Experimental investigation into dynamics of flat and ...past.isma-isaac.be › downloads › isma2012 › papers › isma... · Figure 1: Stable shapes of bistable plate (a) Flat

Experimental investigation into dynamics of flat andbistable plates

A. Shaw, S. A. Neild, D. J. Wagg, P. M. WeaverAdvanced Composites Centre for Science and Innovation (ACCIS)University of Bristol, Queen’s Building, Bristol, BS8 1TR, United Kingdome-mail: [email protected]

AbstractBistable plates have two stable states between which they may ‘snap’, and are attracting interest in the fieldof adaptive structures. This work presents results of an experimental investigation into the vibrations of bothflat and bistable composite plates with free edges, employing a novel video method to capture the full fieldof vibration in 3 dimensions. For flat plates, it is shown that deflection shapes of vibration can transformfrom the mode shape as amplitude is increased, and that at certain frequencies two deflection shapes canbe achieved depending on initial conditions. For bistable plates, it is noted that lower vibration modes canbe linear up to moderate amplitudes, whereas at high amplitude highly nonlinear ‘snapping’ oscillations areshown. We discuss these effects in terms of the anisotropic stiffness of CFRP, and the geometric couplingbetween doubly-curved out of plane deflection shapes and in-plane deformations.

1 Introduction

Multi-stable composite plates can occupy multiple different stable curved configurations, between whichthey may ‘snap’ when forced [1]. They are attracting considerable interest in the field of morphing structures,and for their potential use in actuators, due to their ability to form multiple shapes with no ongoing powerconsumption [2, 3]. An understanding of the dynamics of these plates will be vital to their implementation;for example a resonant response could lead to snapping, intentional or otherwise. A previous work by Arrietashowed that such plates can exhibit nonlinearity in their response, even at moderate amplitudes [4].

This work aimed to expand on the results of Arietta’s work, by gaining insight into the deflection shapesof the oscillations of bistable plates, and also comparing responses with those of flat plates to understandthe source of nonlinearity. It utilised a novel high-speed video camera technique developed by the authorsto capture their entire motions in 3D [5]. Square plates with bistability induced through thermal effects ofasymmetric crossply laminates are investigated, and compared with the nearest equivalent flat plate. Thelayups are varied so as to maintain constant bending and in-plane stiffnesses, whilst varying the couplingbetween these responses. Both small amplitude responses as well as those that include large amplitude,highly nonlinear motions that snap between stable states are shown.

It is found that in some cases of small-amplitude vibrations of bistable plates, the deflection varies along justone dimension of the plate, and a simple model that is directly analogous to classical flexural beam vibrationis applicable. Results are also presented that show snapping oscillations that exhibit a highly elegant period-trebled motion, demonstrating the possible routes into chaotic motion of the plates. These results suggestthat resonance could be used to actuate the plates between states, leading to an efficient switching strategy.

We see further nonlinear effects in the case of the flat plate, such as modeshapes changing with amplitude andmultiple deflection shapes at given frequency, and explain these in qualitative terms the geometric couplingbetween doubly-curved out of plane deflection shapes and in-plane deformations.

2535

Figure 1: Stable shapes of bistable plate (a) Flat plate at cure temperature (b) Saddle shape as plate begins tocool (c) Two possible singly curved stable shapes at room temperature.

2 Experimental Method

2.1 Plates

Plate Stacking Sequence B11 = −B22 (N) Approximate depth of curvature (∆zbetween points B and E in Figure 3)

(mm)A [90◦, 90◦, 90◦, 90◦, 0◦, 0◦, 0◦, 0◦]T 1.676 29B [90◦, 90◦, 90◦, 0◦, 90◦, 0◦, 0◦, 0◦]T 1.466 19C [90◦, 90◦, 0◦, 90◦, 0◦, 90◦, 0◦, 0◦]T 1.047 8D [90◦, 0◦, 0◦, 90◦, 0◦, 90◦, 90◦, 0◦]T 0 0

Table 1: Plate properties

Three bistable plates, labelled A-C, and a further flat plate were studied with stacking sequences and nominalstiffness values shown in Table 1. All were made from 8 plies of IM7/8552 Intermediate Modulus CarbonFibre pre-preg, cut to 280mm squares (if held flat) using a diamond wheel cutter. They were vacuum baggedusing tool plates on both sides, to ensure the most symmetrical properties in terms of resin distributionthrough thickness. They were cured using the manufacturers recommended cycle in an autoclave at 180◦Cfor 2 hours.

A 5mm hole was drilled at the centre, to accommodate the bolt for the shaker. To create points to be tracedby the image capture routine, a template with a 5mm square grid of holes was used as a mask whilst theplates were sprayed with between 4 and 6 coats of ordinary spray paint. Moisture ingress can significantlyaffect the bistable plate properties [6]. Therefore the plates were dried in an oven at 110◦C for 2 hours afterthey were cut on a diamond wheel cutter (which uses a water spray to cool and lubricate the blade), and werestored in a sealed cupboard with desiccant whenever they were not being used for experiments, to minimisethese effects.

The assumed properties of the material plies are: E1 = 1.64105 N mm−2, E2 = 1.20104 N mm−2, G12 =4.60103 N mm−2, ν12 = 0.3, t = 1.1710−1 mm , α1 = 0 K−1, α2 = 310−5 K−1, ρ = 1570 kg m−3. Wecan derive stiffness properties for the complete laminates using Classical Laminate Analysis, allowing us tocalculate in-plane force intensity as:

{N} = [A] {ε}+ [B] {κ} (1)

and moment intensity as{M} = [B] {ε}+ [D] {κ} (2)

2536 PROCEEDINGS OF ISMA2012-USD2012

where {N} =[Nx Ny Nxy

]T , {M} =[Mx My Mxy

]T , {ε} =[εx εy εxy

]T , and {κ}approximates mid-plane curvatures and is given as {κ} =

[−∂2w

∂x2 −∂2w∂y2 −2 ∂2w

∂x∂y

]T. The only non-

zero [B] matrix terms areB11 andB22, listed in Table 2.1. All laminates have the same [A] and [D] matrices:

[A] =

8.291 0.339 00.339 8.291 0

0 0 0.431

× 104 N mm−1 , [D] =

6.053 0.248 00.248 6.053 0

0 0 0.314

× 103 N mm

The asymmetric layups of plates A-C induce thermal moments as they cool from cure temperature. LinearClassical Laminate Analysis predicts that these plates will therefore form the saddle shape as shown inFigure 1 (b). However nonlinear analysis shows that double curvature induces in plane stretching of theplate, and consequent high elastic strain energy [1]. Therefore at room temperature, the saddle configurationis unstable, and instead the plates adopt one of two singly curved stable configurations shown in Figure 1 (c)to minimise strain energy, as modelled by Hyer [1]. The varying layups of the plates affect the amount ofbistable curvature, as shown in Table 2.1.

2.2 Dynamic Excitation and Signal Conditioning

The plates were mounted on a piezo-electric force transducer at their centre, with the transducer mountedon the shaft of a Ling shaker. The plates were excited sinusoidally. Due to their nonlinear response, and thepossibility of internal shaker interactions, a controller was developed using DSpace and Simulink to allowmanual conditioning of the force input to the plate. This allowed the tuning out of unwanted harmonics ofthe required sinusoidal input signal. Figure 2 shows a schematic of this system. A Fourier series calculationwas performed on the force input signal, with the required driving frequency as the fundamental frequency.The user interface displayed the amplitude at up to the 5th harmonic, and allowed the user to manuallyspecify harmonic corrections to the voltage signal sent to the shaker amplifier. It was typically possible totune unwanted frequencies to an amplitude two orders of magnitude below that of the fundamental.

Note that the force input into the centre constitutes an additional boundary condition on the plate, and thatthe resulting deflections are therefore not simply ‘free-free’ responses, although clearly free-free modes maybe excited.

2.3 Motion Capture

The motion of the plates was recorded using a novel video method developed by the authors [5]. Briefly, ahigh-speed video camera recorded both the plate motion, and its reflection in an angled mirror. By trackingeach marked point’s position in both the true image and the reflection it was possible to triangulate its motionin 3D, thus allowing large angular deflections of the plate to be recorded. Figure 3 shows a single image fromthe camera, showing the plate mounting and the mirror arrangement. As can be seen, just over half the platewas filmed, with the other half inferred by symmetry or asymmetry as appropriate, and it is this portion ofthe plate that is shown in results. When presenting full-field results from this method, we fit a 5th orderpolynomial surface to the data collected, as this gives a much clearer view of the shape.

3 Results

3.1 Low amplitude motion of Plates A-C

It was seen that for plates A and B at small amplitudes, the deflection showed negligible variation in the ydirection for the first two mode shapes as shown in Figure 4. This was not true of plate C, which showedvariation along both x and y for these modes, as shown in Figure 5.

NON-LINEARITIES: IDENTIFICATION AND MODELLING 2537

Figure 2: Schematic of control arrangement

Figure 3: Single frame of image capture showing plate (left), reflection (right) and reference points A-F.


Figure 4: Vertical deflections from stable configuration of 1st (left) and 2nd (right) mode shape for Plate Ashowing negligible variation along y-axis. Plate B shows a similar behaviour.

Figure 5: Vertical deflections from stable configuration of 1st (left) and 2nd (right) mode shapes for Plate C.


Figure 6: Snapping response of Plate C excited at 1st mode frequency, showing time traces of verticalposition. Letters indicate positions shown in Figure 3.

3.2 Snapping responses of plate C

It was found that attempting to excite Plates A and B at high amplitude, so that snapping occurred as part ofthe response, resulted in a violent, seemingly chaotic oscillation that could not be controlled. However PlateC allowed a more controllable response to be captured. Figures 6 and 7 show the effect of increasing theexcitation amplitude until snapping occurs, at the 1st and 2nd modes respectively. In each case, the responsehas a period three times greater than the forcing period, which is indicated by the motion of the centre (pointB).

3.3 Simple responses of Plate D

We note two simple motions of interest for plate D. At lower frequencies, the plate deformed in an antisym-metric twisting motion, which appeared as a diagonally symmetrical corner flapping motion when combinedwith the plates rigid body motion, as illustrated in 8. A fully symmetric ‘dish’ resonance was encountered at44.2Hz as shown in Figure 9, where the nodal lines pass through the shaker bolt.

3.4 Dual responses of plate D

It was found that at certain frequencies, the plate could respond in two different ways, depending on initialconditions. If the frequency was increased from low frequency to 35Hz, a twisting response as shown inFigure 10 would be found. However, if the fully symmetrical response of Figure 9 was activated, thenfrequency was steadily reduced to 35Hz, a symmetric response would be encountered at this frequency asshown in Figure 11.


Figure 7: Snapping response of Plate C excited at 2nd mode frequency, showing time traces of verticalposition. Letters indicate positions shown in Figure 3.

Figure 8: Twist/corner flap motion of Plate D Frequency 22.5 Hz Forcing amplitude 5N.

Figure 9: Doubly symmetric ‘dish’ resonance of Plate D Frequency 44.2 Hz Forcing amplitude 5N.


Figure 10: Corner flap/twisting motion Plate D Frequency 35 Hz Forcing amplitude 5N.

Figure 11: Symmetric motion of Plate D Frequency 35 Hz Forcing amplitude 5N.

3.5 Changing modeshape of plate D

If amplitude was increased for the 44.2 Hz mode shown in Figure 11, the response would change to showan additional subharmonic response with a twisting characteristic. Figure 12 shows the extremes of positionover 2 forcing periods. Over the 1st period the plate moves from its upper position (A) to one of the lowertwisted shapes (B) and then returns to its upper position, then over the second cycle it moves to the otherlower twisted position (C) before returning and repeating the whole sequence.

A

B

C

Figure 12: Amplitude effect on response; twisting at 1/2 subharmonic Plate D Frequency 44.2 Hz Forcingamplitude 10N. Plate repeats sequence of positions ABAC over 2 forcing periods.


4 Discussion

4.1 Equation of motion

We start with the nonlinear equation of motion, including bending and membrane effects but neglecting shearand rotary inertia, for a shallow-curved shell [7]:

∂2Mx

∂x2+ 2

∂2Mxy

∂x∂y+∂2My

∂y2− Nx

Rx− Ny

Ry

+∂

∂x

(Nx

∂w

∂x+Nxy

∂w

∂y

)+

∂

∂y

(Nxy

∂w

∂x+Ny

∂w

∂y

)= ρh

∂2w

∂t2(3)

and the in-plane strain/displacement compatibility equation:

∂2εx∂y2

− ∂2εxy

∂x∂y+∂2εy∂x2

=(∂2w

∂x∂y

)2

− ∂2w

∂x2

∂2w

∂y2+

1Rx

∂2w

∂y2+

1Ry

∂2w

∂x2(4)

We can substitute equations (1) and (2) into (3) , use the Airey stress function so that we satisfy equation (4),and use the structure of our particular stiffness matrices to eliminate redundant terms to obtain:

B11a12

(∂4Φ∂x4− ∂4Φ∂y4

)−(D∗

11

∂4w

∂x4+ 2 (D∗

12 + 2D∗66)

∂4w

∂x2∂y2+D∗

11

∂4w

∂y4

)−(

1Rx

∂2Φ∂y2

+1Ry

∂2Φ∂x2

)+(∂2Φ∂y2

∂2w

∂x2− 2

∂2w

∂x∂y

∂2Φ∂x∂y

+∂2Φ∂2x2

∂2w

∂y2

)= ρh

∂2w

∂t2(5)

where starred terms are coefficients from the reduced flexural stiffness matrix given by:

[D∗] = [D]− [B] [A]−1 [B] (6)

which shows that the presence of bend-twist coupling reduces the effective flexural stiffness of a laminate[8].

4.2 Small amplitude motions of bistable plates

If we assume that the plate in one of its stable positions is a singly curved shell, and that deflection onlyvaries significantly in the direction of this single curvature as suggested by our observations of plates A andB, we can eliminate all y-derivatives from our equation of motion. Furthermore, since no double curvatureoccurs, there is justification for assuming that no in-plane force develops because Gaussian curvature remainsat zero, and therefore Φ = 0. This means that in these cases the equation of motion can reduce to:

−D∗11

∂4w

∂x4= ρh

∂2w

∂t2(7)

The form of this equation is identical to that of the Euler-Bernoulli equation for the transverse vibration ofan isotropic beam with Young’s modulus E, 2nd moment of area I and cross sectional area A:

− EI∗∂4w

∂x4= ρA

∂2w

∂t2(8)

the solutions of which are widely known.

To apply this to mode 1, we assume that the nodal line along the line of symmetry at x = 0, acts like aclamped boundary condition. Therefore, we apply the solution for a clamped-free beam to one half of theplate, and infer the other half with symmetry. For mode 2, we model the whole length of the plate in x as afree-free beam, and adapt the relevant beam solution.


35.0

43.1 40.338.443.8

51.252.5

63.1 62.561.2

69.7

82.7

0

10

20

30

40

50

60

70

80

90

A B C

Hz

Plate

Mode 1 Experimental

Mode 1 Predicted

Mode 2 Experimental

Mode 2 Predicted

Figure 13: Comparison of predicted and experimental frequencies.

Figure 14: Comparison of data points on Plate A near line BE to beam-like model (left) 1st mode (right) 2ndmode.

Figure 13 shows that this form of model makes reasonable frequency predictions for plates A and B whenone allows for factors such as variation in material properties and unmodelled features such as the mountingbolt. The Plate C prediction is poor, as expected because Plate C has deflection that varies in both x and y,contrary to the assumptions of the model. The difference in frequencies between plates A and B demonstratesthe importance of the [B] matrix in reducing flexural stiffness, as both plates have identical dimensions and[D] matrices. Figure 9 compares the deflection profile predicted by the beam-like model with data pointsfrom experiment showing excellent agreement, in particular the location of the node for Mode 2.

4.3 Snapping response of plate C

Referring to Figures 6 and 7 we notice that the nature of snapping oscillations can depend on the frequencyat which snapping is induced. In the case of the 1st mode, by following the corner Point D it can be seen thatthe plate performs a ‘bounce’ lasting one forcing period, before a snap occurs over 1

2 a forcing period. Theplate then ‘bounces’ once more before snapping back in a similar manner, and then the sequence repeats. In


the case of the 2nd mode, it can be seen that the corner (point D) performs a simpler motion, performing justa single oscillation over three forcing periods.

We see that in both cases ‘snapping’ can be regarded as a smooth process that occurs in timescales com-parable to the natural frequencies of the plate, rather than a near-instantaneous change of state. As such, itsuggests that the process may be readily controlled. These results also raise the possibility that a small os-cillator could be designed to actuate snapping of a bistable plate, by exciting a natural mode to an amplitudethat causes a snap. Such an oscillator could potentially be much smaller and lighter than one that relied onstatic force alone for actuation.

4.4 Responses of plate D

With the absence of a strong initial curvature, Plate D can form diagonal principle curvature directions duringdeflection, hence the diagonal motion which has no equivalent in plates A-C. The possibility of multipleresponses at the same frequency, as shown by Figures 10 and 11 is a nonlinear phenomenon; it appears thatinstead of responses being able to superimpose as in a linear system, a stable response is formed of onedeflection shape or the other in this situation.

We propose that the primary source of nonlinearity in the plate is geometric; at significant amplitudes suchas these, any double curvature of the plate will necessitate in plane stretching, and incur high potentialstrain energies. Therefore, the plate will seek different deflections to reduce the stored potential energy. Anexample of this is the change to the 44.2Hz modeshape seen initially in Figure 9, which features strongdouble curvature. As amplitude increases, the oscillatory motion changes radically to that seen in Figure 12despite being a resonant response at lower amplitude.

5 Conclusions

We have experimentally demonstrated a simple model for small scale vibrations of bistable plates. We havealso demonstrated 2 period-trebled forced oscillation cycles of a plate during which it snaps between bothof its stable configurations. These results indicate a possible strategy for actuation of the bistable plates.We have also shown how nonlinearity due to amplitude is easily encountered in flat plates with free edges,and demonstrated some of the effects that this nonlinearity can have on deflection shapes, including dualresponses to the same excitation, modeshape changes due to amplitude and subharmonic response.

Acknowledgements

The authors would like to thank the EPSRC for funding this work.

Nomenclature

[A] Laminate in-plane stiffness matrixA Cross sectional area of beam[B] Laminate bend-twist coupling matrix[D∗] Laminate reduced flexural stiffness matrix[D] Laminate flexural stiffness matrixE , E1 , E2 Young’s Modulus, for isotropic material, ply fibre and transverse directions respec-

tivelyG12 In-plane shear modulus for a ply


h Laminate thicknessI Second moment of area of beamMx , My , Mxy Moment intensities; moment per unit distance. Single subscripts indicate direction of

induced curvature, xy indicates twist.Nx , Ny , Nxy In-plane loading intensities; force per unit distance. Single subscripts indicate direc-

tion, xy indicates in-plane shear.Rx , Ry Radius of plate curvatures in x and y directions respectivelyt Thickness of a ply and also used as time variablew Vertical displacement from stable configuration of bistable platex , y , z Spatial dimensions; x is in the plan form of the plate, in direction of curvature when

in stable configuration. y is second plan form variable and z is vertical.α1 , α2 Thermal expansion coefficients, subscripts indicate ply fibre and transverse directions

respectivelyεx , εy , γxy In-plane strains in x and y and in-plane shear strain respectivelyν12 In-plane Poisson ratio for a plyρ Mass densityΦ Airey stress function, defined such that Nx = ∂2Φ

∂y2, Ny = ∂2Φ

∂x2, Nxy = − ∂2Φ

∂x∂y

References

[1] M. Dano, M. W. Hyer, Thermally-induced deformation behavior of unsymmetric laminates, InternationalJournal of Solids and Structures, Vol. 35, No. 17, Elsevier (1998), pp. 2101-2120.

[2] S. Daynes, P. M. Weaver, J. A. Trevarthen, A morphing composite air inlet with multiple stable shapes,Journal of Intelligent Material Systems and Structures, Vol. 22, No. 9, Sage (2011), pp. 961-973

[3] C. G. Diaconu, P. M. Weaver, F. Mattioni, Concepts for morphing airfoil sections using bi-stable lami-nated composite structures, Thin-Walled Structures, Vol. 46, No. 6, Elsevier (2008), pp. 689-701

[4] A. Arrieta, S. Neild, D. Wagg, Nonlinear dynamic response and modeling of a bi-stable composite platefor applications to adaptive structures, Nonlinear Dynamics, Vol. 58, No. 1-2, Springer (2009), pp.259-272

[5] A. Shaw, S. Neild, D. Wagg, P. Weaver, Single Source Three Dimensional Capture of Full Field PlateVibrations, Experimental Mechanics , Springer (2011)

[6] J. Etches, K. Potter, P. Weaver, I. Bond, Environmental effects on thermally induced multistability inunsymmetric composite laminates, Composites Part A: Applied Science and Manufacturing, Vol. 40,No. 8, Elsevier (2009), pp. 1240-1247

[7] D. Wagg, S. Neild, Nonlinear Vibration with Control, Springer (1997)

[8] E. Reissner, Bending and stretching of certain types of heterogeneous aeolotropic elastic plates, Journalof applied mechanics, Vol. 28, No. 3, ASME (1961)


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Experimental investigation into dynamics of flat and ...past.isma-isaac.be › downloads › isma2012 › papers › isma... · Figure 1: Stable shapes of bistable plate (a) Flat