HARMONICALLY EXCITED NONLINEAR VIBRATION OF HEATED FUNCTIONALLY GRADED PLATE INTEGRATED
2.3 Solution methodology
2.4.4 PFRC layer on metal/ceramic rich surface of substrate FG plate
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Fig. 2.4 Nonlinear frequency responses of the overall FG plate when the PFRC layer is attached either to the ceramic rich (=2) or to the metal rich (=1) top surface of the FG substrate plate (Tc Tm=300 K, n=2, p = 300 N / m2, kd=100).
2.4.3 Effect of fiber-orientation angle ( ) of PFRC actuator layer
Figure 2.3 shows the variation of peak-point deflection (Wpeakt ) of controlled nonlinear frequency response of the overall FG plate for different values of piezoelectric fiber orientation angle ( ) in the PFRC actuator layer. This figure suggests a better use the PFRC actuator layer when its piezoelectric fiber orientation angle is,45o. But, the negligibly small variation in the magnitude of Wpeakt (Fig. 2.3) indicates insignificant effect of on the control authority of PFRC actuator layer. Thus, the present investigations are carried out considering the piezoelectric fiber orientation angle in the PFRC actuator layer as, 0o.
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Figure 2.5(a) represents the nonlinear frequency responses of the overall FG plate for different values of load-amplitude (p) when the control gain (kd) remains constant in room temperature (Tc Tm300 K). The corresponding variations in the required control voltage across the thickness of the PFRC actuator layer are also illustrated in Fig.
2.5(b). These figures indicate that a significant damping in the overall FG plate can be achieved in expense of feasible applied control voltage across the thickness of PFRC actuator layer. The peak-point deflections (Wpeakt ) of frequency responses and the corresponding required control voltages (Vpeak) are plotted against load-amplitude (p) in Figs. 2.5(c) and 2.5(d), respectively. These figures demonstrate an important fact that a Fig. 2.5 (a) Nonlinear frequency responses of the overall FG plate and (b) the corresponding variations of control voltage, variations of (c) the peak-point deflection (Wpeakt ) and (d) the corresponding control voltage (Vpeak) with the load amplitude (300K, kd=100, n=2, =1).
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higher value of load-amplitude (p) causes more nonlinearity in the frequency response (Fig. 2.5(a)), but the corresponding variations in Wpeakt and Vpeak (Figs. 2.5(c)-(d)) are linear.
2.4.6 Effect of control gain (kd)
Figure 2.6(a) demonstrates the nonlinear frequency responses of the overall FG plate for different values of the feedback control gain (kd) while the load-amplitude (p) and the temperature gradient (Tc Tm 300 K) remain constant. Figure 2.6(a) shows that the PFRC actuator layer is capable to induce more damping in the overall FG plate when a higher value of control gain (kd) are assigned. From Fig. 2.6(b), it is also observed that when the mechanical load-amplitude remains constant, higher values of control gain (kd) do not cause the increase in the maximum value of required control voltage (Vpeak).
An important aspect in the present analysis is to investigate the effect of temperature gradient (Tc 300 K, Tm 300 K) across the thickness of the substrate FG plate on its nonlinear dynamic behavior in the frequency-domain and also on the control authority of PFRC actuator layer.
Fig. 2.6 (a) Nonlinear frequency responses of the overall FG plate for different values of control gain (kd), (b) the corresponding variation of required control voltage (Tc Tm=300 K, n=2, =1, p = 300N / m2).
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2.4.7 Effect of ceramic rich surface temperature (Tc)
Fig. 2.8 Linear and nonlinear frequency responses of the overall FG plate for different ceramic rich surface temperatures (n=1, =1, kd =100, Tm=300 K, p = 100 N / m2for linear responses and p = 400 N / m2for nonlinear responses).
Fig. 2.7 Initial thermal bending deflections of the overall FG plate for different ceramic rich surface
temperatures (n=1, λ=1, Tm=300 K).
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The temperature of the bottom ceramic rich surface of the FG plate causes an initial bending deformation of the overall FG plate along the negative z-direction. It is observed that due to an applied harmonic mechanical load, the variation of this initial bending deflection with the frequency of vibration is negligibly small. Therefore, at a temperature gradient, it is reasonable to assume that the equilibrium position (Ws) of the overall FG plate is constant to its initial thermal bending deflection at all frequencies of vibration. Figure 2.7 illustrates the initial thermal bending deflections of the overall FG plate for different ceramic rich surface temperatures. The linear and the nonlinear frequency responses of the overall FG plate corresponding to different ceramic rich surface temperatures are shown in Fig. 2.8. The linear and the nonlinear frequency responses at a particular temperature gradient are achieved by applying a lower (p100 N/m2) and a higher (p400 N/m2) values of mechanical load-amplitude. It should be noted that although the difficulty in the convergence of
Fig. 2.9 Variation of fundamental frequency (0) of vibration with the ceramic rich surface temperature (Tc) (p= 100 N / m2, =1, kd=100,
n=1, Tm=300 K).
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Fig. 2.10 Difference between the frequencies () corresponding to the peak-points of linear and nonlinear frequency responses of the overall FG plate at each ceramic rich surface temperature (n=1, =1, kd=100, Tm=300 K, p=100 N / m2for linear responses and p=400 N / m2for nonlinear responses).
solutions arises in evaluation of the nonlinear frequency responses under a temperature
gradient, but the peak points of the response curves are distinctly obtained as shown in the results. It may be observed from Fig. 2.8 that the fundamental frequency (o) of vibration of the overall FG plate decreases up to a certain value of increasing ceramic rich surface temperature (Tc375 K). After that, it (o) increases with further increase of temperature (Tc). It should be noted here that while the initial decrease in the fundamental frequency (o) with the increase in Tc is a known fact, the increase in the same (o) for further increase in temperature (Tc375 K) is due to the significant effect of initial thermal bending of the overall FG plate at high temperature (Tc). In Fig. 2.9 the fact of initial thermal bending seen in Fig. 2.8 is also demonstrated by plotting the variation of fundamental frequency (o) with the ceramic rich surface temperature (Tc).
Another important effect of initial thermal bending of the overall FG plate on its nonlinear dynamic behavior in the frequency-domain can also be observed from Fig. 2.8 that the hardening structural behavior of the overall FG plate switches from to softening
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one as the ceramic rich surface temperature increases from room temperature to a higher temperature. In Fig. 2.10, the difference between the frequencies, (
nonlinear linear
) corresponding to the peak-points of linear and nonlinear responses at each ceramic rich surface temperature is plotted in order to show the switching of the structural behavior in a clear manner. It is known that the positive/negative value of this frequency difference indicates hardening/softening structural behavior of the plates. Thus, as demonstrated in Fig. 2.10, the overall FG plate turns to behave as a softening structure when the ceramic rich surface temperature (Tc) exceeds a value about 500 K. Figure 2.11(a) shows the peak-points (Wpeakt ) of nonlinear frequency responses for different ceramic rich surface temperatures (Tc). The effect of ceramic rich surface temperature on the performance of PFRC actuator layer for controlling the harmonically excited nonlinear vibration of the overall FG plate is also significant as can be noticed from Fig. 2.8.It may be observed from Fig. 2.11(a) that for constant values of control gain (kd) and mechanical load-amplitude (p), the magnitude of Wpeakt increases with the increasing ceramic rich surface temperature (Tc). But, the same again decreases after a certain value Fig. 2.11 (a) Variations of the peak point (Wpeakt ) of frequency response and (b) the corresponding control voltage (Vpeak) with ceramic rich surface temperature (Tc) (p = 400
N / m2, =1, kd=100, n=1, Tm=300 K).
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of ceramic rich surface temperature (i.e. about 450 K (Fig. 2.11(a)). Although this result shows more control authority of PFRC actuator layer at very high temperature (Tc), but the corresponding maximum value of required control voltage (Vpeak) (at a constant gain value) significantly increases as can be noticed from Fig. 2.11(b). In fact, since it happens at a high frequency (Fig. 2.8), the corresponding required control voltage reaches to a higher value. Therefore, Fig. 2.11(b) suggests a lesser value of control gain (kd) at higher temperature in view of the permissible applied voltage across the thickness of the PFRC actuator layer.