Lecture 3
Design as an Inverse Problem
and 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 3.1
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 3.2
Why design for mode shapes?
• Resonant MEMS
– Capacitive resonant sensors
– Micro rate gyroscope
• AFM (atomic force microscope)
cantilevers
See: 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 3.3
Resonant-mode micromachined
pressure sensor
Resonant beam
Top view
Capacitance is
measured in this
gap
Side view
Pressure
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 3.4
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 3.5
Rate gyroscope with a micromachined
vibrating polysilicon ring
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.
Two degenerate mode shapes
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.6
Principle of the rate gyroscope
Foucault pendulum
Wine glass
Ring
Plane of oscillation rotates
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.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.7
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 3.8
Design for a desired mode shape of a bar
Area of c/s
Axially deforming bar
A(x )
d 
dv 
EA( x)   A( x)v( x)  0

dx 
dx 
Analysis:
Given:
Synthesis: Given:
A( x),  , E
Find:
Natural frequency
Mode shape
v(x),   
v( x),  , E ,  Find: A(x )

Ev "( x)  v ( x)
A( x) 
A( x)  0
Ev ' ( x)
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.9
Direct and optimal synthesis techniques

Ev "( x)  v ( x)
A( x) 
A( x)  0
Ev ' ( x)
v (x )
Solving this “inverse”
differential equation is the
direct synthesis technique
Designed
error  e(x )
Desired
x
L
Minimize  e( x) 2 dx
A( x )
0
Subject to
L
*

A
(
x
)
dx

W
0

Finding the area profile to
minimize the integrated
error is the optimal
synthesis technique.
0
Eigenequat ion
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.10
Direct synthesis solution

Ev "( x)  v ( x)
A( x) 
A( x)  0
Ev ' ( x)
Solution for area of c/s

  ( x ) dx
A ( x)  Ce
Can

Ev "( x)  v ( x)
 ( x) 
Ev ' ( x)
be specified also?
v( x)  0
decides what  should be!
Furthermore, boundary conditions decide what mode
shapes are possible; so, we cannot ask whatever we wish.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.11
Some examples
Desired mode
shape
sin  
 2L 
 ( x  1)  1
2
xx
2
Frequency
must be…
 (x)
v (x)
 x 
(to cancel off the denominator in  (x ) )



E  4L2 sin  2xL    sin  2xL 
2
E 2L cos 2xL 
2   / E  x2  x 
2( x  1)

Area of c/s
 2   / E x  x 2
(1  2 x)

A (x)
E 2 /
(4 L )
2
e 
 0 dx
C
2E / 
Ce
8E / 
Ce 2( x x
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
x  12 x 2
2
)
Slide 3.12
Desired mode shape for a beam
d 2  d 2w 
 EI 2   Aw  0,
2 
dx 
dx 
  2
Assume thatI  A as before.
Inverse eigenproblem for the beam:

wA  2 wA  ( wiv 
w) A  0
E
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 3.13
Discretized model
Using finite-difference derivatives…for a cantilever beam:

wA  2wA  ( w 
w) A  0  Kw  Mw
E
iv
 a1
 b
 1
 c1

K 0
 

 0
 0

 b1
c1
0
0

0
a2
 b2
c2
0

0
 b2
a3
 b3
c3

0
c2
 b3
a4
 b4 
0






0
0
0
0

a N 1
0
0
0
0
  bN 1
0 
0 
0 

0 
 

 bN 1 
a N 
k i  4k i 1  k i  2
l2
k k
bi  i 1 2 i  2
l
ki2
ci  2
l
ai 
ki  EAi / l 3
M is a diagonal matrix with M i ,i  Ai l.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.14
Re-arrange the variables…
C11 C12
0 C
22

 

CA  
0
0
0
0

0
 0
where
C1,1 
Eh 2
12l 3
0

C23 C24
0






0
C N _ 2, N  2
C N  2, N 1

0
0
C N 1, N 1

0
0
0
C13
0

 A1 
0 
A 
 2
0 
 A3   0

C N  2, N 
 
 


C N 1, N
 AN 

N 1
C N , N  N  N
0
w1  lw1
Ci ,i 1  6El3 ( wi 1  2 wi  wi 1 )
Ci ,i  2 12El 3 ( wi  2 wi 1  wi  2 )
Ci ,i  12El 3 ( wi  2  2 wi 1  wi )  lwi
C N , N  12El 3 ( wN  2  2 wN 1  wN )  lwN
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.15
Solution and conditions on
frequency and mode shape
From the last row of the previous system of equations:

E ( wN  2  2wN 1  wN )
12 l 4
wN
And then, solve for the areas:
AN 1  
Ai  
C N 1, N AN
C
C N 1, N 1
i ,i 1
Ai 1  C i ,i  2 Ai  2 
C i ,i
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 3.16
Return to the differential equation…

wA  2wA  ( w 
w) A  0
E
iv
For a cantilever, at the free end, i.e., at x  L :
wL  wL  0
iv

E

w
L
( wLiv 
wL ) A L  0   
E
wL
(assuming AL is not zero)
A condition to ensure positive  :
wL wLiv  0
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 3.17
An example: valid mode shapes
Explore which 6th degree polynomials are valid mode shapes for a cantilever:
wx   ao  a1 x  a2 x 2  a3 x 3  a4 x 4  a5 x 5  a6 x 6
With essential and natural boundary conditions imposed:




wx   6a4 L2  20a5 L3  45a6 L4 x 2  2 2a4 L  5a5 L2  10a6 L3 x 3  a4 x 4  a5 x 5  a6 x 6
Two other conditions:



u L  u LIV  24 L2 3a 4  11a5  26a6 L2 a 4  5a5  15a6 L2  0
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 3.18
Some examples of mode shapes and
area profiles
a6 = 2
a6 = 1
a6 = 4
a6 = 0
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 3.19
Now, we are ready for optimal synthesis…
w(x )
Designed
error  e(x )
Desired
x
L
Minimize  e( x) 2 dx
A( x )
0
Subject to
L
*
  A( x) dx  W  0
0
Eigenequat ion
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 3.20
An example
Given (invalid) mode shape
First
derivative
of the mode
shape
Area profile
Rectified
polynomial
mode
shape
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.21
Return to stiff structure design
Minimize

1
strain energy   σ : ε dV or mean compliance   f b  u d   f t  u d
2


Subject to
*
dV

V
0


  σ  fb  0
Force
?
ft
fb  body force
Design variables are in  .
Volume constraint
Equilibrium equation
+ boundary conditions (displacements and
tractions)
σ n  ft on t
u  u specified on u
σ  D: ε
Stress-strain relationship
1
ε  u  uT  Strain-displacement relationship
2
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.22
Stiff structure design
Minimize

strain energy 
1 T
σ ε dV or mean compliance   f b T u d   f t T u d

2


Subject to
*
 dV  V  0

 xx  xy  xz


 f bx  0
x
x
x
 yx  yy  yz


 f by  0
y
y
y
 zx  zy  zz


 fz  0
z
z
z
 xx 
 
 yy 
 zz 
σ 
 xy 
 yz 
 
 zx 


u x
x
u y
y
u z
z
u x
y
u y
z
u z
x




u y
x
u z
y
u x
z












σ  Dε
 xx nx   xy n y   xz nz  f tx
 yx nx   yy n y   yz nz  f ty on  t
Design variables
 zx nx   zy n y   zz nz  f tz
u  u specified on u




ε  1
2
1
 12
 2

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.23
With the discretized model…
Strain energy
Minimize
K
Mean compliance
or  Fi U i
1 T
U KU
2
Subject to
 Ve  V  0 Volume constraint
KU  F
Equilibrium equation
*
Stiffness matrix = K
Displacement vector = U
Design variables are in K .
1
1 T
Strain energy = SE   σ :ε dV  U KU
2
2
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.24
Return to compliant mechanism design
volume   dV
Minimize


Subject to
   σ : ε dV  0

Flexibility (deflection) constraint
1
σ : ε dV  S *  0 Stiffness (strain energy) constraint

2
  σ  f b  0 Equilibrium equations
+ boundary conditions (displacements and
  σ  f b  0 tractions)
Force
σ, ε
?

σ  D: ε
ft
σ, ε
Unit dummy load
ft  1
Design variables are in  .
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.25
Alternatively…
Since nonlinear constraints are more difficult to deal with,
and multiple constraints make optimization harder, the
problem is reformulated as:
Minimize

1



w  σ : ε dV   (1  w)  σ : ε dV 
2



Subject to
or


  σ : ε dV 





  σ : ε dV 


*
dV

V
0


  σ  fb  0
  σ  fb  0
Linear combination or ratio of two
conflicting objectives.
σ  D: ε
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.26
With the discretized model…
Minimize
K

1

w UT KU   (1  w) U T KU
2


or 
Subject to
U
T
KU

1 T

 U KU 
2

 Ve  V  0
*
KU  F
KU  F
Mutual strain energy = MSE   σ :ε dV  U KU
T

1
1 T
Strain energy = SE   σ :ε dV  U KU
2
2
Output
spring
constant
1
2
koutuout
2
MSE uout sign ( MSE ) MSE
Obj. function t o be maximized 

,
2
SE
uin
SE
f inuin
Geometric advantage
Mechanical efficiency
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.27
Modeling the work-piece in the
compliant mechanism design problem
Force
ft
?
?

kout
Output spring to model
the work-piece
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.28
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 3.29
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!
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 3.30