Outline – Flexures

• Definition of Flexures– Monolithic Designs

– Clamped Designs

• Four-bar Link Flexure– Motion Errors

– Two-stage

– Differential

– Multi-blade

• Flexures for Angular Motion

L ki Fl• Locking Flexures

• Transmission Systems

• Application ExamplesApplication Examples

• Rolamite

Chapter 8 ME 551 2

Flexures• A flexure is a bearing system that

ll ti th h b di fallows motion through bending ofits hinge elements.

Also called flexural hinges and elastic /– Also called flexural hinges and elastic /compliant mechanisms

• Flexures do have a large number ofguseful features for precisionengineering:– Light weight

– Low cost

Very low friction (w/o lubrication)– Very low friction (w/o lubrication)

– Maintenance free operation

– Yield high repeatable motion

Chapter 8 ME 551 3

g p

Flexures (Cont’d) ( )

• Some of their disadvantages areg– Limited travel span if compared to the overall size of the

mechanism

V l d i– Very low damping

– Poor load capacity

– Might be difficult to manufactureMight be difficult to manufacture

– Susceptable to fatigue failure

• Their applications range frompp g– Flip-top containers

– Piezo-beams on a scan tunneling microscope (STM)

• Due to bending stresses created in flexures, great careneeds to be exercised when designing such bearings.

Chapter 8 ME 551 4

Monolithic Design1g

• They are very accurate.

• Mechanism’s size is about 20 times the range of motion.

• Abrasive water jet machining is an economical way to cut non• Abrasive water-jet machining is an economical way to cut non-critical areas (links) whereas wire EDM can be used for precisioncuts in critical hinge areas.

– May require localized heat treatment of the material in the flexure.

• Applications include flexural couplings, mirror/lens mounts, STM's,and more.

Chapter 8 ME 551 5

Clamped Designs1p g• Elastic members are clamped to the

stationary- and moving elements via y gbolts.

– Applications include wafer steppers, mirrormounts, and more.

– Mechanism’s size is about 10 times therange of motion.

• Nanometer accuracy can be obtained

• Make sure that residualstresses don't create

yif bolted joint is designed properly:

– Rounded edges

– Each bolt’s cone of action should overlapasymmetries leading toparasitic motions.

– Cut spring metal with EDM or

Each bolt s cone of action should overlap.

• Easy to assemble from annealed parts and hardened spring steel.

B lt t b ti ht d f ll thCut spring metal with EDM orwater jet.

– Use dual lubricated washers underbolt heads and guides for clamp

– Bolts must be tightened carefully as as the torque can twist the flexure.

plates.

Chapter 8 ME 551 6

Features of Clamped Designs p g• Speed and acceleration limits:

– Limited only by yield strength and design.

• Range of motion:T i ll d f ti l th f– Typically used for motions less than a few mms.

– Monolithic designs: flexure length/motion ≥ 20

– Clamped designs: flexure length/motion = 5 10Clamped designs: flexure length/motion 5 ... 10

• Applied loads:– Design goal is to obtain load capacity with minimum springg g p y p g

constant.

• Repeatability:– Axial: limited only by the drive system

– Lateral: < 1 nm for monolithic designs.

Chapter 8 ME 551 7

Features (Cont’d)( )• Axial Resolution:

Limited only by the drive system– Limited only by the drive system.

• Inherently preloaded.

• Stiffness:Stiffness:– The greater the motion and the lower the spring rate, the less the stiffness.

– Theory of elasticity or FEA yields very accurate predictions of performanceperformance.

– Flexures often have low transverse stiffness so they are more susceptible to parasitic forces.

• Good vibration and shock resistance.

• Damping capability:M t i l d i l (2 5%)– Material damping only (2-5%).

– Damping mechanisms (constrained layer dampers) can be used to obtain high damping.

Chapter 8 ME 551 8

Features - Accuracyy• Axial: limited only by the drive system.

L t l b l th t f lithi d i• Lateral: can be less than nanometers for monolithic designs.

• Depends on how well the bearing was assembled or machined.

• Even if there is a small off-axis error motion associated with theEven if there is a small off axis error motion associated with the primary motion:

– The error motion is usually very predictable and highly repeatable.

Fl l b i t tt i f t ti d t• Flexural bearings cannot attain perfect motion due to– Variation in spring strength

• Elastic modulus varies with rolling direction in steels (anisotropy).

Variation in spring geometry– Variation in spring geometry.

– Overall inaccuracies of manufacture.

– Bending of the bearing in an unintended manner.

B di f t t– Bending of structure.

– External applied loads (e.g., gravity and the manner in which the actuation force is applied)

Chapter 8 ME 551 9

Four-bar Linkage Flexure1g

• Leaf springs are clamped to the platforms to create a four-bar linkage systemlinkage system.

– Parasitic motions are present in the bearing system.

• X direction stiffness is equal to two fixed-fixed beams acting together in a side-by-side mode.

• Y direction (vertical) stiffness is equal to that of two columns.

• Buckling effect is negligible for small motions• Buckling effect is negligible for small motions.

Chapter 8 ME 551 10

Motion Errors

• The most common errors in four-bar linkage flexures are the pitch angle and vertical motionflexures are the pitch angle and vertical motion.– They accompany linear motion in a four-bar linkage

flexureflexure.Chapter 8 ME 551 11

Motion Errors (Cont’d)( )The pitch angle (θ) and vertical motion error (δ) of the mechanism canbe simply expressed asbe simply expressed as

2

2 2 2

6( 2 )l a t xθ⎡ ⎤−

= ⋅⎢ ⎥2 2 2

2

3 2 6b l t l at l

x

θ

δ

⎢ ⎥− +⎣ ⎦

≈2l

δ ≈

where x is the travel distance; t is the thickness of the spring plates;where x is the travel distance; t is the thickness of the spring plates;b is the length of the platform.

Notice that there is no pitch error when a = l / 2. If the force isNotice that there is no pitch error when a l / 2. If the force isapplied at a point other than halfway between the platforms, abending moment is generated which causes a pitch error to occur.

Chapter 8 ME 551 12

Motion Errors (Cont’d)( )The pitch errors are also caused by the difference in spring length and difference in platform length respectively That isdifference in platform length respectively. That is

2s xδθ =

22s

pp

l bx

θ

δθ

=

=p bl

where subscripts s and p refer to the quantites associated with thei d l tf ti lspring and platform respectively.

Typical tolerances on the spring- and platform length are 25 μm andTypical tolerances on the spring and platform length are 25 μm and 1...3 μm respectively.

Chapter 8 ME 551 13

Motion Errors (Cont’d)( )

• To minimize parasitic motions,S i l d l f b li k li i– Symmetrical dual four bar linkage eliminates δY error.

• Strains along flexure's length resisted by the frame.

• Stiffness is increased: – Range of motion is decreased as δ = FL3/(192EI)

– Use a two-stage (stacked) four-bar linkage.

Chapter 8 ME 551 14

Two-Stage Linkage1g g

• The parasitic motions of one 4-bar cancel the parasiticmotions of the second 4-bar.

• Lateral and yaw stiffness will not be high due to bucklingeffect.

• Overcome by use of monolithic hourglass type members• Overcome by use of monolithic hourglass-type members.Chapter 8 ME 551 15

Cascaded Linkage2g

• In this design, any change in height of platform A relative to itssupport B will be compensated by an equal and opposite change inheight of B relative to the base.

• In theory this should result in a perfect rectilinear motion of A relativeto the base.to the base.

Chapter 8 ME 551 16

Differential Design2g

Chapter 8 ME 551 17

Multi-blade Flexures1

Many thin blades can be usedto increase lateral stiffness andload capacity while keepingaxial stiffness and stress low:

verticalF ntw∝

3

( )

vertical

ntwEK

L

LK l ibl d

∝

3

3

3

( )

( )( )

axial

axial

K multibladeEnt w

LK monolithic

E nt w

∝

∝

• Beware of alignment and slip problems between the blades.

• Care must be exercised when tightening bolts as the torque can twist the

( )E nt w

• Care must be exercised when tightening bolts as the torque can twist theflexure!

• Rubber placed between the blades can increase damping.

Chapter 8 ME 551 18

Flexures for Angular Motion1g• Device is developed by Polaroid

Corp to adjust pitch- and yawCorp. to adjust pitch and yawangles of a lens.

• Mechanical advantage is providedb th th t h thby the screws that push up on theupper beam using the lower beamas a base.

FL2

I

– If the top beam were 2 mm thick and thebottom one were 1 mm thick, thetransmission ratio would be (2/1)3 = 8.

Lmain= 2L/3

FDeformed upper beam

FL3

3EI=

F2EI= • Another interesting feature is that

the slope at the root of the mainbeam causes an Abbe error

upper beam 3EIcanceling the beam’s displacementat its free end as

δ i = δ - α⋅L i = 0

Chapter 8 ME 551 19

δmain = δ - α⋅Lmain = 0

Cross-strip Flexure1p

• Cross-strip flexure allows angular rotation about an axis.

Chapter 8 ME 551 20

Angle Hinges2g g

Chapter 8 ME 551 21

Locking Flexures1g

• Flexures can be used as one-shot hinges to guide g gcomponents into alignment and then lock them into place.

• X direction stiffness of the clamping system is equal to:– The sum of the X direction stiffness of the anvil and locking

member due to the preload effectmember due to the preload effect.Chapter 8 ME 551 22

Locking Flexures (Cont’d)g ( )

• Small taper keeps component preloadedSmall taper keeps component preloaded in both the X and Y directions.

• To minimize distortion caused by bending, line of force must pass through the support p g ppledge.

• A lens could be kinematically held by two• A lens could be kinematically held by two anvils and a single locking member.

• The geometry is tolerant of manufacturing errors.

Chapter 8 ME 551 23

Locking Flexures (Cont’d)g ( )

• For precise X position location, utilize a grooved t tstructure.

• Numerous permutations are possible.

Chapter 8 ME 551 24

Transmission Systems1y

Chapter 8 ME 551 25

Transmission Systems (Cont’d)y ( )

• A bowed flexure can be used as a high reduction transmission. A downward motion δ causes a lateral motion Δ ≈ 4δlh / lw.

• It is possible to chain together a series of these bowed flexures at right angles to each other.

• This can yield a very high transmission ratio For instance if lh = 2 mmThis can yield a very high transmission ratio. For instance, if lh 2 mm and lw = 20 mm, the transmission ratio becomes 5.

• In series, the ratios become 25, 125, 625... for 2, 3, 4... units respectively.

Chapter 8 ME 551 26

Application – CMM Probe

Touch-trigger probe (TTP) designed/built byMETU-SPARC group employs a diaphragmspring to support the stylus shaft

Chapter 8 ME 551 27

spring to support the stylus shaft.

Application – Diaphragm Flexure3pp p g

• Diaphragm flexures are utilized toguide out of plane motion of a high endguide out-of-plane motion of a high-endmicroscope.

• Axial motion range is 140 μm• Axial motion range is 140 μm.

• Maximum lateral parasitic motion over the full range is about 2 μmthe full range is about 2 μm.

Chapter 8 ME 551 28

Rolamite• Rolamite (a major twist in flexural

bearings) is a technology for very lowbearings) is a technology for very lowfriction bearings:– Developed by Donald F. Wilkes of Sandiap y S

National Lab in 1960s.

– Uses a tensioned metal band and counterrotating rollers within an enclosurerotating rollers within an enclosure.

– Resulting linear bearing loses very littleenergy to friction.

• Effective friction coefficient is as low as 0.0005(an order of magnitude better than ball bearingsat the time)

• Main commercial application is theairbag deployment in passenger cars.

Chapter 8 ME 551 29

References

1 A H Slocum Precision Machine Design SME Press1. A. H. Slocum, Precision Machine Design, SME Press,1992.

• A. H. Slocum, ME 2.075 Course Notes, MIT, 2001.A. H. Slocum, ME 2.075 Course Notes, MIT, 2001.

2. S. T. Smith & D. G. Chetwynd, Foundations of UltraPrecision Mechanism Design, Vol 2, Taylor & Francis,g y2005.

Chapter 8 ME 551 30