Abstract

The effect of differences in nominally equal sectors of a bladed disk (mistuning) is a well-known problem for designers since the forced response may show localized amplification of the blade response with respect to a cyclically symmetric (tuned) configuration. In order to perform a large number of simulations in a reasonable amount of time to characterize the highest blade response, corresponding to the worst mistuning pattern, reduction techniques have been developed, where the mistuning is introduced directly in a compact, reduced order model (ROM) obtained from very large finite element (FE) models. Typically, mistuning is introduced in the ROM in terms of natural frequency perturbations of the blade; nevertheless, a better insight is specifically required in order to correlate mistuning to a specific source (geometrical, material, contact mistuning). In this paper, a reduction technique is presented to take into account mistuning due to the contact uncertainties at the blade root joint, which can be caused by design tolerances, manufacturing process, assembly procedures, wear, etc. The technique takes its basis from the Craig–Bampton component mode synthesis (CB-CMS) applied to the uncoupled blade and disk sector, which is typically included in most of the FE software for easy implementation in standard industrial practice. The full set of master degrees-of-freedom (DOFs) at the random contacts are purposely reduced using an optimal local modal basis based on the Gram–Schmidt interface (GSI) technique developed by the authors. Experimental evidence of actual uncertain contact obtained during joint preloading is used to find an appropriate base to represent typical contact patterns.

References

1.
Castanier
,
M. P.
, and
Pierre
,
C.
,
2006
, “
Modeling and Analysis of Mistuned Bladed Disk Vibration: Status and Emerging Directions
,”
J. Propul. Power
,
22
(
2
), pp.
384
396
.10.2514/1.16345
2.
Feiner
,
D. M.
, and
Griffin
,
J. H.
,
2002
, “
A Fundamental Model of Mistuning for a Single Family of Modes
,”
ASME J. Turbomach.
,
124
(
4
), pp.
597
605
.10.1115/1.1508384
3.
Castanier
,
M. P.
, and
Pierre
,
C.
,
2002
, “
Using Intentional Mistuning in the Design of Turbomachinery Rotors
,”
AIAA J.
,
40
(
10
), pp.
2077
2086
.10.2514/2.1542
4.
Petrov
,
E. P.
,
Sanliturk
,
K. Y.
, and
Ewins
,
D. J.
,
2002
, “
A New Method for Dynamic Analysis of Mistuned Bladed Disks Based on the Exact Relationship Between Tuned and Mistuned Systems
,”
ASME J. Eng. Gas Turbines Power
,
124
(
3
), pp.
586
597
.10.1115/1.1451753
5.
Petrov
,
E. P.
, and
Ewins
,
D. J.
,
2003
, “
Analysis of the Worst Mistuning Patterns in Bladed Disk Assemblies
,”
ASME J. Turbomach.
,
125
(
4
), pp.
623
631
.10.1115/1.1622710
6.
Martel
,
C.
, and
Corral
,
R.
,
2009
, “
Asymptotic Description of Maximum Mistuning Amplification of Bladed Disk Forced Response
,”
ASME J. Eng. Gas Turbines Power
,
131
(
2
), p.
022506
.10.1115/1.2968868
7.
Wei
,
S. T.
, and
Pierre
,
C.
,
1990
, “
Statistical Analysis of the Forced Response of Mistuned Cyclic Assemblies
,”
AIAA J.
,
28
(
5
), pp.
861
868
.10.2514/3.25131
8.
Rivas-Guerra
,
A. J.
, and
Mignolet
,
M. P.
,
2003
, “
Maximum Amplification of Blade Response Due to Mistuning: Localization and Mode Shape Aspects of the Worst Disks
,”
ASME J. Turbomach.
,
125
(
3
), pp.
442
454
.10.1115/1.1506958
9.
Lim
,
S. H.
,
Castanier
,
M. P.
, and
Pierre
,
C.
,
2004
, “
Vibration Modeling of Bladed Disks Subject to Geometric Mistuning and Design Changes
,”
AIAA
Paper No. 2004-1686. 10.2514/6.2004-1686
10.
Sternchuss
,
A.
, and
Balmes
,
E.
,
2006
, “
On the Reduction of Quasi-Cyclic Disk Models With Variable Rotation Speeds
,”
Proceedings of ISMA2006: International Conference on Noise and Vibration Engineering
, Leuven, Belgium, Sept. 18–20, pp.
3925
3939
.https://hal.archives-ouvertes.fr/hal-00266394
11.
Sternchuss
,
A.
,
Balmes
,
E.
,
Jean
,
P.
, and
Lombard
,
J. P.
,
2009
, “
Reduction of Multistage Disk Models: Application to an Industrial Rotor
,”
ASME J. Eng. Gas Turbines Power
,
131
(
1
), p.
012502
.10.1115/1.2967478
12.
Legrand
,
M.
,
Batailly
,
A.
,
Magnain
,
B.
,
Cartraud
,
P.
, and
Pierre
,
C.
,
2012
, “
Full Three-Dimensional Investigation of Structural Contact Interactions in Turbomachines
,”
J. Sound Vib.
,
331
(
11
), pp.
2578
2601
.10.1016/j.jsv.2012.01.017
13.
Craig
,
R. R.
, and
Bampton
,
M. C. C.
,
1968
, “
Coupling of Substructures for Dynamic Analyses
,”
AIAA J.
,
6
(
7
), pp.
1313
1319
.10.2514/3.4741
14.
Siewert
,
C.
,
Panning
,
L.
,
Wallaschek
,
J.
, and
Richter
,
C.
,
2010
, “
Multiharmonic Forced Response Analysis of a Turbine Blading Coupled by Nonlinear Contact Forces
,”
ASME J. Eng. Gas Turbines Power
,
132
(
8
), p.
08250
.10.1115/1.4000266
15.
Tang
,
W.
, and
Epureanu
,
B. I.
,
2017
, “
Nonlinear Dynamics of Mistuned Bladed Disks With Ring Dampers
,”
Int. J. Non-Linear Mech.
,
97
, pp.
30
40
.10.1016/j.ijnonlinmec.2017.08.001
16.
Lassalle
,
M.
, and
Firrone
,
C. M.
,
2016
, “
Nonlinear Forced Response of a Stator Vane With Multiple Friction Contacts Using a Coupled Static/Dynamic Approach
,”
Proceedings of the Seventh European Congress on Computational Methods in Applied Sciences and Engineering
, Crete, Greece, June 5–10, pp.
8612
8626
.10.7712/100016.2437.6969
17.
Pesaresi
,
L.
,
Salles
,
L.
,
Jones
,
A.
,
Green
,
J. S.
, and
Schwingshackl
,
C. W.
,
2017
, “
Modelling the Nonlinear Behaviour of an Underplatform Damper Test Rig for Turbine Applications
,”
Mech. Syst. Signal Process.
,
85
, pp.
662
679
.10.1016/j.ymssp.2016.09.007
18.
Kaptan
,
F.
,
Panning
,
L.
,
Wallaschek
,
J.
, and
Salit
,
V.
,
2016
, “
Forced Response of Shrouded Blades With Variable Operating Points
,”
Proceedings of the Seventh European Congress on Computational Methods in Applied Sciences and Engineering
, Crete, Greece, June 5–10, pp.
5396
5405
.10.7712/100016.2188.4914
19.
Tran
,
D. M.
,
2001
, “
Component Mode Synthesis Methods Using Interface Modes. Application to Structures With Cyclic Symmetry
,”
Comput. Struct.
,
79
(
2
), pp.
209
222
.10.1016/S0045-7949(00)00121-8
20.
Tran
,
D. M.
,
2009
, “
Component Mode Synthesis Methods Using Partial Interface Modes: Application to Tuned and Mistuned Structures With Cyclic Symmetry
,”
Comput. Struct.
,
87
(
17–18
), pp.
1141
1153
.10.1016/j.compstruc.2009.04.009
21.
Guyan
,
R. J.
,
1965
, “
Reduction of Stiffness and Mass Matrices
,”
AIAA J.
,
3
(
2
), pp.
380
380
.10.2514/3.2874
22.
Castanier
,
M. P.
,
Tan
,
Y. C.
, and
Pierre
,
C.
,
2001
, “
Characteristic Constraint Modes for Component Mode Synthesis
,”
AIAA J.
,
39
(
6
), pp.
1182
1187
.10.2514/2.1433
23.
Yang
,
M. T.
, and
Griffin
,
J. H.
,
2001
, “
A Reduced-Order Model of Mistuning Using a Subset of Nominal System Modes
,”
ASME J. Eng. Gas Turbines Power
,
123
(
4
), pp.
893
900
.10.1115/1.1385197
24.
Lim
,
S. H.
,
Bladh
,
R.
,
Castanier
,
M. P.
, and
Pierre
,
C.
,
2007
, “
Compact, Generalized Component Mode Mistuning Representation for Modeling Bladed Disk Vibration
,”
AIAA J.
,
45
(
9
), pp.
2285
2298
.10.2514/1.13172
25.
Vargiu
,
P.
,
Firrone
,
C. M.
,
Zucca
,
S.
, and
Gola
,
M. M.
,
2011
, “
A Reduced Order Model Based on Sector Mistuning for the Dynamic Analysis of Mistuned Bladed Disks
,”
Int. J. Mech. Sci.
,
53
(
8
), pp.
639
646
.10.1016/j.ijmecsci.2011.05.010
26.
Mitra
,
M.
,
Zucca
,
S.
, and
Epureanu
,
B. I.
,
2016
, “
Adaptive Microslip Projection for Reduction of Frictional and Contact Nonlinearities in Shrouded Blisks
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
4
), p.
041016
.10.1115/1.4033003
27.
Pourkiaee
,
S. M.
, and
Zucca
,
S.
,
2019
, “
A Reduced Order Model for Nonlinear Dynamics of Mistuned Bladed Disks With Shroud Friction Contacts
,”
ASME J. Eng. Gas Turbines Power
,
141
(
1
), p.
011031
.10.1115/1.4041653
28.
Pourkiaee
,
S. M.
,
Zucca
,
S.
, and
Parker
,
R. G.
,
2022
, “
Relative Cyclic Component Mode Synthesis: A Reduced Order Modeling Approach for Mistuned Bladed Disks With Friction Interfaces
,”
Mech. Syst. Signal Process.
,
163
, p.
108197
.10.1016/j.ymssp.2021.108197
29.
Benner
,
P.
,
Gugercin
,
S.
, and
Willcox
,
K.
,
2015
, “
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
,”
SIAM Rev.
,
57
(
4
), pp.
483
531
.10.1137/130932715
30.
Beck
,
J. A.
,
Brown
,
J. M.
,
Kaszynski
,
A. A.
,
Cross
,
C. J.
, and
Slater
,
J. C.
,
2015
, “
Geometric Mistuning Reduced-Order Models for Integrally Bladed Rotors With Mistuned Disk-Blade Boundaries
,”
ASME J. Turbomach.
,
137
(
7
), p.
071001
.10.1115/1.4029122
31.
Yan
,
X.
,
Du
,
D.
,
Liu
,
H.
,
Xu
,
K.
, and
Sun
,
W.
,
2022
, “
Nonlinear Vibration Analysis of Coated Blisks in the Presence of Stiffness Mistuning Identification
,”
Mech. Syst. Signal Process.
,
165
, p.
108338
.10.1016/j.ymssp.2021.108338
32.
Schwerdt
,
L.
,
Willeke
,
S.
,
Panning-Von Scheidt
,
L.
, and
Wallaschek
,
J.
,
2019
, “
Reduced-Order Modeling of Bladed Disks Considering Small Mistuning of the Disk Sectors
,”
ASME J. Eng. Gas Turbines Power
,
141
(
5
), p.
052502
.10.1115/1.4041071
33.
Sinclair
,
G. B.
,
Cormier
,
N. G.
,
Griffin
,
J. H.
, and
Meda
,
G.
,
2002
, “
Contact Stresses in Dovetail Attachments: Finite Element Modeling
,”
ASME J. Eng. Gas Turbines Power
,
124
(
1
), pp.
182
189
.10.1115/1.1391429
34.
Sinclair
,
G. B.
, and
Cormier
,
N. G.
,
2003
, “
Contact Stresses in Dovetail Attachments: Alleviation Via Precision Crowning
,”
ASME J. Eng. Gas Turbines Power
,
125
(
4
), pp.
1033
1041
.10.1115/1.1584477
35.
Sinclair
,
G. B.
, and
Cormier
,
N. G.
,
2002
, “
Contact Stresses in Dovetail Attachments: Physical Modeling
,”
ASME J. Eng. Gas Turbines Power
,
124
(
2
), pp.
325
331
.10.1115/1.1415740
36.
Rajasekaran
,
R.
, and
Nowell
,
D.
,
2006
, “
Fretting Fatigue in Dovetail Blade Roots: Experiment and Analysis
,”
Tribol. Int.
,
39
(
10
), pp.
1277
1285
.10.1016/j.triboint.2006.02.044
37.
Ruiz
,
C.
, and
Nowell
,
D.
,
2000
, “
Designing Against Fretting Fatigue in Aeroengines
,”
Eur. Struct. Integr. Soc.
,
26
, pp.
73
95
.10.1016/S1566-1369(00)80043-6
38.
Botto
,
D.
, and
Alinejad
,
F.
,
2017
, “
Innovative Design of Attachment for Turbine Blade Rotating at High Speed
,”
ASME
Paper No. GT2017-64959. 10.1115/GT2017-64959
39.
Brake
,
M. R. W.
,
Krack
,
M.
, and
Schwingshackl
,
C. W.
,
2020
, “
Special Issue: Tribomechadynamics
,”
ASME J. Vib. Acoust.
,
142
(
5
), p.
050301
.10.1115/1.4048185
40.
Odofin
,
A. O.
, and
Epureanu
,
B. I.
,
2020
, “
Frequency-Adaptive Bilinear Reduced-Order Model for Structures With Intermittent Contacts
,”
Nonlinear Dyn.
,
99
(
1
), pp.
461
477
.10.1007/s11071-019-05000-x
41.
Zucca
,
S.
, and
Epureanu
,
B. I.
,
2014
, “
Bi-Linear Reduced-Order Models of Structures With Friction Intermittent Contacts
,”
Nonlinear Dyn.
,
77
(
3
), pp.
1055
1067
.10.1007/s11071-014-1363-8
42.
Saito
,
A.
,
Epureanu
,
B. I.
,
Castanier
,
M. P.
, and
Pierre
,
C.
,
2010
, “
Node Sampling for Nonlinear Vibration Analysis of Structures With Intermittent Contact
,”
AIAA J.
,
48
(
9
), pp.
1903
1915
.10.2514/1.J050061
43.
Battiato
,
G.
,
Firrone
,
C. M.
,
Berruti
,
T. M.
, and
Epureanu
,
B. I.
,
2018
, “
Reduction and Coupling of Substructures Via Gram–Schmidt Interface Modes
,”
Comput. Methods Appl. Mech. Eng.
,
336
, pp.
187
212
.10.1016/j.cma.2018.03.001
You do not currently have access to this content.