Lecture 1b Analysis

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 1b.1

Lecture 1bAnalysis …of MEMS and of structures and compliant mechanisms undergoing small and large deformations.


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


Hierarchical view of MEMS

System

Device 1

Device 2

Device 3

…Device nD

Device 1

Component 2

Component 1

…Component nC

Component 1

Mask 1

Process

…Mask nMMask 2

Flow channel Masks

ProcessValve

Specimen collector

Plumbing system

Reaction chamber Signal transduction

Signal amplifier and processor

Digital readoutLab on a chip

Pump

Ref: Microsystems Design—S. D. Senturia


Modeling challengesIntegration 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.


Modeling at four levels

System

Device

Component(physical)

Artwork of masksand process

Each level involves designThere is “analysis” (forward) problem and “synthesis” (inverse) problem.

Representing as block diagrams of multi-domain subsystems

Redcuced order “macro models” of the components

Multiple, coupled energy behavioral simulations

Defining mask geometry for the process steps

Ref: Microsystems Design—S. D. Senturia


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.


Roark’s formulas

• 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

Roark’s formulas for stress and strain, Raymond J. Roark, Richard G. Budynas, Warren C. Young, McGraw-Hill, 2001.


Example: compliant ortho-planar platform

For details, see: Compliant Mechanisms, Howell, L. L., Wiley, 2003.

Doing FEA for this is an overkill.Instead, think of simple beam analysis.

The platform moves up and down without rotation.

FEncastered-guided beam

EI

Fl12

3

Stiffness = 3

18lEIk

Maximum stress:3

3

3

leEhClEh

Txy

x


Approximate solutions using energy methodsMostly Rayleigh-Ritz and Castiglianos methods.

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.

The membrane of a pressure sensorEven the spherical approximation is used for large deflection analysis because it is simple and suits capacitance calculation.

r


Support boundary conditions can be tricky

• Most processes do not give perfect supports as in encastered beams

• Especially true of surface micromachined structures

A A

A

B

B

B

It is an artifact of the fabrication process.

The compliance of the support is to be modeled properly.


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


Equivalent circuit modeling of electrostatic MEMS structures

Layout schematic

Behavioral schematic

Circuit schematic

3-D model(of a portion)

Nodas, CMU.SUGAR, Berkeley

E.g., electrostatic linear actuator Components: Combs, suspension, shuttle mass, anchor, electrodes

(Gary Fedder and Tamal Mukherjee, CMU)


Electro-thermally actuated MEMSElectrical analysis

Thermal analysis Elastic analysis

Jx Jy

TJ = current densityT = Temperature


How to handle more complicated geometries?

Heavy computational load if FEA is used.


One-dimensional approximation of electro-thermal micro structures

R1

R4 R3

R2

Electrical Model Thermal Model

Narrow arm, seg. 1

End connection,

seg. 2

Wide arm, seg. 3Flexure, seg. 4

Tin

Tout

Elastic ModelBeam1

Beam2

Beam4 Beam3

NA

Encastre supports


Maizel’s theorem: energy method for thermo-elastic deformationsDeformation at a point of interest in a desired direction due to temperature loading

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:

V

kk dVTT )( 0ofinterestpointaat σ Maizel’s theorem

V

ijij dVεσofinterestpointaat

σ = stress tensor due to unit load applied at point of interest in the desired direction


Advantages of equivalent circuit models

• Can be embedded into system-level simulators (SPICE-like)

• Parameterize the model for design refinement or optimization


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.


PRB for a prismatic cantilever beam with a vertical tip load

L

L

F

Burns and Crossley, 1968:

Howell and Midha, 1995:

65

LEIK 25.285.0

Accurate up to…Kinematics:Kinetostatics:

3.64 5.58

Burns, R.H. and Crossley, F.R.E., 1968, “Kinetostatic Synthesis of Flexible Link Mechanisms,” ASME Paper No. 66-Mech-5.

Howell, L.L., and Midha, A., 1995, "Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms," ASME Journal of Mechanical Design, Vol. 117, No. 1, pp. 156-165.


Example: Fully compliant bistable switch (thermally-actuated)

Shuttle Compression beam

Switch Thermal actuator

N. Masters and L. L. Howell, JMEMS, Vol. 12, No. 3, 2003, pp. 273-280

d


Principle of bistability and design issues

dPE

d

F

PE = potential energyF = actuating forceDesign objective:Achieve suitable PE curve with the available actuating force.Stable 1

Stable 2

Unstable

Adjusting geometry with FEA is very time-consuming.


Modeling using PRB approach

Determining suitable spring constants and lengths (and hence the geometry) using kinematic analysis is much easier!


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

Documents

Lecture 1b Analysis