Study on Optimization for Four Bar-Partially Compliant...

J. Eng. Technol. Educ. (2013) 10(4): 433-449 December 2013

Journal of Engineering Technology and Education, ISSN 1813-3851

Study on Optimization for Four Bar-Partially Compliant Mechanism

Shyh-Chour Huang*, Thanh-Phong Dao

Department of Mechanical Engineering,

National Kaohsiung University of Applied Sciences Kaohsiung, Taiwan

*E-mail: [email protected]

Abstract This paper concentrates on the application of fuzzy logic based on Taguchi method (FLTM) for

multiobjective optimization of the small-length flexural pivot (belongs to link 4) in partially four bar-compliant

mechanisms. The input parameters such as the input torque and the crank angle of link 2 were optimized with

considerations of the multiple performance measures as angular displacement and stress of the small-length

flexural pivot. Angular displacement and stress of the small-length flexural pivot are two objective functions were

formulated using pseudo-rigid-body model (PRBM) and principle of virtual work. The results found that the input

torque at level 1 of 3 Nm and crank angular at level 1 of 100 degrees are favorable parameters for compliant

flapper mechanism. Based on ANOVA, the results showed that input torque is the most significant with respond

highest value of F test of 4.1752.

Keywords: Multi-response performance index, Fuzzy logic based on Taguchi method, Pseudo-rigid-body model

1. Introduction Design and analysis of compliant mechanisms have interested in past decades due to their less expensive,

lighter as compared to rigid-link mechanisms. The compliant mechanisms relies on the deflection of flexible

members, thus their maximum deflection and minimum stress have been the important issues in designing flexible

structures in recent years. In 2010, Tanık et al. [1] used the theory of the pseudo-rigid-body model (PRBM) to

analyze compliant variable stroke mechanism. Khatait et al. [2] modified the stiffness of the flexural hinges to

minimize driving torque. Midha et al. [3] utilized PRBM to analyze limit positions of compliant mechanism. Dado

[4] studied large deflections of compliant mechanism using PRBM. In 2012, Hsiang et al. [5] studied the

optimization of extrusion magnesium alloy bicycle carriers by using fuzzy logic based on Taguchi method (FLTM).

According to Venanzi [6], the non-linear position analysis of planar compliant mechanisms was performed by

using an iterative technique. A new method is proposed to easily establish simple and analyze using PRBM for a

variety of beam-based compliant mechanisms performed by Pei [7]. The other study by Tsay [8] presented the

design, fabrication and experiment of fully compliant bistable micro-mechanisms. Pucheta [9] presented a method

is based on the solution of the initial and final unstrained positions for the design of bistable mechanisms. In 2011,

Gupta et al. [10] presented the application of FLTM for multiple output optimization of high speed CNC turning.

In spite of the four-bar compliant mechanisms have been so far received a growth of interest in the robotic

devices and the other growth of specific interest to humanoid robots where human-like manipulation skill is

required as well because their size, weight, reliability, cost and applicability, etc. In order to analyze kinematics,

dynamics, and optimize these compliant mechanisms, they have still been the complicated problems to researchers.

In particular, the structure of the compliant mechanisms is very sensitive to dynamic actions like shock and

Shyh-Chour Huang, Thanh-Phong Dao 434

vibrations that occur during motion process. In order to move to desired positions the fatigue, the stress, and thus

the displacement must be considered in design area simultaneously.

One of the challenges of compliant mechanism is to allow large enough deflections for the mechanism to

perform its function while maintaining stresses below an allowable maximum stress. Therefore, the maximum

displacement and minimum stress have been still interested tasks in recent years for many researchers. There are

many optimization approaches used in engineering such as Taguchi method, fuzzy logic control, genetic algorithm,

and neural network; among them, Taguchi method is not a complicate procedure but it can only optimize for single

objective function. As a result, an attempt to develop approaches for optimization of multi-objective functions has

researched so far, like FLTM, genetic algorithm based on fuzzy logic control, etc. One of the another reasons is

that almost previous studies have not significantly investigated the optimization compliant mechanisms by using

FLTM; they focused on the use of conventional procedures to optimize topology of mechanisms and more recently

years some authors have optimized the various shapes of flexible segments to release the stress concentration

along the flexible beams. As a result, the optimization of process parameters to maximize deflection of flexible

segment and minimize its stress simultaneously was studied in this paper.

There have been many existed effective approaches such as the PRBM, the screw theory, the building-block

based approach, the topological optimization, and the constraint-based design to analyze statics, kinematics and

dynamics of the compliant mechanisms. Among them, the PRBM is a helpful tool and easy to use. So, the PRBM

was applied in this paper.

In addition, the equations of multiobjective functions are formulated by using PRBM and the principle of

virtual work; these have been easy tools to analyze the compliant mechanisms in past decades. In fuzzy-logic

control algorithm, the membership functions setting is very important in order to achieve optimal parameter value,

there are excessive number of experiments; therefore, this research exploited Taguchi method was adopted to

decrease the number of experiments. Furthermore, the adjusting the weight of membership function is also

significantly important to achieve optimal system.

This paper describes the main guidelines for analysis and optimization of four-bar partially compliant

mechanism. It proposes a novel optimization method that is FLTM in order to maximize angular displacement and

minimize stress simultaneously of small-length flexural pivot. This one can further uses in the other structures and

engineering fields.

The rest of the paper is organized as follows. Section 2 analyzes four-bar partially compliant mechanism in

order to formulate the multi-objective function equations. Section 3 describes the optimization of four-bar partially

compliant mechanism. It will present Taguchi method with multiple performance characteristics and the way how

to combine with the fuzzy logic controller in the optimization for angular displacement and the stress

simultaneously. Also in Section 3, this paper will present the results and discussions of the paper. Final section will

be the conclusion of this study.

2. Analysis of Four-Bar Partially Compliant Mechanism In this stage, the paper will apply theory of PRBM and virtual work [11] to model and analyze flexible

beams and segments. And then it will formulate the equations of multiobjective functions including angular

displacement and stress for optimal process.

Study on Optimization for Four Bar-Partially Compliant Mechanism 435

2.1 Modeling

In this paper, the compliant beams are made of polypropylene ( )2/000,200 inlbE = due to its low density and

high strength-to-modulus ratio. Polypropylene is ductile material and thus it is much less likely to result in

catastrophic failure when yielded. The PRBM [11] was utilized to analyze the deflection and the stress of flexible

segment in for-bar partially compliant mechanism. The mathematical models of the deflection and the stress of

flexible segment were formulated as the following.

The partially compliant mechanism as shown in Fig. 1 is a four bar mechanism as a case of a Grashof

mechanism known as triple couplers with input crank is link 2, remaining links are rockers. A functional binary,

fixed-pinned flexible segment is link 3. The output rocker is link 4 that has one small-length flexural pivot.

Deflected position gives in Fig. 2a. The PRBM is shown in Fig. 2b. Using Brushless DC Servo Motor to control flapper compliant mechanism with the following specifications (power: 200 to 4,000 Watts, torque: Nm2 to

Nm115 ). The dimension of links as follows:

- Cross section of rigid links: Width of 0.8 in and thickness of 0.2 .in

- Ground link: .02.31 inr =

- Input crank: .2 inr =

- The long flexible segment: ,97.23 inl = width ,5.03 inb = thickness .04.03 inh =

- The output rocker: Rigid link of ,3in ,31.04 inl = ,6.04 inb = .5.04 inh = This study selected constant values of characteristic radius factor, 85.0=γ and stiffness coefficient,

67.2=ΘK [11] result in a pseudo-rigid coupler with length ,53.233 inlr == γ ( ) .07.32/91.2 44 inlr =+= The moments of inertial of the flexible segments are:

12

333

3hbI = (1)

12

344

4hbI = (2)

The flexible segments always remains initial state where 02 90=θ . The values of 30θ and 40θ can be calculated

by using the closed-form equations [11] as follows: The crank angle measured from 1r is 2θ ′ :

122 θθθ +=′ (3)

where 1θ is the angle between horizontal direction with the line coupled from joint A to joint D.

The law of cosines may be used to determine the length of δ and the internal angles β andλ as follows:

( ) 2/1221

22

21 cos2 θδ ′−+= rrrr (4)

δδβ

1

22

221

2cos

rrra −+

= (5)


δδλ

4

23

224

2cos

rrra −+

= (6)

δδ

α3

24

223

2cos

rrr −+

= (7)

Two possible values exist for each angle for a given .2θ The link angles are calculated as follows:

For πθ ≤′≤ 20

( )13 θβαθ −−= (8)

( )14 θβλπθ −−−= (9)

For πθπ 22 ≤′≤

( )13 θβαθ ++= (10)

( )14 θβλπθ ++−= (11)

Fig. 1 Model of a four-bar partially compliant four bar


Fig. 2 a. Deflected position model, b. Pseudo-rigid body model

2.2 Reaction Forces at Pivot joints

Free body diagrams (Newtonian methods) [11] are used to determine equations of reaction forces at pivots for

static equilibrium of each link. Free body diagrams of links are illustrated in Fig. 3.

Fig. 3 Free body diagram of links: a. Link 2, b. Link 3, c. Link 4

For link 2:

03212 =+ xx FF (12)

03212 =+ yy FF (13)

( ) ( ) 090cos90sin 22322232 =−××−−××− θθ rFrFT xyin (14)

For link 3:


04323 =+ xx FF (15)

04323 =+ yy FF (16)

0sincos 3343334343 =××−××+ θθ rFrFT xy (17)

For link 4:

03414 =+ xx FF (18)

03414 =+ yy FF (19)

( ) ( ) 090cos90sin 443444343414 =−××−−××−+ θθ rFrFTT xy (20)

At a pin joint, the forces on the connected links have equal magnitude but are in opposite directions:

xx FF 3223 −= (21)

yy FF 3223 −= (22)

xx FF 4334 −= (23)

yy FF 4334 −= (24)

The spring constants at pin joints due to the springs [11]:

Due to fixed-pinned segment

334

=

lEIKK Θγ (25)

Due small-length flexural pivot

414

=

lEIK (26)

The torques at pin joints due the as follows [11]:

A PRBM with a linear torsional spring constant, Ki,


iiKT Ψ−= (27)

where Ψ is the Largrangian coordinate

The torque at pin joints 1 and 2:

021 == TT (28)

The torque caused by the torsional spring modeling fixed-pinned segment:

( ) ( )[ ]3034043

33434 θθθθγ −−−

−=Ψ−= Θ l

EIKKT (29)

The torque caused by the torsional spring modeling the small-length flexural pivot:

( ) ( )4044

40441414 θθθθΨ −

−=−=−=

lEIKT (30)

Combining Eqs.15, 16, and 17 results in:

0sincos 3323332343 =××+××− θθ rFrFT xy (31)


( ) ( ) 090cos90sin 22232223 =−××+−××+ θθ rFrFT xyin (32)

From Eq.31 results in:

33

4332323 sin

cotθ

θ×

−×=r

TFF yx

(33)

Substituting Eq.33 into Eq.32 results in:

( )( ) ( )[ ]90coscot90sinsin

sin90cos

232332

33224323 −×+−××

××−−××=

θθθθθθ

rrrTrTF in

y (34)

Substituting Eq.34 into Eq.33 results in

( )[ ][ ]( ) ( )[ ]

×

−−×+−××

×××−−××=

33

43

232332

333224323 9090

90θθθθθ

θθθsinr

Tcoscotsinsinrr

cotsinrTcosrTF inx (35)



0sincos 3334333443 =××+××− θθ rFrFT xy (36)

From Eq.36 results:

33

4333434 sin

cotθ

θ×

−×=r

TFF yx (37)


( )( )( ) ( )[ ]9090

90

434343

4433343334 −×+−××

−×+×+××=

θθθθθθθ

coscotsinsinrrcosrsinrTsinrTF in

y (38)


( )( )[ ]( ) ( )[ ]

×

−−×+−××

−×+×+××=

33

43

434343

34433343334 9090

90θθθθθ

θθθθsinr

Tcoscotsinsinrr

cotcosrsinrTsinrTF inx (39)

2.3 Angular Displacement of Small-Length Flexural Pivot

Virtual work [11] was used to determine the deflection-force relationship. Based on PRBM and virtual work,

the deflection-force relationship for flexible segment was determined. Degrees of freedom of the mechanism:

( ) 21213 JJnF −−−= (40)

where n is number of links ( )4=n

1J is lower pair ( )41 =J

2J is higher pair ( )02 =J

Thus, this study calculated degree of freedom is 1=F

The minimum number of Lagrangian coordinates required for a complete set is equal to the number of

degrees of freedom of system. Because the mechanism has one degree of freedom, it will have one generalized coordinate. Choose 2θ as the generalized coordinate because it is the known input. Considering the system of this

mechanism, there is only an input moment inT that causes input angular displacement, .2δθ A torque, 34T , caused

by the torsional spring causes angular displacement, 3δθ and a torque, 14T caused by the torsional spring causes


angular displacement, .4δθ Thus, total virtual work of this mechanism is [11]:

22

4

2

3 δθδθδθ

δθδθ

δ

++= CBAw

(41)

Applying the principle of virtual work ( 0=wδ ) results in

02

4

2

3 =++δθδθ

δθδθ CBA

(42)

where:

inTA = (43)

( ) ( )[ ]3034043

34 θθθθγ −−−

=−= Θ l

EIKTB (44)

( ) ( )[ ] ( )4044

3034043

1434 θθθθθθγ −

−−−−

−=+= Θ l

EIl

EIKTTC (45)

The kinematics coefficients are:

( )( )433

242

2

3

sinsin

θθθθ

δθδθ

−−

=rr

(46)

The angular displacement of the small-length flexural pivot, δθ4, is derived from Eq.45 as follows:

CAB 2

2

34

δθδθδθδθ

+−= (47)

Substituting Eqs.42, 43, 44, 45and 46 into Eq.47 results in:

( ) ( )[ ] ( )( ) ( ) ( )[ ] ( )

−

+−−−

+

−−

−−−

=

Θ

Θ

4044

3034043

2

432

243303404

34 sin

sin

θθθθθθγ

δθθθθ

θθθθγδθ

lEI

lEIK

Trr

lEIK in (48)

2.4 Maximum Stress of Small-Length Flexural Pivot

The maximum stress for a cantilever flexible beam occurs at the fixed end. The maximum stress may occur at

the top or bottom of the beam, depending on the direction of the force. In this mechanism, a flexible beam has

cross-section with the out-of-plane thickness, w and the in-of-plane thickness, h [11] as follows:


( )whnp

whnpbpa

top −+−

= 2

6σ (49)

( )

whnp

whnpbpa

bot −+

= 2

6σ (50)

There is no axial force and only has bending forces in this case. Thus, considering stress due bending force.

44 cos07.3155.0cos22

θθ +=

++=

lLla (51)

44 sin07.3sin2

θθ =

+=

lLb (52)

( )

23434

whbFaF yx

top

+−=σ (53)

( )

23434

whbFaF yx

bot

+=σ (54)

The maximum stress at fixed-end of small-length flexural pivot is

( ) ( )( )[ ]( ) ( )[ ]

( )( )( ) ( )[ ]

2434343

443334334

33

43

434343

3443334334 90coscot90sinsin

90cossinsinsin07.3sin90coscot90sinsin

cot90cossinsincos07.3155.0

whrr

rrTrTr

Trr

rrTrT inin

−×+−××−×+×+××

+

×

−−×+−××

−×+×+××+

=θθθθθθθ

θθθθθθ

θθθθθ

σ (55)

where ( ) ( )[ ]3034043

34 θθθθγ −−−

−= Θ l

EIKT , 4334 TT −= (56)

The variations in process parameters such as input torque and input crank angular greatly affect the motion

perform of the flapper compliant mechanism as angular displacement and stress. Therefore, proper selection of the

process parameters can result in better performance in motion of this mechanism.

3. Optimization of Four-Bar Partially Compliant Mechanism

3.1 Formulation of the optimization problem

This study determines the optimal process parameters that influence the structure of four-bar partially

compliant mechanism by simultaneous optimizing the maximum angular displacement and minimum stress to

obtain optimal motion performance. Therefore, the higher-the-better angular displacement and the lower-the-better

stress should be selected. The optimization problem for four-bar partially compliant mechanism is formulated as

follows:

Maximize angular displacement of small-length flexural pivot belongs to link 4:


( ) ( ) ( ) ( )( ) ( ) ( ) ( )

3 4 2 21 2 4 40 3 30

3 2 3 44 40 3 30 4 40

3 4

sin,

sinin in

rEIf T K Tl r EI EIK

l l

θ θ δθ γ θ θ θ θ

θ θγ θ θ θ θ θ θ

Θ

Θ

− = − − − + − − − − + − (57)

Minimize the maximum stress at fixed end of small-length flexural pivot belongs to link 4:

( )( )

( )( )( ) ( )

( )( )( )

3 3 34 3 3 4 4 3 3 3 34 3 3 4 4434 4

3 33 4 3 4 3 4 3 4 3 4 3

2 2

sin sin cos 90 cot sin sin cos 900.155 3.07cos 3.07sin

sinsin sin 90 cot cos 90 sin sin 90 cot c,

in in

in

T r T r r T r T r rTrr r r r

f T

θ θ θ θ θ θ θθ θ

θθ θ θ θ θ θ θθ

× × + × + × − × × + × + × − + − +×× × − + × − × × − + × =

( )4

2

os 90

wh

θ

− (58)

subject to the constraints:

2

95100d3

eg 180deginNm NT mθ

≤ ≤ ≤ ≤ (59)

Where Tin is a input torque and θ2 is the input rotational angle of link 2 that must determine to satisfy two

objective functions; while θ30, θ3, θ40, and θ4 are angles that were calculated in terms of θ2 known.

3.2 Optimal Procedure

In order to find optimal process parameter values based on a single quality characteristic, the Taguchi method

is one of the most significant tools because it is an efficient experimental method, and only requires a small

number of experiments to measure the quality and analysis of the optimal process. However, optimal results

obtained using different quality characteristics always contradict each other. However, optimal results obtained

using different quality characteristics always contradict each other. As a result, in an attempt to improve this

contradictory problem, fuzzy logic combined with the Taguchi method (FLTM) was utilized in this paper to find

the combination of process parameters that optimize the multi-response performance index (MRPI). The flow

chart structure of the fuzzy logic controller coupled with the Taguchi method used in the study is shown in Fig. 4.


Fig. 4 Flow chart of Fuzzy logic controller combined with Taguchi method

3.2.1. Taguchi method

Taguchi method applications are concerned with the optimization of a single performance characteristic.

The Taguchi method uses a special design of orthogonal arrays to study an entire parameter space with only a

small number of experiments. The experimental results are then transformed into a signal-to-noise (S/N) ratio.

The S/N ratio can be used to measure performance characteristics deviating from the desired values. Usually,

there are three categories of the performance characteristics in the analysis of the S/N ratio: the lower-the-better,

the higher-the-better and the nominal-the-better. In this study, L9 orthogonal array experiment is used with the

two right columns are ignored because there are two parameters and their three levels. To obtain optimal motion

performance, the maximum angular displacement and the minimum stress are desired. Therefore, the

higher-the-better angular displacement and the lower-the-better stress should be selected. After determining the

orthogonal array experiment and the number of parameters levels, this research performs calculation for S/N of

the angular displacement and the stress of small-flexible segment as the following equations briefly describe:

The higher-the-better angular displacement is:

−= ∑

=

n

i iL yn

NS1

2

11log10/ (60)

The lower-the-better maximum stress is:


−= ∑

=

n

iiS y

nNS

1

21log10/ (61)

where y is the observed data

To consider the two different performance characteristics in the Taguchi method, the S/N ratios

corresponding to the angular displacement and the maximum stress at fixed end of flexure hinge are two inputs

processed by the fuzzy logic control in order to find out optimal parameter values.

3.2.2 Fuzzy Logic Based on Taguchi Method

Using fuzzy logic control, the optimization of multiple performance characteristics can be transformed into

the optimization of a single performance index. Thus, the proposed method is the integration of fuzzy logic

control with the Taguchi method; they are used to simultaneously achieve the optimization of multiple

performance characteristics.

A fuzzy logic unit comprises a fuzzifier, membership functions, a fuzzy rule base, an inference engine and

a defuzzifier. First, the fuzzifier uses member functions to fuzzify the signal-to-noise ratios. Next, the inference

engine performs fuzzy reasoning on fuzzy rules to generate a fuzzy value. Finally, the defuzzifier converts the

fuzzy value into a multi-response performance index. The structure of the two-input-one-output fuzzy logic is

shown in Figure 5. In the following, the concept of fuzzy reasoning is briefly described, based on the

two-input-one-output fuzzy logic unit. The fuzzy rule base consists of a group of if-then control rules, with two

inputs, x1 and x2, and one output, y.

Fig. 5 Structure of the two-input-one-output fuzzy logic control

An orthogonal array, the S/N ratio and MRPI are used to study the performance characteristics of this

mechanism. Matlab 7.1 software is used to solve the roots of nonlinear equations and support for fuzzy logic.


Regardless of the category of the performance characteristic, a larger S/N ratio corresponds to a better

performance characteristic. As a result, the optimal level of the process parameters is the level with the highest

S/N ratio. The loss function corresponding to each performance characteristic is fuzzified; then an MRPI is

achieved through fuzzy reasoning using fuzzy rules.

3.3. Results and Discussions

According to the flow chat structure in Fig.4 an optimal process for for-bar partially compliant mechanism is

performed using FLTM. This research formulated nine fuzzy rules in Table 1 that are directly based on the fact that

the larger the S/N ratio is, the better the performance characteristic in fuzzy logic controller as presented.

There are two input process parameters in this research such as input torque and crank angular. The value of input torque A has the range from 3 to 95 Nm , while the value of crank angle B has the range from 100 to 180

degrees; and the value of each of these process parameters was divided into three levels as shown in Table 2.

Based on Taguchi method, the L9 orthogonal array with nine experiments and detailed values of each level was

given in Table 3. The higher-the-better angular displacement and the lower-the-better stress should be selected in

this study. After that, the membership functions were created by using Matlab software as presented in Fig. 6. The

membership function of the S/N ratio for the stress has the range from 0 to 1 and the membership function of the

S/N ratio of angular displacement also has the range from 0 to 1. The membership functions can be adjusted to

find real optimal parameters value. These membership functions are triangle shapes. Meanwhile, the membership

functions of MRPI have trapezoidal shapes and the range from 0 to 1. Figure 6 also presents the range of the S/N

ratio for the angular displacement from 60 to 110, the range of S/N ratio of the stress from -250 to -190, and the

range of MRPI from 0 to 1. The results of the S/N ratio for the stress, S/N ratio of angular displacement, and the

MRPI were calculated in Table 4. Next, the mean of MRPI of input torque A and crank angle B at each of their

levels was calculated in Table 5. The larger the MRPI is, the smaller the variance of performance characteristics

around the desired value. Based on Fig.7, the predicted optimal parameter levels are input torque at level 1 ( )Nm3

and crank angle at level 1 (100 degrees).

Table 1. Fuzzy rules

MRPI S/N ratio of stress σ

Small Medium Large

S/N ratio of angular displacement

δθ4

Small Very small Small Medium

Medium Small Medium Large

Large Medium Large Very large

Table 2. The values of process parameters and their levels

Symbol Parameter Range Unit Level 1 Level 2 Level 3

A Input torque 3-95 Nm 3 49 95

B Crank angle 100-180 Degree 100 140 180


Table 3. Nine trials with detailed values

Motion parameters

Experiment No. A, Input torque (Nm) B, Crank angle (degrees)

1 3 100

2 3 140

3 3 180

4 49 100

5 49 140

6 49 180

7 95 100

8 95 140

9 95 180

Table 4. Results for S/N ratio and the MRPI

Experiment number

Stress σ (lb/in2)

S/N ratio (dB) of Stress

Angular Displacement δθ4

(degrees)

S/N ratio (dB) of δθ4

Multi-response performance index

1 9.8116e+003 -183.8264 207 106.6544 0.877

2 9.8116e+003 -183.8264 298.4 113.9687 0.927

3 9.8116e+003 -183.8264 19.8 59.7136 0.5

4 1.5087e+005 -238.4835 282.96 112.9061 0.581

5 1.5087e+005 -238.4835 130.68 97.4550 0.431

6 1.5087e+005 -238.4835 104.04 92.8955 0.346

7 2.9192e+005 -251.6847 22.32 62.1097 0.0749

8 2.9192e+005 -251.6847 21.96 61.7845 0.0746

9 2.9192e+005 -251.6847 203.4 106.3035 0.463

Table 5. Mean of MRPI

Symbol Input parameter Mean of MRPI

Level 1 Level 2 Level 3 Max-Min

A Input torque 0.768 0.45 0.48

0.2 0.47

0.568

B Crank angle 0.5 0.03


Fig. 6 Membership functions of angular displacement and stress and MRPI

Fig. 7 MRPI graph

Analysis of Variance (ANOVA) of MRPI An ANOVA is performed to identify the process parameter that is statistically significant affecting

performance characteristics. Table 6 indicates that input torque is the most significant parameter affecting structure

of four-bar partially compliant mechanism with largest value of F test of 4.1752.


Table 6. Results of ANOVA

Symbol Parameters DOF SS V F

A Input torque 2 0.4791 0.2395 4.1752

B crank angle 2 0.0084 0.0042 0.0731

Error 4 0.2295 0.0574

Total 8 0.717

4. Conclusions The paper presents the combination of fuzzy logic controller with Taguchi method as a novel proposed

optimization tool in for the four-bar partially compliant mechanism. The two objective functions such as angular

displacement and stress of the small-length flexural pivot were formulated using theory of PRBM and principle of

virtual work. Through the use of fuzzy logic-Taguchi method, this paper found the optimal process parameters such as input torque at level 1 of 3 Nm and crank angle at level 1 of 100 degrees for maximizing angular

displacement and minimizing the stress simultaneously. The result revealed that input torque is the most

significant parameter affecting structure of for-bar partially compliant mechanism with largest value of F test of

4.1752. This proposed novel optimization method can be further utilized in engineering areas.

Acknowledgments The authors acknowledge and thank the National Science Council of the Republic of China for their financial

support of this study under Contract Number: NSC 99-2221-E-151 -004 -MY2.

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