## 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 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.

## 1. Introduction

Suppose we have an elastic structure, under the influence of a fixed set of loads, that can be modelled as a linear system. The positive stiffness of the structure in resisting the loading generates an equilibrium configuration that is unique and stable, and associated with a minimum of the underlying potential energy. That is, the configuration is unique and robust against small static or dynamic disturbances. But, under the variation of the external loading (especially, if the loads have axial effects), or depending on the geometry, or if disturbances are not small, the linear model may be inadequate and recourse to a nonlinear model may be necessary [1]. In the nonlinear context, equilibria may routinely cease to be stable: buckling occurs, and, furthermore, multiple, or coexisting, equilibria reflect an often convoluted relationship between loading and elastic deformation [2]. The potential energy function will typically have myriad turning points, especially in a higher dimensional system. This generates the following questions:

— in a system with multiple stable equilibrium configurations, how does the system transit between them, given sufficiently large disturbances?

— during this process of transient dynamics are there clues to the location of unstable equilibria?

On a global level it is the presence and location of unstable equilibria that have a profound effect on long-term recurrent behaviour. Equilibrium paths exhibit exchanges of stability and their relative dominance has a strong effect on transient behaviour: unstable states effectively act as watersheds delineating the catchments regions for stable equilibria. A loose comparison can be seen in the field of hydrology, where although the run-off is characterized by rivers in the bottom of valleys, it is the mountain ridges that determine overall separation and distribution of rainfall into catchment regions. Stretching this analogy a little further, we can also consider the paths of least energy in going from one valley to another, given some initial conditions. It is rarely the most efficient route to go straight over an intervening mountain top but rather meander around and through any available mountain passes.

In order to generate the approach used later in this paper, consider the situation depicted schematically in figure 1. In figure 1*a*, a simple single degree of freedom potential energy function is shown. This has a stable equilibrium at the origin and an adjacent unstable equilibrium beyond which another stable equilibrium may be located. If we subject the system (at rest) to an impulse or equivalently an initial velocity, the resulting transient may have enough energy to traverse the unstable (hilltop) equilibrium, or may not. Theoretically, there is an initial velocity that would cause the trajectory to end up exactly at rest on the unstable equilibrium, but typically the trajectory slows down as it approaches the unstable equilibrium position before veering away. This situation is not overly (negatively) influenced by the presence of some energy dissipation (damping). The potential energy shape may morph under the influence of external forcing (for example), with new equilibria appearing and disappearing, and it is the ability to identify all these equilibria from the motion of transient trajectories that is addressed here. Figure 1*c* shows an equilibrium path involving hysteresis and thus incorporating an unstable path (between two saddle-node bifurcations [3]).

Moving to a two-dimensional potential clearly opens up the possibilities for a greater range of transient dynamic behaviour shown as a configuration space plot in figure 1*d* . Transitions between stable equilibria will naturally involve more complicated traverses of the potential energy landscape. However, the evolution of the trajectory, as it moves across the potential energy landscape, will have a tendency to reflect the morphology of the potential energy surface (or hyper-surface) such that there is a characteristic ‘slowing-down’ as a trajectory passes close to an unstable configuration, specifically represented by a saddle point. Similarly, it is anticipated that trajectories would tend to avoid configurations associated with a hill-top in the potential energy. Figure 2 shows three views of a potential energy surface in which colour contouring is used to identify the surface, with yellow for example representing relative minima [4]. Superimposed in black are trajectories starting from within the lower right potential energy minimum. There is no damping in this system and total energy is conserved. In figure 2*a*,*b*, the initial conditions generate trajectories that remain in the general vicinity of the equilibria (there are two adjacent minima in the surface but it is the transition over to the other deeper well that is of primary interest). However, in figure 2*c*, the trajectory passes close to a saddle point as it enters the upper left well. The approach to be described later assesses the speed (and hence kinetic energy) of the trajectory to gives clues about the location of the saddle point. In the experiments to be described later, the initial conditions are the rest state but an impact generates initial velocities. Even in those cases in which the trajectory does not escape, it may very well slow down as it comes close to the saddle. In these examples, the system is conservative (this is actually based on the Müller–Brown potential used in chemical reactions [4]) but light damping does not alter the picture significantly excepting that the trajectory finally ends up at rest at a minimum, and in fact has the benefit of dissipating energy so that the multiple close encounters with a saddle have a tendency to be slower at each subsequent pass. Navigating potential energy landscapes is also a feature of some studies in celestial mechanics [5].

This paper focuses its attention on generating a variety of initial conditions and hence transients that effectively explore the potential energy surface (for a system where the behaviour is dominated by two modes). Given sufficiently repeated impacts some of these trajectories will experience ‘close encounters’ and help to uncover the unstable equilibrium configurations. After the initial input of energy via an impact, the tests are entirely passive and involve a ‘non-contact’ observation of the structural response. A more direct, but experimentally involved, approach to extracting unstable (periodic) behaviour experimentally is based on the notion of using a feedback controller in conjunction with continuation (path-following) to trace out unstable paths [6,7].

A post-buckled beam is used as the test case, as it is representative of many curved (potentially bistable) lightweight aerospace, naval and civil structures [2]. The equilibria and stability of shallow arches have been extensively studied [8–11], including by the current authors [12–14]; however, recent research [15] has shown numerically that a vast set of previously undiscovered unstable equilibria may be present, most of which have never been observed experimentally, and could not be observed with single-parameter force- or displacement-control experiments. Beyond the specific application of post-buckled beams, the framework for obtaining unstable static equilibria from experimental measurements (and without any reference to a model) may prove highly valuable in many areas of research, for example in the study of chemical reactions in chemistry and biology. The potential energy topography (over the possible atomistic configurations) dictates the progression of chemical reactions and may be used to investigate the stability, or the existence of, otherwise unobserved compounds. In these fields, many methods have been developed for uncovering unstable saddle nodes, such as the numerical methods in [16,17].

## 2. The experimental system

The procedure described in this paper involved subjecting a clamped–clamped post-buckled beam (see figure 3*a* for a photo and the beam thickness *t*, width *b*, length *L*, modulus of elasticity *E* and density *ρ*) to a variety of impacts to reveal the presence of unstable equilibria. The beam was buckled mechanically, by applying a transverse load (similar to pulling on a bowstring), then clamping the supports with the beam in a deformed configuration. The beam would relax slightly upon releasing the transverse load, but depending on the magnitude of this load, the beam could be given any arbitrary initial rise. It was hoped that the rise of the buckled beam would be large enough to exhibit both symmetric (limit-point) and asymmetric (bifurcation) buckling, but small enough to ensure that the small impact hammer could deliver enough energy to induce snap-through. The beam was re-clamped several times until a suitable rise of approximately 1.3 mm was obtained. The beam used in the investigation included minor imperfections, and thus the resulting post-buckled beam was not perfectly symmetric. This was not of particular concern in the work, as the goal was to uncover unstable equilibria in a realistic, and thus imperfect, structure.

The disturbances were caused by hand using an instrumented impact hammer (visible in photo). Digital image correlation (DIC), i.e. the use of two cameras in stereo to capture three-dimensional deformations, was used to measure the subsequent full-field displacement of the arch immediately after strikes from the impact hammer. The use of the DIC system was a critical component of the research as it allowed for the characterization of the full-field geometry of the unstable equilibria. For a more detailed review of DIC, we refer the reader to one of the early works on the topic by Chu *et al.* [18]. This technology is becoming more common in vibrations research with several off-the-shelf products available. The work herein was performed using the GOM ARAMIS and IVIEW products, which are discussed in [19]. The DIC system recorded the transverse displacement of the beam at 17 different positions. However, in the following results, the response will instead be represented by an 8th order polynomial fit curve. In all cases, the fit curves faithfully represented the DIC data. Two representative samples are shown in figure 3*b*; they show very good agreement but do not capture some of the very small spatial fluctuations of the DIC data (which could be noise in the DIC system, or small ‘kinks’ in the beam). The DIC system was also capable of measuring beam twist; however, no appreciable torsional motion was detected (which is further evidenced later by the excellent agreement between the experimental and model results when no torsional motion was included).

Unfortunately, it was necessary to apply the impacts to the front of the beam as the fixture supporting the beam did not have sufficient clearance in the back (as seen in figure 3*a*). This meant that a part of the DIC field of view was obscured. The DIC software was also incapable of recapturing any data points that had been obscured at any prior time in a particular test. Thus, it was decided to apply impacts at only one ‘sacrificial’ spot on the beam (the obscured region in figure 3*b*). This meant that the initial condition space was perhaps not as thoroughly sampled as one might prefer; however, the impact magnitudes varied and the impact zone was randomly shifted within the sacrificial zone to overcome this shortcoming.

The exploration of phase space required that a significant amount of data be collected in order to, by coincidence, capture the necessary data while the transient response of the beam was in the vicinity of the unstable equilibria. Furthermore, as the snap-through events were very fast, the DIC system needed to record at a frame rate of 2000 Hz in order to capture those few important frames near snap-through. For this frame rate, and the size of the images (large enough to capture the beam), the DIC cameras were capable of capturing 100 s of dynamic response data prior to filling their internal memory storage. Two tests of this length were performed.

Impacts were nominally applied at a 2 s interval, which was significantly longer than the amount of time required for the snap-through events to cease after each impact. The load cell on the impact hammer made it possible to eliminate any response data for which the beam and hammer were still in contact, this ensured the response was entirely unforced free decay. Unfortunately, many of the snap-through events had to be eliminated due to double-hits where the hammer was not removed fast enough [20]. In several cases, the impacts were also not sufficiently large to induce any snap-through events at all. In total, of the 200 s of data, 24 usable snap-through responses were observed. Several of these responses, however, included many individual snap-through events.

An 18 s sample of the experimentally measured nonlinear free decay is shown in figure 4. The arrows above the plot denote the times that impacts from the hammer (the four labelled impacts will be discussed later) were applied. The subsequent dynamic response typically included a number of snap-through events. In this reference frame, the impacts were always applied ‘upward’ as the mounting frame did not leave space for the impact hammer to allow any ‘downward’ impacts. In some instances, the beam would come to rest after an odd number of snap-through events in the reversed configuration, i.e. buckled toward the clamping fixture (‘up’ or positive *y*_{c} in the plot). When buckled ‘up’ it was very difficult to hit the beam hard enough with an ‘upward’ force (without inducing plastic deformation) to induce snap-through events, as the force from the hammer was orientated away from snap-through. In these cases, the beam was manually returned (dashed green arrows) into ‘down’ configuration. The data from the manual adjustments were not processed as there was no way to determine how long contact with the beam was maintained. The plot shows only the position of the midspan of the beam, *y*_{c}. Single point measurements like those shown in figure 4 are not surprising, or new; however, the full-field information, particularly when captured at 2000 Hz, will be shown later to reveal the unstable equilibria.

## 3. Numerical approach

Although the goal of this work is focused on a useful experimental method for uncovering unstable equilibria, the results of a two-mode reduced order model (ROM) and a finite-element model (FEM) will also be presented. The FEM results help serve as a check on the experimental results, while the two degrees of freedom (2 d.f.) model is especially revealing as it allows for visualization of an approximate potential energy surface of the buckled beam.

### (a) Two-mode model

Buckled beams and shallow curved arches that are loaded laterally typically present two dominant modes of snap-through: symmetric and asymmetric [21]. The symmetric and asymmetric snap-through paths are associated with the classical limit point and bifurcation point, respectively, with the former most commonly encountered in relatively mildly buckled or shallow arches. This is the situation shown schematically in figure 5*a*, in which a softening stiffness leads to a vertical tangency in the (central) deflection–force relation. This is the classic saddle-node bifurcation (point C), and is essentially the same as one of the classical forms of Duffing's equation [22]. At a point prior to snap-through, say point A, the system is in a state of bi-stability, and thus even though the equilibrium is stable, a sufficiently large perturbation might cause the system to move to the remote equilibrium (passing through the unstable (light blue) point). However, it is also possible (and for deeper arches more likely) that the behaviour involves an asymmetric component. For example, figure 5*b* shows two modes characterized by a symmetric component (*x*) and an angle at the centre (*θ*). In this case, the system may experience a pitchfork bifurcation prior to the saddle-node, as shown in figure 5*c*, with the system again exhibiting a snap-through event (point D) but this time with a significant asymmetry. Later, we shall link this scenario with the underlying potential energy.

Motivated by this behaviour, we introduce a 2 d.f. approximation of the beam response using the first two vibration mode shapes of a clamped–clamped beam (light grey curves in figure 6*a*), which are given by [23]
*α*_{n} has been artificially added to the traditional mode shapes in order to ensure they have both have maximum values of unity as shown in figure 6*a*. This step was not necessary but allows quicker comparisons with physical coordinates, for example, the modal coefficient of mode one is identical to the midspan displacement.

Snap-through can frequently include contributions from modes with higher wavenumbers; however, the 2 d.f. results will be shown to reveal useful insights into the beam behaviour. The formulation is done using a Rayleigh–Ritz approach and begins with the assumption that the beam is ‘shallow enough’ such that the axial load within the beam can everywhere be approximated by using the average length change. For the undamped free response, this yields an integro-differential equation [2]
*y*_{0} is the initial unloaded (*y*_{0}≠0 implies imperfect geometry) shape, *ρ* is the material density, *A* is the cross-sectional area, EI is the bending stiffness, AE is the axial stiffness, *P* is the equivalent axial load if the beam were flattened between the pin–pin supports (this could be replaced by a temperature increase term) and *L* is distance between the beam supports (also the nominal beam length). The integral term adjusts the in-plane load due to beam deformation about the unloaded shape and is the result of a Taylor series approximation of the classic arc-length formula. After applying integration by parts (using the fact that the mode shapes satisfy clamped–clamped boundary conditions), the method of virtual work with virtual displacement *w* yields
*y*_{0}) of the form
*w*=*ϕ*_{1}, then *w*=*ϕ*_{2} to yield the nonlinear vector equations
*a*} and {*γ*} are the vectors of the modal and imperfection coefficients, respectively. Note that although all focus was placed on a two-mode solution, this multiple degrees of freedom equation is general to any selected number of modes.

The restoring forces in this multiple degrees of freedom equation are conservative and thus may be integrated to yield the potential energy

Beyond the static configuration, natural frequencies provide another valuable measurement for model validation. Figure 6 also shows the mode shapes corresponding to the first two experimentally measured natural frequencies (using impact testing) and those of the 2 d.f. model. The 2 d.f. natural frequencies were obtained by determining the eigenvalue solutions of the linearized multiple degrees of freedom governing equations
*u*} about the post-buckled shapes {*a**} in figure 6. Despite the differences in the natural frequencies, it will be shown that the 2 d.f. model reveals much about the underlying potential energy of the structure.

The static equilibria, under changing *P* were obtained through a Newton–Raphson solution of the equation (3.7) with the accelerations set to zero. This resulted in the classic pitchfork bifurcation (buckling) with *P* as the control parameter (which could not be measured directly in the experiment). As seen in figure 6, due to the initial imperfections, the natural frequencies and the deformed shapes of the two stable buckled equilibrium configurations were not identical. The axial load, along with the initial imperfections *γ*_{1} and *γ*_{2} were iteratively modified to calibrate (nominally) the model parameters to best match the experimental measurements. As the goal of the 2 d.f. model was to obtain qualitative rather than quantitative information, the calibration was performed with a relatively coarse grid of axial loads and imperfection parameters. This still resulted in reasonably good matching of the natural frequencies and deformed shapes as shown in figure 6. The selected axial load was 184.1 N (significantly higher than the first buckling load of 59.9 N), while the initial imperfections were 0.082 mm and −0.077 mm in the first and second modes, respectively. The primary source of error in the Galerkin ROM is that it was unable to match (due to the choice of mode shapes) the slightly non-zero slope of the clamped support at the right of the beam.

### (b) Co-rotational finite-element model

As already discussed, the 2 d.f. results are valuable for visualization of potential energy surfaces but are less quantitatively accurate. In order to improve on these results an implicit co-rotational finite-element model using 24 uniform beam elements was also implemented. The finite-element model allowed for greater accuracy, and particularly, permitted for a better approximation of the right side of the beam, where the fixture was slightly tilted (visible in figure 6). The co-rotational formulation was obtained from [14], and solved using an arc-length method to apply both in-plane (to induce buckling) and transverse (to explore the stable and unstable static equilibria) loading. Similar to the calibration of the 2 d.f. model, a prestressing method was used to best match the experimental post-buckling configuration. The prestressing involved first rotating on the right side (i.e. a clamped–clamped beam with non-zero rotation on the right) to closely match the experimentally measured slope, then applying end shortening until both the buckled up and down configurations closely matched the experimental measurements. Additional details about the calibration process (on a similar structure) can be found in [12], and details on the overall process from the co-rotational formulation, the arc-length solution and the calibration are available in [13].

Figure 6*b* shows the results of the calibration, along with a comparison of the natural frequencies obtained using FEM and from the experiments. The FEM accurately captures both the static and dynamic behaviour of the beam, with overall improved predictions of the natural frequencies over the ROM. This co-rotational model is used later to predict the unstable equilibria, for comparison with the experimental transient results.

Damping was not included in either model. This is because the primary aspects of interest were the natural frequencies and static equilibrium configurations (for calibration of the numerical models), and the underlying potential energy surface. The dissipation in the experimental system was observed to be predominantly linear viscous damping (particularly no Coulomb or other velocity-independent damping was detected), which has little effect on the natural frequencies, and no effect on static equilibria or potential energy and thus it was not thoroughly investigated.

## 4. Experimentally probing the phase space

The unstable equilibria, either saddles or hilltops, influence phase trajectories (such as the brown paths in figure 1*b*,*d*) which follow energetically favourable routes from one stable equilibrium to another. Even without an expression for the governing equation of motion, the phase portrait of the free decay of a structure may be used to gain a qualitative sense of the underlying potential energy. Figure 7 shows projections of the phase portrait (self-intersections may occur in a projected phase portrait) for the four free decays labelled (*a*) through (*d*) in figure 4. The response is shown according to two pseudo-DOFs, i.e. the midspan displacement *y*_{c} and the difference between the left *y*_{L} and right *y*_{R} quarter point displacements. A ‘ghost’ of the unstable equilibria may loosely be seen as many of the snap-through events (*y*_{c}=0) occur within two bands near asymmetric components of ±0.5. One of these, denoted with an orange symbol, has a slightly lower potential energy as will be discussed later. For a perfect buckled beam the two mirror image asymmetric snap-through shapes (asymmetric component ±0.5) would occur with equal likelihood. The initial imperfections and lack of unbiased sampling (one-sided hammer hits) make it difficult, however, to infer directly from this plot which of the two saddles was of lower potential energy. From this, we can infer that asymmetric snap-through is greatly preferred as very few snap-through events occur with a asymmetric component near zero.

The starting point in each case (red dot) is the negative buckled-down configuration, as the impacts could only be applied in an ‘upward’ direction. The remote stable equilibrium is indicated by the light blue points. The final few oscillations of the free decays (green portion of the curves) may be used to infer the equilibrium configuration in which the system came to rest. For example, for hammer hit figure 7*b*, the beam began buckled down, snapped-through several times and came to rest in the buckled-up configuration. Each crossing of the *y*_{c}=0 plane usually indicates a snap-through occurrence. It is possible that the midspan displacement could cross this plane without the beam actually continuing to complete a snap-through from one stable equilibrium to the other. Nonetheless, this figure shows that many snap-through events occur after each hammer hit. Note that the time frame during which the hammer was in contact with the beam (i.e. forced response) has been removed from the plot, and all subsequent analysis.

We now focus on the time series of the free-decay of case (*b*) from figures 4 and 7. The plot of the total kinetic energy (KE) during the final five snap-through events (labelled 1–5), and one ‘near miss’ (labelled 6) is shown in figure 8. Each data point denotes the experimentally measured KE obtained by numerically differentiating the DIC measured displacement to obtain velocity, and then summing the squared velocities over all of the measurement points. The total KE was not multiplied by the element mass, instead it was simply normalized between 0 and 0.9 for plotting purposes (to show conveniently between 0 and 1). The magnitude of the KE oscillates several times through the snap-through process. For example, starting in configuration S1, which is a snap-through event with a generally ‘upward’ velocity, the beam gains KE as it drops down into the potential energy well toward the ‘up’ static equilibrium. The beam reaches maximum kinetic energy when it makes it nearest passing to the stable equilibrium (lowest PE), then begins to ascend (from a state space standpoint) the potential surface in the direction opposite to snap-through (this is generally much steeper and a ‘stiffening’ ascent). The beam quickly loses KE on this ascent and almost comes to a complete stop reaching (near zero KE) the (locally) maximum displacement of configuration M1 (M indicates maximum). The beam then reverses direction (nominally) and begins to travel downward the stable equilibrium, again reaching a local maximum KE at its nearest passing of the stable equilibrium. Finally, the trajectory continues past the unstable equilibrium toward the saddle reaching another locally minimum KE during the snap-through event at S2.

For a perfect structure, the snap-through occurring near the positive or negative saddle-node would be equally likely for a randomly located and oriented impact load. The imperfections inherent in any experiment mean this will never be the case. Additionally, as seen in figure 8, the peak KEs observed near the ‘up’ static equilibrium (the peaks between S1 and S2) are significantly higher than the KEs near the ‘down’ static equilibrium (the peaks between S2 and S3). This indicates that the ‘up’ static equilibrium has a deeper potential well and is likely to be more stable. If it had been possible to apply impacts both ‘upward’ and ‘downward’ this should have presented itself as a slight preference toward resting in the ‘up’ configuration. Given the bias induced by only applying ‘upward’ impacts this cannot be inferred from the experimental data. In fact, more of the experiments ended up resting in the ‘down’ configuration.

The total energy is not shown in figure 8. It can be inferred, however, that the KE of the system at the points highlighted by the green circle and square is just sufficient to pass over the unstable saddle when departing from near the ‘up’ and ‘down’ stable equilibria respectively. For the red circle, however, the KE is not quite sufficient as configuration NS6 is close to, but not on the separatrix. This can also be seen by looking at the local maximum configurations, after alternating between ‘up’ and ‘down’ there is no reversal between M5 and M6. Thus, the red dashed line approximately denotes the minimum KE needed to snap-through when departing the ‘up’ static equilibrium. The green square could equally be used to approximate the same threshold when departing the ‘down’ equilibrium.

The local temporal minimum KE does not necessarily occur at the saddle-point. This is because, unlike the one-dimensional Duffing double-well case, the orbit can traverse through to the remote stable equilibrium via an infinite set of paths. However, it is generally true that as the total energy of the orbit of a higher-dimensional system decays, the portions of the separatrix (the ridge-line separating stable equilibria) that the system may pass over shrinks toward the saddle-point. Thus, it is hypothesized that the final few snap-through events in a free decay will show a trend toward passing nearer and nearer to the saddle. Damping is a beneficial aspect of the generation of transients in this context. Damping ensures that the response surveys various total energy levels to approximate the minimum KE required to cause snap-through. The exact form of the damping is not especially important, it is merely necessarily that it gradually reduce the total energy of the system.

In figure 9, the final five snap-through events plus near-snap NS6 are shown as white dots in the projected phase space superposed on top of the approximate 2 d.f. potential energy surface. The theoretical stable and unstable equilibria from the 2 d.f. model are also shown as red dots. The white dots clearly trend closer to the theoretical unstable saddle-points of the approximate 2 d.f. model. The near-miss NS6 then moves further away from the saddle.

In this case, each snap-through event occurred via the ‘negative’ saddle point, i.e. *y*_{L}−*y*_{R}<0. As seen in figure 7, it could also have occurred for the mirror-image ‘positive’ saddle point. The potential energy of the 2 d.f. model, however, shows that there is a significant bias. Using the 2 d.f. model, the strain energies of the two stable equilibria are *U*_{1}=0 J (the datum) and *U*_{2}=1.04×10^{−5} J, while the unstable equilibria have strain energies of *U*_{3}=2.46×10^{−5} J and *U*_{4}=5.19×10^{−5} J for the two saddles, and *U*_{5}=5.22×10^{−4} J for the hilltop. Thus, the two saddle-points have a significant difference in their strain energy, with the higher saddle-point being much closer, both spatially, and in strain energy to the hilltop. One would thus expect more snap-through events to occur near the ‘negative’ asymmetric saddle point. Additionally, the close vicinity of the ‘positive’ unstable saddle and the hilltop would indicate that it will be difficult to distinguish between these two in the ensemble of phase trajectories.

## 5. Results

The procedure discussed above was repeated for every experimentally measured snap-through orbit, omitting all invalid orbits (double-hits and hits that resulted in no snap-through). Additionally, the data was down-sampled to only the final snap-through event as it is expected to pass closest to the saddle. The final results are shown in figure 10. Figure 10*a*,*b* shows the theoretical static equilibrium path obtained under a point load applied at the quarter point (from the FEM). There is nothing special about the chosen loading type, rather any load that will access the five *F*=0 crossings (i.e. the unloaded stable and unstable equilibria) would work equally well. These curves were generated using an arc-length procedure. Figure 10*a*,*b*, for the same simulation, the only difference being the displacement that is plotted, with figure 10*b* being somewhat more useful in distinguishing the different components of the path. The numerical ‘negative’ saddle (point ii, red), unstable hilltop (point iii, green), and the ‘positive’ saddle (point iv, blue) configurations are shown in figure 10*c*–*e* with the same colour coding. Of the 24 usable free decays, there were 17 for which the last snap-through event occurred through the ‘negative’ saddle. The remaining seven cases appeared to occur in the vicinity of either the unstable hilltop or the ‘positive’ saddle. As is indicated by the potential energy surface, it was difficult to distinguish these. These two are separated in figure 10 based on their relative comparison with the theoretical curves with four being more similar to the unstable hilltop, and three closer to the unstable saddle. The reality might be closer to a smeared average between the two. The ‘negative’ saddle obtained experimentally however shows excellent agreement with the co-rotational model prediction. The dashed solid line shows the average of the configuration of the final snap-through event in the 17 ‘negative’ saddle snap cases. The grey-shaded region shows the 1 s.d. error region. The experimental average nearly lays directly on top of the theoretical prediction (red dashed curve). For the other two cases, given the low number of observations, each configuration is plotted independently rather than showing the average and standard deviation.

## 6. Summary and conclusion

The unstable static equilibria of a bistable (post-buckled) beam are exposed via the tracking of large-amplitude transients and their slow decay. Analytic models are used, both a 2 d.f. and a co-rotational FEM, to verify and analyse the results; however, the experimental approach uses only experimental data, and requires no modelling whatsoever. The method is shown to yield excellent results in comparison with the theoretical predictions of the FEM.

Although the work is motivated by the global dynamic response of multi-stable geometrically nonlinear systems, it is also seen as a first step towards the modal analysis of unstable equilibria in high-order systems. Further research into the linearized dynamics about the identified unstable equilibria may yield approximate eigenvalues, or equivalently, the degree of instability. However, it is the insight that is gained into global behaviour of highly nonlinear systems that is the key contribution of this approach.

As a final footnote, it is worth mentioning that the high spatial and temporal data resolution facilitated by the DIC system was a crucial component in obtaining these results. During the experiments the naked-eye was unable to distinguish individual snap-through events as they occurred. Furthermore, the velocities involved were such that the snap-through ‘event’ could be easily missed by coarse sampling. It is possible that scanning laser vibrometres may also be suitable for these experiments; however, the methods would need to be adapted to capture aperiodic transient motion over the entire structure at a high-sampling rate.

## Data accessibility

Data are available at https://goo.gl/1o4yRU.

## Authors' contributions

Both authors acquired the experimental data and jointly wrote the paper. The first author conducted the data analysis and developed the numerical model. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

L.N.V. acknowledges Universal Technology Corporation (UTC) contract FA8650-10-D-3037, and AFOSR grant no. FA9550-13-1-0130.

## Acknowledgements

This research was conducted using the facilities of AFRL, Wright-Patterson AFB, Dayton, OH, USA.

- Received March 8, 2016.
- Accepted May 26, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.