Chains of nonlinear shear indeformable beams with distributed mass, resting on movable supports, are considered. To determine the dynamic response of the system, the transfer-matrix approach is merged with the harmonic balance method and a perturbation method, thereby transforming the original space-temporal continuous problem into a discrete one-dimensional map $xk+1=F(xk)$ expressed in terms of the state variables $xk$ at the interface between adjacent beams. Such transformation does not imply any discretization, because it is obtained by integrating the single-element field equations. The state variables consist of both first-order variables, namely, transversal displacement and couples, and second-order variables, which are longitudinal displacement and axial forces. Therefore, while the linear problem is monocoupled, the nonlinear one becomes multicoupled. The procedure is applied to determine frequency-response relationship under free and forced vibrations.

1.
Sen Gupta
,
G.
, 1970, “
Natural Flexural Waves and the Normal Modes of Periodically-Supported Beams and Plates
,”
J. Sound Vib.
0022-460X,
13
, pp.
89
101
.
2.
Faulkner
,
M. G.
, and
Hong
,
D. P.
, 1985, “
Free Vibrations of Mono-Coupled Periodic System
,”
J. Sound Vib.
0022-460X,
99
, pp.
29
42
.
3.
Romeo
,
F.
, and
Luongo
,
A.
, 2002, “
Invariant Representation of Propagation Properties for Bi-coupled Periodic Structures
,”
J. Sound Vib.
0022-460X,
257
, pp.
869
886
.
4.
Romeo
,
F.
, and
Luongo
,
A.
, 2003, “
Vibration Reduction in Piecewise Bi-coupled Periodic Structures
,”
J. Sound Vib.
0022-460X,
268
, pp.
601
615
.
5.
Luongo
,
A.
, and
Romeo
,
F.
, 2005, “
Real Wave Vectors for Dynamic Analysis of Periodic Structures
,”
J. Sound Vib.
0022-460X
279
, pp.
309
325
.
6.
Vakakis
,
A. F.
, and
King
,
M. E.
, 1995, “
Nonlinear Wave Transmission in a Mono-Coupled Elastic Periodic System
,”
J. Acoust. Soc. Am.
0001-4966,
98
, pp.
1534
1546
.
7.
Davies
,
M. A.
, and
Moon
,
F. C.
, 1996, “
Transition From Soliton to Chaotic Motion Following Sudden Excitation of a Nonlinear Structure
,”
ASME J. Appl. Mech.
0021-8936,
63
, pp.
445
449
.
8.
Luongo
,
A.
, 1995, “A Transfer Matrix Perturbation Approach to the Buckling Analysis of Nonlinear Periodic Structures,” 10th ASCE Conference, Boulder, pp.
505
508
.
9.
Yong
,
Y.
, and
Lin
,
Y. K.
, 1989, “
Propagation of Decaying Waves in Periodic and Piecewise Periodic Structures of Finite Length
,”
J. Sound Vib.
0022-460X,
129
, pp.
99
118
.
10.
Luongo
,
A.
,
Rega
,
G.
, and
Vestroni
,
F.
, 1986, “
On Nonlinear Dynamics of Planar Shear Indeformable Beams
,”
ASME J. Appl. Mech.
0021-8936,
53
, pp.
619
624
.