On the experimental identification of unstable static equilibria

Abstract

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 snapthrough 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 closeencounter 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.

A tube dynamics perspective governing stability transitions: An example based on snap-through buckling

“…The selected thermal load corresponds to 184.1 N, while the initial imperfections are γ 1 = 0.082 mm and γ 2 = −0.077 mm. These values match the parameters given in the experimental study [1]. For all the numerical results given in this section, the initial energy of the system is set at 3.68×10 −4 J -above the energy of saddle point S 1 , so that the equilibrium point W 1 is inside the configuration space projection.…”

“…Suppose we load the beam to a value slightly below the snap value at P C , and fix it at that value. In this case there will be the five equilibria mentioned earlier: three equilibria purely in sag (two stable and an unstable one between them), and two saddles, with significant angular components but geometrically opposed [1]. Small geometric imperfections (in A and/or S) will break the symmetry and influence which path is more likely to be followed.…”

“…A Cartesian coordinate system o-xyz is established on the mid-plane of the beam in which o is the origin, x, y the directions along the length and width directions and z the downward direction normal to the mid-plane. Based on Euler-Bernoulli beam theory [1,22], the displacement field (u 1 , u 3 ) of the beam along (x, z) directions can be written as…”

“…Using M x in (12) and N x in (14), the second equation in (10) can be described in terms of the transverse displacement w as [1] ρAẅ…”

“…The nonlinear behavior of slender structures under loading is often dominated by a potential energy function that possesses a number of stationary points corresponding to various equilibrium configurations [1,2]. Some are stable (local minima, or 'well'), some are unstable (local maxima or 'hill-top'), and some correspond to saddle points, i.e., a shape with opposite curvature in different directions, but still unstable, having both stable and unstable directions.…”