deformations where previous works failed to do so. The future work involves the experimental investigation on how in such kind of parametrically excited systems one may bring more stability.
Table 5.4: Quantitative comparison of the present work with previous literature
References Acc.
(ms−2) Vol.
(cm3)
Freq.
(Hz)
Max Power Output
Normalized Power Density (NPD) µW cm−3m−2s4 Daqaq et al. [21] 3.14 9.51 5.37 0.06mW 6.42E - 01 Abdelkefi et al. [181] 12 11.7 − 80mW 4.75E + 01 Friswell et al. [64] 1.19 1.23 2.45 0.2mW 1.15E + 02
Zhu et al. [313] 6.37 2.66 9 − 3.30E - 03
Jia et al. [202] 0.57 1800 3.57 171.5mW 2.93E + 02 Jia and Seshia [207] 20 1.21 20 0.15mW 3.10E - 01
Yildirim et al. [205] 9 − 54.4 − 4.14E - 04
Yan and Hajj [208] 98.3 170.5 8.44 1.4W 8.50E - 01 Yildirim et al. [22] 20 1.12 12.1 2.15mW 1.87E + 00 Mam et al. [314] 57.13 10.92 29.8 0.96mW 2.70E - 02 Panyam et al. [315] 60 1.51 224 0.04mW 1.66E - 02
Searle et al. [309] − 4.1 10.8 0.068mW −
Bahareh et al. [316] 47 15.44 10.9 0.0087mW 2.55E - 04 Garg and Dwivedy [301] 36.9 3.94 15.3 9W 1.67E + 03
Kuang and Zhu [312] 4.9 − 18 3.6mW −
Xia et al. [223] − 0.6 − 0.36W −
Present work 13.38 7.98 7.22 22.7mW 1.59E + 01
Figure 5.14: A vertical beam with unimorph MFC patch and an attached mass
Similar to the previous section here also the beam is attached to a slider crank mechanism based shaker. The MFC patch is attached using 3M 465 film tape.
Unlike the PZT ceramics, the MFC patch can be detached form the substrate for reuse purpose. Using the Oscilloscope the rms voltage Vrms, maximum voltage
Table 5.5: Material and geometric properties of substrate and piezoelectric patch
Property Piezo Patch Sym Substrate-S1 Substrate-S2
Young’s modulus, GPa Ep = 15.86 Eb 190 190
Density, kgm−3 ρp= 5440 ρb 7800 7800
Length, m Lp = 28×10−3 Lb 270×10−3 190×10−3
Width, m bp = 14×10−3 bb 14×10−3 14×10−3
Height, m tp = 0.30×10−3 tb 0.25×10−3 0.25×10−3
Permittivity, nFm−1 eˆ= 19.36 − − −
Piezoelectric constant, Cm−2 e31=−19.84 − − −
L1, m − − 20×10−3 10×10−3
L2, m − − 48×10−3 38×10−3
Subscript p:piezo b:beam
Vmax are determined and plotted in Fig. 5.15 and Fig. 5.16 for various values of attached mass and load resistance, respectively. The experimentally obtained data are shown using the symbols either triangle or the diamond. Using the
curve fitting method the variation is also shown by solid lines. The effect of variation of attached mass m on the output voltage and power is observed in Fig. 5.15 by keeping the load resistance Rl = 500 kΩ and position β = 0.233. It is observed that the maximum power achieved when m = 36 gm. It may be noted from the same figure that the frequency of excitation decreases as m increases.
0 1 2
Vrms Rl = 500 kΩ, β=0.233
0 1 2 3
P (μW)
0 10 20m (gm)30 40 50 0
1 2 3 Vmax
4 5 6
f (Hz)
(V)(V)
Figure 5.15: Variation of Vrms,P andVmaxwith attached
massm
0 1 2
Vrms
0 1 2 3 4
P (μW)
200 400 600 800 1000
Rl (kΩ)
0 2 4
Vmax
Figure 5.16: Variation of Vrms,P and Vmax with load re-
sistanceRl
The output voltage and power is plotted against external load resistance Rl in Fig. 5.16. The voltage increases with Rl while power P achieves its maximum value when Rl = 800 kΩ. Here the frequency of excitation is kept around f = 4.5 Hz. The voltage time response and FFT, corresponds to the specific points in Fig. 5.15 and Fig. 5.16 respectively are shown in Fig. 5.17. Figure 5.17(a) and (b) shows the time response and FFT related to mass value m = 36 gm. Two peaks are observed in FFT, showing that the higher peak is near the excitation frequency (synchronous frequency) and the other one is observed at 2.5 Hz which is similar to the sub-synchronous frequency that are generally observed in rotor bearing systems.
It may be noted that the 2.5 Hz frequency is same as the first mode frequency of the system. Similarly the time response and FFT for Rl = 800 kΩ in Fig. 5.16 is shown in Fig. 5.17(c) and (d) where a maximum power of 2 µW is observed. It is
0 2 4 6 8 10 -4
-2 0 2 4
0 5 10 15
-4 -2 0 2 -4
-2 0 2 4
0 0.5 1
(c) (d)
(a) (b)
Figure 5.17: Voltage time response and FFT for (a) mass value m =36gm in Fig. 5.15 (b) FFT of (a) (c) at Rl= 800 kΩ in Fig. 5.16 (d) FFT of (c)
to be noted that the MFC patch covers only 5.2% of the total surface area of the substrate.
Figure 5.18 shows the effect of the position of attached mass β on output voltage and power by keeping the load resistance Rl = 500 kΩ and mass m = 18 gm. The voltage and power output increases initially then almost remain constant with in- crease in β. One may get an output power of 1.45 µW at β = 0.296. Also as the position of the mass along the beam goes towards the free end, the excitation fre- quency decreases. Figure 5.19 shows the variation output voltage and power with external load resistance Rl. The voltage increases with Rl while power P achieves its maximum value of 0.6 µW when Rl = 700 kΩ. The frequency of excitation in this case is kept nearly 10 Hz.
Figure 5.20(a) and (b) shows the voltage time response and FFT forβ= 0.296 with other system parameters as in Fig. 5.18. Similar to the previous case, two frequency peaks at 2.5 Hz and 4.9 Hz are obtained. Further, in Fig. 5.20(c) and (d), the voltage time response and FFT corresponds toRl = 1000 kΩ is plotted, where other system parameters as in Fig. 5.19. Here several frequency peaks are observed in the FFT, which is due to the energy interaction between participating modes.
0 1 2
Vrms Rl = 500 kΩ
0 1 2 3
P (μW)
0.20 0.25 0.30
0 β
1 2 3
Vmax
4 5 6
f (Hz)
(V)(V)
Figure 5.18: Variation of Vrms, P and Vmax with the po-
sition of attached massβ
0.0 0.5 1.0
Vrms
0.0 0.5 1.0
P (μW)
200 400 600 800 1000
Rl (kΩ)
0 1 2 3
Vmax
Figure 5.19: Variation of Vrms,P and Vmax with load re-
sistanceRl.
0 2 4 6 8 10
-4 -2 0 2 4
0 5 10 15 20
0 0.1 0.2 0.3 0.4 -4
-2 0 2 4
0 0.5 1
(d) (c)
(a) (b)
Figure 5.20: Voltage time response (a) forβ= 0.296 (Fig. 5.18) (b) correspond- ing FFT (c) voltage time response atRl= 1000 kΩ (Fig. 5.19) (d) corresponding
FFT plot.
The effect of length of the substrate and position of the MFC patch are also studied and plotted in Figs. 5.21 to 5.23. As given in Table 5.5 length of the substrate (S2) is lesser than that of substrate (S1). With decrease in the length of the beam the natural frequency of the system increases from 2.5 Hz to 4.3 Hz. Here the mass, its position along the beam and load resistance values are taken as m = 36 gm, β = 0.263 and Rl = 1000 kΩ.
0 2 4 6 8 10
−8
−6
−4
−2 0 2 4 6 8
t(sec)
V(V)
0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5
f(Hz)
|V|
(a) (b)
f= 4.3
f= 8.6
Figure 5.21: (a) Voltage time response and (b) corresponding FFT, for param- etersm= 36 gm, β = 0.263,Rl= 1000 kΩ and frequency of excitation around 8
Hz
Figure 5.21 shows the voltage time response and corresponding FFT when the fre- quency of excitation is kept nearly 8 Hz, which is close to twice of the first natural frequency of the system with substrate S2. As shown in Fig. 5.21(b), which are similar to other figures, here also two peaks are observed with frequency 4.3 Hz and 8.6 Hz and they are in 1:2 ratio. In S2 the MFC patch is close to the fixed end of the beam (see Table 5.5) as compared to that of the system with S1. This leads to higher strain and, consequently more output voltage around 5 V and power of 25 µW, which is significantly higher as compared to the previous case where S1 is considered with longer beam length and position of the MFC patch is slightly far away from the fixed end.
When the excitation frequency increases to that of combination of first two natural frequencies of the system the voltage time response changes (shown in Fig. 5.22a) and higher frequencies dominates the time response which is visible in FFT (in
0 2 4 6 8 10
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
t(sec)
V(V)
0 20 40 60 80
0 0.02 0.04 0.06 0.08 0.1 0.12
f(Hz)
|V|
(a) (b)
f= 14.3
f= 50
Figure 5.22: (a) Voltage time response and (b) corresponding FFT, for param- etersm= 36 gm, β= 0.263, Rl= 1000 kΩ. The frequency of excitation is 13 Hz
which is around the combination of first two modal frequencies.
0 2 4 6 8 10
−2
−1.5
−1
−0.5 0 0.5 1 1.5
t(sec)
V(V)
0 20 40 60
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
f(Hz)
|V|
(a) (b)
f= 4.2 f= 8.5
f= 14.3
f= 50 f= 55.6 f= 28.7
f= 10.1 f= 18.6
Figure 5.23: (a) Voltage time response and (b) corresponding FFT, for parame- tersm= 36 gm,β = 0.263,Rl= 1000 kΩ and frequency of excitation is positively
detuned to that of 13 Hz
5.22b). Further a positive detuning leads to multimodal voltage time response (in Fig. 5.23a) with multiple frequency components in FFT (in Fig. 5.23b).