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Lecture Summary



(unrefereed, unpublished)



prepared by

Brian Trease

Compliant System Design Laboratory

The University of Michigan



ME599


Compliant Mechanisms

Winter 2004






April 30, 2004



Page
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60

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INTRODUCT
ION AND BACKGROUND
................................
................................
................................
.............

4

T
RADITIONAL
(C
ONVENTIONAL
)

J
OINTS VS
.

F
LEXIBLE
J
OINTS

................................
................................
...........

4

B
ENEFITS OF
C
OMPLIANT
J
OINTS

................................
................................
................................
.............................

4

S
HORT
H
ISTORY

................................
................................
................................
................................
..........................

5

S
OME
A
PPLICATION
S

................................
................................
................................
................................
..................

5

K
INEMATICS
C
LASSIFICATION


W
HERE DO FLEXURES FIT
?
................................
................................
................

6

MOTIVATION


CHALLENGES / CRITERI
ON
................................
................................
................................
.

6

R
ANGE OF
M
OTION

................................
................................
................................
................................
.....................

7

A
XIS
D
RIFT
................................
................................
................................
................................
................................
...

7

O
FF
-
A
XIS
S
TIFFNESS

................................
................................
................................
................................
..................

8

S
TRESS
C
ONCENTRATION

................................
................................
................................
................................
..........

8

JOINT SURVEY
................................
................................
................................
................................
............................

8

S
URVEY
R
EFERENCES
................................
................................
................................
................................
.................

8

J
OINT
R
EPLACEMENT
C
LARIFICATION

................................
................................
................................
.....................

9

SURVEY OF ROTATIONAL

JOINTS

................................
................................
................................
................
10

N
OTCH
-
J
OINTS
................................
................................
................................
................................
...........................

11

Spreadsheet Calculations

................................
................................
................................
................................
..
15

Notch Joint Comparisons: leaf hinge, circular, and elliptical

................................
................................
....
17

B
EAM
-
B
ASED
R
EVOLUTE
J
OINTS

................................
................................
................................
...........................

17

Cross
-
strip Pivot

................................
................................
................................
................................
.................
17

Cartwheel Hinge

................................
................................
................................
................................
.................
19

Angled Leaf Springs
................................
................................
................................
................................
............
20

Two
-
axis hinges

................................
................................
................................
................................
...................
20

Passive joints

................................
................................
................................
................................
.......................
22

Q
-
joints

................................
................................
................................
................................
................................
.
22

Torsional Hinges

................................
................................
................................
................................
.................
23

Split Tube Joint

................................
................................
................................
................................
....................
24

Disc Couplings

................................
................................
................................
................................
....................
24

Rotat
ionally Symmetric Leaf Type Hinge

................................
................................
................................
.......
25

C
OMPLIANT
R
EVOLUTE
J
OINT
................................
................................
................................
................................
.

26

CR Joint Range of Motion

................................
................................
................................
................................
.
28

CR Joint Stiffness Design Charts

................................
................................
................................
.....................
29

Open
-
Cross
CR Joint
................................
................................
................................
................................
..........
32

SURVEY OF TRANSLATIO
NAL JOINTS

................................
................................
................................
........
33

A
PPLICATIONS

................................
................................
................................
................................
...........................

33

J
OINT REPLACEMENT CLA
RIFICATION

................................
................................
................................
....................

33

L
EAF
S
PRINGS

................................
................................
................................
................................
............................

33

C
OMPLIANT
T
RANSLATIONAL
J
OINT
................................
................................
................................
......................

34

CT

R
ANGE OF
M
OTION
A
NALYSIS

................................
................................
................................
.........................

35

P
ARAMETRIC
S
TUDIES
/

D
ESIGN
T
OOLS

................................
................................
................................
................

36

Parametric Study
................................
................................
................................
................................
.................
36

CT Joint Design C
harts
................................
................................
................................
................................
......
37

OTHER NEW DESIGNS, V
ARIATIONS, AND CONCE
PTS

................................
................................
......
39

O
THER
C
OMPLIANT
J
OINT
C
ONCEPTS
................................
................................
................................
....................

39

C
OMPLIANT
U
NIVERSAL
J
OINT

................................
................................
................................
...............................

39

E
MBEDDED
S
ENSING
................................
................................
................................
................................
.................

40

S
ERIAL
C
HAINS WITH
C
USTOMIZABLE
D.O.F.

................................
................................
................................
.....

41



Page
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60

MORE ON A GENERAL DE
SIGN METHODOLOGY
................................
................................
..................
42

A
PPLICATIONS
/

E
XAMPLES

................................
................................
................................
................................
.....

43

Motorcycle Suspension
................................
................................
................................
................................
.......
43

Compliant Haptic 2
-
DOF Joystick
................................
................................
................................
...................
45

POTENTIAL APPLICATIO
NS
................................
................................
................................
..............................
47

O
RTHOTIC
D
EVICES

................................
................................
................................
................................
..................

47

R
OBOTICS

................................
................................
................................
................................
................................
...

47

S
TATICALLY
B
ALANCED
M
ECHA
NISMS
................................
................................
................................
.................

48

D
OOR
,

T
RUNK
,

AND
H
OOD
H
INGES
................................
................................
................................
........................

48

P
OSITIONING
G
IMBALS
................................
................................
................................
................................
.............

49

H
ANDMADE
C
OMPLIANT
M
ECHANISMS FOR
D
EVELOPING
C
OUNTRIES

................................
...........................

50

V
IBRATION ISOLATIO
N SYSTEMS

................................
................................
................................
............................

50

B
OAT
D
OCKING
M
ECHANISM
................................
................................
................................
................................
..

50

O
PTICAL
M
IRRORS AND
A
NTENNAS
................................
................................
................................
.......................

50

O
THER
I
DEAS

................................
................................
................................
................................
.............................

51

PREVIOUS STUDENT'S C
OMMENTS/SUGGESTIONS

................................
................................
.............
51

O
PEN
Q
UESTIONS
(F
UTURE
W
ORK
)

................................
................................
................................
.......................

52

FINAL COMMENTS

................................
................................
................................
................................
.................
52

ORGANIZATION OF REFE
RENCES
................................
................................
................................
.................
52

TABLE OF FIGURES

................................
................................
................................
................................
...............
54

REFERENCES
................................
................................
................................
................................
.............................
55

APPENDIX
................................
................................
................................
................................
................................
......

1




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)
)


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v
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.
.


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Rigid mechanical connections, such as hinges, sliders, universal joints, and ball
-
and
-
socket joints, allow diff
erent kinematic degrees of freedom between connected parts.
These are the building blocks of most of the mechanisms used in manufacturing, robotics,
and automobiles, just to name a few technologies.
However, the clearance between
mating parts of rigid jo
ints causes backlash in mechanical assemblies. Further, in all the
above joints there is relative motion causing friction that leads to wear and increased
clearances. A kinematic chain of such joints compounds the individual errors from
backlash and wear
, resulting in poor accuracy and repeatability.


Flexible Joints
(
a.k.a.
Flexures, Couplings, Flexure Pivots, Flex Connectors,

Living Joints, Compliant Joints)


Flexible joints (a.k.a. flexures)
offer an alternative to traditional mechanical joints that

alleviates many of their disadvantages. Flexures
utilize the inherent compliance of a
material rather than restrain such deformation. These joints eliminate the presence of
friction, backlash, and wear. Further benefits include up to sub
-
micron accurac
y due to
their continuous monolithic construction. Such accuracy is important in many micro
-
,
nano
-
, and bio
-
applications. The monolithic construction also simplifies production,
enabling low
-
cost fabrication.


The benefits of compliance in design are li
sted in the introduction section of any
the published papers on compliant mechanisms and are stated again
below.

(
http://www.engin.umich.edu/labs/csdl/pub.html
)


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Co
mpliance

is i
nherent
in all materials



Improved Life

o

Friction, Backlash, Wear



Sub
-
micron accuracy



Monolithic Construction



Very Compatible with Planar
Manufacturing



No Assembly Needed



No Noise



No Lubrication

Required




Fewer Parts



High Precision



Repeatabilit
y



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In the last 50 years, many flexible joints have been researched and developed, most of
which are considered one of two varieties: notch
-
type joints and
leaf springs. Notch
-
type flexible joints (a.k.a. fillet joints)

(
Figure
1
(a, b))
were first analyzed by Paros and
Weisbord
[1]

in 1965 and have since become well understood by many researchers and
designers. Today, notch
-
type joint assemblies are widely used

for high
-
precision, small
-
displacement mechanisms. These joints have also been applied by Howell and Midha
[2]

to develop the field of pseudo
-
rigid
-
body compliant mechanisms.



Leaf springs provide the most ge
neric flexible translational joint, composed of sets of
parallel flexible beams (
Figure
1
(c)). In addition to high
-
precision motion stages, leaf
spring joints are also widely used in medical instrumentation and MEMS
devices

.




(a)
planar notch
joint

(b)

spherical notch joint

(c)

leaf
-
springs

Figure
1
.
Basic Flexible Joint Components


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Small Displacements



Positioning Stage
s



Medical Applications



MEMS Devices



PRBM Compliant
Mechanisms



Installation Misalignment



No Assembly Applications



Instrumentation





www.tribology
-
abc.com




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d
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f
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?
?





It is important that we be clear on where compliant joints stand in

the overall field of
compliant mechanism kinematics, shown above. Flexures are usually analyzed based on
the first category, Rigid
-
Body Kinematics. Thus, the kinematics are already determined
and we are merely doing joint replacement.
Even i
f there is
not an existing

mechanism
,
then at least we already have the RB kinematics design tools
; joint stiffness only needs to
be added.

To say it in another way, mechanical degree
-
of
-
freedom is not influenced by
joint stiffness.




All of the kinematics is already

established for us.



With flexures, we are doing “joint replacement”.



Motions are same as before,
except there are now spring
forces in our systems.


In a way,
this is kind of the
opposite

of
the CSDL’s

current approach
to compliant
mechanism synthesis.


Y
et
it is
still very valuable in understanding the field as a whole
and gaining insight into how flexures work
.


The other types of compliant mechanisms integrate deformation of their links into
kinematics, thus a rigid body model doesn’t work and a new for
mulation is created. This
will not be covered in this report.


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/
/


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Rather than list the disadvantages of compliant mechanisms, it is better to see any
drawbacks as “challenges” and make them the criteria by which we judge g
ood flexures.



Page
7

of
60


Figure
2
.
Flexure Design Criteria


The benefits gained from using flexure joints come at the cost of several disadvantages
that must be taken into account when designing. To overcome these drawbacks and
develop be
tter flexures, a set of criterion must be established for benchmarking. The four
most important criterion are (1) the range of motion, (2) the amount of axis drift, (3) the
ratio of off
-
axis stiffness to axial stiffness, and (4) stress concentration effec
ts.


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All flexures are limited to a finite range of motion, while their rigid counterparts rotate
infinitely or translate long distances. The range of motion of a flexible joint is limited by
the permissible stresses and strains in the mate
rial. When the yield stress is reached,
elastic deformation becomes plastic, after which, joint behavior is unstable and
unpredictable. Therefore, the range of motion is determined by both the material and
geometry of the joint.


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In addition t
o limited range of motion, most flexure joints also undergo imprecise motion
referred to as axis drift or parasitic motion. For notch
-
type joints, the center of rotation
does not remain fixed with respect to the links it connects. With translational flex
ures,
there can be considerable deviation from the axis of straight
-
line motion. For example, a
simple four
-
bar leaf spring experiences curvilinear motion.


Axis drift can be improved by adding symmetry to the design of a joint. However, this
often incr
eases the stiffness of the joint in the desired direction of motion. Further, more
space is required to accommodate any symmetric joint components.



Page
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60

Having minimal axis drift is essential to preserving the kinematics of the original
mechanism when doing co
nventional joint replacement with flexures.


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While most flexure joints deliver some degree of compliance in the desired direction,
they typically suffer from low rotational and translational stiffness in other directions. A
high ratio o
f off
-
axis to axial stiffness is considered a key characteristic of an effective
compliant joint.


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Most notch
-
type joints have areas of reduced cross
-
section through which their primary
deflection occurs. Depending on the shape of the
se reduced cross
-
sections, the joints
may be prone to high stress concentrations and hence a poor fatigue life. Refer again to
Figure
1
(a,

b) for examples of flexures with stress concentrations.


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As ment
ioned, primitive joints previously developed typically fall into one of two
categories: notch joints or leaf spring joints. These joints are often combined in
assemblies and are most commonly used as revolute joints, universal joints, or parallel
four
-
bar

translational joints. Most commercially
-
available flexible joints are such
derivatives of the primitive joints, with the addition of any variety of packaging and
connections to suit particular engineering needs. For a detailed study of traditional
flexu
res, including design methods, material selection, and geometry optimization, please
refer to Lobontinu
[4]
.


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Most of the joints described in the following sections can also be found in the foll
owing
resources. Please refer to these for further analysis, empirical data, and an even wider
array of compliant joints.

Full citations are found in the reference section.




Flexures, by Stuart
Smith
(
Book
)

[3]

o

A source of man
y of the equations provided in this paper



Compliant Mechanisms, Design of Flexure Hinges, by N.
Lobontinu

(
Book
)

[4]


Design and a
nalysis of notch joints



Design of Large
-
Displacement Compliant Joints, by Trease (
Paper)

[10]

o

S
ummary

and
Comparison of many flexure joints




Compliant Mechanisms, by L. Howell (
Book
)

[2]

o

Another survey of several flexures with their basic design equations




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What exactly are we doing when we design with compliant joints?

When learning about compliant joints, it is beneficial to first understand how you will be
using them when designing mechanisms. Usually, an already designed,
traditional,
rigid
-
body

m
echanism is considered. This mechanism may or may not have a spring as
one of its components, as shown in the figure on the left below, in green. If there is a
spring, then any point on the mechanism will seem to also have a stiffness, even though
no spr
ing element is physically located at that point. This
effective output stiffness

is also
shown in the figure below.


The goal is to remove the existing pin joints (or bushings), replacing them with compliant
joints to gain all the benefits listed in the

introduction. Because of the joint compliance,
the resulting system will also have an
effective output stiffness
,
whether it is desired or
not.
If the original mechanism employed a spring, then the designer’s task is to match the
effective output stiffn
ess

of each mechanism. The spring in the original mechanism is
replaced by the stiffness in the joints in the new mechanism.



Figure
3
.
Conversion from Conventional to Compliant Joints in a mechanism


If the original mechanism
contains no spring, then the task is usually to minimize the
effective output stiffness

of the new mechanism.


Calculating the effective output stiffness the original mechanism, requires knowledge of
both the kinematics and the spring stiffnesses. Matchin
g the original effective output
stiffness the original mechanism, requires knowledge of the kinematics and manipulation
of the joint stiffnesses as the design variables.

Once the desired joint stiffnesses are
known, the appropriate compliant joints can be

selected and sized using the information
detailed in the rest of this paper.




Page
10

of
60

With the joint equations, a number of sizing calculations can be run on the joints until all
constraints are met. A single
-
joint can easily be “coded” in a Matlab program or e
ven
somtimes better, an Excel Spreadsheet, as demonstrated in the next section for circular
notch joints.


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Rotational joints are more numerous and
more
commonly used. This section will briefly
describe many revolute flexure joi
nts, provided equations for their stiffness, range of
motion, and axis drift.

Whenever possible, both the functional (a.k.a. desired, axial,
primary) stiffness and the off
-
a
xis stiffness will be given
. Divide the off
-
axis stiffness by
the axial stiffness

to get the
stiffness ratio
. High stiffness ratios indicate effective joints.
Also note that the range
-
of
-
motion equations are based on the linear elastic failure of the
joint material. Thus, these are equations relating maximum yield stress and moment.

Finally, remember that low axis drift is critical to achieving precise kinematics.


Living
H
inge
T
erminology

Existing research and literature use a variety of terms to describe bending
-
based flexures.
These include
cantilever, n
otch
,

l
eaf
,

p
ivot
,

l
iving

hinge
, and more. Each of these terms
has a slightly different shade of meaning. Strict definitions are hard to find and are based
either on length, geometry, function, or means of approximation.


Typically, a
beam

is the basic structure for all these te
rms, defining their long, slender
geometry which is capable of
bending
.


Cantilever

usually refers to a beam that is loaded at its end.


A
leaf

(spring) is a beam specifically being used as a spring. Its kinematic quality
may not be important.


A
pivot

is

a (short) beam that acts as a revolute joint. Its

stiffness may not be
important to the application, but still must be considered.


A
notch joint

is so named for its appearance

(and possibly the way it was
manufactured). Most notch joints are pivots. T
hey may have a rectangular or
elliptical shape.


A
living hinge

is also a pivot, but with a relatively small size. Its stiffness is so much
lower than other stiffnesses in the system that it is considered zero. Thus, a living
hinge can be modeled directe
d as a pin joint

(although it may have considerable axis
-
drift)
. See
Figure
4
.


Shampoo bottle lids are good examples of the definition of a “living hinge”
; they

are
an application where
precision and load
-
bearing capacity are non
-
essential, and low
cost is most important.




Page
11

of
60


According to Howell, a living hinge becomes a small
-
length flexural pivot if it is
sufficiently large enough in all dimensions. A small
-
length flexural pivot becomes a
beam when it
becomes

more than 10% longer

than the link to which it is connected.


More Definitions



Smith, Chapter 4



Howell, Chapter 5.8



Small
-
Length Flexural Pivots, Howell, p.411



Figure
4
:
Examples of living hinges and small
-
length pivots (notch joints)


So, we see
that terminology is mostly a function of size. As a designer, there are several
issues of “s
cale”

we consider at different size regimes. These include:



Method of Mechanics Analysis



Method of Kinematics



Stress Stiffening



Large
-
deflections


N
N
o
o
t
t
c
c
h
h
-
-
J
J
o
o
i
i
n
n
t
t
s
s


No
tch joints have already been discussed and are shown
again in
Figure
5
.
Paros and
Weisbord
[1]

first reported on the mechanical analysis of these

joints in

1965
.

The fou
r
varieties of notch cut (
Circular
,
Elliptical
,
Rectangular
,
Filleted
) are shown in
Figure
6
.



Page
12

of
60


Figure
5
.
Some of the Basic Notch Joints



Figure
6
.
Varieties in the transition from cantilever joints to circular notch joints


Paros and Weisbord first developed the equations

for stiffness and range of motion

of the
circular notch joint
. Smith has derived general equation for any ellipt
ic notch
-
joint.

Smith’s equations approximate Paros’s equations at the limits: eccentricity = 0 →
rectangular joint; eccentricity = 1 → circular joint.



Figure
7
.
Notch Joints for
which
Smith provides equations


Before the gen
eral equation is given, the original equations for the rectangular and
circular joints are shown below
, beginning with the
rectangular joint
.

In all joints, “b” is
the out
-
of
-
plane dimension.



Page
13

of
60



Figure
8
.
Rectangular Notch

Join
t with stiffness and

range
-
of
-
motion equations
. (z
-
axis is out
-
of
-
plane)


The o
ff
-
axis
stiffnesses are the same as those

of a beam
, and can be calculated easily
from most mechanics textbooks.



Circular Notch Joint and Equations


Figure
9
.
Circular Notch Joint


Stiffness

of Circular Notch Joint Calculations





Range of Motion (S
tress
-
R
otation

Equation)





Page
14

of
60


Maximum Load
-
Carrying Capability (Stress
-
Moment Equation)




Elliptical Notch Joint


Generalized Parametric Equations

The next set of equations are for elliptical joints, pictured below.



Figure
10
.
Elliptic Notch Joint


Ellip
tical Joint Stiffness



for small beta










Page
15

of
60

The other 5 off
-
axis stiffnesses are also given:










Note that these equations can also be used for

circular

notch joints by appropriately
setting


= 1.


At this time, the Sm
ith book does not provide range of motion formulas for the elliptical
notch joint.


Spreadsheet Calculations

A methodology for optimizing a design utilizing notched joints has high value in aiding
the design process. For instance, the equations can be org
anized in a spreadsheet analysis
in order to optimize and compare different material/geometry configurations.
Figure
11

below demonstrates a simple example of such an analysis for a circular notched joint.




Page
16

of
60


Figure
11
:
Spreadsheet Analysis of a Circular Notched Joint


The inputs to this spreadsheet analysis are the basic material properties and joint
geometry (both physical and operational parameters). The outputs are the joint stress and
rotational stiff
ness. In addition, the margin of safety to material yield and ultimate are
calculated with user defined factors of safety. Using the goal seek tool within Excel
allows for simple configuration comparisons.


For example, joint stress due to rotation for

a circular notch joint is a function of joint
length L, joint thickness t, material modulus E, and joint rotation Ө. Thus, for a given
material stress any one of these four parameters can be optimized while holding the other
three constant. So, if the d
esigner desires a joint made of 6061
-
T6 Al with a thickness of
0.015” and a maximum rotation angle of 2 degrees they can determine the length using
goal seek while maintaining positive stress margins in the joint.


Again, this is only one simple example
of the type of comparison analyses that can be
performed but it demonstrates the usefulness if there are particular design constraints that
must be satisfied. This spreadsheet can also be expanded to include calculation of off
-
axis stiffness/stress for fu
rther joint performance evaluation.




Page
17

of
60

Notch Joint Comparisons
: leaf hinge, circular, and ellipt
ical


Leaf hinge notch joint

Circular notch joint

Elliptical notch joint

Design for stiffness





Note that all K's are proportional to E and b

Design for stress



N/A in the handout


Note that all

's are proportional to E, but independent of b

E: material modulus

b: out
-
of
-
plane thickness

t: min
imum in
-
plane dimension of the hinge

a
x
: half of the hinge length a
y
: length of minor axis in elliptical joint


For the same E, b, t, and a
x
, the stiffness and maximum stress comparison are as follow:



Stiffness comparison: leaf hinge < elliptical < cir
cular



Stress comparison (for the same range of motion θ
): leaf

hinge < elliptical <
circular

The stress of the elliptical design is estimated from the circular notch hinge, because the
circular design is a degenerate version of elliptical one.


So, when th
e design requires high stiffness and small range of motion, pick the circular
notch joint; for low stiffness and larger range of motion, pick the leaf hinge joint. The
elliptical joint has intermediate stiffness and range of motion (stress). Using the abov
e
table, the notch dimensions (t and a
x
) can be designed to achieve desired stiffness and
reduce stress.


Another Point of View

on Design


The selection of the notch type depends on the type of application it will be used for. If
we assume that, for a typi
cal application, kinematics requirement limits the value of a
x

and the main function of the notch is to produce a desired motion against a small load,
elliptical and rectangular notch will provide lower stiffness, and hence, more suitable for
this applicat
ion than a circular notch. However, a rectangular notch has a stress
concentration at sharp corners, leaving an elliptical notch the most suitable for this
application. On the other hand, if the range of motion is small and the load is large, a
circular no
tch will be more suitable than the others.


B
B
e
e
a
a
m
m
-
-
B
B
a
a
s
s
e
e
d
d


R
R
e
e
v
v
o
o
l
l
u
u
t
t
e
e


J
J
o
o
i
i
n
n
t
t
s
s


Leaf springs can also be used in a variety of ways to create revolute joints, as shown in

Figure
12
,
Figure

15
, and
Figure
16
. 2(c) is also recognized as the well
-
known “free
-
flex” or “cross
-
spring” pivot, commercially available in many forms (See
Figure
12
.)

Cross
-
strip P
ivot

The cross
-
strip pivot is also a very old design, first describ
ed by Haringx

[9]

in 1949. It
was designed to have a b
etter
r
ange of
m
otion than notch joints
. However, it suffers from


Page
18

of
60

considerable axis drift, calculated in the equations below. The center of rotation moves
while the joint
undergoes its large deflection.



Figure
12
.
Commerical

Free
-
Flex


Joint

Bendix Corporation


Figure
13
.
Cross
-
Strip Joint

Range of Motion and Stiffness Equations



;


;

n
=total number of strips


Axis Drift (Center of Rotation Movement)

Note: L is the same as “a” in
Figure
13
.


For alpha =
45

degrees




Figure
14

shows dimensionless axis drift (

p
/L) as a function of angular deflection.




Page
19

of
60


Figure
14
.
Axis Drift in the Cross
-
Strip Pivot


For more information

on the cross
-
strip pivot
, see these references:



Smith, p.192



Howell, p.189


“Cross
-
Axis Flexural Pivot”


Cartwheel

Hinge

Taking the cross
-
strip pivot and “welding”
the strips together results in the planar
cartwheel hinge
. This is generally an
improvement of design, although it may be
more difficult to manufacture. See Smith,
p. 199.



Figure

15
.
Cartwheel Hinge


Design Equations:

Stiffness, Range of Motion, and Axis Drift








Cross
-
strip vs. Cartwheel hinge

For two similar scale joints (L=2R)



C
artwheel hinge is stiffer than cross
-
strip joint



Cartwheel hinge has a smaller range of motion



Cart wheel hinge has a smaller axis drifting

They are both more scalable than notch
-
type hinges, but the cross
-
strip joint may be more
difficult to manufacture.




Page
20

of
60

Angled
L
eaf

Springs

Kyusojin and
Sagawa

[5]

developed several more revolute joints based on leaf springs.
These generally have a good range of motion, but can be bulky and difficult to implement
in a mechanism. The “2R” join
t has
lots of parasitic

motion (axis drift), while the “6R
-
1”
has much less. The “
6R
-
2
” theoretically has
no parasitic

motion.



Figure
16
.
“2R” Angled Leaf Spring



Figure
17
.
“6R
-
1” Angled Leaf
Spring



Figure
18
.
“6R
-
2” Angled Leaf
Spring



A. Kyusojin and D. Sagawa, “Development of Linear and Rotary Movement Mechanism
by Using Flexible Strips,” Bulletin of
Japan Society of Precision Engineering, Vol.
22, No. 4, Dec. 1988, pp. 309
-
314


Two
-
axis hinges

As depicted in the following table, there are also many flexures that allow compliance
about more than one axis. The two axes of compliance are shown in the ad
jacent
schematics. It may be easier to think in term of the schematics first (based on your
design requirements), then find an appropriate hinge. See Christine Vehar’s lecture
summary on Precision Mechanisms for more possibilities in stacking orthogonal
parallelogram mechanisms more multiple axis stages.


Two rotational, orthogonal compliant axes represent a universal
-
joint, of course, shown
in
Figure
21

and
Figure
22
. The joint shown in
Figure
20

may be a universal joint or a
spherical joint, depending on the thinness of the narrowest cross
-
section.


Paros and
Weisbord

studied these 2
-
axis joints. Smith also considered them (p. 206, 212,
217), included effective notch/moment
-
arm
stiffness calculations for the joint in
Figure
21

(p. 217).




Page
21

of
60


Figure
19
.

2 Compliant Axis Hinge from Smith



Figure
20
.

Spherical Notch Joint (3 D.O.F.)



Figure
21
.
Compliant Notch Universal Joint



Figure
22
.
Co
-
linear Notch U
-
Joint





Page
22

of
60

Passive joints

Passive joints (
Figure
23
) are co
ntact/sliding joints, described

by Howell and sometimes
used
in PRBM
-
based compliant mechanisms.

They can be thought of as f
orce
-
closed
conventional hinge joint
s. They are sometimes helpful in design, but not ideal because
they utilize contact forces, which cause friction. Further, with relative motion, it is
pos
sible that the kinematics may change, invalidating your design models.
They can,
however, greatly increase load carrying capability in some applications. So
,
you should
know it’s out there and that
passive joint

have been used to solve some problems
.



Figure
23
.
Passive Joint in a Compliant Crimping Mechanism

Q
-
joints

Thus far, it would be difficult to use any of the described joints where two beams are
intersecting. This problem is solved by the q
uadrilateral
-
joint or Q
-
joint.



2 Types



Parallelogram

(
Fig
ure
25
)



Deltoid
(adds mechanical advantage)



Howell, p.186



Figure
24
.
Examples of rigid segment joined somewhere besides the ends,


including (a) scissors, and (b) a pantograph
.


The parallelogram form (
Fig
ure
25
) constrains the angles of opposite links to be equal,
thus transmitting equal angle across the joint. The deltoid form performs similarly, while
adding mechanical advantage to the joint.



Page
23

of
60



Fig
ure
25
.
(a) Parallelogram Q
-
joint, and (b) example of its use with a compliant pantograph mechanism.


Torsional H
inges



What is the difference between torsional and revolute joints?

The difference is based on
where the compliance i
s coming from.

In
Figure
26

below, we see the 3 degrees
-
of
-
freedom of a beam. Thus far, we have only been
using R1 and R3 in our revolute joints. Torsion is deflection about R2, and can
also be used in revolute joints. For thi
s, the R2 axis must point out of the plane of
the mechanism, and the proper attachment is required. Torsion based joints are
considered to have more
d
istributed compliance

than bending based joints. (See
Howell, p. 62, 190
.)




Conversion from Torsional to

Revolute

Joint



Results in 3
-
D out
-
of
-
plane geometry



Kinematics and Analysis remain planar, 2
-
D



Figure
26
.
The Effective Degrees of Freedom of an Elastic Beam




Closed vs. Open Shells

Not only rectangular beams can serve as tors
ion joints. Square, circular, and
hollow cross
-
sections can also be used. A hollow
-
beam is also called a closed
shell. An open shell is formed by cutting a slit lengthwise along a hollow
beam, resulting in very low R2 compliance while maintaining the hi
gh off
-


Page
24

of
60

axis stiffnesses of a closed shell. This concept has been developed by
Goldfarb, described in the next section.


Split T
ube

Joint

The split
-
tube joint was
developed by Michael Goldfarb

[7]
. It has

the o
ff
-
axis stiffness
of a cylinder

and very little torsional stiffness. Further, it has almost
no center of rotation
drift

when the connected links are fixed along the line of center of rotation, shown in
Figure
27
. (Also see Howell
, p.
193
)



Figure
27
.
Goldfarb Split
-
Tube Joint


Figure
28
.

Center
-
loaded configuration


This joint offers the off
-
axis stiffness of a solid circular tube while having a low tors
ional
stiffness. While the axis drift of a split
-
tube is small, it is not zero. Perfect rigidity would
require infinitely thin line contact between the connecting link and the tube. Further, this
joint exhibits a tradeoff between range of motion and off
-
axis stiffness. Under large
displacements, the gap separation increases and the tube warps out of circular shape,
reducing the off
-
axis stiffness.


D
is
c C
ouplings

Disc coupling
s

are 3 degree
-
of
-
freedom joints, usually used as compliant replacements to
tr
aditional universal joints, while also
adding an axial degree
-
of
-
freedom. (See Smith, p.
291) The purpose of these joints is to transmit torque from one shaft to another, even
when the shafts connect at an angle. The axial degree
-
of
-
freedom allows for s
ome play
during assembly of a system, and is also useful for self
-
alignment applications.


Note: torque transmission performance is often rated in terms of energy efficiency. Any
design using compliant u
-
joints will have lower energy efficiency due to th
e energy
required to deflect the joint. Not all energy is lost though, as that stored energy is used to
return the spring back to its original position.




Page
25

of
60


Figure
29
.

Inner
-
to
-
outer Edge

Disc Coupling



Figure
30
.

Outer
-
edge
Disc
Coupling


Rotationally Symmetric Leaf Type Hinge

The function of this joint is the same as the previous two, but the form is different. The
joint can be created simply by machining

notches in the sides of a tube.


Figure
31
.

Axial Plunge Leaf
-
Spring Universal Tube Joint,
Smith, p.308






Page
26

of
60

C
C
o
o
m
m
p
p
l
l
i
i
a
a
n
n
t
t


R
R
e
e
v
v
o
o
l
l
u
u
t
t
e
e


J
J
o
o
i
i
n
n
t
t


The

Compliant Revolute
(CR)
joint,
developed at the University of Michigan
Error!
Reference source not found
.

and
shown
below
, maintains zero axis drift under moment
-
loading. Of all the flexible revolute joints, it is the only
one to have a large range of
motion combined with a high ratio of off
-
axis stiffness to stiffness

in the desired direction
of motion. In comparison, the rotation axis of a popular cross
-
spring pivot (
Figure
1
(c))
of comparable size to a CR joint (diagonal leaf
-
springs 114mm long) drifts 5.5mm while
rotating thro
ugh 40 degrees. (See Haringx
[9]

for design tables.) Even under typical
axial loading, the proposed compliant joint’s axis of rotation

drifts only nanometers.



Figure
32
.
UM Complian
t Revolute Joint (a.k.a. Center
-
Moment CR, Segmented Cross CR)


Figure
33
.
Primitive Design Form used to create
CR joint (See Smith, p.
204
)




Page
27

of
60

Most of the stiffness components of the CR joint, except for the primary rotational
sti
ffness, can be calculated with standard beam formulas. An empirical formula for the
rotational stiffness of a cruciform hinge, accurate to within 4%, is described by Smith
[3]
.
A cruciform hinge is a torsion bar with a cross
-
s
haped cross
-
section, depicted in

Figure
33
. The CR joint is considered as two cruciform hinges used in parallel
(
Figure
32
),

thus
having twice the axial, bending, and torsional stiffness suggested by Smith
, and 8 times
the bending/rotational stiffness. Due to symmetry and loading at the center, the resulting
6x6 spatial stiffness matrix is purely diagonal. The six diagonal elements, based on the
coordinates of
Figure
38

are given
in
Table
1
. “w” and “t” represent the width and
thickness, as labeled in

Figure
34
.


Figure
34
.
CR Joint Cross
-
Section parameters


Table
1
.

Analytic CR Joint Stiffness Formulas (closed cross)

Torsional
Stiffness

k
66

(M
z
/

z
)


Bending /
Rotational
Stiffness

k
44

(M
x
/

x
),
k
55

(M
y
/

y
)

8 EI/L

Bending
Stiffness

k
11

(F
x
/d
x
),

k
33

(F
y
/d
y
)

24 EI/L
3

Axial
Stiffness

k
22

(F
z
/d
z
)

2

AE/L

Note 1: I
x

= I
y
= I = 1/12*(wt
3

+ tw
3



t
4
); A = 2wt


t
2

Note 2: Displacements, d
i
, are at the joint center



Figure
35
.
Schematic to demonstrate joint
-
replacement using a CR joint




Page
28

of
60

A motorcycle suspension example using
the CR joint, created by Cavin Daniel, is shown
later in this paper (
Figure
61
).






(a)

End
-
Moment CR Joint

(b)

Center
-
Moment CR Joint

Figure
36
.
Cross
-
Type Compliant Revolut
e Joints

(Patents Pending)


It is also possible to use the CR in an “end
-
moment” configuration, shown in
Figure
36
.
This allows for the design of serial chains of compliant joints for customized degrees
-
of
-
freedom, also discussed la
ter in this paper (
Figure
58
).


CR Joint Range of Motion

If

max

(yield strength in shear) is known for a given material (based on its yield strength),
then the last equation below can relate that value to maximum rotation of a CR joint of
known dimensions.



(for circles, Q = J/r;

J = polar

moment of inertia)



(Norton, 2000)
[10]




In practice, a fillet must be used in the CR joint to alleviate the stress concentrations and
increase the range of motio
n. Analysis of a fillet is shown in the next figure. Such a
fillet can increase the range of motion by 30%, while only causing small increases in the
torsional stiffness.

Note that the analytic equations typically
over
-
estimate the range of
motion by 10

to 15%, causing the increase in range of motion to appear
even larger

than
30%.

Motion Axis

Motion Axis

1
st

connecting link

2
nd

connecting link

ribs




Page
29

of
60


Figure
37
.
FEA of fillet used in CR joint


CR Joint Stiffness Design Charts

Many parametric studies were performed on a numerical model of the CR j
oint to create
a

catalog of graphs
that

serves as a quick and effective design tool for sizing new joints.
These charts are presented to the designer to be used in an iterative fashion when
designing joints. In a sense, the catalog is a low
-
level m
etamod
eling

effort: the stiffness
f
unctions
are
too complex to calculate analytically; too slow to calculate numerically
. It
is easier to fit a curve to the answer than to recalculate it at every step.


The
motivation for parametric studies
is more thoroughly

d
escribed
later

in the
C
ompliant Translational

Joint section
, with some demonstrations
. Nine dual
-
parameter
studies were done with the titanium CR joint, represented by 3
-
D surface plots of the
output variables. The nine studies consisted of three groups
to evaluate
:

Torsional
Stiffness (M
z
/

z
), Bending Stiffness (F
x
/d
x
)
,
and
Bending/Rotational

Stiffness
(M
y
/

y
)
.


In
each of these studies, the following three combinations of parameters were i
nspected:


Width

and

Length

Thickness

and

Length

Width

and

the
Ratio of Thickness to Width

(RTW)


Of the 9 studies, several significant ones are included in this report. As with the CT
joints, these parametric studies serve as design charts to aid in creating new joints.
Interested readers may contact
Brian Trease
or the Compliant System Design Lab

to
obtain the complete set of design charts.




Page
30

of
60


Figure
38
.
Parameterization of the CR joint and x
-
y
-
z axes for stiffness calculations


The first quantity considered reflects the desired motion of

the joint: torsional
compliance. To maximize the desired compliance, the torsional stiffness, illustrated in
Figure
39

and
Figure
40
, must be minimized. The first plot indicat
es that stiffness
decreases nonlinearly with respect to width when the RTW is constant and vice versa.



Figure
39
.
z
-
Rotational Stiffness of CR Joint (beam length = 50mm)


Figure
40

shows th
e combined effects of beam length and width on the torsional stiffness.
Beam width has only a linear effect on stiffness for a given beam length. However, beam
length nonlinearly decreases the stiffness for a given width.




Page
31

of
60


Figure
40
. z
-
Rotational Stiffness of CR Joint (thickness = 1mm)


While the first two plots suggest small widths, small thicknesses, and long beams for
minimal torsional stiffness, these conflict with the requirements for maximum off
-
axis
stiffness. This requi
res referring to
Figure
41
, which shows the rotational bending
stiffness. From the plot, it is evident that maximum bending stiffness requires shorter
beams with thicker flanges. This contradiction verifies the need

for a design tool to
balance both objectives.



Figure
41
.
y
-
Bending/Rotational Stiffness of CR Joint (width = 20mm)


Figure
42

shows the stiffness of the CR joint when it is loaded as a fix
ed
-
fixed beam with
a perpendicular force (i.e. x
-
direction) applied at its center. Increased width and reduced
length are required to increase the x
-
axis stiffness. The effect of width is nearly linear for
a given length, but the length has a nonlinear e
ffect for a constant width.




Page
32

of
60


Figure
42
.
x
-
Bending Stiffness of CR Joint (thickness = 1mm)


Open
-
C
ross CR

Joint

The open
-
cross CR joint is designed with the same principles of the original in mind,
while removing all the stress co
ncentrations completely, greatly improving the range of
motion.
It was developed by Brian Trease and Audrey Plinta.
Though it has not yet been
published, the design equations are given below.

Since the structure is now composed
essentially of only recta
ngular beams, modeling is much easier. We have also created a
useful, parametric model of this joint with ADAMS software, which you may contact us
about using.



Figure
43
.
Parametric Model of the Open
-
Cross CR Joint (cross
-
section)


Table
2
:
Analytic Stiffness Table

Torsional [N
-
mm/rad] M
x
/

x

24 EI
1
(w+g)
2
/L
3


+


8 GK/L

Bending [N/mm] F
y
/d
y
, F
z
/d
z

48 E(I
1
+I
2
)

/

L
3

Axial [N/mm] F
x
/d
x

2 AE/L

Range Of Motion [

rad]

0.577*

ys
L
2
Q

/

[2.25(EQt)
2
(w+g)
2

+ 3(KGL)
2
]
1/2

Note
:

I
1
=1/12*wt
3
;

I
2
=1/12*tw
3
;

A=4wt;

K = wt
3
/16 [16/3


3.36 t/w (1
-
t
4
/ (12w
4
))];

Q = w
2
t
2
/[3w+1.8t]

E~Young’s Modulus;

G~Shear Modulus;


ys
~Yield Strength




Thickness (t)



Width (w)



gap





Page
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60

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M
any mechanisms also use translation joints, such as sliders, rails, or linear bearings.

As
with revolute joints, there are many advantages to using compliant joint in these
situations when appropriate.

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Positioning

Mechanisms



Measurement Syste
ms



Optical Alignment



Parallel Kinematic Machines

(Stewart Platforms)



Precision Guides



Tracks



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Once again, the following figure demonstrates what is meant by joint replacement with
compliant joints. Note that the new design

on the right will have an effective output
stiffness, whether or not there was one in the original system.


Figure
44
.
Conversion from Conventional to Compliant Joints in a mechanism


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f


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Most of the existing translatio
nal joints are based on a parallel four
-
bar building block.
Their flexibility is derived from leaf springs (
Figure
46
(
a)) or notch joints (
Figure
46
(b))
.
This is schematically shown in
Figure
45
. The geometry constrains the R2 degrees
-
of
freedom, while the parallelogram shape unites the remaining degrees
-
of
-
freedom together
to create a curvilinear motion.
The compound four
-
bar joints in
Figure
46
(c) and
Figure
46
(d) deliver a larger range of straight
-
line motion. All four joints have acceptable off
-
axis stiffness, but the range of motion is very limited, even for the compound joints.




Page
34

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60


Figure
45
.
One D.O
.F. Configuration for Translation



(a)


(b)


(c)


(d)

Figure
46
.
Conversion from Conventional to Compliant Joints in a mechanism


The application of these joints as stages in high
-
precision mechanisms
is somewhat of a
field in its own. Please refer to the lecture summary by Christine Vehar on Precision
Mechanisms for more information.


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Y. Moon and S. Kota

from the
UM Compliant Systems Design Laboratory design
ed

the
Complia
nt Translational (CT) Joint as an improvement to other translational flexures.

This joint uses redundancy to attain high off
-
axis stiffness ratios and zero axis drift.




Page
35

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60


Figure
47
.
UM Compliant Translational Joint (undeflected

and deflected)


Benefits



Range of Motion increased



Having
multiple

thin beams further increases the range of motion



allows for greater displacements before local joint yielding



Over
-
constrained 5
-
Bar Design



Ensures parallelism



Less Compression in Member
s


Example
Range of Motion of ABS Plastic CT Joint

The joint on the right in
Figure
47

shows the range of motion of an example CT joint.
The parameters are listed below.


Dimensions
:
(w = 10mm, t = 0.8mm,
beam le
ngth

= 35mm)

Material
:
(ABS; E = 2480 MPa,
σ
y

= 34.5 MPa)

Results
: stiffness = 1.8N/mm, range of motion =

11.4mm, maximum load = 39N


The only reliable stiffness equation for the CT joint is the axial stiffness (in the direction
of desired compliance):



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T


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t
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i
i
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n
n


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a
a
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y
y
s
s
i
i
s
s


The range of motion of a single beam is x
b
. The joint’s range of motion, x
t
, is twice this.



;




Page
36

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60


Figure
48
.

aluminum (σ
y
/E = 414/73100 = 0.0057)


The range of
motion is a function of only three parameters: beam length, beam thickness,
and
material (σ
y
/E).
Figure
48

shows a plot of the range of motion for an aluminum CT
Joint.

The maximum load
-
carrying capability of any
joint can also be determined:




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a
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m
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t
r
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i
i
c
c


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t
u
u
d
d
i
i
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s
s


/
/


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Parametric Study

The catalog of graphs obtained from the parametric studies serves as a quick and
effective design tool for sizing new joints. If a maximum axial sti
ffness (F
x
/d
x
) or
minimum lateral stiffness (M
z
/

z
or M
y
/

y
) is specified, that value can be found on the
vertical axis of the corresponding graph. For example, when attempting to meet a given
axial stiffness, the given value corresponds to a horizontal plane cut through the graph.
Many of these plane
s are already shown in the following figures (e.g. the 200N
-
m/deg
line in
Figure
52
). Any point located below this plane indicates the feasible design space
left to meet any other design specifications.




Page
37

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60


Figure
49
.
Parameterization used in CT Joint Parametric Studies


A designer may next wish to go to the lateral stiffness graph and find the greatest lateral
stiffness that can be achieved from the subspace determined by the previous graph
. This
technique requires only a few iterations and can be used to meet stiffness requirements,
spatial limitations, and weight limitations. The slope of a graph at any point also gives
the designer an idea of what changes may be implemented to improve f
uture designs, and
by what degree.



The parameters studied include the interbeam spacing (L
2
), the length of the input/output
arms (L
1
), and the beam dimensions (width, thickness, and length). Variations of these
parameters were analyzed for their effect

o
n axial and lateral stiffness.

CT Joint Design Charts

Four studies were performed with the CT joint model, using aluminum material
properties. Three of these considered parametric effects on the moment
-
loaded lateral
stiffness (N
-
mm/degree) and are show
n

in
Figure
50

through
Figure
52
. For the designs
in
Figure
50
, the cross
-
section is held constant: width = 10mm and thickness

=

2mm. It
is noted here that the gap betwe
en the two halves of the joint (L
3
) does not effect the
moment
-
loaded lateral stiffness of the joint.




Page
38

of
60


Figure
50
.

Lateral Stiffness of CT Joint

(thickness

=

2mm, width = 10mm)


In
Fig
ure
51
, the gap is constant at 30mm, the interbeam spacing is 5mm, and the beam
width is 10mm. In
Figure
52
, the gap and interbeam spacing are held at the same values,
but the beam thickness is instead fixed at 1mm.



Fig
ure
51
.

Lateral Stiffness of CT Joint (width

=

10mm)




Page
39

of
60


Figure
52
.

Lateral Stiffness of CT Joint (thickness

=

1mm)


Due to the assumption of rigid connecting members, L
2

and L
3

have no effect on the axial
s
tiffness of the joint. This is evident by v
isual inspection of the joint.


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t
h
h
e
e
r
r


N
N
e
e
w
w


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n
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,
,


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t
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,
,


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d


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r


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s


The CR joint in
Figure
36
(b) requires a large space in the directi
on of the axis of rotation
(motion axis). While this may be acceptable for some applications, others may be limited
by different size constraints. An alternate CR joint configuration, shown in
Figure
53
,
allows for
the tradeoff of joint footprint in the xy
-
plane and joint depth in the z
-
direction.



Figure
53
.
Alternate CR Joint Conceptual Design

(Patents Pending)

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To further increase the library
of compliant joints for the design of generic mechanisms,
two CR joints are concatenated to create a compliant universal (CU) joint.

x

z

y

2
nd

connecting link

1
st

connecting link



Page
40

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60


Figure
54
.
CU Joint Conceptual Design

(Patents Pending)


The CU joint allows only two rotational

degrees of freedom, as does its traditional
mechanical counterpart. However, a Compliant Spherical (CS) joint with 3 degrees of
freedom can be built by connecting CU and CR joints as demonstrated in
Fi
gure
55
.



Fi
gure
55
.
CS Joint Conceptual Design

(Patents Pending)


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m
b
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d
d
d
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e
d
d


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g


Other work includes the integration of embedded sensors for deformation feedback,
allowing for increased precision and repeatability in the micro
-

and nanomete
r range.




Page
41

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60


Figure
56
.
Position tracking with sensors



Figure
57
.
Close
-
up of CU joint with embedded
sensors


With embedded sensors we t
ake advant
age of
continuous

deformation

of the material,
while other sensors often utilize digital encoding. The above fabrications were created by
Michael Peshkin from Northwestern University
.


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h


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a
a
b
b
l
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D
D
.
.
O
O
.
.
F
F
.
.


Compliant Joints can be used in

a different manner (other than joint replacement) to
achieve c
ustom
izable degrees
-
of
-
freedom in serial chains.

A chain like to one shown on
the left in
Figure
58
, contains two CU joints, one CT joint, and one end
-
moment CR
joint,

for a total of six degrees
-
of
-
freedom. This chain can be designed for specific use in
a parallel kinematic platform, such as the one shown in the figure on the right.



Page
42

of
60



Figure
58
.
Serial Chain of Compliant Joints; used in a p
arallel kinematic platform


This design method was developed by Yong
-
Mo Moon from The University of
Michigan. It is detailed in his dissertation and in this paper:

Moon Y.M,
Design of Compliant Parallel Kinematic Machines
,
DETC2002/MECH
-
34204


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a
a


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a
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l


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n
n


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h
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d
o
o
l
l
o
o
g
g
y
y


Again, our whole approach to using compliant joints is in joint
-
replacement. There are
two main types of design problems. The first is S
pring Replacement, when we wish to
use to stiffness in our joints to replace the stiffness of an external spring.
We are killing
two birds with one stone by removing both the pin joints and the external spring (which
required its own pin joints) from the
system.


This was described in the clarification section
before the survey portion of this paper.
The other general problem is
Minimal
S
tiffness

design, when stiffness is not intended to
be part of the final design. It is desired to keep the stiffness be
low a threshold level, at
least in the desired directions of motion.


Minimal Output Stiffness

Design



No initial spring



Try to force stiffness to “unused” (off
-
axis) directions



Must keep track of h
ow do
input force requirements change



Friction forces elimi
nated

(can eliminate noise; simplifies control scheme)




Page
43

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60

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/
/


E
E
x
x
a
a
m
m
p
p
l
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e
s
s


Motorcycle Suspension

A suspension usually consists of a 4
-
bar mechanism connected to an external spring.
(This type of application is
begging

for compliant joint replacement!
) A cartoon of a
commercial motorcycle suspension is shown in
Figure
59
. The spring and the tire are
both attached to the same link.

This project was the Master’s project of Cavin Daniel
while at the University of Michigan.


Usi
ng both statics and kinematics, the effective stiffness that the tire
feels

must first be
determined. Once this is determined
, the compliant joints of the new system are
appropriately sized to provide them same output stiffness to the tire. This joint si
zing
operation again involves a coupled statics / kinematics analysis. (Kinematics to
understand how much each joint rotates relative to the others, and statics to translate the
joint moments to the output). The links do not change during this process (t
he original
mechanism is considered sufficient).



Figure
59
.
Motorcycle Suspension (4
-
Bar Mechanism with spring)


Degrees of Freedom & Possible Constraints

At this point, there are more design degrees of freedom (i.e. design cho
ices, not kinematic
degrees
-
of
-
freedom) than constraints. Each
joint stiffness is one design choice, for a total
of 4. As is, there is only one goal: output stiffness in the direction of motion.

While this
means that there are many solutions to this pro
blem, an intelligent designer can take
advantage of these extra degrees of freedom. Additional constraints can be added to the
problem description, to either make a cheaper product or a more functional product:


Constraints

that reduce the Degrees of Free
dom

1.

Require
All Joints Same Size