This paper deals with the forced response analysis of chains of thin elastic beams that are subject to periodic external loading and frictionless intermittent contact between the beams. Our study shows that the beams show nonlinear resonances whose frequencies are the same as the linear resonant frequencies if all the beams have the same stiffness. Furthermore, it is also shown that small gaps between the beams and small deviation or mistuning in the stiffness of each beam can cause drastic changes in the nonlinear resonant frequencies of the system dynamics. The system is modeled as a semidiscrete system of piecewise-linear oscillators with multiple degrees-of-freedom (DOF) that are subject to unilateral constraints, which is derived from a finite element discretization of the beams. The resulting equations of motions are solved by a second-order numerical integration scheme, and steady-state solutions are sought for various driving frequencies. Results of parametric studies with respect to the gaps between the beams and the number of beams are presented to discuss how these parameters affect the resonant behavior of the system.

References

References
1.
Shaw
,
S.
, and
Holmes
,
P.
,
1983
, “
A Periodically Forced Piecewise Linear Oscillator
,”
J. Sound Vib.
,
90
(
1
), pp.
129
155
.
2.
Dyskin
,
A. V.
,
Pasternak
,
E.
, and
Pelinovsky
,
E.
,
2012
, “
Periodic Motions and Resonances of Impact Oscillators
,”
J. Sound Vib.
,
331
(
12
), pp.
2856
2873
.
3.
Ing
,
J.
,
Pavlovskaia
,
E.
,
Wiercigroch
,
M.
, and
Banerjee
,
S.
,
2008
, “
Experimental Study of Impact Oscillator With One-Sided Elastic Constraint
,”
Philos. Trans. R. Soc. London A: Math., Phys. Eng. Sci.
,
366
(
1866
), pp.
679
705
.
4.
Melcher
,
J.
,
Champneys
,
A. R.
, and
Wagg
,
D. J.
,
2013
, “
The Impacting Cantilever: Modal Non-Convergence and the Importance of Stiffness Matching
,”
Philos. Trans. R. Soc. London A: Math., Phys. Eng. Sci.
,
371
(
1993
), p.
20120434
.
5.
Saito
,
A.
,
Castanier
,
M. P.
, and
Pierre
,
C.
,
2009
, “
Estimation and Veering Analysis of Nonlinear Resonant Frequencies of Cracked Plates
,”
J. Sound Vib.
,
326
(
3–5
), pp.
725
739
.
6.
Saito
,
A.
, and
Epureanu
,
B. I.
,
2011
, “
Bilinear Modal Representations for Reduced-Order Modeling of Localized Piecewise-Linear Oscillators
,”
J. Sound Vib.
,
330
(
14
), pp.
3442
3457
.
7.
Jung
,
C.
,
D'Souza
,
K.
, and
Epureanu
,
B. I.
,
2014
, “
Nonlinear Amplitude Approximation for Bilinear Systems
,”
J. Sound Vib.
,
333
(
13
), pp.
2909
2919
.
8.
Tien
,
M.-H.
, and
D'Souza
,
K.
,
2017
, “
A Generalized Bilinear Amplitude and Frequency Approximation for Piecewise-Linear Nonlinear Systems With Gaps or Prestress
,”
Nonlinear Dyn.
,
88
(
4
), pp.
2403
2416
.
9.
Burlayenko
,
V.
, and
Sadowski
,
T.
,
2012
, “
Finite Element Nonlinear Dynamic Analysis of Sandwich Plates With Partially Detached Facesheet and Core
,”
Finite Elem. Anal. Des.
,
62
, pp.
49
64
.
10.
Burlayenko
,
V.
, and
Sadowski
,
T.
,
2014
, “
Nonlinear Dynamic Analysis of Harmonically Excited Debonded Sandwich Plates Using Finite Element Modelling
,”
Compos. Struct.
,
108
, pp.
354
366
.
11.
Dauksevicius
,
R.
,
Briand
,
D.
,
Lockhart
,
R.
,
Quintero
,
A. V.
,
de Rooij
,
N.
,
Gaidys
,
R.
, and
Ostasevicius
,
V.
,
2014
, “
Frequency Up-Converting Vibration Energy Harvester With Multiple Impacting Beams for Enhanced Wideband Operation at Low Frequencies
,”
Procedia Eng.
,
87
, pp.
1517
1520
.
12.
Vijayan
,
K.
,
Friswell
,
M.
,
Khodaparast
,
H. H.
, and
Adhikari
,
S.
,
2015
, “
Non-Linear Energy Harvesting From Coupled Impacting Beams
,”
Int. J. Mech. Sci.
,
96–97
, pp.
101
109
.
13.
Jiang
,
D.
,
Pierre
,
C.
, and
Shaw
,
S.
,
2004
, “
Large-Amplitude Non-Linear Normal Modes of Piecewise Linear Systems
,”
J. Sound Vib.
,
272
(
3 − 5
), pp.
869
891
.
14.
Uspensky
,
B.
, and
Avramov
,
K.
,
2014
, “
On the Nonlinear Normal Modes of Free Vibration of Piecewise Linear Systems
,”
J. Sound Vib.
,
333
(
14
), pp.
3252
3265
.
15.
Uspensky
,
B.
, and
Avramov
,
K.
,
2014
, “
Nonlinear Modes of Piecewise Linear Systems Under the Action of Periodic Excitation
,”
Nonlinear Dyn.
,
76
(
2
), pp.
1151
1156
.
16.
Casini
,
P.
, and
Vestroni
,
F.
,
2011
, “
Characterization of Bifurcating Non-Linear Normal Modes in Piecewise Linear Mechanical Systems
,”
Int. J. Non-Linear Mech.
,
46
(
1
), pp.
142
150
.
17.
Giannini
,
O.
,
Casini
,
P.
, and
Vestroni
,
F.
,
2011
, “
Experimental Evidence of Bifurcating Nonlinear Normal Modes in Piecewise Linear Systems
,”
Nonlinear Dyn.
,
63
(
4
), pp.
655
666
.
18.
Dyskin
,
A. V.
,
Pasternak
,
E.
, and
Shufrin
,
I.
,
2014
, “
Structure of Resonances and Formation of Stationary Points in Symmetrical Chains of Bilinear Oscillators
,”
J. Sound Vib.
,
333
(
24
), pp.
6590
6606
.
19.
Pierre
,
C.
, and
Dowell
,
E.
,
1987
, “
Localization of Vibrations by Structural Irregularity
,”
J. Sound Vib.
,
114
(
3
), pp.
549
564
.
20.
Bendiksen
,
O. O.
,
2000
, “
Localization Phenomena in Structural Dynamics
,”
Chaos, Solitons Fractals
,
11
(
10
), pp.
1621
1660
.
21.
Simo
,
J. C.
, and
Laursen
,
T. A.
,
1992
, “
An Augmented Lagrangian Treatment of Contact Problems Involving Friction
,”
Comput. Struct.
,
42
(
1
), pp.
97
116
.
22.
Laursen
,
T. A.
, and
Maker
,
B. N.
,
1995
, “
An Augmented Lagrangian Quasi-Newton Solver for Constrained Nonlinear Finite-Element Applications
,”
Int. J. Numer. Methods Eng.
,
38
(
21
), pp.
3571
3590
.
23.
Bertsekas
,
D. P.
,
1996
,
Constrained Optimization and Lagrange Multiplier Methods
,
Athena Scientific
, St. Belmont, MA, pp.
96
124
.
24.
Chaudhary
,
A. B.
, and
Bathe
,
K. J.
,
1986
, “
A Solution Method for Static and Dynamic Analysis of Three-Dimensional Contact Problems With Friction
,”
Comput. Struct.
,
24
(
6
), pp.
855
873
.
25.
Saito
,
A.
,
Castanier
,
M. P.
,
Pierre
,
C.
, and
Poudou
,
O.
,
2009
, “
Efficient Nonlinear Vibration Analysis of the Forced Response of Rotating Cracked Blades
,”
ASME J. Comput. Nonlinear Dyn.
,
4
(
1
), p.
011005
.
You do not currently have access to this content.