Nonlinear Vibration Analysis of Silver Electroded IPMC Actuator Subjected to Alternating Voltage
5.4 Results and Discussions
5.4.1 Numerical Results
An IPMC actuator is numerically studied and non-linear response is investigated. A silver electroded IPMC actuator of length (L) = 0.02 m, width (wd) = 0.005 m and thickness (h) = 0.0002 m has been considered for the numerical analysis. The density (ρ) and Young’s modulus (E) are taken 2125 kg/m3 and 82 MPa (same as previous chapter), respectively. Both book-keeping parameter
( )
ε and scaling factor( )
r are taken as 0.1 for simulation. All the results in this section are shown considering dehydration. Variation of non-dimensional force term with input voltages is given in Table 5.1.Table 5.1 Variation of non-dimensional force term with voltage.
Voltage (V) Fnd
(positive dehydration)
0
Fn
(zero dehydration)
0.2 0.6516 0.7087
0.4 1.4092 1.5226
0.6 3.1714 3.3333
0.8 5.8895 6.1022
1.0 8.2612 8.8769
1.2 15.2365 16.2023
Equation (5.31) is solved and the results are verified with the results by solving the temporal equation of motion (5.15). The multiple equilibrium points obtained in the frequency response curve is validated by plotting the basin of attraction. As evident, due to the presence of various non-linear terms in the temporal equation of motion (5.15), the system yields a typical non-linear behavior. For simple resonance case, the frequency response curve is obtained by using equation (5.31) which is shown in Figures 5.4-5.7 for different system parameters. While the solid line represents the stable equilibrium points the dashed line represents the unstable state.
Figure 5.4 Frequency response curve for input voltage of 0.2V.
Figure 5.5 Frequency response curve for input voltage of 0.4V.
It is observed from the results that the system does not possess any trivial state response. It is thus, anticipated that the actuator always vibrates about its equilibrium position with an amplitude equal to the non-trivial response as shown in Figure 5.4, 5.5, 5.6 and 5.7. In Figure 5.4 the frequency response curve is plotted for 0.2V and it is observed an isolated unstable regime in addition to a stable branch.
With increase in voltage further these two branches merge to form a frequency response curve similar to Figure 5.5-5.7. It is clearly observed that frequency range
XC (Figure 5.5) demarcates the unstable response of the system while frequency range AX and CD shows the stable regime. When the actuator is activated (e.g. at point A), with increase in frequency of excitation, the amplitude of response increases and it reaches a critical value at point X. At this point, further increase in frequency, leads the system to unstable. Further, increase in frequency drives to the critical point C, beyond which the system again exhibits stable condition. It is further observed that, frequency beyond point C, the amplitude decreases to point D.
This observation is consistent with the results presented by Nemat-Nasser and Wu (2006) and Cilingir and Papila (2010). If the system is driven by a voltage with frequency in between point X and C the response will be thus unstable, and may results in failure in this range. Hence, to operate the system with moderate amplitude of response, either the applied voltage should be lowered or the operating frequency should be kept well below the bifurcation point X or above of the bifurcation point C. It is also observed that as the input voltage increases the response amplitude also increases as demonstrated in Figure 5.5, Figure 5.6 and Figure 5.7(a,b).
Figure 5.6 Frequency response curve for input voltage of 0.6V.
(a) (b)
Figure 5.7 Frequency response curve for input voltage of (a) 1.0V (b) 1.2V.
Similarly, when the excitation frequency is swept down, i.e., from point D, with decrease in frequency the response amplitude increases and it reaches a critical value at point C, which is a saddle node bifurcation point. At this point with further increase in frequency, the system experiences a jump up phenomenon i.e. a sudden jump from point C to point B which leads to a sudden increase of amplitude and may lead to the system failure. Table 5.2 lists the variation of bifurcation points C and B, and the jump length for different input voltages. It is observed that with increase in the value of excitation voltage, the jump length increases. Table 5.3 lists the variation of bifurcation points C and X with input voltage. Further, it is clearly observed that the unstable zone, CX increases with increase of input voltage.
As shown in Figure 5.5, due to the presence of bi-stable region between the frequency ranges B-B1, the initial conditions in this region plays an important role to find the appropriate system response. Hence, in order to know the influence of initial condition, basin of attraction is plotted in the ‘a−γ’plane corresponding to σ =0.5 for 0.4V as shown in Figure 5.8. It clearly shows two stable solutions corresponding to point (P1, P3) and one unstable solution corresponding to point (P2) as marked in Figure 5.5. Figure 5.9 shows the time response and phase portrait, obtained by solving the equation (5.27) and (5.28) for applied electric potential of 0.4V. These results are in good agreement with the results presented in Figure 5.5 at point P1.
Table 5.2 Variation of bifurcation points C and B with excitation voltage.
At critical point C At critical point B Voltage
(V)
Detuning parameter
( )
σResponse amplitude
( )
aDetuning parameter
( )
σResponse amplitude
( )
aJump length
0.4 0.2681 3.779 0.2681 6.579 2.8
0.6 0.4955 4.747 0.4955 9.21 4.463
0.8 0.7616 5.744 0.7616 11.48 5.736
1.0 0.9589 6.537 0.9589 12.83 6.293
1.2 1.447 7.989 1.447 15.75 7.761
Table 5.3 Variation of the critical points X and C with excitation voltage.
Detuning parameter
( )
σVoltage
(V) At critical point X At critical point C
0.4 0.5786 0.2681
0.6 0.4701 0.4955
0.8 0.3126 0.7616
1.0 0.1899 0.9589
1.2 -0.2079 1.447
Figure 5.8 Basin of attraction forσ =0.5, 0.4 V, Figure 5.5.
Figure 5.9 Time response and phase portrait at point,σ =0.5, 0.4V, Figure 5.5.
Figure 5.10 shows the vibration response and phase portrait at a point, where σ =0.5 as shown in Figure 5.5. The results are obtained by solving the temporal equation of motion (5.15 (a)). It is observed that, the transient response (Figure 5.10 (i),(ii)) of the system produces a beating type phenomenon, and the steady-state response of the system is periodic (Figure 5.10 (iii),(iv)). It is observed that the steady-state response obtained is in good agreement with the results obtained by using the method of multiple scales as shown in Figure 5.5. Figure 5.11 shows the time response forσ =0.5, 1.0V, which is found to be unstable by solving the
temporal equation of motion and can be well verified with the results presented in Figure 5.7 (a).
Figure 5.10 Vibration response and phase portrait forσ =0.5, obtained by solving temporal equation of motion, (Key as in Figure 5.5), where, (i), (ii) transient
response and (iii), (iv) steady-state response.
Figure 5.11 Unstable response forσ =0.5, obtained by solving temporal equation of motion for an input voltage 1.0 V (Key as in Figure 5.6 (a)).