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Transcript of 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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.2
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.3
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, …
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.4
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.5
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.6
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.7
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.8
Design the spokes for improved mode shapes (and better sensitivity)
(Lai and Ananthasuresh, 1999)
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.9
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.10
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.11
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.12
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.13
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?
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.14
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.15
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.16
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.17
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.18
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.19
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.20
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.21
An example
Given (invalid) mode shape
Rectified polynomial mode shape
First derivative of the mode shape
Area profile
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.22
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.23
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.24
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.25
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,
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.26
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.27
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.28
Modeling the work-piece in the compliant mechanism design problem
?
Force tf
Output spring to model the work-piece
?
outk
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.29
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 3.30
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!