Flexures
Lecture Summary
(unrefereed, unpublished)
prepared by
Brian Trease
Compliant System Design Laboratory
The University of Michigan
ME599 – Compliant Mechanisms
Winter 2004
April 30, 2004
TABLE OF CONTENTS
INTRODUCTION AND BACKGROUND ................................................................................................. 4
TRADITIONAL (CONVENTIONAL) JOINTS VS. FLEXIBLE JOINTS ................................................................... 4
BENEFITS OF COMPLIANT JOINTS ................................................................................................................ 4
SHORT HISTORY.......................................................................................................................................... 4
SOME APPLICATIONS .................................................................................................................................. 5
KINEMATICS CLASSIFICATION – WHERE DO FLEXURES FIT? ....................................................................... 5
MOTIVATION – CHALLENGES / CRITERION ...................................................................................... 6
RANGE OF MOTION ..................................................................................................................................... 6
AXIS DRIFT ................................................................................................................................................. 6
OFF-AXIS STIFFNESS .................................................................................................................................. 7
STRESS CONCENTRATION ........................................................................................................................... 7
JOINT SURVEY ........................................................................................................................................... 7
SURVEY REFERENCES ................................................................................................................................. 7
JOINT REPLACEMENT CLARIFICATION ........................................................................................................ 7
SURVEY OF ROTATIONAL JOINTS ...................................................................................................... 8
NOTCH-JOINTS .......................................................................................................................................... 10
Spreadsheet Calculations .................................................................................................................... 13
Notch Joint Comparisons: leaf hinge, circular, and elliptical ............................................................ 15
BEAM-BASED REVOLUTE JOINTS .............................................................................................................. 15
Cross-strip Pivot ................................................................................................................................. 15
Cartwheel Hinge ................................................................................................................................. 17
Angled Leaf Springs ............................................................................................................................ 17
Two-axis hinges ................................................................................................................................... 18
Passive joints ....................................................................................................................................... 19
Q-joints ................................................................................................................................................ 20
Torsional Hinges ................................................................................................................................. 21
Split Tube Joint.................................................................................................................................... 21
Disc Couplings .................................................................................................................................... 22
Rotationally Symmetric Leaf Type Hinge ............................................................................................ 23
COMPLIANT REVOLUTE JOINT .................................................................................................................. 23
CR Joint Range of Motion ................................................................................................................... 25
CR Joint Stiffness Design Charts ........................................................................................................ 26
Open-Cross CR Joint........................................................................................................................... 29
SURVEY OF TRANSLATIONAL JOINTS ............................................................................................ 29
APPLICATIONS .......................................................................................................................................... 29
JOINT REPLACEMENT CLARIFICATION ....................................................................................................... 30
LEAF SPRINGS ........................................................................................................................................... 30
COMPLIANT TRANSLATIONAL JOINT ......................................................................................................... 31
CT RANGE OF MOTION ANALYSIS ............................................................................................................ 32
PARAMETRIC STUDIES / DESIGN TOOLS .................................................................................................... 32
Parametric Study ................................................................................................................................. 32
CT Joint Design Charts ....................................................................................................................... 33
OTHER NEW DESIGNS, VARIATIONS, AND CONCEPTS .............................................................. 34
OTHER COMPLIANT JOINT CONCEPTS ....................................................................................................... 34
COMPLIANT UNIVERSAL JOINT ................................................................................................................. 35
EMBEDDED SENSING ................................................................................................................................. 36
SERIAL CHAINS WITH CUSTOMIZABLE D.O.F. .......................................................................................... 36
MORE ON A GENERAL DESIGN METHODOLOGY ........................................................................ 37
APPLICATIONS / EXAMPLES ...................................................................................................................... 37
Motorcycle Suspension ........................................................................................................................ 37
Compliant Haptic 2-DOF Joystick ...................................................................................................... 39
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POTENTIAL APPLICATIONS ................................................................................................................ 41
ORTHOTIC DEVICES .................................................................................................................................. 41
ROBOTICS ................................................................................................................................................. 41
STATICALLY BALANCED MECHANISMS .................................................................................................... 41
DOOR, TRUNK, AND HOOD HINGES .......................................................................................................... 42
POSITIONING GIMBALS ............................................................................................................................. 43
HANDMADE COMPLIANT MECHANISMS FOR DEVELOPING COUNTRIES .................................................... 43
VIBRATION ISOLATION SYSTEMS............................................................................................................... 43
BOAT DOCKING MECHANISM ................................................................................................................... 44
OPTICAL MIRRORS AND ANTENNAS.......................................................................................................... 44
OTHER IDEAS ............................................................................................................................................ 44
PREVIOUS STUDENT'S COMMENTS/SUGGESTIONS .................................................................... 44
OPEN QUESTIONS (FUTURE WORK) .......................................................................................................... 45
FINAL COMMENTS ................................................................................................................................. 45
ORGANIZATION OF REFERENCES .................................................................................................... 46
TABLE OF FIGURES ................................................................................................................................ 47
REFERENCES............................................................................................................................................ 48
APPENDIX .................................................................................................................................................... 1
Page 3 of 53
Introduction and Background
Traditional (Conventional) Joints vs. Flexible Joints
Rigid mechanical connections, such as hinges, sliders, universal joints, and ball-and-socket joints, allow
different 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 joints 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
accuracy 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 listed 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)
Benefits of Compliant Joints





Compliance is inherent in all materials
Improved Life
o Friction, Backlash, Wear
backlash
Sub-micron accuracy
Monolithic Construction
Very Compatible with Planar
Manufacturing
No Assembly Needed
No Noise
No Lubrication Required



Fewer Parts
High Precision
Repeatability



wear
friction
Conventional Joint (BAD!)
Short History
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, smalldisplacement mechanisms. These joints have also been applied by Howell and Midha [2] to develop the field of
pseudo-rigid-body compliant mechanisms.
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Leaf springs provide the most generic 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
.
x
z
(a) planar notch
(b) spherical notch joint
joint
(c) leaf-springs
Figure 1. Basic Flexible Joint Components
www.tribology-abc.com
Some Applications
–
–
–
–
–
Small Displacements
Positioning Stages
Medical Applications
MEMS Devices
PRBM Compliant
Mechanisms
– Installation Misalignment
– No Assembly Applications
– Instrumentation
Kinematics Classification – Where do flexures fit?
Rigid Body
Kinematics
Pseudo-Rigid
Body Model
Fully
Distributed
Compliant
Mechanisms
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 if 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.
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In a way, this is kind of the opposite of the CSDL’s current approach to compliant mechanism synthesis. Yet 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 formulation is created. This will not be covered in this report.
Motivation – Challenges / Criterion
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 good flexures.
360 ?
•
•
•
•
Range of Motion
Minimal Axis Drift
Off-axis Stiffness
Stress Concentration
• Size / Compactness
• Manufacturability
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 better 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 effects.
Range of Motion
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
material. 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.
Axis Drift
In addition to 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 flexures, there can be considerable deviation from the axis of straight-line
motion. For example, a simple four-bar leaf spring experiences curvilinear motion.
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Axis drift can be improved by adding symmetry to the design of a joint. However, this often increases the
stiffness of the joint in the desired direction of motion. Further, more space is required to accommodate any
symmetric joint components.
Having minimal axis drift is essential to preserving the kinematics of the original mechanism when doing
conventional joint replacement with flexures.
Off-Axis Stiffness
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 of off-axis to axial stiffness is
considered a key characteristic of an effective compliant joint.
Stress Concentration
Most notch-type joints have areas of reduced cross-section through which their primary deflection occurs.
Depending on the shape of these 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.
Joint Survey
As mentioned, 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 flexures, including design methods, material
selection, and geometry optimization, please refer to Lobontinu [4].
Survey References
Most of the joints described in the following sections can also be found in the following 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 many of the equations provided in this paper
Compliant Mechanisms, Design of Flexure Hinges, by N. Lobontinu (Book) [4]
Design and analysis of notch joints
Design of Large-Displacement Compliant Joints, by Trease (Paper) [10]
o Summary and Comparison of many flexure joints
Compliant Mechanisms, by L. Howell (Book) [2]
o Another survey of several flexures with their basic design equations
Joint Replacement Clarification
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 mechanism 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
Page 7 of 53
spring 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 stiffness of each mechanism. The spring in the original
mechanism is replaced by the stiffness in the joints in the new mechanism.
ss
tiffne
s
”
t
u
“outp
s
ffnes
t” sti
utpu
“o
Pin
Pin
?
?
Pin
?
Pin
Compliant
Joints
with torsional
stiffness
?
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. Matching 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.
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 even somtimes better, an Excel Spreadsheet, as
demonstrated in the next section for circular notch joints.
Survey of Rotational Joints
Rotational joints are more numerous and more commonly used. This section will briefly describe many
revolute flexure joints, 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-axis 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.
Page 8 of 53
Living Hinge Terminology
Existing research and literature use a variety of terms to describe bending-based flexures. These include
cantilever, notch, leaf, pivot, living 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 terms, 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. They 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 directed 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.
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)
Page 9 of 53
So, we see that terminology is mostly a function of size. As a designer, there are several issues of “scale” we
consider at different size regimes. These include:
 Method of Mechanics Analysis
 Method of Kinematics
 Stress Stiffening
 Large-deflections
Notch-Joints
Notch 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 four varieties of notch cut (Circular, Elliptical,
Rectangular, Filleted) are shown in Figure 6.
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 elliptic notch-joint. Smith’s equations approximate Paros’s
equations at the limits: eccentricity = 0 → rectangular joint; eccentricity = 1 → circular joint.
Page 10 of 53
Figure 7. Notch Joints for which Smith provides equations
Before the general 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.
ax
L
t
ax
Figure 8. Rectangular Notch Joint with stiffness and
range-of-motion equations. (z-axis is out-of-plane)
L
The off-axis stiffnesses are the same as those of a beam, and can be calculated easily from most mechanics
textbooks.
t
Circular Notch Joint and Equations
ax
D
L
t
Figure 9. Circular Notch Joint
Stiffness of Circular Notch Joint Calculations

Page 11 of 53
t
t

2a x 2 L
Range of Motion (Stress-Rotation Equation)
f ( ) 
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
Elliptical Joint Stiffness
for small beta
f ( ) 
Page 12 of 53
3
(2 )5 / 2
3
(2 )5 / 2
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 Smith 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 organized 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 13 of 53
Inputs
Material Properties
Type
Modulus of Elasticity
Yield Stress
Ultimate Stress
Description
6061-T6
Units
psi
psi
psi
10000000
39900
45000
Joint length
Joint thickness
Flexible hinge thickness
Applied moment to hinge
Desired hinge rotation angle
Units
in
in
in
in-lb
deg
0.531
0.015
0.500
0.5
2.00
Desired hinge rotation angle
Circular joint radius
rad
in
0.028226676
12446.5
0.03
0.266
Joint rotational stiffness
Joint stress due to rotation
in-lb/rad
psi
19
35644
%
%
1.10
1.25
1.8
1.0
ax
D
L
t
Joint Geometry
L
t
b
M
Theta
Outputs
Calculated Parameters
Beta
f(Beta)
Theta
ax
Joint properties
Krot
Stressrot
Margins
Factor of Safety to Yield
Factor of Safety to Ultimate
Margin of Safety to Yield
Margin of Safety to Ultimate
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 stiffness. 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 designer 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 further joint performance evaluation.
Page 14 of 53
Notch Joint Comparisons: leaf hinge, circular, and elliptical
Leaf hinge notch joint
Design for stiffness
Circular notch joint
3
t
K
ax
K
t
Elliptical notch joint
5
2
ax
K
t
5
2
ay
ax
Note that all K's are proportional to E and b
Design for stress
E: material modulus
b: out-of-plane thickness

t
ax

t
ax
N/A in the handout
Note that all 's are proportional to E, but independent of b
t: minimum in-plane dimension of the hinge
ax: half of the hinge length ay: length of minor axis in elliptical joint
For the same E, b, t, and ax, the stiffness and maximum stress comparison are as follow:
 Stiffness comparison: leaf hinge < elliptical < circular
 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 the 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 above table, the notch dimensions (t and ax) 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
typical application, kinematics requirement limits the value of ax 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 application 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 notch will be more suitable than the others.
Beam-Based Revolute Joints
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 Pivot
The cross-strip pivot is also a very old design, first described by Haringx [9] in 1949. It was designed to have a
better range of motion than notch joints. However, it suffers from considerable axis drift, calculated in the
equations below. The center of rotation moves while the joint undergoes its large deflection.
Page 15 of 53
Figure 12. Commerical “Free-Flex” Joint
Bendix Corporation
Figure 13. Cross-Strip Joint
Range of Motion and Stiffness Equations
 max 
2L
t
 max 
Et
24 L
;
KM  n
EI EI tot

L
L
;
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.
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”
Page 16 of 53
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
k M 
4 EI
R
 max 
M max t Et
 
12
R
p
R

2 2
30
Cross-strip vs. Cartwheel hinge
For two similar scale joints (L=2R)
 Cartwheel 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.
Angled Leaf 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” joint has lots of
parasitic motion (axis drift), while the “6R-1” has much less. The “6R-2” theoretically has no parasitic motion.
Figure 17. “6R-1” Angled Leaf
Spring
Figure 18. “6R-2” Angled Leaf
Spring
Figure 16. “2R” 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
Page 17 of 53
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 adjacent 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).
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Figure 19. 2 Compliant Axis Hinge from Smith
x
z
Figure 20. Spherical Notch Joint (3 D.O.F.)
Figure 21. Compliant Notch Universal Joint
Figure 22. Co-linear Notch U-Joint
Passive joints
Passive joints (Figure 23) are contact/sliding joints, described by Howell and sometimes used in PRBM-based
compliant mechanisms. They can be thought of as force-closed conventional hinge joints. They are sometimes
helpful in design, but not ideal because they utilize contact forces, which cause friction. Further, with relative
motion, it is possible that the kinematics may change, invalidating your design models. They can, however,
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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 quadrilateral-joint or Q-joint.
• 2 Types
– Parallelogram (Figure 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 (Figure 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.
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Figure 25. (a) Parallelogram Q-joint, and (b) example of its use with a compliant pantograph mechanism.
Torsional Hinges
•
What is the difference between torsional and revolute joints?
The difference is based on where the compliance is 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 this, 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 distributed 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
R1
R2
R3
3 Degrees of Freedom
Figure 26. The Effective Degrees of Freedom of an Elastic Beam
•
Closed vs. Open Shells
Not only rectangular beams can serve as torsion 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 high
off-axis stiffnesses of a closed shell. This concept has been developed by Goldfarb, described in the
next section.
Split Tube Joint
The split-tube joint was developed by Michael Goldfarb [7]. It has the off-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)
Page 21 of 53
Figure 28. Center-loaded configuration
Figure 27. Goldfarb Split-Tube Joint
This joint offers the off-axis stiffness of a solid circular tube while having a low torsional 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.
Disc Couplings
Disc couplings are 3 degree-of-freedom joints, usually used as compliant replacements to traditional 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-offreedom allows for some 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 the 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.
Figure 29. Inner-to-outer Edge Disc Coupling
Page 22 of 53
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
Compliant Revolute Joint
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 through 40
degrees. (See Haringx [9] for design tables.) Even under typical axial loading, the proposed compliant joint’s
axis of rotation drifts only nanometers.
Page 23 of 53
Motion Axis
Figure 32. UM Compliant Revolute Joint (a.k.a. Center-Moment CR, Segmented Cross CR)
Motion Axis
Cruciform Hinge
Figure 33. Primitive Design Form used to create CR joint (See Smith, p. 204)
Most of the stiffness components of the CR joint, except for the primary rotational stiffness, 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-shaped 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.
width
Figure 34. CR Joint Cross-Section parameters
Table 1. Analytic CR Joint Stiffness Formulas (closed cross)
Page 24 of 53
Torsional
Stiffness
k66 (Mz/z)
w
 4G t
  0.373
t
 3L
4
Bending /
k44 (Mx/x),
Rotational
8 EI/L
k55 (My/y)
Stiffness
Bending
k11 (Fx/dx),
24 EI/L3
Stiffness
k33 (Fy/dy)
Axial
k22 (Fz/dz)
2 AE/L
Stiffness
3
3
Note 1: Ix = Iy = I = 1/12*(wt + tw – t4); A = 2wt – t2
Note 2: Displacements, di, are at the joint center
Figure 35. Schematic to demonstrate joint-replacement using a CR joint
A motorcycle suspension example using the CR joint, created by Cavin Daniel, is shown later in this paper
(Figure 61).
1st connecting link
Motion Axis
Motion Axis
ribs
2nd connecting link
(a) End-Moment CR Joint
(b) Center-Moment CR Joint
Figure 36. Cross-Type Compliant Revolute 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 later 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.
Page 25 of 53
 max 
Q
T
Q
(for circles, Q = J/r; J = polar moment of inertia)
U 2t 2
2 w 2t 2

3U  1.8t 3w  0.9t
 max 
 max Q
k

 max
(Norton, 2000) [10]
L
2Gt (1  0.3 t )(1  0.373 t )
w
w
In practice, a fillet must be used in the CR joint to alleviate the stress concentrations and increase the range of
motion. 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 overestimate the range of motion by 10 to 15%, causing the increase in range of motion to appear even larger than
30%.
However, fillets are required
to increase range of motion
30% increase in R.O.M.
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 joint 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 metamodeling effort:
the stiffness functions 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 described later in the Compliant 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 (Mz/z), Bending Stiffness (Fx/dx), and Bending/Rotational Stiffness (My/y). In each of
these studies, the following three combinations of parameters were inspected:
Width and Length
Thickness and Length
Width and the Ratio of Thickness to Width (RTW)
Page 26 of 53
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.
Figure 38. Parameterization of the CR joint and x-y-z axes for stiffness calculations
3600-4000
3200-3600
2800-3200
2400-2800
2000-2400
1600-2000
1200-1600
800-1200
400-800
0-400
3
Width (mm)
9
15
27
4000
3600
3200
2800
2400
2000
1600
1200
800
400
0
21
Torsional Stiffness
(N-m/deg)
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 indicates that stiffness decreases nonlinearly with respect to width when the RTW is constant and vice
versa.
10
30
50
9
70 0
Percent
Ratio t/d
Figure 39. z-Rotational Stiffness of CR Joint (beam length = 50mm)
Figure 40 shows the 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 27 of 53
45
2.4-2.7
2.1-2.4
1.8-2.1
1.5-1.8
1.2-1.5
0.9-1.2
0.6-0.9
0.3-0.6
0-0.3
Width
(mm)
5
10
0
80
Length (mm)
60
40
25
20
Torsional Stiffness
(N-m/deg)
2.7
2.4
2.1
1.8
1.5
1.2
0.9
0.6
0.3
0
Figure 40. z-Rotational Stiffness of CR Joint (thickness = 1mm)
8
7
6
5
4
3
2
1
0
7-8
6-7
5-6
4-5
3-4
2-3
1-2
0-1
20
30
40
50
60
70
80
90
100
y-axis Rotational Stiffness
(kN-m/deg)
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 requires 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.
Length (mm)
9
5
1
13
17
Thickness
(mm)
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 fixed-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 effect for a
constant width.
Page 28 of 53
105-120
90-105
75-90
60-75
45-60
30-45
15-30
0-15
120
105
90
60
45
30
15
0
20
30
40
50
60
70
80
90
100
x-Axis Stiffness
(kN/mm)
75
Length (mm)
45
35
25
15
Width
5
(mm)
Figure 42. x-Bending Stiffness of CR Joint (thickness = 1mm)
Open-Cross CR Joint
The open-cross CR joint is designed with the same principles of the original in mind, while removing all the
stress concentrations 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 rectangular 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.
Thickness (t)
Width (w)
gap
Figure 43. Parametric Model of the Open-Cross CR Joint (cross-section)
Torsional [N-mm/rad] Mx/x
Bending [N/mm] Fy/dy, Fz/dz
Axial [N/mm] Fx/dx
Table 2: Analytic Stiffness Table
24 EI1(w+g)2/L3 + 8 GK/L
48 E(I1+I2) / L3
2 AE/L
Range Of Motion [rad]
0.577*ysL2Q / [2.25(EQt)2(w+g)2 + 3(KGL)2]1/2
3
3
Note: I1=1/12*wt ;
I2=1/12*tw ;
A=4wt;
K = wt3/16 [16/3 – 3.36 t/w (1-t4/ (12w4))];
Q = w2t2/[3w+1.8t]
E~Young’s Modulus; G~Shear Modulus; ys~Yield Strength
Survey of Translational Joints
Many 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.
Applications
•
Positioning Mechanisms
• Measurement Systems
Page 29 of 53
•
•
•
• Optical Alignment
Parallel Kinematic Machines (Stewart Platforms)
Precision Guides
Tracks
Joint replacement clarification
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.
Pin
?
Pin
Slider
?
?
Figure 44. Conversion from Conventional to Compliant Joints in a mechanism
Leaf Springs
Most of the existing translational 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.
R1
R2
R3
fixed ends
translation
FigureFigure
1. One45.
DoF
Configuration
for Translation
One
D.O.F. Configuration
for Translation
Page 30 of 53
(c)
(a)
(b)
(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.
Compliant Translational Joint
Y. Moon and S. Kota from the UM Compliant Systems Design Laboratory designed the Compliant
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.
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 Members
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.
Page 31 of 53
Dimensions: (w = 10mm, t = 0.8mm, beam length = 35mm)
Material: (ABS; E = 2480 MPa, σy = 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):
k axial( spatial)
Et 3 w
6 3
LB
CT Range of Motion Analysis
The range of motion of a single beam is xb. The joint’s range of motion, xt, is twice this.
1 L2  y
3 t E
xt  2 xb 
;
2 L2  y
3 t E
18-20
16-18
14-16
12-14
10-12
8-10
6-8
4-6
2-4
0-2
3
Beam Thickness (mm)
2.5
2
1.5
1
20
18
16
14
12
10
8
6
4
2
0
0.5
Joint Range of Motion
(mm)
xb 
45
35
25
15 Beam Length
5
(mm)
Figure 48. aluminum (σy/E = 414/73100 = 0.0057)
The range of motion is a function of only three parameters: beam length, beam thickness, and material (σy/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:
Fmax( spatial)  12
wt 2
y
3L
Parametric Studies / Design Tools
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 stiffness (Fx/dx) or minimum lateral stiffness (Mz/z or My/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
Page 32 of 53
planes 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.
y
x
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 future designs, and
by what degree.
The parameters studied include the interbeam spacing (L2), the length of the input/output arms (L1), and the
beam dimensions (width, thickness, and length). Variations of these parameters were analyzed for their effect
on 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 shown 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 between the two halves of the joint (L3) does not effect the
moment-loaded lateral stiffness of the joint.
Lateral Stiffness (N-m/deg)
2500
2000
1500
1000
500
0
0
10
20
30
Space between Beams (mm)
Figure 50. Lateral Stiffness of CT Joint
(thickness = 2mm, width = 10mm)
Page 33 of 53
40
600
540
480
420
360
300
240
180
120
60
0
4
5
540-600
480-540
420-480
360-420
300-360
240-300
180-240
120-180
60-120
0-60
3
Beam Length (mm)
2
90
30
40
50
60
70
80
Lateral Stiffness
(N-m/deg)
In Figure 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.
1
Beam
Thickness
(mm)
Figure 51. Lateral Stiffness of CT Joint (width = 10mm)
360
320-360
280-320
240-280
200-240
160-200
120-160
80-120
40-80
0-40
80
60
70
Beam Length (mm)
50
30
40
240
200
160
120
80
40
0
20
Lateral Stiffness
(N-m/deg)
320
280
45
35
25
15
Beam Width
5
(mm)
Figure 52. Lateral Stiffness of CT Joint (thickness = 1mm)
Due to the assumption of rigid connecting members, L2 and L3 have no effect on the axial stiffness of the joint.
This is evident by visual inspection of the joint.
Other New Designs, Variations, and Concepts
Other Compliant Joint Concepts
The CR joint in Figure 36(b) requires a large space in the direction 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.
Page 34 of 53
1st connecting link
z
x
y
2nd connecting link
Figure 53. Alternate CR Joint Conceptual Design
(Patents Pending)
Compliant Universal Joint
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.
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 Figure 55.
Figure 55. CS Joint Conceptual Design
(Patents Pending)
Page 35 of 53
Embedded Sensing
Other work includes the integration of embedded sensors for deformation feedback, allowing for increased
precision and repeatability in the micro- and nanometer range.
Figure 56. Position tracking with sensors
Figure 57. Close-up of CU joint with embedded
sensors
With embedded sensors we take advantage of continuous deformation of the material, while other sensors often
utilize digital encoding. The above fabrications were created by Michael Peshkin from Northwestern
University.
Serial Chains with Customizable D.O.F.
Compliant Joints can be used in a different manner (other than joint replacement) to achieve customizable
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 36 of 53
Figure 58. Serial Chain of Compliant Joints; used in a parallel 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
More on a General Design Methodology
Again, our whole approach to using compliant joints is in joint-replacement. There are two main types of
design problems. The first is Spring 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 Stiffness design, when stiffness is not intended to be part of the final design. It is desired to
keep the stiffness below 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 how do input force requirements change
 Friction forces eliminated (can eliminate noise; simplifies control scheme)
Applications / Examples
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
Page 37 of 53
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.
Using 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 sizing 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 (the 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 choices, 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 problem, 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 Freedom
1. Require All Joints Same Size
2. 3 Joints are a “Standard” Size, 4th is “Custom”
Constraints the increase the number of total constraints
3. Stiffness Goals in Multiple Directions
4. Stiffness Goals at Multiple Points
5. Maximize Off-axis Stiffness
6. Minimizing Size of the Joints
7. Tuning joints for desired natural frequency of the system
8. Minimize the output sensitivity to variations in any single joint
9. Non-linear load/deflection curve matching
10. Can you find any others? Go find others!
Choice number 6 was used in the design of the motorcycle suspension. Data was attained from the manufacture
for the actual stiffness-deflection relationships in commercial suspensions that provide the best
feel/performance. This curve is shown as a black dotted line in Figure 60.
Page 38 of 53
Stiffness
From Analysis of Original System
or
From Manufacturer’s Requirements
Swingarm Displacement
Figure 60. Possible Stiffness-Displacement Relationships
The other curves in the above figure show some of the curves that Cavin was able to achieve by iteratively
adjusting the 4 joint stiffnesses. After generating enough curves, it was easy to choose the curve (and
parameters) that best meet the desired performance curve. Figure 61 shows the final prototype of the new
suspension, fabricated in ABS plastic. The white arrows show the four centerlines of the CR joints.
Figure 61. Prototype of Compliant Suspension (located in CSDL)
Compliant Haptic 2-DOF Joystick
The Compliant Haptic Mechanism is an example of the other type of design: Minimal Stiffness. The original
system was a two-degree-of-freedom five-bar mechanism, with no springs attached. A haptic device is a
hardware/human user interface that provides force feedback to the human. With two degrees of freedom, the
user can move a point around in a plane, while a computer monitors the motion and can provide arbitrary forces
on the user. It is then possible to create a virtual environment. Virtual environments have many applications,
including: tele-robotics, rehabilitation, and human training.
Page 39 of 53
In addition to simplifying manufacture, part count, and lowering cost, compliant joints provide two other
important benefits in a haptic device. First, with feedback via sensors, nearly infinite precision is now possible.
Second, and more importantly, the friction from the conventional joints has now been replaced with the
torsional spring forces. Spring forces are much easier to model than friction (non-conservative, nonlinear), and
thus make it much easier to implement a controller.
This application provides many opportunities to put our compliant joint design knowledge to good use. It is
easier to write a parametric software code that easily shows the motor torques required to move the device into
any position and hold it there. Similiarly, it is very easy to evaluate the kinematics and determine what range of
motion (in 2-D space) the mechanism can move without yielding any of the joints. Both of these types of data
can easily be plotted in two-dimensions representing the x-y location of the input port.
Figure 62. Original 5-Bar Haptic Mechanism
Figure 63. Compliant 5-Bar Haptic Mechanism
Figure 64. Schematic of Compliant 5-Bar Haptic Mechanism
A schematic of the 5-bar is shown above. Two motors control the two degrees of freedom, acting on the two
links that are connected to the ground. The bottom most point is the human interface. Position sensing is
Page 40 of 53
currently done with optical encoding, but could be done with embedded sensing, as already described in this
paper.
Potential Applications
Orthotic Devices
“Using flexure hinges for prosthetic or orthotic devices
is possible, but the range of motion and required
stiffness should be considered. A device for ankle is a
good choice, since the range of motion is smaller,
compared to other joints in the limbs. Notch hinges
have been used in Ankle Foot Orthoses (AFO). They
are used in patients with reduced or no muscle activity
around their ankle (causing ‘drop foot’, i.e. instability of
an individual to lift their foot). Planar notch hinges are
good options because of the planar feature (split tubes
and CR joints have larger out-of-plane dimensions and
may not fit into patients’ shoes). The figure on the right
shows one design found in the literature, where the
notch hinge and the main brace could easily be made as
one part.”
Figure 65. Compliant Ankle-Foot
Orthosis
Robotics
“Flexure hinges can also be used in robotics applications. The undesirable friction in traditional revolute joints
can be eliminated if friction-free compliant joints are used instead. Flexures with zero-axis-drifting, such as CR
joints and CT joints, are the best candidates for this application, because the location of the end effecter depends
greatly on the kinematics of each joint. A robot arm incorporating cross-strip hinges is shown below. The
reference mentions improved stiffness in cross-strip compared to notched joints, but they did not mention offaxis drifting, which I think is an important factor. In addition, if the end effecter will be subjected to large
external load, such as handling heavy weight, the hinges might fail due to low off-axis stiffness. In this case, CR
and CT joints should be considered.”
Figure 66. Multi-D.O.F. Camera Manipulator
Figure 67. Close-up of Flexure used in
Manipulator
Statically Balanced Mechanisms
Just Herder introduced the concept of statically-balanced mechanisms. Such mechanisms use the strain energy
of springs in a novel fashion to maintain a constant potentional energy level of an entire system. Therefore, the
Page 41 of 53
system has no preferred position; all are equal. While Herder has done this with linear springs to cancel out the
force of gravity, we in the CSDL have also accomplished this using four-bar mechanisms with our compliant
revolute joints. In the device shown below, the weight of the suspended mass is statically-balance by the spring
forces in the two CR joints. The balancing holds for rotations of the mass arm from minus to plus 45 degrees.
F=mg
Figure 68. Compliant Four-Bar Statically-Balanced Gravity Compensator Mechanism
For more information, please see the project report by Brian Trease and Ercan Dede. For more information on
statically-balanced mechanisms, please see Herder’s dissertation [8] or his website:
(http://www.wbmt.tudelft.nl/mms/wilmer/herder.htm).
Door, Trunk, and Hood Hinges





Includes automobile applications and airline overhead compartments.
Custom performance could be generated, such as stiffness as a function of displacement. This could be
done to create the same feel as opening a real car trunk.
Four bar linkages are often used for hood and sometimes trunk mechanisms. Likewise, overhead
compartments in newer airplanes also employ four bars. Springs (and dampers) are used to help the user
open or close these mechanisms. The problem would be interesting in every case due to the size
constraint. We’d also have to consider the range of motion through which these mechanisms must travel.
For the range of motion that we’d need, the compliant revolute joint would seem to fit the best. The only
issue would be the out of plane space that the joint would occupy.
The hood and trunk mechanisms would require the joint to be stressed in the rest position. Also, you
would need to consider creep (if you used plastics) since the temperature could vary greatly.
The airplane overhead bin would be unstressed at the rest position. You would probably still have to use
some kind of damping element, though. Actually, I’m not sure if this mechanism is spring assisted, but I
think that it very well could be and the use of the compliant joints would be interesting.
Page 42 of 53
Positioning Gimbals
“A 2-axis split tube gimbal is one application for split tube flexures that takes advantage of their large range of
motion, zero-axis drift, and good off-axis stiffness. While similar mechanisms have been developed in the field
of MEMS, larger versions could be employed in the aerospace industry in such applications as the pointing of
measurement sensors and antennas. Figure 69 below is a solid model of the basic design concept for a 2-axis
gimbal.”
Instrument or antenna mounted to
output plate
R2
Linear actuator to create
second stage/axis rotation
R2
Base plate
- fixed to
ground
Vertical displacement
using a linear
actuator to create first
stage/axis rotation R1
R1
2 center-loaded
split tube
flexures (2X)
Figure 69: Design Concept for 2-Axis Gimbal
In this design there are two axes of rotation, R1 and R2, which are used to realize the desired motion at the
output plate. Both stages of flexures utilize a center-loaded configuration in order to achieve greater off-axis
stiffness. For the purposes of clarity the stages have been separated in order to visualize the basic design
mechanisms. However, with attention to geometry and the placement of the linear actuators it should be
possible to optimize the envelope of the mechanism such that the first and second stages of split-tube flexures
lie within the same plane. Design challenges would include assuring adequate stiffness of the overall assembly
and proper vibration isolation.
Handmade Compliant Mechanisms for Developing Countries
“I always like to focus on developing community needs where materials and tools are not as accessible. With
the simplicity of the notch joints and the living hinges, skilled carvers can turn scraps of soft material, such as
plastic and rubber, into hinges for their doors, windows, etc. When I was in India, I was impressed by the
number of skilled carvers who made Hindu god figurines out of rock; therefore, I think this idea is very
feasible.” The design challenge would be to find the right material that meet the application’s need, and then
make the parts as to only require hand tools, not machining.
Vibration isolation systems
“Another type of application that flexure joints may possibly be suitable for is vibration isolation systems. In
most cases, the amplitudes of undesired disturbances are small and the small range of motion provided by
flexure joints is sufficient. The system may consist of several flexure joints, connected together to produce
desired direction of motion. The stiffness can be predetermined and calculated based on pseudo rigid body
Page 43 of 53
models. The flexibility in the design for off-axis stiffness in flexures will provide advantages over the use of
conventional leaf or coil springs.”
Boat Docking Mechanism
“A four-bar parallelogram compliant mechanism could find new life as a protection for docked boats. In this
capacity it would allow boats to move forward and backwards slightly against the dock, but without allowing
the boats to actually rub against them and damage their hulls.”
Optical Mirrors and Antennas
“Some applications can be found in optical
mirrors and antennas. Figure 70 shows a
mirror bender to increase resolution. Two
piezo actuators located below the mirror
expand and push against the 4-bar
mechanisms (with notched hinges) on both
ends. The motion of the coupler links then
bends the center bridge, where the mirror
will be attached.”
Figure 70. Four-bar Mechanism Mirror Bender
Other Ideas






compliant bike derailleur
compliant locking mechanism for pocket knives
pump mechanisms
o monolithic construction leaves less opportunity for leakage
windshield wipers
compliant building blocks to fit into toys such as Legos, Connex, etc.
o provide engineers and even children an easy way to experiment with all of the flexure joints
discussed in this paper
car transmission controller
o in a drive-by-wire situation, the manual shift could be mounted to a compliant universal joint
Previous Student's Comments/Suggestions

“In contrast to distributed compliant mechanisms, flexure joints are locally compliant. The development
of pseudo rigid body model will provide a tool for designing complicated motions of an output point. In
this situation, the analysis of distributed compliant mechanisms is difficult because the only tool
available so far is a numerical finite element method.”

Many students point out that the shifting axis of rotation found in most flexures is an important design
consideration that the uneducated might easily neglect.

“Monolithic compliant flexible joints have an improved life because of reduced friction, backlash, wear,
and a general sense of pride just from being compliant.”
Page 44 of 53

Notch joints are purported to have moving centers of rotation, yet those who use the pseudo-rigid-body
method for compliant mechanism design treat the centers as fixed. There is still debate over this point.

Stiffness Comparisons
o It can be very difficult to compare off-axis stiffness. Torsional and linear stiffness have different
units and can’t be normalized. Comparison requires a moment arm, and would be different for
every application.

Visco Elastic Damping
o “At some point we should try to see if we can use the visco-elastic properties of certain materials
to incorporate damping into the design of our compliant joints. We may lose on stiffness, but I
think that it is worth looking into. The design problems would certainly get more complicated,
b/c we’d have to look into dynamic effects of the joints, but I think that’s a challenge that we can
handle with appropriate modeling.”

Finally, since we are using flexures to replace bearings, it would be very helpful if there existed a
catalog for designing flexures similar to those for sizing bearings.
Open Questions (Future Work)

Is it possible to create flexures with variable stiffness control?

What are the equations for filleted notch joints?

What is the range of motion for elliptic notch joints?

It seems that much of flexure design is either toward optimization of stiffness ratios or toward maximum
precision capability. Can two lists of flexures be made, ranking them best to worst in each of these
categories?
Final Comments
Now that you know what flexures are, I want you always thinking of them as a possibility. Flexures should be
the first, maybe easiest way, of generating a compliant mechanism from a conventional design. Keep an eye
out for which design criterion will be important to your application and remember to look through a catalog
(such as this paper) to find the best match. If the kinematics are already known, then the task is going to be
joint replacement.
Also, in the field of compliant mechanisms, distributed compliance is often considered the best approach. (At
least we here at UM think so.) This is not to discredit the use of compliant joints, though, as understanding
flexures in important to understanding compliance in mechanism design in general.
Imagine looking at a fully-distributed compliant mechanism (such as one of the designs synthesized here at
UM) and adding just one revolute joint to the topology… what would happen? What if that joint was
compliant? How would its stiffness affect the rest of the system? You can see how this opens the door to a
much bigger design space when creating compliant mechanism.
Page 45 of 53
Organization of References
To learn more about flexures, see these resources first.
Smith’s “Flexures” Book
 Analysis of many of the joints in this paper, including numerical error calculations and empirical data
The Paros and Weisbord Paper
 The original notch joint paper from 1965.
Howell’s “Compliant Mechanisms” Book
 Many of the flexures in this paper are covered in depth in Chapter 5.
Trease’s Paper on “Design of Large-displacement Compliant Joints”
 Comparisons of flexures and development of the Compliant Revolute and Translational Joints.
Lobontiu’s “Compliant Mechanisms” Book
 Extremely thorough analysis of all type of notch joints, including advanced topics such as buckling and
dynamics.
Chironis Mechanisms Sourcebook
 Provides many brief examples of flexures in use, including many not covered in this report.
Page 46 of 53
Table of Figures
FIGURE 1. BASIC FLEXIBLE JOINT COMPONENTS ............................................................................................ 5
FIGURE 2. FLEXURE DESIGN CRITERIA ........................................................................................................... 6
FIGURE 3. CONVERSION FROM CONVENTIONAL TO COMPLIANT JOINTS IN A MECHANISM .............................. 8
FIGURE 4: EXAMPLES OF LIVING HINGES AND SMALL-LENGTH PIVOTS (NOTCH JOINTS) .................................. 9
FIGURE 5. SOME OF THE BASIC NOTCH JOINTS ............................................................................................. 10
FIGURE 6. VARIETIES IN THE TRANSITION FROM CANTILEVER JOINTS TO CIRCULAR NOTCH JOINTS .............. 10
FIGURE 7. NOTCH JOINTS FOR WHICH SMITH PROVIDES EQUATIONS ............................................................. 11
FIGURE 8. RECTANGULAR NOTCH JOINT WITH STIFFNESS AND RANGE-OF-MOTION EQUATIONS. (Z-AXIS IS OUT-OF-PLANE)
11
FIGURE 9. CIRCULAR NOTCH JOINT .............................................................................................................. 11
FIGURE 10. ELLIPTIC NOTCH JOINT............................................................................................................... 12
FIGURE 11: SPREADSHEET ANALYSIS OF A CIRCULAR NOTCHED JOINT ........................................................ 14
FIGURE 12. COMMERICAL “FREE-FLEX” JOINT BENDIX CORPORATION ......................................................... 16
FIGURE 13. CROSS-STRIP JOINT .................................................................................................................... 16
FIGURE 14. AXIS DRIFT IN THE CROSS-STRIP PIVOT ..................................................................................... 16
FIGURE 15. CARTWHEEL HINGE.................................................................................................................... 17
FIGURE 16. “2R” ANGLED LEAF SPRING ....................................................................................................... 17
FIGURE 17. “6R-1” ANGLED LEAF SPRING .................................................................................................... 17
FIGURE 18. “6R-2” ANGLED LEAF SPRING .................................................................................................... 17
FIGURE 19. 2 COMPLIANT AXIS HINGE FROM SMITH .................................................................................... 19
FIGURE 20. SPHERICAL NOTCH JOINT (3 D.O.F.) .......................................................................................... 19
FIGURE 21. COMPLIANT NOTCH UNIVERSAL JOINT ...................................................................................... 19
FIGURE 22. CO-LINEAR NOTCH U-JOINT ....................................................................................................... 19
FIGURE 23. PASSIVE JOINT IN A COMPLIANT CRIMPING MECHANISM ............................................................ 20
FIGURE 24. EXAMPLES OF RIGID SEGMENT JOINED SOMEWHERE BESIDES THE ENDS, INCLUDING (A) SCISSORS, AND (B) A PANTOGRAPH.
............................................................................................................................................................. 20
FIGURE 25. (A) PARALLELOGRAM Q-JOINT, AND (B) EXAMPLE OF ITS USE WITH A COMPLIANT PANTOGRAPH MECHANISM.
21
FIGURE 26. THE EFFECTIVE DEGREES OF FREEDOM OF AN ELASTIC BEAM .................................................. 21
FIGURE 27. GOLDFARB SPLIT-TUBE JOINT.................................................................................................... 22
FIGURE 28. CENTER-LOADED CONFIGURATION .............................................................................................. 22
FIGURE 29. INNER-TO-OUTER EDGE DISC COUPLING .................................................................................... 22
FIGURE 30. OUTER-EDGE DISC COUPLING ................................................................................................... 23
FIGURE 31. AXIAL PLUNGE LEAF-SPRING UNIVERSAL TUBE JOINT, SMITH, P.308 ....................................... 23
FIGURE 32. UM COMPLIANT REVOLUTE JOINT (A.K.A. CENTER-MOMENT CR, SEGMENTED CROSS CR) ..... 24
FIGURE 33. PRIMITIVE DESIGN FORM USED TO CREATE CR JOINT (SEE SMITH, P. 204) ................................ 24
FIGURE 34. CR JOINT CROSS-SECTION PARAMETERS ................................................................................... 24
FIGURE 35. SCHEMATIC TO DEMONSTRATE JOINT-REPLACEMENT USING A CR JOINT ................................... 25
FIGURE 36. CROSS-TYPE COMPLIANT REVOLUTE JOINTS .............................................................................. 25
FIGURE 37. FEA OF FILLET USED IN CR JOINT .............................................................................................. 26
FIGURE 38. PARAMETERIZATION OF THE CR JOINT AND X-Y-Z AXES FOR STIFFNESS CALCULATIONS ........... 27
FIGURE 39. Z-ROTATIONAL STIFFNESS OF CR JOINT (BEAM LENGTH = 50MM) .............................................. 27
FIGURE 40. Z-ROTATIONAL STIFFNESS OF CR JOINT (THICKNESS = 1MM) ..................................................... 28
FIGURE 41. Y-BENDING/ROTATIONAL STIFFNESS OF CR JOINT (WIDTH = 20MM).......................................... 28
FIGURE 42. X-BENDING STIFFNESS OF CR JOINT (THICKNESS = 1MM) ........................................................... 29
FIGURE 43. PARAMETRIC MODEL OF THE OPEN-CROSS CR JOINT (CROSS-SECTION) .................................... 29
FIGURE 44. CONVERSION FROM CONVENTIONAL TO COMPLIANT JOINTS IN A MECHANISM .......................... 30
FIGURE 45. ONE D.O.F. CONFIGURATION FOR TRANSLATION ...................................................................... 30
FIGURE 46. CONVERSION FROM CONVENTIONAL TO COMPLIANT JOINTS IN A MECHANISM .......................... 31
FIGURE 47. UM COMPLIANT TRANSLATIONAL JOINT (UNDEFLECTED AND DEFLECTED) .............................. 31
FIGURE 48. ALUMINUM (ΣY/E = 414/73100 = 0.0057) ................................................................................... 32
FIGURE 49. PARAMETERIZATION USED IN CT JOINT PARAMETRIC STUDIES ................................................. 33
FIGURE 50. LATERAL STIFFNESS OF CT JOINT (THICKNESS = 2MM, WIDTH = 10MM).................................... 33
FIGURE 51. LATERAL STIFFNESS OF CT JOINT (WIDTH = 10MM) ................................................................... 34
FIGURE 52. LATERAL STIFFNESS OF CT JOINT (THICKNESS = 1MM) .............................................................. 34
FIGURE 53. ALTERNATE CR JOINT CONCEPTUAL DESIGN ............................................................................. 35
FIGURE 54. CU JOINT CONCEPTUAL DESIGN ................................................................................................. 35
FIGURE 55. CS JOINT CONCEPTUAL DESIGN .................................................................................................. 35
FIGURE 56. POSITION TRACKING WITH SENSORS ........................................................................................... 36
Page 47 of 53
FIGURE 57. CLOSE-UP OF CU JOINT WITH EMBEDDED SENSORS .................................................................... 36
FIGURE 58. SERIAL CHAIN OF COMPLIANT JOINTS; USED IN A PARALLEL KINEMATIC PLATFORM ................. 37
FIGURE 59. MOTORCYCLE SUSPENSION (4-BAR MECHANISM WITH SPRING) ................................................ 38
FIGURE 60. POSSIBLE STIFFNESS-DISPLACEMENT RELATIONSHIPS ............................................................... 39
FIGURE 61. PROTOTYPE OF COMPLIANT SUSPENSION (LOCATED IN CSDL) .................................................. 39
FIGURE 62. ORIGINAL 5-BAR HAPTIC MECHANISM ...................................................................................... 40
FIGURE 63. COMPLIANT 5-BAR HAPTIC MECHANISM ................................................................................... 40
FIGURE 64. SCHEMATIC OF COMPLIANT 5-BAR HAPTIC MECHANISM ........................................................... 40
FIGURE 65. COMPLIANT ANKLE-FOOT ORTHOSIS ......................................................................................... 41
FIGURE 66. MULTI-D.O.F. CAMERA MANIPULATOR .................................................................................... 41
FIGURE 67. CLOSE-UP OF FLEXURE USED IN MANIPULATOR ......................................................................... 41
FIGURE 68. COMPLIANT FOUR-BAR STATICALLY-BALANCED GRAVITY COMPENSATOR MECHANISM ........ 42
FIGURE 69: DESIGN CONCEPT FOR 2-AXIS GIMBAL ....................................................................................... 43
FIGURE 70. FOUR-BAR MECHANISM MIRROR BENDER ................................................................................. 44
REFERENCES
Paros, J.M. and Weisbord, L., 1965, “How to Design Flexure Hinges”, Machine Design, pp. 151-156
Howell, L.L., 2001, Compliant Mechanisms, John Wiley & Sons, Inc., New York, NY
Smith, S., 2000, Flexures, Elements of Elastic Mechanisms, Taylor & Francis, London, England
Lobontinu, N., 2002, Compliant Mechanisms: Design of Flexure Hinges, CRC Press, Boca Raton, FL
Kyusojin, A., and Sagawa, D., 1988, “Development of Linear and Rotary Movement Mechanism by Using Flexible
Strips,” Bulletin of Japan Society of Precision Engineering, 22(4), pp. 309-314
[6] Moon Y.M, Design of Compliant Parallel Kinematic Machines, DETC2002/MECH-34204
[7] Goldfarb, M. and Speich, J., 2000, “The Development of a Split-Tube Flexure”, Proceedings of the ASME
Dynamics and Control Div., 2, pp. 861-866
[8] Herder, J. (2001) Energy-free Systems. Theory, conception, and design of statically balanced spring mechanisms.
PhD-thesis. Delft University of Technology, Delft, The Netherlands.
[9] Haringx, J.A., 1949, “The Cross Spring Pivot as a Constructional Element,” Applied Scientific Research, A1(5-6)
[10] Y.M. Moon, B. Trease, and S. Kota, Design of Large Displacement Compliant Joints, Proceedings of DETC 2002,
27th Biannual Mechanisms and Robotics Conference, DETC2002/MECH-34207
[11] Norton, R., 2000, Machine Design, An Integrated Approach, 2nd Edition, Prentice Hall, Upper Saddle River, NJ
[1]
[2]
[3]
[4]
[5]
Page 48 of 53
Appendix
Comparison of various Revolute Flexures .....................................................1, 2
Comparison of Translational Flexures .............................................................. 3
More Comparisons (some repeated) from Trease’s paper ............................ 4, 5
Appendix 1
Appendix 2
Appendix 3
Range of
Motion
Axis Drift
Stress
Concentration
Off-Axis
Stiffness
Compactness
Benchmarked Flexible Translational Joints
(–: poor, 0: normal, +: good)
(a)
–
–
0
0
+
(b)
–
–
–
0
+
(c)
–
–
–
0
+
(d)
–
+
–
0
+
(e)
+
+
+
+
+
Appendix 4
Stress
Concentratio
n
Off-Axis
Stiffness
Compactness
z
Axis Drift
(b)
Range of
Motion
(–: poor, 0: normal, +: good)
Benchmarked Flexible
Revolute Joints
(a)
–
–
–
–
+
0
–
+
–
0
+
–
+
–
–
(d)
–
–
0
–
+
(e)
–
0
–
0
0
(f)
+
+
+
–
–
(g)
–
+
–
–
–
(h)
–
0
–
–
0
(i)
+
0
+
0
0
(j)
+
+
+
+
0
x
(c)
Appendix 5