The propagation constant technique has previously been used to predict band gap regions in linear oscillator chains by solving an eigenvalue problem for frequency in terms of a wave number. This paper describes a method by which selected design parameters can be separated from the eigenvalue problem, allowing standard uncertainty propagation techniques to provide closed form solutions for the uncertainty in frequency. Examples are provided for different types of measurement or environmental uncertainty showing the varying robustness of a band gap region to changes in parameters of the same or different order. The system studied in this paper is comprised of repelling magnetic oscillators using a dipole model. Numerical simulation has been performed to confirm the accuracy of analytical solutions up to a certain level of base excitation amplitude after which nonlinear effects change the predicted band gap regions to low energy chaos.

References

References
1.
Mead
,
D. J.
,
1996
, “
Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton
,”
J. Sound Vib.
,
190
(
3
), pp.
495
524
.10.1006/jsvi.1996.0076
2.
Nesterenko
,
V. F.
,
2000
,
Dynamics of Heterogeneous Materials
,
Springer
,
New York
.
3.
Nesterenko
,
V. F.
,
1983
, “
Propagation of Nonlinear Compression Pulses in Granular Media
,”
J. Appl. Mech. Tech. Phys.
,
24
(
5
), pp.
733
743
.10.1007/BF00905892
4.
Friesecke
,
G.
, and
Wattis
,
J. A.
,
1994
, “
Existence Theorem for Solitary Waves on Lattices
,”
Commun. Math. Phys.
,
161
(
2
), pp.
391
418
.10.1007/BF02099784
5.
Lazaridi
,
A.
, and
Nesterenko
,
V. F.
,
1985
, “
Observation of a New Type of Solitary Waves in a One-Dimensional Granular Medium
,”
J. Appl. Mech. Tech. Phys.
,
26
, pp.
405
408
.10.1007/BF00910379
6.
Coste
,
C.
,
Falcon
,
E.
, and
Fauvre
,
S.
,
1997
, “
Solitary Waves in a Chain of Beads Under Hertz Contact
,”
Phys. Rev. E
,
56
(
5
), pp.
6104
6117
.10.1103/PhysRevE.56.6104
7.
MacKay
,
R. S.
,
1999
, “
Solitary Waves in a Chain of Beads Under Hertz Contact
,”
Phys. Lett. A
,
251
, pp.
191
192
.10.1016/S0375-9601(98)00867-6
8.
Doney
,
R. L.
, and
Sen
,
S.
,
2005
, “
Impulse Absorption by Tapered Horizontal Alignments of Elastic Spheres
,”
Phys. Rev. E
,
72
, p.
041304
.10.1103/PhysRevE.72.041304
9.
Doney
,
R.
, and
Sen
,
S.
,
2006
, “
Decorated, Tapered, and Highly Nonlinear Granular Chain
,”
Phys. Rev. Lett.
,
97
, p.
155502
.10.1103/PhysRevLett.97.155502
10.
Ponson
,
L.
,
Boechler
,
N.
,
Lai
,
Y. M.
,
Porter
,
M. A.
,
Kevrekidis
,
P. G.
, and
Daraio
,
C.
,
2010
, “
Nonlinear Waves in Disordered Diatomic Granular Chains
,”
Phys. Rev. E
,
82
, p.
021301
.10.1103/PhysRevE.82.021301
11.
Harbola
,
U.
,
Rosas
,
A.
,
Esposito
,
M.
, and
Lindenberg
,
K.
,
2009
, “
Pulse Propagation in Tapered Granular Chains: An Analytic Study
,”
Phys. Rev. E
,
80
, p.
031303
.10.1103/PhysRevE.80.031303
12.
Daraio
,
C.
,
Nesterenko
,
V.
,
Herbold
,
E.
, and
Jin
,
S.
,
2006
, “
Tunability of Solitary Wave Properties in One-Dimensional Strongly Nonlinear Phononic Crystals
,”
Phys. Rev. E
,
73
, p.
026610
.10.1103/PhysRevE.73.026610
13.
Starosvetsky
,
Y.
, and
Vakakis
,
A. F.
,
2010
, “
Traveling Waves and Localized Modes in One-Dimensional Homogeneous Granular Chains With No Precompression
,”
Phys. Rev. E
,
82
, p.
026603
.10.1103/PhysRevE.82.026603
14.
Brillouin
,
L.
,
1953
,
Wave Propagation in Periodic Structures
,
Dover
,
New York
.
15.
Martinsson
,
P.
, and
Movchan
,
A.
,
2003
, “
Vibrations of Lattice Structures and Phononic Band Gaps
,”
Q. J. Mech. Appl. Math.
,
56
, pp.
45
54
.10.1093/qjmam/56.1.45
16.
Jensen
,
J.
,
2003
, “
Phononic Band Gaps and Vibrations in One- and Two-Dimensional Mass-Spring Structures
,”
J. Sound Vib.
,
266
, pp.
1053
1078
.10.1016/S0022-460X(02)01629-2
17.
Poulton
,
C.
,
Movchan
,
A.
,
McPhedran
,
R.
,
Nicorovici
,
N.
, and
Antipov
,
Y.
,
2000
, “
Eigenvalue Problems for Doubly Periodic Elastic Structures and Phononic Band Gaps
,”
Proc. R. Soc. London
,
456
, pp.
2543
2559
.10.1098/rspa.2000.0624
18.
Jayaprakash
,
K.
,
Starosvetsky
,
Y.
,
Vakakis
,
A. F.
,
Peeters
,
M.
, and
Kerschen
,
G.
,
2011
, “
Nonlinear Normal Modes and Band Zones in Granular Chains With No Pre-Compression
,”
Nonlinear Dyn.
,
63
, pp.
359
385
.10.1007/s11071-010-9809-0
19.
Romeo
,
F.
, and
Rega
,
G.
,
2006
, “
Wave Propogation Properties in Oscillatory Chains With Cubic Nonlinearities
,”
Chaos, Solitons Fractals
,
27
, pp.
606
617
.10.1016/j.chaos.2005.04.087
20.
Narisetti
,
R. K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2010
, “
A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures
,”
ASME J. Vib. Acoust.
,
132
, pp.
1
11
.10.1115/1.4000775
21.
Boechler
,
N.
,
Daraio
,
C.
,
Narisetti
,
R.
,
Ruzzene
,
M.
, and
Leamy
,
M.
,
2009
, “
Analytical and Experimental Analysis of Bandgaps in Nonlinear One-Dimensional Periodic Structures
,”
Proceedings of the IUTAM Symposium on Recent Advances of Acoustic Waves in Solids
, Taipai, Taiwan, May 25–28, Vol.
26
,
Springer
,
New York
, pp.
209
219
.10.1007/978-90-481-9893-1_20
22.
Vakakis
,
A. F.
, and
King
,
M. E.
,
1995
, “
Nonlinear Wave Transmission in a Monocoupled Elastic Periodic System
,”
J. Acoust. Soc. Am.
,
98
(
3
), pp.
1534
1546
.10.1121/1.413419
23.
Greene
,
J. B.
, and
Karioris
,
F. G.
,
1970
, “
Force on a Magnetic Dipole
,”
Am. J. Phys.
,
39
, pp.
172
175
.10.1119/1.1986086
24.
Boyer
,
T. H.
,
1988
, “
The Force on a Magnetic Dipole
,”
Am. J. Phys.
,
56
(
8
), pp.
688
692
.10.1119/1.15501
25.
Yung
,
K. W.
,
Landecker
,
P. B.
, and
Villani
,
D. D.
,
1998
, “
An Analytic Solution for the Force Between Two Magnetic Dipoles
,”
Magn. Electr. Sep.
,
9
, pp.
39
52
.10.1155/1998/79537
26.
Fornasini
,
P.
,
2008
,
The Uncertainty in Physical Measurements: An Introduction to Data Analysis in the Physics Laboratory
,
Springer
,
New York
.
27.
Furlani
,
E. P.
,
2001
,
Permanent Magnet and Electromechanical Devices
,
Academic Press
,
New York
.
28.
Mann
,
B. P.
,
Barton
,
D. A.
, and
Owens
,
B. A.
,
2012
, “
Uncertainty in Performance for Linear and Nonlinear Energy Harvesting Strategies
,”
J. Intell. Mater. Syst. Struct.
,
23
(
13
), pp. 1451–1460.10.1177/1045389X12439639
29.
Lazarov
,
B. S.
, and
Jensen
,
J. S.
,
2007
, “
Low-Frequency Band Gaps in Chains With Attached Non-Linear Oscillators
,”
Int. J. Nonlinear Mech.
,
42
, pp.
1186
1193
.10.1016/j.ijnonlinmec.2007.09.007
30.
Farzbod
,
F.
, and
Leamy
,
M. J.
,
2011
, “
Analysis of Bloch's Method and the Propagation Technique in Periodic Structures
,”
ASME J. Vib. Acoust.
,
133
, p.
031010
.10.1115/1.4003202
You do not currently have access to this content.