In this section, first the FE formulation for the smart annular plate is verified.
Next, the control capability of the SPFC/CPFC actuator is substantiated through the numerical evaluation of frequency responses of the simply-supported smart annular plate. The geometrical properties of the overall annular plate are considered as, ri 0.25 m, ro 1 m, h4 mm, hp 250 µm, 60oand 30o . The substrate annular plate is considered to be made of Aluminum (E = 70 GPa,
= 0.3, = 2700 kg/m3) while the material properties of the smart actuator patches are given in Table 2.3. The density of actuator patches is taken according to the rule of mixture (c 3666 kg/m3). For the linear relation between the electric field (E) and the applied voltage (V ) (Fig. 2.15), the parameters Gr, G and Gz have constant values which are taken from section 2.6.2. For the constant values of these parameters, the damping coefficient matrix (Ct) is independent of the velocities (wq) at sensing-points, and it yields linear governing equations of motion (Eq. (3.42)) of the overall smart annular plate. In order to excite first two bending modes (Fig. 3.7) of the overall annular plate, the transversely distributed harmonic mechanical load is considered as,
( , , ) 1 cos( ) ej t
p r t p , j 1 (3.43)
where, p indicates the intensity of the transversely distributed load and is the driving frequency. For the steady-state linear vibration of the overall plate under this mechanical excitation (Eq. (3.43)), the nodal displacement vector (
p) can be written as (Meirovitch, 2007),j t
p e
, (R jXI) (3.44)
where, is a complex nodal displacement vector and R/XI is its real/imaginary counterpart. Substituting Eq. (3.44) in the linear form of governing equations of motion (Eq. (3.42)), the following algebraic equations can be obtained,
2 jCt t
PM0 (3.45)The complex nodal displacement vector ( ) can be obtained by solving Eq. (3.45), and the absolute value of the same ( ) is the nodal displacement-amplitude vector for the linear steady-state vibration of the smart annular plate.
Table 3.2 Comparison of the first two dimensionless natural frequencies ( 0 0 0r h D , 1 1 0r h D ) of the simply-supported annular plate (
p 0
h ) with the similar results given in (Chakraverty et al., 2001) ( = 1/3, r ri / 0 = 0.4, h r/ 0 = 0.001), DEh3/ 12(1
2)).Source 0 1
Chakraverty et al., 2001 28.08 30.09 Present FE results 28.35 30.43
In order to verify the present FE model of the smart annular plate, the dimensionless natural frequencies (0, 1) for first two bending modes of vibration of the simply-supported annular plate (hp 0) are computed and compared with the similar results available in the literature (Chakraverty et al., 2001). This comparison is illustrated in Table 3.2. Table 3.2 shows a good agreement of present results with similar results reported in an earlier study (Chakraverty et al., 2001). This comparison verifies the accuracy of the present FE model for the annular plate. Next, the modelling of electro-elastic coupling is verified. Since similar smart composite actuator is not available in the literature, this verification is carried out considering a circular substrate plate integrated with a vertically poled monolithic piezoelectric layer. A negligibly small thickness (h0) of substrate plate is considered, and the transverse deflections at different radial locations of the simply-supported overall smart circular plate are computed for an applied electric field across the top and bottom electrode-surfaces of the piezoelectric layer. These results are then compared with those for a similar smart circular plate studied by Dong et al. (Dong et al., 2007). Figure 3.8 illustrates this comparison, and it shows an excellent agreement of present results with the similar published results (Dong et al., 2007). Thus, the present FE formulation is verified for handling the electro-elastic coupling in piezoelectric actuator.
Chapter 3: Control capability of SPFC/CPFC
Fig. 3.8 Verification of FE formulation for handling electro-elastic coupling in piezoelectric actuator.
In order to present the frequency responses of the overall annular plate, the transverse displacement-amplitude (Aw) of flexural vibration at a point (
(rori) / 2,/ 4,0) on the plate is computed at every frequency and represented by, ( w/ )
W A h . The present SFC/CPFC actuators are utilized to reduce the amplitude of vibration of the overall annular plate by inducing smart damping within it (overall plate). The maximum reduction of amplitude due to smart damping occurs at the resonant frequency for any mode of vibration. So, for every mode of vibration, the efficiency of actuators (SPFC and CPFC) in inducing smart damping within the overall plate is measured in terms of the change of transverse displacement-amplitude (Wpeak) at the resonant frequency. All actuator-patches are considered to be activated by a uniform value of the control gain (kdq kd, q 1, 2, 3, 4). But, the velocity-amplitudes corresponding to the locations of velocity- sensors may be of different values. So, the control-voltages for four actuator- patches are computed separately at any frequency of vibration, and the maximum one (Vm) is taken at that frequency for presenting the numerical results. Figure 3.9(a) demonstrates the frequency responses of the smart annular plate when the
actuator-patches are either made of SPFC or made of CPFC. The corresponding variations of control-voltage (Vm) are also illustrated in Fig. 3.9(b).
Fig. 3.9(a) Controlled frequency responses of the annular plate integrated with the patches of SPFC/CPFC actuator, (b) variations of corresponding control voltage (p= 0.3 N/m2, kd 100, 200).
It may be observed from Fig. 3.9 that the SPFC/CPFC induces significant damping within the overall annular plate in the expense of reasonable control voltage. The smart damping in the overall plate can also be increased by increasing the value of the control-gain (kd). But, the corresponding required control-voltage (Vm, Fig. 3.9(b)) remains almost the same because of the constant
Chapter 3: Control capability of SPFC/CPFC
value of load-parameter (p). For a particular value of the control-gain (kd), Fig.
3.9 illustrates an important observation that the induced damping in the overall plate by CPFC actuator is a little lesser than that by SPFC actuator in the expense of more control-voltage (Vm). In order to exemplify this difference for every mode (first and second modes) of vibration, the variations of peak amplitude (Wpeak) either with the load parameter (p, a constant value of kd) or with the control gain (kd, a constant value of p) are presented in Figs. 3.10 and 3.11.
Fig. 3.10 Variations of (a) the peak-amplitude (Wpeak) and (b) the corresponding control voltage (Vpeakm ) with the load-parameter (p) for first two bending modes of vibration of the overall annular plate (kd 100).
Fig. 3.11 Variations of (a) the peak-amplitude (Wpeak ) and (b) the corresponding control voltage (Vpeakm ) with control gain kd for first two bending modes of vibration of the overall annular plate (p= 0.3 N/m2).
Figure 3.10 illustrates the variations of peak amplitude (Wpeak) and the corresponding control voltage (Vpeakm ) with the load-parameter (p) for a constant value of the control gain (kd 100). Figure 3.11 demonstrates the variations of the same parameters (Wpeak,Vpeakm ) with the control gain (kd) for a constant value of the load-parameter (p0.3 N/m2). In comparison to CPFC actuator, it may be observed from Figs. 3.10 and 3.11 that the SPFC actuator induces more damping in the expense of lesser control-voltage. This difference also increases as the value of Wpeak increases either by increasing the value of load-parameter (p, a constant value of kd) or by reducing the value of control gain (kd, a constant value ofp).
However, from these observations (Figs. 3.9-3.11), it may be concluded that the control capability of PFC actuator with the cylindrical principal material system can be improved by the use of short/discontinuous piezoelectric fibers instead of the similar fibers in continuous form. The use of the short/discontinuous fibers instead of the continuous fibers also facilitates to have greater flexibility and conformability of the smart actuator.
For constant values of load-parameter (p) and control gain (kd), Fig. 3.12 illustrates the variations of peak-amplitude (Wpeak) and the corresponding control voltage (Vpeakm ) with the thickness (hp) of the actuator-patches for first bending mode of vibration of the overall annular plate. For every value of hp, Wpeak is computed at the corresponding resonant frequency (for the first mode). The variation of Wpeak at a resonant frequency infers the change of damping within the overall annular plate. So, Figs. 3.12(a) and 3.12(b) indicate that a significant damping within the overall annular plate in the expense of a low value of control voltage can be achieved by increasing the thickness of the SPFC/CPFC actuator- patches. But, for constant values of kd and p, this increase of damping continues up to the certain value of increasing hp. Beyond that value of hp, the negligibly small rate of decrease of Wpeak with increasing hp (Fig. 3.12(a)) appears, and it indicates insignificant effect of hp on the smart damping within the overall plate.
Chapter 3: Control capability of SPFC/CPFC
Fig. 3.12 Variations of (a) the peak-amplitude (Wpeak) and (b) the corresponding control voltage (Vpeakm ) with the thickness of (hp) actuator-patches (p0.3 N/m2, kd 100 or 200) for first bending mode of vibration of the smart annular plate.