Ioannis T. Georgiou

DYNAMICS OF LARGE SCALE COUPLED STRUCTURAL/ MECHANICAL SYSTEMS: A SINGULAR PERTURBATION/ PROPER ORTHOGONAL DECOMPOSITION APPROACH

We have combined the theories of geometric singular perturbation and proper orthogonal decomposition to study systematically the dynamics of coupled systems in mechanics involving coupling between continuous structures and nonlinear oscillators. Here we analyze a prototypical structural/mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. We cast the equations of motion in a singularly perturbed set of oscillators and compute analytic approximations to an attractive global invariant manifold in phase space of the coupled system. The invariant manifold, two-dimensional for the unforced system and three-dimensional for the forced system, carries a continuum of slow motions. For a sufficiently stiff rod, a proper orthogonal decomposition of any long time motion extracts a single structure for the spatial coherence of the dynamics, which is a realization of the slow invariant manifold. As the flexibility of the rod increases, the energy of...

Interaction Between Slow and Fast Oscillations in an Infinite Degree-of-Freedom Linear System Coupled to a Nonlinear Subsystem: Theory and Experiment

The interaction dynamics of a cantilever linear beam coupled to a nonlinear pendulum, a prototype for linear/nonlinear coupled structures of infinite degrees-of-freedom, has been studied analytically and experimentally. The spatio-temporal characteristics of the dynamics is analyzed by using tools from geometric singular perturbation theory and proper orthogonal decompositions. Over a wide range of coupling between the linear beam and the nonlinear pendulum, the coupled dynamics is dominated by three proper orthogonal (PO) modes. The first two dominant PO modes stem from those characterizing the reduced slow free dynamics of the stiff/soft (weakly coupled) system. The third mode appears in all interactions and stems from the reduced fast free dynamics. The interaction creates periodic and quasi-periodic motions that reduce dramatically the forced resonant dynamics in the linear substructure. These regular motions are characterized by four PO modes. The irregular interaction dynamics consists of low-dimensional and high-dimensional chaotic motions characterized by three PO modes and six to seven PO modes, respectively. Experimental tests are also carried out and there is satisfactory agreement with theoretical predictions.

On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.

Instant chaos and hysteresis in coupled linear-nonlinear oscillators

Abstract A novel bifurcation to high-dimensional hyperchaos is observed in a driven coupled pendulum-flexible rod system. When the rod is in resonance with the pendulum, the system changes from a low-dimensional periodic attractor to a high-dimensional chaotic attractor abruptly. The bifurcation, which is hysteretic, is conjectured to stem from an unstable global invariant manifold of slow frequency motions.

THE SLOW INVARIANT MANIFOLD OF A CONSERVATIVE PENDULUM-OSCILLATOR SYSTEM

We analyze the motions of a conservative pendulum-oscillator system in the context of invariant manifolds of motion. Using the singular perturbation methodology, we show that whenever the natural frequency of the oscillator is sufficiently larger than that of the pendulum, there exists a global invariant manifold passing through all static equilibrium states and tangent to the linear eigenspaces at these equilibrium states. The invariant manifold, called slow, carries a continuum of slow periodic motions, both oscillatory and rotational. Computations to various orders of approximation to the slow invariant manifold allow analysis of motions on the slow manifold, which are verified with numerical experiments. Motion on the slow invariant manifold is identified with a slow nonlinear normal mode.

Inariant manifolds and chaotic vibrations in singularly perturbed nonlinear oscillators

Abstract This work concerns the forced nonlinear vibrations of a dissipative soft–stiff structural dynamical system consisting of a soft nonlinear oscillator coupled to a linear stiff oscillator. The equations of motion are cast into a set of singularly perturbed ordinary differential equations, with the ratio of linear frequency like quantities as the singular parameter. Then, using the theory of invariant manifolds, it is shown that, for sufficiently small coupling, the forced system possesses a 3-dimensional slow invariant manifold. The invariant manifold is a regular perturbation of a global invariant manifold for the conservative system. It is shown that the conservative system possesses a homoclinic orbit on the slow invariant manifold. Numerical simulations reveal that the forced system undergoes a period doubling cascade of bifurcations. The cascade of bifurcations gives rise to a weak strange attractor which undergoes a metamorphosis into a strong strange attractor as the forcing amplitude increases. Using Melnikovs method, it is shown that the strong strange attractor stems from transverse intersections of the invariant manifolds of a saddle-type periodic motion carried by the slow invariant manifold.

An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators

We consider a one-dimensional linear spring-mass array coupled to a one-dimensional array of uncoupled pendula. The principal aim of this study is to investigate the non-linear dynamics of this large-scale system in the limit of weak non-linearities, i.e. when the (fast) non-linear pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providing small perturbations to the motions on the invariant manifold. By restricting the motion on the slow invariant manifold and performing asymptotic analysis we prove that the non-linear large-scale system possesses propagation and attenuation zones (PZs and AZs) in the frequency domain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible.

Extreme parametric uncertainty and instant chaos in coupled structural dynamics

Bifurcation to high-dimensional hyperchaos is observed in a driven coupled pendulum-flexible rod system. When the rod is in resonance with the pendulum, the system changes from a low dimensional periodic attractor to a high-dimensional chaotic attractor abruptly. It is shown that high-dimensional chaotic dynamics is hysteretic, and exhibits extreme sensitivity with respect to small parameter changes.

Dynamics of Singularly Perturbed Nonlinear Systems with Two Degrees-of-Freedom

In engineering applications, complex structural systems are usually composed of simpler substructures with widely varying elasticities and damping properties. This broad diversity in flexibilities present in a complex structural system prompts us to view its motion in terms of the dynamics of interacting stiff and soft substructures. More specifically, we are here interested in soft-stiff structuralmechanical systems with multiple equilibrium states. A fundamental question to be asked is: how is the dynamics of a soft-stiff structural system related to the dynamics of a simpler structure obtained in the limit as its stiff substructures become essentially rigid? In this work, we present a systematic analytic-geometric methodology, by combining the singular perturbation theory with invariant manifolds and symbolic and numeric computation, to study the nonlinear dynamics of a soft-stiff two degrees-of-freedom system.