Abstract
Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 373-386 |
| Number of pages | 14 |
| Journal | Chaos |
| Volume | 14 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jun 2004 |
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ASJC Scopus subject areas
- Applied Mathematics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics
Cite this
Multi-scale continuum mechanics : From global bifurcations to noise induced high-dimensional chaos. / Schwartz, Ira B.; Morgan, David S.; Billings, Lora; Lai, Ying-Cheng.
In: Chaos, Vol. 14, No. 2, 06.2004, p. 373-386.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Multi-scale continuum mechanics
T2 - From global bifurcations to noise induced high-dimensional chaos
AU - Schwartz, Ira B.
AU - Morgan, David S.
AU - Billings, Lora
AU - Lai, Ying-Cheng
PY - 2004/6
Y1 - 2004/6
N2 - Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.
AB - Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.
UR - http://www.scopus.com/inward/record.url?scp=3042687129&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=3042687129&partnerID=8YFLogxK
U2 - 10.1063/1.1651691
DO - 10.1063/1.1651691
M3 - Article
C2 - 15189066
AN - SCOPUS:3042687129
VL - 14
SP - 373
EP - 386
JO - Chaos (Woodbury, N.Y.)
JF - Chaos (Woodbury, N.Y.)
SN - 1054-1500
IS - 2
ER -