A hybrid approach which combines the reduced sequential quadratic programing (SQP) method with the shooting method is proposed to search the worst resonance response of nonlinear systems. The shooting method is first employed to construct the nonlinear equality constraints for the constrained optimization problem. Then, the complex optimization problem is simplified and solved numerically by the reduced SQP method. By virtue of the coordinate basis decomposition scheme which exploits the gradients of nonlinear equality constraints, the nonlinear equality constraints are eliminated, resulting in a simple optimization problem subject to bound constraints. Moreover, the second-order correction (SOC) technique is adopted to overcome Maratos effect. The novelty of the approach described lies in the capability to efficiently handle nonlinear equality constraints. The effectiveness of the proposed algorithm is demonstrated by two benchmark examples seen in the literature.

References

References
1.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
,
Wiley
,
New York
.
2.
Akhavan
,
H.
, and
Ribeiro
,
P.
,
2017
, “
Geometrically Non-Linear Periodic Forced Vibrations of Imperfect Laminates With Curved Fibres by the Shooting Method
,”
Comp. Part B: Eng.
,
109
, pp.
286
296
.
3.
Sinou
,
J. J.
,
Didier
,
J.
, and
Faverjon
,
B.
,
2015
, “
Stochastic Non-Linear Response of a Flexible Rotor With Local Non-Linearities
,”
Int. J. Non-Linear Mech.
,
74
, pp.
92
99
.
4.
Gong
,
G.
, and
Dunne
,
J. F.
,
2011
, “
Efficient Exceedance Probability Computation for Randomly Uncertain Nonlinear Structures With Periodic Loading
,”
J. Sound Vib.
,
330
(
10
), pp.
2354
2368
.
5.
Dou
,
S.
, and
Jensen
,
J. S.
,
2015
, “
Optimization of Nonlinear Structural Resonance Using the Incremental Harmonic Balance Method
,”
J. Sound Vib.
,
334
, pp.
239
254
.
6.
Xiong
,
H.
,
Kong
,
X.
,
Li
,
H.
, and
Yang
,
Z.
,
2017
, “
Vibration Analysis of Nonlinear Systems With the Bilinear Hysteretic Oscillator by Using Incremental Harmonic Balance Method
,”
Commun. Nonlinear Sci. Numer. Simul.
,
42
, pp.
437
450
.
7.
Attili
,
B. S.
, and
Syam
,
M. I.
,
2008
, “
Efficient Shooting Method for Solving Two Point Boundary Value Problems
,”
Chaos, Solitons Fractals
,
35
(
5
), pp.
895
903
.
8.
Ardeh
,
H. A.
, and
Allen
,
M. S.
,
2016
, “
Multiharmonic Multiple-Point Collocation: A Method for Finding Periodic Orbits of Strongly Nonlinear Oscillators
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
4
), p.
041006
.
9.
Wang
,
X. F.
, and
Zhu
,
W. D.
,
2015
, “
A Modified Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Broyden's Method
,”
Nonlinear Dyn.
,
81
(
1–2
), pp.
981
989
.
10.
Hoang
,
T.
,
Duhamel
,
D.
,
Foret
,
G.
,
Yin
,
H. P.
, and
Argoul
,
P.
,
2017
, “
Frequency Dependent Iteration Method for Forced Nonlinear Oscillators
,”
Appl. Math. Modell.
,
42
, pp.
441
448
.
11.
Javed
,
U.
,
Abdelkefi
,
A.
, and
Akhtar
,
I.
,
2016
, “
An Improved Stability Characterization for Aeroelastic Energy Harvesting Applications
,”
Commun. Nonlinear Sci. Numer. Simul.
,
36
, pp.
252
265
.
12.
Peeters
,
M.
,
Viguié
,
R.
,
Sérandour
,
G.
,
Kerschen
,
G.
, and
Golinval
,
J. C.
,
2009
, “
Nonlinear Normal Modes—Part II: Toward a Practical Computation Using Numerical Continuation Techniques
,”
Mech. Syst. Signal Process
,
23
(
1
), pp.
195
216
.
13.
Wang
,
F.
,
2015
, “
Bifurcations of Nonlinear Normal Modes Via the Configuration Domain and the Time Domain Shooting Methods
,”
Commun. Nonlinear Sci. Numer. Simul.
,
20
(
2
), pp.
614
628
.
14.
Lee
,
K. H.
,
Han
,
H. S.
, and
Park
,
S.
,
2017
, “
Bifurcation Analysis of Coupled Lateral/Torsional Vibrations of Rotor Systems
,”
J. Sound Vib.
,
386
, pp.
372
389
.
15.
Kim
,
S.
, and
Palazzolo
,
A. B.
,
2017
, “
Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
3
), p.
031003
.
16.
Habib
,
G.
,
Detroux
,
T.
,
Viguié
,
R.
, and
Kerschen
,
G.
,
2015
, “
Nonlinear Generalization of Den Hartog' s Equal-Peak Method
,”
Mech. Syst. Signal Process
,
52–53
, pp.
17
28
.
17.
Detroux
,
T.
,
Habib
,
G.
,
Masset
,
L.
, and
Kerschen
,
G.
,
2015
, “
Performance, Robustness and Sensitivity Analysis of the Nonlinear Tuned Vibration Absorber
,”
Mech. Syst. Signal Process
,
60–61
, pp.
799
809
.
18.
Förster
,
A.
, and
Krack
,
M.
,
2016
, “
An Efficient Method for Approximating Resonance Curves of Weakly-Damped Nonlinear Mechanical Systems
,”
Comput. Struct.
,
169
, pp.
81
90
.
19.
Liao
,
H.
, and
Sun
,
W.
,
2013
, “
A New Method for Predicting the Maximum Vibration Amplitude of Periodic Solution of Non-Linear System
,”
Nonlinear Dyn.
,
71
(
3
), pp.
569
582
.
20.
Liao
,
H.
,
2015
, “
Optimization Analysis of Duffing Oscillator With Fractional Derivatives
,”
Nonlinear Dyn.
,
79
(
2
), pp.
1311
1328
.
21.
Dednam
,
W.
, and
Botha
,
A. E.
,
2015
, “
Optimized Shooting Method for Finding Periodic Orbits of Nonlinear Dynamical Systems
,”
Eng. Comput.
,
31
(
4
), pp.
749
762
.
22.
Nocedal
,
J.
, and
Wright
,
S. J.
,
2006
,
Numerical Optimization, 2nd ed.
,
Springer
,
New York
.
23.
Nocedal
,
J.
, and
Wright
,
S. J.
,
2006
,
Sequential Quadratic Programming
,
Springer
,
New York
.
24.
Byrd
,
R. H.
, and
Nocedal
,
J.
,
1990
, “
An Analysis of Reduced Hessian Methods for Constrained Optimization
,”
Math. Prog.
,
49
(
1–3
), pp.
285
323
.
25.
Schmid
,
C.
,
1994
, “
Reduced Hessian Successive Quadratic Programming for Large-Scale Process Optimization
,” Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA.
26.
Biegler
,
L. T.
,
Nocedal
,
J.
,
Schmid
,
C.
, and
Ternet
,
D.
,
2000
, “
Numerical Experience With a Reduced Hessian Method for Large Scale Constrained Optimization
,”
Comput. Opt. Appl.
,
15
(
1
), pp.
45
67
.
27.
Biegler
,
L. T.
,
Nocedal
,
J.
, and
Schmid
,
C.
,
1995
, “
A Reduced Hessian Method for Large-Scale Constrained Optimization
,”
SIAM J. Opt.
,
5
(
2
), pp.
314
347
.
28.
Bonis
,
I.
, and
Theodoropoulos
,
C.
,
2012
, “
Model Reduction-Based Optimization Using Large-Scale Steady-State Simulators
,”
Chem. Eng. Sci
,
69
(
1
), pp.
69
80
.
29.
Zhang
,
S. Y.
, and
Deng
,
Z. C.
,
2006
, “
Group Preserving Schemes for Nonlinear Dynamic System Based on RKMK Methods
,”
Appl. Math. Comput.
,
175
(
1
), pp.
497
507
.
30.
Lazarus
,
A.
, and
Thomas
,
O.
,
2010
, “
A Harmonic-Based Method for Computing the Stability of Periodic Solutions of Dynamical Systems
,”
C. R. Méc.
,
338
(
9
), pp.
510
517
.
31.
Boggs
,
P. T.
, and
Tolle
,
J. W.
,
1995
, “
Sequential Quadratic Programming
,”
Acta Numer.
,
4
, pp.
1
51
.
32.
Ugray
,
Z.
,
Lasdon, L.
,
Plummer, J.
,
Glover, F.
,
Kelly, J.
, and
Marti, R.
,
2007
, “
Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization
,”
INFORMS J. Comput.
,
19
(
3
), pp.
328
340
.
You do not currently have access to this content.