Nonlinear dynamics and chaotic motions of a singularly perturbed nonlinear viscoelastic beam

Ioannis Theodorou Georgiou, Purdue University

Abstract

The dynamics of 1 or 2-dimensional structures such as rods, beams, plates, etc., is usually described by 1 or 2-dimensional nonlinear partial or integro-partial differential equations (PDEs). The equations of motion of such structural systems, with the associated boundary conditions, are amenable to numerical study and to some extent to analytical study. In practice there are many complex structural systems composed of simpler structural members or subsystems with well diverse flexibilities. A simple example is that of a frame comprised of a flexible beam mounted on two relatively stiff columns. The objective of this investigation is to relate the dynamics of a given structure to the dynamics of a simpler structure which is obtained in the limit when the much stiffer substructure of the system is assumed to be perfectly rigid. Without loss of generality, we study both analytically and numerically the dynamics of a representative complex structure, the frame. More precisely, it consists of a nonlinear viscoelastic beam hinged at its ends to two linear viscoelastic columns (supports). We view the equations of motion of the frame structure with sufficiently stiff supports as a singular perturbation of the simpler structure: the nonlinear beam pinned on rigid supports. By applying the Galerkin reduction method, we approximate the equations of motion (coupled PDEs) by a system of a finite number of coupled nonlinear oscillators interacting with a finite number of linear oscillators. Under certain conditions on system parameters, this finite system of ordinary differential equations (ODEs) constitutes the core or "backbone" of the coupled integro-partial differential equations of motion, an infinite dimensional nonlinear dynamical system. By applying the singular perturbation methodology and the theory of invariant manifolds, we show that for sufficiently stiff linear oscillators (stiff supports) the steady state behavior including nonlinear vibrations and chaotic motions of the above finite dynamical system (full order system) is described by a system of lower dimensions. This reduced system is defined on a nonlinear invariant manifold of the full order system and it possesses symmetry properties that are exactly the same as those of the degenerate reduced system, i.e., the system obtained in the limit when the linear oscillators are assumed to be perfectly rigid.

Degree

Ph.D.

Advisors

Bajaj, Purdue University.

Subject Area

Aerospace materials|Mechanical engineering|Civil engineering

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