HARMONICALLY EXCITED NONLINEAR VIBRATION OF HEATED FUNCTIONALLY GRADED PLATE INTEGRATED
Chapter 3 Chapter 3
3.3 Numerical results and discussions
3.3.1 Validation of the present incremental finite element model
For verification of present nonlinear incremental finite element model, the linear and the nonlinear free vibration
responses of the FG substrate plate are evaluated without ACLD layer (hp0, hv0, V0). It is observed that the computed fundamental natural frequency parameter () and the nonlinear frequency- amplitude relations do not deviate from the similar results for previous verification (Table
2.1 and Table 2.2). However, for verifying the implementation of GHM method, the forced linear frequency response of the overall plate are evaluated by modeling the material properties of the viscoelastic layer according to the complex stiffness method. In Figure
Chapter 3:Piezo-viscoelastically damped .... heated plate-surface
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Fig. 3.3 Nonlinear frequency responses of the overall FG plate (Tc Tm=300 K, n=2, p = 500 N / m2).
3.2, the obtained frequency response is compared with that when the GHM method is used for modeling the viscoelastic layer. It may be observed from Fig. 3.2 that the results using GHM method are in good agreement with those obtained using complex stiffness method thus verifying the implementation of GHM method in the present formulation.
3.3.2 Effect of control gain (kd)
Figure 3.3 illustrates the nonlinear frequency response curves when the PFRC layer is either a passive constraining layer (PCLD,kd0) or an active constraining layer (ACLD,
d 0
k ). The same figure also shows the frequency response curve when the active PFRC actuator layer is directly attached to the substrate plate-surface.
From the PCLD and ACLD responses, it may be observed that the active (kd0) PFRC constraining layer significantly increases the damping in the
overall FG plate compared to that when it is a passive (kd0) constraining layer. This result implies a potential use of PFRC actuator as the active constraining layer for ACLD treatment for controlling the nonlinear vibration of FG plates in the frequency-domain. It may also be observed from Fig. 3.3 that the actuation-capability of active PFRC layer increases significantly when it acts as an active constraining layer instead of a simple actuator layer directly attached to the substrate plate-surface. So, Fig. 3.3 indicates to employ the PFRC actuator in the form of ACLD layer for effective control of vibration of FG plates.
Chapter 3:Piezo-viscoelastically damped .... heated plate-surface
Figure 3.4(a) demonstrates the nonlinear frequency responses of the overall FG plate for different values of feedback control gain (kd) when the mechanical load-amplitude (p) remains constant in the absence of thermal effect. The corresponding variations of required control voltage are also illustrated in Fig. 3.4(b). It may be observed from these figures that the damping in the overall plate can be increased by assigning a higher value of control gain (kd). Although a higher value of control gain raises the maximum value of required control voltage across the thickness of the PFRC constraining layer, but it is within the reasonable range (Fig. 3.4(b)). In case of the direct integration of PFRC actuator layer over the surface of substrate plate, similar results in Fig. 2.6 show that the maximum value of control voltage remains almost constant for any value of control gain (kd) (with a constant value of, p). However, it should be noted here that the value of control gain (kd) would be assigned so that the corresponding voltage across the thickness of the PFRC actuator layer would not exceed its permissible value.
3.3.3 Effect of substrate FG plate-surface temperature (Tc)
Figures 3.5-3.8 illustrate the effect of ceramic-rich surface temperature (Tc) on the nonlinear vibration characteristics of the overall smart FG plate in the frequency-domain and also, on the corresponding damping caused by the ACLD layer. The overall plate
Fig. 3.4(a) Nonlinear frequency responses of the overall FG plate and (b) the corresponding variations of control voltage (Tc Tm 300 K, n=2, p= 500 N / m2).
Chapter 3:Piezo-viscoelastically damped .... heated plate-surface
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Fig. 3.5 Variations of thermal bending deflections of the overall FG plate with increasing bottom ceramic rich substrate-plate-surface temperature (Tc) (n=1,
Tm=300 K, p= 500 N / m2).
undergoes thermal bending deformations along the negative z-direction due to the bottom ceramic-rich surface temperature. Similar to the previous analysis (Chapter 2), this initial thermal bending deflection of the overall plate at a temperature remains almost constant during its vibration under an applied harmonic mechanical load of any frequency. So, the initial thermal bending deflection of the overall plate due to a ceramic- rich surface temperature of substrate plate may be considered as the equilibrium position of the overall plate during its vibration at any frequency. Figure 3.5 illustrates the initial thermal bending deflections of the FG plate integrated with ACLD layer for different bottom ceramic-rich surface temperatures of substrate plate. For the same temperatures, the corresponding linear and nonlinear frequency responses are demonstrated in Figs.
3.6(a)-(b). It could be noted here that the difficulty in the convergence of solutions arises due to the sharp turn of different solution components (harmonic terms). But, the peak- point deflection of every frequency response curve is distinctly obtained as shown in the results.
Chapter 3:Piezo-viscoelastically damped .... heated plate-surface
Fig. 3.7 Variation of fundamental frequency (0) of the overall FG plate with the ceramic rich substrate- plate-surface temperature (Tc) (p = 100
N / m2,n=1, Tm=300 K, kd 50).
Fig. 3.6 Linear and nonlinear frequency responses of the overall FG plate for different ceramic rich substrate-plate-surface temperatures (n=1,kd=50, Tm=300 K, p = 100
N / m2for linear responses and p = 500 N / m2for nonlinear responses).
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Fig. 3.8 Difference between the frequencies (peak) corresponding to the peak-points of linear and nonlinear frequency responses of the overall FG plate (Fig. 3.6) at each ceramic rich surface temperature (Tc) (n=1, kd=50, Tm=300 K, p = 100 N / m2for linear responses and p = 500 N / m2for nonlinear responses).
Figures 3.6(a)-(b) illustrate the same observations as those are obtained from Figs.
(2.8). In the present case (ACLD), the variation of fundamental frequency (o) of the overall plate with temperature (Tc) is presented in Fig. 3.7. In comparison to Fig. 2.9, Fig.
3.7 shows that the limiting value of temperature (Tc) for alteration of nature of variation of fundamental frequency (o) remains almost the same (Tc375 K) for both the cases (ACLD layer and PFRC layer). For the FG substrate plate integrated with ACLD layer, the difference between the frequencies (Δωpeak(ωpeak nonlinear) (ωpeak linear) ) corresponding to the peak-points of linear and nonlinear frequency response curves for every temperature (Tc) is plotted in Fig. 3.8. A comparison of this results (Fig. 3.8) with similar results in Fig. 2.10 implies that the limiting temperature (Tc) corresponding to alteration of structural behavior of the overall smart FG plate does not depend on the type of smart layer (ACLD layer and PFRC layer).
Figure 3.9(a) shows the peak-point deflections (Wpeakt ) of the nonlinear frequency responses (Fig. 3.6) for different
temperatures (Tc). The corresponding values of required control voltage (Vpeak) are also illustrated in Fig. 3.9(b). It may be observed from Fig. 3.9(a) that for constant values of control gain (kd) and load- amplitude (p), the magnitude of Wpeakt increases with the increasing temperature (Tc). Thus, the increased initial thermal bending deflection at a higher temperature (Tc) causes lesser damping in the overall FG plate in expense of higher required control voltage (Fig.
3.9(b)) across the thickness of PFRC
constraining layer. It may be noted by comparing the results in Fig. 3.9 with those in Fig.
Chapter 3:Piezo-viscoelastically damped .... heated plate-surface
2.11 that the nature of variation of the magnitude of Wpeakt within a range of temperature (Tc) differs when the ACLD layer is used instead of PFRC layer even though the nature of variation of corresponding control voltage remains almost the same.