Parameter sweeps are commonly used to explore the behavior of dynamical systems. This paper derives exact solutions for the instances in time to stroboscopically sample the response of a dynamical system subject to varying input excitations. This work will enable more accurate bifurcation diagrams and Poincaré sections in parameter regimes where numerical approaches may lead to incorrect behavior characterization. The simplest case of a linear frequency sweep is first considered before generalizing the results to include more complex functions with nonlinear sweep rates and arbitrary phase shifts.

References

References
1.
Meirovitch
,
L.
,
2010
,
Fundamentals of Vibrations
,
Springer, Berlin
.
2.
Stanton
,
S. C.
,
McGehee
,
C. C.
, and
Mann
,
B. P.
,
2010
, “
Nonlinear Dynamics for Broadband Energy Harvesting: Investigation of a Bistable Piezoelectric Inertial Generator
,”
Phys. D
,
239
(
10
), pp.
640
653
.
3.
Mann
,
B.
,
2013
, “
Broadband Energy Harvesting From a Bistable Potential Well
,”
Advances in Energy Harvesting Methods
,
Springer
,
New York
, pp.
91
115
.
4.
Masana
,
R.
, and
Daqaq
,
M.
,
2012
, “
Energy Harvesting in the Super-Harmonic Frequency Region of a Twin-Well Oscillator
,”
J. Appl. Phys.
,
111
(
4
), p.
044501
.
5.
Arrieta
,
A.
,
Neild
,
S.
, and
Wagg
,
D.
,
2009
, “
Nonlinear Dynamic Response and Modeling of a Bi-Stable Composite Plate for Applications to Adaptive Structures
,”
Nonlinear Dyn.
,
58
(
1–2
), pp.
259
272
.
6.
Virgin
,
L.
,
1987
, “
The Nonlinear Rolling Response of a Vessel Including Chaotic Motions Leading to Capsize in Regular Seas
,”
Appl. Ocean Res.
,
9
(
2
), pp.
89
95
.
7.
Todd
,
M.
, and
Virgin
,
L.
,
1997
, “
An Experimental Impact Oscillator
,”
Chaos, Solitons Fractals
,
8
(
4
), pp.
699
714
.
8.
Thuraisingham
,
R.
,
2007
, “
Enhancing Poincare Plot Information Via Sampling Rates
,”
Appl. Math. Comput.
,
186
(
2
), pp.
1374
1378
.
9.
Azeez
,
M. F. A.
, and
Vakakis
,
A. F.
,
1999
, “
Numerical and Experimental Analysis of a Continuous Overhung Rotor Undergoing Vibro-Impacts
,”
Int. J. Nonlinear Mech.
,
34
(
3
), pp.
415
435
.
10.
Arrieta
,
A.
,
Hagedorn
,
P.
,
Erturk
,
A.
, and
Inman
,
D.
,
2010
, “
A Piezoelectric Bistable Plate for Nonlinear Broadband Energy Harvesting
,”
Appl. Phys. Lett.
,
97
(
10
), p.
104102
.
11.
Thompson
,
J.
, and
Soliman
,
M.
,
1991
, “
Indeterminate Jumps to Resonance From a Tangled Saddle-Node Bifurcation
,”
Proc. R. Soc. A
,
432
(
1884
), pp. 101–111.
12.
Feigenbaum
,
M. J.
,
1980
, “
The Metric Universal Properties of Period Doubling Bifurcations and the Spectrum for a Route to Turbulence
,”
Ann. New York Acad. Sci.
,
357
(
1
), pp.
330
336
.
13.
Kappagantu
,
R.
, and
Feeny
,
B.
,
2000
, “
Part 2: Proper Orthogonal Modal Modeling of a Frictionally Excited Beam
,”
Nonlinear Dyn.
,
23
(
1
), pp.
1
11
.
You do not currently have access to this content.