The linearized stability analysis of dynamical systems modeled using finite element-based multibody formulations is addressed in this paper. The use of classical methods for stability analysis of these systems, such as the characteristic exponent method or Floquet theory, results in computationally prohibitive costs. Since comprehensive multibody models are “virtual prototypes” of actual systems, the applicability to numerical models of the stability analysis tools that are used in experimental settings is investigated in this work. Various experimental tools for stability analysis are reviewed. It is proved that Prony’s method, generally regarded as a curve-fitting method, is equivalent, and sometimes identical, to Floquet theory and to the partial Floquet method. This observation gives Prony’s method a sound theoretical footing, and considerably improves the robustness of its predictions when applied to comprehensive models of complex multibody systems. Numerical and experimental applications are presented to demonstrate the efficiency of the proposed procedure.

1.
Bolotin
,
V. V.
, 1963,
Nonconservative Problems of the Theory of Elastic Stability
,
Pergamon Press
, Oxford, England.
2.
Goodwin
,
M. J.
, 1989,
Dynamics of Rotor-Bearing Systems
,
Unwinm Hyman
, London.
3.
Lalane
,
M.
, and
Ferraris
,
G.
, 1990,
Rotordynamics Prediction in Engineering
,
Wiley
, New York.
4.
Hochstadt
,
H.
, 1964,
Differential Equations
,
Dover Publications
, New York.
5.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
, 1979,
Nonlinear Oscillations
,
Wiley
, New York.
6.
Cardona
,
A.
, 1989, An Integrated Approach to Mechanism Analysis, Ph.D. thesis, Université de Liège.
7.
Cardona
,
A.
, and
Géradin
,
M.
, 1989, “
Time Integration of the Equations of Motion in Mechanism Analysis
,”
Comput. Struct.
0045-7949,
33
(
3
), pp.
801
820
.
8.
Bauchau
,
O. A.
,
Bottasso
,
C. L.
, and
Nikishkov
,
Y. G.
, 2001, “
Modeling Rotorcraft Dynamics With Finite Element Multibody Procedures
,”
Math. Comput. Modell.
0895-7177,
33
, pp.
1113
1137
.
9.
Hsu
,
C. S.
, 1972, “
Impulsive Parametric Excitation: Theory
,”
ASME J. Appl. Mech.
0021-8936,
39
, pp.
551
558
.
10.
Hsu
,
C. S.
, 1974, “
On Approximating a General Linear Periodic System
,”
J. Math. Anal. Appl.
0022-247X,
45
, pp.
234
251
.
11.
Peters
,
D. A.
, and
Hohenemser
,
K. H.
, 1971, “
Application of the Floquet Transition Matrix to Problems of Lifting Rotor Stability
,”
J. Am. Helicopter Soc.
0002-8711,
16
, pp.
25
33
.
12.
Friedmann
,
P. P.
,
Hammond
,
C. E.
, and
Woo
,
T. H.
, 1977, “
Efficient Numerical Treatment of Periodic Systems With Application to Stability Problems
,”
Int. J. Numer. Methods Eng.
0029-5981,
11
, pp.
1171
1136
.
13.
Bauchau
,
O. A.
, and
Nikishkov
,
Y. G.
, 2001, “
An Implicit Floquet Analysis for Rotorcraft Stability Evaluation
,”
J. Am. Helicopter Soc.
0002-8711,
46
, pp.
200
209
.
14.
Bauchau
,
O. A.
, and
Nikishkov
,
Y. G.
, 2001, “
An Implicit Transition Matrix Approach to Stability Analysis of Flexible Multibody Systems
,”
Multibody Syst. Dyn.
1384-5640,
5
, pp.
279
301
.
15.
Wang
,
Xin
, and
Peters
,
D. A.
, 1997, “
Floquet Analysis in the Absence of Complete Information on States and Perturbations
,” in
Proc. of Seventh Int. Workshop on Dynamics and Aeroelasticity Stability Modeling, St. Louis
, Oct. 14–16, pp.
237
248
.
16.
Peters
,
D. A.
, and
Wang
,
X.
, 1998,
Generalized Floquet Theory for Analysis of Numerical or Experimental Rotor Response Data
, in
Proc. of 24th European Rotorcraft Forum, Marseilles, France
, Sept. 15–17, pp.
653
661
.
17.
Quaranta
,
G.
,
Mantegazza
,
P.
, and
Masarati
,
P.
, 2004, “
Assessing the Local Stability of Periodic Motions for Large Multibody Non-Linear Systems Using Proper Orthogonal Decomposition
,”
J. Sound Vib.
0022-460X,
271
, pp.
1015
1038
.
18.
Murphy
,
K. D.
,
Bayly
,
P. V.
,
Virgin
,
L. N.
, and
Gottwald
,
J. A.
, 1994, “
Measuring the Stability of Periodic Attractors Using Perturbation Induced Transients: Applications to Two Nonlinear Oscillators
,”
J. Sound Vib.
0022-460X,
172
, pp.
85
102
.
19.
Trickey
,
S. T.
,
Virgin
,
L. N.
, and
Dowell
,
E. H.
, 2002, “
The Stability of Limit Cycle Oscillations in a Nonlinear Aeroelastic System
,”
Proc. R. Soc. London, Ser. A
1364-5021,
458
, pp.
2203
2226
.
20.
Ibrahim
,
S. R.
, and
Mikulcik
,
E. C.
, 1977, “
A Method for the Direct Identification of Vibration Parameters From the Free Response
,”
Shock Vibration Bulletin
,
47
, pp.
183
198
.
21.
Ewins
,
D. J.
, 1984,
Modal Testing: Theory and Practice
,
Wiley
, New York.
22.
Hammond
,
C. E.
, and
Doggett
,
R. V.
, Jr.
, 1975, “
Determination Subcritical Damping by Moving-Block/Randomdec Applications
,” in
NASA Symp. on Flutter Testing Techniques
, NASA SP-415, pp.
59
76
.
23.
Bousman
,
W. G.
, and
Winkler
,
D. J.
, 1981, “
Application of the Moving-Block Analysis
,” in
Proc. of 22th Structures, Structural Dynamics, and Materials Conf.
, Dallas, April 17–20.
24.
Golub
,
G. H.
, and
Van Loan
,
C. F.
, 1989,
Matrix Computations
,
Second Edition
,
Johns Hopkins University Press
, Baltimore.
25.
Dahlquist
,
G.
, and
Björck
,
Å
, 1974,
Numerical Methods
,
Prentice Hall
, Englewood Cliffs, NJ.
26.
Press
,
W. H.
,
Flannery
,
B. P.
,
Teutolsky
,
S. A.
, and
Vetterling
,
W. T.
, 1990,
Numerical Recipes: The Art of Scientific Computing
,
Cambridge University Press
, Cambridge.
27.
Goland
,
M.
, 1945, “
The Flutter of a Uniform Cantilever Wing
,”
Applied Mechanics
,
12
(
4
), pp.
A197
A208
.
28.
Peters
,
D. A.
,
Karunamoorthy
,
S.
, and
Cao
,
W. M.
, 1995, “
Finite State Induced Flow Models, Part I: Two-dimensional Thin Airfoil
,”
J. Aircr.
0021-8669,
32
, pp.
313
322
.
29.
Peters
,
D. A.
, and
He
,
C. J.
, 1995, “
Finite State Induced Flow Models, Part II: Three-dimensional Rotor Disk
,”
J. Aircr.
0021-8669,
32
, pp.
323
333
.
30.
Bisplinghoff
,
R. L.
,
Ashley
,
H.
, and
Halfman
,
R. L.
, 1955,
Aeroelasticity
,
Second Edition
,
Addison-Wesley
, Reading, MA.
31.
Stol
,
K.
,
Bir
,
G.
, and
Balas
,
M.
, 1999, “
Linearized Dynamics and Operating Modes of a Simple Wind Turbine Model
,” in
Proc. of 37th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 11-14, Reno
, AIAA, Washington, DC, NICH Report No. 32548, pp.
135
142
.
32.
Nixon
,
M. W.
,
Langston
,
C. W.
,
Singleton
,
J. D.
,
Piatak
,
D. J.
,
Kvaternik
,
R. G.
,
Corso
,
L. M.
, and
Brown
,
R. K.
, 2003, “
Aeroelastic Stability of a Four-Bladed Semi-Articulated Soft-Inplane Tiltrotor Model
,” in
American Helicopter Society 59th Annual Forum Proc.
, Phoenix, May 6–8, pp.
11
25
.
33.
Bauchau
,
O. A.
,
Rodriguez
,
J.
, and
Bottasso
,
C. L.
, 2001, “
Modeling Of Unilateral Contact Conditions With Application to Aerospace Systems Involving Backlash, Freeplay, and Friction
,”
Mech. Res. Commun.
0093-6413,
28
(
5
), pp.
571
599
.
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