Compliant Displacement Amplifier ISSN 2395-1621 and Analysis of Two Stage... · flexure beams 1, 2,...

www.ierjournal.org International Engineering Research Journal (IERJ) Special Issue 2 Page 3045-3051, 2015, ISSN 2395-1621

© 2015, IERJ All Rights Reserved

ISSN 2395-1621

Design and Analysis of Two Stage

Compliant Displacement Amplifier

#1S. B. Hebli,

#2S. P. Deshmukh,

#3K. C. More

[email protected] [email protected] [email protected]

#

#13Flora Institute of Technology, Pune, India

#2Sinhgad Academy of Engineering, Pune, India

ABSTRACT

ARTICLE INFO

The compliant mechanism is a mechanism which gets its motion by the elastic

deformation of its links. Paper presents design, analysis and development of XY

scanning mechanism using compliant displacement amplifier. Piezoelectric actuators

which are used in engineering applications have the disadvantage of having relatively

short displacement of 2 mm. This displacement is not sufficient for many engineering

applications. This disadvantage can be overcome by the use of a displacement

amplification mechanism in connection with the piezoelectric actuators. A compliant

displacement amplifier, when used with piezoelectric actuator achieves a long range

displacement. The effects of the change in the geometrical dimensions of the compliant

displacement amplifier on the amplification ratio are determined in this paper. Finally,

the optimum design giving high amplification ratio and comparatively low input force is

developed.

Keywords— Compliant mechanism, Micro-electro-mechanical system, Micro-leverages

mechanism, Piezoelectric actuator, Two stage displacement amplifier.

Article History

Received :18th

November

2015

Received in revised form :

19th

November 2015

Accepted : 21st

November ,

2015

Published online :

22nd

November 2015

I. INTRODUCTION

A compliant mechanism is a mechanism which gains its

mobility from the elastic deformation of its members called

flexures. Elastic deformations are within elastic limit. The

compliant mechanisms are popular for their advantages like

high precision, a lack of friction and wear at the joints, no

need for lubrication and it has monolithic structure.

Compliant micro-leverage mechanisms are used in micro-

electro-mechanical system (MEMS) to transfer an input

force/displacement to an output force/displacement to

achieve mechanical (force) and/or geometrical

(displacement) advantage.

Agarwal et al. described working principles, design steps,

synthesis process of compliant mechanism, its advantages

and disadvantages, as in [11]. Author stated that the synthesis

process was in two steps. First step in synthesis was a

‗topology synthesis‘. In this, a kinematic geometry was

required to be constructed that will give the required

displacement and force at output. The second step was ‗size

and shape optimization‘. Author also stated two types of

formulation. First was flexibility stiffness formulation and

second energy efficiency formulation. Kota et al. presented a

generalized methodology for designing a compliant

mechanism. For this, author gave example of design of

MEMS multiplier, as in [14]. Lin and Shih proposed

technique of the mechanical flexure hinge design. To

illustrate this, author gave an example of a compliant micro

gripper as in [1].

Su elaborated on compliant micro leverage mechanism for

MEMS and its governing laws (force and moment balance),

classification (1st, 2nd, 3rd kind and S, D subtype), as in

[15]. Author presented design theory of the force amplifier of

2nd kind. Author described about construction of single

stage, two stage and multi stage micro leverage mechanisms.



Author also presented detail synthesis of resonant output

accelerometer.

Many literatures discussed about the topology design but

not the size and shape optimization, so I decided to work on

design and synthesis of a two stage compliant displacement

amplifier connected to XY mechanism. Objective of the

paper was to design a compliant amplifier in two stages for

geometrical advantage around 6. It was optimized for its

maximum amplification factor.

Paper presents design, analysis and development of

compliant amplifier based XY mechanism. Section 2

presents concept of displacement amplification and design of

compliant displacement amplifier with XY mechanism.

Section 3 presents synthesis of mechanism and results of

analytical analysis are compared with FEA analysis. Section

4 presents experimental implementation and its validation.

Section 5 presents conclusion and futures scope of work.

II. XY COMPLIANT MECHANISM

A XY compliant mechanism design is shown in Fig.1.

Flexure units 1, 2, 3 and 4 are the beams. There are four

stages in this mechanism. These are (i) fixed stage, (ii) the

motion stage, and (iii) (iv) two intermediate stages. The

necessary force (Fx and Fy) is applied at two intermediate

stages by the actuators. This gives motion to the motion

stage in two perpendicular directions (as X and Y directions).

Fig. 1 XY mechanism

2.1 Displacement Amplifier

Compliant displacement amplifier consists of lever arm, a

pivot with a pivot beam, output system with output

connection beam and input system. The lever is a rigid

member while pivot beam and connection beam are

compliant members. They are further classified as

1) Single stage

2) Two stage

3) Multi stage.

Fig. 2 Single stage displacement amplifier

If L is distance between pivot and input and l is distance

between pivot and output The amplification factor of single

stage displacement amplifier as shown in fig. 2, is given by

y/x =l/L. The selected amplifier is a two stage displacement

amplifier connected to XY mechanism as shown in

photograph in fig. 3. The amplification factor of two stage

displacement amplifier is given by y/x = (l1/L1) X (l2/L2). It

has lever length fixed to 90 mm. L = 25 mm, l = 65 mm and

amplification ratio of 2.6 per stage (l/L), giving overall

theoretical amplification ratio as 2.6 X 2.6 = 6.76. For

amplifier, pivot beams and output connection beams are

around 0.5 mm thick, 10 mm in width, length 25 mm with

circular arc cut (radius 18 mm) along length and giving

minimum width at center of length as 4 mm.

2.2 Design of XY Mechanism

XY mechanism is connected with two stage displacement

amplifier at X and Y directions as shown in fig. 3. The

flexure beams 1, 2, 3, 4 are of 10 mm width and around 0.5

mm thick.

Fig. 3 Photograph of the model

First maximum output displacement (δout) at motion stage

and required force ‗W‘ for XY mechanism i.e. output force

Fout is calculated then calculations are done from output to

input in reverse direction. For amplifier design, first, second

stage (top) is designed. Then first stage (bottom) is designed.

Finally input force and input displacement is found out.

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I. 3. SYNTHESIS

Synthesis of a compliant mechanism is different than a traditional mechanism and also it is difficult.

3.1 Displacement Amplifier

In a compliant mechanism, the total strain energy is the

summation of the strain energies of the individual parts of

the entire mechanism. The work done (Uin(Total)) by the input

force (Fin1) is equal to the tensile/compression strain energy

(Uvo, Uvp, Uxy) at output connection beam, pivot beam and

beams of the XY mechanism and the moment-bending strain

energy at the output connection beam, pivot beam, xy

mechanism (Uθc, Uθp , Uxy). It is shown in the following

equations.

Uvo = Fout*δout, / 2

(1)

Uvp = Kvp*δ2/2

(2)

Uθp= Kθp*θ2 /2

(3)

Uxy = Kθo* θxy(actual)2 /2

(4)

Uin (Total) = Uvo1+Uvp1+Uθp1+Uθc1+ Uvo+Uvo2+Uvp2 +Uθp2 +

Uθc2 + Uxy (5)

Fin1=2*Uin (Total) / δin,1

(6)

Where,

δin,1 = Input displacement at first stage

δout = Output displacement

δ = Elongated/compressed length of pivot beam

θ = The rotation angle of the lever arm when loaded

θxy = The rotation angle of the beam of the xy mechanism

K = stiffness of pivot beam (Kvp, Kθp)/output connection

beam (Kvc, Kθc )/output system (Kθo), either vertical or

rotational

3.1.1 Calculation of input displacement δin,1 and output

displacement δout,2 ( For inward direction of input force)

δx p1= lp1-x p1

(7)

Similar calculations for Second stage pivot beam P2.

For certain output displacement, input displacement is given

by,

δin, 1 (actual) = 90/65(δx p1+25*δx p2/65) + δin, 1 (theoretical)

(8)

3.1.2 Calculation of input displacement δin,1 and output displacement δout,2 ( For outward direction of input force)

In this case output connection beam C1 is in compression. Due to compression and bending (considering cantilever loading) its length is reduced and so the actual displacements are reduced. Beams under tension are ignored.

Input displacement is given by,

δin, 2 (actual) = δout, 2 (actual)*L2 / l2

(9)

δout,1 (actual) = δin,2 (actual) + ( Reduction in length of output

connection beam C1)

(10)

δin, 1 (actual) = δout, 1 (actual)*L1 / l1

(11)

3.2 XY Mechanism

First maximum load (W(max)) and maximum deflection

(δxy(max)) on XY mechanism is calculated. Then vertical (kvo)

and rotational (kθo) spring constants of XY mechanism i.e.

external output system are calculated.

Vertical spring constant and bending spring constant of XY mechanism is given by

kvo = W (max) /δxy (max)

(12)

kθo 1 = E*Ixy1 / lxy1 (13)

kθo 2 = E*Ixy2 / lxy2 (14)

kθo = kθo 1 + kθo 2 (15)

3.3 Results

For the proposed displacement amplifier connected to XY

mechanism, the analytical calculations are done for requited

output displacement and corresponding input displacement,

input force and maximum stress is calculated as presented in

Table I.

In further analytical analysis, the lever ratio l / L i.e. lever

ratio 65 / 25 was kept constant. Only the sizes of both pivot

beams and both connection beams i.e. length, thickness and

width were varied one by one. Analysis was done for three

parameters that are output displacement means amplification

ratio, input force and maximum stress. For all this analysis,

the output displacement was kept constant and equal to 2

mm. Analysis was done for both inward and outward

direction of input force. The graphs were plotted for each and

every calculation.

TABLE I

ANALYTICAL RESULTS FOR INWARD DIRECTION OF INPUT

FORCE

δout

(mm)

δin

(mm)

Input

Force

(N)

Amplification

Ratio

Max

Stress

(N/mm2)

0.17 0.03 3.03 5.16 16.19

0.68 0.13 12.05 5.12 64.77

1.09 0.21 19.24 5.09 103.83

2.17 0.43 38.17 5.01 206.72

3.17 0.64 55.96 4.94 302.01

4.16 0.85 74.17 4.87 396.38

5.02 1.04 90.71 4.82 478.39

5.8 1.22 106.46 4.76 552.81



Amplifier size was optimized for maximum amplification

ratio, minimum input force and minimum maximum stress.

Table II presents the optimum amplifier sizes and some

of the properties that the various beams should possess for

the maximum output displacement and minimum input force.

TABLE III

OPTIMUM SIZE OF THE AMPLIFIER

Size

(mm)

Out-

ward

Remark In-

ward

Remark Property

lc2 50 High 50 High

compliant tc2 0.1 Low 0.4 Low

bc2 0.5 Low 1 Low

lp2 30 High 40 High

compliant tp2 0.1 Low 0.4 Low

bp2 0.5 Low 3 Low

lc1 30

Low /

Medium 200 High Medium

stiff –

outward

compliant

– inward tc1 1.5

High /

Medium 0.1 Low

bc1 10

High /

Medium 0.5 Low

lp1

any

(25) any 15

Low/

Medium Medium

stiff –

inward

compliant

–

outward

tp1 0.4 Low 1

High /

Medium

bp1

any

(10) any 20

High /

Medium

3.4 FEA Results

FEA modeling and analysis was done on ansys software.

It was done only on actual model for applying input force as

a variable. Force was applied in both directions i.e. inward

and outward. The parameters to be checked were input

displacement, output displacement, amplification ratio,

maximum stresses (Von Misses stresses) developed and its

location. The following readings were observed on ansys

and are presented in Table III. The readings are exactly

same for both inward and outward direction of input force.

TABLE IIIII

ANSYS RESULTS FOR BOTH DIRECTIONS OF INPUT FORCE

δout

(mm)

δin

(mm)

Amplific-

ation

ratio

Input

Force

(N)

Max

Stress

(N/mm2)

0.1 0.02 5.76 4 15

0.52 0.09 5.76 20 75

1.02 0.18 5.76 40 150

1.55 0.27 5.76 60 225

2.05 0.36 5.76 80 301

2.6 0.45 5.76 100 376

3.1 0.54 5.76 120 451

3.4 0.59 5.76 130 489

Ansys analysis shows the same values of input

displacement, amplification ratio and maximum stress for

both inward as well as outward direction of input force.

Ansys also shows that maximum output displacement

possible is 3.4 mm only.

Fig. 4 Amplification ratio Vs input displacement graph for

FEA results

Graph in fig. 4 shows that amplification ratio remains

constant and equals to 5.76, for both directions of input

force.

III. EXPERIMENTATION

The model was manufactured from material stainless steel.

It was fabricated as a monolithic structure and cut from a

single metal plate of 10 mm thick on wire cutting electro

discharge machine.

4.1 Experimental set up

Force was applied in both directions i.e. inward and

outward directions. For loading in outward direction, the

pan was attached to input point by string and weights were

added in it as input force as shown in Fig. 5. The input

displacement was measured by dial gauge and output

displacement was measured by micrometer head. For

inward direction of input force, force measurement was not

taken. Only input and output displacements were measured.

Input displacement was given by micrometer head and also

measured by the same. Output displacement was measured

by dial gauge.



Fig. 5 Photograph of experimental set up

4.1 Results

Experimental observations for outward direction of input

force are presented in Table IV.

TABLE IVV

EXPERIMENTAL OBSERVATIONS FOR OUTWARD DIRECTION OF

INPUT FORCE

δout

(mm)

δin

(mm)

Input

force

(N)

Amplification

ratio

0.08 0.02 3.53 3.91

0.55 0.14 11.38 3.89

1.03 0.27 27.07 3.87

1.53 0.4 40.22 3.84

1.78 0.46 52 3.83

The following graphs show the comparison between

analytical, ansys and experimental results.

4.1.1 Comparison between input force (For outward

direction of input force)

Fig. 6 Comparison between input force (For outward


Graph in fig. 6 shows that analytical values of input force

are at lower side and values by ansys are at higher side of

that of experimental results.

4.1.2 Comparison between amplification ratio (For outward


Fig. 7 Comparison between amplification ratio (For outward

direction of input force). (Graphs of analytical and

experimental coincide)

Graphs of analytical and experimental values of

amplification ratios in fig. 7 coincide and are around 3.9 and

are constant. Ansys shows high amplification ratio and is

5.76.

4.1.3 Comparison between input force (For inward


Graph in fig. 8 shows that ansys gives higher values than

analytical one and both graphs are linearly increasing.

Experimental reading for input force was not taken.



Fig. 8 Comparison between input force

(For inward direction of input force)

4.1.4 Comparison between amplification ratio (For

inward direction of input force)

Fig. 9. Comparison between amplification ratio

(For inward direction of input force)

shown in fig. 9, the graph of experimental values of

amplification ratio is at lower side at the beginning. It

coincides with analytical values after 2 mm of output

displacement. It may be due to predominant bending of the

pivot beams at the beginning.As

IV. CONCLUSIONS

The investigations have been done for two stage

displacement amplifier with XY mechanism. The simulated

and analytical results are in agreement with experimental

results. The ideal amplification ratio was 6.76 and was

found to be 4.85 by experiment for inward direction of input

force and 3.86 by experiment for outward direction of input

force. The amplification ratio was found to be 5.0 and 3.9

for inward and outward direction of input force respectively

analytically, which is 1% to 3% greater than the

experimental one. It also shows that the required input force

is more than the theoretical one. Maximum stress was found

at 3.4 mm of output displacement by FEA and 5 mm of

output displacement by analytical method and at the same

location i.e. output connection beam of second stage.

It concludes that the first stage must be stiff than the

second stage to have maximum amplification. At the same

time the first stage should be compliant enough to reduce

the input force requirement. The first stage geometry is

affecting severely on the input force, amplification ratio and

maximum stress of the amplifier.

In this paper length, width, thickness of beams are varied

and its effect on amplification ratio, input force, maximum

stress is studied but not of the lever ratio (l/L). In future, the

effect of lever ratio (l/L) can be studied and optimum lever

ratio (l/L) can be found out. Pivot beam and connection

beam have radius and they are acting as a hinges. Hinge

design also required to be studied.

ACKNOWLEDGMENT

I am thankful to Dr. Atul S. Padalkar, Principal, Flora

Institute of Technology, for guiding me for the technical

structure of the paper and helping me for the successful

completion of this paper. At the same time I express my

gratefulness towards Prof. Sudheendra Subramanya, Head

of Department, Mechanical Engineering Department, Flora

Institute of Technology, for the cooperation.

I have to mention my sincere thanks to Prof Sushant

Mulay and Prof Sujeetkumar Patil of Cad Cam Guru, for

giving me software support. Many thanks go to Mr. Vijay

Jadhav for helping me in fabrication of project model on

wire cutting electro discharge machine. Also thanks goes to

Mr. Pravin Hawaldar who helped me in constructing

experimental setup of this dissertation.

Finally I am grateful to all those who have assisted me in

accomplishing this paper, directly or indirectly.

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Compliant Displacement Amplifier ISSN 2395-1621 and Analysis of Two Stage... · flexure beams 1, 2,...

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