1
2
11 11
1
k k
h k
k h
B C zdz
, 11 2
1 2 11
1
k k
h k
k h
D z C dz
(3.14)Equations (3.12a-3.12c) are the equilibrium equations corresponding to the geometrically nonlinear electro-elastic bending of a simply-supported beam integrated with a layer of SPFC/CPFC actuator. For numerical evaluation of the bending responses of the overall smart beam, these simultaneous nonlinear algebraic equations are solved using direct iteration method.
Chapter 3: Control capability of SPFC/CPFC
m), and its electro-elastic deflections corresponding to different values of the applied electric field (Ex) are computed. These results are illustrated in Table 3.1 along with the similar results obtained by developing an FE model of the same piezoelectric beam in ANSYS software. It may be observed from Table 3.1 that the present analytical results are in good agreement with the similar results obtained from the model developed in ANSYS software. These comparisons (Fig. 3.2 and Table 3.1) verify the accuracy of the present electro-elastic variational formulation for analyzing the electro-elastic bending of a smart beam.
Fig. 3.2 Comparison of deformed shape of the overall simply-supported beam (
p 0
h , V 0) with that of an identical beam analyzed in (Fallah and Ebrahaminejad, 2014).
Table 3.1 Transverse deflection (
/2
wx l ) of the simply-supported smart beam (h0,hp 0.002 m, l0.2 m) due to the applied electric field (Ex).
Ex
(×103 volt/m)
/2
wx l (×10-7 m) ANSYS Present 88.92 0.3938 0.4025 177.84 0.7876 0.8049 266.75 1.1814 1.2074 355.67 1.5752 1.6098 444.60 1.9690 2.0123
The magnitude of actuation (Ma) by the smart actuator layer at a particular applied load (Q) is measured in terms of the change in transverse deflection (W ) due to an applied voltage (Ma
W V0
W V0 ). For the linear bending deformation of the overall smart structure as well as linear piezoelectric actuator, it is well known that the magnitude of actuation by the actuator remains constant for any value of applied load (Q). The variation in the actuation-capability of the linear piezoelectric actuator with the load (Q) appears when the overall structure starts to deform nonlinearly. So, the boundary value of load (Q) between the linear and nonlinear deformations of the smart beam is first identified to study the performance of SPFC actuator layer for controlling both the linear and nonlinear deformations.Figure 3.3 illustrates the variation of the dimensionless transverse deflection (W ) of the simply-supported smart beam with the mechanical load (Q) in the absence of the external voltage. It may be observed from this figure that the overall beam undergoes nonlinear deformation as the applied load (Q) exceeds its value of 4. It may also be observed from the same figure that the deflection of the overall beam at any applied load (Q) does not change indicatively if the deactivated SPFC
Fig. 3.3 Variation of dimensionless transverse deflections (W ) at the middle point of the overall smart beam with the mechanical load (Q).
Chapter 3: Control capability of SPFC/CPFC
layer (V 0) is replaced by the deactivated CPFC layer (V 0). Thus, for both the smart composite actuators, this value (Q4) of the applied load can be referred to study the linear and nonlinear deformation characteristics of the smart beam.
Figure 3.4 illustrates the variation of the magnitude of actuation (Ma) caused by SPFC/CPFC layer with the applied mechanical load (Q). For an applied voltage ( V 300 volt or 500 volt), it may be observed from this figure that the magnitude of actuation caused by any of the actuators decreases with the increased mechanical load. This may be due to the fact that the rate of change of electrically induced actuation-force is lesser than that of the overall nonlinear stiffness of the smart beam even though both the parameters increase with the increasing mechanical load. For a higher value of the applied voltage, the same figure shows a greater rate of decrease of the magnitude of actuation with the increasing load (
Q). So, an insignificant change in the magnitude of actuation corresponding to an increase of the applied voltage (V ) can be observed at a higher mechanical load (Q). Therefore, any of the actuators (SPFC or CPFC) can be used for controlling small or moderate deflection of the overall smart beam.
Fig. 3.4 Variations of the magnitude of actuation (Ma ) by the SPFC/CPFC actuator with the applied mechanical load (Q) for different values of applied voltage (V ).
It is interesting to observe from the same figure (Fig. 3.4) that the magnitude of actuation caused by SPFC actuator is more than that caused by CPFC actuator for any value of the applied voltage (V). Although a small increase of Ma occurs due to the use of the SPFC actuator instead of the CPFC actuator, but this observation is important for the usefulness of the SPFC in comparison to that of the CPFC. As discussed in the previous chapters (Chapter 1 and Chapter 2), the present SPFC actuator is designed mainly to have a flexible smart composite actuator over the CPFC actuator. Along with this advantage, the analysis shows an additional advantage of greater actuation capability in the use of SPFC actuator.
Figure 3.5 demonstrates the variations of the magnitude of actuation caused by SPFC/CPFC actuator with the applied voltage (V ) for linear (Q1) and nonlinear (Q20) deformations of the overall beam. It may be observed from this figure that the magnitude of actuation linearly increases with the increasing applied voltage (V ) for both the linear and nonlinear deformations of the overall beam. Similar to the previous results (Fig. 3.4), Fig. 3.5 also shows greater actuation capability of SPFC actuator as compared to that of CPFC actuator although it is a small difference in the quantitative measure.
Fig. 3.5 Variations of the magnitude of actuation (Ma ) by the SPFC/CPFC actuator with the applied voltages (V ) for linear (Q1) and nonlinear (Q20 ) deformations of the overall beam.
Chapter 3: Control capability of SPFC/CPFC