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Research Papers

Symmetry Properties in the Symmetry-Breaking of Nonsmooth Dynamical Systems

[+] Author and Article Information
Alessio Ageno, Anna Sinopoli

Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Via Gramsci 53, Roma 00195, Italy

J. Comput. Nonlinear Dynam 5(1), 011001 (Oct 09, 2009) (9 pages) doi:10.1115/1.3201924 History: Received March 04, 2008; Revised February 27, 2009; Published October 09, 2009

In this paper, the block simply supported on a harmonically moving ground is assumed as a system well representing a typical nonsmooth dynamical behavior. The aim of the work is to carry out the existence conditions of asymmetric responses; an analysis that comes first in any stability investigation. By using simple definitions belonging to the symmetry group theory, it is possible to completely clarify the relationships between the various initial conditions that allow simple asymmetric responses, and to develop tools, which will be very useful in the stability analysis of more complex asymmetric responses.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

The block simply supported on a rigid ground

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Figure 2

Symmetric response in the phase plane

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Figure 3

Asymmetric response in the phase plane

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Figure 4

Asymmetric response: displacement time-history

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Figure 5

Family of curves f(t0,t1) for different values of k

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Figure 6

f(t0,t1)=0 for ω=5, β=25, and e=0.912

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Figure 7

Solutions curves of Eq. 10 for ω=2, β=2.5, and e=0.7

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Figure 8

Rotated plot for ω=2, β=2.5, and e=0.7

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Figure 9

Asymmetric response displacement time-history

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Figure 10

T0=I and T1 permutations and related simbology

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Figure 11

Relations between t0(1), t1(0) and t0(0), t1(1)

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Figure 12

The points whose coordinates t0 and t1 both correspond to a time translation of 2nπ/ω

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Figure 13

Comparison between permutations and the solutions curves of Fig. 7

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Figure 14

Comparison between T0(2nπ) and T1[(2n−1)x]∗, and the solutions curves of Fig. 7

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Figure 15

Comparison between all the points Ti(nπ) and Ti(nπ)∗, and the solutions curves of Fig. 7

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Figure 16

The time-histories of the two basic distinct solutions

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