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TECHNICAL PAPERS

Towards Chaos in Vibrating Damaged Structures—Part I: Theory and Period Doubling Cascade

[+] Author and Article Information
Alberto Carpinteri, Nicola Pugno

Department of Structural Engineering and Geotechnics,  Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

J. Appl. Mech 72(4), 511-518 (Apr 08, 2005) (8 pages) doi:10.1115/1.1934582 History: Received December 27, 2002; Revised April 08, 2005

The aim of the present paper is to evaluate the complex oscillatory behavior, i.e., the transition toward deterministic chaos, in damaged nonlinear structures under excitation. In the present paper (Part I), we show the developed theoretical approach and how it allows us to capture not only the super-harmonic and offset components (predominant for moderate nonlinear systems) but also the subharmonics of the structural dynamic response, describing complex and highly nonlinear phenomena, like the experimentally observed period doubling. Moreover, a period doubling cascade with a route to chaos seems to emerge from our considerations.

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Copyright © 2005 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

(a) Structure I—Damaged structure and characteristics of the excitation (a=2.4mm,F=5N,f=ω∕2π=25Hz). (b) Structure I—Time history of the free end displacement and of the applied force. (c) Structure I—Relative error as a function of the iteration number, for each jth harmonic (j=0,1,…,16). (d) Structure I—Zero- (offset), sub- and super-harmonic components for the free end displacement (i.e., A20j2+B20j2 for j=0,1,…,16). (e) Structure I—Dimensionless phase diagram of the response (free-end displacement).

Grahic Jump Location
Figure 2

(a) Structure II—Damaged structure and characteristics of the excitation (a1=4.25mm,a2=4.25mm,F=2N,f=ω∕2π=18.9Hz). (b) Structure II—Time history of the free end displacement and of the applied force. (c) Structure II—Relative error as a function of the iteration number, for each jth harmonic (j=0,1,…,16). (d) Structure II—Zero- (offset), sub- and super-harmonic components for the free end displacement (i.e., A20j2+B20j2 for j=0,1,…,16). (e) Structure II—Dimensionless phase diagram of the response (free-end displacement).

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