Theoretical and Experimental Investigation of Non-linear Vibration Response of an IPMC Actuator...

JoMA (2014) 11-26 © STM Journals 2014. All Rights Reserved 1

Journal of Mechatronics and Automation

Volume 1, Issue 1

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Theoretical and Experimental Investigation of Non-linear

Vibration Response of an IPMC Actuator Subjected to

Alternating Electric Potential

Dillip Kumar Biswal, Dibakar Bandopadhya*, Santosha Kumar Dwivedy Department of Mechanical Engineering, Indian Institute of Technology Guwahati,

Guwahati, Assam, India

Abstract The precise predication of stable and unstable zone for an applied excitation voltage and

frequency is of importance in many micromechanical systems utilizing ionic polymer

metal composite (IPMC) as the actuator. A linear dynamic model thus will no longer be valid for IPMC actuator having low thickness and thus more flexible. In this article, non-

linear vibration characteristics of a silver (Ag) based IPMC actuator is studied under alternating electric potential in inextensible condition. The IPMC actuator of silver

electrode is fabricated first using Nafion as the base polymer following the chemical

decomposition method. A theoretical model of motion has been derived using experimental bending data and following the D’Alembert’s principle that describes the

nonlinear vibration characteristics of the actuator. Generalized Galerkin’s method is

then utilized to discretize the equation of motion. Method of Multiple Scales has been used to solve the non-linear equation of motion. Simulations have been performed taking

into account the experimental data and by solving the temporal equation of motion of the system. Several experiments are conducted with an IMPC actuator in fixed-free

configuration and vibration response is studied applying the alternating electric

potential. Both transient and steady-state response of the system is studied and the theoretical results are validated and correlated with the experimental results.

Keywords: Ionic polymer-metal composites, non-linear vibration, D’Alembert’s

principle, Galerkin’s method, method of multiple scales

*Author for Correspondence E-mail: [email protected], [email protected]

INTRODUCTION Polymeric active material technologies are

being developed; that closely matches with the

properties of natural muscle systems; such as

produce similar strains and higher stresses,

shrinks and swells using electrostatic forces,

electrostriction, ion insertion, and sometimes

molecular conformational changes.

Particularly, ionic polymer metal composite

(IPMC) a class of electro-active polymers

(EAPs) have received much attention

nowadays both for sensing and actuation

related application. When an external electric

potential of magnitude 1–3V is applied, IPMC

shows considerable bending deformation;

conversely a measurable output voltage is

obtained across its surface when it is deformed

mechanically. These unique properties make

them suitable for being used both as actuator

and sensor [1,2]. In addition, attractive

features such as large bending deformation

with low driving voltage, light weight,

actuation capability in water as well as in air,

make IPMC suitable for application such as in

the field of micro-robotics, biomedical,

marine, artificial muscles, aerospace etc.

[3–6]. It is well accepted that the mobility of

ion-water cluster causes swelling near the

cathode electrode and equivalent contraction

on the other side results bending of IPMC

actuator. Further, it has been shown that, the

bending deformation depends on various

factors such as magnitude and direction of

applied electric potential, moisture content in

the base polymer etc. [7]. IPMC usually

consists of an ion-exchange polymer (IEP)

membrane (Nafion/Flemion), coated with

electrodes made up of high conducting pure

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Non-linear Vibration Response of an IPMC Actuator Biswal et al.

JoMA (2014) 11-26 © STM Journals 2014. All Rights Reserved Page 12

metals like platinum (Pt) or gold (Au) [2, 8,9];

however, it incurs high manufacturing cost. To

reduce production cost, non-precious metals

such as silver (Ag) as electrode material was

proposed for fabrication of IPMC material

[10–12].

When a low magnitude alternative electric

potential (1-2V) is applied to a fully hydrated

IPMC actuator, it shows a considerable

bending vibration at that applied frequency.

An analytical model has been developed for an

IPMC actuator using Euler-Bernoulli’s beam

theory and the vibration response was studied

[13]. Euler-Bernoulli beam theory and the

concept of hydrodynamic functions were used

to develop a theoretical model [14] to find out

the vibration response of an IPMC actuator in

a fluid medium and the results were validated

experimentally. Nonlinear response of IPMC

actuator has been observed under step input

due to various factors such as permanent

strain, [15] or remnant deformation [16] or due

to moisture loss in open environment during

working condition, [17]. Further, an

experimental investigation was carried out of

IPMCs in different solvents forms in

cantilever link configuration to characterize

the nonlinearities in actuation response using

the Volterra series [18]. In application to

walking robots, [19] a model has been

developed applying Hammerstein technique to

identify nonlinearity in IPMC materials. A

two-stage nonlinear electromechanical model

was developed to characterize the actuation

behavior of the IPMC [20]. A nonlinear black-

box model, based on the recurrent multi-layer

perceptron neural network and self-adjustable

learning mechanism was developed [21] to

measure the tip displacement. A simulink

model was developed for an elastic cantilever

beam using nonlinear equation assuming large

angle deflection that demonstrates the static

and dynamic deflection [22,23].

It is experimentally observed that an IPMC

actuator displays considerable mechanical

vibration during actuation. Where, with small

scale displacement and motion, effect of

dehydration and non-linearity of the system in

response become significant aspect when

accurate measurement is required. In micro-

scale applications, small change in amplitude

of vibration or excitation input may bring the

system into non-linear regime, and this

motivates to utilize the non-linear analysis to

predict the behavior of the system more

accurately. However, till date very limited

research has been carried out on non-linear

response and to develop a suitable model that

addresses the nonlinear vibration

characteristics of IPMC actuator, excited

under a low magnitude alternating electric

potential. Though development of a

comprehensive model is still going on, factors

such as dynamic response and nonlinearities of

IPMC have received low attention in the

previous work. Thus, it is relevant to address

and investigate the nonlinear response of

IPMC actuator. In this work, an IPMC actuator

is analyzed in cantilever mode configuration

under alternating electric potential to study the

nonlinear-vibration characteristics. Initially,

IPMCs are fabricated using Nafion as the base

polymer while non-precious metal silver as the

surface electrode following the chemical

decomposition method. The actuator is

modeled following the Euler-Bernoulli

approach treating it as flexible distributed

parameter system. The governing equation of

motion has been developed using

D’Alembert’s principle and the generalized

Galerkin’s method has been used to derive the

temporal equation of motion. A closed form

expression for the response amplitude and the

detuning parameter has been developed and is

used to obtain the nonlinear frequency

response curve for the IPMC actuator. This

simplified expression can be used for finding

the response of the system instead of solving

the temporal equation of motion of the system

which is time consuming and requires more

memory space. Method of multiple-scales has

been applied to solve the temporal equation of

motion and subsequently obtain the steady-

state response and stability of the system. The

influence of various system parameters such as

amplitude, and frequency of the applied

electric potential on the frequency response

curves have also been investigated for simple

resonance condition. Time and frequency

response along with phase portrait have been

plotted to study the system response. The

result obtained from perturbation analysis has

been compared with those obtained by solving

the temporal equation of motion and are found

to be in good agreement. Further, the

numerical results are compared and verified by


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performing experiment. The theoretical results

clearly predict the stable and unstable zone

and the range of the frequencies of the regime.

The results are validated experimentally for a

number of frequencies both in stable, unstable

and steady-state and transient response regime.

BENDING CHARACTERISTICS The current state-of-the-art procedure to

fabricate this smart material of superior

performance is expensive as the surface

electrode material used mostly is Platinum or

Gold. It is anticipated that by optimizing

process and operating parameters and using

non-precious metal as the surface electrode

may reduce the needs of costly precious

material. In the present study, non-precious

metal silver (Ag) is used for electrode material

that reduces the cost of fabrication and at the

same time is shown to be improving the

performance of IPMC.

IPMCs are first fabricated following the

chemical decomposition method. A Nafion

membrane with an equivalent weight (EW) of

1100 g/mol and 0.183 mm thick, (purchased

from Ion Power, Inc, New Castle, DE 19720

USA) is used as the base polymer. The

fabrication process comprises multi-steps

including pretreatment, adsorption, reduction

and developing. Silver nitrates GR (AgNO3),

Ammonia solution (NH3), Sodium hydroxide

(NaOH), Dextrose anhydrous GR (C6H12O6),

(purchased from Merck Specialities Private

Limited, India) are used for fabrication. The

pre-treated Nafion membrane is immersed in

NaOH, 0.5 mol/L solution followed by

Diamminesilver (I) hydroxide [Ag (NH3)2OH],

0.15 mol/L solution to allow Na+ and Ag

(NH3)2+ diffuse into the membrane via ion-

exchange process. C6H12O6, 0.088mol/L is

used as the reducing agent for depositing silver

particles over the Nafion membrane surface.

Finally, the silver coated membrane is

trimmed from all sides to avoid any shorting

between two surfaces. Figure 1 shows the

photograph of a fabricated Ag-IPMC sample.

In the current method of fabrication of silver

electrode IPMCs, several steps of trial-and-

error method were carried out for optimizing

the process parameters as detailed procedure

was not outlined in the previously published

work [10]. Further, the composition of

chemicals used in the current method is

different from those used in the previous work

[10]. Also, the developed IPMC exhibits

superior performance in terms of bending

actuation, tip force including low power

consumption. Further, morphological studies

of the fabricated IPMC show no crack

formation in the micro-structure (with 1.17kX

magnification). It is observed that the average

operating life of the IPMC is around 90–100s.

The membrane can work again upon full

hydration by putting it into deionized/distilled

water. The fabricated IPMC exhibits superior

performance also shows no oxidation problem.

Before starting the experiment, IPMC is

immersed into ammonium hydroxide solution

for considerable time to prevent oxidation

problem, if any, as the ammonium hydroxide

dissolves silver oxide.

Experiment is conducted with an IPMC

actuator of size 20 x 5 x 0.2 (mm3)

in fixed-free configuration under varying input

voltage. Water is used as polar solvent. Copper

strips are used across the fixed end and voltage

is applied from a DC power supply (0–32 V,

0–2 A) and subsequently bending of IPMC is

measured. The IPMC has been subjected to

input voltage from 0.2 to 1.2 V with an

increment of 0.2 V each step. For each input

voltage after 30s, tip deflection of the IPMC is

measured to avoid any back relaxation. Figure

2 shows the bending configuration of the

IPMC actuator subjected to input of 1.2 V.

Fig. 1: Photograph of the Fabricated IPMC.

Non-linear Vibration Response of an IPMC Actuator Biswal et al. __________________________________________________________________________________________


Fig. 2: A Fabricated IPMC is being actuated

under DC Electric Potential of 1.2 V.

Figure 3(a) shows the X and Y coordinates of

tip position for various input voltages. Figure

3(b) shows the relationship between voltage

(V) and the radius of curvature ( cR ) obtained

by 3rd

order curve fitting approximation

technique with a quality factor (R2) 0.9997 and

is given in Eq. (1).

(1)

Bending moment at the tip of the actuator can

be expressed as, , where, is the

X-coordinate of tip position of the actuator.

Assuming pure bending, i.e., satisfy the

relationship as,

Popov [24]; the tip

force produced due to input voltage (V) can

be expressed as:

(2)

NON-LINEAR MODEL OF

ACTUATOR Figure 4(a) shows insight of an IPMC actuator

that describes the bending mechanism with

input potential. When an electric potential is

applied across the thickness of IPMC, mobile

cations combined with the surrounding water

molecules move towards the oppositely

charged electrode. The process results in re-

distribution of water molecules as shown in

Figure 4(b). The well accepted mechanism for

actuation of IPMC is considered to be the

migration of mobile cations that initiates

swelling near the cathode electrode and

shrinkage near the anode that leads to bending

deformation towards anode. However, the

nonlinear characteristics of the IPMC actuator

is observed to be due to the structural

flexibility of the system and that depends on

electrode layer (type, area, thickness); change

in the applied potential difference, polymer

property(type of polymer, counter ion,

hydration level/moisture/water content), and as

well as other environmental conditions.

(a) (b)

Fig. 3: (a) X and Y Coordinate of Tip Position (b) Radius of Curvature Obtained for

Different Input Voltage.


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Fig. 4: (a) Schematic of an IMPC before Potential is applied (b) its Actuation Mechanism after

Potential is on.

Fig. 5: Fixed-free Bending Configuration of an IPMC Actuator for an Input Voltage V.

Governing Equation of Motion

Figure 5 shows the schematic diagram of

bending configuration of IPMC actuator

subjected to alternating electric potential at the

fixed end across its thickness. The actuator is

modeled as uniform flexible distributed

parameter system following the Euler-

Bernoulli beam theory as shown in Figure 2.

The actuator is operated by applying electric

potential at the fixed end and allows it to bend

at the free end. Shear deformation and rotary

inertia effects if any, are neglected. The

bending moment ( ) at a distance from the

fixed end along the link length, as shown in

Figure 5 can be expressed as [25,26]:

( ) [

] (3)

where, is the transverse displacement of the

link, ( ) is the first derivative with respect to

curvature ( ). is the modulus of elasticity

and is the area moment of inertia. Assuming

that length of the actuator does not change

during bending, i.e. preserve inextensible

condition, the longitudinal displacement (X-

direction) ( )and the transverse

displacement (Y-direction) ( ) can be

expressed as [26]:

( )

( ) ∫ ( )

(4)



where, are the integration variables. Inertia forces per unit length of the actuator are and

in longitudinal and transverse directions, respectively. where, is the density and A is the cross

sectional area. The bending moment in longitudinal ( ) and transverse ( ) directions due to

the inertia forces at a distance from the fixed end can be expressed as:

( ) ∫

( ) ∫ ( )

(5)

( ) ∫

( ) ( ) ∫ ( )

(6)

The external moment developed for an applied voltage V can be given by:

( ) ∫ ( ) ( )

∫ ( )

(7)

where, is the frequency of the applied electric potential. Hence, applying D’ Alembert’s principle,

( ) ( ) ( ) ( ) (8)

Substituting Eqs. (3–7) into (8) and differentiating the resulting equation twice with respect to s and

using the Leibniz’s rule, the equation is transformed into the following equation of motion:

(

) [∫ (

)

] [∫ ∫ (

)

]

(

) [∫ ( )

] ( ) ( )

The nonlinear term associated with forcing is neglected. Generalized Galerkin’s method is utilized to

discretize the governing equation of motion (9) into its temporal form by substituting:

( ) ( ) ( ) (10)

where, ( ) and ( ) are the scaling factor, admissible function and time modulation. The

admissible function ( ) is considered to be same as the Eigen-function of a cantilever link and can

be expressed as:

( ) ( )

( )( ) ( ) (11)

where, for the first mode of vibration. Introducing the following non-dimensional

parameters,

(12)

and on substitution Eq. (12) into Eq. (9), the resulting non-dimensional temporal equation of motion is

derived as:

( ) (13)

where, is the book keeping parameter. The expressions for the coefficients

( ) are given in the appendix. It is observed that the non-linear temporal Eq. (13)

contains linear force term , cubic geometric term , nonlinear inertia terms

and

nonlinear damping term . As the closed form solution of Eq. (13) is difficult to obtain, an

approximate solution is obtained by using perturbation technique and thus, method of multiple scales

[27,28] has been used to solve the equation of motion.

Method of Multiple Scales

Displacement is expressed in terms of different time scales ( ) and a book keeping parameter

as follows:

( ) ( ) ( ) ( ) (14)

where, Taking the first and second time derivatives of Eq. (14) and substituting into

Eq. (13) and equating the coefficients of the power terms of , one can obtain the following

expressions.

Order

(15)

Order 1 :


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( )

(

) (

)

( ) (16)

The general solution of the Eq. (15) is of the form:

( ) ( )

(17)

Substituting the value of 0q into Eq. (16), and expressing ( ) in exponential form:

(

) (

)

(18)

where, stands for the complex conjugate of preceding terms. It is observed that any solution of Eq.

(18) contains both secular and mixed-secular terms when frequency of the electric potential is

nearly equal to 1. This gives rise to simple resonance case and is discussed in the following section.

For the simple resonance case, a detuning parameter is used to express the near-ness of to 1

as: ( ), and ( ). Eliminating the secular or small divisor terms of Eq. (18), yields,

(19)

Substituting,

, and

and separating the real and imaginary parts of the resulting

equation and introducing into the Eq. (19), the modulation frequency and amplitude are

obtained as:

( ) (20)

( ) (21)

For steady-state motions, . Thus, the steady-state frequency response of the system can be

obtained by solving Eqs. (20, 21) simultaneously. Eliminating from Eqs. (20, 21), one may find the

relation between and a as:

√(

) (

)

(22)

where, {

} Eq. (22) is an implicit form for amplitude of the response as a function

of excitation frequency, damping ratio and other system parameters. It is observed that, Eq. (22)

does not possess any trivial solution. The stability of the steady-state frequency response can be

determined, by perturbing the Eqs. (20) and (21), and substituting and .

where, ( ) are the singular points, and further investigating the Eigen-values of the Jacobian

matrix (J) which is obtained as:

[

{

}

] (23)

For simple resonance condition, the system with steady-state motion will be stable, if and only if all

the real parts of the Eigen-values are negative. As on Eq. (17), the first-order non-trivial approximate

steady-state solution is given as:

( ) (24)



RESULTS AND DISCUSSIONS Numerical Results

An IPMC actuator is numerically studied and

non-linear response is investigated. The

various parameters and physical properties are

shown in Table 1. Modulus of elasticity of the

actuator in hydrated state is obtained by

conducting micro-tensile test (maker: Deben

UK Ltd., model: Microtest 5 kN). Anticipating

low amplitude vibration, book-keeping

parameter is taken 0.1 and scaling factor

r is selected as 0.1 for simulation. The

scaling factor ‘r’ is the dimensional parameter

of length that depends on thickness, width, and

the length of the ‘centre of mass’ of the IPMC

from the base. Eq. (22) is solved and the

results are verified with the results obtained by

solving the temporal equation of motion (13).

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 Eq. (13), the

system yields a typical non-linear behavior.

For simple resonance case and in hydrated

condition, the frequency response is obtained

by using the closed form expression (Eq. (22))

and is shown in Figure 6. It is observed from

the frequency response 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 6. In Figure 6(a), the

frequency response is plotted for 0.4 V where

one may clearly observe a frequency range BC

at which system response is unstable and in

the frequency range of AB and CD the system

response is stable. When the actuator is

activated, (e.g., at point A), with increase in

frequency of excitation, the amplitude of

response increases and reaches a critical value

at point B. At this point, further increase in

frequency, brings the system into unstable

regime. Further increase in frequency, drives

the system 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.

Table 1: Physical Properties and Dimensions of the IPMC Actuator.

Elastic modulus ( E ) Length ( l ) Width (b ) Thickness ( h ) Density ( )

0.081877 GPa 0.02 m 0.005 m 0.0002 m 2125 kg/m3

(a) (b)


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(c) (d)

Fig. 6: Frequency Response Curve for Various Input Voltage of (a) 0.4V (b) 0.6V (c) 1.0 V (d) 1.2 V

and Frequency 12.5 Hz.

This observation is consistent with the results

presented by Nemat-Nasser and Wu [7] and

Cilingir and Papila [29]. If the system is driven

by a voltage with frequency in between point

B and C the response will thus be 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 B or

above of the bifurcation point C. It is also

observed that as the input voltage increases the

response amplitude also increases as shown in

Figures 6(a)–(d).

Table 2: Variation of the Bifurcation Points

with Various Input Voltages.

Input

( V )

Bifurcation point (σ)

Point B Point C

0.4 0.5786 0.268

0.6 0.4671 0.4954

0.8 0.3119 0.7616

1.0 0.1395 0.9586

1.2 -0.2079 1.447

Fig. 7: Basin of Attraction for (Key

as in Fig. 6 (a) for 0.4V).

As shown in Figure 6(a), due to the presence

of bi-stable region before the bifurcation point

C, the initial conditions in this region plays an

important role to find the appropriate steady-

state response. Hence, in order to know the

influence of initial condition, basin of

attraction is plotted in the ‘ ’plane

corresponding to for 0.4 V as shown

in Figure 7. Figure 7 clearly shows two stable

solutions corresponding to point (P1, P3) and

one unstable solution corresponding to point



(P2) as marked in Figure 6(a). Figure 8 shows

the time response and phase portrait, obtained

by solving the Eqs. (20) and (21) for applied

electric potential of 0.4V. These results are in

good agreement with the results presented in

Figure 6(a) at point P1. Table 2 lists the

variation of bifurcation points B and C with

input voltage. Further, it is clearly observed

that the unstable zone BC increases with

increase of input voltage. Figure 9 shows the

vibration response and phase portrait at a

point, where as shown in Figure 6(a).

The results are obtained by solving the

temporal equation of motion (13). 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 6(a). The steady-

state response of the system as shown in

Figure 9 (iii) is periodic which is in good

agreement with the amplitude obtained using

method of multiple scales. Figure 10 shows

the time response for which is found

to be unstable by solving the temporal

equation of motion and can be well verified

with the results presented in Figure 6(c).

(a) (b)

Fig. 8: (a) Time Response and (b) Phase Portrait at Point, , 0.4 V, Corresponds to Fig.6 (a).

(i) (ii)


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(iii) (iv)

Fig. 9: Vibration Response and Phase Portrait for , obtained by Solving Temporal Equation

of Motion, (Key as in Fig. 6 (a)). Where, (i), (ii) Transient Response and (iii), (iv) Steady-state

Response.

Fig. 10: (a) Time Response for ,

obtained by Solving Temporal Equation of

Motion for an Input Voltage 1.0 V (Key as in

Figure 6 (c)).

Experimental Results

An IPMC actuator of size 20 x 5 x 0.2 (mm3)

is prepared and tested under sinusoidal input

voltage with different frequencies. Transient

and steady-state vibration characteristics are

studied and compared with the theoretical

results. The experimental setup is shown in

Figure 11. One end of the control module is

connected to the Rotational Laser Vibrometer

(RVL-5500; maker: Polytec, GmbH) and other

end to the computer where the data are

processed using Bruel and Kjaer PULSE Lab

Shop, Version 13.1.0.246 software. The

function generator (Sony/Tektronix AFG 320)

is used to apply AC voltage at the fixed end of

the sample while Laser is directed on tip of the

free end. Before the experiment, IPMC sample

is boiled in deionized water for 10 min and

kept within deionized water at room

temperature for 24 h to ensure full hydration.

The entrapped surface moistures are removed

by keeping the actuator in open environment

for 5 min prior to the experiment. Voltage is

applied for 30s and subsequently response of

the free end is measured using Rotational

Laser Vibrometer.



Fig. 11: Experimental Setup for Vibration Response Analysis.

Fig. 12: Experimental Vibration Response at 0.4V (a) 12.5 Hz (b) 16.5 Hz.

All the experiments are conducted at room

temperature. The experimental results are

presented in Figure 12–15. Taking the physical

properties and dimensions of the IPMC

actuator as given in Table 1 (same for the

experiment) and using the expressions for

and given in the

Appendix, the coefficients of the nonlinear

equation of motion (13) have been determined.

For a given frequency and using Eq. (22), the

amplitude of the response is determined and

further used in Eq. (24) and Eq. (10) to obtain

the response of the system. Figures 12(a) and

(b) demonstrate the vibration response of the

IPMC actuator for an applied voltage of 0.4 V

at different frequency. It is observed that as the

input frequency changes from 12.5 Hz

(corresponds to ) to 16.5 Hz ( )

the vibration amplitude decreases. For an input

of 0.4 V, corresponding to a frequency of 12.5

Hz ( ) the nontrivial amplitude of time

modulation of steady-state response ( a ) is

found to be around 0.004 mm. As shown in

Figure 6(a), the transverse displacement

(steady-state response amplitude)

corresponding to is found to be

( )

( ) ( ) . Hence,


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the response obtained from the perturbation

analysis is found to be in good agreement with

the experimental value. Similarly for the point

D ( ) in Figure 6(a), the amplitude is

found to be 0.00352 mm. Also, for the

corresponding point D in Figure 12(b), the

transverse displacement is found to be around

0.003 mm, which is in good agreement with

the numerical results as shown in Figure 6(a).

Figures 13(a) and (b) show the vibration

response corresponding to different input

voltages with constant frequency of 15.251 Hz

( ). It is observed that as the input

voltage increases the amplitude of the

vibration response also increases validating the

simulation results shown in Figures 6(a) and

(b).

Fig. 13: Experimental Vibration Response at 15.25 Hz (a) 0.4 V (b) 0.6 V.

Fig. 14: Experimental Vibration Response for an Input 1.0V (a) Unstable (b) Stable



Fig. 15: FFT of the Vibration Response at 0.4

V, 12.5 Hz.

Figure 14(a) shows the unstable vibration

response of the actuator for an input 1.0 V and

frequency of 14.5 Hz corresponds to .

The results clearly show that with input

voltage the amplitude increases gradually and

become unbounded. The results also validate

the simulation results as shown in Figure 6(c)

and Figure 10. Figure 14(b) shows the steady-

state response (after the voltage is withdrawn)

at 1.0 V with an applied frequency 8.0 Hz

corresponding to . Figure 15

demonstrates the FFT of the vibration

response as shown in Figure 12(a). It is

observed that band-width of vibration response

remains within 2–30 Hz.

CONCLUSION In this work, a closed form expression for the

response amplitude and detuning parameter

has been developed that can be used to obtain

the nonlinear frequency response curve for any

IPMC actuator. This simplified expression can

be used for finding the response of the system

instead of solving the temporal equation of

motion which is time consuming and requires

more memory space. The result obtained from

perturbation analysis has been compared with

those obtained by solving the temporal

equation of motion and are found to be in good

agreement. The numerical results are

compared and verified by performing

experiment. It is concluded that in small-scale

application, amplitude of vibration could

increase considerably owing to the small

change in excitation of amplitude occurs.

Further, in light-scale applications, the

amplitude of vibration can be easily increased

and it may bring the system into non-linear

regime, and therefore, it is useful to utilize the

non-linear analysis to predict the behavior of

the system more accurately.

ACKNOWLEDGEMENT The ‘corresponding author’ thankful for the

partial financial support received from

Department of Science and Technology

(DST), Government of India-under SERC

FAST Track Scheme (SR/FTP/ETA-

076/2009) for carrying out this research work.

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http://dx.doi.org/10.1088/0964-1726/19/4/045026



Appendix

The system fundamental frequency, 11

4 2

1 1

n

EIh

Al h

.

The coefficients of the temporal equation (13) are given by the following expressions:

Damping ratio, 6

1 1

0.11012 n

ch

A h

.

Coefficient of the non-linear geometric term (3q ),

2 1213 14

1 4 2

1 1

32

0.0818

hEIr h h

Al h

.

Coefficient of the non-linear inertia term 2q q ,

2 52 3 4

2

1

20.2143

hr h h h

h

.

Coefficient of the non-linear inertia term 2q q ,

2 87

3

1 1

20.0054

hcr h

A h

.

Coefficient of the non-linear inertia term 2qq , 2

9 10

4

1

0.4597r h h

h

.

Coefficient of the linear force term, 15

2

1 1

n

F hF

Arl h

Here the expression for 1 2 3 15, , ,..............,h h h h are given below:

1

1

0

( ) ( )wh s s ds , 1

2

2

0 0

( ) ( ) ( )wh s s d s ds

,

1 1

3

0

( ) ( ) ( ) ( )w

s

h s s s d s ds 1 1

2

4

0 0

( ) ( ) ( )w

s

h s s d d s ds

,

1

2

5

0

( ) ( ) ( )wh s s s ds ,

1

6

0

( ) ( )wh s s ds , 1 1

7

0

( ) ( ) ( ) ( )w

s

h s s s d s ds ,

1

2

8

0

( ) ( ) ( )wh s s s ds ,

1

2

9

0 0

( ) ( ) ( )wh s s d s ds

,

1 1

2

10

0

( ) ( ( ) ) ( )w

s o

h s s d d s ds

1

2

11

0

( )h s ds , 1

2

12

0

( ) ( ) ( )iv

wh s s s ds , 1

3

13

0

( ) ( )wh s s ds

1

14

0

( ) ( ) ( ) ( )wh s s s s ds ,

1

15

0

( )wh s ds

Theoretical and Experimental Investigation of Non-linear Vibration Response of an IPMC Actuator...

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