A method is proposed for analyzing the periodic stick-slip motion of a single degree of freedom model of frictionally damped turbine blades. The method of finite elements in the time domain (FET) is based on a Hamilton’s weak principle, paralleling the variational methods in elastostatics. It permits a complete determination of the hysteretic friction force, and results in a set of highly structured system equations. The method has a number of unique features, which are utilized to provide a simple yet efficient approach for predicting the steady-state response. When applied to a number of example problems, including systems with static friction and the excitation of multiple discrete frequencies, the FET method is demonstrated to be an efficient and reliable alternative technique for nonlinear dynamic response analysis.

Issue Section:
Research Papers
Topics:
Airfoils, Finite element analysis, Stick-slip, Turbines, Degrees of freedom, Dynamic response analysis, Excitation, Friction, Steady state, Stiction, Turbine blades, Variational techniques
1.
Bauchau
O. A.
, and
Hong
C. H.
,
1988
, “
Nonlinear Response and Stability Analysis of Beams Using Finite Elements in Time
,”
AIAA Journal
, Vol.
26
, No.
9
, pp.
1135
–
1142
.
2.
Borri
M.
,
1986
, “
Helicopter Rotor Dynamics by Finite Element Time Approximation
,”
Computers and Mathematics with Applications
, Vol.
12A
, No.
1
, pp.
149
–
160
.
3.
Borri
M.
,
Bottasso
C.
, and
Mantegazza
P.
,
1992
, “
Basic Features of the Time Finite Element Approach for Dynamics
,”
Meccanica
, Vol.
27
, pp.
119
–
130
.
4.
Borri
M.
, et al.,
1985
, “
Dynamic Response of Mechanical Systems By a Weak Hamiltonian Formulation
,”
Computers and Structures
, Vol.
20
, No.
1–3
, pp.
495
–
508
.
5.
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 Journal of Applied Mechanics
, Vol.
56
, pp.
149
–
154
.
6.
Den Hartog
J. P.
,
1931
, “
Forced Vibrations With Combined Coulomb and Viscous Friction
,”
Transactions of ASME
, Vol.
53
, pp.
107
–
115
.
7.
Griffin
J. H.
,
1980
, “
Friction Damping of Resonant Stresses in Gas Turbine Engine Airfoils
,”
ASME Journal of Engineering for Power
, Vol.
102
, pp.
329
–
333
.
8.
Hodges
D. H.
, and
Bless
R. R.
,
1991
, “
Weak Hamiltonian Finite Element Method for Optimal Control Problems
,”
Journal of Guidance
, Vol.
14
, No. 1, pp.
148
–
156
.
9.
Kim
Y. B.
, and
Noah
S. T.
,
1991
, “
Stability and Bifurcation Analysis of Oscillators With Piecewise-Linear Characteristics: A General Approach
,”
ASME Journal of Applied Mechanics
, Vol.
58
, pp.
545
–
553
.
10.
Kundert, K. S., and Sangiovanni-Vinventelli, A., 1986, “Simulation of Nonlinear Circuits in the Frequency Domain,” IEEE Transactions on Computer Aided Design, Vol. CAD-5, No. 4, pp. 521–535.
11.
Lau
S. L.
,
Cheung
Y. K.
, and
Wu
S. Y.
,
1982
, “
Variable Parameter Incrementation Method for Dynamic Instability of Linear and Nonlinear Systems
,”
ASME Journal of Applied Mechanics
, Vol.
49
, pp.
849
–
852
.
12.
Ling
F. H.
, and
Wu
X. X.
,
1987
, “
Fast Galerkin Method and its Application to Determine Periodic Solutions of Non-Linear Oscillators
,”
International Journal of Non-linear Mechanics
, Vol.
22
, pp.
89
–
98
.
13.
MATLAB, 1992, MATLAB Reference Guide, The Math Works, Inc., Natick, Mass.
14.
Menq
C.-H.
,
Chidamparam
P.
, and
Griffin
J. H.
,
1991
, “
Friction Damping of Two-Dimensional Motion and Its Application in Vibration Control
,”
Journal of Sound and Vibration
, Vol.
144
, No.
3
, pp.
427
–
447
.
15.
Peters
D. A.
, and
Izadpanah
A. P.
,
1988
, “
Hp-Version Finite Elements For the Space-Time Domain
,”
Computational Mechanics
, Vol.
3
, pp.
73
–
88
.
16.
Pierre
C.
,
Ferri
A. A.
, and
Dowell
E. H.
,
1985
, “
Multi-Harmonic Analysis of Dry Friction Damped Systems Using an Incremental Harmonic Balance Method
,”
ASME Journal of Applied Mechanics
, Vol.
52
, pp.
958
–
964
.
17.
Powell
J. A.
, and
Wiercigroch
M.
,
1992
, “
Influence of Non-Reversible Coulomb Characteristics on the Response of a Harmonically Excited Linear Oscillator
,”
Machine Vibration
, Vol.
1
, pp.
94
–
104
.
18.
Pratt
T. K.
, and
Williams
R.
,
1981
, “
Non-Linear Analysis of Stick/Slip Motion
,”
Journal of Sound and Vibration
, Vol.
74
, No.
4
, pp.
531
–
542
.
19.
Shaw
S. W.
,
1986
, “
On the Dynamic Response of a System With Dry Friction
,”
Journal of Sound and Vibration
, Vol.
108
, No.
2
, pp.
305
–
325
.
20.
Sinha
A.
, and
Griffin
J. H.
,
1983
, “
Friction Damping of Flutter in Gas Turbine Engine Airfoils
,”
Journal of Aircraft
, Vol.
20
, No.
4
, pp.
372
–
376
.
21.
Sinha
A.
, and
Griffin
J. H.
,
1984
, “
Effects of Static Friction on the Forced Response of Frictionally Damped Turbine Blades
,”
ASME Journal of Engineering for Gas Turbines and Power
, Vol.
106
, pp.
65
–
69
.
22.
Wang
J. H.
, and
Chen
W. K.
,
1993
, “
Investigation of the Vibration of a Blade with Friction Damper by HBM
,”
ASME Journal of Engineering for Gas Turbine and Power
, Vol.
115
, pp.
294
–
299
.
23.
Wang
Y.
,
1996
, “
An Analytical Solution for Periodic Response of Elastic-Friction Damped Systems
,”
Journal of Sound and Vibration
, Vol.
189
, pp.
299
–
313
.
24.
Wang
Y.
,
1995
, “
Dynamics of Unsymmetric Piecewise-Linear/Nonlinear Systems Using Finite Elements in Time
,”
Journal of Sound and Vibration
, Vol.
185
, pp.
155
–
170
.
25.
Yeh
G.
,
1966
, “
Forced Vibrations of a Two-Degree-Of-Freedom System With Combined Coulomb and Viscous Damping
,”
Journal of Acoustical Society of America
, Vol.
39
, pp.
14
–
24
.
26.
Zangwill, W. I., and Garcia, C. B., 1981, Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall, Englewood Cliffs, NJ.
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