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TECHNICAL PAPERS

# A Transfer-Matrix-Perturbation Approach to the Dynamics of Chains of Nonlinear Sliding Beams

[+] Author and Article Information
Angelo Luongo

Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno, Università di L’Aquila, 67040 Monteluco di Roio (L’Aquila), Italyluongo@ing.univaq.it

Francesco Romeo1

Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza,” Via Gramsci 53, 00197 Rome, Italyfrancesco.romeo@uniroma1.it

1

Corresponding author.

J. Vib. Acoust 128(2), 190-196 (Nov 08, 2005) (7 pages) doi:10.1115/1.2159034 History: Received September 04, 2004; Revised November 08, 2005

## Abstract

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.

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## Figures

Figure 1

(a) Chain of nonlinear beams and (b) inextensible and shear indeformable single beam

Figure 2

First four natural frequencies-amplitude relationship for a four four-bay beam: thick lines, nonlinear; thin lines, linear

Figure 3

Second four natural frequencies-amplitude relationship for a four-bay beam: thick lines, nonlinear; thin lines, linear

Figure 4

First natural frequencies-amplitude relationship for a chain with number of beams increasing from 1 to 4 (Ω, ω: nonlinear and linear frequency)

Figure 5

First and second mode of a four-bay beam: (a) first mode (β=π)a=0.04π, (b) first mode (β=π)a=0.08π, (c) second mode (β=3.393)a=0.04π, and (d) second mode (β=3.393)a=0.08π; thick lines, nonlinear; thin lines, linear

Figure 6

Third and fourth mode of a 4-bay beam: (a) third mode (β=3.926)a=0.04π, (b) third mode (β=3.926)a=0.06π, (c) fourth mode (β=4.464)a=0.02π, and (d) fourth mode (β=4.464)a=0.035π; thick lines, nonlinear; thin lines, linear

Figure 7

Frequency-response relationship for a four-bay beam with m0=0.0,0.001,0.005,0.01: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode (Ω, ω: nonlinear and linear frequency)

Figure 8

Frequency-response relationship in the first linear pass band for a four-bay beam; m0=0.01

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