Lecture 1b
Analysis
…of MEMS and of structures and compliant mechanisms
undergoing small and large deformations.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.1
Contents
• Analysis and simulation of MEMS
• Deformation and stress analysis of
deformable structures
• Pseudo rigid-body model-based analysis of
elastic structures undergoing large
deformations
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.2
Ref: Microsystems Design—S. D. Senturia
Hierarchical view of MEMS
Lab on a chip
Specimen collector
System
Plumbing system
Device 3
Device 1
…Device nD
Device 2
Reaction
chamber
Digital readout
Signal amplifier
and processor
Signal transduction
Pump
Valve
Process
Flow channel
Masks
Component 1
Device 1
Component 1
Component 2
Process
Mask 1
Mask 2
…Component nC
…Mask nM
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.3
Modeling challenges
Integration of sensor, actuator, mechanism,
processor, power, and communication makes system
level tasks challenging
-- common representation for multiple energy
domains is needed.
Device level too has multiple energy domains
-- “macromodels” are necessary.
Component (physical) level
-- coupled energy domain equations need to be
solved.
Mask level
-- geometric modeling has its own difficulties.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.4
Modeling at four levels
System
Ref: Microsystems Design—S. D. Senturia
Representing as block diagrams of multi-domain subsystems
Device
Redcuced order “macro models” of the components
Component
Multiple, coupled energy behavioral simulations
(physical)
Artwork of masks Defining mask geometry for the
process steps
and process
Each level involves design
There is “analysis” (forward) problem and “synthesis” (inverse) problem.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.5
Structural analysis of MEMS
• Roark’s formulas
• Energy methods
• Finite element and boundary element
analyses
– Commercial packaged software are now
available exclusively for MEMS
–
–
–
–
Intellisuite
CoventorWare
Memscap
Etc.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.6
Roark’s formulas
Roark’s formulas for stress and strain, Raymond J. Roark, Richard G. Budynas, Warren
C. Young, McGraw-Hill, 2001.
• These are widely used by MEMS designers
• They are very accessible to people with any
engineering/science background
• Reasonably accurate
• Well suited for back-of-the-envelope
calculations, which most situations demand
in the initial stages
• Disadvantage: Large deformations and
residual stresses require special attention
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.7
Example: compliant ortho-planar
platform
The platform moves up and down without rotation.
Doing FEA for this is an overkill.
Instead, think of simple beam analysis.
d
d
F
Encastered-guided beam
d
Fl 3
d
12 EI
18 EI
Stiffness = k  3
l
Maximum stress:
For details, see: Compliant Mechanisms, Howell, L. L., Wiley, 2003.
3Eh
l3
deEhCT

l3
x 
 xy
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.8
Approximate solutions using energy
methods
Mostly Rayleigh-Ritz and
Castiglianos methods.
The membrane of a pressure sensor
Even the spherical approximation is used for
large deflection analysis because it is simple
and suits capacitance calculation.
Assume a polynomial deflection
profile for the beam and obtain
coefficients by minimizing the
potential energy.
Axial stretching is also
accounted for.
Residual stress effect is also
considered.
q
r
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.9
Support boundary conditions can
be tricky
• Most processes do not give perfect supports as in
encastered beams
• Especially true of surface micromachined
structures
It is an artifact of the fabrication process.
B
A
A
B
B
A
The compliance of the support is to be
modeled properly.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.10
Finite/boundary element analysis
of MEMS structures
• Several energy domains are coupled and self-consistent
solutions need to be obtained.
• Aspect ratios (thickness to lateral dimensions) poses
problems in meshing.
• What commercial MEMS-CAD software do:
– Enable model construction from mask layouts and process
description to get realistic geometry
– Hide FEA related details from the user (e.g., type of elements,
imposing boundary conditions, etc.)
– Include “wrappers” that communicate between different solvers
and the user’s model
– Finally, they show cool animations
• Lately, some also provide “macromodeling” capability and
circuit simulation
– Automatic extraction of reduced order models
– Simulation of dynamic behavior with equivalent circuit models
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.11
Equivalent circuit modeling of
electrostatic MEMS structures
(Gary Fedder and Tamal Mukherjee, CMU)
E.g., electrostatic linear actuator
Layout
schematic
Components: Combs, suspension,
shuttle mass, anchor, electrodes
3-D model
(of a portion)
Nodas, CMU.
SUGAR, Berkeley
Behavioral
schematic
Circuit
schematic
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.12
Electro-thermally actuated MEMS
Electrical analysis
Jx
J = current density
T = Temperature
Jy
T
Thermal analysis
Elastic analysis
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.13
How to handle more complicated
geometries?
Heavy computational load if FEA is used.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.14
One-dimensional approximation of
electro-thermal micro structures
R1
Tin
Narrow arm, seg. 1
End
connection,
seg. 2
R2
Tout
R4
R3
Flexure,
seg. 4
Electrical Model
Thermal Model
Beam1
Encastre
supports
NA
Beam2
Beam4
Wide arm, seg. 3
Elastic Model
Beam3
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.15
Maizel’s theorem: energy method
for thermo-elastic deformations
Deformation at a point of interest in a desired direction
due to temperature loading
 at a pointofinterest   σkk (T  T0 )dV
Maizel’s theorem
V
σ = stress tensor due to unit load applied at point of
interest in the desired direction
Maizel’s theorem is similar to the unit dummy load method
used for computing deflection at a point (in a given
direction) due to mechanical loads:
 at a pointofinterest   σijε ij dV
V
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.16
Advantages of equivalent circuit
models
• Can be embedded into system-level
simulators (SPICE-like)
• Parameterize the model for design
refinement or optimization
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.17
Pseudo rigid-body (PRB) modeling
• Approximating an elastic structure using
rigid bodies connected with joints and
springs.
• Reasonable accuracy over large
deformations.
• Can use the simpler analysis and synthesis
techniques of rigid bodies.
• Good reduced order models can be
obtained.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.18
PRB for a prismatic cantilever
beam with a vertical tip load
Burns and Crossley, 1968:
5
g
6
Burns, R.H. and Crossley, F.R.E., 1968, “Kinetostatic Synthesis of Flexible Link Mechanisms,” ASME Paper
No. 66-Mech-5.
L
KQ
gL
Q
Howell and Midha, 1995: g  0.85
F
Accurate up to…
Kinematics: Q  64.3
Kinetostatics: Q  58.5
K Q  2.25
EI
L
Howell, L.L., and Midha, A., 1995, "Parametric Deflection Approximations for End-Loaded, LargeDeflection Beams in Compliant Mechanisms," ASME Journal of Mechanical Design, Vol. 117, No. 1, pp.
156-165.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.19
Example: Fully compliant bistable
switch (thermally-actuated)
N. Masters and L. L. Howell, JMEMS, Vol. 12, No. 3, 2003, pp. 273-280
d
Thermal actuator
Switch
Shuttle
Compression beam
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.20
Principle of bistability and design
issues
PE = potential energy
PE
d
Design objective:
Achieve suitable PE curve
with the available
actuating force.
Unstable
Stable 1
F = actuating force
Stable 2
Adjusting geometry with FEA is
very time-consuming.
F
d
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.21
Modeling using PRB approach
Determining suitable spring constants and lengths (and
hence the geometry) using kinematic analysis is much
easier!
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.22
Main points
• Hierarchical view of analyzing MEMS
– System level
– Circuit simulation at device level
– Detailed domain level simulation
• Methods of analysis
–
–
–
–
Roark’s formulas
Energy methods
Finite element analysis
Pseudo rigid-body analysis
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 1b.23