Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

References

1.
Bolotin
,
V. V.
,
1964
,
The Dynamic Stability of Elastic Systems
,
Holden-Day
,
San Francisco, CA
.
2.
Johnson
,
W.
,
1980
,
Helicopter Theory
,
Princeton University Press
,
Princeton, NJ
.
3.
Roseau
,
M.
,
1987
,
Vibration in Mechanical Systems: Analytical Methods and Applications
,
Springer-Verlag
,
New York
.
4.
Streit
,
D. A.
,
Krousgrill
,
C. M.
, and
Bajaj
,
A. K.
,
1989
, “
Nonlinear Response of Flexible Robotic Manipulators Performing Repetitive Tasks
,”
ASME J. Dyn. Syst.
,
111
(
3
), pp.
470
479
.
5.
Sanchez
,
N. E.
, and
Nayfeh
,
A. H.
,
1990
, “
Nonlinear Rolling Motion of Ships in Longitudinal Waves
,”
Int. Shipbuild. Prog.
,
37
(
411
), pp.
247
272
.https://trid.trb.org/view.aspx?id=407297
6.
Mingori
,
D. L.
,
1969
, “
Effects of Energy Dissipation on the Attitude Stability of Dual Spin Satellite
,”
AIAA J.
,
7
(
1
), pp.
20
27
.
7.
Glass
,
L.
,
1991
, “
Cardiac Arrhythmias and Circle Maps—A Classical Problem
,”
Chaos
,
1
(
1
), pp.
13
19
.
8.
Powell
,
J. L.
, and
Crasemann
,
B.
,
1961
,
Quantum Mechanics
,
Addison-Wesley
,
Boston, MA.
9.
Floquét
,
M. G.
,
1883
, “
Sur Les Équations Différentielles Linéaires à Coefficients Périodiques
,”
Ann. Sci. Ec. Norm. Superieue
,
12
(
2
), pp.
47
88
.
10.
Hsu
,
C. S.
,
1974
, “
On Approximating a General Linear Periodic System
,”
J. Math. Anal. Appl.
,
45
(
1
), pp.
234
251
.
11.
Sinha
,
S. C.
,
Chou
,
C. C.
, and
Denman
,
H. H.
,
1979
, “
Stability Analysis of Systems With Periodic Coefficients: An Approximate Approach
,”
J. Sound Vib.
,
64
(
4
), pp.
515
527
.
12.
Hammond
,
C. E.
,
1974
, “
An Application of Floquet Theory to Prediction of Mechanical Instability
,”
J. Am. Helicopter Soc.
,
19
(
4
), pp.
14
23
.
13.
Friedmann
,
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.
,
11
(
7
), pp.
1117
1136
.
14.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
Wiley
,
New York
.
15.
Sanders
,
J. A.
, and
Verhulst
,
F.
,
1985
,
Averaging Methods in Nonlinear Dynamical Systems
,
Springer
,
New York
.
16.
Sinha
,
S. C.
, and
Wu
,
D. H.
,
1991
, “
An Efficient Computational Scheme for the Analysis of Periodic Systems
,”
J. Sound Vib.
,
151
(
1
), pp.
91
117
.
17.
Sinha
,
S. C.
, and
Butcher
,
E. A.
,
1997
, “
Symbolic Computation of Fundamental Solution Matrices for Time Periodic Dynamical Systems
,”
J. Sound Vib.
,
206
(
1
), pp.
61
85
.
18.
Kirkland
,
G. W.
, and
Sinha
,
S. C.
,
2016
, “
Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
4
), p.
041022
.
19.
Sinha
,
S. C.
,
Ravindra
,
B.
, and
Henrichs
,
J. T.
,
2000
, “
A General Approach in the Design of Active Controllers for Nonlinear Systems Exhibiting Chaos
,”
Int. J. Bifurcation Chaos
,
10
(
1
), pp.
165
178
.
20.
Dávid
,
A.
, and
Sinha
,
S. C.
,
2003
, “
Bifurcation Control of Nonlinear Systems With Time-Periodic Coefficients
,”
ASME J. Dyn. Syst.
,
125
(
4
), pp.
541
548
.
21.
Ko
,
W.
, and
Stockie
,
J. M.
,
2016
, “
Parametric Resonance in Spherical Immersed Elastic Shells
,”
SIAM J. Appl. Math.
,
76
(
1
), pp.
58
86
.
22.
Ko
,
W.
, and
Stockie
,
J. M.
,
2015
, “
An Immersed Boundary Model of the Cochlea With Parametric Forcing
,”
SIAM J. Appl. Math.
,
75
(
3
), pp.
1065
1089
.
23.
Karavitaki
,
K. D.
, and
Mountain
,
D. C.
,
2007
, “
Evidence for Outer Hair Cell Driven Oscillatory Fluid Flow in the Tunnel of Corti
,”
Biophys. J.
,
92
(
9
), pp.
3284
3293
.
24.
Robles
,
L.
,
Ruggero
,
M. A.
, and
Rich
,
N. C.
,
1997
, “
Two-Tone Distortion on the Basilar Membrane of the Chinchilla Cochlea
,”
J. Neurophysiol.
,
77
(
5
), pp.
2385
2399
.http://jn.physiology.org/content/77/5/2385.long
25.
Ruggero
,
M. A.
,
Robles
,
L.
,
Rich
,
N. C.
, and
Recio
,
A.
,
1992
, “
Basilar Membrane Responses to Two-Tone and Broadband Stimuli
,”
Philos. Trans. R. Soc. London, Ser. B
,
336
(
1278
), pp.
307
315
.
26.
Ben-Haim
,
S.
,
Darvish
,
N.
,
Mika
,
Y.
,
Rousso
,
B.
,
Felzen
,
B.
, and
Malonek
,
D.
,
2002
, “
Local Cardiac Motion Control Using Applied Electrical Signals
,” U.S. Patent No.
6,442,424
.https://www.google.ch/patents/US6442424
27.
Matheny
,
R. G.
, and
Taylor
,
C. S.
,
1997
, “
Method of Using Vagal Nerve Stimulation in Surgery
,” Cardiothoracic Systems, Inc., Cupertino, CA, U.S. Patent No.
5651378A
.https://www.google.tl/patents/US5651378
28.
Davis
,
S. H.
, and
Rosenblat
,
S.
,
1980
, “
A Quasiperiodic Mathieu-Hill Equation
,”
SIAM J. Appl. Math.
,
38
(
1
), pp.
139
155
.
29.
Johnson
,
R. A.
, and
Moser
,
J.
,
1982
, “
The Rotation Number for Almost Periodic Potentials
,”
Commun. Math. Phys.
,
84
(
3
), pp.
403
438
.
30.
Zounes
,
R. S.
, and
Rand
,
R. H.
,
1998
, “
Transition Curves for the Quasi-Periodic Mathieu Equation
,”
SIAM J. Appl. Math.
,
58
(
4
), pp.
1094
1115
.
31.
Broer
,
H.
, and
Simó
,
C.
,
1998
, “
Hill’s Equation With Quasi-Periodic Forcing: Resonance Tongues, Instability Pockets and Global Phenomena
,”
Bull. Braz. Math. Soc.
,
29
(
2
), pp.
253
293
.
32.
Puig
,
J.
, and
Simó
,
C.
,
2011
, “
Resonance Tongues in Quasi-Periodic Hill-Schrödinger Equation With Three Frequencies
,”
Regular Chaotic Dyn.
,
16
(
1
), pp.
61
78
.
33.
Waters
,
T. J.
,
2010
, “
Stability of a 2-Dimensional Mathieu Type System With Quasiperiodic Coefficients
,”
Nonlinear Dyn.
,
60
(
3
), pp.
341
356
.
34.
Grimshaw
,
R.
,
1990
,
Nonlinear Ordinary Differential Equations
,
Blackwell Scientific Publications
,
Boston, MA
.
35.
Yakubovich
,
V. A.
, and
Starzhinski
,
V. M.
,
1975
,
Linear Differential Equations With Periodic Coefficients—Part I
,
Wiley
,
New York
.
36.
Butcher
,
E.
, and
Sinha
,
S. C.
,
1998
, “
Symbolic Computation of Local Stability and Bifurcation Surfaces of Time-Periodic Nonlinear Systems
,”
Nonlinear Dyn.
,
17
(
1
), pp.
1
21
.
37.
Bogoljubov
,
N. N.
,
Mitropoliskii
,
J. A.
, and
Samoilenko
,
A. M.
,
1976
,
Methods of Accelerated Convergence in Nonlinear Mechanics
,
Hindustan Publishing
,
Delhi, India
.
38.
Murdock
,
J. A.
,
1978
, “
On the Floquet Problem for Quasiperiodic Systems
,”
Proc. Am. Math. Soc.
,
68
(
2
), pp.
179
184
.
39.
Broer
,
H.
, and
Levi
,
M.
,
1995
, “
Geometrical Aspects of Stability Theory of Hill’s Equations
,”
Arch. Ration. Mech. Anal.
,
131
(
3
), pp.
225
240
.
40.
Broer
,
H.
, and
Simó
,
C.
,
2000
, “
Resonance Tongues in Hill’s Equations: A Geometric Approach
,”
J. Differ. Equations
,
166
(
2
), pp.
290
327
.
41.
Sato
,
C.
,
1966
, “
Correction of Stability Curves in Hill-Meissner’s Equation
,”
Math. Comput.
,
20
(
93
), pp.
98
106
.
42.
Brogan
,
W. L.
,
1991
,
Modern Control Theory
,
Prentice Hall
,
Upper Saddle River, NJ
.
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