Abstract

Underplatform dampers (UPDs) are widely used in bladed disks to mitigate blades' vibration amplitude. Such devices introduce localized nonlinearities, whose modeling requires nonlinear solution techniques. When the harmonic balance method (HBM) is used to compute the nonlinear forced response of blades with UPDs, two different implementations are available: (i) the uncoupled approach, where the static equilibrium is determined in advance and used as input to determine the dynamic equilibrium; (ii) the coupled approach, where the static and dynamic equilibria are determined simultaneously. A common issue for both approaches is the variability in the static tangential contact forces, when the Coulomb friction model is used, that can result in multiple static equilibria and in multiple vibration levels. In this paper, the two approaches are used to determine the variability in the response levels of blades with UPDs and results are compared and discussed. Due to the large computation times associated with optimization algorithms implemented to compute the response limits, lumped parameter models are used; nevertheless, the main findings of the paper can be considered general. In particular, results show that the uncoupled approach systematically overestimates the uncertainty with respect to the coupled approach and that UPDs geometry affects the range of variability of the response.

References

1.
Cowles
,
B.
,
1996
, “
High Cycle Fatigue in Aircraft Gas Turbines—An Industry Perspective
,”
Int. J. Fract.
,
80
(
2–3
), pp.
147
163
.10.1007/BF00012667
2.
Sanliturk
,
K. Y.
,
Ewins
,
D. J.
, and
Stanbridge
,
A. B.
,
2001
, “
Underplatform Dampers for Turbine Blades: Theoretical Modeling, Analysis, and Comparison With Experimental Data
,”
ASME J. Eng. Gas Turbines Power
,
123
(
4
), pp.
919
929
.10.1115/1.1385830
3.
Cigeroglu
,
E.
,
An
,
N.
, and
Menq
,
C.-H.
,
2009
, “
Forced Response Prediction of Constrained and Unconstrained Structures Coupled Through Frictional Contacts
,”
ASME J. Eng. Gas Turbines Power
,
131
(
2
), p.
022505
.10.1115/1.2940356
4.
Pesaresi
,
L.
,
Salles
,
L.
,
Jones
,
A.
,
Green
,
J.
, and
Schwingshackl
,
C.
,
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
5.
Laxalde
,
D.
,
Thouverez
,
F.
, and
Lombard
,
J.-P.
,
2010
, “
Forced Response Analysis of Integrally Bladed Disks With Friction Ring Dampers
,”
ASME J. Vib. Acoust.
,
132
(
1
), p.
011013
.10.1115/1.4000763
6.
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
7.
Szwedowicz
,
J.
,
Secall-Wimmel
,
T.
, and
Dünck-Kerst
,
P.
,
2008
, “
Damping Performance of Axial Turbine Stages With Loosely Assembled Friction Bolts: The Nonlinear Dynamic Assessment
,”
ASME J. Eng. Gas Turbines Power
,
130
(
3
), p.
032505
.10.1115/1.2838998
8.
Ferhatoglu
,
E.
,
Zucca
,
S.
,
Botto
,
D.
,
Auciello
,
J.
, and
Arcangeli
,
L.
,
2022
, “
Nonlinear Vibration Analysis of Turbine Bladed Disks With Midspan Dampers
,”
ASME J. Eng. Gas Turbines Power
,
144
(
4
), p.
041021
.10.1115/1.4053107
9.
Petrov
,
E. P.
, and
Ewins
,
D. J.
,
2003
, “
Analytical Formulation of Friction Interface Elements for Analysis of Nonlinear Multi-Harmonic Vibrations of Bladed Disks
,”
ASME J. Turbomach.
,
125
(
2
), pp.
364
371
.10.1115/1.1539868
10.
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.
082501
.10.1115/1.4000266
11.
Petrov
,
E. P.
, and
Ewins
,
D. J.
,
2006
, “
Effects of Damping and Varying Contact Area at Blade-Disk Joints in Forced Response Analysis of Bladed Disk Assemblies
,”
ASME J. Turbomach.
,
128
(
2
), pp.
403
410
.10.1115/1.2181998
12.
Siewert
,
C.
,
Panning
,
L.
,
Gerber
,
C.
, and
Masserey
,
P.
,
2008
, “
Numerical and Experimental Damping Prediction of a Nonlinearly Coupled Low Pressure Steam Turbine Blading
,”
ASME
Paper No. GT2008-51073.10.1115/GT2008-51073
13.
Griffin
,
J. H.
,
1980
, “
Friction Damping of Resonant Stresses in Gas Turbine Engine Airfoils
,”
ASME J. Eng. Power
,
102
(
2
), pp.
329
333
.10.1115/1.3230256
14.
Cameron
,
T. M.
,
Griffin
,
J. H.
,
Kielb
,
R. E.
, and
Hoosac
,
T. M.
,
1990
, “
An Integrated Approach for Friction Damper Design
,”
ASME J. Vib. Acoust.
,
112
(
2
), pp.
175
182
.10.1115/1.2930110
15.
Petrov, E. P., 2008, “Explicit Finite Element Models of Friction Dampers in Forced Response Analysis of Bladed Disks,”
ASME J. Eng. Gas Turbines Power
, 130(2), p.
022502
.10.1115/1.2772633
16.
Sanliturk
,
K.
, and
Ewins
,
D.
,
1996
, “
Modelling Two-Dimensional Friction Contact and Its Application Using Harmonic Balance Method
,”
J. Sound Vib.
,
193
(
2
), pp.
511
523
.10.1006/jsvi.1996.0299
17.
Yang
,
B.
, and
Menq
,
C.
,
1998
, “
Characterization of 3D Contact Kinematics and Prediction of Resonant Response of Structures Having 3D Frictional Constraint
,”
J. Sound Vib.
,
217
(
5
), pp.
909
925
.10.1006/jsvi.1998.1802
18.
Cardona
,
A.
,
Coune
,
T.
,
Lerusse
,
A.
, and
Geradin
,
M.
,
1994
, “
A Multiharmonic Method for Non-Linear Vibration Analysis
,”
Int. J. Numer. Methods Eng.
,
37
(
9
), pp.
1593
1608
.10.1002/nme.1620370911
19.
Krack
,
M.
,
Salles
,
L.
, and
Thouverez
,
F.
,
2017
, “
Vibration Prediction of Bladed Disks Coupled by Friction Joints
,”
Arch. Comput. Methods Eng.
,
24
(
3
), pp.
589
636
.10.1007/s11831-016-9183-2
20.
Firrone
,
C. M.
,
Zucca
,
S.
, and
Gola
,
M. M.
,
2011
, “
The Effect of Underplatform Dampers on the Forced Response of Bladed Disks by a Coupled Static/Dynamic Harmonic Balance Method
,”
Int. J. Non-Linear Mech.
,
46
(
2
), pp.
363
375
.10.1016/j.ijnonlinmec.2010.10.001
21.
Zucca
,
S.
,
Firrone
,
C. M.
, and
Gola
,
M. M.
,
2012
, “
Numerical Assessment of Friction Damping at Turbine Blade Root Joints by Simultaneous Calculation of the Static and Dynamic Contact Loads
,”
Nonlinear Dyn.
,
67
(
3
), pp.
1943
1955
.10.1007/s11071-011-0119-y
22.
Ferhatoglu
,
E.
,
Gastaldi
,
C.
,
Botto
,
D.
, and
Zucca
,
S.
,
2022
, “
An Experimental and Computational Comparison of the Dynamic Response Variability in a Turbine Blade With Under-Platform Dampers
,”
Mech. Syst. Signal Process.
,
172
, p.
108987
.10.1016/j.ymssp.2022.108987
23.
Yang
,
B. D.
, and
Menq
,
C. H.
,
1998
, “
Characterization of Contact Kinematics and Application to the Design of Wedge Dampers in Turbomachinery Blading: Part 1—Stick-Slip Contact Kinematics
,”
ASME J. Eng. Gas Turbines Power
,
120
(
2
), pp.
410
417
.10.1115/1.2818138
24.
Yang
,
B. D.
, and
Menq
,
C. H.
,
1998
, “
Characterization of Contact Kinematics and Application to the Design of Wedge Dampers in Turbomachinery Blading: Part 2—Prediction of Forced Response and Experimental Verification
,”
ASME J. Eng. Gas Turbines Power
,
120
(
2
), pp.
418
423
.10.1115/1.2818139
25.
Zucca
,
S.
,
Firrone
,
C. M.
, and
Gola
,
M.
,
2013
, “
Modeling Underplatform Dampers for Turbine Blades: A Refined Approach in the Frequency Domain
,”
J. Vib. Control
,
19
(
7
), pp.
1087
1102
.10.1177/1077546312440809
26.
Ferhatoglu
,
E.
,
Groß
,
J.
, and
Krack
,
M.
,
2023
, “
Frequency Response Variability in Friction-Damped Structures Due to Non-Unique Residual Tractions: Obtaining Conservative Bounds Using a Nonlinear-Mode-Based Approach
,”
Mech. Syst. Signal Process.
,
201
, p.
110651
.10.1016/j.ymssp.2023.110651
27.
Ferhatoglu
,
E.
, and
Zucca
,
S.
,
2021
, “
On the Non-Uniqueness of Friction Forces and the Systematic Computation of Dynamic Response Boundaries for Turbine Bladed Disks With Contacts
,”
Mech. Syst. Signal Process.
,
160
, p.
107917
.10.1016/j.ymssp.2021.107917
28.
Zara
,
G.
,
Ferhatoglu
,
E.
,
Berruti
,
T. M.
, and
Zucca
,
S.
,
2022
, “
Multiple Response Levels of Structures With Wedge Friction Dampers
,”
Proceedings of ISMA2022 International Conference on Noise and Vibration Engineering
, KU Leuven, Leuven, Belgium, Sept. 12–14, pp.
975
989
.https://www.researchgate.net/publication/373108013_Multiple_response_levels_of_structures_with_wedge_friction_dampers
29.
Cameron
,
T. M.
, and
Griffin
,
J. H.
,
1989
, “
An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems
,”
ASME J. Appl. Mech.
,
56
(
1
), pp.
149
154
.10.1115/1.3176036
You do not currently have access to this content.