Fig. 4.3 Schematic diagram of a differential patch of PFRC removed surrounding a typical test point
z
Viscoelastic layer
h
vPFRC layer
FG substrate plate
a h
ph x
The differential patch of ACLD removed surrounding a typical test point
Y
b
Fig. 4.4 Schematic diagram of a differential patch of ACLD removed surrounding a typical test point
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Since the removal of differential PFRC patch from an important location/point significantly affects the actuation-capability of PFRC layer, the corresponding Wpeakt would have more value. Thus, the surface plot for the variation of Wpeakt in Fig. 4.5(a) implies that the area surrounding the middle point on the top surface of substrate FG plate is the most important location of PFRC patch for effective control of vibration of the overall FG plate. This location is also elaborated in Fig. 4.5(b) by the corresponding Fig.4.5(a) Surface plot for the variation of (Wpeakt ) with the different test points on the top surface of FG substrate plate, (b) corresponding contour plot,(c)-(d) variations of (Wpeakt ) with the different test points along x- direction (y b /2) and y direction(xa/2) (p = 400 N / m2,
=1, kd=100, Tc=Tm=300K, n=2).
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contour plot. Along with this best location of the PFRC patch, other important parameters like its dimensions along x and y directions (Fig. 4.1) can also be estimated from Figs. 4.5(c)-(d). Figures 4.5(c)-(d) represent the variations in the value of Wpeakt along x-direction (at yb / 2) and y-direction (at xa / 2), respectively.
These figures show that the points at the middle zone (xa / 2, yb /2) are almost equally important but the importance of the points towards the edges of the plate decreases at a greater rate. Thus, the dimensions of the PFRC patch along x and y Fig. 4.6 (a) Surface plot for the variation of (Wpeakt ) with the different test points on the top surface of FG substrate plate, (b) corresponding contour plot,(c)-(d) variations of Wpeakt with the different test points along x-direction (y b /2) and ydirection (xa/2) (PFRC patch,
p= 400 N / m2, =1,kd=100, Tm= 300 K, Tc= 500 K, n=1).
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directions can be estimated within the area defined by, x0.1 m, x0.3 m, y0.1 m and y0.3 m (Figs. 4.5(c)-(d)). In the presence of a ceramic-rich surface temperature
(Tc500 K), the location of PFRC patch does not alter as it is indicated in Figs. 4.6(a)-(b).
But, Figs. 4.6(c)-(d) show that the dimensions (size) of the patch along x and y directions would not be the same as those estimated in earlier case (Figs. 4.5(c)-(d)). In this case, a larger stretch of the patch along each of x and y-directions may be considered for estimating the effective size/dimensions of the patch. In case of the use of ACLD patch in the absence of temperature (Tc), Fig. 4.7(a) shows the surface plot of
t
Wpeak with test points (from which a differential ACLD patch is removed) and 4.7(b) Fig. 4.7 (a) Surface plot for the variation of (Wpeakt ) with the test points on the top surface of FG substrate plate, (b) corresponding contour plot,(c)-(d) variations of Wpeakt
with the test points along x-direction (y b /2) and ydirection (xa/2) (ACLD patch,
p=500 N / m2, kd=50, Tc Tm=300 K, n=1).
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shows the corresponding contour plot. Figures 4.7(c)-(d) show the variation of Wpeakt along x and y directions, respectively indicating the sensitivity of ACLD location on the
t
Wpeak. Similarly in the presence of temperature (Tc),
Fig. 4.8(a) shows the surface plot of Wpeakt with test points and 4.8(b) shows the corresponding contour plot. Figures 4.8(c)-(d) show the variation of Wpeakt along x and y directions, respectively indicating the sensitivity of ACLD locations on the Wpeakt . Also, from Figs. 4.7(c)-(d) and 4.8(c)-(d), the same estimation for effective dimensions of
Fig. 4.8(a) Surface plot for the variation of (Wpeakt ) with the test points on the top surface of FG substrate plate, (b) corresponding contour plot, (c)-(d) variations of
t
Wpeak with the test points along x-direction (y b /2) and ydirection (xa/2) (ACLD patch, p=400 N / m2,kd=50, Tm 300 K, Tc 500 K, n=1).
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ACLD patch could be made as that is done for PFRC patch in the absence of temperature (Tc) (Figs. 4.5(c)-(d)). It should be noted that the above mentioned location and size of PFRC actuator/ACLD patch are determined for effective control of nonlinear vibration of FG plates corresponding to its fundamental mode of vibration. For other modes of vibration of the same, these effective parameters (size and location) may be different and can be determined following the same numerical procedure as described in the present work. However, in the present estimations of size and location of PFRC/ACLD patch over the surface of FG substrate plate for effective control of a particular mode of vibration of overall plate, none of the available optimization algorithms is utilized. So, the estimated size and location of PFRC/ACLD patch are called at present as “effective size and location of patch” instead of “optimal size and location of patch”.
4.3 Conclusions
In this chapter, a numerical procedure is described for estimating size and location of PFRC/ACLD patch over the top surface of substrate FG plate for effective control of vibration of overall FG plate. This procedure is implemented in conjunction with the incremental nonlinear finite element models of the overall FG plate developed in the preceding chapters. For a particular mode of vibration of the overall FG plate, the numerical results reveal that the effective location of PFRC/ACLD patch over the top surface of FG substrate plate does not alter due to the presence of ceramic-rich surface temperature. But, the effective size of the same (patch) may alter in effect of the presence of temperature. An indicative change of effective size of PFRC patch due to the existence of temperature gradient across the thickness of substrate FG plate is observed while similar temperature gradient has no significant effect on the same (effective size) of ACLD patch.
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