Richard Wiebe

On the experimental identification of unstable static equilibria.

This paper shows how the presence of unstable equilibrium configurations of elastic continua is reflected in the behaviour of transients induced by large perturbations. A beam that is axially loaded beyond its critical state typically exhibits two buckled stable equilibrium configurations, separated by one or more unstable equilibria. If the beam is then loaded laterally (effectively like a shallow arch) it may snap-through between these states, including the case in which the loading is applied dynamically and of short duration, i.e. an impact. Such impacts, if applied at random locations and of random strength, will generate an ensemble of transient trajectories that explore the phase space. Given sufficient variety, some of these trajectories will possess initial energy that is close to (just less than or just greater than) the energy required to cause snap-through and will have a tendency to slowdown as they pass close to an unstable configuration: a saddle point in a potential energy surface, for example. Although this close-encounter is relatively straightforward in a system characterized by a single degree of freedom, it is more challenging to identify in a higher order or continuous system, especially in a (necessarily) noisy experimental system. This paper will show how the identification of unstable equilibrium configurations can be achieved using transient dynamics.

Inconsistent Stability of Newmark's Method in Structural Dynamics Applications

The stability of numerical time integrators, and of the physical systems to which they are applied, are normally studied independently. This conceals a very interesting phenomenon, here termed inconsistent stability, wherein a numerical time marching scheme predicts a stable response about an equilibrium configuration that is, in fact, unstable. In this paper, time integrator parameters leading to possible inconsistent stability are first found analytically for conservative systems (symmetric tangent stiffness matrices), then several structural arches with increasing complexity are used as numerical case studies. The intention of this work is to highlight the potential for this unexpected, and mostly unknown, behavior to researchers studying complex dynamical systems, especially through time marching of finite element models. To allow for direct interpretation of our results, the work is focused on the Newmark time integrator, which is commonly used in structural dynamics.

Co-existing Responses in a Harmonically-Excited Nonlinear Structural System

A key feature of many nonlinear dynamical systems is the presence of co-existing solutions, i.e, nonlinear systems are often sensitive to initial conditions. While there have been many studies to explore this behavior from a numerical perspective, in which case it is trivial to prescribe initial conditions (for example using a regular grid), this is more challenging from an experimental perspective. This paper will discuss the basins of attraction in a simple mechanical experiment. By applying both small and large stochastic perturbations to steady-state behavior, it is possible to interrogate the initial condition space and map-out basins of attraction as system parameters are changed. This tends to provide a more complete picture of possible behavior than conventional bifurcation diagrams with their focus on local steady-state behavior.

On Euler Buckling and Snap-Through

Bistable structures have seen significant attention in recent years for their potential uses in switching and energy harvesting. The behavior of these structures, however, is very sensitive to boundary conditions and initial geometry making their calibration for various applications a difficult task. Additionally, obtaining the force-deformation behavior of these highly (geometrically) nonlinear structures often requires computationally expensive continuation methods. This paper presents a very simple closed-form method which estimates several important characteristics of classical snap-through curves of transversely loaded beams. The estimation is based on a relationship between classic Euler buckling of beams under axial load and the snap-through of post-buckled and curved beams and arches under transverse loading that has recently been investigated by the authors.

Stability of a nonlinear second order equation under parametric bounded noise excitation

The motivation for the following work is a structural column under dynamic axial loads with both deterministic (harmonic transmitted forces from the surrounding structure) and random (wind and/or earthquake) loading components. The bounded noise used herein is a sinusoid with an argument composed of a random (Wiener) process deviation about a mean frequency. By this approach, a noise parameter may be used to investigate the behavior through the spectrum from simple harmonic forcing, to a bounded random process with very little harmonic content. The stability of both the trivial and non-trivial stationary solutions of an axially-loaded column (which is modeled as a second order nonlinear equation) under parametric bounded noise excitation is investigated by use of Lyapunov exponents. Specifically the effect of noise magnitude, amplitude of the forcing, and damping on stability of a column is investigated. First order averaging is employed to obtain analytical approximations of the Lyapunov exponents of the trivial solution. For the non-trivial stationary solution however, the Lyapunov exponents are obtained via Monte Carlo simulation as the stability equations become analytically intractable.