STATIC AND DYNAMIC ANALYSIS OF A BISTABLE PLATE FOR APPLICATION IN MORPHING STRUCTURES

STATIC AND DYNAMIC ANALYSIS OF A BISTABLE PLATE FOR APPLICATION IN MORPHING STRUCTURES A. Carrella 1, F. Mattioni 1, A.A. Diaz 1, M.I. Friswell 1, D.J. Wagg 1 and P.M. Weaver 1 1 Department of Aerospace Engineering University of Bristol, Bristol BS8 1TR, UK e-mail: A. Carrella@bristol.ac.uk Keywords: Morphing, bistable plate, Duffing ABSTRACT The need for better aircraft performance is increasingly prompting designers towards the realisation of morphing or shape-adaptable structural systems. One way to achieve this is to employ multistable composites elements. In this paper a square, curved bistable plate is studied. The finite element analysis provides the load-deflection characteristic of the bistable plate. A cubic expression is then fitted to this curve, and the equation of motion of the plate is reduced to a single-degree-of-freedom (SDOF) system which has the form of the Duffing equation for a double-well potential system. The dynamics of the plate are investigated using bifurcation diagrams and shows that the qualitative behaviour given by the measured response is predicted well by the simple SDOF model. 1. INTRODUCTION Morphing or shape-adaptable structural systems are increasingly being considered as a solution to the always present need for better aircraft performance. Such systems should simultaneously fulfil the contradictory requirements of flexibility and stiffness. So far the solutions adopted consist of complex assemblies of rigid bodies hinged to the main structure and actuated. To enhance the performance of the aircraft as a system, multistable composites could provide an interesting alternative to traditional designs, thanks to their multiple equilibrium configurations. Unsymmetric laminates exhibit out-of-plane displacements at room temperature even if cured flat. These displacements are caused by residual stress fields induced during the cool-down process between the highest curing temperature ( 16 C) and room temperature ( 2 C). The thermal stresses are mainly caused by the mismatch of the coefficients of thermal expansion of constituent layers [1 5]. The main feature of a bistable structure is the snap-through mechanism which marks the passage from one stable position to the other. It is clear that an analytical dynamic analysis of the bistable plate would reveal very complicated behaviour. However, the analysis of an accurate Finite Element (FE) model would have a high computational cost. Therefore, in this paper it is proposed to model the dynamics of the snap-through mechanism

with a SDOF system, as suggested by the work of Mattioni et al.[6] and Arrieta et al. [7]. The change from one stable state to the other can be observed from the static force-displacement characteristic which can been readily obtained with a commercially available Finite Element (FE) package (in this case ABAQUS). The curve shown in Fig. 1 (dashed line) is obtained by pinning the four corners of the plate and applying a quasi-static load in the centre. It shows a displacement range in which the stiffness is negative, and where an oscillatory dynamic response is not possible. It is likely that the plate is able to exhibit a strongly nonlinear dynamic response with the potential for chaotic behaviour. Performing a numerical nonlinear analysis of the FE model with all its degrees of freedom would be computationally expensive. In this paper the response of a square, curved bistable plate is analysed by focusing only on the snap-through mechanism. This study assumes the plate can be modelled with a SDOF model. This can be done if the motion of the plate is referred to the motion of the application point of the load and only one mode of deformation is considered. The restoring force is obtained numerically by FE analysis (shown as the dashed line in Fig. 1) and can be approximated by a cubic polynomial with a negative linear coefficient and a positive cubic term. In doing so, the dynamics of the snap mechanism is modelled by a system with a double-well potential. This is mathematically expressed by the Duffing equation which has been extensively studied [8 1] and it is known to exhibit chaotic behaviour. By writing the equation of motion in non-dimensional form, the system dynamics with sinusoidal forcing are determined by two parameters of the excitation force, namely the amplitude and the frequency, and one parameter of the system, namely the damping coefficient. As expected, the system behaves linearly for small amplitudes of excitation, but gives chaotic regions as the amplitude is increased. Finally, experimental results are shown, and a qualitative match between the SDOF model predictions and the measurements is observed. 2. STATIC ANALYSIS The dashed line in Fig. 1 represents the force-deflection curve of a square plate (3 3 cm) with 8-ply asymmetric laminate [ 4 9 4 ] T obtained with finite element analysis. It is possible to approximate the numerical load-deflection characteristic with the analytical expression f k (y) = k 1 y + k 3 y 3 (1) where the coefficients k 1 and k 3 can be determined by ensuring the function (1) is symmetric and, for example, the values of the maximum and minimum of the analytical function are the same as for the numerical curve. In the example shown the coefficients were estimated to be k 1 1.667 kn/m and k 3 1.2 MN/m 3. The analytical curve, Eqn. (1) is plotted as solid line in Fig. 1. It should be noted that the slope of the curve between the peak and the trough is negative, which means that the stiffness is negative. A SDOF mechanical model that has a similar restoring force is shown in Fig. 2. Moon and Carrella et al. [1, 11] showed that the restoring force for this system is h 2 + a 2 f = 2k o (h x) (h x) 2 + a 1 (2) 2 which, by introducing the variable y = h x and expanding in Taylor s series about y =, can be approximated by the cubic expression of Eqn. (1).

2 FEA Analytical 1 force [N] 1 2 4 2 2 4 displacement [mm] Figure 1: Force-deflection characteristic of a bistable square plate obtained with finite element analysis (dashed) and the fitted analytical approximation (solid). f k o k o x h a y Figure 2. Single-degree-of-freedom mechanical model for the bistable plate. 3. DYNAMIC ANALYSIS Having approximated the restoring force of the bistable plate with a cubic polynomial and introduced a SDOF mechanical model of the snap mechanism, its dynamics can be described by the nonlinear equation of motion m m ÿ + c ẏ k 1 y + k 3 y 3 = F cos(ω t) (3) where m m is an equivalent mass, that is a coefficient that expresses only that part of the mass which contributes to the mode considered, c is the viscous damping coefficient and k 1 and k 3 are the coefficients determined above. Eqn. (3) can be written in non-dimensional form as where ŷ + h ŷ ŷ + α ŷ 3 = Â cos(ωτ) (4)

h = c m mω n ω 2 n = k 1 m m α = k 3 d 2 k 1  = F k 1 d Ω = ω ω n τ = ω n t ŷ = ÿ ω 2 n d ŷ = ẏ ω n d ŷ = y/d and where the denotes differentiation with respect to the non-dimensional time τ and d is an opportune reference length of the system. In particular, if this length is chosen 1 so that then Eqn. (4) can be written as d = k1 k 3 α = 1, (5) ŷ + h ŷ ŷ + ŷ 3 =  cos(ωτ) (6) The Duffing equation, Eqn. (6), is often used to demonstrate nonlinear dynamics of systems with a double-well potential and has been extensively studied [8, 1, 12 15]. It is well known that a system with a double well potential can exhibit chaotic behaviour, and here we wish to predict the onset and the boundaries of the chaotic region. Moon [1] illustrated different tests to ascertain the presence of a chaotic region, such as Poincare maps, spectral analysis, and the determination of Lyapunov coefficients. In this paper, the dynamic behaviour of the system is characterised with bifurcation diagrams. A bifurcation plot is obtained by changing one of the system parameters and keeping the others fixed. Because there are two main parameters, the forcing amplitude  and the forcing frequency Ω, both of these cases will be considered. For all the numerical simulations presented, the damping coefficient is assumed to be h =.3, which corresponds to a damping ratio of 15% and the initial conditions are ŷ = ˆẏ =. The equation of motion (6) is solved numerically at discrete times. For this study the time is varied between and 3 times the period T = 2 π/ω with one output sample at every period, i.e. t = : T : 3 T. To ensure that the initial transient has decayed, the first 1 time periods are ignored. The bifurcation diagram shows the amplitude of the response Ŷ at each time sample, and is equivalent to a projection of the Poincare map of the system. After the transients have decayed, a periodic system response at the forcing frequency (period-1) gives a single response point on the bifurcation diagram. If the response period is twice that of the excitation (period-2) the bifurcation diagram has two points. A chaotic motion results in a large number of response points. 3.1 Bifurcation diagram for the forcing amplitude If the excitation frequency is kept constant, it is possible to vary the amplitude of the force to study its effect on the system s dynamics. A bifurcation diagram for the forcing amplitude parameter is shown in Fig. 3. Here, the forcing frequency is Ω = 1.2, whilst the forcing amplitude  changes between.2 and.8 with a step of  =.2. The value of the frequency has been chosen because in reference [8] the same case is considered and thus the simulations can be verified. Fig. 3 shows that there is a forcing amplitude range,  =.27.73, within which transient chaos alternates with periodic solutions, and this agrees with the discussion presented in reference [8]. It can be concluded that, for a forcing frequency of Ω = 1.2, period- 1 solutions do not exist if.27 <  <.73. 1 Note that setting α = 1 is equivalent to setting the stable position as the reference length, i.e. the distance of the equilibrium position from the origin of the reference system.

Ŷ 1.5 1.5.5 1.27 1.5.2.3.4.5.6.7.8  Figure 3: Bifurcation diagram of the response amplitude Ŷ for the amplitude of excitation force of the Duffing equation (6). The excitation frequency is Ω = 1.2.73 Fig. 4 shows the bifurcation diagram for a frequency of excitation significantly smaller than the linear resonance frequency, Ω =.3. Even for such a low frequency of excitation, chaotic regions exist if the forcing amplitude is high. 2 Ŷ 1.5 1.5.5 1.2.3.4.5.6.7.8  Figure 4: Bifurcation diagram of the response amplitude Ŷ for the amplitude of excitation force  of the Duffing equation (6). The excitation frequency is Ω =.3

3.2 Bifurcation diagram for the excitation frequency It can be argued that a more interesting bifurcation diagram, from a practical point of view, is one in which the forcing amplitude is kept constant whilst the frequency is varied. This is because, during measurements, the input signal generally has a fixed amplitude and the frequency is changed. Fig. 5 shows the case with  =.1, where the frequency, Ω, changes from.5 to 1.5 with a step of Ω =.5. The figure shows that when the amplitude of the applied force is small the system oscillates periodically at the excitation frequency within one potential well (this is denoted by the amplitude of the response Ŷ ). Furthermore, the maximum amplitude occurs at Ω > 1 which denotes a hardening system. When the amplitude of excitation is increased 1.2 1.15 1.1 1.5 Ŷ 1.95.9.85.5.75 1 1.25 Ω 1.5 Figure 5: Bifurcation diagram of the response amplitude Ŷ for the excitation frequency Ω of the Duffing equation (6). The amplitude of excitation is  =.1. cross well motion is expected to occur. This can be observed in Fig. 6 for  =.4. 4. EXPERIMENTAL RESULTS In this section, experimental results are presented. It is shown that the snap-mechanism of the bistable, curved plate is well described by the SDOF mechanical model. A 3 by 3 mm carbon-fibre epoxy [ 4 9 4 ] T square bi-stable laminate (with a mass of.128 kg) is used as the experimental specimen. As a result of the curing process, the laminate is curved as shown in Fig. 7. The structure was attached to a Ling shaker in order to induce vibration in the plate. A differential laser vibrometer measured the relative displacement between the centre and a given point on the laminate. The experimental arrangement was mounted on a steel table with a large mass in order to ensure there is no interaction between the plate and its surroundings, as shown in Fig. 7. Note that the plate has the same geometry and material properties as the FE model discussed above. However, the force at which the plate changed from one stable state to the other was experimentally determined to be F snap 12 N. Furthermore, the displacement of the point

1.5 1 Ŷ.5.5 1 1.5.5.75 1 1.25 Ω 1.5 Figure 6: Bifurcation diagram of the response amplitude Ŷ for the excitation frequency Ω of the Duffing equation (6). The amplitude of excitation is  =.4 Figure 7. Experimental Assembly. Ling shaker V45, vibrometer OFV-552 of application of the force (centre) was measured to be 3.2 cm (i.e. the reference length is d = 1.6 cm). Using this data, a cubic fit was performed to approximate the load-deflection characteristic, Eqn. (1). The coefficients of the polynomial were estimated to be k 1 1.95 kn/m and k 3 7.62 MN/m 3. In order to determine the value of the equivalent mass for the SDOF model, the plate was excited with a harmonic force of low amplitude about one of its stable states. Fig. 8(a) shows the measured response of the plate for a fixed amplitude of excitation as the frequency is varied. The experimental frequency response of the plate was obtained by stroboscopic sampling of the time series for the measured displacement data. This procedure is a conventional analogue to the Poincare map technique, [13]; further details on the experimental procedure can be found in [7]. It can be seen that the response is approximatively linear with a resonance frequency of 3 Hz. In order to match this frequency, in the following analysis the value for the equivalent mass is set to m m =.64 kg, which is approximately half of the total

(a) F =.5 N - measured.3 displacement Y [mm].2.1.1.2 1 15 2 25 3 35 4 45 frequency [Hz] (b) F =.5 N - analytical model Figure 8: (a) Measured response and (b) numerical simulation of the SDOF mechanical model of the experimental plate for an amplitude of the excitation force of.5 N. mass of the plate. Note that, at this stage of major interest is the frequency, e.g. resonance onset of chaos, rather than the amplitude. Thus, the damping ratio, which needs to be measured, is taken from the numerical examples illustrated earlier, i.e. ζ = 15%. Fig. 8(b) shows the bifurcation diagram obtained by numerically solving the equation of

motion given in the previous section when the amplitude of the excitation force was.5 N, i.e. the nondimensional amplitude was  =.16, and initial conditions were y = 1 and ẏ =. The y-axis gives the displacement from the stable position, and the x-axis gives the frequency of excitation. 1 (a) F = 6 N - measured displacement X [mm] 1 2 3 4 2 25 3 35 frequency [Hz] (b) F = 12 N - analytical model Figure 9: (a) Measured response for an amplitude of the excitation force of 6 N; (b) numerical simulation of the SDOF mechanical for an amplitude of the excitation force of 12 N. Fig. 9(a) shows the experimental frequency response when the amplitude of the excitation force is 6 N, and shows three different types of behaviour. Between 2-24 Hz the system behaves linearly, since there is a single point at each frequency, and hence the response is

periodic. In the frequency range 25-3 Hz, for each driving frequency the displacement of the reference point takes many different values, illustrating a chaotic behaviour. Finally, to the right of the chaotic range the system response is nonlinear, showing a period-2 response. Fig. 9(b) shows the results of the numerical solution of the equation of motion when the amplitude of the excitation force is approximately equal to the value of the measured static snapping load, F = 12 N (i.e. Â =.38) 5. CONCLUDING REMARKS To enhance the performance of the aircraft as a system, multistable composites could provide an interesting alternative to traditional designs, thanks to their multiple equilibrium configurations. In this paper a square composite bistable plate has been considered. The plate can be designed and studied using a finite element model, although a refined model requires a large number of the degrees of freedom. Although a static analysis for this model can be readily performed, a nonlinear dynamic analysis would prove computationally expensive. Therefore, a single degree of freedom model of the plate has been proposed. The dynamics of the plate has been reduced to Duffing s equation with a double potential well. This equation has been extensively studied and is known to exhibit chaotic behaviour under certain conditions. Experimental results have confirmed a region of chaos. Interestingly, the simple SDOF model has predicted, at least qualitatively, the dynamic response of the plate tested. Future work will focus on the design and manufacturing of a bistable plate, together with the test rig, to enable a quantitative validation of the proposed mathematical model. REFERENCES [1] M. L. Dano and M. W. Hyer. Thermally-induced deformation behavior of unsymmetric laminates. Int. J. Solids Struct., 35:211 212, 1998. [2] M. L. Dano M.W. Hyer. Snap-through of unsymmetric fiber-reinforced composite laminates. J. Composite Materials, 39:175 198, 22. [3] K. Iqbal and S. Pellegrino. Bi-stable composite shells. In 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference and Exhibit, Atlanta, GA, 3-6 April 2. [4] L. Kroll W. Hufenbach, M. Gude. Design of multistable composites for application in adaptive structures. Composites Structures, 55:319 327, 22. [5] C. S. Hong W. J. Jun. Effect of residual shear strain on the cured shape of unsymmetric cross-ply thin laminates. Composites Science and Technology, pages 55 67, 199. [6] F. Mattioni, P.M. Weaver, K. Potter, and M.I. Friswell. The analysis of cool-down and snap-through of cross-ply laminates used as multistable structures. In ABAQUS UK group conference, 26. [7] A.F. Arrieta, F. Mattioni, S.A. Neild, P.M. Weaver, D.J. Wagg, and K. Potter. Nonlinear dynamics of a bi-stable composite laminate plate with applications to adaptive structures. In 2nd European Conference for Aero-Space Sciences, 27.

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