Slide 3.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc,...

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.1

Lecture 3Design as an Inverse Problemand its Pitfalls “What is right to ask”, an important thing in computational and optimal design, illustrated with the “design for desired mode shapes” problem.


Contents

• Design for desired mode shapes– What is wrong with the optimal synthesis

formulation?– Direct synthesis technique

• of a bar• of a beam

– Analytical solutions and insights– Solution using discretized models

• Stiff structure and compliant mechanism design problem formulations—a summary– Continuous model– Discretized model


Why design for mode shapes?

• Resonant MEMS– Capacitive resonant sensors– Micro rate gyroscope

• AFM (atomic force microscope) cantileversSee: Pedersen,N., “Design of Cantilever Probes for Atomic Force Microscopy (AFM),” Engineering Optimization, Vol. 32, No. 3, 2000, 373-392.

• Swimming and flying mechanisms• Acoustics, …


Resonant-mode micromachined pressure sensor

Pressure

Top view

Side view

Resonant beam

Capacitance is measured in this gap

The mode shape of the beam influences the sensitivity of the sensor.


The cantilever in atomic force microscopy (AFM) (in the resonant mode)

Laser Detector

When AFM operates in the resonant mode, it helps to shape the cantilever to have a mode shape that has larger slope towards the tip.

Pedersen,N., “Design of Cantilever Probes for Atomic Force Microscopy (AFM),” Engineering Optimization, Vol. 32, No. 3, 2000, 373-392.


Rate gyroscope with a micromachined vibrating polysilicon ring

Two degenerate mode shapes

M. Putty and K. Najafi, “A Micromachined Vibrating Ring Gyroscope,” Tech. Digest of the 1994 Solid State Sensors and Actuators workshop, Hilton Head Island, SC, pp. 213-220.


Principle of the rate gyroscopeFoucault pendulum

All of the above have degenerate pairs of mode shapes.When one mode shape is excited, the rotation of the base causes energy-transfer to the other mode due to Coriolis force.

Wine glass Ring

Plane of oscillation rotates


Design the spokes for improved mode shapes (and better sensitivity)

(Lai and Ananthasuresh, 1999)


Design for a desired mode shape of a bar

0)()()(

xvxAdx

dvxEA

dx

d

Axially deforming bar

Analysis: Given: ExA ,),( Find: ),(xv

Mode shape

Natural frequency

Given: ,,),( ExvSynthesis: Find: )(xA

0)(

)('

)()(")(

xA

xvE

xvxvExA

)(xA

Area of c/s


Direct and optimal synthesis techniques

error

Designed

Desired)(xv

x

Finding the area profile to minimize the integrated error is the optimal synthesis technique.

0)(

)('

)()(")(

xA

xvE

xvxvExA

Solving this “inverse” differential equation is the direct synthesis technique

ionEigenequat

0)(

toSubject

)(Minimize

*

0

0

2

)(

WdxxA

dxxe

L

L

xA

)(xe


Direct synthesis solution

dxx

CexA)(

)(

)('

)()(")(

xvE

xvxvEx

0)( xv decides what should be!Furthermore, boundary conditions decide what mode shapes are possible; so, we cannot ask whatever we wish.

Solution for area of c/s

Can be specified also?

0)(

)('

)()(")(

xA

xvE

xvxvExA


Some examples

)(xv )(x )(xA

L

x

2sin

LxL

Lx

Lx

L

E

E

22

224

cos

sinsin2

2

)4(

/2

2

L

E

Cedx 0

1)1( 2 x

)1(2

2/2

x

xxE/2E 2

21 xxCe

2xx )21(

/2 2

x

xxE

/8E )(2 2xxCe

Desired mode shape

Frequency must be…

Area of c/s

(to cancel off the denominator in ))(x


Desired mode shape for a beam

22

2

2

2

,0

Aw

dx

wdEI

dx

d

Assume that as before.AI

Inverse eigenproblem for the beam:

0)(2 AwE

wAwAw iv

Solution?What are the conditions on the frequency to make a mode shape valid?


Discretized model

NN

NN

ab

ba

babc

cbabc

cbab

cba

1

11

4432

33321

2221

111

00000

00000

000

00

000

0000

K

4

22

221

221

l

kc

l

kkb

l

kkka

ii

iii

iiii

3/ lAEk ii

Using finite-difference derivatives…for a cantilever beam:

MwKw

0)(2 AwE

wAwAw iv

., lAM iii is a diagonal matrix withM


Re-arrange the variables…

NNNNlE

NN

iiiilE

ii

iiilE

ii

iiilE

ii

lEh

NN

NNNN

NNNN

NNNNNN

lwwwwC

lwwwwC

wwwC

wwwC

lwwC

A

A

A

A

C

CC

CCC

CCC

CCC

)2(

)2(

)2(

)2(

where

00000

0000

000

0

000

000

1212,

1212,

21122,

1161,

11121,1

1

3

2

1

,

,11,1

,21,22,2_

242322

131211

3

3

3

3

3

2

0CA


Solution and conditions on frequency and mode shape

N

NNN

w

www

l

E )2(

1212

4

From the last row of the previous system of equations:

ii

iiiiiii

NN

NNNN

C

ACACA

C

ACA

,

22,11,

1,1

,11

And then, solve for the areas:

For details, see: Lai, E. and Ananthasuresh, G.K., “On the Design of Bars and Beams for Desired Mode Shapes,” Journal of Sound and Vibration, Vol. 254, No. 2, 2002, pp. 393-406.


Return to the differential equation…

0ivLLww

0)(2 AwE

wAwAw iv

For a cantilever, at the free end, i.e., at :Lx

L

ivL

LLivL

LL

w

wEAw

Ew

ww

0)(

0

(assuming is not zero)LA

A condition to ensure positive :

Another condition due to Gladwell:The number of sign changes in the mode shape and its first derivatives must be the same.See: Inverse Vibration Problems, G. M. L. Gladwell, 1986.


An example: valid mode shapes

66

55

44

33

221 xaxaxaxaxaxaaxw o

66

55

44

336

254

246

35

24 1052245206 xaxaxaxLaLaLaxLaLaLaxw

01552611324 2654

2654

2 LaaaLaaaLuu IVLL

Explore which 6th degree polynomials are valid mode shapes for a cantilever:

With essential and natural boundary conditions imposed:

Two other conditions:

The number of sign changes in the mode shape and its first derivatives must be the same.

Valid 1st mode shapes Valid 1st and 2nd mode shapes


Some examples of mode shapes and area profiles

a6 = 0

a6 = 1

a6 = 2

a6 = 4

a6 = 3

For details, see: Lai, E. and Ananthasuresh, G.K., “On the Design of Bars and Beams for Desired Mode Shapes,” Journal of Sound and Vibration, Vol. 254, No. 2, 2002, pp. 393-406.


Now, we are ready for optimal synthesis…

error

Designed

Desired)(xw

x

ionEigenequat

0)(

toSubject

)(Minimize

*

0

0

2

)(

WdxxA

dxxe

L

L

xA

)(xe

Now, given a mode shape, we check if it is valid. If it is not, we can give the closest valid polynomial (or other) mode shape and get a solution.


An example

Given (invalid) mode shape

Rectified polynomial mode shape

First derivative of the mode shape

Area profile


Return to stiff structure design

0

0

toSubject

:2

1energystrainMinimize

*

b

VdV

dV

fσ

εσ

Volume constraintEquilibrium equation+ boundary conditions (displacements and tractions)

εDσ :

?

Force tf

dd tb ufufcompliancemeanor

Tuuε 2

1 Strain-displacement relationship

forcebodybftt onfnσ

uspecified onuu

Design variables are in .

Stress-strain relationship


0

toSubject

2

1energystrainMinimize

*

VdV

dVTεσ

εDσ

dd Tt

Tb ufufcompliancemeanor

zu

xu

yu

z

ux

u

yu

zuy

uxu

xz

zy

yx

z

y

x

21

21

21

ε

Stiff structure design

0

0

0

fzzzz

fyyy

fxxx

zzzyzx

byyzyyyx

bxxzxyxx

zx

yz

xy

zz

yy

xx

σ

uspecified onuu

tzzzzyzyxzx

ttyzyzyyyxyx

txzxzyxyxxx

fnnn

fnnn

fnnn

on

Design variables


With the discretized model…

FKU

KUUK

0

toSubject2

1Minimize

*VVe

T iiUFor

Stiffness matrix =K

Strain energy = KUUεσ TdVSE

2

1:

2

1

Displacement vector =U

Equilibrium equationVolume constraint

Strain energy Mean compliance

Design variables are in .K


Return to compliant mechanism design

0

0

0:2

1

0:

toSubject

volumeMinimize

*

b

b

SdV

dV

dV

fσ

fσ

εσ

εσ Flexibility (deflection) constraint

Stiffness (strain energy) constraintEquilibrium equations+ boundary conditions (displacements and tractions)

εDσ : Design variables are in .

?

Force tf

1tfUnit dummy loadεσ,εσ,


Alternatively…

Since nonlinear constraints are more difficult to deal with, and multiple constraints make optimization harder, the problem is reformulated as:

0

0

0

toSubject

:)1(:2

1Minimize

*

b

b

VdV

dVwdVw

fσ

fσ

εσεσ

dV

dV

εσ

εσ

:

:or

εDσ :

Linear combination or ratio of two conflicting objectives.


With the discretized model…

FUK

FKU

KUUKUUK

0

toSubject

)1(2

1Minimize

*VV

ww

e

TT

KUU

KUU

T

T

21

or

Mutual strain energy = KUUεσ TdVMSE

:

Strain energy = KUUεσ TdVSE

2

1:

2

1

inin

outout

in

out

uf

uk

SE

MSEMSEsign

u

u

SE

MSE2

221

)(,maximized be ofunction tObj.

Geometric advantage Mechanical efficiency

Output spring constant


Modeling the work-piece in the compliant mechanism design problem

?

Force tf

Output spring to model the work-piece

?

outk


Main points

• An optimal design problem, as posed, should make sense.

• Design for desired mode shapes problem– Restrictions on “desired” mode shapes and

frequencies

• Stiff structure design problem statement revisited

• Compliance design problem statement revisited– Flexibility and stiffness requirements should be

optimally balanced– Work-piece can be modeled as an output spring


Let’s make up some specifications…

a)For a stiff structure

b)For a compliant mechanism

… so that we can compare designs given by the optimization program (PennSyn) and designs conceived by You!

Slide 3.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc,...

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