Flexures - METU | Department of Mechanical Engineering

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 Examples
Rolamite
Chapter 8
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2
Flexures
• A flexure is a bearing system that
allows
ll
motion
ti
th
through
h bending
b di
off
its hinge elements.
– Also called flexural hinges and elastic /
compliant mechanisms
• Flexures do have a large
g number of
useful features for precision
engineering:
–
–
–
–
–
Chapter 8
Light weight
Low cost
Very low friction (w/o lubrication)
Maintenance free operation
Yield high
g repeatable
p
motion
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Flexures (Cont’d)
(
)
• Some of their disadvantages
g are
– Limited travel span if compared to the overall size of the
mechanism
– Very
V
llow d
damping
i
– Poor load capacity
– Might be difficult to manufacture
– Susceptable to fatigue failure
• Their applications
pp
range
g from
– Flip-top containers
– Piezo-beams on a scan tunneling microscope (STM)
• Due to bending stresses created in flexures, great care
needs to be exercised when designing such bearings.
Chapter 8
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Monolithic Design
g 1
•
•
•
They are very accurate.
Mechanism’s size is about 20 times the range of motion.
Abrasive water-jet
water jet machining is an economical way to cut nonnon
critical areas (links) whereas wire EDM can be used for precision
cuts 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
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Clamped
p Designs
g 1
•
Elastic members are clamped to the
stationaryy and moving
g elements via
bolts.
– Applications include wafer steppers, mirror
mounts, and more.
– Mechanism’s size is about 10 times the
range of motion.
•
•
Make sure that residual
stresses
don't
create
asymmetries
leading
to •
parasitic motions.
– Cut spring metal with EDM or
water jet.
– Use dual lubricated washers under
bolt heads and guides for clamp
plates.
Chapter 8
Nanometer accuracyy can be obtained
if bolted joint is designed properly:
– Rounded edges
– Each bolt
bolt’ss cone of action should overlap.
overlap
Easy to assemble from annealed
parts and hardened spring steel.
– B
Bolts
lt mustt b
be titightened
ht
d carefully
f ll as as the
th
torque can twist the flexure.
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Features of Clamped
p Designs
g
• Speed and acceleration limits:
– Limited only by yield strength and design.
• Range of motion:
– T
Typically
i ll used
d ffor motions
ti
lless th
than a ffew mms.
– Monolithic designs: flexure length/motion ≥ 20
– Clamped designs: flexure length/motion = 5 ... 10
• Applied loads:
– Design
g g
goal is to obtain load capacity
p
y with minimum spring
p g
constant.
• Repeatability:
– Axial: limited only by the drive system
– Lateral: < 1 nm for monolithic designs.
Chapter 8
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Features (Cont’d)
(
)
• Axial Resolution:
– Limited only by the drive system
system.
• Inherently preloaded.
• 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
performance.
performance
– Flexures often have low transverse stiffness so they are more susceptible
to parasitic forces.
• Good vibration and shock resistance.
• Damping capability:
– M
Material
t i ld
damping
i only
l (2
(2-5%).
5%)
– Damping mechanisms (constrained layer dampers) can be used to obtain
high damping.
Chapter 8
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Features - Accuracy
y
•
•
•
•
Axial: limited only by the drive system.
L t l can be
Lateral:
b lless th
than nanometers
t
ffor monolithic
lithi d
designs.
i
Depends on how well the bearing was assembled or machined.
Even if there is a small off
off-axis
axis error motion associated with the
primary motion:
– The error motion is usually very predictable and highly repeatable.
•
Fl
Flexural
l bearings
b i
cannott attain
tt i perfect
f t motion
ti due
d tto
– Variation in spring strength
•
–
–
–
–
–
Chapter 8
Elastic modulus varies with rolling direction in steels (anisotropy).
Variation in spring geometry
geometry.
Overall inaccuracies of manufacture.
Bending of the bearing in an unintended manner.
B di off structure.
Bending
t t
External applied loads (e.g., gravity and the manner in which the actuation
force is applied)
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Four-bar Linkage
g Flexure1
•
Leaf springs are clamped to the platforms to create a four-bar
linkage system.
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
motions.
Chapter 8
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Motion Errors
• The most common errors in four-bar linkage
flexures are the pitch angle and vertical motion.
motion
– They accompany linear motion in a four-bar linkage
flexure.
flexure
Chapter 8
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Motion Errors (Cont’d)
(
)
The pitch angle (θ) and vertical motion error (δ) of the mechanism can
be simply expressed as
⎡ 6(l − 2a )t 2 ⎤ x
⋅
θ =⎢ 2
2
2⎥
3
b
l
−
2
t
l
+
6
at
⎣
⎦ l
x2
δ≈
2l
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 is
applied at a point other than halfway between the platforms, a
bending moment is generated which causes a pitch error to occur.
Chapter 8
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Motion Errors (Cont’d)
(
)
The pitch errors are also caused by the difference in spring length and
difference in platform length respectively.
respectively That is
θs =
θp =
δ s x2
2l 2b
δpx
bl
where subscripts s and p refer to the quantites associated with the
spring
i and
d platform
l tf
respectively.
ti l
Typical tolerances on the springspring and platform length are 25 μm and
1...3 μm respectively.
Chapter 8
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Motion Errors (Cont’d)
(
)
• To minimize parasitic motions,
– Symmetrical
S
i ld
duall ffour b
bar lilinkage
k
eliminates
li i
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
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Two-Stage
g Linkage
g 1
• The parasitic motions of one 4-bar cancel the parasitic
motions of the second 4-bar.
• Lateral and yaw stiffness will not be high due to buckling
effect.
• Overcome by use of monolithic hourglass
hourglass-type
type members.
members
Chapter 8
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Cascaded Linkage
g 2
•
•
In this design, any change in height of platform A relative to its
support B will be compensated by an equal and opposite change in
height of B relative to the base.
In theory this should result in a perfect rectilinear motion of A relative
to the base.
Chapter 8
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Differential Design
g 2
Chapter 8
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Multi-blade Flexures1
Many thin blades can be used
to increase lateral stiffness and
load capacity while keeping
axial stiffness and stress low:
Fvertical ∝ ntw
K vertical ∝
ntwE
L
L3
Ent 3 w
L3
K axial (monolithic) ∝
E (nt )3 w
K axial (multiblade
l ibl d ) ∝
•
•
•
Beware of alignment and slip problems between the blades.
Care must be exercised when tightening bolts as the torque can twist the
flexure!
Rubber placed between the blades can increase damping.
Chapter 8
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Flexures for Angular
g
Motion1
•
•
– If the top beam were 2 mm thick and the
bottom one were 1 mm thick, the
transmission ratio would be (2/1)3 = 8.
2
Deformed
upper beam
Chapter 8
L ma
in
L/
=2
FL
= 2E I
3
F
=
Device is developed by Polaroid
Corp to adjust pitch
Corp.
pitch- and yaw
angles of a lens.
Mechanical advantage is provided
b the
by
th screws that
th t push
h up on the
th
upper beam using the lower beam
as a base.
•
FL3
3EI
Another interesting feature is that
the slope at the root of the main
beam causes an Abbe error
canceling the beam’s displacement
at its free end as
δmain
i = δ - α⋅Lmain
i = 0
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Cross-strip
p Flexure1
• Cross-strip flexure allows angular rotation about an axis.
Chapter 8
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Angle
g Hinges
g 2
Chapter 8
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Locking
g Flexures1
• Flexures can be used as one-shot hinges
g to g
guide
components 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 effect.
effect
Chapter 8
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Locking
g Flexures ((Cont’d))
• Small taper keeps component preloaded
in both the X and Y directions.
• To minimize distortion caused by bending,
pass through
g the support
pp
line of force must p
ledge.
• A lens could be kinematically held by two
anvils and a single locking member.
• The geometry is tolerant of manufacturing
errors.
Chapter 8
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Locking
g Flexures ((Cont’d))
• For precise X position location, utilize a grooved
structure.
t t
• Numerous permutations are possible.
Chapter 8
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1
Transmission Systems
y
Chapter 8
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Transmission Systems
y
(Cont’d)
(
)
• 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
ratio. For instance,
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
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Application – CMM Probe
Touch-trigger probe (TTP) designed/built by
METU-SPARC group employs a diaphragm
spring to support the stylus shaft.
shaft
Chapter 8
ME 551
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Application
pp
– Diaphragm
p g Flexure3
• Diaphragm flexures are utilized to
guide out-of-plane
out of plane motion of a high-end
high end
microscope.
• Axial motion range is 140 μm.
μm
• Maximum lateral parasitic motion over
the full range is about 2 μm.
μm
Chapter 8
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Rolamite
• Rolamite (a major twist in flexural
bearings) is a technology for very low
friction bearings:
– Developed
p byy Donald F. Wilkes of S
Sandia
National Lab in 1960s.
– Uses a tensioned metal band and counter
rotating rollers within an enclosure.
enclosure
– Resulting linear bearing loses very little
energy to friction.
• Effective friction coefficient is as low as 0.0005
(an order of magnitude better than ball bearings
at the time)
• Main commercial application is the
airbag deployment in passenger cars.
Chapter 8
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References
1 A
1.
A. H.
H Slocum,
Slocum Precision Machine Design,
Design SME Press,
Press
1992.
• A. H. Slocum, ME 2.075 Course Notes, MIT, 2001.
2. S. T. Smith & D. G. Chetwynd, Foundations of Ultra
g Vol 2, Taylor
y
& Francis,
Precision Mechanism Design,
2005.
Chapter 8
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