## Abstract

We consider nonlinear elastic deformations of a magneto-elastic beam, using a combined experimental and theoretical approach. In the experiments, a beam had one end clamped with a magnet attached at its free end. When it was placed in an external magnetic field, it was susceptible to Euler beam buckling. However, the classic supercritical bifurcation associated with this buckling became subcritical when an attracting magnet was introduced in close proximity to the beam. To understand these experiments, we develop a model that couples the Euler elastica and dipole magnetic interactions with a uniform external field. The numerical model captures the observed behaviour well and shows that the supercritical magnetic field strength depends almost exclusively on elastic properties of the beam and strength of the permanent magnet, whereas the subcritical behaviour also depends on the separation distance between the attracting pair of magnets. We examine the bifurcation behaviour of the nonlinear system and show that for sufficiently small inter-magnet separation distances, other buckled states coexist with the fundamental mode.

## 1. Introduction

Recent advances in material sciences, particularly in the field of polymeric materials, have led to a growing interest in novel composite materials that generate elastic deformations by coupling the mechanical strain with an external excitation source such as an electric [1–3] or magnetic field [4–9].

Magnetic forcing introduces the possibility for remote control of structures, which can lead to interesting applications. For example, Dreyfus *et al*. [4] used microchains of paramagnetic particles with elastic links in an external oscillating field to induce swimming in a filament. The phenomenon involves a buckling instability which is entirely dependent on the relative strengths of the magnetic and elastic properties [10]. Forward motion requires longitudinal symmetry breaking, which can arise naturally through defects [10] or can be precisely tailored by anisotropic material distribution as shown by Kimura *et al*. [5]. In this example, an inchworm walker has steel micro-wires embedded in a polymeric material: the wires in the unstrained configuration are oriented along magnetic dipole field lines, and subjected to a uniform external magnetic field. The wires experience differential magnetic torques, and this magneto-elastic coupling leads to the required deformations of the beam.

Advances in magnetic materials, in particular high strength rare-earth neodymium magnets [11], have led to a range of magneto-elastic applications. These were exploited by Han [12] to propel a two link dumbbell swimmer comprising three spheres of equal diameter with embedded neodymium magnets linked by asymmetric elastic beams. Neodymium magnets have also been exploited to deform two-dimensional structures. Tipton *et al*. [6] embedded magnets in a periodic elastic structure comprising soft polymeric rubber that has four circular holes of equal diameter and magnets arranged symmetrically around the holes. For small fields, the structure compresses in a linear fashion, whereas above a critical field, the inter-magnet attraction is sufficient to buckle the elastic matrix and the two-dimensional arrangement undergoes a coupled twist–buckle behaviour. The unique properties of this structure, specifically the rapid switching between the trivial undeformed and buckled states, and its repeatability, suggest the scope for a magneto-elastic actuator with a number of practical applications.

In the above, applications of specific importance are the relative magnetic and elastic strengths of the material, magneto-elastic anisotropy and the boundary conditions. Here, we address these three aspects in order to generate a fundamental understanding into nonlinear elastic deformations induced in a magneto-elastic beam. Motivated by the experiments in the two-dimensional magneto-elastic array of Tipton *et al*. [6], we study the simplest element of the configuration that can become bistable: an elastic beam, which is held vertically by a clamp at its upper end and has a single neodymium magnet attached at its free end. This beam buckled above a critical value of an externally applied, uniform, anti-aligned, magnetic field. Further control of the buckling was exploited by introducing a second attracting neodymium magnet at controlled distances from the end of the beam, and this was found to enrich the bifurcation structure. In §2, we introduce the experimental configuration and the materials and methods used, and the theoretical model is developed in §3. The experimental results for different magnet configurations are examined in §4 in comparison with those in the theoretical model. The nonlinearities in the structure are found to have qualitative similarities with other nonlinear systems, and in §5, we exploit insights from studies on bounded Taylor–Couette flows and find the quartic bifurcation that corresponds to the change in bifurcation type.

## 2. Experiments

### (a) General configuration

In figure 1, we show a schematic of the experimental arrangement used to analyse the magneto-elastic buckling of the beam. The beam was clamped rigidly at the fixed end and held vertically. The free end of the beam had a small permanent magnet (*m*_{1}) attached to it, with the magnetic moment-oriented tangential to the beam tip. The arrangement was placed in a pair of large Helmholtz coils, for a uniform magnetic field (*B*_{c}) aligned anti-parallel to the magnetic moment orientation of *m*_{1}. The current through the coils was varied quasi-statically, and the deformation of the beam was monitored. For small fields, elastic forces dominate, and the beam remained undeformed. As the applied field was increased, magnetic forces began to grow, and above a critical field, the beam buckled as a result of the magnetic torque acting at the free end of the beam. This sequence of events is illustrated in figure 2.

Next, another neodymium magnet (*m*_{f}) was introduced into the set-up and positioned a vertical distance *R*_{o} below the beam magnet as shown in figure 1. The axes of the magnets were aligned with each other and with the longitudinal axis of the undeformed beam in order that the experiment commenced such that the poles of both magnets were aligned with each other. Magnet *m*_{f} was placed on a vertical travelling stage, the height of which was varied and the buckling response was monitored.

### (b) Material and methods

In the experimental configuration developed in §2*a*, first, an elastic sheet of required thickness was moulded from a silicone rubber mixture, EPS580 (supplied by ACC Silicones) that had a quoted Young's modulus of 1.4 MPa at 23^{°}C. The sheet was manufactured by pouring a two part fluid mixture into a custom-built rectangular mould. A beam of dimensions (18.3±0.1) mm×(3.60±0.07) mm×(1.18±0.03) mm was carved out of the sheet. This two-step process ensured the beam properties were uniform to a good approximation.

A permanent magnet was attached to the end of the beam by a small bead of glue, and an identical magnet was mounted on the travelling stage with the alignment such that the poles of both magnetic were aligned. The neodymium magnets were sourced from Yunsheng (USA) and were selected for their magnetic hardness (specifications indicate a magnetic remanence of 1.19±0.05 T and coercivity of ≤867 kA m^{−1}). These cylindrical magnets had a diameter 2.50±0.01 mm and length 3.00±0.01 mm. The magnetic moment was estimated to be *m*=0.0140±0.0006 A m^{2}, and was effectively independent of the applied magnetic field in the range of field strengths used.

The uniform field was provided by a GMW 5451 uniform field electromagnet, driven by a regulated power supply. The Helmholtz coils were 300 mm in diameter, and the field was measured to be uniform to better than 1 per cent.

## 3. Theory

Here, we develop a model of the system described in §2. The magnets are deemed to be point dipoles, the applied field is assumed uniform and the beam is modelled as an Euler–Bernoulli beam [13]. The magneto-elastic model is developed as follows: in §3*a*, we first develop the general equations for a nonlinear beam bending under the effect of distributed and discrete forces and torques. In §3*b*, magnetic interactions between a point dipole mounted at the beam tip and an external uniform repelling field is modelled; in §3*c*, the model is extended to include the attraction between two magnetic dipoles a prescribed distance apart.

### (a) Elastic beam deformation model

The elastic beam is modelled as a nonlinear beam of length *L*, stiffness *EI*, density *ρ*_{b} and cross-sectional area *A*, which deforms as the result of an applied external magnetic field (figure 3). We apply elastica theory [14] for an inextensible beam where the natural variable to express beam deformations is the tangent angle *θ*. Thus, a point on the beam, identified by its Lagrangian coordinate *s* (or position vector **r**=*x*(*s*)**e**_{1}+*y*(*s*)**e**_{2}), makes an angle with the global **e**_{1}-axis and the local tangent basis {** τ**(

*s*),

**n**(

*s*)} at

*s*. With this definition, the beam naturally satisfies the inextensibility condition:

**r**′=

**(where (.)′≡d(.)/ d**

*τ**s*). In the experiments, the external field strength was varied quasi-statically, correspondingly transient effects associated with deforming the beam from one state to another were allowed to die out before the next deformation was applied. Thus, in the model, we neglect transient and other dynamic effects.

The equations of equilibrium for an elastic beam subjected to an external force and torque distribution, **f**^{e} and **l**^{e}, respectively, are
3.1and
3.2The internal resistance **N** has the tension and shear force as tangential and normal components, respectively; the bending moment is given by the standard relation, **M**=*EIθ*′(*s*)**e**_{3}. To establish the contributions to the force and torque in the above expression, we next consider the key sources. The dominant contribution is from the magnetic effects, acting on the beam through the interactions between the tip magnet and the external magnetic field. Additional effects arise from the weights of the beam and magnet. The beam density is assumed to be uniform and can be accounted as a force distribution (*ρ*_{b}*Ag***e**_{1}), whereas the magnet's weight is concentrated at the tip of the beam. Observing that the forces and torques associated with the magnet are concentrated at the beam tip, these can be conveniently included as boundary conditions at the beam free end
3.3where *w*_{o} is the mass of the magnet and **F**_{m} and **L**_{m} are the magnetic force and torque, respectively. Note that we will derive expressions for these magnetic interactions in parts (*b*) and (*c*) of this section.

Next, we integrate (3.1) from *L* to *s*, apply the boundary condition (3.3) and obtain an expression for the internal resistance,
3.4Substituting this expression into equation (3.2), we obtain the magneto-elastic equation
3.5Non-dimensionalization with respect to the characteristic length (*L*), mass (*ρ*_{b}*L*^{3}) and time (*L*^{2}[*ρ*_{b}*L*/*EI*]^{1/2}) yields the following non-dimensional variables: , , , . For convenience, we drop the overbars to obtain the complete non-dimensional magneto-elastic equation with boundary conditions:
3.6Here, the orientation of the magnet *β* is governed by the beam tangent angle through the relation *β*=*θ*(1).

### (b) The uniform field case

In the arrangement shown in figure 2, the cylindrical magnet is fixed at the end of the flexible beam. We model the magnet as a dipole of magnetic strength, **m**_{1}=*m*_{1}**b**_{1}, where the local magnetic basis {**b**_{1},**b**_{2}} is assumed to be aligned with the beam axis, so **b**_{1}=** τ**(1). The magnetic and elastic properties of the system can be combined to define a dimensionless parameter that represents the relative strength of the magnetic to elastic forces,
3.7We term this parameter the magneto-elastic number, noting that the magnetic contribution represents the interactions between the magnetic dipole and the external field.

Next, we observe that the magnet mounted on the beam experiences no force in a uniform external field, however, it is subject to a net torque. For the external field aligned with the undeformed beam axis, (**B**_{c}=−*B*_{c}**e**_{1}), the dimensional magnetic torque experienced by magnet *m*_{1} is given by **L**_{m}=**m**_{1}∧**B**_{c}. Thus, in dimensionless terms, we have **f**_{m}=0 and
3.8Substituting (3.8) into (3.6) yields the magneto-elastic equation for a dipole in a uniform field:
3.9The nonlinear nature of the model can be clearly discerned from (3.9), noting that for small deformations and the equations reduce to the familiar linear Euler-beam equation [13], the behaviour of which is examined in specific detail in §4*b*.

### (c) Influence of dipole–dipole attraction

Next, we introduce an attracting permanent magnet (strength **m**_{f}=*m*_{f}**e**_{1}), a distance *r*_{o} from the undeformed beam tip (figure 3). Using the law of magneto-static interactions [15], the magnetic flux density, experienced at position vector **r** from the dipole, is **B**=*μ*_{0}/4*π***r**^{3}(3**p**(**p**⋅**m**_{f})−**m**_{f}), where **p**=**r**/*r*. The vector distance between the two magnets may be defined in terms of angle *φ* made by the line of action **p** with the fixed dipole *m*_{f} (see figure 3). Then, as , the flux density experienced by magnet *m*_{1} is
3.10The dipole introduces spatial non-uniformity in the external field, and the beam magnet (*m*_{1}) experiences a force and torque . Non-dimensionalization yields a second magneto-elastic number for the attracting pair of dipoles,
3.11which is essentially a measure of the strength of attraction between the dipole pair relative to the elastic resistance of the beam. The non-dimensionalized expressions for force and torque can now be determined and these are given by
3.12and
3.13where
3.14and
3.15We now collate terms to obtain an expression for the force and torque experienced by the beam–magnet combination. The magnetic interaction terms comprise magnetic repulsion of a dipole in an external field, given by (3.8), and attraction between two dipoles an initial distance *r*_{o} apart, given by (3.12)–(3.15). Substituting these terms into (3.6), we obtain the equilibrium equation for the magneto-elastic beam:
3.16For small deformations, the assumptions: *r*_{m}=*r*_{o} and *φ*=*π*, and the above nonlinear equation reduces to the linear Euler-beam equation.

Two points need a little consideration before we proceed. First, the advantage of defining a magneto-elastic number is that not only do we collate several quantities in the model into a single dimensionless parameter, we can also compare the effect of the different magnetic interactions on the beam response. For example, in the experiments, we have , whereas critical above which the beam buckles supercritically is ≈0.5. Thus, the magnetic attraction is almost an order of magnitude less than the critical repulsion strength; nonetheless, it is responsible for the interesting physical behaviour observed experimentally and examined in §4. Second, the dipole approximation limits the magnets to a certain inter-magnet spacing, as indicated in the recent experiments of Vokoun *et al*. [16]. In §4, we will determine the limits of this approximation in the context of our elastic beam.

## 4. Results

### (a) Supercritical bifurcation: experiments versus the perfect model

The system of nonlinear equations developed in §3 was solved by numerical continuation using AUTO 07p [17] that revealed the bifurcation structure. In figure 4, we plot the numerical solutions of the system of equations (3.9) where the experimental configuration corresponds to beam and magnet weight coefficients *g*_{m}=0.52, *g*_{b}=0.49, respectively. Assuming the system has perfect symmetry we plot with as the control parameter. A supercritical pitchfork bifurcation is found such that above the critical magnetic field , there is an exchange of stability between the trivial solution and the pair of solutions corresponding to the buckled states.

Imperfections are inevitably present in the experiments and the result of cumulative effects of small misalignments of the field, magnets and beam, as well as non-uniformities within the beam. These result in the buckling of a beam in a preferred direction, and for this configuration the preferred direction was along −**e**_{2}, which is to the left-hand side in figure 1. In the experiments, the field was applied to an initially nominally straight beam with its axis aligned with the vertical **e**_{1}. The applied field, ranging from 0 to 4.25 mT, was varied quasi-statically in steps of 0.25 mT. At each step, the field was held constant for 30 s to eliminate transient effects, and the beam was photographed; four such cases are reproduced in figure 2.

The photographs were analysed using MATLAB's image processing toolbox, and the beam coordinates were extracted for each field strength. In figure 4, we superimpose the experimentally measured results for *y*_{cg}, and we see that above the critical field strength the beam buckles in its preferred direction, where the chosen branch depends on system imperfections; the unfavoured branch is acquired by giving the beam an initial push to the right-hand side. When the field is decreased the deflections reduce, and we see an identical response for increasing and decreasing fields, indicating the supercritical nature of the nonlinear bifurcation. This supercriticality was confirmed numerically; however, the discrepancy between experiments and the model is apparent; in §4*b* we explain the origin of the discrepancy.

### (b) Understanding imperfections with the linear model

The assumption of perfect symmetry in the theoretical model developed in §3 is now investigated to reconcile differences between experimental and numerical results of figure 4. We note that this mismatch has two main effects: (i) the existence of a preferred branch and (ii) an over-prediction of the critical magneto-elastic number. Noting that these inconsistencies may be first examined from the linearized equations, we next analyse the onset of the instability. Applying the small deflection assumption, , the nonlinear model (3.9) simplifies to give
4.1that may be reformulated to give the Airy equation [18]. To gain an intuitive understanding, we further simplify the analysis by neglecting *g*_{b}, therefore (4.1) reduces to a linear eigenvalue problem that has the solution , and the free boundary condition gives a critical magneto-elastic number: . In figure 5, the predicted critical magneto-elastic number (solid curve) is calculated for the experimentally relevant range (*g*_{m}∈[0,1]). The linearized solution with *g*_{b}=0.49 (dotted curve) confirms the result from the nonlinear computation.

Examination of the experimental set-up suggests that the most likely cause for loss of symmetry is a violation of the assumption of perfect alignment between the undeformed beam axis and polarity axis of the beam magnet. Noting that the experiments show that the two branches are virtually symmetric, a very small misalignment between axes is sufficient to break the symmetry of the configuration. This misalignment, *β*_{0}, may now be construed as the imperfection parameter in the supercritical configuration. In terms of the over-prediction of the critical magneto-elastic number, we conjecture that differences between manufacturer specifications and the actual material properties, as well as non-uniformities in the beam could cumulatively contribute to reducing the critical magneto-elastic number by a factor of almost two. We therefore introduce an empirical material correction factor, *α*, to account for these terms.

Combining these factors, we now adjust the model developed in §3 to include the misalignment correction, *θ*(1)+*β*_{0}, and the material correction factor, *α*, and for *g*_{b}=0, we get a closed form expression for the critical magneto-elastic number,
4.2We adjust these parameters in order to match with the experiments, and find that a misalignment of *β*_{0}=2.8^{°} and material correction factor of *α*=0.48 allow us to recover the experimentally measured values for this critical magneto-elastic number. Intriguingly, when the beam weight is included (obtained by solving (4.1) numerically) the critical condition becomes virtually independent of *g*_{m}. Henceforth, we use the adjusted model,
4.3for the remainder of the computations.

### (c) Supercritical bifurcation and the adjusted model

Introducing the adjustments (4.3) in the nonlinear model, we now compare the results obtained using the adjusted numerical model with the experimental data, and from figure 6, we see that the agreement extends over all of the nonlinear regime explored experimentally. The model accurately captures the nonlinear beam deformation for high magnetic field strengths for both the favoured and unfavoured branches. Furthermore, the numerical continuation algorithm is able to identify both the stable (solid curves) and unstable (dashed curves) branches of the pitchfork bifurcation. It is worth emphasizing that the predicted misalignment is less than 3^{°}; yet, this small imperfection is sufficient to yield a significantly disconnected pitchfork bifurcation. It is worth observing that an imperfection that induces asymmetry is perfectly consistent with an Euler buckling beam, interestingly in this model, an imperfection in the tip angle is sufficient to produce the necessary disconnection in the bifurcation curve.

### (d) Established model for bifurcation unfolding and cusp coalescence

When an attracting permanent magnet was introduced into the experiments at a controlled distance from the beam free end (as shown in the schematic in figure 3) we found a codimension-2 bifurcation. This codimension-2 bifurcation corresponds to the switch in primary bifurcation from super- to subcritical, and is observed by varying the control parameter, *r*_{o} [19]. An example of a subcritical bifurcation in these experiments is seen in figure 7, the bistability over a range of is clearly seen for a pole-to-pole distance, *r*_{o}=0.88±0.07 (the error corresponds to the uncertainty arising from the finite dimensions of the magnetic ‘dipoles’). In figure 7*a*, the direction of change in field strength is illustrated by arrows: the external field is increased quasi-statically from *A* to *B* above which the beam buckles, and we see an exchange of stability from the trivial state to the favoured lower branch as the beam jumps to point *C*. The beam continues to deform smoothly to *D* as the field is increased further. On reducing the field strength, the beam deformation decreases gradually as it returns to its trivial state, *E*, at a field strength much lower than its critical value for an increasing field (at *B*); in other words, we see significant hysteresis. Here, nominally identical magnets have been used and for the experimental configuration examined, the attracting magneto-elastic number is calculated to be .

In figure 7*b*, superimposed on the experimental results is the outcome from the numerical model (3.16) using the adjusted parameters indicated in (4.3). For an inter-magnet separation distance of *r*_{o}=0.7, the model reproduces the experimentally observed subcritical pitchfork bifurcation, with its associated hysteresis. The unfolding of the bifurcation with the stable (solid curves) and unstable (dashed curves) branches is clearly visible, and the correspondence between experiments and numerical results is in excellent agreement.

The results in figure 7*b* indicate that the codimension-2 behaviour is revealed when the attracting magnets are brought sufficiently close to each other. The details are uncovered by varying the distance between magnets, *r*_{o}, and seeking the corresponding critical magneto-elastic number, . The results are plotted in figure 8, and we find that above a critical magnetic separation distance (*r*_{C2}≈1.35), there exists a unique critical magnetic strength, which corresponds to the supercritical bifurcation. This critical distance *r*_{C2} corresponds to the codimension-2 bifurcation point. For smaller separation distances than *r*_{C2} bistability in the form of a subcritical bifurcation is observed, yielding two values for . The experimental results, superimposed on figure 8, show excellent agreement with the numerics. However, we see larger discrepancies between experiments and numerics, as the inter-magnet separation is reduced. The numerical results show that when magnets are brought sufficiently close (*r*_{o}<0.7) higher-order beam buckling modes are observed. In order to understand the origin of the discrepancies and the higher-order beam buckling deformations, in §5, we attempt to uncover the fundamental nonlinear behaviour.

## 5. Discussion: interpreting the nonlinear structure

The experiments indicate that the distance between magnets is a crucial parameter that affects a change in the nonlinear beam response yielding a switch between supercritical and subcritical symmetry breaking bifurcations. We have seen that as the external repelling field strength is increased; there is an exchange of stability between the trivial and deformed states, and depending upon the separation distance from the additional small magnet, there will be either one, three or five equilibrium points. Qualitatively, similar phenomenon was reported by Moon & Holmes [7] with a ferromagnetic beam in close proximity to two magnets of equal strength, placed equidistant on either side of the beam, a controlled distance away from its tip. In this work, the authors used a qualitative energy method to show a ‘butterfly cusp’ that explains the existence of experimentally observed multiple equilibria.

Although the earlier-mentioned work gives some insights into the nonlinearities in the system, for a deeper understanding of the unfolding of the bifurcation structure, we examined studies on a canonical nonlinear system in bounded Taylor–Couette flows in small aspect ratio cylinders, where the aspect ratio corresponds to the ratio of cylinder height to radius [20]. The most interesting result from this work is that as the aspect ratio is increased from very small values a transition from a symmetric two-cell state to a pair of single-cell states is observed above a critical Reynolds number. The nonlinear analysis revealed that this exchange of stability between symmetric and asymmetric states corresponds to a switch from a super- to subcritical bifurcation through a quartic bifurcation [21,22].

With this insight, we rerun the numerics, specifically seeking bifurcation and limit points and transitions in the beam buckling state. These critical points are now plotted in figure 9*a* with the experimental data from figure 8, where 1/*r*_{o} is the control parameter. The segments of the curves are labelled to indicate the different bifurcation regimes. Curve *AB* is the line of supercritical bifurcation points corresponding to buckling of the beam, where the buckling mode is indicated by the snapshots in figure 2. In figure 9*b*, we portray the two beam buckling states that have been observed, and define these as modes-1 and -2. As , the critical magneto-elastic number asymptotes to a constant value (). As *r*_{o} is decreased, increases and at the quartic bifurcation point *B*, the supercritical bifurcation switches to subcritical and the curve splits into a line of bifurcation points (*BE*) and limit points (*BF*). The hysteretic behaviour associated with the subcritical bifurcation has been portrayed in figure 7, and at point *C*, the higher-order (mode-2) buckling mode is observed. Experimentally, this state is achieved when the two magnets are brought very close to each other, and the end of the beam is effectively pinned as a result of dipole–dipole attraction. In figure 9*d*, we show a snapshot of the deformed beam indicating this second mode.

Decreasing *r*_{o} further, we see a continuous connection for mode-2 buckling, and a line of symmetry breaking bifurcation points corresponding to curve *CD*. We find that in this range of *r*_{o} the fundamental beam mode is truly disconnected, and experimentally, buckling is possible only by forcing the beam to deform. Indeed, this would explain the large error we see between experiments and theory in this region, and is a consequence of the delicate procedure associated with locating the bifurcation points in this region. Curve *BF* indicates the limit points of stability for mode-1 buckling, and the experimental measurements are in good agreement with the model. It is significantly easier to identify the limit points experimentally, which suggests that the theoretical model is valid in this range of *r*_{o},

The insights gained from the numerical investigation, led us to seek the second mode from the experiments, and the data point corresponding to this experiment is superimposed on figure 9*a*. Here, we must note that in the experiments this mode is quite elusive, and we are not convinced that this is the bifurcation point. Reducing *r*_{o} further we proceed along curve *DE*, which yields an increasingly complex bifurcation structure, and the numerical computations become increasingly difficult. We also start to run into the limits of validity of the dipole assumptions, so we have confined the model to the range, .

We now have the key elements to piece together a bifurcation sequence for an experimental configuration that includes imperfections. As noted by Mullin *et al*. [23], and sketched in figure 10, imperfection induced symmetry breaking pitchfork bifurcations correspond to two asymmetric states with a small disconnection. The resulting impact on bifurcation structure is realized by a secondary pitchfork bifurcation associated with the symmetric state, which induces the observed disconnection in the supercritical and subcritical bifurcation. For sufficiently small *r*_{o}, a pitchfork bifurcation corresponding to a pair of pinned buckled (or mode-2) states emerges at finite , and the subcritical and secondary bifurcations coalesce, leaving a pair of disconnected branches. Thus, the observed transition from a mode-1 to mode-2 buckling arises resulting from an exchange of stability between the buckling modes, and can be explained through the existence of successive pitchfork bifurcations.

It is worth bearing in mind that here we have chosen to focus on the evolution of a pitchfork from super- to subcritical, which is only part of the bifurcation sequence found in small aspect ratio Taylor–Coutte flow [23]. Similarly, we have seen that the bifurcation sequence for the magneto-elastic system becomes more complex as the distance between magnets is reduced further than those presented in this work. Nevertheless, for purposes of clarity, we have chosen to focus on the tractable aspects of the nonlinear regime.

We note that the theory underlying disconnected bifurcations may be studied formally applying singularity theory [24]. Indeed, in their review on bifurcation problems, Cliffe *et al*. [25] demonstrate how the theory may be applied in bounded Taylor–Couette flow. In applying their insights to our system, we have been able to explain the emergence of higher modes of buckling of the beam, induced by an effective pinning of the free end as the attracting magnets are brought sufficiently close to each other. It is especially satisfying that based on these insights, which were confirmed by the numerical model, we sought and found the higher buckling mode experimentally. Thus, in probing the nonlinearities, we believe we have a better understanding of the aspects of this simple magneto-elastic system that results in multiple buckling states.

## 6. Conclusions

A magneto-elastic system was studied experimentally, based on which a theoretical model was developed. The simplest magneto-elastic system consists of an elastic beam with a permanent magnet mounted on its tip, placed in an external repelling field, with an attracting magnet positioned a finite distance away. The model accounts for localized attracting and uniform repelling fields; far enough away from the attracting field, the beam undergoes classical Euler buckling through a supercritical bifurcation. The linearized model was used to show how imperfections in the system, combined with the tension induced in the beam as a result of the weight of the beam and magnet, can destabilize the structure and cause it to buckle at smaller magnetic fields than predicted for a perfect system. As the magnets are brought closer to each other, the beam undergoes a switch from a supercritical to subcritical bifurcation. The model is able to replicate this observed sequence of events. Using insights from a canonical nonlinear system of a bounded Taylor–Couette flow, we sought out and found higher modes. We then used the model to understand how nonlinear mode-switching may be induced during buckling. The possibility for triggering mode-switching in a multiple-equilibrium configuration, induced by controlling the external field strength, introduces the possibility of controlled pattern switching. Thus, remote actuation makes such magneto-elastic systems especially attractive from an applications perspective.

## Acknowledgements

The authors thank Finn Box for his invaluable help in the laboratory.

- Received February 15, 2013.
- Accepted April 4, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.