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Unsymmetric composite laminates morphing via piezoelectric actuators
Composites: Part A 42 (2011) 748–756
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Unsymmetric composite laminates morphing via piezoelectric actuators Samer A. Tawfik a,⇑, D. Stefan Dancila b, Erian Armanios b a b
School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332, United States Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, 500 West First Street, Arlington, TX 76019, United States
a r t i c l e
i n f o
Article history: Received 25 September 2010 Accepted 1 March 2011 Available online 4 March 2011 Keywords: B. Buckling C. Finite element analysis (FEA) D. Mechanical testing
a b s t r a c t A composite panel, with a [0/90] unsymmetric layup, exhibits bi-stable anticlastic equilibrium configurations. One way to provide morphing capabilities to such a panel is attained by attaching piezoelectric actuators to its surface. In such a design an electric field is applied to the piezoelectric actuators in order to trigger snap-through behavior of the panel/actuator assembly. A simple cantilever beam configuration is discussed to perform actuator assessment. Results are compared to with predictions from Extended Classical Lamination Theory and finite element analysis. Guidelines are proposed to design a morphing unsymmetric panel/actuator assembly. The panel/actuator(s) assembly is manufactured accordingly. Snap-through experiments are conducted and the ABAQUS finite element software is used to predict snap-through behavior. Analysis predictions are in good agreement with test results. Published by Elsevier Ltd.
1. Introduction Observations of birds and/or flying insects and contrasting these with current technologies inspired researchers to depart from rigid airframe structures. Rigid airframe structure limit optimized aircraft performance when multiple missions are required. The wing as the major lifting structure received most focus in terms of morphing. Several attempts to provide morphing wings, in an abstract sense, have been recorded by the research community. Among these attempts are NASA’s efforts [1–3] focusing on biologically inspired morphing wings. McGowan et al. [2] investigated several morphing wing designs such as the Gull wing, the Shark wing and the Hyper-Elliptic Cambered Span (HECS) wing as shown in Fig. 1. As underscored by McGowan et al. [2] biologically inspired morphing wings are not to be understood as straightforward mimicking of birds and/or flying insects. Rather, they should be the outcome of learning from nature’s elegant designs in order to foster more robust and versatile enhanced engineering designs. For example a morphing wing from the structural perspective should not entail additional complexity due to the actuation process, while providing adequate stiffness as a lifting surface and being adaptable with optimal shape over a broader range of flight missions. Different linkage mechanism systems were proposed to provide the structural design of the HECS morphed wing as presented by ⇑ Corresponding author. Tel.: +1 678 462 2597. E-mail address: samer.tawfi[email protected] (S.A. Tawfik). 1359-835X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.compositesa.2011.03.001
Stubbs [4] and Wiggins [5], Fig. 2, or tendon actuated compliant cellular structure, as presented by Ramrakhyani [6], and shown in Fig. 3. Such mechanical systems provide several challenges due to the complexity of their design, actuation and maintenance. Incorporation of these mechanical systems in design results in weight penalty that may render the design unfeasible especially in the case of Uninhabited Aerial Vehicles (UAVs). On the other hand thin unsymmetric laminated composites possess bi-stable configurations [7]. This feature makes them attractive candidates in building morphing structures. Tawfik and Armanios [8] investigated the design concept of an UAV morphing wing utilizing bi-stable composites. This investigation provided an understanding to the design of actuated bi-stable panels of different planform shapes suitable for an UAV wing. Schultz and Hyer [9] investigated triggering snap-through in an unsymmetric panel using Macro-Fiber Composite (MFC) actuators. Also other researcher [10–12] investigated the same option of using MFC actuators to trigger snap-through in an unsymmetric panel. MFC actuators are stiff and found to decrease the panel curvature when bonded its surface. The current research investigates using different options of piezoelectric actuators. This investigation is performed for both MFC and Polyvinylidene Fluoride (PVDF) piezoelectric actuators. PVDF is commercially available in various film thicknesses; hence more suitable for fabrication into refined designs. Typical properties of a piezoelectric PVDF film are provided in Table 1. While properties for MFC actuators are provided in Table 2.
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Fig. 1. Biologically inspired morphing wings suggested by NASA [2] (a) Gull wing, (b) Shark wing and (c) Hyper-Elliptic Cambered Span (HECS) wing.
Fig. 2. General topology of a quaternary-binary cross linked mechanism [4,5].
2. Analysis and modeling
Fig. 3. Bench-top demonstration model of compliant cellular structure [6].
Table 1 Piezoelectric (PVDF) film properties.
Thickness Piezo strain constants Young’s modulus Max operating voltage Breakdown voltage
d31 d33 YE Vmax
Property
Units
9, 28, 52, or 110 23 1012 33 1012 2–4 30 80
lm m/V m/V GPa V/lm V/lm
A simple model is developed to assess actuator performance. In this model piezoelectric actuator(s) are used to control the tip displacement of an unsymmetric composite cantilever beam. This allows identifying the controlling parameters that govern the morphing panel design. Guidelines in terms of actuator parameters are extracted and used to form a basis to design a morphing panel. Piezoelectric actuators are used to provide extension/contraction as shown in Fig. 4. An actuator will shrink or expand depending on the polarity of the applied electric field through the thickness. PVDF actuators are transversely isotropic in terms of material properties, and their electrodes are aligned parallel to one direction only, referred to as actuation direction. On the other hand MFC actuators possess orthotropic mechanical and piezoelectric properties. A thermal analogy approach [13,14] is used to model piezoelectric behavior. In this analogy an applied electric field is modeled as thermal load while equivalent piezoelectric constants are used to replace the thermal expansion coefficients. Equivalent piezoelectric constants can be easily derived in the case of an extension actuation mechanism, i.e. where the electric field and poling is through the thickness. The three-dimensional thermo-elastic strain–stress relations are
Table 2 Macro-Fiber Composite (MFC) piezoelectric actuators properties. Property
Units
300
lm
d31 d33
460 1012 210 1012
m/V m/V
d31 d33
m/V m/V GPa
Thickness High-field (|E| > 1 kV/mm) Piezo strain Constants Low-field (|E| < 1 kV/mm) Piezo strain constants Young’s moduli
Y E1
400 1012 170 1012 30.336
Y E2
15.857
GPa
Shear modulus
GE12
5.515
GPa
Poisson’s ratio Max operating voltage
m12
0.31 1500 500
V V
Vmax
Fig. 4. PVDF actuator in an extension actuation mechanism.
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feg ¼ ½Sfrg þ fagDT
ð1Þ
where {e} are the total strains, [S] is the compliance matrix, {r} are the mechanical stresses, {a} are the thermal expansion coefficients and DT is the temperature difference. A similar equation can be written in the case of extension actuation mechanism, i.e. only the electric potential w3 across the thickness is nonzero
feg ¼ ½Sfrg þ fdg
w3 t
ð2Þ
where {d} is the piezoelectric strain coefficient vector and t is the piezoelectric actuator thickness. Relationships establishing a thermal analogy of piezoelectric actuator response are achieved by comparing Eqs. (1) and (2). For a given actuator, the piezoelectric strain coefficient vector can be used to provide the equivalent thermal expansion coefficients
a1 ¼ Fig. 5. Cured shapes of a rectangular laminate with unsymmetric stacking.
d31 ; t
a2 ¼
d32 ; t
a12 ¼ 0; DT ¼ w3
ð3Þ
This approach is well suited for finite element formulation. Modeling electric actuation of an orthotropic piezoelectric actuator based on the thermal analogy approach can be easily implemented in ABAQUS. 2.1. Cantilever beam configuration
Fig. 6. Laminate/actuator structure as cantilever beam.
Fig. 7. Cantilever beam cross-section detail.
The stability characteristics of unsymmetric cross-ply rectangular panel, shown in Fig. 5, were investigated in [15]. ABAQUS finite element results [15] were in good agreement with those of the Extended Classical Lamination Theory (ECLT). ABAQUS finite element results, in agreement with observations, also reported a critical sidelength-to-thickness ratio below which the rectangular laminate ceases to possess two equilibrium configurations, i.e. become mono-stable. In this case, the laminate will cease to maintain the equilibrium configuration identified by being curved along the shorter side, and retain only the equilibrium configuration associated with being curved along the longer side. A mono-stable panel, curved along its length, is used in a cantilever beam setup. Its curvature can be controlled by attaching two layers of actuators to both the top and bottom surfaces of the laminate. This cantilever beam setup will be used to study actuation requirements and compare predictions from different analysis methods. A schematic of an unsymmetric cross-ply rectangular cantilever laminate with actuators bonded to its top and bottom surfaces is shown in Fig. 6. The laminate/actuator beam has a length L = 200 mm, a width w = 10 mm and a total thickness t. The unsymmetric cross-ply laminate has two plies each of thickness tlayer while each piezoelectric actuator has a thickness tpiezo as shown in the cross-section detail of Fig. 7. The curvature j of the cured laminate before attaching the actuators is associated to a tip deflection di. Upon bonding the actuators
Fig. 8. (a) Cured cantilever laminate with unsymmetric stacking. (b) Bonded laminate/actuator assembly (no actuation).
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ing thermal expansion coefficients, Ep the Young’s modulus of the PVDF actuators, ap the equivalent thermal expansion coefficient, Ac the cross-sectional area of a single composite lamina, and Ap the cross-sectional area of a single PVDF layer. For effective actuation, the moments in the unsymmetric laminate due to thermal loads associated with the cure cycle should be balanced by the moment introduced by the piezoelectric layers upon actuation.
j¼ Fig. 9. Laminate/actuator structure in direct actuation.
to the laminate its initial curvature will decrease and consequently the tip deflection will be reduced from di to do, as shown in Fig. 8. The deformed shape of the laminate/actuator structure is assumed to conform to a portion of a cylindrical shape. Hence throughout the calculations in this section, the curvature j will be used to evaluate actuator performance. Fig. 9 shows a laminate/actuators assembly with the electric potential being applied through the thickness of each actuator to cause the top actuator to contract and the bottom one to extend. Actuation in this sense, termed direct actuation, results in reducing the beam curvature and minimizes the tip deflection or even cause the beam to acquire an opposite curvature. Two factors control the beam curvature namely the actuator thickness and the level of actuation, i.e. applied electric potential. Causing the assembly to curve in the opposite direction is of no significance and successful actuation will be based on reducing its curvature to zero. This problem is studied initially utilizing PVDF actuators. They will be subsequently replaced by MFC actuators for performance comparison. 2.2. Beam Theory (BT) model The engineering beam theory is used to calculate the beam curvature due to thermal loads representing the curing cycle and actuation potential, respectively. From equilibrium of forces acting on the composite laminate layers and piezoelectric actuator layers, as shown in Fig. 10
P1 P2 P3 þ P4 ¼ 0
ð4Þ
Using thermal analogy and assuming perfect bonding the tip extension d can be written as
P3 L P1 L d ¼ d3 d3 ¼ ap DVL ¼ d1 þ d1 ¼ a1 DTL þ Ep Ap E11 Ac P2 L P4 L ¼ d2 d2 ¼ a2 DTL ¼ d4 þ d4 ¼ ap DVL þ E22 Ac Ep Ap
ð5Þ
where E11 and E22 denote Young’s moduli of the composite ply in the principal material directions respectively, a1, a2 the correspond-
MT þ ME EI
ð6Þ
for MT is the bending moment due to curing, ME is the bending moment due to actuation and EI is the total flexural stiffness of the bonded laminate/actuators assembly. The mechanical properties of Hexcel IM7/8551-7 graphite/ epoxy prepreg are used throughout the analysis and provided in Table 3. The maximum curing temperature is 177 °C and the room temperature is 21 °C. The curvature is obtained using beam theory at values of curing cycle temperature and applied electric potential. Two limiting cases provide physical understanding of actuator response, namely no actuation and full actuation. The case at no actuation represents the stiffening effect associated with bonding the actuators to the unsymmetric laminate with no applied electric potential. While in the full actuation case the maximum operating voltage is applied after bonding. 2.3. Extended Classical Lamination Theory (ECLT) The Rayleigh–Ritz energy based extension to the Classical Lamination Theory accounting for geometrically nonlinear strain–displacement model is adopted. This analysis is developed by Schultz and Hyer in [9] and its details can be found in their work. In this section, ECLT predictions are based on using a 4-term displacement field. 2.4. ABAQUS finite element analysis The beam/actuator assembly is modeled in the ABAQUS finite element code for two cases. In the first case the nominal thickness of a composite ply is approximated as 100 lm and in the second case the thickness is chosen to be 300 lm. These correspond to sidelength-to-thickness ratios of 50 and 16.67, respectively. Therefore neither one of those cases represents a bistable panel, as the critical sidelength-to-thickness ratio above which a panel possesses two stable configurations is 82.88 as provided in [16]. The finite element model used a total of 240 shell elements S4R to represent the three layers, i.e. the unsymmetric panel and the top and bottom actuators. A comparison of the actuator stiffening effect predicted using BT, ECLT and ABAQUS models at tlayer = 100 lm is
Fig. 10. Forces and displacements in the beam due to thermal and electric loads.
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Table 3 Elastic properties IM7/8551-7 graphite/epoxy prepreg. Material
E11 (GPa)
m12 E22 G12 (GPa) (GPa)
141.18 7.20 IM7/8551-7 graphite/epoxy prepreg
4.45
a1
a2
t (lm)
(106/°C) (106/°C)
0.30 0.14
30.98
138.75
Fig. 13. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 100 lm, full actuation.
Fig. 11. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 100 lm, no actuation.
Fig. 14. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 300 lm, full actuation.
Fig. 12. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 300 lm, no actuation.
provided in Fig. 11. The results obtained from all three methods are in good agreement with each other. Similar trends among all three models for the case of a moderate thickness lamina, tlayer = 300 lm are observed as shown in Fig. 12. An electric potential at Vmax is applied in both cases, tlayer = 100, 300 lm, in order to study the effect of actuation and the results are shown in Figs. 13 and 14, respectively. In both cases BT and ABAQUS results are in good agreement. While values of curvature obtained from the ECLT are in good agreement with the general trend except in the vicinity of zero curvature. Two steps in predicted ECLT curvature are observed at both the beginning and the end of a zero curvature plateau. Inspecting the curvatures predicted by the 4-term ECLT in both cases it is found that at the first step the curvature along the length of the beam, jy, changes from 1 m1 to nearly zero. The associated curvature in the width direction, jx, changes from being nearly zero to 1 m1 indicating snap-through from one equilibrium shape to the other. The zero curvature plateau represents the
Fig. 15. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 100 lm, full actuation. Corresponding beam structure shapes illustrated.
beam is in its second equilibrium configuration (flat along the length and curved in the width direction). While the second step represents the case when the moment developed by the actuators forces the beam to go through same sense bending similar to a retractable tape or tape springs. Same sense bending of tape springs is extensively studied in [17,18]. For illustration purpose, the curvature as a function of PVDF thickness obtained using ECLT from Fig. 13 and regenerated in Fig. 15, is divided into three zones with the corresponding beam structural shapes provided schematically. Further details regarding this comparison are provided in [16].
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In conclusion, the 4-term ECLT lamination theory suffered from a numerical anomaly when studying a mono-stable unsymmetric cross-ply laminated. In literature and to the best knowledge of the authors the ECLT was not used to predict the minimum or critical sidelength-to-thickness ratio responsible for losing one of the stable configurations of a bistable panel. Also, Dano documented in [19] that ECLT predictions deviate from test data in the vicinity of the snap-through in the case of bistable panels. The deviation associated to ECLT predictions in the vicinity of snap-through in the case of bistable panels is paralleled by predicting a zero curvature plateau in the case of mono-stable laminate. Table 4 provides the thickness of the PVDF layers required for actuation, i.e. associated with bringing the curvature of the beam/actuators assembly to zero. The table also includes the initial curvature of the unsymmetric laminate, before bonding the PVDF layers to its surfaces, and the curvature of the beam/actuators assembly at no actuation, i.e. V = 0. It can be concluded that successful actuation of an unsymmetric laminate with thickness 2t requires two PVDF actuator layers each of thickness approximately 3t. Bonding such thick actuators to the unsymmetric laminate leads to nearly 90% loss of its initial curvature. Using thicker actuator layers may lead to causing the laminate to lose its curvature entirely. Consequently using PVDF actuators for this configuration requires the electric potential to be always at Vmax in order to keep the actuator thickness to a minimum and consequently decrease the stiffening effect. A comparison between actuator types is subsequently conducted using ABAQUS finite element analysis in which it was taken into consideration that the MFC actuators are available commercially only in 300 lm thickness. Stiffening and actuation effects are studied for the case of bonding MFC actuators to cantilever beams of ply thicknesses 100 and 300 lm and presented in Table 5. Actuator stiffening effect is more pronounced when using MFC actuators. For the case of the 100 lm ply thick the curvature is reduced by 98.7% when using MFC actuators versus 89.7% when using PVDF actuators although both actuator types are nearly of the same thickness. In the case of 300 lm ply thick the curvature is reduced by 84.3% when using MFC actuators versus 89.3% when using PVDF actuators whereas the MFC actuator thickness is approximately one third of its PVDF counterpart. Stiffening effect can be explained in terms of actuator stiffness as the Young’s modulus of MFC actuators is one order of magnitude higher than the PVDF counterpart. On the other hand for both 100 and 300 mm
PVDF thickness (lm) Initial Curvature (m1) Curvature upon bonding (m1)
Ply thickness 300 lm
BT
ECLT
ABAQUS
BT
ECLT
ABAQUS
306 18.44 2.1
239 17.69 3.1
360 18.56 1.9
931 6.15 0.681
788 5.89 1.91
1060 6.13 0.65
Table 5 Actuator comparison for a composite beam with [90/0] layup. Ply thickness 100 lm PVDF Actuator thickness (lm) Initial curvature (m1) Curvature upon bonding (m1) Curvature upon actuation (m1)
360 1.9 0
MFC 300 18.56 0.23 4.56
Ply thickness 300 lm PVDF 1060 0.65 0
Actuator thickness (lm) Initial curvature (m1) Curvature upon bonding (m1) Curvature upon actuation (m1)
Ply thickness 100 lm
Ply thickness 300 lm
M=0
M=0
M–0
300 18.56 0.23 2.17 4.56 2.62
M–0
300 6.13 0.96 1.24 0.94 0.396
thicknesses, MFC actuators have higher actuation authority or effectiveness than PVDF actuators. This is reflected in the resulting curvature upon actuation as MFC actuators can bring the curved beam to an opposite curvature. One can conclude therefore that MFC actuators are more suitable to induce snap-through behavior provided their high stiffening effect is reduced. Due to their high actuation effectiveness they are not required to be of the same size as the actuated panel. Therefore an actuator with a fractional surface area of the actuated panel is adequate, leading to a decrease in its stiffening effect. Further reduction could be achieved through the means of bonding the actuator to the bistable panel. Two options are evaluated. The first is to force the curved panel into a flat configuration then bond the actuator to its surface. While the other option is to bend the actuator causing it to conform to the panel’s curvature then bond them together. So far all presented numerical results are associated with the first option. The second option can be simulated in ABAQUS by applying moments at the edges of the actuators to conform to the panel curvature. Considering only MFC actuators, a comparison between both bonding options is presented in Table 6 for cantilever beams with thickness of 100 and 300 lm. Predictions corresponding to the second option of bonding show 88.3% and 79.7% loss of curvature for 100 and 300 lm ply thickness, respectively. The loss of curvature in this case is less than the corresponding values associated to the first option of bonding. Also, a higher loss of curvature is predicted for a thicker composite panel. Considering the gain attained using the second option of bonding it will be adopted to design and manufacture morphing bistable panel/actuators assembly using MFC actuators in subsequent sections of this work.
3. Experimental and analytical results
Table 4 PVDF actuator requirements for a composite beam with [90/0] layup. Ply thickness 100 lm
Table 6 MFC Actuator bonding options comparison.
MFC 300 6.13 0.96 0.94
In this section the design of an unsymmetric rectangular laminate/actuator assembly is considered based on the actuator assessment results. A rectangular unsymmetric panel of width 65 mm and length 85 mm is manufactured from Hexcel IM7/8551-7 graphite/epoxy prepreg with a stacking sequence of [0/90]T. The description of its stability behavior follows the schematic of Fig. 5. Two MFC actuators of dimensions 40 mm 10 mm are centrally bonded to the top and bottom surfaces of the unsymmetric panel using Loctite’s E120 HP Epoxy. The major axes of both actuators are aligned along the shorter dimension of the composite panel. The actuators are bent to conform to the curved surface of the cured panel and the adhesive is applied. The panel/actuators assembly is placed in a vacuum bag to maintain its curvature and placed in an autoclave for three hours at 60 °C with the vacuum on to ensure better bonding. Figs. 16 and 17 show side-byside panels with and without assembled actuators for a qualitative comparison of the bonding method. ABAQUS finite element simulation was used to predict the equilibrium shapes of the panel/actuators assembly and the
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Fig. 16. Bonded panel assembly versus identical dimension bistable panel (first equilibrium shape).
Fig. 17. Bonded panel assembly versus identical dimension bistable panel (second equilibrium shape).
voltage required to trigger snap-through behavior. Four-noded, reduced integration, doubly curved shell elements (S4R) are used to model both the unsymmetric laminate and the actuators. The unsymmetric panel is modeled using 884 elements, while for each actuator 64 elements are used to model its active portion and 156 elements to model its casing. The thermal analogy approach is used to model piezoelectric performance. Moreover, the nonlinear piezoelectric properties of the MFC actuator are incorporated in the thermal analogy model accounting for different ranges of piezoelectric constants as provided in Table 2. Moments are calculated and applied at the actuator edges to simulate bonding in the panel’s curved configuration. The three layers of elements representing the unsymmetric panel and the two actuators are tied together making sure to offset their respective neutral axis based on their position in the assembly. The panel/actuators assembly is held fixed in space at the midpoint of the panel by constraining all its degrees of freedom. The displacements of the four corner points are monitored throughout the analysis. The finite element simulation is carried out in three linear/nonlinear analysis steps. The first step is a linear eigenvalue buckling problem performed to obtain the mode shapes of the unsymmetric panel corresponding to the temperature loads of the cure cycle. A second step, nonlinear analysis, is carried out to account for the geometric imperfections of the panel expressed in percentage of the mode shapes obtained in the first step. Also moments are applied at the actuators’ edges to simulate the second option of bonding and hence predict the cured/bonded shape of the panel/ actuators assembly. The final step, nonlinear analysis, employing the thermal analogy approach is used to predict the voltage required by the actuators to trigger snap-through or snap-back. Further details of the finite element simulation are provided in [16]. Finite element predictions of the cured shapes of an unsymmetric panel and panel/actuators assembly are obtained as one quantitative measure for the manufacturing method. jx is the curvature of panel/actuators assembly in the first equilibrium shape and jy is its curvature in the second shape. Accordingly, these curvatures are predicted for the panel/actuators assembly. A reference case is cho-
sen to be an identical unsymmetric cross-ply panel with no bonded actuators. Comparisons of center curvatures between the actuated and the reference panels are provided in Table 7. The loss of curvature due to bonding the actuators, as a percentage of the reference panel curvature, is provided in parenthesis in Table 7. The efficiency of the bonding method is indicated by the percentage loss in curvature. Parallel to the actuators, the loss of curvature is 25.56% due to the stiffening effect of the narrow side of the actuators. While normal to the actuators, where the stiffening effect is expected to be a maximum, the loss of curvature is only 30.56%. Therefore bonding the actuators in the curved configuration contributes to minimizing the effect of stiffening on curvature. The equilibrium shapes of the panel/actuators assembly predicted by ABAQUS finite element analysis are shown in Fig. 18. The equilibrium shapes of the manufactured panel/actuators assembly are shown in Fig. 19 for qualitative comparison. The experimental setup, shown in Fig. 20, is used to measure the required voltage to trigger snap-through behavior. It includes three posts, one is used to suspend the panel/actuators assembly by a thread and the other two are used to support the actuators wiring. The setup consists of a variable voltage DC power supply, two miniature DC to High Voltage (HV) converters, a voltmeter and the manufactured panel/actuators assembly. A variable voltage DC power supply, Agilent E3648A, is used to generate a ramp signal. This ramp signal is fed into each of the miniature HV-DC converters, G05 and G15 obtained from AMCO. The isolated output of these HV-DC converters is linearly proportional to the input. The G05 amplifies a 12 V signal to 500 V while the G15 amplifies the
Table 7 Curvature comparison between actuated and reference panels.
jx (m1) jy (m1)
Reference panel
Actuated panel
11.995 11.20
8.18 (25.56%) 7.78 (30.56%)
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Fig. 18. Predicted equilibrium shapes of the panel/actuators assembly.
Fig. 19. Equilibrium shapes of the manufactured panel/actuators assembly.
12 V signal to 1500 V, according to the G series datasheet [20]. Each of these converters weighs 43 g and has dimensions of 38.1 mm 38.1 mm 16.0 mm. The voltmeter is used to check the amplified voltage applied to the actuators. According to observations in [9,19] it is noted that an unsymmetric panel undergoes a loss of curvature during the post-cure stage. However this loss of curvature ceases after a limited period of time. On the other hand an unsymmetric panel/actuators assembly is not stress free in either equilibrium configuration due to actuator bonding. In the current work, the panel/actuators assembly was snapped back-and-forth several times before performing the experiment. These pre-tests were carried out to eliminate any bias resulting from maintaining the assembly for some period of time in one equilibrium shape rather than the other. The experiment was repeated four times, for both snap-through and snapback, and the required voltages were averaged and are provided in Table 8. Percentage deviations from the averaged values are shown in the same table.
Table 8 Required voltage to trigger snap-through behavior. Required voltage (V) Top actuator
Snap-through Snap-back
Percentage difference (%) Bottom actuator
Experiment
FEA
Experiment
FEA
460 2217
495.5 2416
1380 739
1489 805.2
7.89 8.97
It can be seen from the results in Table 8 that trends in predicting the required voltage to trigger snap-through/snap-back behavior are captured by the numerical finite element analysis predictions. Another observation is that the predicted values using the finite element analysis represent an upper bound to the experimentally obtained ones. The differences between the finite elements results and the experiment could be due to the post-cure loss of curvature in unsymmetric panels, the assumption of perfect bonding between the panel and the actuators, the exact alignment of the actuators along the x-direction of the laminate, and/or uncoupled electrical, mechanical and thermal effects during bonding and actuation. In case when modeling combined thermal and piezoelectric behavior is required a user shell element can be developed within ABAQUS. Moreover, the finite element simulation considers all material properties independent of the cure cycle temperature and does not include any chemical reaction or conservation of heat during the cure simulation part.
4. Summary and concluding remarks
Fig. 20. Experimental setup.
A comprehensive study for design, analysis and manufacturing of morphing composites via piezoelectric actuators has been
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presented. The developed simplified beam theory model was validated by comparison of results with ABAQUS finite element and the Extended Classical Lamination Theory. The behavior associated with the Extended Lamination Theory for zero curvature was isolated. A comparison between PVDF and MFC actuators is performed leading to pertinent design, manufacturing and analysis guidelines for a rectangular unsymmetric panel/actuators assembly. In terms of manufacturing it is recommended to bond the actuators with their axis along the shorter side of the rectangular panel in its curved configuration. This allows bonding of actuators to an unsymmetric bistable panel while minimizing the loss of its bistable characteristics and enhancing sensitivities of snap-through/ snap-back response. In terms of analysis, the previously developed finite element methodology [15] was extended to include actuation via a thermal analogy approach using the ABAQUS code. The developed methodology provides a suitable tool for both design and analysis of morphing unsymmetric structures. It accurately models actuator bonding to a flat or curved panel, accounting for the actuator nonlinear properties as well as the requirements for inducing snapthrough. In terms of experimental effort, a morphing panel is manufactured and its predicted stability characteristics verified. The panel successfully snapped back-and-forth utilizing two MFC actuators that represented, collectively, only 14% of the panel surface area. The successful actuation of the manufactured panels points to their viability for morphing wing applications to Unmanned Aerial Vehicles (UAV). References [1] McGowan AR, Washburn AE, Horta LG, Bryant RG, Cox DE, Siochi EJ, Padula SL, Holloway NM. Recent results from NASA’s morphing project. In: Proceedings of SPIE’s 9th annual international symposium on smart structures and materials; 2002. p. 97–111. [2] McGowan AR, Cox DE, Lazos BS, Waszak MR, Raney DL, Siohci EJ, Pao SP. Biologically-inspired technologies in NASA’s morphing project. In: Proceedings of SPIE’s 10th annual international symposium on smart structures and materials; 2003. p. 1–13.
[3] McGowan AR. AVST morphing project research summaries in fiscal year 2001, NASA/TM-2002-211769; 2002. [4] Stubbs M. Kinematic design and analysis of a morphing wing. Master thesis, Blacksburg, Virginia: Virginia Polytechnic Institute and State University; 2003. [5] Wiggins L. Structural design and analysis of a kinematic mechanism for a morphing Hyper-Elliptic Cambered Span (HECS) wing. Master thesis, Blacksburg, Virginia: Virginia Polytechnic Institute and State University; 2003. [6] Ramrakhyani DS. Morphing aircraft structures using tendon actuated compliant cellular truss. Ph.D. dissertation, Pennsylvania: Pennsylvania State University; 2005. [7] Hyer MW. Some observations on the cured shapes of thin unsymmetric laminates. J Compos Mater 1981;15:175–94. [8] Tawfik S, Armanios E. Preliminary investigation of a morphing wing design concept employing unsymmetric cross-ply composites stability characteristics. In: Proceedings of 21st annual technical ASC conference, Dearborn, MI: University of Michigan-Dearborn; 2006. [9] Schultz MR, Hyer MW. A morphing concept based on unsymmetric composite laminates and piezoceramic MFC actuators. In: 45th AIAA/ASME/ASCE /AHS/ ASC structures, structural dynamics & materials conference, Palm Springs, California; 2004. [10] Hufenbach W, Gude M, Czulak A. Actor-initiated snap-through of unsymmetric composites with multiple deformation states. J Mater Process Technol 2006;175:225–30. [11] Portela PM, Camanho PP, Weaver PM, Bond IP. Analysis of morphing, multi stable structures actuated by piezoelectric patches. Comput Struct 2008;86:347–56. [12] Bowen CR, Butler R, Jervis R, Kim HA, Salo AIT. Morphing and shape control using unsymmetrical composites. J Intell Mater Syst Struct 2007;18:89–98. [13] Soderkvist J. Similarities between piezoelectric, thermal and other internal means of exciting vibrations. J Micromech Microeng 1993;3:24–31. [14] Dong X, Meng G. Dynamic analysis of structures with piezoelectric actuators based on thermal analogy method. Int J Adv Manuf Technol 2006;27:841–4. [15] Tawfik S, Tan X, Ozbay S, Armanios E. Anticlastic stability modeling for crossply composites. J Compos Mater 2007;41:1325–38. [16] Tawfik S. Stability and morphing characteristics of bi-stable laminates. Ph.D. dissertation, Atlanta, Georgia: School of Aerospace Engineering, Georgia Institute of Technology; 2008. [17] Yee JCH, Soykasap O, Pellegrino S. Carbon fiber reinforced plastic tape springs. In: 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and material conference and exhibit, Palm Springs, CA; 2004. [18] Silver MJ, Hinkle JD, Peterson LD. Controlled displacement snap-through of tape springs: modeling and experiment. In: 46th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and material conference and exhibit, Austin, Texas; 2005. [19] Dano M. SMA-Induced Deformations in General Unsymmetric Laminates. Ph.D. Dissertation, Blacksburg, Virginia: Virginia Polytechnic Institute and State University; 1997. [20] Miniature DC to HV DC converters – G series datasheet. EMCO High Voltage Corp.; 2008
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Unsymmetric composite laminates morphing via piezoelectric actuators Samer A. Tawfik a,⇑, D. Stefan Dancila b, Erian Armanios b a b
School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332, United States Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, 500 West First Street, Arlington, TX 76019, United States
a r t i c l e
i n f o
Article history: Received 25 September 2010 Accepted 1 March 2011 Available online 4 March 2011 Keywords: B. Buckling C. Finite element analysis (FEA) D. Mechanical testing
a b s t r a c t A composite panel, with a [0/90] unsymmetric layup, exhibits bi-stable anticlastic equilibrium configurations. One way to provide morphing capabilities to such a panel is attained by attaching piezoelectric actuators to its surface. In such a design an electric field is applied to the piezoelectric actuators in order to trigger snap-through behavior of the panel/actuator assembly. A simple cantilever beam configuration is discussed to perform actuator assessment. Results are compared to with predictions from Extended Classical Lamination Theory and finite element analysis. Guidelines are proposed to design a morphing unsymmetric panel/actuator assembly. The panel/actuator(s) assembly is manufactured accordingly. Snap-through experiments are conducted and the ABAQUS finite element software is used to predict snap-through behavior. Analysis predictions are in good agreement with test results. Published by Elsevier Ltd.
1. Introduction Observations of birds and/or flying insects and contrasting these with current technologies inspired researchers to depart from rigid airframe structures. Rigid airframe structure limit optimized aircraft performance when multiple missions are required. The wing as the major lifting structure received most focus in terms of morphing. Several attempts to provide morphing wings, in an abstract sense, have been recorded by the research community. Among these attempts are NASA’s efforts [1–3] focusing on biologically inspired morphing wings. McGowan et al. [2] investigated several morphing wing designs such as the Gull wing, the Shark wing and the Hyper-Elliptic Cambered Span (HECS) wing as shown in Fig. 1. As underscored by McGowan et al. [2] biologically inspired morphing wings are not to be understood as straightforward mimicking of birds and/or flying insects. Rather, they should be the outcome of learning from nature’s elegant designs in order to foster more robust and versatile enhanced engineering designs. For example a morphing wing from the structural perspective should not entail additional complexity due to the actuation process, while providing adequate stiffness as a lifting surface and being adaptable with optimal shape over a broader range of flight missions. Different linkage mechanism systems were proposed to provide the structural design of the HECS morphed wing as presented by ⇑ Corresponding author. Tel.: +1 678 462 2597. E-mail address: samer.tawfi[email protected] (S.A. Tawfik). 1359-835X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.compositesa.2011.03.001
Stubbs [4] and Wiggins [5], Fig. 2, or tendon actuated compliant cellular structure, as presented by Ramrakhyani [6], and shown in Fig. 3. Such mechanical systems provide several challenges due to the complexity of their design, actuation and maintenance. Incorporation of these mechanical systems in design results in weight penalty that may render the design unfeasible especially in the case of Uninhabited Aerial Vehicles (UAVs). On the other hand thin unsymmetric laminated composites possess bi-stable configurations [7]. This feature makes them attractive candidates in building morphing structures. Tawfik and Armanios [8] investigated the design concept of an UAV morphing wing utilizing bi-stable composites. This investigation provided an understanding to the design of actuated bi-stable panels of different planform shapes suitable for an UAV wing. Schultz and Hyer [9] investigated triggering snap-through in an unsymmetric panel using Macro-Fiber Composite (MFC) actuators. Also other researcher [10–12] investigated the same option of using MFC actuators to trigger snap-through in an unsymmetric panel. MFC actuators are stiff and found to decrease the panel curvature when bonded its surface. The current research investigates using different options of piezoelectric actuators. This investigation is performed for both MFC and Polyvinylidene Fluoride (PVDF) piezoelectric actuators. PVDF is commercially available in various film thicknesses; hence more suitable for fabrication into refined designs. Typical properties of a piezoelectric PVDF film are provided in Table 1. While properties for MFC actuators are provided in Table 2.
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Fig. 1. Biologically inspired morphing wings suggested by NASA [2] (a) Gull wing, (b) Shark wing and (c) Hyper-Elliptic Cambered Span (HECS) wing.
Fig. 2. General topology of a quaternary-binary cross linked mechanism [4,5].
2. Analysis and modeling
Fig. 3. Bench-top demonstration model of compliant cellular structure [6].
Table 1 Piezoelectric (PVDF) film properties.
Thickness Piezo strain constants Young’s modulus Max operating voltage Breakdown voltage
d31 d33 YE Vmax
Property
Units
9, 28, 52, or 110 23 1012 33 1012 2–4 30 80
lm m/V m/V GPa V/lm V/lm
A simple model is developed to assess actuator performance. In this model piezoelectric actuator(s) are used to control the tip displacement of an unsymmetric composite cantilever beam. This allows identifying the controlling parameters that govern the morphing panel design. Guidelines in terms of actuator parameters are extracted and used to form a basis to design a morphing panel. Piezoelectric actuators are used to provide extension/contraction as shown in Fig. 4. An actuator will shrink or expand depending on the polarity of the applied electric field through the thickness. PVDF actuators are transversely isotropic in terms of material properties, and their electrodes are aligned parallel to one direction only, referred to as actuation direction. On the other hand MFC actuators possess orthotropic mechanical and piezoelectric properties. A thermal analogy approach [13,14] is used to model piezoelectric behavior. In this analogy an applied electric field is modeled as thermal load while equivalent piezoelectric constants are used to replace the thermal expansion coefficients. Equivalent piezoelectric constants can be easily derived in the case of an extension actuation mechanism, i.e. where the electric field and poling is through the thickness. The three-dimensional thermo-elastic strain–stress relations are
Table 2 Macro-Fiber Composite (MFC) piezoelectric actuators properties. Property
Units
300
lm
d31 d33
460 1012 210 1012
m/V m/V
d31 d33
m/V m/V GPa
Thickness High-field (|E| > 1 kV/mm) Piezo strain Constants Low-field (|E| < 1 kV/mm) Piezo strain constants Young’s moduli
Y E1
400 1012 170 1012 30.336
Y E2
15.857
GPa
Shear modulus
GE12
5.515
GPa
Poisson’s ratio Max operating voltage
m12
0.31 1500 500
V V
Vmax
Fig. 4. PVDF actuator in an extension actuation mechanism.
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feg ¼ ½Sfrg þ fagDT
ð1Þ
where {e} are the total strains, [S] is the compliance matrix, {r} are the mechanical stresses, {a} are the thermal expansion coefficients and DT is the temperature difference. A similar equation can be written in the case of extension actuation mechanism, i.e. only the electric potential w3 across the thickness is nonzero
feg ¼ ½Sfrg þ fdg
w3 t
ð2Þ
where {d} is the piezoelectric strain coefficient vector and t is the piezoelectric actuator thickness. Relationships establishing a thermal analogy of piezoelectric actuator response are achieved by comparing Eqs. (1) and (2). For a given actuator, the piezoelectric strain coefficient vector can be used to provide the equivalent thermal expansion coefficients
a1 ¼ Fig. 5. Cured shapes of a rectangular laminate with unsymmetric stacking.
d31 ; t
a2 ¼
d32 ; t
a12 ¼ 0; DT ¼ w3
ð3Þ
This approach is well suited for finite element formulation. Modeling electric actuation of an orthotropic piezoelectric actuator based on the thermal analogy approach can be easily implemented in ABAQUS. 2.1. Cantilever beam configuration
Fig. 6. Laminate/actuator structure as cantilever beam.
Fig. 7. Cantilever beam cross-section detail.
The stability characteristics of unsymmetric cross-ply rectangular panel, shown in Fig. 5, were investigated in [15]. ABAQUS finite element results [15] were in good agreement with those of the Extended Classical Lamination Theory (ECLT). ABAQUS finite element results, in agreement with observations, also reported a critical sidelength-to-thickness ratio below which the rectangular laminate ceases to possess two equilibrium configurations, i.e. become mono-stable. In this case, the laminate will cease to maintain the equilibrium configuration identified by being curved along the shorter side, and retain only the equilibrium configuration associated with being curved along the longer side. A mono-stable panel, curved along its length, is used in a cantilever beam setup. Its curvature can be controlled by attaching two layers of actuators to both the top and bottom surfaces of the laminate. This cantilever beam setup will be used to study actuation requirements and compare predictions from different analysis methods. A schematic of an unsymmetric cross-ply rectangular cantilever laminate with actuators bonded to its top and bottom surfaces is shown in Fig. 6. The laminate/actuator beam has a length L = 200 mm, a width w = 10 mm and a total thickness t. The unsymmetric cross-ply laminate has two plies each of thickness tlayer while each piezoelectric actuator has a thickness tpiezo as shown in the cross-section detail of Fig. 7. The curvature j of the cured laminate before attaching the actuators is associated to a tip deflection di. Upon bonding the actuators
Fig. 8. (a) Cured cantilever laminate with unsymmetric stacking. (b) Bonded laminate/actuator assembly (no actuation).
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ing thermal expansion coefficients, Ep the Young’s modulus of the PVDF actuators, ap the equivalent thermal expansion coefficient, Ac the cross-sectional area of a single composite lamina, and Ap the cross-sectional area of a single PVDF layer. For effective actuation, the moments in the unsymmetric laminate due to thermal loads associated with the cure cycle should be balanced by the moment introduced by the piezoelectric layers upon actuation.
j¼ Fig. 9. Laminate/actuator structure in direct actuation.
to the laminate its initial curvature will decrease and consequently the tip deflection will be reduced from di to do, as shown in Fig. 8. The deformed shape of the laminate/actuator structure is assumed to conform to a portion of a cylindrical shape. Hence throughout the calculations in this section, the curvature j will be used to evaluate actuator performance. Fig. 9 shows a laminate/actuators assembly with the electric potential being applied through the thickness of each actuator to cause the top actuator to contract and the bottom one to extend. Actuation in this sense, termed direct actuation, results in reducing the beam curvature and minimizes the tip deflection or even cause the beam to acquire an opposite curvature. Two factors control the beam curvature namely the actuator thickness and the level of actuation, i.e. applied electric potential. Causing the assembly to curve in the opposite direction is of no significance and successful actuation will be based on reducing its curvature to zero. This problem is studied initially utilizing PVDF actuators. They will be subsequently replaced by MFC actuators for performance comparison. 2.2. Beam Theory (BT) model The engineering beam theory is used to calculate the beam curvature due to thermal loads representing the curing cycle and actuation potential, respectively. From equilibrium of forces acting on the composite laminate layers and piezoelectric actuator layers, as shown in Fig. 10
P1 P2 P3 þ P4 ¼ 0
ð4Þ
Using thermal analogy and assuming perfect bonding the tip extension d can be written as
P3 L P1 L d ¼ d3 d3 ¼ ap DVL ¼ d1 þ d1 ¼ a1 DTL þ Ep Ap E11 Ac P2 L P4 L ¼ d2 d2 ¼ a2 DTL ¼ d4 þ d4 ¼ ap DVL þ E22 Ac Ep Ap
ð5Þ
where E11 and E22 denote Young’s moduli of the composite ply in the principal material directions respectively, a1, a2 the correspond-
MT þ ME EI
ð6Þ
for MT is the bending moment due to curing, ME is the bending moment due to actuation and EI is the total flexural stiffness of the bonded laminate/actuators assembly. The mechanical properties of Hexcel IM7/8551-7 graphite/ epoxy prepreg are used throughout the analysis and provided in Table 3. The maximum curing temperature is 177 °C and the room temperature is 21 °C. The curvature is obtained using beam theory at values of curing cycle temperature and applied electric potential. Two limiting cases provide physical understanding of actuator response, namely no actuation and full actuation. The case at no actuation represents the stiffening effect associated with bonding the actuators to the unsymmetric laminate with no applied electric potential. While in the full actuation case the maximum operating voltage is applied after bonding. 2.3. Extended Classical Lamination Theory (ECLT) The Rayleigh–Ritz energy based extension to the Classical Lamination Theory accounting for geometrically nonlinear strain–displacement model is adopted. This analysis is developed by Schultz and Hyer in [9] and its details can be found in their work. In this section, ECLT predictions are based on using a 4-term displacement field. 2.4. ABAQUS finite element analysis The beam/actuator assembly is modeled in the ABAQUS finite element code for two cases. In the first case the nominal thickness of a composite ply is approximated as 100 lm and in the second case the thickness is chosen to be 300 lm. These correspond to sidelength-to-thickness ratios of 50 and 16.67, respectively. Therefore neither one of those cases represents a bistable panel, as the critical sidelength-to-thickness ratio above which a panel possesses two stable configurations is 82.88 as provided in [16]. The finite element model used a total of 240 shell elements S4R to represent the three layers, i.e. the unsymmetric panel and the top and bottom actuators. A comparison of the actuator stiffening effect predicted using BT, ECLT and ABAQUS models at tlayer = 100 lm is
Fig. 10. Forces and displacements in the beam due to thermal and electric loads.
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Table 3 Elastic properties IM7/8551-7 graphite/epoxy prepreg. Material
E11 (GPa)
m12 E22 G12 (GPa) (GPa)
141.18 7.20 IM7/8551-7 graphite/epoxy prepreg
4.45
a1
a2
t (lm)
(106/°C) (106/°C)
0.30 0.14
30.98
138.75
Fig. 13. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 100 lm, full actuation.
Fig. 11. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 100 lm, no actuation.
Fig. 14. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 300 lm, full actuation.
Fig. 12. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 300 lm, no actuation.
provided in Fig. 11. The results obtained from all three methods are in good agreement with each other. Similar trends among all three models for the case of a moderate thickness lamina, tlayer = 300 lm are observed as shown in Fig. 12. An electric potential at Vmax is applied in both cases, tlayer = 100, 300 lm, in order to study the effect of actuation and the results are shown in Figs. 13 and 14, respectively. In both cases BT and ABAQUS results are in good agreement. While values of curvature obtained from the ECLT are in good agreement with the general trend except in the vicinity of zero curvature. Two steps in predicted ECLT curvature are observed at both the beginning and the end of a zero curvature plateau. Inspecting the curvatures predicted by the 4-term ECLT in both cases it is found that at the first step the curvature along the length of the beam, jy, changes from 1 m1 to nearly zero. The associated curvature in the width direction, jx, changes from being nearly zero to 1 m1 indicating snap-through from one equilibrium shape to the other. The zero curvature plateau represents the
Fig. 15. Curvature, jx, obtained from BT, ECLT and ABAQUS for tlayer = 100 lm, full actuation. Corresponding beam structure shapes illustrated.
beam is in its second equilibrium configuration (flat along the length and curved in the width direction). While the second step represents the case when the moment developed by the actuators forces the beam to go through same sense bending similar to a retractable tape or tape springs. Same sense bending of tape springs is extensively studied in [17,18]. For illustration purpose, the curvature as a function of PVDF thickness obtained using ECLT from Fig. 13 and regenerated in Fig. 15, is divided into three zones with the corresponding beam structural shapes provided schematically. Further details regarding this comparison are provided in [16].
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In conclusion, the 4-term ECLT lamination theory suffered from a numerical anomaly when studying a mono-stable unsymmetric cross-ply laminated. In literature and to the best knowledge of the authors the ECLT was not used to predict the minimum or critical sidelength-to-thickness ratio responsible for losing one of the stable configurations of a bistable panel. Also, Dano documented in [19] that ECLT predictions deviate from test data in the vicinity of the snap-through in the case of bistable panels. The deviation associated to ECLT predictions in the vicinity of snap-through in the case of bistable panels is paralleled by predicting a zero curvature plateau in the case of mono-stable laminate. Table 4 provides the thickness of the PVDF layers required for actuation, i.e. associated with bringing the curvature of the beam/actuators assembly to zero. The table also includes the initial curvature of the unsymmetric laminate, before bonding the PVDF layers to its surfaces, and the curvature of the beam/actuators assembly at no actuation, i.e. V = 0. It can be concluded that successful actuation of an unsymmetric laminate with thickness 2t requires two PVDF actuator layers each of thickness approximately 3t. Bonding such thick actuators to the unsymmetric laminate leads to nearly 90% loss of its initial curvature. Using thicker actuator layers may lead to causing the laminate to lose its curvature entirely. Consequently using PVDF actuators for this configuration requires the electric potential to be always at Vmax in order to keep the actuator thickness to a minimum and consequently decrease the stiffening effect. A comparison between actuator types is subsequently conducted using ABAQUS finite element analysis in which it was taken into consideration that the MFC actuators are available commercially only in 300 lm thickness. Stiffening and actuation effects are studied for the case of bonding MFC actuators to cantilever beams of ply thicknesses 100 and 300 lm and presented in Table 5. Actuator stiffening effect is more pronounced when using MFC actuators. For the case of the 100 lm ply thick the curvature is reduced by 98.7% when using MFC actuators versus 89.7% when using PVDF actuators although both actuator types are nearly of the same thickness. In the case of 300 lm ply thick the curvature is reduced by 84.3% when using MFC actuators versus 89.3% when using PVDF actuators whereas the MFC actuator thickness is approximately one third of its PVDF counterpart. Stiffening effect can be explained in terms of actuator stiffness as the Young’s modulus of MFC actuators is one order of magnitude higher than the PVDF counterpart. On the other hand for both 100 and 300 mm
PVDF thickness (lm) Initial Curvature (m1) Curvature upon bonding (m1)
Ply thickness 300 lm
BT
ECLT
ABAQUS
BT
ECLT
ABAQUS
306 18.44 2.1
239 17.69 3.1
360 18.56 1.9
931 6.15 0.681
788 5.89 1.91
1060 6.13 0.65
Table 5 Actuator comparison for a composite beam with [90/0] layup. Ply thickness 100 lm PVDF Actuator thickness (lm) Initial curvature (m1) Curvature upon bonding (m1) Curvature upon actuation (m1)
360 1.9 0
MFC 300 18.56 0.23 4.56
Ply thickness 300 lm PVDF 1060 0.65 0
Actuator thickness (lm) Initial curvature (m1) Curvature upon bonding (m1) Curvature upon actuation (m1)
Ply thickness 100 lm
Ply thickness 300 lm
M=0
M=0
M–0
300 18.56 0.23 2.17 4.56 2.62
M–0
300 6.13 0.96 1.24 0.94 0.396
thicknesses, MFC actuators have higher actuation authority or effectiveness than PVDF actuators. This is reflected in the resulting curvature upon actuation as MFC actuators can bring the curved beam to an opposite curvature. One can conclude therefore that MFC actuators are more suitable to induce snap-through behavior provided their high stiffening effect is reduced. Due to their high actuation effectiveness they are not required to be of the same size as the actuated panel. Therefore an actuator with a fractional surface area of the actuated panel is adequate, leading to a decrease in its stiffening effect. Further reduction could be achieved through the means of bonding the actuator to the bistable panel. Two options are evaluated. The first is to force the curved panel into a flat configuration then bond the actuator to its surface. While the other option is to bend the actuator causing it to conform to the panel’s curvature then bond them together. So far all presented numerical results are associated with the first option. The second option can be simulated in ABAQUS by applying moments at the edges of the actuators to conform to the panel curvature. Considering only MFC actuators, a comparison between both bonding options is presented in Table 6 for cantilever beams with thickness of 100 and 300 lm. Predictions corresponding to the second option of bonding show 88.3% and 79.7% loss of curvature for 100 and 300 lm ply thickness, respectively. The loss of curvature in this case is less than the corresponding values associated to the first option of bonding. Also, a higher loss of curvature is predicted for a thicker composite panel. Considering the gain attained using the second option of bonding it will be adopted to design and manufacture morphing bistable panel/actuators assembly using MFC actuators in subsequent sections of this work.
3. Experimental and analytical results
Table 4 PVDF actuator requirements for a composite beam with [90/0] layup. Ply thickness 100 lm
Table 6 MFC Actuator bonding options comparison.
MFC 300 6.13 0.96 0.94
In this section the design of an unsymmetric rectangular laminate/actuator assembly is considered based on the actuator assessment results. A rectangular unsymmetric panel of width 65 mm and length 85 mm is manufactured from Hexcel IM7/8551-7 graphite/epoxy prepreg with a stacking sequence of [0/90]T. The description of its stability behavior follows the schematic of Fig. 5. Two MFC actuators of dimensions 40 mm 10 mm are centrally bonded to the top and bottom surfaces of the unsymmetric panel using Loctite’s E120 HP Epoxy. The major axes of both actuators are aligned along the shorter dimension of the composite panel. The actuators are bent to conform to the curved surface of the cured panel and the adhesive is applied. The panel/actuators assembly is placed in a vacuum bag to maintain its curvature and placed in an autoclave for three hours at 60 °C with the vacuum on to ensure better bonding. Figs. 16 and 17 show side-byside panels with and without assembled actuators for a qualitative comparison of the bonding method. ABAQUS finite element simulation was used to predict the equilibrium shapes of the panel/actuators assembly and the
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Fig. 16. Bonded panel assembly versus identical dimension bistable panel (first equilibrium shape).
Fig. 17. Bonded panel assembly versus identical dimension bistable panel (second equilibrium shape).
voltage required to trigger snap-through behavior. Four-noded, reduced integration, doubly curved shell elements (S4R) are used to model both the unsymmetric laminate and the actuators. The unsymmetric panel is modeled using 884 elements, while for each actuator 64 elements are used to model its active portion and 156 elements to model its casing. The thermal analogy approach is used to model piezoelectric performance. Moreover, the nonlinear piezoelectric properties of the MFC actuator are incorporated in the thermal analogy model accounting for different ranges of piezoelectric constants as provided in Table 2. Moments are calculated and applied at the actuator edges to simulate bonding in the panel’s curved configuration. The three layers of elements representing the unsymmetric panel and the two actuators are tied together making sure to offset their respective neutral axis based on their position in the assembly. The panel/actuators assembly is held fixed in space at the midpoint of the panel by constraining all its degrees of freedom. The displacements of the four corner points are monitored throughout the analysis. The finite element simulation is carried out in three linear/nonlinear analysis steps. The first step is a linear eigenvalue buckling problem performed to obtain the mode shapes of the unsymmetric panel corresponding to the temperature loads of the cure cycle. A second step, nonlinear analysis, is carried out to account for the geometric imperfections of the panel expressed in percentage of the mode shapes obtained in the first step. Also moments are applied at the actuators’ edges to simulate the second option of bonding and hence predict the cured/bonded shape of the panel/ actuators assembly. The final step, nonlinear analysis, employing the thermal analogy approach is used to predict the voltage required by the actuators to trigger snap-through or snap-back. Further details of the finite element simulation are provided in [16]. Finite element predictions of the cured shapes of an unsymmetric panel and panel/actuators assembly are obtained as one quantitative measure for the manufacturing method. jx is the curvature of panel/actuators assembly in the first equilibrium shape and jy is its curvature in the second shape. Accordingly, these curvatures are predicted for the panel/actuators assembly. A reference case is cho-
sen to be an identical unsymmetric cross-ply panel with no bonded actuators. Comparisons of center curvatures between the actuated and the reference panels are provided in Table 7. The loss of curvature due to bonding the actuators, as a percentage of the reference panel curvature, is provided in parenthesis in Table 7. The efficiency of the bonding method is indicated by the percentage loss in curvature. Parallel to the actuators, the loss of curvature is 25.56% due to the stiffening effect of the narrow side of the actuators. While normal to the actuators, where the stiffening effect is expected to be a maximum, the loss of curvature is only 30.56%. Therefore bonding the actuators in the curved configuration contributes to minimizing the effect of stiffening on curvature. The equilibrium shapes of the panel/actuators assembly predicted by ABAQUS finite element analysis are shown in Fig. 18. The equilibrium shapes of the manufactured panel/actuators assembly are shown in Fig. 19 for qualitative comparison. The experimental setup, shown in Fig. 20, is used to measure the required voltage to trigger snap-through behavior. It includes three posts, one is used to suspend the panel/actuators assembly by a thread and the other two are used to support the actuators wiring. The setup consists of a variable voltage DC power supply, two miniature DC to High Voltage (HV) converters, a voltmeter and the manufactured panel/actuators assembly. A variable voltage DC power supply, Agilent E3648A, is used to generate a ramp signal. This ramp signal is fed into each of the miniature HV-DC converters, G05 and G15 obtained from AMCO. The isolated output of these HV-DC converters is linearly proportional to the input. The G05 amplifies a 12 V signal to 500 V while the G15 amplifies the
Table 7 Curvature comparison between actuated and reference panels.
jx (m1) jy (m1)
Reference panel
Actuated panel
11.995 11.20
8.18 (25.56%) 7.78 (30.56%)
755
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Fig. 18. Predicted equilibrium shapes of the panel/actuators assembly.
Fig. 19. Equilibrium shapes of the manufactured panel/actuators assembly.
12 V signal to 1500 V, according to the G series datasheet [20]. Each of these converters weighs 43 g and has dimensions of 38.1 mm 38.1 mm 16.0 mm. The voltmeter is used to check the amplified voltage applied to the actuators. According to observations in [9,19] it is noted that an unsymmetric panel undergoes a loss of curvature during the post-cure stage. However this loss of curvature ceases after a limited period of time. On the other hand an unsymmetric panel/actuators assembly is not stress free in either equilibrium configuration due to actuator bonding. In the current work, the panel/actuators assembly was snapped back-and-forth several times before performing the experiment. These pre-tests were carried out to eliminate any bias resulting from maintaining the assembly for some period of time in one equilibrium shape rather than the other. The experiment was repeated four times, for both snap-through and snapback, and the required voltages were averaged and are provided in Table 8. Percentage deviations from the averaged values are shown in the same table.
Table 8 Required voltage to trigger snap-through behavior. Required voltage (V) Top actuator
Snap-through Snap-back
Percentage difference (%) Bottom actuator
Experiment
FEA
Experiment
FEA
460 2217
495.5 2416
1380 739
1489 805.2
7.89 8.97
It can be seen from the results in Table 8 that trends in predicting the required voltage to trigger snap-through/snap-back behavior are captured by the numerical finite element analysis predictions. Another observation is that the predicted values using the finite element analysis represent an upper bound to the experimentally obtained ones. The differences between the finite elements results and the experiment could be due to the post-cure loss of curvature in unsymmetric panels, the assumption of perfect bonding between the panel and the actuators, the exact alignment of the actuators along the x-direction of the laminate, and/or uncoupled electrical, mechanical and thermal effects during bonding and actuation. In case when modeling combined thermal and piezoelectric behavior is required a user shell element can be developed within ABAQUS. Moreover, the finite element simulation considers all material properties independent of the cure cycle temperature and does not include any chemical reaction or conservation of heat during the cure simulation part.
4. Summary and concluding remarks
Fig. 20. Experimental setup.
A comprehensive study for design, analysis and manufacturing of morphing composites via piezoelectric actuators has been
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presented. The developed simplified beam theory model was validated by comparison of results with ABAQUS finite element and the Extended Classical Lamination Theory. The behavior associated with the Extended Lamination Theory for zero curvature was isolated. A comparison between PVDF and MFC actuators is performed leading to pertinent design, manufacturing and analysis guidelines for a rectangular unsymmetric panel/actuators assembly. In terms of manufacturing it is recommended to bond the actuators with their axis along the shorter side of the rectangular panel in its curved configuration. This allows bonding of actuators to an unsymmetric bistable panel while minimizing the loss of its bistable characteristics and enhancing sensitivities of snap-through/ snap-back response. In terms of analysis, the previously developed finite element methodology [15] was extended to include actuation via a thermal analogy approach using the ABAQUS code. The developed methodology provides a suitable tool for both design and analysis of morphing unsymmetric structures. It accurately models actuator bonding to a flat or curved panel, accounting for the actuator nonlinear properties as well as the requirements for inducing snapthrough. In terms of experimental effort, a morphing panel is manufactured and its predicted stability characteristics verified. The panel successfully snapped back-and-forth utilizing two MFC actuators that represented, collectively, only 14% of the panel surface area. The successful actuation of the manufactured panels points to their viability for morphing wing applications to Unmanned Aerial Vehicles (UAV). References [1] McGowan AR, Washburn AE, Horta LG, Bryant RG, Cox DE, Siochi EJ, Padula SL, Holloway NM. Recent results from NASA’s morphing project. In: Proceedings of SPIE’s 9th annual international symposium on smart structures and materials; 2002. p. 97–111. [2] McGowan AR, Cox DE, Lazos BS, Waszak MR, Raney DL, Siohci EJ, Pao SP. Biologically-inspired technologies in NASA’s morphing project. In: Proceedings of SPIE’s 10th annual international symposium on smart structures and materials; 2003. p. 1–13.
[3] McGowan AR. AVST morphing project research summaries in fiscal year 2001, NASA/TM-2002-211769; 2002. [4] Stubbs M. Kinematic design and analysis of a morphing wing. Master thesis, Blacksburg, Virginia: Virginia Polytechnic Institute and State University; 2003. [5] Wiggins L. Structural design and analysis of a kinematic mechanism for a morphing Hyper-Elliptic Cambered Span (HECS) wing. Master thesis, Blacksburg, Virginia: Virginia Polytechnic Institute and State University; 2003. [6] Ramrakhyani DS. Morphing aircraft structures using tendon actuated compliant cellular truss. Ph.D. dissertation, Pennsylvania: Pennsylvania State University; 2005. [7] Hyer MW. Some observations on the cured shapes of thin unsymmetric laminates. J Compos Mater 1981;15:175–94. [8] Tawfik S, Armanios E. Preliminary investigation of a morphing wing design concept employing unsymmetric cross-ply composites stability characteristics. In: Proceedings of 21st annual technical ASC conference, Dearborn, MI: University of Michigan-Dearborn; 2006. [9] Schultz MR, Hyer MW. A morphing concept based on unsymmetric composite laminates and piezoceramic MFC actuators. In: 45th AIAA/ASME/ASCE /AHS/ ASC structures, structural dynamics & materials conference, Palm Springs, California; 2004. [10] Hufenbach W, Gude M, Czulak A. Actor-initiated snap-through of unsymmetric composites with multiple deformation states. J Mater Process Technol 2006;175:225–30. [11] Portela PM, Camanho PP, Weaver PM, Bond IP. Analysis of morphing, multi stable structures actuated by piezoelectric patches. Comput Struct 2008;86:347–56. [12] Bowen CR, Butler R, Jervis R, Kim HA, Salo AIT. Morphing and shape control using unsymmetrical composites. J Intell Mater Syst Struct 2007;18:89–98. [13] Soderkvist J. Similarities between piezoelectric, thermal and other internal means of exciting vibrations. J Micromech Microeng 1993;3:24–31. [14] Dong X, Meng G. Dynamic analysis of structures with piezoelectric actuators based on thermal analogy method. Int J Adv Manuf Technol 2006;27:841–4. [15] Tawfik S, Tan X, Ozbay S, Armanios E. Anticlastic stability modeling for crossply composites. J Compos Mater 2007;41:1325–38. [16] Tawfik S. Stability and morphing characteristics of bi-stable laminates. Ph.D. dissertation, Atlanta, Georgia: School of Aerospace Engineering, Georgia Institute of Technology; 2008. [17] Yee JCH, Soykasap O, Pellegrino S. Carbon fiber reinforced plastic tape springs. In: 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and material conference and exhibit, Palm Springs, CA; 2004. [18] Silver MJ, Hinkle JD, Peterson LD. Controlled displacement snap-through of tape springs: modeling and experiment. In: 46th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and material conference and exhibit, Austin, Texas; 2005. [19] Dano M. SMA-Induced Deformations in General Unsymmetric Laminates. Ph.D. Dissertation, Blacksburg, Virginia: Virginia Polytechnic Institute and State University; 1997. [20] Miniature DC to HV DC converters – G series datasheet. EMCO High Voltage Corp.; 2008