Modeling and Evaluation of Dehydration Effect on Tip Position and Bending Response of IPMC Actuator
3.4 Modeling of Bending Characteristics
3.4.2 Pseudo Rigid Body Modeling Approach
3.4.2.1 Single Patch IPMC Actuator
Fixed Parameter Pseudo-Rigid Body Modeling
Pseudo-rigid body modeling of IPMC is dominated by the bending moment generated due to the applied voltage. Bending moment generated in IPMC can be expressed by M = Κθ where, Mis the bending moment, Κis the torsional spring constant, and θ is the corresponding pseudo-rigid body angle [Bandopadhya, (2009)]. The spring constant is directly obtained as for the model as shown in Figure 3.15 [Howell, (2001)]
1.5164EI
Κ = l (3.21)
End point coordinates of the link as shown in Figure 3.13 at dehydrated condition (experimented results) can be obtained as [Bandopadhya, (2009)]
(1 ) cos
px =l −γ +γl θ (3.22)
y sin
p =γl θ (3.23)
where, γ is the characteristic radius factor.
Figure 3.13 Pseudo-rigid body models for the deflected IPMC.
Figure 3.14 Change in tip position of the IPMC actuator following fixed pseudo- rigid body modelling for (a) positive (b) both positive and negative input voltage.
Figure 3.14(a) and (b) demonstrate the effect of dehydration on the tip position of the IPMC actuator for various input voltages using fixed parameter pseudo-rigid body modeling technique.
Variable Parameters Pseudo-Rigid Body Modeling (VPPRBM)
The experimental results obtained in the previous section are used to model the bending characteristic of the IPMC following the variable parameter pseudo-rigid body modelling (VPPRBM) technique. The technique validates the material properties and assesses the bending resistance offered by the IPMC during actuation, [Howell, (2001)].
Figure 3.15 Equivalent bending model of IPMC for an input voltage V.
Figure 3.15 shows the initial (OB1)and bending (OB) configuration of the actuator.
Figure 3.13 shows the dual schematic representation of the bending configuration of IPMC with its pseudo-rigid body model. The IPMC is replaced by a link with a characteristic pivot located at a length γvlfrom the free end. A torsional spring at the pivot measures the resistance to its bending. Bending moment developed by the actuator can be expressed as Md = Κvθv where, Κv,θvare the variable spring constant and variable pseudo-rigid body angle respectively. The main objective of the analysis is to establish relationships among the parameters such as variable spring constant
(
Κv)
, variable pseudo-rigid body angle( )
θv and variable characteristic radius factor( )
γv with positive dehydration. End point coordinates of the link as shown in Figure 3.13 at dehydrated condition (experimented results) can be obtained as [Bandopadhya and Njuguna (2009)]:(1 ) cos
x v v v
p =l −γ +γ l θ (3.24)
y v sin v
p =γ l θ (3.25)
where, px,pyare the end point coordinates at dehydrated condition.
The product γvl(AB), the variable characteristic radius, is the radius of the circular deflection path traversed by the end-tip of the pseudo-rigid body link and (1l −γv) represents the length of the characteristic pivot located from the fixed end
‘O’[Bandopadhya and Njuguna (2009)]. Further, as shown in Figure 3.13
[ ]
2 2(1 v) x (1 v) y
OA+AB=l ⇒l −γ + p −l −γ + p =l (3.26)
Simplifying the equation (3.26)
(
2 2 2)
(1 )
2( )
x y
v
x
p p l
l −γ = +p −l
− (3.27)
Therefore, one can establish the relationship for characteristic radius factor as:
(
2 2 2)
1 2 ( )
x y
v
x
p p l
l p l γ
+ −
= − −
(3.28)
Subsequently, pseudo-rigid body angle (θv) is obtained as:
1 1
sin y sin y
v
v
p p
AB l
θ γ
− −
= =
(3.29)
Treating IPMC as continuous distributed parameter system with large deflection bending problem, the bending moment generated by the actuator is obtained as,
(1 cos )
v y
M EI
p ϕ
= − , (equation (3.9)), and subsequently in terms of pseudo-rigid body angle can be expressed as:
(1 cos )
v sin
v v
M EI
l ϕ
γ θ
= − (3.30)
Figure 3.16 Effect of dehydration on the characteristic radius factor of IPMC with various input voltages.
The variable spring constant for the IPMC in dehydrated condition can be expressed as:
v v
v
M
Κ = θ (3.31)
Figure 3.16 shows the variation of characteristics radius factor for both hydrated and dehydrated condition, while Figure 3.17 shows the change in the variable pseudo- rigid body angles. It is observed that with the input voltage, the variable pseudo- rigid body angle increases with dehydration and comes closer to a value of hydrated condition with increase in voltage. It is anticipated that the percentage differences in y-deflection between hydrated and dehydrated condition increases with input voltage while the characteristics radius factor decreases gradually. As the pseudo- rigid body angle depends both on y-deflection and the radius factor as given in equation (3.30),thus at low operating voltage, pseudo-rigid body angle increases with dehydration compared to hydrated state as shown in Figure 3.17.
Figure 3.17 Change in pseudo-rigid body angle due to dehydration for various input voltage.
Figure 3.18 Variation of spring constant of IPMC with various input voltages.
Figure 3.18 is showing the change in spring constant with input voltage. The spring constant measures the resistance offered by IPMC during bending. It is observed that both spring constant and bending moment decreases with dehydration and comes to a closer value after certain time interval. Thus the proposed model validates the results pertaining to the assumptions made and the material properties taken.
(a) (b)
Figure 3.19 Path followed by the IPMC actuator using variable parameter pseudo- rigid body modelling for (a) positive (b) both positive and negative input voltages.
Figure 3.20 Effect of dehydration following variable parameter pseudo-rigid body modelling for (a) positive (b) both positive and negative input voltages.
Figure 3.19(a) and (b) show the changes in tip position for a single patch IPMC actuator for various input voltages, while Figure 3.20(a) and (b) demonstrate the effect of dehydration on the tip position using VPPRBM technique. It is observed that accurate tip position can be modeled by VPPRBM technique compared to Euler-Bernoulli and fixed parameter pseudo-rigid body modeling technique.