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The Pennsylvania State University
The Graduate School
College of Engineering
COMPLIANT MEDICAL TOOL DESIGN
FOR SOFT TISSUE CUTTING
A Thesis in
Mechanical Engineering
by
Justin A. Jones
2017 Justin A. Jones
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
May 2017
ii
The thesis of Justin A. Jones was reviewed and approved* by the following:
Jason Z. Moore
Assistant Professor of Mechanical Engineering
Thesis Advisor
Reuben Kraft
Shuman Early Career Professor
Karen Thole
Distinguished Professor of Mechanical Engineering
Head of the Department of Mechanical Engineering
*Signatures are on file in the Graduate School
iii
ABSTRACT
Needles are some of the most commonly used medical devices in minimally invasive
procedures. Examples of these procedures include drug delivery, biopsies and brachytherapy
cancer treatment. The efficacy of these procedures is highly dependent on the ability for precise
placement of the needle. However, high insertion forces can lead to complications resulting in
poor needle placement and possibly a poor outcome for the patient. Ongoing concerns for
physician include the needle diverging from the target path and deformation or rotation of the soft
tissue, both the result of high insertion forces. Several studies have been carried out to reduce the
insertion force in the needle insertion procedure. These include geometry studies, the speed of
insertion, the use of vibrational insertion, and more recently the use of a novel compliant hinge to
act as a cutting mechanism, however the work with the novel compliant hinge was limited to the
exploration of the concept. This research advances knowledge in tissue cutting using a novel
compliant needle design. The properties of the compliant needle design are empirically derived
and used to determine the cutting motion of a dynamically inserted needle. By understanding how
the compliant mechanism aids in transverse cutting motion an improved needle can be achieved.
Finite element analysis is used to determine empirical compliant equations for varying
hinge geometries. Polynomial regression equations are generated to create the empirical equations
for each directional stiffness. The limitation to the empirical equations are determined to be a
function of the distance between the hinges and the thickness of the hinge. It was found that if the
distance between the hinges is less than 1.68 times the thickness than the cross hinge is a primary
flexible member. After this, its contribution to the compliance quickly diminishes. To verify the
compliant equations the models are compared to FEA results. The percent error for a distance
ratio of 1.04, 1.57, 3.15, and 4.72 was 4.7, 23, 164, and 438 percent respectively. This research
allows for the design of the compliant mechanisms to be simplified for these ranges. Outside of
iv
these ranges it is possible to use compliant equations developed by Lobontiu et al.[1] found in
Appendix A.
v
TABLE OF CONTENTS
List of Figures .......................................................................................................................... v
List of Tables ........................................................................................................................... vi
Acknowledgements .................................................................................................................. vii
Chapter 1 Introduction ............................................................................................................. 1
Motivation ........................................................................................................................ 1 Needle Cutting Background ............................................................................................. 2 Role of Compliant Mechanism ........................................................................................ 5 Research Outline .............................................................................................................. 6
Chapter 2 Flexure Hinge Design Theory ................................................................................. 7
Compliant Mechanisms .................................................................................................... 7 Pseudo-Rigid Body Method ............................................................................................. 11 Finite Element Method ..................................................................................................... 14
Chapter 3 Empirical Compliant Equations for Asymmetric Transition ................................... 16
Variable Geometric Study ................................................................................................ 17 Finite Element Analysis Setup ......................................................................................... 17 Results .............................................................................................................................. 19
FEA Empirical results for d/t study .......................................................................... 20 FEA Empirical Results for h/t study ........................................................................ 22 FEA Empirical Results for t/w Study ....................................................................... 24
Validation of FEA Based Empirical Equations................................................................ 25 Limitation of Lumped Hinge Model ................................................................................ 29
Chapter 4 Conclusions ............................................................................................................. 33
Limitations and Future Work ........................................................................................... 33
References ................................................................................................................................ 35
Appendix Selected Compliant Equations ....................................................................... 40
vi
LIST OF FIGURES
Figure 1-1. Common Complications from (a) needle deflection and (b) soft tissue
deflection. ......................................................................................................................... 2
Figure 1-2. Fracture Component Forces for Needle Insertion. ................................................ 3
Figure 1-3. (a) Novel compliant hinge design and (b) the transverse cutting motion
perpendicular to insertion direction. ................................................................................ 5
Figure 2-1. Rigid-link vise grip with its compliant mechanism counterpart [51]. ................... 7
Figure 2-2. Common Flexural Hinge Geometries for Planer Motion. ..................................... 8
Figure 2-3. (a) Flexible hinge connecting two rigid links and (b) its equivalent discrete
spring model counterpart [1] ............................................................................................ 9
Figure 2-4. (a) Circular Flexure Hinge and (b) transversely asymmetric hinge [1]. ................ 10
Figure 2-5. Transition of the (a) circular symmetric hinge to a (b) transversely
asymmetric hinge. ............................................................................................................ 11
Figure 2-6. (a) A flexible cantilever beam with end load, and (b) along with its PRBM
counterpart [51] ................................................................................................................ 12
Figure 3-1. Geometric parameters for the asymmetric hinge design. ...................................... 16
Figure 3-2. Free Body Diagram of Simulation Setup. ............................................................. 18
Figure 3-3. Example of meshing for FEA simulation of flexure hinge with rigid body A
and B. ............................................................................................................................... 19
Figure 3-4. Rotational Stiffness Empirical Model for d/t study. .............................................. 21
Figure 3-5. Rotation of the symmetric hinge into two aysmmetric hinges with short center
element. ............................................................................................................................ 22
Figure 3-6. Rotational Stiffness Empirical Model for h/t study. .............................................. 23
Figure 3-7. Rotational Stiffness Model for t/w study ............................................................... 25
Figure 3-8. Verification Simulation Set Parameters. ............................................................... 26
Figure 3-9. Rotation Angle at End of Beam for d/t of 1.04. .................................................... 27
Figure 3-10. Rotation Angle at End of Beam for d/t of 1.57. .................................................. 27
Figure 3-11. Rotation Angle at End of Beam for d/t of 3.15. .................................................. 28
Figure 3-13. AStrain Energy of compliant mechanism from FEA. ......................................... 30
vii
Figure 3-14. Analytical Strain Energy calculation to determine transition period. ................. 31
Figure 4-1. (a) Flexible hinge connecting two rigid links and (b) its equivalent discrete
spring model counterpart.................................................................................................. 34
viii
LIST OF TABLES
Table 3-1. Design Parameters for Geometric Study. .............................................................. 17
Table 3-2. Coefficients for Empirical Stiffness Equations for d/t ratio. ................................. 21
Table 3-3. Coefficients for Empirical Stiffness Equations for h/t ratio. ................................. 23
Table 3-4. Coefficients for Empirical Stiffness Equations for t/w ratio.................................. 24
Table 3-5. Needle Geometric Parameters for Verification. .................................................... 26
Table 3-6. Verification Results. Rotation from M = -50 Nmm load ........................................ 29
ix
ACKNOWLEDGEMENTS
To my Wife.
1
Chapter 1
Introduction
Motivation
Needles are among the mostly widely used medical devices. Some minimally invasive
procedures include tissue biopsy, blood sampling, drug delivery, and the cancer treatment of
interstitial brachytherapy [2]. The ability to accurately place the needle with respect to the
location of the target is paramount for the procedures to be successful. For example, a poor needle
placement during tissue biopsy can lead to a false negative diagnosis or unwanted tissue damage
and result in a poor prognosis. Another example is that of internal radiation therapy, also known
as brachytherapy treatment, a common treatment plan for many types of cancer. During
interstitial brachytherapy radioactive seeds are placed inside of the body in a precise pattern near
or inside of the tumor. It can be seen then that if the placement is poor, the outcome may not be
optimal.
One such type of cancer commonly treated with interstitial brachytherapy is that of
prostate cancer. Prostate cancer is the most frequently diagnosed cancer in men with an estimated
238,590 new cases of prostate cancer to arise in the United States in 2013 [3]. In permanent
brachytherapy 50-70 radiation seeds, roughly the size of a grain of rice, are placed by hand using
thin gauged needles into and around the tumor using ultrasound guidance [4, 5]. The accuracy of
the needle placement is typically between 2 to 3 mm inside the tissue [6]. The level of accuracy is
constrained due to the chance of edema from multiple needle repositioning. An edema may
further lead to complications such as shifting of anatomy or the seed position after the procedure
[7]. With this knowledge, physicians will commonly implant more seeds than necessary to
2
compensate [8]. This type of over seeding will lead to a larger dose of radiation and can lead to
complications and a lower quality of life for the patient [4, 9]. A needle that can improve the
accuracy during these minimally invasive procedures is needed to improve patient outcomes.
Needle Cutting Background
In many of these minimally invasive procedures, small diameter needles are chosen in
order to lessen pain and reduce recovery time [10]. However, with a thinner needle, there is an
increased chance of deflection from insertion forces [11, 12]. This type of needle deflection can
lead to inaccurate needle placement as seen in Figure1-1(a). Additionally, the soft tissue target is
free to rotate and deflect as seen in Figure1-1(b). This soft tissue deflection is hard to predict due
to the heterogeneous nature of soft tissue. Prostate tissue is especially prone to tissue deflection.
One study showed that the prostate is capable of rotations up to a maximum of 13.8° [13]. This
high degree of rotation would greatly affect the dosage of radiation in brachytherapy [14].
Figure 1-1. Common Complications from (a) needle deflection and (b) soft tissue deflection.
(a)
(b)
3
Previous studies have investigated ways to improve the needle insertion process. Some of
these methods included a predictive and correction method that predicts the deformation and
needle trajectories by modeling the needle-tissue interaction [15, 16], measuring needle insertion
forces [17-20], and the use of robotic-assisted tools [21-24]. However, these types of predictive
methods are limited in use due to the large variability found in soft tissue.
The cause of poor needle placement can be attributed to the high tissue cutting force that
acts parallel to the direction of needle insertion as seen in Figure 1-2. The reduction of high tissue
cutting force that acts parallel to the needle insertion could allow for increased accuracy which
would greatly improve the outcomes of medical procedures.
Figure 1-2. Fracture Component Forces for Needle Insertion.
Previous work has investigated how to reduce these insertion forces. One method was to
look at the effects of needle tip geometry on insertion forces. Studies have shown that by
changing the needle tip geometry the needle can cut with less force [25, 26]. There has been work
on optimizing the standard hypodermic grind angles of the three plane bevel tip [27, 28].
Additionally, a five plane bevel-tipped needle has been shown to cut with less force [29-31]
Further investigation in reducing insertion force include reducing the diameter of the
needle. Reducing the gauge of a needle from 30 gauge to 20 gauge have shown to reduce patient
pain as well as a reduction in insertion force by 20% [10]. A large body of work has been
investigating microneedles. Microneedles have sub-micron diameters and have shown to
4
significantly reduce insertion force as well as pain [32-35]. However, microneedles are limited in
that they cannot allow for larger particulates to be passed through the sub-micron diameters [36].
This would exclude them from many minimally invasive procedures including brachytherapy
seeding.
Insertion speed and vibratory insertion have been another area of research to reduce
insertion forces. A study by Heverly et al. [37] showed when the use of high insertion speed is
used there is a reduction in insertion force needed to puncture soft tissue. This has also been
demonstrated in porcine liver [38] and turkey breast tissue as well [39]. Conversely, Koelsman et
al. [40] showed an increase in insertion force of 12% when increasing the insertion speed into
skin like simulant, a tougher tissue type.
Vibrational cutting has been adopted in the medical field as well. For example harmonic
scalpels have been used to cut tissue and bone. These devices use high frequency vibration to
improve the safety, precision, and accuracy of the cutting operation [41]. Vibrational cutting with
needles has been shown to decrease insertion forces in excised animal skin [42], reduce tissue
deformation in chicken breast [43] as well as silicone gel [21]. One study by Huang et al. showed
that ultrasonic vibrations reduce needle insertion force by 28% [44]. Other work has shown a
reduction of insertion force of up to 70% with the use of microneedles and axial vibrations [45].
However, none of these studies investigates vibration using a needle designed to generate cutting
motions. It was shown by Barnet et al. [46] that cutting force is reduced when a compliant needle
is coupled with vibration. This type of design allows for transverse cutting, however, it was left to
future work to fully explore how compliance affects the cutting motion.
5
Role of Compliant Mechanism
As mentioned in traditional needle insertion the cutting force acts parallel to the insertion
direction. A novel compliant hinge initially developed by Barnet et al. [46] can be seen in Figure
1-3(a). This compliant needle design will allow for the high cutting parallel force to be redirected
to a more perpendicular direction relative to the target tissue and the insertion direction as seen in
Figure 1-3(b). This cutting motion will reduce the forces acting on the needle, thus reducing the
chance of the needle to deflect, as well as the forces pushing on the soft body target and in turn
reducing inaccurate placement of the needle with respect to the target. In all other ways the
needle geometry, grind angles and gauge, can remain the same as a traditional needle allowing for
this design to be available for a large number of applications.
Figure 1-3. (a) Novel compliant hinge design and (b) the transverse cutting motion perpendicular
to insertion direction.
Cutting force direction
d D
t
h
(a)
(b)
r
6
The use of compliance in needle design is not solely novel. There have been several
studies that have investigated the use of built in compliance to allow for steerable needle
insertions and path prediction. Chen and Chen [47] used a multi-hinged design to generate an
algorithm to predict the path direction of a beveled needle during a traditional needle insertion
procedure. Swaney et. al [48] used a flexural hinge coupled with an internal actuator to reduce
tissue damage but maintain steerability of a needle compared to the more traditional kinked
bevel-tip needle. Again, there is little research on how a compliant mechanism could be used to
generate transverse cutting motion and improve the cutting mechanics. The work presented here
investigates the compliance for several asymmetric designs and will allow for the cutting motion
to be optimized using standard mechanism design practices.
Research Outline
The purpose of this research is to study the utility of a compliant hinge coupled with
dynamic insertion to reduce cutting forces. In order to design the needle, it is necessary to know
the compliance of the hinge. In Chapter 2 background on compliant mechanisms is provided. In
Chapter 3 a geometric design study is carried out using finite element analysis to generate
empirical compliant equations for a lumped asymmetric flexure hinge. The empirical equations
are then compared to FEA results to determine the error. Further, the range of when it is
appropriate to use the empirical model vs other models is explored. Chapter 4 provides a
summary of the findings as well as limitations and potential future works.
7
Chapter 2
Flexure Hinge Design Theory
Compliant Mechanisms
The general definition of a mechanism is a mechanical device that is used to transfer
energy, force or motion [49, 50]. This can further be broken down into the more traditional rigid-
body mechanism, and compliant mechanisms. Where the rigid-body mechanism is one that is
made of rigid links and kinematic pairs, such as revolute joints, a compliant mechanism uses
flexible members to transfer input forces, energy or displacements using stored strain energy in
the flexible members of the mechanism [51].
Compliant mechanisms have gained favor recently over their rigid-body mechanisms
counterpart due to their cost reduction, increased performance, and reduction in parts, wear,
weight and maintenance [51]. For these reasons, compliant mechanisms are widely used in
microscale mechanisms, such as Micro-Electro-Mechanical Systems (MEMS). An example of
how a compliant mechanism can replace a traditional joint pair is shown in Figure 2-1.
Figure 2-1. Rigid-link vise grip with its compliant mechanism counterpart [51].
Flexible
members
Rigid body vice
grip
8
One commonly used compliant mechanism is the flexure hinge, typically used to replace
a revolute joint. Like its rigid-body counterpart it acts as a kinematic pair that restricts movement
in the translation directions, however, unlike a the revolute joint, a flexural hinge can only create
a limited angular motion about its rotational axis due to the lack of freedom to fully rotate without
failure [52]. For a flexural hinge to be effective it needs to provide increased compliance about its
favored axis, but remain relatively rigid about the remaining axes to minimize any unwanted
motion and stay within the materials yield stress [52].
Generally, flexural hinges are classified within two categories; primitive flexural hinges
and complex flexural hinges. Complex flexure hinges provide multiple degrees of freedom and
are typical for three-dimensional applications. Primitive flexural hinges, the more common of the
two, predominantly focus on two-dimensional applications [53]. A few examples of the primitive
flexural hinge, which includes the circular flexural hinge, corner filleted, and elliptical, can be
seen in Figure 2-2. The monolithic design of these flexural hinges allows for high reproducible
motion, reduced friction and reduced wear [51].
Figure 2-2. Common Flexural Hinge Geometries for Planer Motion.
Circular Hinge Elliptical Hinge
Right Fillet
Hinge
Axially Symmetric
Transversely
Symmetric
Hyperbolic
Hinge
9
For any flexural hinge designs, the defining characteristic is the compliance of the hinge,
which allows for the relationship between displacement and force to be determined. In Figure 2-3
a flexure hinge with a discrete spring model can be seen. If only two dimensional planer motion is
assumed, common in primitive flexure hinges, then this relationship is defined as:
M
F
F
C
u
u
u
y
x
y
x
][ (2.1)
where ux, uy, uϑ are the displacements in the axial, transverse and rotational direction, Fx,, Fy, Fϑ
are the applied forces in the axial, transverse or rotational direction. The compliance, C is defined
as:
zy
zy
x
MF
MyFy
Fx
CC
CC
C
C
0
0
00
(2.2)
Here, the axial, transverse, and rotational compliances are denoted by the Cx, Cy, and Cθ
respectively. It is common to report either the compliance or the stiffness of the hinge. Here the
stiffness will be used to describe the hinge where stiffness K is defined as:
1 CK (2.3)
Figure 2-3. (a) Flexible hinge connecting two rigid links and (b) its equivalent discrete spring model
counterpart [1]
Flexible Hinge
Member Rigid
Link
Rigid
Link Kx
Ky
Kϑ Rigid
Link Rigid
Link
(a)
(b)
10
There has been considerable research on analyzing and designing compliant mechanisms.
In 1965 Paros and Weisbord [54] published the first work on a compliance-based approach to
flexure hinges. They provided the compliance equations for a symmetric circular and right
circular flexure hinge. Smith et al. [53] used similar methods to provide the closed form
compliance equations for the elliptical flexure hinge. Compliant equations were further provided
by Lobontiu et al. who, using the reciprocity principle and Castigliano’s displacement theorem,
formulated closed-form equations for the symmetric corner-filleted [55], parabolic and hyperbolic
[56], and the symmetric circular [57] hinge types. Lobontiu’s compliance equations for a
symmetric circular hinge as well as the transversely asymmetric hinge, Figure 2-4, are based on
the capacity of rotation about its sensitive axis and is a function of the thickness of the hinge. For
the compliant needle design either of these equations could be used if d is zero, fully symmetric
hinge, or far enough apart, transversely asymmetric, that there would be no coupling effects
between the two hinges. As previously stated, the purpose of this study is to determine a
compliance model for this transition phase as seen in Figure 2-5.
Figure 2-4. (a) Circular Flexure Hinge and (b) transversely asymmetric hinge [1].
t
t(x) t(x)
t
x
(a) (b)
11
Figure 2-5. Transition of the (a) circular symmetric hinge to a (b) transversely asymmetric hinge.
Pseudo-Rigid Body Method
The Pseudo-Rigid-Body Method (PRBM) was developed to simplify the design and
analysis of compliant mechanisms. Previously a common method to analyze the deflection for
flexible mechanisms was to use elliptic integrals. However, this can be exceedingly difficult for
anything but simple geometries and not provide intuitive results [51]. The PRBM creates a
parametric approximation model of the deflection of flexible members using rigid-body
components that have equivalent force-deflection characteristics [51]. Additionally, the PRBM
can be used to compute the forces required for a given deflection and works for linear as well as
non-linear applications. A straight cantilever beam PRBM with a vertical end load developed by
Howell et al. [58] can be seen in Figure 2-6 and has been provided to give context to the theory.
(a)
(b)
Transition as d increases
12
Figure 2-6. (a) A flexible cantilever beam with end load, and (b) along with its PRBM counterpart
[51]
Figure 2-6 (b) shows the PRBM of a simple cantilever beam with and end load. The
PRBM consists of two rigid links, connected by a characteristic pivot point, γ. The characteristic
pivot is the fraction of the beam length where the path of the rigid beam matches closely to the
path of the compliant beam. Once the location of the characteristic pivot is determined, the
deflection path may be described in terms of the pseudo-rigid-body angle, Θ, as described by
equations (2.6) – (2.8). The distance from the end of the beam to the pivot point is known as the
EI
F
Undeflected Position
Torsional
Spring, K
y
x
y
x0
F
y0
x
γl
Θ
x0
y0
(a)
(b)
L
Characteristic
pivot, γ
ϑ0
F Fn
Ft
13
characteristic radius, γl. The angle of rotation, ϑ0, of the compliant beam can also be determined
using equation (2.9), where Cϑ is the angle coefficient equal to 1.2385 [51, 59]. The PRBM has
been shown to be able to predict the deflection of the end of the beam within 0.5% of the closed-
form elliptic integral solutions for quite large deflections by using a γ of 0.8517 allowing a
maximum Θ of 64.3 degrees[59]. A limitation of this method is that the accuracy decreases as the
relative length of the hinge increases [51].
)cos1(10 l
x (2.6)
sin0 l
y (2.7)
)1(tan
0
01
lx
y (2.8)
C0 (2.9)
The PRBM also incorporates a torsional spring located at the characteristic pivot location
that is used to model the stiffness, K, in the beam. From Figure 2.6(b) the initial force causing the
beam to bend tangential component can be described by equation 2.10.
l
KFFt
)sin( (2.10)
This transverse force creates the torque, T, at the characteristic pivot point.
lFT t (2.11)
Substituting equation (2.11) into (2.12) it is then possible to write the equation for torque as a
function of the spring stiffness and pseudo rigid body angle.
KT (2.12)
Howell presented a simplified way for calculating the consent spring stiffness, K. Using the beam
geometry, I/l, and Young’s Modulus, E, as well as the PRBM constants γ and Kϑ the spring
14
stiffness can be calculated using equation (2.13). Howell et al. [59] presented an average Kϑ of
2.61 for load angles of 11.3 ͦ < φ < 174.3 ͦ.
l
EIKK (2.13)
Finally the equation (2.13) can be used to accurately predict the forces.
2l
EIKFt
(2.14)
Finite Element Method
The use of numerical methods such as the finite element method (FEM) have been
utilized more due to the inherent difficulties of using the previously mentioned analytical
methods. Where analytical methods can typically only be used for simple geometries and require
complicated derivations, FEM allows for more general compliant mechanisms to be modeled
more efficiently. There have been several research groups that have used FEM to derive empirical
compliant equations and compared the results to known analytical solutions. Yong et al. used
FEM to create empirical stiffness equations for a circular flexural hinge where the ratio of the
thickness of the hinge with respect to the radius of the hinge was iterated through [60]. They then
compared analytically derived equations of the rotational and translational compliance to the
FEM predictions. They showed the percentage error of the empirical equations was less than 3%.
Meng et al. [61] used FEM to explore the existing stiffness equations for corner filleted flexural
hinges. They created three empirical stiffness equations for corner-filleted flexural hinges based
on finite element analysis by iterating over the thickness with respect to the length of the hinge. In
this study, they found that the empirical models were a closer approximation, 6% error, than the
15
analytical methods developed by other research groups. A similar approach is used to formulate
the empirical compliant equations for the lumped asymmetric flexure hinge design.
16
Chapter 3
Empirical Compliant Equations for Asymmetric Transition
It is necessary to design an asymmetric flexure hinge so when an axially vibration is
applied a transverse cutting motion will be induced. This type of hinge can be seen in Figure 3-1.
As previously mentioned in order to design and analyze a compliant mechanism the compliance
of the flexure hinge is required. For a simple longitudinal asymmetric hinge we could potentially
use an equation similar to the one described by Lobontiu et al. [1], however, it is necessary to
fully understand the interactions of the two hinges when they are in close proximity.
A geometric study is performed using the finite analysis method. The resulting empirical
stiffness equations are then presented. It is shown that there is a transition phase between when a
lumped model is appropriate and the more traditional compliant equations can be used. To better
determine this transition period the elastic strain energy of the mechanisms is investigated.
Figure 3-1. Geometric parameters for the asymmetric hinge design.
D
h
L
Rigid
Body B
Rigid Body A
t
d
* *
x
y
RP1 RP2
17
Variable Geometric Study
Three design parameters were chosen to explore due to their physical relationship to
beam bending and can be seen in Figure 3-1 and described in Table 3-1. The first design explored
was how the distance between the two hinges affected the compliance of the lumped hinge
mechanism. This was done by iterating the distance between the hinges such that the ratio was
0.05 ≤ d/t ≤ 8, where d is the distance between the hinges and, t is the thickness of the hinges. The
second design explored was how the thickness, t, with respect to the length, h, of the hinge
affected the compliance of the hinge mechanism. This was done by iterating the thickness and the
length of the hinge between 0.1 ≤ h/t ≤ 0.5 with a fixed d of 1/6th the total length, L. For the third
study, the effect of the thickness of the hinge, t with respect to the thickness of the rod, w was
explored. This was done by iterating the thickness of the hinge from 0.1 ≤ t/w ≤ 0.9.
Table 3-1. Design Parameters for Geometric Study.
Study d(mm) t(mm) L(mm) h(µm) w(mm) Number of iterations
d/t 0.032 – 5.08 0.635 5-10.08 90 1.27 25
h/t 0.8 0.635 5 63.5- 318 1.27 25
t/w 0.8 0.127 - 1.14 5 90 1.27 25
Finite Element Analysis Setup
ABAQUS software was used to model the hinge designs. To calculate the stiffness for
the parametric studies, the hinges were modeled as three-dimensional cantilever beams fully fixed
on the left end and free on the right end as seen in Figure 3-2. To exclude any deformation caused
by either end of the beam, both the areas left and right of the hinge were modeled as rigid bodies.
18
For the loading conditions either an end axial load, Fx, of 1N, or end moment load, Mϑ, of 1N*D
was applied at the reference point, RP1 seen in Figure 3-2. For transverse loading, Fy, a load of
1N was applied perpendicular to the beam at reference point RP2 in the negative direction. To
isolate the stiffness in the transverse direction a negative moment was applied to the non-loaded
hinge equal to:
dFM yfy (3.1)
The displacement in the axial, transverse and rotational directions were recorded from
RP2 and used to calculate the stiffness in the respective direction using equation 2.1. The material
properties chosen for the study were that of AISI 304 stainless steel with a Modulus of Elasticity
of 200 GPa, density of 8000 kg/m3 and a Poisson’s ratio of 0.29.
Figure 3-2. Free Body Diagram of Simulation Setup.
The mesh chosen for this study were quadratic hexahedral elements (C3D8R) and an
example of the mesh can be seen in Figure 3-3. Since the right and the left end of the geometry
are modeled as rigid bodies, it is not critical to have a fine mesh. However, it is critical that the
mesh for the hinge is adequate to capture an accurate approximation of the material response. A
mesh convergence study was performed to determine the optimal seeding for the geometry. It was
determined that after a mesh density of approximately 700 elements/mm3 there is no appreciable
improvement, so this seeding was chosen for all hinges design models.
Fy Fx
Mz
Mfy RP2 RP1
* *
19
Figure 3-3. Example of meshing for FEA simulation of flexure hinge with rigid body A and B.
Results
For all design studies, the recorded displacements were used to calculate the stiffness for
each design iteration. This stiffness was then plotted versus the design ratio. A polynomial
regression line was then fit to determine the function of stiffness, R2 > 0.9. It was found that the
rotational stiffness was the dominating factor of the three principle directions, as expected. The
rotational stiffness plots from each study can be seen in Figures 3-4, 3-6 and 3-7. Similar graphs
were produced for the stiffness’s in each of the remaining direction to determine the polynomial
function. The form of the empirical stiffness equations are presented in Equations 3.2(a)-(d):
n
i
in
inx a
Ew
K
0
)( (3.2a)
n
i
in
inbEw
K
0
)( (3.2b)
n
i
in
in
yc
Ew
K
0
)( (3.2c)
n
i
in
in
fyd
Ew
K
0
)(
(3.2d)
20
Where the Young’s modulus (E), and the diameter (w) of the rod are divided out in order to create
a non-dimensional function. The design ratio, α, is multiplied by its corresponding polynomial
coefficients that can be found in Tables 3-2 to - 3-4.
FEA Empirical results for d/t study
Table 3-2 and Figure 3-4 show the empirical stiffness equations for the study that
explores how the hinge stiffness changes with respect to the length between the hinges. It can be
seen that Figure 3-4 is broken into two distinct phases. In Phase A, the distance between the two
hinges is very small with respect to the hinge thickness and the loading conditions on both hinges
are nearly equivalent. The stiffness of these hinge configurations are a product of the hinge
thickness, t and the cross hinge thickness, Tc, between the two hinges as seen in Figure 3-5. As d
is increased further, the cross hinge thickness increases until the mechanism becomes two distinct
asymmetric hinges with a short center section that can be considered a short rod element. For
some distance this rod elements contribution to stiffness can be ignored. This can be seen in
Euler-Bernoulli’s beam equation for constant cross section beams:
EI
Ml (3.3)
It is then easy to see that as the length goes to zero, the rotational displacement for the rod
section will go to zero. However, the cross thickness Tc is still equivalent to the hinge thickness, t
during this distance. In this way it is possible to consider the hinges as a lumped mechanism. In
Phase B, the distance between the two hinges starts to have a more dominating effect on the
overall stiffness. Therefore, in Phase B the previous assumption is not valid and the section of the
rod in-between the hinges must be considered using a different function.
21
Table 3-2. Coefficients for Empirical Stiffness Equations for d/t ratio.
Figure 3-4. Rotational Stiffness Empirical Model for d/t study.
0.00E+00
5.00E-09
1.00E-08
1.50E-08
2.00E-08
2.50E-08
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00
Rota
tional
Sti
ffnes
s (Kϑ/E
w)
Non-Dimensional Design Parameter (d/t)
Phase BPhase A
n a b c d
7 1.032E-04 --- 6.331E-05 ---
6 -0.0031 6.139E-12 -0.002 9.829E-10
5 0.0386 -1.365E-10 0.0253 -2.098E-08
4 -0.2451 1.036E-09 -0.1679 1.450E-07
3 0.8491 -2.482E-09 0.6163 -2.099E-07
2 -1.5479 -4.694E-09 -1.2128 -1.517E-06
1 1.2451 2.590E-08 1.1034 5.376E-06
0 0.0140 -4.092E-09 -0.2597 -8.696E-07
22
Figure 3-5. Rotation of the symmetric hinge into two aysmmetric hinges with short center element.
FEA Empirical Results for h/t study
In Figure 3-6 the rotational stiffness is shown to be nearly linear, R2 of 0.96, with the
change in length of the hinge. Table 3-3 provides the coefficients for a polynomial regression
with an R2 of 0.99. As the hinge length increases the rotational stiffness reduces. However, it
should be noted that the scale of the rotational stiffness does not change as much as any of the
other studies. It has a maximum stiffness of 5.6 Nm/rad and a minimum stiffness of roughly 4.8
Nm/rad. It can be seen that the length of the hinge has little to no effect on the overall stiffness of
the lumped hinge design. The reason for this can be seen in Castigliano’s theorem for bending:
l
dxEI
QMM
0
(3.4)
It can be seen that for a constant flexural rigidity, EI, a constant moment, M, for any arbitrary
load, Q, the deflection. Δ, will not increase sizably due to the short length and the even smaller
d
Tc
t
Short Rod Element
d
Increases
23
increment. This would obviously not be the case for much larger lengths, but due to design
restrictions is outside the scope of this study due to long hinges having a potential of causing
unwanted tissue damage.
Table 3-3. Coefficients for Empirical Stiffness Equations for h/t ratio.
n a b c d
7 -776.197 --- --- ---
6 1.995E03 --- -1.147E03 ---
5 -2.053E03 -4.436E-08 2.0087E03 ---
4 1.0918E03 -2.867E-07 -1.383E03 -4.285E-05
3 -322.317 3.541E-07 475.895 4.594E-05
2 52.63 -1.258E-07 -86.1929 -1.433E-05
1 -4.4513 6.2121E-09 7.7681 -4.8345E-07
0 0.4168 2.2069E-08 -0.1837 4.0978E-06
Figure 3-6. Rotational Stiffness Empirical Model for h/t study.
1.85E-08
1.90E-08
1.95E-08
2.00E-08
2.05E-08
2.10E-08
2.15E-08
2.20E-08
0.10 0.20 0.30 0.40 0.50 0.60
Rota
tional
Sti
ffnes
s (K
ϑ/E
w)
Non-Dimensional Design Parameter (h/t)
24
FEA Empirical Results for t/w Study
Rotational stiffness is shown to greatly increase as the hinge becomes thicker as seen in
Figure 3-7. There is an increase of over 500 times as the thickness increases from 10% up to 90%
of the total diameter of the rod. Again, referring back to Castigliano’s theorem it is apparent that
the deflection is dependent on the second moment of area, I:
R
dAI 2 (3.5)
Where ρ is the distance from the origin of the axis to the differential area. It is then
readily apparent that as the cross-sectional area becomes smaller, the deflection will increase.
However, for very thin hinges the stress experienced would cause failure in the part so care must
be taken. Table 3-4 provides the coefficients for the polynomial regression fits for Equations
3.2(a)-(d). All fits had an R2 greater than 0.97.
Table 3-4. Coefficients for Empirical Stiffness Equations for t/w ratio
n a b c d
7 --- --- 3.313E03 ---
6 52.107 --- -8.649E03 ---
5 -153.631 --- 8.390E03 ---
4 173.620 -3.511E-07 -3.651E03 -6.480E-05
3 -88.528 7.241E-07 627.645 1.335E-04
2 23.241 -3.397E-07 11.9023 -6.289E-05
1 -2.899 7.741E-08 -13.8646 1.445E-05
0 0.1325 -5.397E-09 0.9536 -1.0214E-06
25
Figure 3-7. Rotational Stiffness Model for t/w study
Validation of FEA Based Empirical Equations
To validate the empirical compliant equations the end displacement and rotation angle of
several needle designs were calculated using the empirical equations. The results were then
compared to FEA results. Figure 3-8 shows the design parameters and loading condition for the
simulation. The values of parameters for each needle, N1-N4, can be found in Table 3-5. For
validation, only the distance with respect to hinge thickness is looked at. AISI 304 stainless steel
with a Modulus of Elasticity of 200 GPa, density of 8000 kg/m3 and a Poisson’s ratio of 0.29 was
used for the FEA simulation. For loading conditions an end moment, M, was incrementally
increased by 5 Nmm from 0 to 100 Nmm at the end of the needle. The left end of the needle was
fixed in all directions. The end displacement, as well as the angle of rotation at the right end of
the needle, were recorded.
0.00E+00
1.00E-08
2.00E-08
3.00E-08
4.00E-08
5.00E-08
6.00E-08
7.00E-08
8.00E-08
9.00E-08
1.00E-07
0 0.2 0.4 0.6 0.8 1
Rota
tional
Sti
ffnes
s (K
ϑ/E
w)
Non-Dimensional Design Parameter (t/w)
26
Table 3-5. Needle Geometric Parameters for Verification.
D1 (mm) D2 (mm) r (mm) t (mm) h (µm)
N1 1 0.33 0.3175 0.3175 90
N2 1 0.5 0.3175 0.3175 90
N3 1 1.0 0.3175 0.3175 90
N4 1 1.5 0.3175 0.3175 90
Figure 3-8. Verification Simulation Set Parameters.
It can be seen in Figures 3-9 to 3-12 that the lumped model starts to deviate from the FEA
results as the distance between the hinges increases as well as when the load is increased. As the
distance, D2, is increased the empirical model over predicts the stiffness of the flexure hinge.
This is to be expected, due to the added deflection of the rod section in between the hinges starts
to become the dominating contributor to the overall rotational deflection. Table 3-6 shows the end
rotation angle percent error increases from an average of 4.7% for a ratio of 1.04 to an average of
438% for the ratio of 4.72. This increasing error highlights that there is a limited range that the
* D2
D1
Mz
y
x z
r t
27
empirical compliance model can be used for. It is necessary to know when the transition from
Phase A to Phase B occurs in order to know when it is appropriate to use the empirical model.
Figure 3-9. Rotation Angle at End of Beam for d/t of 1.04.
Figure 3-10. Rotation Angle at End of Beam for d/t of 1.57.
0 20 40 60 80 100
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
End Moment (Nmm)
Angula
r R
ota
tion (
rad)
N1
FEA
Empircal
0 20 40 60 80 100
-0.2
-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
End Moment (Nmm
Angula
r R
ota
tion (
rad)
N2
FEA
Empirical
28
Figure 3-11. Rotation Angle at End of Beam for d/t of 3.15.
Figure 3-12. Rotation Angle at End of Beam for d/t of 4.72.
0 20 40 60 80 100
-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
End Moment (Nmm
Angula
r R
ota
tion (
rad)
N3
Empircal
FEA
0 20 40 60 80 100
-0.25
-0.2
-0.15
-0.1
-0.05
0
End Moment (Nmm)
Angula
r R
ota
tion (
rad)
N4
FEA
Empirical
29
Table 3-6. Verification Results. Rotation from M = -50 Nmm load
FEA – ϑ (rad) Empirical – ϑ (rad) Average Percent Error(%)
N1 -0.126703 -0.12077 4.97
N2 -0.097389 -0.10869 23.97
N3 -0.083813 -0.02878 164.6
N4 -0.121683 -0.02077 438
Limitation of Lumped Hinge Model
To determine when the transition from Phase A and Phase B occurs elastic strain energy
can be used. This allows the mechanism to be decomposed into its individual components then
the individual strain energy values can be compared to the total strain energy of the system. As
previously described, if the mechanism is considered to be a lumped model as the distance
between the centers of the hinges is increased, the new configuration can be considered a single
hinge that is rotating and increasing in thickness, Tc. At some distance, d, the lumped hinge is no
longer valid since the distance, t, will be much less than the Tc, and it will be more appropriate to
model it as two separate hinges. This is due to the fact that at some point, the solid rod sections
ability to store elastic strain energy will outpace that of the lumped hinge model and cannot be
neglected. By determining the strain energy in the total mechanism while the degree of
asymmetry is increased then comparing it to numerically determined strain energy of a single
hinge model versus two hinges and a rod model it will be possible to determine the transition.
To determine the total elastic strain energy of the mechanism an FEA simulation was
performed using ABAQUS software. To simplify the numerical calculations the hinges were
loaded under a pure end moment of 1 Nmm. As before quadratic hexahedral elements (C3D8R)
30
were used to mesh the hinge geometry. The left end of the hinge was fixed in all directions while
the moment end was applied to the right end of the hinge. As for the design iteration, the degree
of asymmetry was increased by from 0.02 ≤ d/t ≤ 10. After the simulation completed the total
strain energy was recorded and can be seen in Figure 3-13. The phases can clearly be seen, with
Phase A transitioning to Phase B approximately around d/t of two.
Figure 3-13. AStrain Energy of compliant mechanism from FEA.
For the numerical calculations, the only loading condition for this set up is a moment
about the z axis. To calculate the strain energy for both models the strain from pure bending was
used.
L
bending dsEI
MU
2
(3.6)
As for the two hinges with a rod it is numerically solved as two constant hinges with an
increasing rod length starting from 0 ≤ L < 8 mm. Then using equation 3.6 the individual strain
1.00E-04
1.00E-03
1.00E-02
1.00E-01
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Str
ain E
ner
gy (
Nm
m)
Distance Ratio (d/t)
Phase B
Phase A
31
energy was calculated. For the cross hinge design a function for the second moment of area (I)
was generated by fitting a polynomial to several calculated geometry’s and can be seen in
equation 3.7.
10-4E - 09t(x)-3E
...09t(x)-6E - 09t(x)-6E 09t(x)--3E 234
I (3.7)
Where t(x) is defined as:
x)-x(2r-rt xt (3.8)
Figure 3-14. Analytical Strain Energy calculation to determine transition period.
The elastic strain energy for both models can be seen in Figure 3-14. It can be seen that at
a distance ratio of 1.68 the strain energy of a rod model starts to quickly outpace the lumped
hinge model. This is also supported by the FEA determined total strain energy seen previously in
Figure 3-13. It stands to reason then that when the distance between the hinges is less than 1.68
the cross hinge is a primary flexible member. Within this range, the empirical compliant models
allow for a simplified way to determine the transverse motion of the needle tip. For ranges greater
than the 1.68 a rod section, seen in Figure 3-5, can no longer be ignored or modeled as a rigid
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
0.00 2.00 4.00 6.00 8.00 10.00
Str
ain E
ner
gy (
Nm
m)
Distance Ratio (d/t)
Cross Hinge
Constant Rod Element
32
element since its contribution to the overall elastic strain energy quickly outpaces that of a cross
hinge configuration. After a ratio of five, there is a difference of an order of magnitude between
the two models. At this point, it is best to model the hinge mechanism as a multilink-hinge
mechanism using previously mention compliant equations.
33
Chapter 4
Conclusions
Empirical compliance models for an asymmetric flexural hinge were presented in this
paper. It was shown that the asymmetric hinge design, modeled as a single lumped element, has a
limited scope. Since the length between the hinges will eventually become a dominating factor in
the overall stiffness. However, can greatly simplify the design of a compliant needle within a
given range. It was also shown that the cross-sectional area of the hinge, as well as the distance
between the hinges, affect the stiffness of the hinges greater than the length of the hinge. It is
recommended that if the distance between the flexure hinges is less than 1.68 times the thickness
of the hinge that the empirical compliance model can be used for the mechanism design.
However, if the distance is larger it should be modeled using non-symmetric compliance
equations, such as the ones developed by Lobontiu et al. [1], and a multilink approach.
Limitations and Future Work
This work presented an empirical compliance model for a hinge to be coupled with
ultrasonic axial vibration. Previous work demonstrated the feasibility of this type of hinge to
generate transverse cutting motion. With the current compliance models it will be possible to
optimize the needle design. This future work will have to incorporate stress considerations into
the optimization, due to the compliance model not taking into account fatigue failure from the
cyclic loading conditions. Further work should be done to improve the models prediction around
the transition between Phase A and Phase B.
34
A PRB kinematic model can be used to analyze the motion of the cutting path of the
needle tip. It can be seen in Figure 4-1 that the needle can be modeled as a double link
manipulator with two torsion springs with a stiffness Kϑ connecting two rigid links or if link 2 is
ignored a single rigid link with a torsion spring. If a single lumped mechanism is used the
stiffness of the torsion springs can be determined by the empirical equations presented in chapter
3. The driving torque, M(t), can be applied to point 1 and should be a sinusoidal function of time
to mimic the loading conditions of the vibratory needle.
Figure 4-1. (a) Flexible hinge connecting two rigid links and (b) its equivalent discrete spring model
counterpart.
Y
X x
y
y2 x2
x1
y1
θ1
θ2
k1
k2
L2
L1
M(t)
35
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40
Appendix
Selected Compliant Equations
Lobontui et al. compliance equations
In-plane compliance for symmetric right-circular flexure hinge:
2
41arctan
4
221
t
r
trt
tr
EwFxxC
t
r
trt
trtrtr
trt
trttrtrrtrttr
trt
trtrrtr
trEwFyyC
41arctan
4
4628
4
214238248042
4
52844822
24
3
55
224
55
432234
22
223
t
rtrttrr
trtrtrt
trtrEwt
r
MzyC 4
1arctan426
464
42
242
22
33
2
r
MzyC
MzC
41
Lobontui et al. compliance equations
In-plane compliance for nonsymmetric circular flexure hinge:
1
22arctan
2
221
trt
tr
trt
r
trt
tr
EwFxxC
22arctan
2
2432
2
312 223
22
3
trt
r
trt
trtrtr
tr
r
trt
trr
EwFyyC
22
arctan3
2243
)2(
122
22
55
2
trt
rtrr
trttrtr
trttrEw
r
MzyC
r
MzyC
MzC
