An original method for calculating the maximum vibration amplitude of the periodic solution of a nonlinear system is presented. The problem of determining the worst maximum vibration is transformed into a nonlinear optimization problem. The shooting method and the Floquet theory are selected to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness and ability of the proposed approach are illustrated through two numerical examples. Numerical examples show that the proposed method can give results with higher accuracy as compared with numerical results obtained by a parameter continuation method and the ability of the present method is also demonstrated.

References

References
1.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1995
,
Nonlinear Oscillations
,
John Wiley and Sons
,
New York
.
2.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
1995
,
Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods
,
Wiley-Interscience
,
New York
.
3.
Chen
,
S. H.
,
Yang
,
X. M.
, and
Cheung
,
Y. K.
,
1998
, “
Periodic Solutions of Strongly Quadratic Non-Linear Oscillators by the Elliptic Perturbation Method
,”
J. Sound Vib.
,
212
(
5
), pp.
771
780
.10.1006/jsvi.1997.1411
4.
Okabe
,
T.
,
Kondou
,
T.
, and
Ohnishi
,
J.
,
2011
, “
Elliptic Averaging Methods Using the Sum of Jacobian Elliptic Delta and Zeta Functions as the Generating Solution
,”
Int. J. Non-Linear Mech.
,
46
(
1
), pp.
159
169
.10.1016/j.ijnonlinmec.2010.08.004
5.
Liao
,
S. J.
,
2003
,
Beyond Perturbation: Introduction to Homotopy Analysis Method
,
Chapman and Hall/CRC Press
,
Boca Raton
FL.
6.
Liu
,
L.
,
Thomas
,
J. P.
,
Dowell
,
E. H.
,
Attar
,
P.
, and
Hall
,
K. C.
,
2006
, “
A Comparison of Classical and High Dimensional Harmonic Balance Approaches for a Duffing Oscillator
,”
J. Comput. Phys.
,
215
(
1
), pp.
298
320
.10.1016/j.jcp.2005.10.026
7.
Cochelin
,
B.
and
Vergez
,
C.
,
2009
, “
A High Order Purely Frequency-Based Harmonic Balance Formulation for Continuation of Periodic Solutions
,”
J. Sound Vib.
,
324
(
1–2
), pp.
243
262
.10.1016/j.jsv.2009.01.054
8.
Thomsen
,
J. J.
,
2003
,
Vibrations and Stability: Advanced Theory, Analysis, and Tools
,
Springer-Verlag
,
Berlin/Heidelberg
.
9.
Groll
,
G.
and
Ewins
,
D. J.
,
2001
, “
The Harmonic Balance Method With Arc-Length Continuation in Rotor/Stator Contact Problems
,”
J. Sound Vib.
,
241
(
2
), pp.
223
233
.10.1006/jsvi.2000.3298
10.
Ribeiro
,
P.
,
2004
, “
Nonlinear Forced Vibrations of Thin/Thick Beams and Plates by the Finite Element and Shooting Methods
,”
Comput. Struct.
,
82
, pp.
1413
1423
.10.1016/j.compstruc.2004.03.037
11.
Sundararajan
,
P.
, and
Noah
,
S. T.
,
1997
, “
Dynamics of Forced Nonlinear Systems Using Shooting/Arclength Continuation Method—Application to Rotor Systems
,”
ASME J. Vibr. Acoust.
,
119
, pp.
9
20
.10.1115/1.2889694
12.
Dimitriadis
,
G.
,
2008
, “
Continuation of Higher-Order Harmonic Balance Solutions for Nonlinear Aeroelastic Systems
,”
J. Aircr.
,
45
(
2
), pp.
523
537
.10.2514/1.30472
13.
Ribeiro
,
P.
,
2008
, “
Non-Linear Free Periodic Vibrations of Open Cylindrical Shallow Shells
,”
J. Sound Vib.
,
313
, pp.
224
245
.10.1016/j.jsv.2007.11.029
14.
Stoykov
,
S.
, and
Ribeiro
,
P.
,
2008
, “
Periodic Geometrically Nonlinear Free Vibrations of Circular Plates
,”
J. Sound Vib.
,
315
, pp.
536
555
.10.1016/j.jsv.2008.02.001
15.
Kerschen
,
G.
,
Peeters
,
M.
,
Golinval
,
J. C.
, and
Vakakis
,
A. F.
,
2009
, “
Nonlinear Normal Modes—Part I: A Useful Framework for the Structural Dynamicist
,”
Mech. Syst. Signal Process.
,
23
(
1
), pp.
170
194
.10.1016/j.ymssp.2008.04.002
16.
Peeters
,
M.
,
Viguié
,
R.
, and
Sérandour
,
G.
,
2009
, “
Nonlinear Normal Modes—Part II: Toward a Practical Computation Using Numerical Continuation Techniques
,”
Mech. Syst. Signal Process.
,
23
(
1
), pp.
195
216
.10.1016/j.ymssp.2008.04.003
17.
Georgiades
,
F.
, and
Peeters
,
M.
,
2009
, “
Modal Analysis of a Nonlinear Periodic Structure With Cyclic Symmetry
,”
AIAA J.
,
47
, pp.
1014
1025
.10.2514/1.40461
18.
Lazarus
,
A.
, and
Thomas
,
O.
,
2010
, “
A Harmonic-Based Method for Computing the Stability of Periodic Solutions of Dynamical Systems
,”
C. R. Mec.
,
338
(
9
), pp.
510
517
.10.1016/j.crme.2010.07.020
19.
Grolet
,
A.
, and
Thouverez
,
F.
,
2011
, “
Vibration Analysis of a Nonlinear System With Cyclic Symmetry
,”
ASME J. Eng. Gas Turbines Power
,
133
(
2
), p.
022502
.10.1115/1.4001989
20.
Sarrouy
,
E.
,
Grolet
,
A.
, and
Thouverez
,
F.
,
2011
, “
Global and Bifurcation Analysis of a Structure With Cyclic Symmetry
,”
Int. J. Non-Linear Mech.
,
46
(
5
), pp.
727
737
.10.1016/j.ijnonlinmec.2011.02.005
21.
Allgower
,
E. L.
, and
Georg
,
K.
,
2003
,
Introduction to Numerical Continuation Methods
,
Springer-Verlag
,
Berlin
.
22.
Petrov
,
E. P.
,
2009
, “
Analysis of Sensitivity and Robustness of Forced Response for Nonlinear Dynamic Structures
,”
Mech. Syst. Signal Process.
,
23
(
1
), pp.
68
86
.10.1016/j.ymssp.2008.03.008
23.
Petrov
,
E. P.
,
2007
, “
Direct Parametric Analysis of Resonance Regimes for Nonlinear Vibrations of Bladed Discs
,”
ASME J. Turbomach.
,
129
, pp.
495
502
.10.1115/1.2720487
24.
Chan
,
Y. J.
, and
Ewins
,
D. J.
,
2010
, “
A Comprehensive Set of Procedures to Estimate the Probability of Extreme Vibration Levels Due to Mistuning
,”
ASME J. Eng. Gas Turbines Power
,
132
, p.
112505
.10.1115/1.4001065
25.
Ugray
,
Z.
,
Lasdon
,
L.
,
Plummer
,
J.
,
Glover
,
R.
,
Kelly
,
J.
,
Marti
,
R.
,
2007
, “
Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization
,”
Informs J. Comput.
,
9
(
3
), pp.
328
340
.10.1287/ijoc.1060.0175
26.
Nocedal
,
J.
, and
Wright
,
S. J.
,
1999
,
Numerical Optimization
, Springer Series in Operations Research,
Springer
,
New York
.
27.
Fletche
,
R.
,
1987
,
Practical Methods of Optimization
,
John Wiley and Sons
,
New York
.
28.
Liao
,
H.
,
2011
, “
The Vibration Characters of Mistuned Bladed Disks
,” Ph.D. thesis,
Beihang University
,
Beijing, China
(in Chinese).
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