30120140502011

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 101-109, © IAEME
101
DEHYDRATION EFFECT ON NON-LINEAR DYNAMICS OF AN IONIC
POLYMER-METAL COMPOSITE ACTUATOR
Dillip Kumar Biswal*
, Saroj Kumar Dash
Department of Mechanical Engineering, Eastern Academy of Science and Technology, Phulnakhara,
Bhubaneswar, Odissa-754001, India,
ABSTRACT
In this paper effort has been given to study the non-linear dynamics of an IPMC actuator with
dehydration during actuation. As the IPMC actuator experiences dehydration in open environment, a
model has been proposed to estimate the loss due to dehydration following Cobb-Douglas
production method. The governing equation of motion of the system has been derived using
D’Alembert’s principle. Generalized Galerkin’s method has been followed to reduce the governing
equation to the second-order temporal differential equation of motion. Method of multiple scales has
been followed to solve the non-linear equation of motion of the system and numerically the effect of
dehydration on the vibration response has been demonstrated.
Key Words: Ionic Polymer-Metal Composites; Dehydration Factor; D’Alembert’s Principle;
Method of Multiple Scales.
I. INTRODUCTION
Recently, artificial muscle materials that mimicking the biological actuating devices are
attracted much attention as an alternative means of actuator for various applications. Due to the
similarity in mechanical properties, and easiness of actuation compared to that of biological systems,
Electro-active polymers (EAPs) are considered to be a new class of active materials for actuation. In
the quest for advanced artificial muscle actuators, Ionic Polymer Metal Composites (IPMCs) are new
breed of active materials belonging to the class of ionic electro-active polymers (EAP). The
advantages make them particularly attractive for application anywhere, where a muscle-like response
is desirable, including in biomedical devices, prostheses, micro-robotics, toys, aerospace, and
micro/nanoelectromechanical systems etc. [1]
The typical two types of base polymers that employed for fabrication of IPMC are Nafion®
and Flemion®
and electrodes made up of high conducting pure metals like gold or platinum plated on
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 2, February (2014), pp. 101-109
© IAEME: www.iaeme.com/ijmet.asp
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IJMET
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102
both faces [2]. IPMC responds to the external electric stimulation (usually < 3V) by producing large
bending deflection; conversely a measurable output voltage is obtained across its surface when it is
deformed mechanically. Thus, IPMC can be used as both actuator and sensor [3-4]. Actual actuation
mechanism of IPMC is still under investigation, but it seems to be the mobility of ion-water cluster
causes swelling near the cathode electrode and equivalent contraction on the other side.
Nemat-Nasser and Wu [5] reported that, when a small (1-2V) alternative electric potential is
applied to a fully hydrated IPMC strip, it shows a considerable bending vibration at that applied
frequency. The effect of tip displacement for various cation forms in IPMC has been investigated
experimentally. Zhang and Yang [6] developed an analytical model to show the vibration response of
a simply supported IPMC beam resting on an elastic foundation subjected to an alternative electric
potential. Euler-Bernoulli’s beam theory has been used to derive the governing equation of motion.
An experimental investigation was carried out by Kothera and Leo [7] to characterize the
nonlinearity observed in the actuation response of cantilever ionic polymer benders. For this,
Volterra series has been employed for nonlinearity identification. In application to walking robots,
Yamakita et al [8] developed a model following Hammerstein technique to identify nonlinearity in
IPMC materials. Kothera and Robertson [9] reported nonlinearity in IPMC is caused due to loss of
moisture content in working condition. The effect of dehydration on the vibration characteristics of
silver-electrode IPMC actuator by considering the average moisture loss during the working period is
shown by Biswal et al. [10, 11].
II. AIMS & OBJECTIVES
IPMC actuator dehydrates continuously when it is operated in an open environment. It is
anticipated that the deformation of the IPMC actuator not only depends on the mechanical/physical
properties and geometric configuration but also with the amount of dehydration during the working
period. Therefore, it is important to estimate the loss due to dehydration corresponding to an applied
electric potential and working period. As the IPMC actuator shows nonlinearity with gradual
dehydration and vibration characteristics in a working condition, it is important to study the non-
linear vibration characteristics of IPMC actuator. In the past, no work has been carried out to obtain
the non-linear vibration response of an IPMC actuator, excited by a small electric potential. The
model presented in this paper helps one to control the tip position accurately by compensating the
loss by applying the proportionate input voltage. The objectives of the present analysis include:
1. Investigation and calculation of water loss due to dehydration in working condition.
2. Analyze the non-linear dynamics of an IPMC actuator under small electric potential.
3. To demonstrate the effect of dehydration on non-linear vibration response.
As the IPMC actuator experiences continuous dehydration in working medium, a model has
been proposed in terms of input voltage and time to estimate the loss due to dehydration following
the Cobb-Douglas production method. For analyzing non-linear vibration response, the IPMC
actuator has been modeled following the Euler-Bernoulli approach treating it as a continuous
distributed flexible system. The governing equation of motion of the system has been derived using
D’Alembert’s principle. Generalized Galerkin’s method has been followed to reduce the governing
equation of motion to the second-order temporal differential equation of motion. Method of multiple
scales has been followed to solve the non-linear equation of the system and numerically the effect of
dehydration on the vibration response has been demonstrated. The influence of various system
parameters such as amplitude and frequency of the applied electric potential on the frequency
response curves has also been investigated for simple resonance condition.
The paper is organized by first calculating the loss due to dehydration in terms of input
voltage and time, followed by mathematical model for non-linear vibration analysis. Simulation
results have been discussed subsequently and finally the conclusion is drawn.

103
III. ESTIMATION OF LOSS DUE TO DEHYDRATION
The objective is to obtain dehydration factor κ from the loss in tip angle for each input
voltage. Taking the experimental tip deflection data [12] as given in Table 1, the tip angles of IPMC
at hydrated and dehydrated condition have been calculated. The dehydration factor ( )κ can be
expressed as:
0 0
1 1
M
M
φ
κ
φ
= − −= (1)
where,φ , 0φ and M , 0M are the tip angles and bending moment with dehydration and without
dehydration respectively. However, it is observed that dehydration depends on the input applied
voltage and actuation time; therefore, the dehydration factor( )κ can be expressed as a function of
both input voltage( )V and actuation time( )t i.e.
x y
V tκ α= (2)
where,α , x , y are the constants and depend on the material properties. A set of equations have been
developed using equation (2) and are solved by using Cobb-Douglas production method to obtain the
dehydration factor( )κ in terms of applied input voltage ( )V and time( )t . The resulting expression
for the dehydration factor is thus obtained as:
0.142 0.594
1.433V tκ −
= (3)
Table 1. Experimental tip deflection data [12]
Input Voltage (V)
0.5 1.5 2.5 3.5 4.5
X- Coordinate( )xp (mm) 38 36 34 31 26
Y-Coordinate( )yp (mm) 3.0 9.5 15.5 20.5 25
Figure 1. Variation of dehydration factor with time for different input voltages

104
Figure 1 shows the variation of dehydration factor with applied voltage and actuation time. It
is observed that the dehydration factor increases as the input voltage increases. For a particular input
voltage dehydration factor decreases with increment of time. This is because of the moisture content
of the IPMC decreases in the working medium due to continuous dehydration.
IV. MATHEMATICAL MODELING
Derivation of the temporal equation of motion: The non-linear vibration characteristics of an
IPMC actuator has been assessed for a flexible IPMC actuator by applying alternating electric
potential in cantilever configuration. Figure 2 shows the schematic of bending configuration of the
IPMC actuator subjected to alternating electric potential at the fixed end across its thickness. The
IPMC is modeled as an Euler-Bernoulli beam. Extended Hamilton’s principle has been followed and
the governing differential equation of motion of the system is derived and can be expressed as,
[ ]
2 3 2 2
0 0
2
1
3 ( ) ( )
2
1
(1 ) ( ) (1 )cos
2
L
ssss s ssss s ss sss ss s s s s ss s s s
s
L
s s ss
s
EI v v v v v v v Av v v v d v A v v v d d
v Av Cv v v Av Cv d f t
ξ ξ
ρ η ρ ξ η
ρ ρ ξ κ
    
+ + + + + − +    
     
 
+ − + + + = − Ω 
 
∫ ∫ ∫
∫
& && & &&
&& & && &
(4)
where, v is the transverse displacement of the beam. ( )s
is the first derivative with respect to time
t along the link elastic line s , ρ is the mass density per unit length of the beam , A is the cross-
section area , andη ξ are the variables of integration,C is the damping coefficient, f is the
amplitude and Ω is the frequency of the applied electric potential.
To discretize the governing equation of motion (4) the following assumed mode expression is used.
( , ) ( ) ( )v s t r s q tψ= (5)
where, r is the scaling factor, ( )q t is the time modulation , and ( )sψ is the admissible function.
Following non-dimensional parameters are used in the analysis.
, , , ,
s
s t
L L L
ξ η
τ ω ξ η ω
ω
Ω
= = = = = (6)
ξ
dξ
( , )Av tρ ξ&&
( , )Au tρ ξ&&
( , )cv tξ&
θθθθ
u
v
Figure 2. Schematic diagram of a flexible IPMC actuator subjected to bending moment due to input
voltage

105
Substituting equation (5) and (6) in equation (4) and using the generalized Galerkin’s method
the resulting non-dimensional temporal equation of motion is obtained and can be expressed as:
( )2 3 2 2 2
1 2 3 42 (1 )cosn nq q q q q q q q qq fω ξω εα εα εα εα κ ωτ+ = − − − − − + −&& & && & & (7)
One may observe that the non-linear temporal equation (7) contains a linear force
term ( )(1 )cosf κ ωτ− , cubic geometric term 3
1qα , and the inertia ( )2 2 2
2 3 4q q q q qqα α α+ +&& & & which is
non-linear. Hence, it is observed that the temporal equation of motion (7) contains many nonlinear
terms and it is very difficult to obtain the exact solution. Therefore, the solution of the above
equation is carried out following the perturbation method.
Solution of Temporal Equation of Motion: Method of multiple scales [13] has been followed to
solve the temporal equation of motion (7). In this analyzing technique the displacement q can be
represented in terms of different time scales ( )0 1,T T and a book keeping parameter ε as follows,
2
0 0 1 1 0 1 2 0 1( ; ) ( , ) ( , ) ( , ) ......q q T T q T T q T Tτ ε ε ε= + + + (8)
where, 2
0 1 2, , ...T T Tτ ετ ε τ= = = , and the transformation of the first and second time derivatives are
obtained as:
20 1
0 1 2
0 1
2
2 2 2
0 0 1 1 0 22
.......... ......
2 ( 2 ) .....
dT dTd
D D D
d d T d T
d
D D D D D D
d
ε ε
τ τ τ
ε ε
τ
∂ ∂
= + + = + + +
∂ ∂
= + + + +
(9)
Substituting equation (8) and (9) into equation (7) and equating the coefficients ofε and its
power terms, one can obtain the following expressions.
Order 0
ε : 2 2
0 0 0 0D q qω+ = (10)
Order 1
ε :
( )
( ) ( ) ( )
2 2 3
0 1 1 0 1 0 1 0 0 0 1 0
22 2 2 2
2 0 0 0 3 0 0 0 4 0 0 0
(1 )cos 2 2D q q f T T D D q D q q
q D q q D q q D q
ω κ ω σ ξ α
α α α
+ = − + − − −
− − −
(11)
The general solution of the equation (10) is given by:
0 0*
0 1 1( ) ( )i T i T
q A T e A T eω ω−
= + (12)
Substituting the value of 0q from equation (12) in to equation (11), and expressing ( )0 1cos T Tω σ+ in
complex form:

106
( )
01
0
2 2 2 * 2 * 2 * 2 *
0 1 1 1 1 2 3 4
33 3 3 3
1 2 3 4
1
2 2 3 3 (1 )
2
i Ti T
i T
D q q D i A i A A A A A iA A A A f e e
A A iA A e cc
ωσ
ω
ω ω ξ ω α α α α κ
α α α α
 
+ = − − − + − − + − 
 
+ − + − + +
(13)
where, ccstands for the complex conjugate of preceding terms. It is observed that any solution of
equation (13) contains secular terms when 1ω ≈ .
Simple resonance case( )1ω ≈
For the simple resonance case, one may represent detuning parameter σ to express the nearness of
ω to 1, as:
( )1 , and 1Oω ε σ σ= + = (14)
Eliminating the secular or small divisor term from equation (11) yields,
12 * 2 * 2 * 2 *
1 1 2 3 4
1
2 2 3 3 (1 ) 0
2
i T
D i A i A A A A A iA A A A f eσ
ω ξ ω α α α α κ
 
− − − + − − + − = 
 
(15)
To analyze the solution of equation (15), one may substitute A in the polar form i.e.
1
2
i
A ae β
= and
* 1
2
i
A ae β−
= and separating the real and imaginary parts of the resulting equation, one can obtain,
331 1 (1 )
sin
8 2
f
a a a
α κ
ξ γ
ω ω
−
′ = − − + (16)
3 1 (1 )
cos
2
f
a a Ka
κ
γ σ γ
ω
−
′ = − + (17)
where, 1 2 43 3 1
8 8 8
K
α α α
ω ω ω
 
= − + 
 
and 1Tγ σ β= − . For steady-state response( )0 0,a γ , a′ and γ ′equal
to zero. Eliminating γ from equations (16, 17), one may find the relation between σ and a as:
22
2 231 (1 ) 1
2 8
f
Ka a a
a
ακ
σ ξ
ω ω
−   
= ± − +   
   
(18)
Equation (18) is an implicit equation for amplitude of the response as a function of the
excitation frequency due to the applied electric potential, and damping factorξ . One may obtain the
response ( ),a γ by solving equation (18) numerically. It may be observed from equation (18) that the
system does not possess any trivial solution.

107
V. RESULTS AND DISCUSSION
Results obtained from numerical simulation are based on the physical properties of an IPMC
actuator as given in Table 2 and using the experimental data as given in Table1.The tip deflections
data are taken for 30 seconds for each input voltage. It is observed that system exhibits a typical
nonlinear behavior due to the presence of various non-linear terms in the temporal equation (7). The
non-linear response of the actuator is determined for various applied electric potential. In the
following section the responses for primary resonance condition has been discussed.
Table 2. Physical properties of IPMC material
Property
Elastic
modulus
( E )
Length
( l )
Width
(b )
Thickness
( h )
Density
( ρ )
Book
keeping
parameter
(ε )
Scaling
parameter
( )r
IPMC 1.2 GPa 0.04 m
0.005
m
0.001 m
3385
kg/m3 0.1 0.01
,σ,σ,σ,σ
,a
With
dehydration
Without
dehydration
2 V
A
B
C
D
,σ,σ,σ,σ
,a
Figure 3. Frequency response curve for simple Figure 4. Frequency response curve for
resonance case for 1V input voltage simple resonance case for 2V input voltage
Figure 3 shows the frequency response curve with and without dehydration for simple
resonance condition for 1V input voltage. The stable and unstable responses of the system have been
represented by solid and dotted line respectively. It is observed from the frequency response curves
that the system does not possess any trivial state response. When the actuator starts to respond at a
frequency corresponding to point A (Figure 3), it is observed that decrease in frequency results in
drop off σ although the amplitude of the actuator slowly increases. Subsequently, the response
reaches a critical value at point B, which is considered as saddle node bifurcation point. At this point,
with further decrease in frequency, the system becomes unstable and experiences a jump
phenomenon. Thus, a sudden jump from point B to C leads to a sudden increase of amplitude as a
result the system may fail. Figure 6 shows the frequency response curve for 2V input voltage. Table
3 lists the variation of the bifurcation points B and C for different input voltages. From Figures 3, 4
and Table 3, it is observed that with decrease in input voltage, critical point B shifts towards 0σ = .

108
The length BC (i.e., Jump length) also decreases with decrease in input voltage. It is also observed
that with dehydration the critical point shifts towards 0σ = and jump length decreases compared to
the zero dehydration condition.
Table 3. Variation of the bifurcation points B and C with various input voltage
Voltage
At critical point B At point C
Jump lengthDetuning parameter
( )σ Response amplitude ( )a
Detuning parameter
( )σ Response amplitude ( )a
With
dehydration
Without
dehydration
With
dehydration
Without
dehydration
With
dehydration
Without
dehydration
With
dehydration
Without
dehydration
With
dehydration
Without
dehydration
1V 0.5659 0.6596 2.022 2.163 0.5659 0.6596 3.955 4.263 1.933 2.1
2V 0.8981 1.039 2.492 2.679 0.8981 1.039 4.985 5.347 2.493 2.668
VI. CONCLUSION
In this paper a model has been developed to estimate the loss due to dehydration in IPMC
following the Cobb-Douglas production method. Non-linear response of an IPMC actuator subjected
to varying electric potential is investigated following method of multiple scales. The simplified
expression can be used for finding the response of the system instead of solving the temporal
equation of motion which is found to be time consuming and requires more memory space. The
estimation of loss due to dehydration helps one to understand the amount of input voltage or
moisture quantity that has to be added with time to compensate the loss to accurately control the tip
position of the IPMC actuator.
VII. ACKNOWLEDGEMENTS
The authors express their keen indebitness to Dr. Dibakar Bandopadhya, Assistant Professor,
IIT Guwahati for providing the necessary information and valuable data for successful completion of
the present study. The authors also wish to acknowledge Prof. Santosha Kumar Dwivedy, Professor,
IIT Guwahati for his help regarding successful completion of this study.
REFERENCES
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[2] K.J.Kim and M.Shahinpoor, Ionic polymer-metal composites: II. Manufacturing
Techniques, Smart Materials and Structures, 12, 2003, 65-79.
[3] M.Shahinpoor, and K.J.Kim , Ionic polymer-metal composites: I. Fundamentals, Smart
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[4] Y.Bar-Cohen, Electric flex, IEEE Spectrum, 41, 2004, 29-33.
[5] S.Nemat-Nasser, and Y.Wu, Tailoring the actuation of ionic polymer-metal composites
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[6] L.Zhang, and Y.Yang, Modeling of an ionic polymer-metal composite beam on human
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109
[7] C.S. Kothera, and D.J.Leo, Characteristics of the solvent-induced nonlinear response of
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[8] M.Yamakita, N.Kamanichi, Y.Kaneda, K. Asaka, and Z.W.Luo, Development of an
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