Unsteady Flows in Turbomachinery Using the Linearized Harmonic Euler Equations
,”
ASME J. Turbomach.
,
115
(
4
), pp.
800
809
.10.1115/1.2929318
10.
Bendiksen
,
O.
, and
Friedmann
,
P.
,
1980
, “
Coupled Bending-Torsion Flutter in Cascades
,”
AIAA J.
,
18
(
2
), pp.
194
201
.10.2514/3.50748
11.
Bendiksen
,
O. O.
,
1984
, “
Flutter of Mistuned Turbomachinery Rotors
,”
ASME J. Eng. Gas Turbines Power
,
106
(
1
), pp.
25
33
.10.1115/1.3239546
12.
Bendiksen
,
O. O.
, and
Friedmann
,
P. P.
,
1982
, “
The Effect of Bending-Torsion Coupling on Fan and Compressor Blade Flutter
,”
ASME J. Eng. Power
,
104
(
3
), pp.
617
623
.10.1115/1.3227324
13.
White
,
J. F. I.
, and
Bendiksen
,
O. O.
,
1986
, “
Aeroelastic Behavior of Low Aspect Ratio Metal and Composite Blades
,”
ASME
Paper No. 86-GT-243.10.1115/86-GT-243
14.
Dugundji
,
J.
, and
Bundas
,
D. J.
,
1984
, “
Flutter and Forced Response of Mistuned Rotors Using Standing Wave Analysis
,”
AIAA J.
,
22
(
11
), pp.
1652
1661
.10.2514/3.8832
15.
Doi
,
H.
, and
Alonso
,
J. J.
,
2002
, “
Fluid/Structure Coupled Aeroelastic Computations for Transonic Flows in Turbomachinery
,”
ASME
Paper No. GT2002-30313.10.1115/GT2002-30313
16.
Liu
,
F.
,
Cai
,
J.
,
Zhu
,
Y.
,
Tsai
,
H. M.
, and
Wong
,
A. S. F.
,
2001
, “
Calculation of Wing Flutter by a Coupled Fluid-Structure Method
,”
J. Aircr.
,
38
(
2
), pp.
334
342
.10.2514/2.2766
17.
Schmid
,
P. J.
,
2007
, “
Nonmodal Stability Theory
,”
Annu. Rev. Fluid Mech.
,
39
(
1
), pp.
129
162
.10.1146/annurev.fluid.38.050304.092139
18.
Schmid
,
P. J.
, and
Brandt
,
L.
,
2014
, “
Analysis of Fluid Systems: Stability, Receptivity, Sensitivity: Lecture Notes From the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013
,”
Appl. Mech. Rev.
,
66
(
2
), p.
021003
.10.1115/1.4026375
19.
Bi
,
Q.
,
Wang
,
X.
,
Chen
,
H.
,
Zhu
,
L.
, and
Ding
,
H.
,
2018
, “
Non-Normal Dynamic Analysis for Predicting Transient Milling Stability
,”
ASME J. Dyn. Syst., Meas., Control
,
140
(
8
), p.
084501
.10.1115/1.4039033
20.
Wang
,
X.
,
Bi
,
Q.
,
Chen
,
T.
,
Zhu
,
L.
, and
Ding
,
H.
,
2019
, “
Transient Vibration Analysis Method for Predicting the Transient Behavior of Milling With Variable Spindle Speeds
,”
ASME J. Manuf. Sci. Eng.
,
141
(
5
), p.
051009
.10.1115/1.4043265
21.
Lee
,
I.
,
Chung
,
C.
,
Shin
,
S. J.
, and
Kim
,
Y.
,
2007
, “
Flutter and Forced Response Analysis of an Intentionally Mistuned Bladed Disk
,”
AIAA
Paper No. 2007-2193.10.2514/6.2007-2193
22.
Feiner
,
D. M.
, and
Griffin
,
J.
,
2002
, “
A Fundamental Model of Mistuning for a Single Family of Modes
,”
ASME J. Turbomach.
,
124
(
4
), pp.
597
605
.10.1115/1.1508384
23.
Malzacher
,
L.
,
Geist
,
S.
,
Motta
,
V.
,
Peitsch
,
D.
, and
Hennings
,
H.
,
2019
, “
A Low-Speed Compressor Test Rig for Flutter Investigations
,”
ASME J. Turbomach.
,
141
(
5
), p.
051009
.10.1115/1.4041817
24.
Lee
,
I.
,
Shin
,
S.
, and
Kim
,
Y.
,
2013
, “
Mistuned Bladed Disk Forced Vibration Analysis Based on Standing Wave Formulation
,”
Aerosp. Sci. Technol.
,
24
(
1
), pp.
275
282
.10.1016/j.ast.2011.12.001
25.
Kaza
,
K. R. V.
, and
Kielb
,
R. E.
,
1982
, “
Flutter and Response of a Mistuned Cascade in Incompressible Flow
,”
AIAA J.
,
20
(
8
), pp.
1120
1127
.10.2514/3.51172
26.
Taira
,
K.
,
Brunton
,
S. L.
,
Dawson
,
S. T.
,
Rowley
,
C. W.
,
Colonius
,
T.
,
McKeon
,
B. J.
,
Schmidt
,
O. T.
,
Gordeyev
,
S.
,
Theofilis
,
V.
, and
Ukeiley
,
L. S.
,
2017
, “
Modal Analysis of Fluid Flows: An Overview
,”
AIAA J.
,
55
(
12
), pp.
4013
4041
.10.2514/1.J056060
27.
Kielb
,
R. E.
,
Feiner
,
D. M.
,
Griffin
,
J. H.
, and
Miyakozawa
,
T.
,
2004
, “
Flutter of Mistuned Bladed Disks and Blisks With Aerodynamic and FMM Structural Coupling
,”
ASME
Paper No. GT2004-54315.10.1115/GT2004-54315
You do not currently have access to this content.