design and analysis of an oscillating mechanism for
TRANSCRIPT
Design and Analysis of an OscillatingMechanism for Applications in a Bone Saw
by
Sara DeVore
A Thesis
Submitted to the Faculty
of
TUFTS UNIVERSITY
in partial fulfillment of the requirements for the
Degree of Master of Science
in
Mechanical Engineering
February 2015
APPROVED BY:
Dr. Thomas P. James, Advisor
Dr. Anil Saigal, Committee Member
Dr. Dan Hannon, Committee Member
Dr. Eric Smith, Tufts Medical Center, Outside Committee Member
Abstract
An oscillating bone saw is the primary surgical power tool used for
resection of bone during total joint replacement of the knee and hip.
For several decades, a pistol-grip configuration of the saw has prevailed,
where the motor is positioned orthogonal to the oscillating mechanism.
This design evolved as a compromise between using the power head
as either a drill or a saw, thus eliminating the need for a second tool
during surgery. However, after direct observation of bone sawing and
through feedback from orthopedic surgeons, it became apparent that
the preferred ergonomics for drilling were significantly different from
sawing. Furthermore, studies of similar oscillating saws used in wood-
working applications revealed that in contemporary designs the motor
and mechanism are aligned along the same axis, creating a body-grip
configuration, rather than a pistol-grip. The aim of this thesis is to
design an in-line oscillating mechanism to accommodate a body-grip
design for improved ergonomic handling of a surgical bone saw.
A computer model of an in-line mechanism is first developed by re-
verse engineering a commercially available woodworking tool. Kinemat-
ics of the computer generated solid model are validated by two methods:
(1) a motion study is performed where blade velocity as a function of
motor speed is recorded with a laser vibrometer, and (2) an analytical
model is developed by using a vector loop method. Components of the
virtual assembly are then refined and mass properties are added such
that a motion study with the computer model generates results that
compare reasonably well with both vector kinematics and experimental
measurements.
After validating the computer model for the existing mechanism, the
design is modified to accommodate specifications for an in-line surgical
bone saw. The updated model is then used to perform a computational
ii
study of the mechanism kinetics to determine resultant forces on the
oscillating components and related bearings. Simulations are run for a
common surgical blade oscillating through a 5◦ angle at 10,000 cycles
per minute, which is representative of a typical surgical bone saw. To
reduce peak loads and to minimize tool vibration, the computational
model is used to counterbalance the new in-line oscillating mechanism.
At full speed prior to counterbalancing, oscillating components generate
a peak load that varies between 9 N and 91 N at the front motor shaft
bearing. While it is not possible to completely balance an oscillating
load with a rotating mass, the peak and alternating load is reduced by
adding an offset mass to the motor shaft, resulting in a more balanced
load that varies between 48 N and 53 N.
A natural frequency analysis is performed to confirm that the op-
erating frequency of the saw does not excite any natural frequencies of
the mechanism. Forces generated by the counterbalanced mechanism
are used to size the bearings, achieving an end of life criterion that
exceeded 8000 hours. In addition, the peak force is used to conduct a
linear elastic finite element analysis on the cam fork, which is the pri-
mary component responsible for converting rotary motion of the motor
cam into oscillating motion of the blade shaft. Considering a ductile
yield criterion, the fork is designed such that the maximum stress results
in a factor of safety of at least 2.0. Mechanism grease is specified to
lubricate contact points. Finally, a push button mechanism is designed
to provide a means of quickly changing blades without secondary tools
while wearing surgical gloves. Using a Goodman failure criterion, the
push button spring is designed to have infinite life. Following mecha-
nism design and analysis, a gear case is designed and a demonstration
prototype of the new in-line system is developed.
iii
Acknowledgments
Completing this thesis part-time has been a long road that would
not have been possible without the support of many wonderful people in
my life. Thank you to my friends in Boston, especially Jim, for taking
care of me when work, school, and life became too much for me to
juggle alone. Thank you to my family for motivating me. To my Mom,
Nancy, for talking to me for hours on end no matter if I was happy,
excited, or frustrated. You are my support in everything I do. To my
brother, Thomas, for encouraging me. To my grandparents, Carolyn,
Don, Eileen, and Harold, for teaching me the value of education and for
always supporting me.
More than anyone else, I would like to thank my Dad, Mike. You
inspired me at every stage of this journey by watching you work on
your Ph.D. part-time. You have always pushed me to do better and be
better by educating myself. Even though I am getting my Masters, I
can still call you for questions on my homework and I know you will
always help me find the answer. I am one lucky girl to have you behind
me.
Thank you to my committee members Dr. Saigal, Dr. Hannon, and
Dr. Smith. I have truly enjoyed working with you. Thank you to Dr.
White for his assistance with the experimental work. Thank you to my
colleagues at Gillette for their help and guidance, specifically Mike Cav-
alear, Devan Spellman, and Brian Guerette. I would like to thank Dr.
James, my advisor, from the bottom of my heart. I truly appreciated
all of your help and advice. Your quick responses and willingness to
work with my difficult schedule made this possible.
iv
Contents
1 Introduction 2
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Survey of the Literature 8
3 Research Objectives 15
4 Methods and Approach 16
5 Model Creation and Validation 20
5.1 CAD Model Creation . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 CAD Model Motion Data . . . . . . . . . . . . . . . . . . . . 24
5.3 Analytical MATLAB Model Creation . . . . . . . . . . . . . . 26
5.4 Comparison of Analytical and Numerical Models . . . . . . . . 32
5.5 Experiment to Validate Models . . . . . . . . . . . . . . . . . 34
5.5.1 Microphone Experiments . . . . . . . . . . . . . . . . . 34
5.5.2 Strobe Light Experiments . . . . . . . . . . . . . . . . 36
5.5.3 Laser Vibrometer Experiments . . . . . . . . . . . . . . 37
5.6 Comparison of Experimental Data to Models . . . . . . . . . . 42
6 Mechanism Selection 43
7 Saw Blade and Motor 45
8 Counterbalancing 46
8.1 Top Shaft Counterbalance . . . . . . . . . . . . . . . . . . . . 47
8.1.1 Other Stryker Blades . . . . . . . . . . . . . . . . . . . 50
8.2 Motor Shaft Counterbalance . . . . . . . . . . . . . . . . . . . 53
v
9 Finite Element Analysis of Fork 59
10 Natural Frequency 65
11 Bearings 69
12 Blade Clamp Design 76
13 Gear Case Design 81
14 Prototype 89
15 Conclusion 91
Appendices 95
A References 95
B MATLAB Codes 99
B.1 Vector Loop Analysis of Bosch Saw . . . . . . . . . . . . . . . 99
B.2 Top Shaft Forces of 3 Different Blades . . . . . . . . . . . . . 101
B.3 Basic Motor Shaft Force Calculation . . . . . . . . . . . . . . 103
B.4 Motor Shaft Counterbalance Angular Position . . . . . . . . . 106
B.5 Motor Shaft Counterbalance Mass Determination . . . . . . . 109
vi
List of Tables
1 Parameters used to define CAD motion analysis . . . . . . . . 24
2 Microphone and strobe frequency data for each blade speed set-
ting on the saw . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Comparison of average blade frequency from strobe and micro-
phone experiments prior to modifying the blade vs the laser
vibrometer frequency after modifying the blade . . . . . . . . 42
4 Needs analysis of mechanism options for converting rotary to
oscillating motion in the saw, rated on a scale of 1-5 where 1 is
poor and 5 is ideal. . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Parameter changes made to the saw model to align it with a
bone saw design . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Parameters defining the oscillation angle for the Bosch saw and
the new saw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7 Dimensions of three Stryker Dual Cut blades, images of the
blades are shown in Figure 43 with corresponding letter labels. 51
8 Location of center of mass and the mass of the assembly with
each blade; center of mass is shown in Figure 44. . . . . . . . . 52
9 Material properties for 4340 steel used in fork FEA model . . 60
10 Applied forces on the system. . . . . . . . . . . . . . . . . . . 73
11 Calculated forces on the bearings, where P refers to Equations
20 and 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12 Bearing life parameters and calculated life for each bearing po-
sition (see Figure 64). . . . . . . . . . . . . . . . . . . . . . . . 75
13 Bearing dimensions for chosen bearing in each position. . . . . 75
14 SKF suggested fits for shafts and housings for bearings selected. 76
vii
15 Spring parameters for clamp spring C0240-024-0620 from Asso-
ciated Spring Raymond. . . . . . . . . . . . . . . . . . . . . . 78
16 Spring positions and forces for clamp spring. . . . . . . . . . . 79
17 American Ring and Tool retaining ring 5001-086 dimensions;
see sketch in Figure 76. . . . . . . . . . . . . . . . . . . . . . . 86
18 Typical physical characteristics of Shell Alvania EP2 Lithium
Grease. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
19 Materials used to fabricate prototype components. . . . . . . . 89
viii
List of Figures
1 Pistol grip saws currently on the market (Solomon et al. 2014) 3
2 Surgeon hand posture issues while using a pistol-grip saw (Solomon
et al. 2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Stryker System 7 Saw . . . . . . . . . . . . . . . . . . . . . . . 4
4 Alternative in-line configurations for sagittal saws (Solomon et
al. 2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5 Milwaukee M18 cordless multi-tool in use . . . . . . . . . . . . 5
6 Stryker blade, Dual Cut 4125-127-100 . . . . . . . . . . . . . . 5
7 Bosch saw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
8 Partially disassembled Bosch saw . . . . . . . . . . . . . . . . 7
9 Patent showing the counterbalancing elements on the drive shaft,
labeled as 246 and 248. . . . . . . . . . . . . . . . . . . . . . . 11
10 Crank-rocker mechanism . . . . . . . . . . . . . . . . . . . . . 12
11 Slider-crank mechanism . . . . . . . . . . . . . . . . . . . . . 12
12 Scotch-yoke mechanism . . . . . . . . . . . . . . . . . . . . . . 13
13 Rotary-to-liner mechanism . . . . . . . . . . . . . . . . . . . . 13
14 Overlay of eccentric bearing on a slider-crank mechanism . . . 14
15 Brown’s mutilated pinion mechanism . . . . . . . . . . . . . . 14
16 Exploded and un-exploded views of the typical drive train com-
ponents of an oscillating saw: top shaft, aluminum gear case,
and motor shaft. . . . . . . . . . . . . . . . . . . . . . . . . . 22
17 Top shaft of the saw showing the blade mount and the fork that
drives the oscillation of the blade. . . . . . . . . . . . . . . . . 22
18 Motor shaft of the saw showing the eccentric bearing, bearing
spacer, and motor. . . . . . . . . . . . . . . . . . . . . . . . . 22
ix
19 NX model of the drive train assembly showing the global co-
ordinate system used in dynamic analysis, the point of interest
(POI), and the drive train components. . . . . . . . . . . . . . 23
20 Simulated displacement data for the point of interest (POI) on
the blade (see Figure 19) for constant oscillation frequency of
167 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
21 Simulated velocity data for the POI on the blade for constant
oscillation frequency of 167 Hz. . . . . . . . . . . . . . . . . . 25
22 Simulated acceleration data for the POI on the blade for con-
stant oscillation frequency of 167 Hz. . . . . . . . . . . . . . . 26
23 Overlay of eccentric bearing and modified slider-crank mecha-
nism where r2 is the input link and is equal to the eccentric
offset, r3 is equal to the bearing radius, r5 is always vertical but
changes length, and r4 is always horizontal but changes length
defining the fork offset and output of the model. . . . . . . . . 28
24 Dimensions of the top sub-mechanism for the analytical model
where p is the distance from the axis of rotation to the point
of interest, h is the distance from the axis of rotation to the
contact point of the fork and the bearing, and γ is the angle of
blade rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
25 Comparison of angular position for NX and MATLAB models
measured at the POI for a constant oscillation frequency of 167
Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
26 Comparison of angular velocity for NX and MATLAB models
measured at the POI for a constant oscillation frequency of 167
Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
x
27 Comparison of angular acceleration for NX and MATLAB mod-
els measured at the POI for a constant oscillation frequency of
167 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
28 Setup for experimentally determining the oscillation frequency
of the blade using a microphone. . . . . . . . . . . . . . . . . . 35
29 Microphone experiment data showing the sound frequency peaks
for each speed dial setting, corresponding to blade frequency of
oscillation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
30 Strobe light used to determine the frequency of oscillation of
the saw blade. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
31 Blade modified by adding a nut and bolt with the bolt surface
parallel to the side of the blade to provide a target surface for
the laser vibrometer. . . . . . . . . . . . . . . . . . . . . . . . 38
32 Laser aligned perpendicular to the nut on the blade: notice the
visible laser dot on the nut. . . . . . . . . . . . . . . . . . . . 39
33 Oscilloscope plot of blade speed in the X direction showing volt-
age vs time for speed setting 1, where 1 V = 1000 mm/s. . . . 40
34 Oscilloscope plot demonstrating uneven curves for blade speed
in X direction on speed setting 5, where 1V = 1000 mm/s. . . 41
35 Fatigued and fractured saw blade due to added weight of nut
and bolt after running on speed setting 6. . . . . . . . . . . . 41
36 Comparison of experimental and predicted maximum velocities
of the point of interest (POI) for speed settings 1,2, and 3 . . 43
37 (a) Comparison of Bosch OSC118F blade, and (b) Stryker Dual
Cut 4125-127-100 blade. . . . . . . . . . . . . . . . . . . . . . 45
38 Stryker motor to be used in new saw design. . . . . . . . . . . 46
xi
39 CAD model of the top shaft assembly used to balance the fork
and blade as they oscillate together around the Z-axis. . . . . 47
40 Top shaft assembly counterbalanced solution. . . . . . . . . . 49
41 Top shaft assembly center of mass, indicated by cross-hairs, for
(a) Bosch fork and Bosch blade, (b) Bosch fork and Stryker
blade, and (c) new fork and Stryker blade. . . . . . . . . . . . 49
42 SLA prototype of fork from various angles. . . . . . . . . . . . 50
43 Images of the three Stryker Dual Cut blades: (a) balanced 4125-
127-100 blade, (b) 4111-147-075, and (c) 4125-064-075. . . . . 51
44 The center of mass of the top shaft assembly is denoted by cross-
hairs for the (a) 4125-127-100 balanced blade, (b) 4111-147-075
blade, and (c) 4125-064-075 blade. . . . . . . . . . . . . . . . . 52
45 Comparison of vibration forces on the top shaft assembly with
the (a) 4125-127-100 balanced blade, (b) 4111-147-075 blade,
and (c) 4125-064-075 blade. . . . . . . . . . . . . . . . . . . . 53
46 Top view free body diagram sketch of motor shaft showing co-
ordinate system, the centrifugal reaction force of the eccentric
bearing, Fc, and the reaction force on the shaft to accelerate
the fork, Ff , with the shaft rotating at an angular velocity, ω. 54
47 Example of Bosch saw motor shaft counterbalance. . . . . . . 55
48 Side view of CAD model of motor shaft counterbalance solution
with and without eccentric bearing shown. . . . . . . . . . . . 57
49 Top view of CAD model of motor shaft counterbalance solution
with and without eccentric bearing shown. . . . . . . . . . . . 57
50 Force on motor shaft with and without counterbalance. . . . . 58
51 X and Z components of the forces on motor shaft with and
without counterbalance. . . . . . . . . . . . . . . . . . . . . . 58
xii
52 Plot of the angle of the resultant force as the motor spins; the
horizontal axis represents the position of the eccentric bearing
as the motor rotates. . . . . . . . . . . . . . . . . . . . . . . . 59
53 Fork with mesh, constraint, and applied load for FEA simulation. 61
54 Displacement magnitude result of FEA simulation showing max-
imum deflection is 0.0079 mm at the end of the leg where the
force was applied. . . . . . . . . . . . . . . . . . . . . . . . . . 62
55 Von Mises stress result of FEA simulation showing maximum
stress is 21.94 x 103 kPa at the hole for the shaft and the inside
corner of the leg. . . . . . . . . . . . . . . . . . . . . . . . . . 63
56 Maximum deflection of the fork for varying mesh sizes used in
the FEA simulation. . . . . . . . . . . . . . . . . . . . . . . . 63
57 Boundary conditions for FEA of 2 mm wide applied load on the
fork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
58 Deflection of fork under 100 N load applied over a 2 mm wide
surface, showing the max deflection of 0.0086 mm. . . . . . . . 64
59 Boundary conditions on the top shaft assembly for the natural
frequency analysis. . . . . . . . . . . . . . . . . . . . . . . . . 66
60 Top shaft assembly shown with 10 node quadratic tetrahedron
mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
61 The first natural frequency at 96.235 Hz. . . . . . . . . . . . . 67
62 The second natural frequency at 575.92 Hz. . . . . . . . . . . 68
63 The third natural frequency at 780.94 Hz. . . . . . . . . . . . 68
64 Mechanism shown with all bearings that must be sized. . . . . 71
65 Free body diagram of top shaft assembly for bearing force anal-
ysis; FC is the applied cutting force, FW is the weight of the
assembly, and FRA and FRB are the bearing reaction forces. . . 72
xiii
66 Free body diagram of motor shaft assembly for bearing force
analysis; Fapp is the applied force from the fork and counterbal-
ance and FRC and FRD are the bearing reaction forces. . . . . 72
67 Cutting force, FC , is a combination of the applied thrust force,
FT and the sawing force, FS due to the blade teeth cutting bone. 73
68 Blade clamp mechanism side view (i) and cross-section view (ii),
showing the push-button (a), clamp spring (b), spring pin (c),
main shaft (d), blade (e), clamp (f), and clamp shaft (g). . . . 77
69 Blade clamp mechanism opened to release the blade by pressing
the push-button. . . . . . . . . . . . . . . . . . . . . . . . . . 78
70 Blade clamp locates blade side to side with a slot in the blade
receiver on the main shaft. . . . . . . . . . . . . . . . . . . . . 81
71 The blade is located within the clamp by the clamp shaft. . . 82
72 Outside of the gear case design to support the saw mechanism. 83
73 Inside of the gear case design to support the saw mechanism. . 83
74 Side view of the main shaft, showing the gear case shoulder (a)
and retaining ring (b) to capture the shaft axially. . . . . . . . 84
75 Angled views of the gear case shoulder (a) and retaining ring
(b) to capture the shaft axially. . . . . . . . . . . . . . . . . . 85
76 American Ring and Tool retaining ring 5001-086 catalog sketch;
dimensions shown in Table 17. . . . . . . . . . . . . . . . . . . 85
77 O-ring providing sealing around the push button. . . . . . . . 88
78 O-ring pocket in gear case. . . . . . . . . . . . . . . . . . . . . 88
79 New mechanism prototype shown with the gear case cover off
and on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
xiv
Design and Analysis of an Oscillating
Mechanism for Applications in a Bone
Saw
1
1 Introduction
Total Joint Arthroplasty (TJA), commonly known as joint replacement surgery,
is a surgical procedure to eliminate joint pain, typically resulting from os-
teoarthritis. The prevalence of TJA is increasing in the United States and
other developed countries. Aging populations and improved orthopedic im-
plant device technology are contributing to the high growth rate. For exam-
ple, Kurtz et al. (2007) projects that by the year 2030, demand for primary
Total Knee Arthroplasty (TKA) will increase 673% to 3.48 million procedures
annually.
In addition to a growing number of primary joint replacement surgeries,
revision surgeries are also on the rise due to implant wear and failure. Ac-
cording to the NIH, the current rate for revision knee arthroplasty is 10% at
ten years after the original TKA and 20% at twenty years post-implantation
(Rankin et al. 2003). Typical causes of knee revisions are inaccurate fitting
or alignment of the joint replacement during the original surgery, infection,
aseptic loosening, and loss of bone density and integrity at the implant site.
Even if the success rate of primary TKA surgeries improves, the number of re-
vision procedures will likely continue to increase. This is due to younger TKA
patients, increased life expectancy, and higher activity levels among patients,
leading to the expectation that orthopedic implants will remain in the body
for longer periods of time and under more strenuous motion (Rankin et al.
2003).
An oscillating bone saw, or sagittal saw, is the primary cutting tool used in
both primary and revision TKA. Contemporary sagittal bone saws, shown in
Figure 1, typically have a pistol-grip form. Unfortunately, a pistol grip design
is not ideal when considering the ergonomics of handling the saw during knee
surgery. For example, consider images from a recent knee replacement surgery
2
Figure 1: Pistol grip saws currently on the market (Solomon et al. 2014)
at Tufts Medical Center, Boston, MA, Figure 2. Here, the surgeon can be seen
using the saw upside down and in several awkward postures during resection of
the tibia. Addressing the strain and pressure issues due to awkward handling
of the saw would increase surgeon satisfaction and blade control, leading to
more accurate cuts and improved surgical outcomes.
An alternative to the pistol-grip design is an in-line saw, which is er-
gonomically preferable (Solomon et al. 2014). In order to enable this new,
more ergonomic, sagittal bone saw design, the power train must be redesigned.
The aim of the current research is to design a new oscillating mechanism to
facilitate an in-line motor and handle configuration, Figure 4.
Figure 2: Surgeon hand posture issues while using a pistol-grip saw (Solomonet al. 2014)
3
1.1 Background
The currently available bone saw that is the starting point for the redesign is
the Stryker System 7 saw (Stryker, Kalamazoo, MI). Like all other common
sagittal bone saws, it has a pistol-grip form, as shown in Figure 3. The al-
ternative configurations being proposed for in-line sagittal saws are shown in
Figure 4 (Solomon et al. 2014). The in-line configuration is commonly found
in oscillating saws sold for home improvement projects, such as the Milwaukee
M18 cordless multi-tool (Milwaukee Electric Tool, Brookfield, WI), which is
shown being used in Figure 5.
Figure 3: Stryker System 7 Saw
Figure 4: Alternative in-line configurations for sagittal saws (Solomon et al.2014)
The Stryker saw oscillates the saw blade at 10,000 cycles per minute on the
“low” setting or at 12,000 cycles per minute on the “high” setting. The blade
oscillates through a 5◦ arc (Conversation with Stryker Technical Department
4
Figure 5: Milwaukee M18 cordless multi-tool in use
June 2011). These specifications are used for the redesigned in-line saw for
vibration studies and subsequent counterbalancing.
The blades commonly used in the Stryker saw for knee revisions have a
cutting depth of 80-100 mm. These are the longest blades available and are
used in order to reach through the surgical guides and still have enough length
to cut through the bones (Stryker Blade Catalog). This same blade, shown in
Figure 6, is also used in the redesigned saw.
Figure 6: Stryker blade, Dual Cut 4125-127-100
The first step in a redesign of the oscillating blade mechanism is to under-
stand current designs. This is accomplished by disassembling and 3D modeling
an existing in-line saw. The Bosch MX25E (Bosch, Munich, Germany) saw,
shown in Figure 7, is disassembled and examined to understand the internal
mechanism configuration, which works on the same principle as the surgical
saws.
Despite the Stryker saw being designed for bone and the Bosch saw being
5
designed for home-improvement use, the two can be confidently compared for
a number of reasons. The Bosch saw can oscillate over a range of frequencies
of 8,000 - 20,000 oscillations per minute, which covers the speed of the Stryker
saw, and oscillates through a similar blade angle at 4◦. The Bosch saw is an
in-line configuration so it is attractive for comparative purposes.
Figure 7: Bosch saw
In order to attach the Stryker blade to the new in-line sagittal saw, a
keyless blade clamp is designed as part of this thesis. It will allow surgeons
or other operating room personnel to change saw blades quickly and without
the use of a separate tool. The design of the outer housing for the in-line
configuration is outside the scope of this thesis. An internal support structure
is also designed to accommodate and support the mechanism design inside the
ergonomic shell. There are also a number of internal saw components that
are examined. A partially disassembled Bosch saw is shown in Figure 8, with
a few key components labeled. The fork, as shown in Figure 8, is a critical
piece of the mechanism and is one focus of the redesign. To ensure its design
is sound, a Finite Element Analysis (FEA) of the dynamically loaded fork is
performed by using input loads from a computational model.
The Bosch saw features two counterweights, as shown in Figure 8, in or-
der to minimize vibration of the tool during use. The redesigned mechanism
has a different mass distribution so a new set of counterweights needs to be
6
Figure 8: Partially disassembled Bosch saw
designed and optimized. Additionally, the size of the blade will affect tool
vibration. The counterweights are optimized for the longest Stryker blade,
which is commonly used in TKA surgeries, but the change in vibration re-
sulting from different blades is also examined. There are 20 Dual Cut Stryker
blades available that vary in length from 75 to 100 mm and also vary in width
and thickness. A representative subset of these is chosen to demonstrate the
effect on the vibration.
1.2 Research Goal
The goal of this research is to design a high frequency in-line oscillating mech-
anism for use in a bone saw and to analyze the dynamics of the mechanism
in order to minimize vibration through counterbalancing. The motivation
behind this research is to provide a new mechanism configuration that can
accommodate an in-line body geometry for improved ergonomics and reduced
user fatigue as compared to contemporary pistol-grip designs.
7
2 Survey of the Literature
It is widely recognized among tool manufacturers that comfort is a key selling
point that can also reduce the risk of occupational injury (Kujit-Evers et al.
2004), which suggests an opening in the sagittal saw market for a more er-
gonomically designed tool. Surgeon discomfort and poor posture can impact
job satisfaction, productivity, and can potentially lead to medical errors or
patient harm. At the very least, this often leads to time-consuming and costly
interruptions during the surgery (Hallbeck et al. 2008). According to OSHA,
the most effective way to address these issues is through engineering controls,
changes to the task, and changes to the tool (OSHA Fed. Reg. 2001).
In addition to the previously mentioned concerns, Eksioglu (2008) states
that awkward postures, repetitive motions, and biomechanical stress and strain
can lead to Musculoskeletal Disorders (MSDs) and Cumulative Trauma Dis-
orders (CTDS), including nerve entrapment, Carpal Tunnel Syndrome (CTS),
epicondylitis, peritendinitis of the forearm, and tenosynovitis in the wrist and
fingers. These risks are shown by the fact that 9-10% of all compensatable
injuries in the USA occur while using hand tools (Lewis & Narayan 1993).
Risk factors for MSDs and CTDs are further elaborated upon by Punnet &
Wegman (2004) as occupations in nursing facilities and patient care, repet-
itive motion patterns, forceful manual exertions, non-neutral body postures,
mechanical pressure concentrations, segmental or whole-body exposure to vi-
bration, insufficient recovery times, exposure to cold, and highly demanding
psychosocial work environments.
These concerns are clearly applicable to surgeons, as shown in a 2012 study
that found that 16.7% or surgeons who performed open surgery self-reported
experiencing hand pain after surgery and 12.5% reported experiencing wrist
pain after surgery (Santos-Carreras et al. 2012). Another survey showed that
8
35.6% of surgeons surveyed in Hong Kong reported “working through the pain
so that the quality of their work would not suffer” (Szeto et al. 2009).
Another risk factor for surgeons, in addition to extreme wrist posture
and repetitive motions, is tool vibration, which has been shown to lead to
vascular, neurological, and musculoskeletal disorders, collectively known as
Hand-Arm Vibration Syndrome (HAVS) (Adewusi et al. 2013). Vascular
disorders of this type are caused by damage to the blood vessels, which is
most often manifested in Reynaud’s Phenomenon, also called vibration white
finger. Other health problems resulting from vibration include a decrease
in finger dexterity, numbness, tingling sensations, loss of hand-grip strength,
pain, and stiffness (Edwards & Holt 2005).
These problems are most commonly seen in construction workers who use
vibrating hand tools for prolonged periods of time, but a concern still exists
for surgeons who must operate with high precision while fully focused at all
times. Even though surgeons may not report discomfort due to vibration,
Vergara et al. (2008) found that workers are not aware of the levels of vibration
transmitted to their hands and that workers can develop HAVS in spite of
short daily exposure times. Regulating bodies have recognized the danger of
vibrating hand tools.
The European Union Physical Agents (Vibration) Directive (2002) lim-
its the vibration exposure for a worker in a single working day and specifies
the standard for measuring vibration exposure as ISO 5349 (2001). This was
matched in the US with ANSI S.2.70 (2006). These standards specify the mag-
nitude of vibration a worker should experience per day in terms of an eight hour
equivalent frequency weighted root mean square (Rimell et al. 2007). Many
researchers offer alternative, perhaps more accurate, methods for measuring
and limiting hand vibrations for workers, but it is universally understood that
9
exposure to hand-transmitted vibration is harmful to one’s health (Dong et
al. 2007).
In addition to the health concerns associated with vibrating hand tools,
there is the obvious concern that vibration of the sagittal saw can cause the
surgeon to make inaccurate cuts. Unstable vibration affects machining or
cutting accuracy and operation efficiency of many machine tools and hand
tools (Wang 2012). Vibration can also be undesirable because of decreased
reliability of the tool and increased noise during tool operation. In order to
reduce the multitude of negative effects of vibrations, the new sagittal saw
mechanism design will include an improved counterbalance. According to
Arakelian & Dahan (2001), “one of the most effective means used for the
reduction of the vibratory activity of the high speed machines is the balancing
of shaking force and shaking moment of linkages, full or partial, by internal
mass redistribution or counterweight addition”.
There are three main types of linkage balancing techniques: link mass
distribution (such as adding counterweights), the addition of moving links
(adding a mirror-image linkage), and the addition of counter rotating or oscil-
lating masses (A comparison of techniques for balancing planar linkages 2006).
Due to the sagittal saw mechanism configuration and the size of the tool, the
addition of counterweights will likely be the best balancing approach in this
case.
A US Patent, tiled ”Counterbalance for Eccentric Shafts” (Bernardi 2011)
shows the counterbalance and the counterbalancing calculations for a Bosch
oscillating saw, similar to the one disassembled in this study. The patent
states that the ”counterbalance arrangement [is] positioned and configured to
offset forces generated by the output drive pin when eccentrically driven.”
It then shows the addition of counterweights on the drive shaft, as shown
10
in Figure 9 where the counterbalance weights are labeled as 246 and 248.
The patent also gives equations for calculating the size and position of the
counterbalance based on the bending moments and centrifugal forces due to
a constant rotational velocity. A similar technique is used in this thesis to
calculate the new saw’s counterbalances.
Figure 9: Patent showing the counterbalancing elements on the drive shaft,labeled as 246 and 248.
The sagittal saw mechanism converts the rotational motion of the motor
into an oscillating motion of the blade. There are many types of mechanisms
that can achieve this conversion, with varying degrees of usability in a saw.
The following discussion of types of mechanisms comes from the Mechanisms
and Mechanical Devices Sourcebook, 5th edition. The categories of mecha-
nisms that are available for converting continuous rotation into oscillation are
linkages, gear trains, and cam mechanisms. Due to number of components
required, mechanism size, and mechanism complexity, a linkage will be the
preferred type of mechanism for the sagittal saw, so that category is explored
11
in more depth.
A crank-rocker mechanism is a four-bar linkage that allows full rotation of
the input link to generate an oscillating motion of the output link, Figure 10. A
slider-crank mechanism, Figure 11, converts rotary motion to linear oscillating
motion by restricting the output point to a linear motion profile. A scotch-
yoke mechanism works like a slider-crank mechanism to create linear sinusoidal
output motion, Figure 12. Part (a) of Figure 12 shows the mechanism when
the roller is at 270◦ and part (b) of the figure shows the roller at 0 ◦. A
rotary-to-linear mechanism is shown in Figure 13 that turns rotary motion
into intermittent reciprocating motion.
Figure 10: Crank-rocker mechanism: Link AB rotates 360 ◦ causing link CDto oscillate
Figure 11: Slider-crank mechanism: Link AB rotates causing Point C to oscil-late along the line
12
Figure 12: Scotch-yoke mechanism: wheel A rotates causing the sliding bar tomove through the sleeves
Figure 13: Rotary-to-linear mechanism: the rotor is the input, creating inter-mittent linear motion of the frame and bars
There are many variations and combinations of these mechanisms. The
existing Bosch saw mechanism is a version of the slider-crank mechanism in
which the input link AB and the connecting rod BC are replaced by an ec-
centric bearing, where the eccentric offset distance acts as length AB and the
radius of the eccentric bearing acts as length BC. This variation is shown in
Figure 14.
13
Figure 14: Overlay of eccentric bearing on a slider-crank mechanism. D isthe eccentric bearing, where the center of the bearing is at B and the rotationpoint is at A. The oscillating part of the mechanism will interact at point Cwhere a linear motion is generated.
An extensive number of creative versions of these mechanisms can be
found in Five Hundred and Seven Mechanical Movements by Henry T. Brown
(1884) and other such handbooks. For example, in Figure 15, Brown gives
an interesting version of the rotary-to-linear mechanism shown previously in
Figure 13. He describes it as a mechanism to change “uniform circular motion
into reciprocating rectilinear motion, by means of mutilated pinion, which
drives alternately the top and bottom rack”.
Figure 15: Brown’s mutilated pinion mechanism
14
The current literature search supports the need for this study and the
potential for improvements for the design of a pistol-grip sagittal bone saw.
The main issues revealed in the literature are the ergonomics of the saw for the
surgeon and vibration of the saw. The pistol-grip design forces awkward hand
and arm postures for the surgeon which can lead to musculoskeletal disorders
and cumulative trauma disorders. Vibration of the saw transmitted to the
surgeon’s hand can also cause medical issues such as Reynaud’s Phenomenon
and other vascular disorders. Additionally, vibration can lead to a decreased
life of the saw, unwanted noise during operations, and a lack of control of the
saw, causing inaccurate cuts. There are many different types of mechanisms to
choose from in literature that will convert rotary motion to oscillating motion.
This new mechanism will allow the new in-line grip for the sagittal saw to be
created, which will decrease the ergonomic and potential health issues for the
surgeon.
3 Research Objectives
To accomplish the research goal of a new in-line bone saw mechanism design,
the following objectives were completed:
1. Create and verify through experiments both computer and numerical
models of an existing oscillating saw;
2. Modify verified models to obtain desired form and function for the new
in-line configuration;
3. Design and optimize counterweights (computational dynamics) to mini-
mize vibration;
15
4. Perform linear elastic Finite Element Analysis of the oscillating fork
component to confirm structural integrity during saw operation;
5. Perform all necessary bearing size and life calculations to choose proper
bearings;
6. Design blade clamp mechanism and integrate into gear case/clamshell;
7. Design gear case and internal clamshell support structures, to interface
with an external housing for an in-line sagittal saw design; and
8. Develop a rapid prototype of the new in-line oscillating mechanism to
demonstrate feasibility of the proposed design.
These design and analysis objectives were accomplished using a variety
of methods described in the next sections. The end objective, or final deliv-
erable, of this thesis is a verified Computer Aided Design (CAD) model of a
new mechanism arrangement including the engineering to support the design
decisions made and the structure to support the mechanism. In addition, a
full scale rapid prototype of the in-line mechanism is fabricated to demonstrate
component clearances and functionality of the blade clamp.
4 Methods and Approach
The first step in the design process was to create a solid CAD model of an
existing oscillating saw mechanism and to verify its accuracy experimentally.
The saw modeled was the Bosch MX25E due to its availability and its likeness
to the Styker saw. The CAD model was created in Unigraphics NX 8.5. This
software is used widely in the machine design industry and has capability to
create solid models and use them in dynamic motion simulations. It also has
16
an FEA functionality that will be utilized. Being able to perform all of these
functions within one software aids in the design process.
A Bosch MX25E saw was disassembled, its components were measured
using calipers, and a model was created. Once the solid model was created, a
numerical motion simulation was created to approximate the dynamic forces
on the mechanism. A constant speed was prescribed to the motor and the
resulting displacement, velocity, and acceleration of a given point on the blade
over time was determined. This data can then be compared to an analytical
model and verified experimentally.
An analytical model of the existing Bosch mechanism was developed using
MATLAB. This MATLAB code used the Vector-Loop Method to analyze an
analogous linkage to the Bosch mechanism. Again, a constant input speed was
applied to the motor and the resulting displacement, velocity, and acceleration
over time of a given point on the blade was determined. Finally, results from
the analytical model were confirmed by computational dynamics of the CAD
model.
When both the CAD model and the analytical model are producing the
same results, they can be compared experimentally to the real Bosch saw.
This was done using a laser vibrometer to measure the speed of a specific
point on the blade during oscillation. The saw was tested through a range
of speeds and data from all speeds was compared to data from the models.
The oscillation speed of the blade was measured by the laser vibrometer, but
was also confirmed by two separate measurements using a microphone and
a strobe. The experimental speeds can then be input into the CAD and
analytical models and the velocity of the blade can be output for comparison
to the laser vibrometer measurements.
Once the CAD model of the current Bosch saw was verified, a new design
17
could be created based on the model with high confidence that the new design
will behave as predicted. The design and FEA was completed using NX 8.5.
The target oscillation speed and blade angle for the new design matches the
specifications of the current Stryker System 7 saw.
The following list outlines the project steps:
• Model Bosch MX25E saw
– Disassemble saw
– Measure components using calipers
– Model components and specify densities
– Create assembly of components and numerical motion simulation
that outputs position, velocity, and acceleration of a given point on
the blade
• Create analytical model of Bosch mechanism using Vector-Loop Method
to analyze position, velocity, and acceleration of a given point on the
blade
• Verify that numerical and analytical models give the same results
• Experimentally verify models
– Use strobe light to determine oscillation frequency of Bosch saw
– Use microphone to verify strobe oscillation measurements of Bosch
saw
– Measure velocity of a given point on the blade using a laser vibrom-
eter
∗ A nut and bolt was fixed through the blade to create a flat
surface for laser to measure
18
∗ Oscillation frequency was verified after addition of nut and bolt
to ensure they did not alter the blade motion
– Using experimentally measured oscillation frequency as a model
input, verify that the analytical and numerical models produce the
same position, velocity, and acceleration data as was experimentally
measured with the laser vibrometer
• Basic Design Tasks
– Change Bosch OSC118F blade used in the model and experiments
to Stryker Dual Cut blade 4125-127-100 that would be used for the
surgery
– Alter mechanism to obtain Stryker blade oscillation angles
– Adjust mechanism as required by current understanding of outer
shell
– Determine baseline vibration in X-Y plane (plane of blade) without
any counterbalance, using NX motion simulation
• Design Counterweights in NX
– One weight opposite the fork to counter the combined mass of the
fork and blade
– Second weight on motor shaft to balance eccentric bearing and
forces due to the fork
– Counterweights were optimized in NX to minimize vibration in X-Y
plane (plane of the blade) for the longest Stryker blade only, Dual
Cut 4125-127-100
– Effect of different length blades on vibration of counterbalanced
mechanism is shown graphically
19
• Perform linear elastic finite element analysis of oscillating fork in mech-
anism
– Adjust design as necessary based on FEA results, factor of safety,
and applicable yield criterion
• Perform resonant frequency analysis
• Bearing calculations for all 5 bearings in assembly
– Bearing fit calculations
– Bearing life calculations - size for “infinite life” if possible
• Design blade clamp mechanism
– A Keyless clamp mechanism was designed based on Stryker’s blade
clamp design
• Design gear case and internal clamshell support structures
• Create prototype of saw mechanism that can be manually rotated to
demonstrate the feasibility of the proposed in-line design
5 Model Creation and Validation
The first step in the design process is to create an experimentally verified
CAD model of the existing Bosch saw. This will provide a known starting
point for the new bone saw design so that there is confidence that the final
design will behave as predicted by the model. In order to create this model, a
Bosch saw was disassembled and modeled in NX to create a motion simulation
that generates position, velocity, and acceleration data for the saw blade at
any given input motor speed. An analytical model of the saw’s motion was
20
also created using Vector Loop Method. The output of the analytical model,
the position, velocity, and acceleration of the blade, was compared to the
CAD model’s output to show agreement between the theoretical and simulated
motions. The velocity of the saw blade was then determined experimentally
and compared to the predicted velocity. By verifying the model in these ways,
it can be reasonably assumed that the final design proposed as a CAD model
will physically behave as predicted.
5.1 CAD Model Creation
In order to create the CAD model as a basis for this redesign, a Bosch saw
was disassembled and the geometry of each component was measured using
digital calipers (Mitutoyo 500-196-20, Aurora, Illinois). A picture of the saw
before and during disassembly, showing the top shaft, top gear case, and the
motor shaft, is shown in Figure 16. A close up of the top shaft, Figure 17,
shows where the blade mounts and the fork that transmits motion from the
motor shaft to the top shaft. Figure 18, a close up of the motor shaft, shows
the eccentric bearing that interacts with the fork, the bearing spacer, and the
motor. These components were then modeled in NX 8.5 and an assembly was
created that defined the moving components of the saw, shown in Figure 19.
21
Figure 16: Exploded and un-exploded views of the typical drive train compo-nents of an oscillating saw: top shaft, aluminum gear case, and motor shaft.
Figure 17: Top shaft of the saw showing the blade mount and the fork thatdrives the oscillation of the blade.
Figure 18: Motor shaft of the saw showing the eccentric bearing, bearingspacer, and motor.
22
The components were also weighed on an Ohaus Trooper Count scale
(Ohaus TC15RS, Pine Brook, NJ). The mass and material density of each
component were subsequently defined in NX. Using the motion simulation
function in NX, a dynamic model of the saw was generated and calculations
were performed using the built-in RecurDyn Solver. A coordinate system was
defined as shown below in Figure 19, with the blade in the x-y plane, the length
of the blade and the motor axis in the Y direction, and the z-axis defined by
the top shaft axis of rotation.
Figure 19: NX model of the drive train assembly showing the global coordinatesystem used in dynamic analysis, the point of interest (POI), and the drivetrain components.
A constant angular velocity was prescribed to the motor shaft. Initially,
an input speed of 10,000 oscillations per minute, or 167 Hz, was chosen because
that is the speed of the low Stryker saw setting. A set of line contacts was
23
defined between the eccentric bearing and the fork. A point was defined as part
of the saw blade, located in the plane of the blade, in line with the axis of the
top shaft, and 70 mm from the top shaft axis. This point, called the point of
interest (POI), Figure 19, moves with the saw blade. This point was designated
in order to have comparable data between the model and experiments designed
to validate the model, to be described in a later section. The experiments
require modifying the blade to attach a surface perpendicular to its motion
to measure its velocity. In order to attach this surface, some blade material
is required on either side so a point midway up the blade is ideal to use as a
reference point. The motion study parameters are shown in Table 1.
Table 1: Parameters used to define CAD motion analysis
Parameter Value
Motor Speed 10,000 oscillations/min (167 Hz)
Distance top shaft to POI 70.0 mm
Length of Fork 30.1 mm
Radius of eccentric bearing 8.0 mm
Eccentric offset 1.0 mm
5.2 CAD Model Motion Data
Measurements were taken at the POI during the NX motion simulation to
determine the x and y displacement, velocity, and acceleration components at
the POI. This data defines the motion of the saw blade as an output of the
model relative to constant motor speed as an input to the model. The plots
of displacement, velocity, and acceleration are shown in Figures 20, 21, 22.
24
Figure 20: Simulated displacement data for the point of interest (POI) on theblade (see Figure 19) for constant oscillation frequency of 167 Hz.
Figure 21: Simulated velocity data for the POI on the blade for constantoscillation frequency of 167 Hz.
25
Figure 22: Simulated acceleration data for the POI on the blade for constantoscillation frequency of 167 Hz.
Figure 20 shows a smooth, sinusoidal shape for the displacement data.
The Y values are very small compared to the X values. This is expected given
the coordinate system chosen and the overall blade motion of approximately
4◦. Likewise, the velocity and acceleration data shown in Figure 21 and Figure
22 is mostly sinusoidal with a relatively small Y component.
5.3 Analytical MATLAB Model Creation
To begin verifying the CAD model, an analytical model was created using the
Vector Loop Method to show the blade’s theoretical position, velocity, and
acceleration. This is a technique in which a vector is assigned to each link in
a mechanism to fully define the position of each link, given a specific rotation
of the input link (Stanisic 2014). The saw mechanism was broken up into
two sub-mechanisms for this analysis. The first sub-mechanism considered
was the motor shaft, eccentric bearing, and fork. This mechanism study had
the constant angular velocity of the motor shaft as the input and the linear
26
displacement of the end of the fork that interacts with the eccentric bearing
as the output. The linear displacement of the end of the fork could then be
used as an input to the second sub-mechanism, which consisted of the fork,
top shaft, blade, and point of interest. With some trigonometry, the linear
displacement of the end of the fork could be translated to the displacement
of the POI. The same analysis can be performed to determine the velocity
and acceleration of the POI using the Vector Loop Method, as will be shown
below.
The program to perform this vector loop analysis was written in MAT-
LAB, and is shown in Appendix B.1. It takes an input constant angular
velocity of the motor and makes time steps through the motion, calculating
the position, velocity, and acceleration of each link at each time step. An
initial position is assumed of the blade being in the vertical position, about
to start moving in the positive X direction, as previously defined in Figure
19. This is necessary to note in order to phase the analytical model with the
numerical model when comparing the data.
The motor shaft sub-mechanism takes a constant angular velocity of the
motor shaft and turns it into a cyclical linear displacement of the fork. This
is a similar function to the one performed by a crank-slider mechanism. An
analogous crank-slider mechanism was drawn to describe the eccentric bearing
- fork arrangement. A sketch of this mechanism is shown in Figure 23. It is
assumed that there is constant contact between both sides of the fork and
the bearing due to a transition fit specification on the width of the fork legs.
Therefore, only one side of the fork is mathematically considered. The contact
point between the bearing and the fork can be considered a block sliding
horizontally while the vertical distance between the contact point and the
motor shaft axis of rotation is a link, r5, of varying length. The center of
27
the motor shaft can be considered a grounded pivot point. In Figure 23, r2
is the input link, defined as the eccentric offset distance, e, or the distance
between the motor shaft and the shaft the eccentric bearing is mounted on;
θ2 is the angle that r2 rotates through and is equal to the angular rotation
of the motor shaft; r3 is always horizontal and defines the distance from the
center of the bearing to the contact point with one side of the fork and is
equal to the diameter of the bearing, L; r4 is the output distance from the
center of the motor shaft to the contact point between the fork and bearing,
and is also always horizontal. As r2 rotates at a constant angular velocity,
r4 changes length, defining the fork offset and r5 changes length, defining the
contact point’s offset from center.
Figure 23: Overlay of eccentric bearing and modified slider-crank mechanismwhere r2 is the input link and is equal to the eccentric offset, r3 is equal tothe bearing radius, r5 is always vertical but changes length, and r4 is alwayshorizontal but changes length defining the fork offset and output of the model.
From the analogous mechanism described in Figure 23, the following vector
loop equation can be written.
28
~r2 + ~r3 − ~r5 − ~r4 = ~0 (1)
This can be decomposed into X and Y component equations,
x : r2cosθ2 + r3 − r4 = 0 (2)
y : r2sinθ2 − r5 = 0 (3)
From the x component equation, it can be shown that r4 is defined as
follows:
r4 = r2cosθ2 + r3 (4)
The equation for r4 can be differentiated to obtain the following velocity
equation.
r4 = −r2θ2sinθ2 (5)
This velocity equation can be differentiated once more to obtain the ac-
celeration equation.
r4 = −r2θ22cosθ2 − r2θ2sinθ2 (6)
This set of three vector loop equations for the displacement, velocity, and
29
acceleration of the fork in the X direction has the following known values,
measured on the Bosch saw during disassembly.
r2 = e = 1.0mm (7)
r3 = L = 8.0mm (8)
The value of θ3 is a constant 0◦, θ4 is a constant 0◦, and θ5 is a constant
90◦. Since the motor is spinning at a constant angular velocity, θ2 is the known
input value for the calculations, θ2 = 0, and θ2 can easily be calculated with
the following equation, where t is the time the motor has been running and
θ2,0 is the initial rotational position of r2 (Meriam et al. “Dynamics” 2007).
θ2 = θ2,0 + θ2t (9)
The unknown values in the vector loop equations are r4, r4, and r4. The
three equations can be solved for these three unknowns at each step of θ2. This
completes the analysis of the first sub-mechanism. Hence with a given angular
velocity, the position, velocity, and acceleration of the fork can be determined
at any time.
The top, blade axis, sub-mechanism has a few dimensions, as defined in
Figure 24, where p is the distance from the axis of blade rotation to the point
of interest, γ is the angle of blade rotation, and h is the distance from the axis
of blade rotation to the center of the eccentric bearing, where the fork contacts
the bearing. The value of h was measured when the saw was disassembled and
the distance p is defined by the experimenter.
To determine the angle of rotation of the blade from vertical, γ, angular
30
Figure 24: Dimensions of the top sub-mechanism for the analytical modelwhere p is the distance from the axis of rotation to the point of interest, h isthe distance from the axis of rotation to the contact point of the fork and thebearing, and γ is the angle of blade rotation.
velocity of the blade, ω, and the angular acceleration of the blade, α, the
following geometry calculations were performed during each time step of the
program.
γ = sin−1(r4 − L
h
)(10)
ω =r4cosγ
h(11)
α =r4cosγ
h(12)
Using this description of blade rotation, the displacement, d, velocity, v,
and tangential acceleration, a of the point of interest can be calculated.
31
d = psinγ (13)
v = pω (14)
a = pα (15)
The X and Y components of the position, velocity, and acceleration can
easily be determined using the angle of blade rotation.
5.4 Comparison of Analytical and Numerical Models
To verify agreement between the analytical and numerical models, the angular
position, angular velocity, and angular acceleration of the point of interest were
compared. These three comparison plots are shown below in Figures 25, 26,
27.
Figure 25: Comparison of angular position for NX and MATLAB modelsmeasured at the POI for a constant oscillation frequency of 167 Hz.
32
Figure 26: Comparison of angular velocity for NX and MATLAB models mea-sured at the POI for a constant oscillation frequency of 167 Hz.
Figure 27: Comparison of angular acceleration for NX and MATLAB modelsmeasured at the POI for a constant oscillation frequency of 167 Hz
It can be seen in these plots that the CAD model matches the mathe-
matical model very closely for all three variables. This high level of agreement
indicates that the numerical model is behaving in a predictable, accurate man-
ner.
33
5.5 Experiment to Validate Models
In order to validate numerical and analytical predictions, an experiment was
designed to gather data on blade velocity for comparison to the models. The
velocity data was recorded over multiple speed settings on the Bosch saw using
a laser vibrometer. Two experiments were conducted to determine baseline
speeds for the Bosch saw to establish inputs for the models. Modifications
were made to enable laser vibrometer experiments, but they did not impact
the saw’s baseline speed. The saw has six speed settings. Blade oscillation
frequency was measured by two methods at each speed, as described below.
5.5.1 Microphone Experiments
A microphone was used to determine the oscillation frequency of the saw blade
at each speed setting by measuring the frequency and amplitude of the sound
generated by the saw. For each speed setting, a peak in amplitude indicates the
frequency of the primary tone, which corresponds to frequency of oscillation.
A Bruel & Kjaer (Naerum, Denmark) Free-Field 1/2” microphone, Type 4190,
was used with a Bruel & Kjaer Nexus conditioning amplifier and an NI data
acquisition card. Labview 2009 was used to collect the data, utilizing a code
written by Professor Robert White (Tufts University) to look at the average
power spectrum of the signal. For this experiment, only the frequency data
and the relative magnitude of the signal was important, so there was no need
to calibrate the system. The saw was held about 6 inches from the microphone
with the blade oscillating perpendicular to the microphone. The sampling rate
was 100 kHz. An image of the setup for blade frequency measurements with
a microphone is shown in Figure 28.
The data from Labview is plotted in Figure 29, where both axes use a log
scale and the y-axis displays the root mean square voltage magnitude at each
34
Figure 28: Setup for experimentally determining the oscillation frequency ofthe blade using a microphone.
frequency. The highest peak for each speed setting indicates the primary tone
in the sound captured by the microphone. This primary tone is the frequency
of blade oscillation. A data point is labeled in the plot, showing the peak of
the data for speed dial setting 6, 331.1 Hz.
35
Figure 29: Microphone experiment data showing the sound frequency peaksfor each speed dial setting, corresponding to blade frequency of oscillation.
There are six peaks shown in Figure 29 that represent the frequency of the
six speed dial settings. Four of the peaks occur at frequencies when only one
speed setting peaks. Two of the peaks, for settings 2 and 5, occur at frequencies
where every speed setting peaks. These overlapping peaks are likely due to
natural harmonics in the motor. However, it can still be determined that
those peak frequencies belong with settings 2 and 5 due to an expectation of
increasing speed for each setting.
5.5.2 Strobe Light Experiments
A strobe light was used to determine oscillation frequency of the blade. A
Nova-Strobe DAX by Monarch Instrument (Amherst, NH), shown in Figure
30, was used for this test.
The lights in the room were turned down and the strobe was held above
36
Figure 30: Strobe light used to determine the frequency of oscillation of thesaw blade.
the oscillating saw blade. The strobe frequency was adjusted until visually the
saw blade appeared to stand still in the middle of its stroke, even though it
was still oscillating. This strobe frequency was then recorded as the speed of
the blade at the particular setting. One half of the actual frequency will also
have this same effect because it will show the blade passing through center as
it moves forward and backward instead of showing it once per cycle. There-
fore, 1/2 frequencies were also tested to make sure the true frequency of the
blade was recorded. The resulting frequencies from the microphone and strobe
experiments are shown in Table 2.
This data shows a high level of agreement, indicating the measurements
are accurate within a small human and sampling error. The average of these
speeds is used as inputs to the analytical and numerical models for comparison
with the experimentally determined blade velocities.
5.5.3 Laser Vibrometer Experiments
In order to verify the analytical and numerical models experimentally, the ac-
tual motion of the saw blade must be determined. This was accomplished by
37
Table 2: Microphone and strobe frequency data for each blade speed settingon the saw
Setting Mic Frequency (Hz) Strobe Frequency (Hz)
1 132.8 133.0
2 180.1 179.5
3 220.5 221.1
4 257.9 259.4
5 299.1 302.8
6 331.1 331.8
using a laser vibrometer to measure the velocity of the blade on the Bosch
saw as it oscillates at different speed settings. The laser vibrometer was aimed
perpendicular to the blade to measure the blade’s linear velocity in the X direc-
tion, as defined in Figure 19. As shown by the models, the X component of the
motion is significantly larger than the Y component due to the small angle of
blade rotation so the X component of velocity is the appropriate measurement.
In order to have a target surface for the laser that is perpendicular to the X
direction, the blade was modified by drilling a hole in the blade and attaching
a nut and bolt through the hole. The flat side of the nut was aligned with the
edge of the blade to provide a target surface for the laser. This modification
is shown in Figure 31.
Figure 31: Blade modified by adding a nut and bolt with the bolt surfaceparallel to the side of the blade to provide a target surface for the laser vi-brometer.
38
A Polytec (Irvine, CA) OFV-511 laser vibrometer sensor head was used
with a Polytec OFV-3001 signal conditioning box. The velocity filter was set to
5 kHz and the velocity range was set to 1000 mm/s/V. The data was collected
using an Agilent Technologies (Santa Clara, CA) oscilloscope, DS03062A. Care
was taken to align the laser perpendicular with the nut on the blade, as shown
in Figure 32, where the red laser dot is visible on the nut.
Figure 32: Laser aligned perpendicular to the nut on the blade: notice thevisible laser dot on the nut.
The resulting data from each run was plotted by the oscilloscope. A
sample of an oscilloscope plot is shown in Figure 33, which shows the data
for the saw on speed setting 1. From these plots for each speed setting, the
blade’s frequency and maximum velocity was determined.
39
Figure 33: Oscilloscope plot of blade speed in the X direction showing voltagevs time for speed setting 1, where 1 V = 1000 mm/s.
Usable data was obtained for speed settings 1, 2, and 3. Data from speed
4 showed an unexpectedly large maximum velocity. During measurements on
speeds 5 and 6, uneven waves were seen in the data, instead of the expected
sinusoidal shape. An example of measurements taken on setting 5 is shown
in Figure 34. The test was repeated multiple times on settings 5 and 6, in an
attempt to understand the data, until during a test on setting 6, the blade
fatigued and broke. This is likely due to the extra mass of the bolt on the
blade. The broken blade is shown in Figure 35.
40
Figure 34: Oscilloscope plot demonstrating uneven curves for blade speed inX direction on speed setting 5, where 1V = 1000 mm/s.
Figure 35: Fatigued and fractured saw blade due to added weight of nut andbolt after running on speed setting 6.
After this event, it was clear that the uneven data on settings 5 and 6 was
the blade crack expanding and fracturing. It is also possible that the unusually
high velocity on setting 4 was the beginning of blade fatigue. Therefore, the
data from those tests was discarded. The blade’s frequency was determined
from the oscilloscope plots and compared to the microphone and strobe fre-
quency data to ensure the extra weight of the nut and bolt did not alter the
41
blade’s frequency. This data is shown in Table 3 and indicates that the blade
continued to oscillate at the same frequency with and without the nut and
bolt. Most likely, this is due to a speed controller in the saw.
Table 3: Comparison of average blade frequency from strobe and microphoneexperiments prior to modifying the blade vs the laser vibrometer frequencyafter modifying the blade
Setting Average Mic & Strobe Frequency (Hz) Laser Frequency (Hz)
1 132.9 133.5
2 179.8 176.7
3 220.8 223.1
5.6 Comparison of Experimental Data to Models
In order to validate the models, the predicted speed of the blade must match
the experimentally determined blade speed. The MATLAB Vector Loop Anal-
ysis, shown in Appendix B.1, was re-run with each speed setting’s measured
frequency as the input. The point where the laser shone on the nut is analo-
gous to the point of interest programmed in the MATLAB code, measured in
the CAD model, and shown as POI in Figure 19. This generated maximum
predicted velocity values at the POI that could be compared to those measured
using the laser vibrometer, as shown in Figure 36.
42
Figure 36: Comparison of experimental and predicted maximum velocities ofthe point of interest (POI) for speed settings 1,2, and 3
Figure 36 shows very strong agreement between the experimental and an-
alytical data, within the 95% confidence error bars shown on the plot. The
slight differences between the experimental and predicted data is likely due to
inaccurate alignment of the laser to the nut on the blade. This validates the
analytical and numerical models that were previously shown to be in agree-
ment. Since the models are validated by the experimental data, the models can
reasonably be expected to yield accurate results when changes to the design
are made.
6 Mechanism Selection
A needs analysis was performed to choose an appropriate mechanism for this
redesign to convert the rotary motion of the motor shaft to oscillating motion
of the blade. Table 4 summarizes a few mechanism options and their merits
relative to applicable selection criteria. These mechanisms are discussed in the
Survey of the Literature portion of this study and are shown in Figures 10 to
43
15. The table rates each option on a scale of 1-5, with 1 being poor and 5
being ideal.
The mechanism needs to be durable to survive for the life of the saw
while operating at a high frequency. This causes any mechanism with impact
or sliding motions to be ranked lower. Compactness is necessary to fit the
mechanism inside a small gear case that a surgeon can get their hands around.
Simplicity and Cost are listed together because often the more complicated a
mechanism and the more components it has, the more expensive it is. The
Sum Total column is a sum of the rankings to help guide selection of the best
mechanism.
Table 4: Needs analysis of mechanism options for converting rotary to oscil-lating motion in the saw, rated on a scale of 1-5 where 1 is poor and 5 isideal.
Mechanism Durability Compactness Simplicity/Cost Sum Total
Crank-RockerLinkage
5 2 3 10
EccentricBearing
version ofSlider-Crank
5 5 4 14
Scotch-YokeMechanism
3 3 3 9
Rotary-to-Linear
Mechanism
1 1 2 4
Gear Train 3 3 2 8
With 14 points, the Eccentric Bearing version of Slider-Crank mechanism
is the recommended choice according to the needs analysis. Thinking qual-
itatively about the problem can confirm this choice. Both the Bosch word
working tool and the Stryker bone saw uses an eccentric bearing mechanism
44
to generate the blade’s oscillatory motion.
7 Saw Blade and Motor
To begin the redesign of the saw, a few changes were made to the Bosch model
to make it more like a bone saw. First, the blade was changed from the Bosch
OSC118F woodworking blade to the Stryker Dual Cut 4125-127-100 bone saw
blade. These two blades are shown in Figure 37.
Figure 37: (a) Comparison of Bosch OSC118F blade, and (b) Stryker DualCut 4125-127-100 blade.
Additionally, some of the mechanism motion parameters had to be changed.
These are listed in Table 5 below.
Table 5: Parameter changes made to the saw model to align it with a bonesaw design
Parameter Bosch Saw Bone Saw
Blade Speed8,000-20,000
oscillations/min10,000
oscillations/min
Blade Angle Oscillation 4◦ 5◦
The change in blade oscillation angle was made by adjusting the fork
length and the eccentric bearing offset distance. These distances were further
45
refined during the counterbalancing exercise. The Bosch open-architecture
motor, with a separately mounted rotor and stator, was replaced with the
canned motor used in the Stryker saw to take advantage of its smaller packag-
ing size. The Stryker saw motor is shown in Figure 38. With these changes,
the in-line saw model can function in the same manner as the Stryker pistol
grip bone saw.
Figure 38: Stryker motor to be used in new saw design.
8 Counterbalancing
It is desirable to have the internal mechanism of the saw counterbalanced
in order to minimize vibration. Decreased vibration will improve surgeon
control and comfort. The saw mechanism has two axis of motion that must be
balanced. The first axis, the top shaft shown in Figure 39, must balance the
oscillating fork and blade. This axis can theoretically be perfectly balanced
by placing the center of mass of the whole assembly along the axis of rotation.
The second axis of motion, the motor shaft, must balance the combination
of two forces: (1) the centripetal force of the eccentric bearing as it rotates,
and (2) the oscillating force of the fork as it is accelerated in the X-direction
46
by the eccentric bearing. The top shaft counterbalance must be done first
because the mass of the fork is a required input to the calculation for motor
shaft counterbalancing.
8.1 Top Shaft Counterbalance
The top shaft assembly is shown in Figure 39 with the Stryker blade attached.
The assembly oscillates about the Z-axis as the motor spins.
Figure 39: CAD model of the top shaft assembly used to balance the fork andblade as they oscillate together around the Z-axis.
The top shaft assembly is balanced when the center of mass is on the axis
of rotation. All of the components are cylindrical about this axis except for
the blade and the fork. For the center of mass of the assembly to lie on the
Z-axis, Equation 16 must be satisfied, where mb is the mass of the blade, rb is
the radius to the center of mass of the blade, mf is the mass of the fork, and
rf is the radius to the center of mass of the fork (Meriam et al. “Dynamics”
2007).
47
mbrb = mfrf (16)
The Stryker blade was weighed using an Ohaus Trooper Count scale, so
that mb is a known value. The blade dimensions were measured using calipers.
The blade was modeled in NX with the correct density specified. NX could
then generate the center of mass of the blade, so that rb was also a known value.
This led to Equation 17, which can be used to define the fork’s geometry.
mbrb = (0.022 kg)(55.647mm) = 1.224 kg mm = mfrf (17)
With the desired relationship between the fork’s mass and location of
center of mass, the model was manually manipulated until a solution was
found. The final fork shape is shown in Figure 40. This design places the
center of mass of the entire assembly on the axis of rotation, meaning it is
a balanced system. When the height of the fork was changed to achieve this
solution, a corresponding change in the eccentric bearing offset distance had
to be made to maintain a 5◦ oscillation angle. The parameters defining the
oscillation angle in the Bosch saw and the new saw are listed in Table 6.
Table 6: Parameters defining the oscillation angle for the Bosch saw and thenew saw.
Parameter Bosch Saw Stryker Bone Saw
Length of Fork 30.10 mm 33.50 mm
Radius of eccentric bearing 8.00 mm 8.00 mm
Eccentric offset 1.00 mm 1.46 mm
48
Figure 40: Top shaft assembly counterbalanced solution.
An image showing the center of mass of the top shaft assembly is shown
in Figure 41. The final design solution with its center of mass on the axis of
rotation is shown in image (c) of Figure 41.
Figure 41: Top shaft assembly center of mass, indicated by cross-hairs, for (a)Bosch fork and Bosch blade, (b) Bosch fork and Stryker blade, and (c) newfork and Stryker blade.
The new fork is slightly more massive than the Bosch fork. The Bosch fork
was in fact not balanced to the OSC118F blade. In reality, the center of mass
of the top assembly with the Bosch blade and fork was above the Z-axis in
the direction of the blade, as seen in Figure 41, image (a). It is likely that the
Bosch tool was balanced to a smaller blade that is used more frequently with
49
that tool, such as a semi-circular blade. When the larger Stryker blade was
added, the center of mass moved even farther away from the axis of rotation,
as shown in Figure 41, image (b). Hence, the new fork is larger than the
Bosch fork because it is balancing a significantly larger and longer blade, the
Stryker blade. A prototype fork was made using stereo lithography (SLA)
to demonstrate the actual size of the component. Images of the SLA fork as
shown in Figure 42.
Figure 42: SLA prototype of fork from various angles.
With the fork defined, its geometry can then be used as an input to the
motor shaft counterbalancing.
8.1.1 Other Stryker Blades
The top shaft mechanism was balanced for a common blade used in hip replace-
ment surgeries at Tufts Medical Center, the Stryker Dual Cut 4125-127-100
blade. However, other blades may be used that would change the balanced
nature of the top shaft. Two other blades were chosen to examine the ef-
fect of different blades on the vibration of the counterbalanced mechanism.
Table 7 shows the dimensions of the 4125-127-100 blade that was used to orig-
inally counterbalance the mechanism and two other blades that may be used
in surgery. Figure 43 shows these three blades as well. All three blades are
from the Stryker Dual Cut series.
50
Table 7: Dimensions of three Stryker Dual Cut blades, images of the bladesare shown in Figure 43 with corresponding letter labels.
Blade Thickness Width Length
4125-127-100 (a - balanced) 1.27 mm 25 mm 100 mm
4111-147-075 (b) 1.47 mm 11 mm 75 mm
4125-064-075 (c) 0.64 mm 25 mm 75 mm
Figure 43: Images of the three Stryker Dual Cut blades: (a) balanced 4125-127-100 blade, (b) 4111-147-075, and (c) 4125-064-075.
These blades were all modeled, the mass was determined, and the location
of the center of mass of the assembly for each blade was determined. The radius
from the centerline of the top shaft to the center of mass of the assembly for
each blade is shown in Table 8 along with the mass of each assembly. The
negative radius values indicate the center of mass is below the axis of rotation,
towards the fork. The location of the center of mass in each case is shown in
Figure 44 with a set of cross-hairs.
51
Table 8: Location of center of mass and the mass of the assembly with eachblade; center of mass is shown in Figure 44.
Blade in Assembly Radius to center of mass Mass
4125-127-100 (a - balanced) 0 mm 0.15 kg
4111-147-075 (b) -5.82 mm 0.14 kg
4125-064-075 (c) -6.26 mm 0.14 kg
Figure 44: The center of mass of the top shaft assembly is denoted by cross-hairs for the (a) 4125-127-100 balanced blade, (b) 4111-147-075 blade, and (c)4125-064-075 blade.
A MATLAB code, shown in Appendix B.2, was used to calculate the
force on the top shaft at each point in the saw’s motion for each blade using
F = mrω2 (Meriam et al. “Dynamics” 2007). The resulting forces are plotted
in Figure 45. This plot shows that changing to the shorter blades induced a
small vibration force in the top shaft by shifting the center of mass off of the
axis of rotation.
52
Figure 45: Comparison of vibration forces on the top shaft assembly with the(a) 4125-127-100 balanced blade, (b) 4111-147-075 blade, and (c) 4125-064-075blade.
8.2 Motor Shaft Counterbalance
Unlike the top shaft, the motor shaft cannot be perfectly counterbalanced
because it has forces in competing directions. The first force that must be bal-
anced is the centrifugal reaction force on the motor shaft due to the eccentric
bearing rotating about the shaft. If this were the only force, it could be bal-
anced by an equal and opposite mass on the motor shaft. However, the motor
shaft also experiences forces due to accelerating the fork as it oscillates. This
force does not rotate with the shaft like the centrifugal force of the bearing,
it is applied only in the X direction. This means the motor shaft cannot be
perfectly balanced by adding an eccentric mass to the shaft. As such, the goal
of the counterbalance is to minimize the force magnitude, or the square root
of the sum of the squares of the forces in the X and Z directions. A free body
diagram of the motor shaft forces, including a coordinate system, is shown in
Figure 46. Note that the fork only oscillates in the X direction.
53
Figure 46: Top view free body diagram sketch of motor shaft showing coordi-nate system, the centrifugal reaction force of the eccentric bearing, Fc, and thereaction force on the shaft to accelerate the fork, Ff , with the shaft rotatingat an angular velocity, ω.
The reactive centrifugal force is the force the motor shaft sees and is equal
and opposite to the centripetal force being applied to the eccentric bearing.
Centripetal force was calculated using F = mrω2 (Meriam et al. “Dynam-
ics” 2007). The centrifugal force was accounted for at the center of mass of
the motor shaft assembly, including the shaft, the eccentric bearing, and the
applied counterbalance. This center of mass was calculated using NX.
In order to combine the force of the fork and the centrifugal force, the
centrifugal force was decomposed to its X and Z components at each point
in the axis rotation, using a MATLAB code, shown in Appendix B.3. The
MATLAB code also calculated the x-direction force being applied to the fork
at each point in the motor shaft rotation. It used the Vector Loop Method
(Stanisic 2014), described previously, to find the fork’s acceleration and then
used F = ma to calculate the force on the fork (Meriam et al. “Statics” 2007).
The forces could then be summed and combined to determine the force
magnitude, Fmag, that the motor shaft experiences at each point in its rotation,
as shown in Equations 18 and 19, where Fcx and Fcz are the components of
54
the centrifugal force at a given motor shaft rotation angle, in the x and z
directions, respectively.
Fx = Fcx + Ffx (18)
Fmag =√F 2x + F 2
cz (19)
To balance these forces, a counterbalance will be pressed onto the motor
shaft, looking something like the Bosch saw’s counterbalance, shown in Figure
47.
Figure 47: Example of Bosch saw motor shaft counterbalance.
There are a few parameters of the counterbalance that can be manipulated
to balance the assembly. The angle around the shaft at which the center of
mass is placed will control the phase of the balancing forces. The mass of
the counterbalance and the radius to its center of mass combine, through the
equation for centripetal force, to control the magnitude of the balancing force.
Therefore, there is an optimal angle to place the mass at and an optimal mr
55
value to control the force magnitude.
First, the angle at which to place the center of mass was determined.
Another MATLAB code, shown in Appendix B.4, was written to iterate the
angular position of the center of mass of a “dummy counterbalance” from 0◦
to 360◦, where the position of the eccentric bearing is considered 0◦. This
dummy counterbalance had a small, constant mass and radius to the center
of mass so that when it is placed opposite the centrifugal and fork forces it
slightly decreases them, but cannot be stronger than them. The MATLAB
code then calculated the total force magnitude, Equation 19, at every point in
the motor shaft’s rotation for each possible counterbalance angular position.
The counterbalance angular position with the lowest peak force was chosen
as the optimal position to place the counterbalance. This optimal position
was at 180◦, directly opposite of the eccentric bearing. This makes sense from
a rotational balance standpoint, but the rigor to check the other points was
desirable to verify that the addition of the variable forces in the X direction
due to the fork’s motion did not change this ideal position.
With the optimal position of the counterbalance determined, the radius
to the center of mass was kept small and constant while the mass of the
counterbalance was varied in MATLAB, as shown in Appendix B.5. When
the force magnitude was minimized through varying the mass, the optimal mr
ratio was determined to be 0.048 kg mm. The CAD model was then manually
varied until an mr value of 0.048 kg mm was obtained. The counterbalance
solution is shown in Figures 48 and 49.
For the counterbalance solution described, the force magnitude on the
motor shaft with and without the balance is shown in Figure 50. The balanced
solution shows a much smoother force is being applied to the shaft. Because of
the speed of shaft rotation, this would cause a very slight, constant vibration.
56
Figure 48: Side view of CAD model of motor shaft counterbalance solutionwith and without eccentric bearing shown.
Figure 49: Top view of CAD model of motor shaft counterbalance solutionwith and without eccentric bearing shown.
The highly variable force shown for the unbalanced solution would cause a more
jerky vibration of the saw, which would be much more difficult to control.
Figure 51 shows the force magnitude decomposed into X and Z components
for the case with and without the counterbalance. It can be seen that in order
to smooth out the force magnitude, the added counterbalance increased the Z
component force and decreased the X component force. The balanced system
has the X and Z components counteracting each other to smooth out the sum
of their squares.
To further illustrate the smoothing effect of the counterbalance, the resul-
tant force angle is plotted in Figure 52. This is showing the angle of the total
summed force on the motor shaft as the motor spins. It was calculated using
57
Figure 50: Force on motor shaft with and without counterbalance.
Figure 51: X and Z components of the forces on motor shaft with and withoutcounterbalance.
trigonometry between the X and Z components of force at each point in the
motor’s rotation. The horizontal axis shows the steps as the motor rotates,
representing the position of the eccentric bearing around the shaft. The verti-
58
cal axis shows the angle of the resultant force at that point in the rotation. A
straight line, as seen in the balanced case, means the forces are rotating evenly
with rotation of the motor. This would represent a small, constant vibration
of the tool. The line for the unbalanced case has plateaus followed by steep
changes. This represents the resultant force quickly switching its angle as the
tool jerks itself back and forth along the X direction due to the fork motion.
Figure 52: Plot of the angle of the resultant force as the motor spins; thehorizontal axis represents the position of the eccentric bearing as the motorrotates.
It can be seen from these plots that although the motor shaft could not
be perfectly balanced to remove all vibration forces, the addition of the coun-
terbalance effectively smoothed out the forces. This creates a manageable,
typical amount of consistent vibration that is often felt in power tools.
9 Finite Element Analysis of Fork
The fork is a critically loaded component so a Finite Element Analysis (FEA)
was performed to verify that it is structurally sound. The FEA was performed
59
using the Advanced Simulation function of NX. The built-in solver, NX NAS-
TRAN, was used to complete this structural analysis.
The fork would be made by a powder metal process from a 4000 series
steel, such as AISI 4340, and hardened to 60 Rc. Powder metallurgy would
allow the component to be produced at an acceptable price point for a mass-
produced medical device. AISI 4340 was chosen because of a high degree of
hardenability and toughness. High hardness is necessary to prevent prema-
ture wear at the contact surface with the eccentric bearing. The appropriate
material properties for AISI 4340 were applied to the FEA model. They were
obtained from “Materials Science and Engineering, An Introduction” (Callis-
ter Jr., W. D. 2007) and are listed in Table 9.
Table 9: Material properties for 4340 steel used in fork FEA model
Parameter 4340 Steel
Density (kg/mm3) 7.85 x 10−6
Young’s Modulus (kPa) 207 x 106
Poisson’s Ratio 0.30
Yield Strength (kPa) 472 x 103
Ultimate Tensile Strength (kPa) 745 x 103
To create a mesh, a 10-nodal 3D tetrahedral mesh, called “CTETRA(10)”
in NX, was used. An element size of 1 mm was chosen. The other mesh
parameters were set automatically by the software: the max Jacobian was 10
and the small feature tolerance was 10% of the element size. To constrain the
fork, a fixed constraint was placed on the inside of the hole for the top shaft,
to mimic the fork being pressed into place on the top shaft. According to
the previous motor shaft counterbalancing exercise, the maximum force being
applied to the fork from the motor shaft when it is balanced is about 52 N
and the maximum force when the motor shaft is unbalanced is about 91 N.
60
To demonstrate the robustness of the fork design, the unbalanced situation
was considered for the FEA simulation and a distributed load of 100 N was
applied to one leg of the fork. The applied load was distributed over the entire
internal fork leg surface to eliminate force concentration errors in the model.
The appropriateness of this choice will be explored subsequently. An image of
the fork with its mesh, constraint, and applied load is shown in Figure 53.
Figure 53: Fork with mesh, constraint, and applied load for FEA simulation.
The material model being used is elastic because the design should exist
well below the yield strength of the part. The simulation took 1 minute to
run on a Dell Precision M6700, 64 bit computer. The first result examined
was the nodal displacement magnitude, shown in Figure 54. The resulting
maximum displacement was 0.0079 mm, occurring at the end of the leg where
the force was applied. This result is on the same order of magnitude as a beam
deflection estimate of the situation using δ = PL3/3EI for a cantilever beam,
meaning the simulation results for stress can be viewed with some confidence
(Meriam et al. “Statics” 2007).
The resulting stress was reported as von Mises stress, which is commonly
used in the analysis of yielding for ductile materials, such as metals. This crite-
rion states that yielding occurs when the elastic energy of distortion, calculated
61
Figure 54: Displacement magnitude result of FEA simulation showing maxi-mum deflection is 0.0079 mm at the end of the leg where the force was applied.
as von Mises stress, reaches a critical value, the material’s yield strength. The
von Mises stress result from the FEA is shown in Figure 55. The highest stress
areas can be seen near the hole for the top shaft and the inside corner of the
leg where the force was applied. The simulation maximum stress was 21.94 x
103 kPa, compared to the yield strength of 472 x 103 kPa. This means the fork
will not yield and will remain safely in the elastic deformation region during
its life in the saw.
To verify the mesh size chosen in the simulation, a sensitivity study was
performed. The mesh size was varied from 0.5 mm to 6 mm. The maximum
deflection of the fork for each mesh size is plotted in Figure 56. The results
reported previously were for a mesh size of 1 mm. It can be seen that at 1
mm, the solution has indeed converged meaning that is an appropriate mesh
size to use. Additionally, it took about 1 minute to run the simulation with
1 mm meshes, it took less than a minute to run the larger meshes, but it
took 21 minutes to run it with a 0.5 mm mesh. So it is also most efficient,
computationally, to use a 1 mm mesh.
62
Figure 55: Von Mises stress result of FEA simulation showing maximum stressis 21.94 x 103 kPa at the hole for the shaft and the inside corner of the leg.
Figure 56: Maximum deflection of the fork for varying mesh sizes used in theFEA simulation.
To verify the choice to distribute the 100 N applied load along the inside
leg of the fork, another variation was explored. Instead of applying the load
over the full length of the fork leg, approximately 23.5 mm, the load was
applied to a 2 mm length centered on the motor shaft axis to simulate contact
with the offset bearing. This set of boundary conditions is shown in Figure
57.
63
Figure 57: Boundary conditions for FEA of 2 mm wide applied load on thefork.
Running the simulation with an applied load length of 2 mm and a mesh
size of 1 mm with all other parameters the same, showed a slight increase in
resulting deflection. The maximum deflection of the fork for this scenario is
0.0086 mm, as shown in Figure 58.
Figure 58: Deflection of fork under 100 N load applied over a 2 mm widesurface, showing the max deflection of 0.0086 mm.
This deflection result shows that greatly varying the area of the applied
load only slightly changes the resulting deflection. Because the variation is
64
less than 10% of the deflection value, it is safe to say that the FEA results
are reliable within typical error and factor of safety considerations. Therefore,
it can be confidently concluded that the fork stress will not reach the yield
strength and the deflection values during saw operation will be acceptably
small.
10 Natural Frequency
A natural frequency extraction analysis was performed to verify that the oper-
ating frequency of the saw, 10,000 oscillations/min, would not excite the mech-
anism’s natural frequency. This analysis was performed using the NX CAD
model, imported into Abaqus 6.13-3 simulation software. Brian Guerette, a
modeling and simulation expert at P&G Gillette assisted with the Abaqus sim-
ulation (Conversation with Gillette Modeling and Simulation Group November
2014). It was only necessary to analyze the top assembly of the mechanism
because the motor shaft is completely constrained at the motor by two bear-
ings. The stiffness of the motor assembly does not contribute to a natural
frequency near the operating frequency.
Material properties for steel, listed in Table 9, were used for the whole
assembly in this analysis as a reasonable simplification. The assembly was also
taken to be one piece, so clearances within the assembly were not considered.
This is due to the limitation of the software. The maximum frequency tested
was 300,000 cycles/min. The shaft was fixed at a point in the center of each
bearing in the X and Y directions only. This allows the shaft to bend and to
twist in torsion. The inner surfaces of the fork that contacts the eccentric
bearing was also fixed because of the stiffness of the motor shaft. These
boundary conditions are shown in Figure 59.
65
Figure 59: Boundary conditions on the top shaft assembly for the naturalfrequency analysis.
A 10 node quadratic tetrahedron mesh, called C3D10 in Abaqus, was
applied with a maximum size of 2 mm. Local seeding forced 2 elements to be
present through the thickness of the blade. The meshed assembly is shown in
Figure 60.
This analysis showed the first natural frequency of the mechanism to be
96.235 Hz, or 5774 cycles/min. The shape of this mode is a bending of the
blade, similar to a diving board. This is reasonable, as the blade is the least
stiff portion of this mechanism. The second natural frequency occurs at 575.92
Hz, or 34,555 oscillations per minute. Again, the shape of this mode is a
bending of the blade. The third natural frequency is at 780.94 Hz, or 46,856
oscillations/min. The shape of this mode is a twist of the blade around the
Y-axis. Figures 61, 62, and 63 show these first three natural frequencies, where
the colored scales show a unitless measure of displacement, with 1 being the
peak of the part mode shape of vibration.
66
Figure 60: Top shaft assembly shown with 10 node quadratic tetrahedronmesh.
Figure 61: The first natural frequency at 96.235 Hz.
67
Figure 62: The second natural frequency at 575.92 Hz.
Figure 63: The third natural frequency at 780.94 Hz.
For the driven mechanism oscillation of 10,000 oscillations/min, there
should be no excitation of the natural frequencies of this mechanism. Even
within ± 20% of the calculated natural frequency, the highest likely error
range according to Mr. Guerette’s experience, there is still no danger of caus-
ing resonance. Additionally, the fact that the first three natural frequencies are
68
confined to the blade confirms the assumption that the rest of the mechanism
will not contribute the first few natural frequencies.
11 Bearings
The bearings in the mechanism must be sized to survive for the life of the
saw. Because medical devices such as the sagittal saw must be sterilized in
an autoclave, they have sealed covers that are not designed to be accessed to
repair the device. Thus all components must be engineered to last the full life
of the saw. The SKF Bearing catalog (2002) gives two guidelines that could be
used for the basic life rating of the saw. It suggests that “technical apparatus
for medical use” should have an L10h life of 300-3,000 operating hours and
that “machines used for short periods or intermittently [such as] electric hand
tools” should have an L10h life of 3,000-8,000 hours.
To find another estimate of saw life, some basic assumptions can be made.
Estimating that a saw is used at maximum for a 10 year period is reasonable
because technology often becomes obsolete in less than that time frame. If
the saw is used continuously for 10 minutes in each surgery and is used for 3
surgeries per day, the tool would be used 30 minutes per day. Three surgeries
per day leaves time for cleaning the tool between each surgery. If surgeries take
place 5 days per week for 50 weeks each year, leaving time for holidays, then
the saw is used for 125 hours per year. With these assumptions, the saw would
see 1250 hours of use in a 10 year period. However, to be conservative, the
bearings in this design are designed to last at least the 8,000 hours suggested
by the SKF catalog.
The basic bearing life calculation is given by Equation 20, where L10 is
the fatigue life of the bearing in millions of revolutions with 90% reliability, C
69
is the dynamic load rating of the bearing, P is the applied load, and n is the
exponent which equals 3 for ball bearings and 10/3 for roller bearings (Norton
2006). For 99% reliability, L1, the L10 life must be multiplied by a factor, as
shown in Equation 21 (Norton 2006).
L10 =
(C
P
)n
(20)
L1 = 0.21L10 (21)
The top shaft assembly, which includes bearings A and B, as shown in
Figure 64, is oscillating, instead of making full revolutions. The life calcula-
tions for the top shaft bearings must take this into account with Equation 22,
where θ is half of the oscillation angle (IKO Bearings 2004). In this application,
θ = 2.5◦.
L10 =90
θ
(C
P
)n
(22)
The mechanism, with all bearings, is shown in Figure 64. Bearings C and
D were added to the mechanism to take up all of the forces from the fork
and counterbalance so that the shaft does not apply any external forces to the
motor and its internal bearings.
70
Figure 64: Mechanism shown with all bearings that must be sized.
The first step in sizing the bearings is determining the forces on them
using a free body diagram. The mechanism was once again split up into the
top shaft assembly and the motor shaft assembly for these calculations. The
free body diagram of the top shaft is shown in Figure 65 and the free body
diagram of the motor shaft is shown in Figure 66. All vectors are shown in
the positive sense. The weight and location of center of mass was determined
using the 3D model in NX.
In the top shaft assembly, the cutting force and weight must be specified.
The cutting force, FC , is actually a combination of two forces, as shown in
Figure 67: FT is the thrust force applied by the surgeon, and FS is the sawing
force due to the blade teeth cutting bone. The thrust force typically applied
to a sagittal saw is 10.0-30.0 N (James et al. 2013). The sawing force was
measured to be 0.5-6.5 N during a senior thesis by Matthew Kelly (James
2014). To be conservative, a thrust force of 30.0 N and a sawing force of 10.0
71
Figure 65: Free body diagram of top shaft assembly for bearing force analysis;FC is the applied cutting force, FW is the weight of the assembly, and FRA andFRB are the bearing reaction forces.
Figure 66: Free body diagram of motor shaft assembly for bearing force anal-ysis; Fapp is the applied force from the fork and counterbalance and FRC andFRD are the bearing reaction forces.
N was used for this analysis. Combining these values, a cutting force of 31.6
N is obtained.
In the motor shaft assembly, the combined force of the fork and the coun-
terbalance, Fapp, is applied at the eccentric bearing to correspond with the
maximum force calculated previously in the “Counterbalance” section to be
72
Figure 67: Cutting force, FC , is a combination of the applied thrust force, FT
and the sawing force, FS due to the blade teeth cutting bone.
52.0 N. Using the free body diagrams, forces and moments were summed to
calculate the bearing reaction forces. These forces applied to the system are
shown in Table 10 and the calculated bearing reaction forces are shown in
Table 11.
Table 10: Applied forces on the system.
Force Value (N)
FC 31.6
FW 1.4
Fapp 52.0
Table 11: Calculated forces on the bearings, where P refers to Equations 20and 22.
Force P (N)
FRA 59.5
FRB 26.6
FRC 120.7
FRD 68.7
Specific bearings of an appropriate physical size were chosen for each lo-
cation and their life was calculated using Equations 20, 21, and 22. All the
bearings selected are SKF brand (Kulpsvile, PA). Bearings A, C, D, and E, the
73
eccentric, are deep groove ball bearings. Bearing B is a needle roller bearing.
Bearings C and D were chosen to be the same to simplify the assembly.
The bearing chosen for the eccentric position has a cylindrical outer race.
To accommodate the fork’s rotation around that bearing, a crowned outer
race is necessary. None of the bearing manufacturers queried make a crowned
bearing that small as a stock component. The closest crowned bearing found
available in a catalog was the IKO bearing NART-5UUR. It had the desired in-
ner and outer diameters, but had a thickness of 12 mm when a 5 mm width was
required for this design (IKO 2004). However, a conversation with commercial
component supplier Motion Industries (Woburn, MA 2014) revealed that the
bearing manufacturers or a supplier like Motion Industries would modify the
bearings to create a crowned outer race as a special order item. This modi-
fication would not change the load rating of the bearing, so the following life
calculations would still apply. The bearing chosen for each location, their life
parameters, and their calculated life in hours, assuming a speed of 10,000 rpm,
are shown in Table 12. The dimensions of each bearing chosen are shown in
Table 13 (SKF USA 2002).
The saw’s maximum speed is 12,000 rpm so the bearings chosen have a
speed rating within a factor of safety from 1.4 to 3.6. Also, the static loads
on the bearings are negligible compared to their Static Load Ratings, C0.
The bearings have an L10 life of greater than the assumed saw life of 8,000
hours. All the bearings except the eccentric even have an L1 life greater than
8,000 hours. The eccentric bearing life limit is shown with Equation 23, which
applies the factor for 96% reliability (Norton, 2006).
L4 = 0.53L10 = 8, 592 hrs (23)
74
Table 12: Bearing life parameters and calculated life for each bearing position(see Figure 64).
Position Bearing
DynamicLoad
RatingC (N)
StaticLoad
RatingC0 (N)
SpeedRating(rpm)
L10 (hrs),Equa-
tions 20and 22
L1 (hrs),Equation
21
ASKF
61900-2z1950 750 34,000 2.24 x 106 4.70 x 105
BSKF
hk-08103690 4050 17,000 8.34 x 108 1.75 x 108
CSKF
6000-2z4620 1960 30,000 9.35 x 104 1.96 x 104
DSKF
6000-2z4620 1960 30,000 5.07 x 105 1.06 x 105
ESKF
625-2z1110 380 43,000 1.62 x 104 3.40 x 103
Table 13: Bearing dimensions for chosen bearing in each position.
Position BearingInner
Diameter(mm)
OuterDiameter
(mm)
Width(mm)
Seal /Shield
A SKF 61900-2z 10 +0−0.008 22 +0
−0.009 6 +0−0.12
DoubleShielded
(non-rubbing)
B SKF hk-0810 8 +0.031+0.013 12 +0
−0.008 10 +0−0.3
OpenEnds
C & D SKF 6000-2z 10 +0−0.008 26 +0
−0.009 8 +0−0.12
DoubleShielded
(non-rubbing)
E SKF 625-2z 5 +0−0.008 16 +0
−0.008 5 +0−0.12
DoubleShielded
(non-rubbing)
75
The 8,000 hour life estimate is likely higher than any real saw experiences.
In addition, the forces were all taken at worst case scenario. Therefore, the
eccentric bearing having an L4 life greater than 8,000 hours and the other
bearings having an L1 life greater than 8,000 hours is an acceptable design.
The fits of the bearings to the shafts and to the housings must also be
specified. The SKF catalog (2002) recommends a shaft fit of js4 with a housing
fit of H7 for ball bearings and a shaft fit of h5 with a housing fit of N6 for needle
roller bearings. The shaft and housing sizes used in this saw design, with their
tolerances, are shown in Table 14 (ISO 286-2 2010). This recommendation
applies for bearings with a high demand for accuracy and light loads, P ≤
0.05C. The forces on each bearing shown in Table 11 are less than 5% of the
Dynamic Load Ratings in Table 12 so this situation applies.
Table 14: SKF suggested fits for shafts and housings for bearings selected.
PositionSymbolicShaft Size
Shaft ToleranceSymbolic
Housing SizeHousing
Tolerance
A 10 js4 +0.002−0.002 22 H7 +0.021
−0
B 8 h5 +0−0.006 12 N6 −0.009
−0.020
C & D 10 js4 +0.002−0.002 26 H7 +0.021
−0
E 5 js4 +0.002−0.002 16 H7 +0.018
−0
These fits will be used for the top shaft, motor shaft, and the gear case
housings that support the bearings.
12 Blade Clamp Design
A blade clamp was designed to hold the blade in the saw during operation and
to allow tool-less blade change by the surgeons and operating room staff. The
clamp mechanism is shown in Figure 68. The clamp, (f), is threaded onto the
76
clamp shaft, shown as component (g). The clamp shaft can slide inside the
main shaft, component (d). The clamp spring, (b), is a compression spring
captured between the main shaft and the push-button, component (a). This
spring applies the force to keep the clamp tight against the blade, holding the
blade against the main shaft assembly. The push-button is attached to the
clamp shaft with a 3x12 mm spring pin (c), inserted through holes in both the
button and the clamp shaft. The push-button can also slide relative to the
main shaft, along its axis. When the surgeon or operating room staff press the
push-button, the spring is compressed and the button-clamp assembly slides
forward relative the main shaft, releasing the blade. This is shown in Figure
69.
Figure 68: Blade clamp mechanism side view (i) and cross-section view (ii),showing the push-button (a), clamp spring (b), spring pin (c), main shaft (d),blade (e), clamp (f), and clamp shaft (g).
77
Figure 69: Blade clamp mechanism opened to release the blade by pressingthe push-button.
The clamp spring chosen for this application is Associated Spring Ray-
mond C0240-024-0620. The spring parameters given in the Associated Spring
Raymond catalog (2013) are shown in Table 15.
Table 15: Spring parameters for clamp spring C0240-024-0620 from AssociatedSpring Raymond.
Parameter Value (English) Value (metric)
Outside Diameter 0.240 in 6.10 mm
Wire Diameter 0.024 in 0.61 mm
Free Length 0.620 in 15.75 mm
Spring Rate 9 lb/in 1.65 N/mm
Material Music Wire -
End Shape Squared and Ground -
This spring was chosen for its size and its spring rate. Ideally, as the
surgeon presses the push-button, they would feel the same force throughout
the travel of the button. This would lead the designer to choose a spring with
a low spring rate. However, this application requires a high clamping force
in a small package size to hold the blade in place during the saw’s opera-
tion, a requirement that is easier to accomplish with a high spring rate. The
recommended maximum force for push-button switches used infrequently is
78
2.8N-11.0 N, with a maximum button travel of 6 mm (Rahman et al. 1998).
Following this recommendation, the force when the clamp was open was tar-
geted at 11 N. The button travel was chosen to be 2.5 mm to allow enough
space to slide the blades in and out without requiring a long button push. The
forces applied to the clamp by the spring at each end of the clamp travel are
detailed in Table 16.
Table 16: Spring positions and forces for clamp spring.
Clamp Position Spring length (mm) Force (N) Force (lb)
Clamp “at rest”,Blade clamped
12 6.17 1.39
Clamp open 9.5 10.29 2.31
The spring life was determined using the Goodman failure criteria. The
equations for this calculation were taken from “Fundamentals of Machine El-
ements” (Hamrock et al. 2005). For Equations 24 through 34, the symbols
mean the following: Pa is the alternating force, Pmax is the maximum spring
force, Pmin is the minimum spring force, Pm is the mean force, D is the mean
coil diameter, OD is the spring outside diameter, d is the wire diameter, C is
the spring index which is a measure of coil curvature, Kw is the Wahl factor,
τa is the alternating stress in the spring, τm is the mean stress, Sut is the ul-
timate strength of the spring material, Ssu is the shear ultimate strength, S ′se
is the torsional endurance limit, and ns is the factor of safety of the spring
against failure for “infinite” life using the Goodman criteria. Sut is calculated
from the coefficients Ap and m that are obtained from a table based on the
spring material - music wire. S ′se is also given in a table based on the surface
treatment of the spring - unpeened in this case.
79
Pa =Pmax − Pmin
2=
10.23 − 6.17
2= 2.06N (24)
Pm =Pmax + Pmin
2=
10.23 + 6.17
2= 8.23N (25)
D = OD − d = 0.0061 − 0.00061 = 0.0055m (26)
C =D
d=
0.0055
0.00061= 9 (27)
Kw =4C − 1
4C − 4+
0.615
C=
4 · 9 − 1
4 · 9 − 4+
0.615
9= 1.16 (28)
τa =8DKwPa
πd3=
8 · 0.0055 · 1.16 · 2.06
π0.000613= 1.47 × 108 Pa (29)
τm =8DKwPm
πd3=
8 · 0.0055 · 1.16 · 8.23
π0.000613= 5.89 × 108 Pa (30)
Sut =Ap
dm=
2170
0.000610.146= 2332.39MPa (31)
Ssu = 0.60Sut = 0.60 · 2332.39 = 1399.44MPa (32)
S ′se = 310MPa (33)
ns =S ′seSsu
τaSsu + τmS ′se=
310 · 1399.44
1.47 × 108 · 1399.44 + 5.89 × 108 · 310
= 1.12 (34)
Equation 34 concludes that the chosen spring has a factor of safety of 1.12
for infinite life. While this is very close to 1, it is acceptable in this application
because the number of working cycles of the spring will be very low. The spring
will only be cycled when blades are being changed, perhaps on the order of
10 times per day. Also, the spring’s restive state is the extended position, or
lower stress state, which makes it less susceptible to early failure.
When the clamp engages the blade, the blade is located side to side by
the blade receiver on the main shaft assembly, as shown in Figure 70. The
blade is 0.27 mm thicker than the sides of the blade receiver slot, such that
the clamp will always contact the blade and capture it with the force discussed
previously. The slot on the receiver has a sliding clearance to allow accurate
80
Figure 70: Blade clamp locates blade side to side with a slot in the bladereceiver on the main shaft.
location of the blade without making it too difficult to change the blades.
The blade is located within the receiver slot by the clamp shaft. There is
a larger-diameter portion of the clamp shaft with lead-in chamfers that will
engage the larger-diameter, circular end of the blade slot. The surgeon or
operating room staff would insert the blade into the clamp until the clamp
engages the end of the blade slot. This blade position is shown in Figure 71,
with the receiver and main shaft hidden in the image to provide clarity. To
prevent the larger-diameter shaft end and the slot in the blade receiver from
over-constraining the blade, the shaft tolerance is on the minus side. This
allows the shaft to capture the blade, but provide enough clearance that it can
be located right to left in the receiver slot.
13 Gear Case Design
The gear case is the structure that holds the mechanism in place, by contacting
the bearing outer races, and provides the connection to the ergonomic shell
that the surgeon would hold. Both the gear case and the previously discussed
blade clamp are starting points that need to be integrated into the ergonomic
shell. An iterative process, not undertaken in this study, between the shell
81
Figure 71: The blade is located within the clamp by the clamp shaft.
design and the gear case design would be required to find the optimal solution.
The exterior gear case design is shown in Figure 72. It is a clamshell design,
with two halves being held together by 6 socket head cap screws, size #6-32
x 0.375 in. The main shaft and blade clamp protrude out of the front and
the blade clamp push-button protrudes out of the back of the shell. The push
button extends 7.25 mm from the gear case to allow it to protrude through
the ergonomic shell that would fit over the gear case. This gear case would be
made from aluminum to allow the saw to be sterilized at high temperatures.
Aside from the gear case material choice, the effects of sterilizing on the saw
are outside the scope of this design.
The inside of the gear case is shown in Figure 73. The contact surfaces
with the bearing outer races have the tolerances previously specified in Table
14. There is clearance for the counterbalance and the fork to move inside the
shell. Bosses were added where necessary to capture the fasteners. The spring
pin on the push-button is inside the clamshell so that it cannot collect debris
during surgery.
In Figure 73, the motor is shown captured in a pocket in the gear case.
This is represented as the canned motor used in the Stryker saw, as discussed
previously. It is reasonable to assume that the same motor used in the Stryker
82
Figure 72: Outside of the gear case design to support the saw mechanism.
Figure 73: Inside of the gear case design to support the saw mechanism.
83
saw, or one of a similar package size, would be suitable for this new saw design.
While motor design is beyond the scope of this thesis, it was assumed that the
motor is sufficiently stiff to maintain the maximum blade speed under load,
which is the worst case scenario that was employed in the kinetic analysis.
To prevent mechanism movement under the cutting and vibration loads,
the entire oscillating shaft assembly must be captured axially within the gear
case. This is accomplished with a shoulder in the gear case located behind the
ball bearing on the main shaft and with a retaining ring in front of the bearing.
This arrangement is shown in Figures 74 and 75. The retaining ring chosen is
American Ring and Tool 5001-086. This is a bowed internal retaining ring to
provide initial axial pressure on the bearing, preventing axial movement due
to tolerance gaps. A catalog image of the retaining ring is shown in Figure
76 and the ring’s dimensions are given in Table 17 (American Ring and Tool
2007).
Figure 74: Side view of the main shaft, showing the gear case shoulder (a) andretaining ring (b) to capture the shaft axially.
84
Figure 75: Angled views of the gear case shoulder (a) and retaining ring (b)to capture the shaft axially.
Figure 76: American Ring and Tool retaining ring 5001-086 catalog sketch;dimensions shown in Table 17.
85
Table 17: American Ring and Tool retaining ring 5001-086 dimensions; seesketch in Figure 76.
Property Value
Application Diameter (B) 22 mm
Groove Diameter (G) 23.37 +0.08−0.10 mm
Groove Width (W) 1.57 +0.08−0.10 mm
Groove Depth 0.69 mm
Ring Free Diameter (D) 24.41 +0.38−0.38 mm
Ring Thickness (t) 1.07 +0.05−0.05 mm
Overall Bow Height (V) 1.85 mm
Clearance Diameter in Bore (C) 13.72 mm
Clearance Diameter in Groove (C1) 14.99 mm
Contact between the eccentric bearing and the fork will see the most
friction and wear in this design, making it a likely failure point. To combat
this, grease could be added to those surfaces prior to the gear case being
sealed. Based on a recommendation from an expert high-speed machinery
assembler, Shell Alvania EP2 Lithium Grease is a good fit for this application
(Conversation with Gillette Machine Assembly Shop July 2014). Based on
company tests, this grease has provided the longest service life of cam follower
devices under similar loading geometry and forces as considered here for the
sagittal saw mechanism. The physical characteristics of this grease are shown
in Table 18 (Shell 2007).
86
Table 18: Typical physical characteristics of Shell Alvania EP2 LithiumGrease.
Property Value
Soap Type Lithium
Base Oil Mineral
Kinematic Viscosity
at 40◦C cSt 189
at 100◦C cSt 15.6
(IP 71/ASTM-D445)
Dropping Point◦C 180
(IP 132)
Core Penetration
Worked at 25◦C 0.1 mm 265-295
(IP 50/ASTM-D217)
To prevent grease or other contaminants from getting out of the saw and to
prevent contaminants from surgeries or the sterilization process from getting
into the saw, it must be sealed. The ergonomic shell that is the topic of a
parallel thesis is where the medical-grade sealing would occur. However, the
mechanism and gear case has a basic level of sealing as well. At the bottom
of the saw handle, the motor will act as a seal. Towards the blade, the gear
case shoulder, front bearings, and retaining ring will seal off the mechanism.
Towards the push button, an o-ring was added to provide a contaminant seal
and to smooth out the motion of pushing the button. A close up of the o-ring
is shown in Figure 77.
87
Figure 77: O-ring providing sealing around the push button.
The O-ring chosen is an Ace Seal 12x1.5 mm ring, which means it has an
ID of 12 mm and a thickness of 1.5 mm. The material is Buna/Nitrile with
a standard hardness of 10 on the Durometer scale, chosen for its resistance
to oil, air, water, silicone greases, and alcohols (Ace Seal and Rubber 2014).
The pocket in the gear case for the o-ring was designed based on a dynamic
o-ring seal with no backup ring (Parker 2007). A sketch of the pocket with
the chosen dimensions is shown in Figure 78.
Figure 78: O-ring pocket in gear case.
88
14 Prototype
In order to demonstrate the feasibility of the in-line sagittal bone saw design,
a prototype was created. The prototype is shown in Figure 79 with the gear
case cover off and on. Table 19 lists the materials each component was made
from. These prototype components were fabricated by James E. Hoffman
and Vincent J. Miraglia of Tufts University. For the bearings and O-ring the
manufacturers and part numbers specified in previous sections of this paper
were used. The bowed retaining ring previously specified was replaced with
a straight retaining ring and the specified spring for the push-button was re-
placed with a much lighter spring for the prototype to avoid damaging the
rapid prototyped plastic components. A standard knurled knob was threaded
onto the end of the motor shaft in place of the motor. This allows the mech-
anism to be turned by hand with the gear case open or closed to demonstrate
the motion.
Table 19: Materials used to fabricate prototype components.
Component Material
Gear Case Rapid Prototype Plastic
Fork Rapid Prototype Plastic
Counterbalance Rapid Prototype Plastic
Blade Receiver Rapid Prototype Plastic
Top Shaft Steel
Motor Shaft Steel
Clamp Shaft Steel
Blade Clamp Aluminum
Push-Button Aluminum
Building the prototype demonstrated that the design was sound. The
components fit and worked together as expected. The blade clamp was shown
89
Figure 79: New mechanism prototype shown with the gear case cover off andon.
to be a viable design for tool-less blade changes. The motion of the blade tip
resulting from rotating the motor shaft was as designed.
The prototype made it obvious that tight tolerances are required for this
design to work. Due to the materials used and the nature of the component
manufacturing, the fits between the shafts, bearings, and housings were not
very exact. Also the fit between the fork and the top shaft allowed play
between the two. These errors lead to the fork moving in a complex fashion
with both axial rotation and twist perpendicular to the shaft, rather that the
pure rotation it is supposed to undergo. In addition to better materials and
fabrication, this twist could also be minimized by increasing the axial length
that the fork interacts with the shaft.
After working with the prototype, one improvement that could be made to
90
this design is to increase the distance that the blade clamp opens up to allow
blade changes. In order to minimize the package size, a small translation was
chosen for the clamp shaft. However, when considering operating room staff
handling this tool with gloves on, a larger opening that allows a greater margin
of error when loading the blade would be desirable.
15 Conclusion
This paper outlines a new mechanism design for a bone saw that allows an
in-line body geometry for improved ergonomics over the traditional pistol-
grip designs. To match the operating parameters of contemporary oscillating
bones saws, the new in-line mechanism is designed to oscillate a saw blade at
a maximum of 12,000 cycles per minute through a 5◦ arc.
To create an experimentally verified CAD model to work from, a Bosch
saw was disassembled and modeled in NX, creating a motion simulation that
output the sinusoidal position, velocity, and acceleration of the saw blade for
a given motor speed. This was compared, with a high level of agreement, to
an analytical Vector Loop model of the mechanism. These models were then
verified experimentally so that there is a high level of confidence that the final
design will behave as predicted by the model. The speed of the Bosch saw
was determined using a microphone to measure the frequency and amplitude
of the sound of the saw and confirmed using a strobe light experimental setup.
These tests confirmed that the operating speeds of the Bosch saw covered the
operating speeds in the design specification. The velocity of the blade was
then measured using a laser vibrometer. The maximum experimental velocity
of the blade at each speed setting was compared to the predicted velocities
and showed strong agreement, thus validating the models.
91
To minimize vibration of the saw, the mechanism was counterbalanced.
The top shaft axis must balance the fork and the blade as they oscillate. This
is accomplished by adjusting the fork design such that the center of mass of
the assembly is placed on the axis of rotation. Using a different blade than the
one the saw is balanced for, the Stryker Dual Cut 4125-127-100, will change
the position of the center of mass and cause a slight force imbalance. The
motor shaft must balance both the rotating centripetal force of the eccentric
bearing and the oscillating force of the fork. Since the force of the fork does not
rotate around the motor shaft, this part of the mechanism cannot be perfectly
balanced; instead, the square root of the sum of the squares of the forces must
be minimized. This was accomplished with the addition of a counterbalance
pressed onto the motor shaft. The balanced motor shaft sees a varying force
between 48 N and 53 N as it rotates while the unbalanced motor shaft sees
9 N to 91 N. The smoother, balanced force creates a manageable, consistent
vibration that is commonly felt in power tools.
To verify that the steel fork design is structurally sound under critical
loads, an FEA analysis was performed. A fixed constraint was applied to the
inside of the hole for the top shaft while a load of 100 N was distributed over
one leg of the fork. The force was chosen to be 100 N to cover the balanced
shaft applied load of 52 N with a factor of safety. The resulting maximum
displacement of the fork leg was 0.0079 mm. The maximum von Mises stress
from the simulation was 21.94 x 103 kPa, well below the material yield strength
of 472 x 103 kPa. The FEA showed that the fork will remain safely in the elastic
deformation region and will not yield during the life of the saw.
The bearing life of all bearings in the saw was also considered. The bear-
ings must last for the life of the saw, which can be estimated at 1250 hours
of use. The SKF bearing catalog suggested generically that medical devices
92
should have a life of 8,000 hours. The forces being applied to the system are the
cutting force and the 52 N motor shaft force from the counterbalance discus-
sion. The cutting force is a combination of the sawing force, due to the blade
teeth cutting bone, and the thrust force applied by the surgeon. This cutting
force was estimated to be 31.6 N using the conservative end of ranges for the
sawing force and the thrust force found in literature. Calculating the reaction
force at each bearing and the resulting life of each bearing showed that all
bearings have an L10 life greater than 8,000 hours. In fact, all bearings except
the eccentric have an L1 life greater than 8,000 hours. The eccentric bearing
has an L4 life of 8,592 hours. Knowing that 8,000 hours is an unrealistically
high life limit, the bearings should have infinite life as compared to the life of
the saw.
A blade clamp was also designed to provide a way to change the blades
without using an extra tool. The clamp is actuated by a spring-loaded button.
Following recommendations found in literature, the button travel is 2.5 mm
with a 6.17 N pre-load and a 10.29 N force when fully opened. The spring
life for this scenario was calculated using the Goodman failure criteria. This
showed the spring to have an infinite life with a factor of safety of 1.12. The
number of working cycles of this spring will be very low and its restive state
is its lower stress state, therefore this factor of safety is acceptable.
A gear case was designed to hold the mechanism in place inside the er-
gonomic shell by contacting the bearings. It includes a bowed internal retain-
ing ring to keep the top shaft in place axially and to take up tolerance gaps in
that direction.
A grease such as Shell Alvania EP2 Lithium Grease would be applied inside
the gear case between the eccentric bearing and the fork to reduce wear. An
O-ring was also added to the gear case around the blade clamp button to
93
prevent grease leakage.
A prototype was designed and built using rapid prototyping that success-
fully demonstrated the feasibility of the new in-line mechanism design. It
successfully showed the oscillating motion of the blade that can be generated
by turning the motor shaft.
The design outlined above accomplishes the research goal of designing a
balanced drive mechanism for a new, in-line oscillating bone saw. It covers the
design as well as the engineering including model validation, counterbalance
optimization, component structural analysis, bearing life, blade clamp design
with spring fatigue life, and gear case design.
94
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98
B MATLAB Codes
B.1 Vector Loop Analysis of Bosch Saw
% Calcu la te po s i t i on , v e l o c i t y , and a c c e l e r a t i o n o fpo int o f i n t e r e s t on blade us ing vec to r loop methodand d i f f e r e n t i a t i n g with time . This f i l e accounts f o r
an r 5 , v e r t i c a l d i sp lacement between po int o fcontact o f f o rk and bear ing and motor a x i s . r 5 = r 2∗ s i n ( the ta 2 ) in y d i r e c t i o n only . r 3 i s now a l s o x−d i r e c t i o n only . THIS MATCHES NX DATA.
c l e a r a l lc l o s e a l l
%osc =10000; %o s c i l l a t i o n s /min , speed o f input l i n k
%w2=osc /60∗2∗ pi ; %rad/s , o s c i l l a t i o n speed , speed o finput l i n k
w2=166.6∗2∗ pi ; %rad/s , o s c i l l a t i o n speed o f input l i n ka2=0; %rad/s , angular a c c e l e r a t i o n o f input l i n ki =1; %indexr =8; %mm, rad iu s o f e c c e n t r i c bear inge=1; %mm, e c c e n t r i c d i s t anc er2=e ; %mm, length o f input l i n kr3=r ; %mm, length o f connect ing l i n kh=30.1; %mm, he ight from cente r o f e c c e n t r i c bear ing to
a x i s o f blade r o t a t i o np=70;%mm, d i s t anc e from a x i s o f blade r o t a t i o n to
l o c a t i o n o f acce l e romete r / po int o f i n t e r e s talpha (1 ) =0; %rad , i n i t i a l i z i n g angular v e l o c i t y
f o r t =0 :0 . 0002 : 0 . 1 %sectime ( i )=t ; %sectheta2 ( i )=w2∗t−pi /2 ; %rad , ang le o f e c c e n t r i c and
ang le o f input l ink , −90 deg to match NX model
i f theta2 ( i )>2∗pi %r e s o l v e theta to between 0 and 2p i
x=f l o o r ( theta2 ( i ) /(2∗ pi ) ) ;theta2 ( i )=theta2 ( i )−2∗pi ∗x ;
end
r4 ( i )=r2∗ cos ( theta2 ( i ) )+r3 ; %mm, x p o s i t i o n o fs l i d e r r e l a t i v e to ground p ivot
99
gamma( i )=atan ( ( r4 ( i )−r3 ) /h) ; %rad , ang le o f r o t a t i o no f blade
gammad( i )=gamma( i ) ∗180/ p i ; %deg , ang le o f r o t a t i o no f blade
disp lacement ( i )=p∗ s i n (gamma( i ) ) ; %mm, disp lacementmagnitude o f po int o f i n t e r e s t
x d i sp ( i )=disp lacement ( i )∗ s i n ( p i/2−gamma( i ) ) ; %mm, xdisp lacement o f po int o f i n t e r e s t
y d i sp ( i )=−disp lacement ( i )∗ cos ( p i/2−gamma( i ) ) ; %mm,y disp lacement o f po int o f i n t e r e s t
v4 ( i )=−r2∗ s i n ( theta2 ( i ) )∗w2 ; %mm/s , speed o f s l i d e r
v tan ( i )=v4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f v e l o c i t y to blade l e g
omega ( i )=v tan ( i ) /h ; %rad/s , angular v e l o c i t y o fa x i s o f blade r o t a t i o n
v e l o c i t y ( i )=p∗omega ( i ) ; %mm/s , v e l o c i t y magnitude atpo int o f i n t e r e s t
x v e l ( i )=v e l o c i t y ( i )∗ cos (gamma( i ) ) ; %mm/s , xv e l o c i t y at po int o f i n t e r e s t
y v e l ( i )=−v e l o c i t y ( i )∗ s i n (gamma( i ) ) ; %mm/s , yv e l o c i t y at po int o f i n t e r e s t
a c c e l 4 ( i )=−r2∗ cos ( theta2 ( i ) )∗w2ˆ2−r2∗ s i n ( theta2 ( i ) )∗a2 ; %mm/ s ˆ2 , a c c e l e r a t i o n o f s l i d e r
a c c e l t a n ( i )=a c c e l 4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f a c c e l e r a t i o n to blade l e g
alpha ( i )=a c c e l t a n ( i ) /h ; %rad/ s ˆ2 , angulara c c e l e r a t i o n o f a x i s o f blade r o t a t i o n
a c c e l ( i )=p∗alpha ( i ) ; %mm/ s ˆ2 , a c c e l e r a t i o n magnitudeat po int o f i n t e r e s t
x a c c e l ( i )=a c c e l ( i )∗ cos (gamma( i ) ) ; %mm/ s ˆ2 , xa c c e l e r a t i o n at po int o f i n t e r e s t
y a c c e l ( i )=a c c e l ( i )∗ s i n (gamma( i ) ) ; %mm/ s ˆ2 , ya c c e l e r a t i o n at po int o f i n t e r e s t
i=i +1;end
100
B.2 Top Shaft Forces of 3 Different Blades
% Calcu la te f o r c e s on top s h a f t assembly f o r 3 d i f f e r e n tStryker b lades .
c l e a r a l lc l o s e a l l
osc =10000; %o s c i l l a t i o n s /min , speed o f input l i n k
w2=osc /60∗2∗ pi ; %rad/s , o s c i l l a t i o n speed , speed o finput l i n k
a2=0; %rad/s , angular a c c e l e r a t i o n o f input l i n ki =1; %indexr =8; %mm, rad iu s o f e c c e n t r i c bear inge =1.46; %mm, e c c e n t r i c d i s t ancer2=e ; %mm, length o f input l i n kr3=r ; %mm, length o f connect ing l i n kh=33.5; %mm, he ight from cente r o f e c c e n t r i c bear ing to
a x i s o f blade r o t a t i o np=60;%mm, d i s t anc e from a x i s o f blade r o t a t i o n to
l o c a t i o n o f acce l e romete r / po int o f i n t e r e s talpha (1 ) =0; %rad , i n i t i a l i z i n g angular v e l o c i t y
f o r ang le =0:1:360 %degtheta2d ( i )=ang le ; %degtheta2 ( i )=theta2d ( i ) ∗( p i /180)−(p i /2) ; %rad , ang le o f
e c c e n t r i c and ang le o f input l ink , −90 deg tomatch NX model
i f theta2 ( i )>2∗pi %r e s o l v e theta to between 0 and 2p i
x=f l o o r ( theta2 ( i ) /(2∗ pi ) ) ;theta2 ( i )=theta2 ( i )−2∗pi ∗x ;
end
r4 ( i )=r2∗ cos ( theta2 ( i ) )+r3 ; %mm, x p o s i t i o n o fs l i d e r r e l a t i v e to ground p ivot
gamma( i )=atan ( ( r4 ( i )−r3 ) /h) ; %rad , ang le o f r o t a t i o no f blade
gammad( i )=gamma( i ) ∗180/ p i ; %deg , ang le o f r o t a t i o no f blade
101
disp lacement ( i )=p∗ s i n (gamma( i ) ) ; %mm, disp lacementmagnitude o f po int o f i n t e r e s t
x d i sp ( i )=disp lacement ( i )∗ s i n ( p i/2−gamma( i ) ) ; %mm, xdisp lacement o f po int o f i n t e r e s t
y d i sp ( i )=−disp lacement ( i )∗ cos ( p i/2−gamma( i ) ) ; %mm,y disp lacement o f po int o f i n t e r e s t
v4 ( i )=−r2∗ s i n ( theta2 ( i ) )∗w2 ; %mm/s , speed o f s l i d e r
v tan ( i )=v4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f v e l o c i t y to blade l e g
omega ( i )=v tan ( i ) /h ; %rad/s , angular v e l o c i t y o fa x i s o f blade r o t a t i o n
v e l o c i t y ( i )=p∗omega ( i ) ; %mm/s , v e l o c i t y magnitude atpo int o f i n t e r e s t
x v e l ( i )=v e l o c i t y ( i )∗ cos (gamma( i ) ) ; %mm/s , xv e l o c i t y at po int o f i n t e r e s t
y v e l ( i )=−v e l o c i t y ( i )∗ s i n (gamma( i ) ) ; %mm/s , yv e l o c i t y at po int o f i n t e r e s t
a c c e l 4 ( i )=−r2∗ cos ( theta2 ( i ) )∗w2ˆ2−r2∗ s i n ( theta2 ( i ) )∗a2 ; %mm/ s ˆ2 , a c c e l e r a t i o n o f s l i d e r
a c c e l t a n ( i )=a c c e l 4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f a c c e l e r a t i o n to blade l e g
alpha ( i )=a c c e l t a n ( i ) /h ; %rad/ s ˆ2 , angulara c c e l e r a t i o n o f a x i s o f blade r o t a t i o n
a c c e l ( i )=p∗alpha ( i ) ; %mm/ s ˆ2 , a c c e l e r a t i o n magnitudeat po int o f i n t e r e s t
x a c c e l ( i )=a c c e l ( i )∗ cos (gamma( i ) ) ; %mm/ s ˆ2 , xa c c e l e r a t i o n at po int o f i n t e r e s t
y a c c e l ( i )=a c c e l ( i )∗ s i n (gamma( i ) ) ; %mm/ s ˆ2 , ya c c e l e r a t i o n at po int o f i n t e r e s t
i=i +1;end
f o r c e a =0.1518∗0.∗omega . ˆ 2 ;f o r c e b =0.1411∗ .0058218.∗ omega . ˆ 2 ;f o r c e c =0.1385∗ .0062572.∗ omega . ˆ 2 ;
p l o t ( theta2d , f o r c e a , ’ b ’ )hold onp lo t ( theta2d , f o r c e b , ’ r ’ )p l o t ( theta2d , f o r c e c , ’ g ’ )
102
B.3 Basic Motor Shaft Force Calculation
% Cal cu l a t e s f o r c e magnitude o f c e n t r i p e t a l and fo rkf o r c e s us ing vec to r
% loop method .
c l e a r a l lc l o s e a l l
osc =10000; %o s c i l l a t i o n s /min , speed o f input l i n k
w2=osc /60∗2∗ pi ; %rad/s , o s c i l l a t i o n speed , speed o finput l i n k
a2=0; %rad/s , angular a c c e l e r a t i o n o f input l i n ki =1; %indexr =8; %mm, rad iu s o f e c c e n t r i c bear inge =1.46; %mm, e c c e n t r i c d i s t ancer2=e ; %mm, length o f input l i n kr3=r ; %mm, length o f connect ing l i n kh=33.5; %mm, he ight from cente r o f e c c e n t r i c bear ing to
a x i s o f blade r o t a t i o np=60;%mm, d i s t anc e from a x i s o f blade r o t a t i o n to
l o c a t i o n o f acce l e romete r / po int o f i n t e r e s talpha (1 ) =0; %rad , i n i t i a l i z i n g angular v e l o c i t y
n=1;
m fork =0.0628; %kg , mass o f f o rk
m nobal =0.01538; %kg , mass o f s h a f t and e c c e n t r i cbear ing , no balance
x nc =0.5704; %mm, x p o s i t i o n o f cent ro id , no balancez nc =0; %mm, z p o s i t i o n o f centro id , no balancer nc=s q r t ( x nc ˆ2 + z nc ˆ2) ; %mm, rad iu s to c en t r o id o f
motorshaft , no balance
F nc=m nobal∗ r nc /1000∗w2ˆ2 ; %N, c e n t r i f u g a l r e a c t i o nf o r c e o f bear ing ( no balance ) r o t a t i n g about motora x i s
c e n t r a n g l e=mod( atan2 (1∗ z c−x c ∗0 ,1∗ x c+0∗ z c ) ,2∗ pi )∗180/ p i ; %ang le between e c c e n t r i c bear ing andcen t r o i d
103
f o r ang le =0:1:360 %degtheta2d ( i )=ang le ; %degtheta2 ( i )=theta2d ( i ) ∗( p i /180)−(p i /2) ; %rad , ang le o f
e c c e n t r i c and ang le o f input l ink , −90 deg tomatch NX model
i f theta2 ( i )>2∗pi %r e s o l v e theta to between 0 and 2p i
x=f l o o r ( theta2 ( i ) /(2∗ pi ) ) ;theta2 ( i )=theta2 ( i )−2∗pi ∗x ;
end
r4 ( i )=r2∗ cos ( theta2 ( i ) )+r3 ; %mm, x p o s i t i o n o fs l i d e r r e l a t i v e to ground p ivot
gamma( i )=atan ( ( r4 ( i )−r3 ) /h) ; %rad , ang le o f r o t a t i o no f blade
gammad( i )=gamma( i ) ∗180/ p i ; %deg , ang le o f r o t a t i o no f blade
disp lacement ( i )=p∗ s i n (gamma( i ) ) ; %mm, disp lacementmagnitude o f po int o f i n t e r e s t
x d i sp ( i )=disp lacement ( i )∗ s i n ( p i/2−gamma( i ) ) ; %mm, xdisp lacement o f po int o f i n t e r e s t
y d i sp ( i )=−disp lacement ( i )∗ cos ( p i/2−gamma( i ) ) ; %mm,y disp lacement o f po int o f i n t e r e s t
v4 ( i )=−r2∗ s i n ( theta2 ( i ) )∗w2 ; %mm/s , speed o f s l i d e r
v tan ( i )=v4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f v e l o c i t y to blade l e g
omega ( i )=v tan ( i ) /h ; %rad/s , angular v e l o c i t y o fa x i s o f blade r o t a t i o n
v e l o c i t y ( i )=p∗omega ( i ) ; %mm/s , v e l o c i t y magnitude atpo int o f i n t e r e s t
x v e l ( i )=v e l o c i t y ( i )∗ cos (gamma( i ) ) ; %mm/s , xv e l o c i t y at po int o f i n t e r e s t
y v e l ( i )=−v e l o c i t y ( i )∗ s i n (gamma( i ) ) ; %mm/s , yv e l o c i t y at po int o f i n t e r e s t
a c c e l 4 ( i )=−r2∗ cos ( theta2 ( i ) )∗w2ˆ2−r2∗ s i n ( theta2 ( i ) )∗a2 ; %mm/ s ˆ2 , a c c e l e r a t i o n o f s l i d e r
a c c e l t a n ( i )=a c c e l 4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f a c c e l e r a t i o n to blade l e g
104
alpha ( i )=a c c e l t a n ( i ) /h ; %rad/ s ˆ2 , angulara c c e l e r a t i o n o f a x i s o f blade r o t a t i o n
a c c e l ( i )=p∗alpha ( i ) ; %mm/ s ˆ2 , a c c e l e r a t i o n magnitudeat po int o f i n t e r e s t
x a c c e l ( i )=a c c e l ( i )∗ cos (gamma( i ) ) ; %mm/ s ˆ2 , xa c c e l e r a t i o n at po int o f i n t e r e s t
y a c c e l ( i )=a c c e l ( i )∗ s i n (gamma( i ) ) ; %mm/ s ˆ2 , ya c c e l e r a t i o n at po int o f i n t e r e s t
F fork ( i )=a c c e l 4 ( i ) /1000∗m fork ; %N, f o r c e o f f o rkin x
F nz ( i )=s i n ( theta2 ( i ) )∗F nc ; %N, no balanceF nx ( i )=cos ( theta2 ( i ) )∗F nc+F fork ( i ) ; %N, no
balanceF nmag( i )=s q r t ( F nz ( i )ˆ2+F nx ( i ) ˆ2) ; %N, no balance
n r e s u l t a n g l e ( i )=atan2 ( F nz ( i ) , F nx ( i ) ) ∗180/ p i ; %deg, r e s u l t a n t f o r c e ang le with no balance
i f i <120 %r e s o l v e r e s u l t i n g ang lei f n r e s u l t a n g l e ( i )<0
n r e s u l t a n g l e ( i )=n r e s u l t a n g l e ( i ) +360;endend
i=i +1;end
f i g u r e ;p l o t ( theta2d , F nmag , ’ b ’ )x l a b e l ( ’ Motor Rotation ( deg ) ’ )y l a b e l ( ’ Force (N) ’ )
f i g u r e (2 ) ;p l o t ( theta2d , F nx , ’ g ’ )hold onp lo t ( theta2d , F nz , ’m’ )legend ( ’ F nx ’ , ’ F nz ’ )
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B.4 Motor Shaft Counterbalance Angular Position
% Calcu la te optimal ang le f o r motor s h a f t counterba lance.
c l e a r a l lc l o s e a l l
osc =10000; %o s c i l l a t i o n s /min , speed o f input l i n k
w2=osc /60∗2∗ pi ; %rad/s , o s c i l l a t i o n speed , speed o finput l i n k
a2=0; %rad/s , angular a c c e l e r a t i o n o f input l i n ki =1; %indexr =8; %mm, rad iu s o f e c c e n t r i c bear inge =1.46; %mm, e c c e n t r i c d i s t ancer2=e ; %mm, length o f input l i n kr3=r ; %mm, length o f connect ing l i n kh=33.5; %mm, he ight from cente r o f e c c e n t r i c bear ing to
a x i s o f blade r o t a t i o np=60;%mm, d i s t anc e from a x i s o f blade r o t a t i o n to
l o c a t i o n o f acce l e romete r / po int o f i n t e r e s talpha (1 ) =0; %rad , i n i t i a l i z i n g angular v e l o c i t y
n=1;
m fork =0.0628; %kg , mass o f f o rk
m shaft =.026857; %kg , mass o f e c c e n t r i c motorshaft ,e c c e n t r i c bear ing , and motorshaft counterba lance
x c =−1.78896; %mm, x p o s i t i o n o f c en t r o id o f motorshaftz c =0; %mm, z p o s i t i o n o f c en t r o id o f motorshaftr c=s q r t ( x c ˆ2 + z c ˆ2) ; %mm, rad iu s to c en t r o id o f
motorshaft
F c=m shaft∗ r c /1000∗w2ˆ2 ; %N, c e n t r i f u g a l r e a c t i o nf o r c e o f bear ing / counterba lance r o t a t i n g about motora x i s
f o r c e n t r a n g l e =0:1:360 %deg , ang le between e c c e n t r i cbear ing and cen t r o i d
i =1;c l e a r F mag ;
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f o r ang le =0:1:360 %degtheta2d ( i )=ang le ; %degtheta2 ( i )=theta2d ( i ) ∗( p i /180)−(p i /2) ; %rad , ang le o f
e c c e n t r i c and ang le o f input l ink , −90 deg tomatch NX model
i f theta2 ( i )>2∗pi %r e s o l v e theta to between 0 and 2p i
x=f l o o r ( theta2 ( i ) /(2∗ pi ) ) ;theta2 ( i )=theta2 ( i )−2∗pi ∗x ;
end
r4 ( i )=r2∗ cos ( theta2 ( i ) )+r3 ; %mm, x p o s i t i o n o fs l i d e r r e l a t i v e to ground p ivot
gamma( i )=atan ( ( r4 ( i )−r3 ) /h) ; %rad , ang le o f r o t a t i o no f blade
gammad( i )=gamma( i ) ∗180/ p i ; %deg , ang le o f r o t a t i o no f blade
disp lacement ( i )=p∗ s i n (gamma( i ) ) ; %mm, disp lacementmagnitude o f po int o f i n t e r e s t
x d i sp ( i )=disp lacement ( i )∗ s i n ( p i/2−gamma( i ) ) ; %mm, xdisp lacement o f po int o f i n t e r e s t
y d i sp ( i )=−disp lacement ( i )∗ cos ( p i/2−gamma( i ) ) ; %mm,y disp lacement o f po int o f i n t e r e s t
v4 ( i )=−r2∗ s i n ( theta2 ( i ) )∗w2 ; %mm/s , speed o f s l i d e r
v tan ( i )=v4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f v e l o c i t y to blade l e g
omega ( i )=v tan ( i ) /h ; %rad/s , angular v e l o c i t y o fa x i s o f blade r o t a t i o n
v e l o c i t y ( i )=p∗omega ( i ) ; %mm/s , v e l o c i t y magnitude atpo int o f i n t e r e s t
x v e l ( i )=v e l o c i t y ( i )∗ cos (gamma( i ) ) ; %mm/s , xv e l o c i t y at po int o f i n t e r e s t
y v e l ( i )=−v e l o c i t y ( i )∗ s i n (gamma( i ) ) ; %mm/s , yv e l o c i t y at po int o f i n t e r e s t
a c c e l 4 ( i )=−r2∗ cos ( theta2 ( i ) )∗w2ˆ2−r2∗ s i n ( theta2 ( i ) )∗a2 ; %mm/ s ˆ2 , a c c e l e r a t i o n o f s l i d e r
a c c e l t a n ( i )=a c c e l 4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f a c c e l e r a t i o n to blade l e g
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alpha ( i )=a c c e l t a n ( i ) /h ; %rad/ s ˆ2 , angulara c c e l e r a t i o n o f a x i s o f blade r o t a t i o n
a c c e l ( i )=p∗alpha ( i ) ; %mm/ s ˆ2 , a c c e l e r a t i o n magnitudeat po int o f i n t e r e s t
x a c c e l ( i )=a c c e l ( i )∗ cos (gamma( i ) ) ; %mm/ s ˆ2 , xa c c e l e r a t i o n at po int o f i n t e r e s t
y a c c e l ( i )=a c c e l ( i )∗ s i n (gamma( i ) ) ; %mm/ s ˆ2 , ya c c e l e r a t i o n at po int o f i n t e r e s t
F fork ( i )=a c c e l 4 ( i ) /1000∗m fork ; %N, f o r c e o f f o rkin x
F z ( i )=s i n ( theta2 ( i )+c e n t r a n g l e ∗ pi /180)∗F c ; %NF x ( i )=cos ( theta2 ( i )+c e n t r a n g l e ∗ pi /180)∗F c−F fork (
i ) ; %N, −F fork b/c l ook ing at r e a c t i o n f o r c e s onmotorshaft
F mag( i )=s q r t ( F z ( i )ˆ2+F x ( i ) ˆ2) ; %N
r e s u l t a n g l e ( i )=atan2 ( F z ( i ) , F x ( i ) ) ∗180/ p i ; %deg ,r e s u l t a n t f o r c e ang le with balance
i f r e s u l t a n g l e ( i )<0 %r e s o l v e ang le o f r e s u l t a n tf o r c e
r e s u l t a n g l e ( i )=r e s u l t a n g l e ( i ) +360;endi f i <120 && r e s u l t a n g l e ( i )<100
r e s u l t a n g l e ( i )=r e s u l t a n g l e ( i ) +360;end
i=i +1;end
c e n t r a n g l e d e g (n)=c e n t r a n g l e ; %capture ang le o fc en t r o i d
max force (n)=max(F mag) ; %capture maximum f o r c e on s h a f tat t h i s ang le
n=n+1;end
f i g u r e (1 ) ;p l o t ( c en t r ang l e deg , max force ) ; %p lo t a l l max f o r c e s
f o r each c en t r o i d ang le
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B.5 Motor Shaft Counterbalance Mass Determination
% Calcu la te optimal mr r a t i o f o r motor s h a f tcounterba lance .
c l e a r a l lc l o s e a l l
osc =10000; %o s c i l l a t i o n s /min , speed o f input l i n k
w2=osc /60∗2∗ pi ; %rad/s , o s c i l l a t i o n speed , speed o finput l i n k
a2=0; %rad/s , angular a c c e l e r a t i o n o f input l i n ki =1; %indexr =8; %mm, rad iu s o f e c c e n t r i c bear inge =1.46; %mm, e c c e n t r i c d i s t ancer2=e ; %mm, length o f input l i n kr3=r ; %mm, length o f connect ing l i n kh=33.5; %mm, he ight from cente r o f e c c e n t r i c bear ing to
a x i s o f blade r o t a t i o np=60;%mm, d i s t anc e from a x i s o f blade r o t a t i o n to
l o c a t i o n o f acce l e romete r / po int o f i n t e r e s talpha (1 ) =0; %rad , i n i t i a l i z i n g angular v e l o c i t y
n=1;
m fork =0.0628; %kg , mass o f f o rk
x c =−0.8; %mm, x p o s i t i o n o f c en t r o id o f motorshaftz c =0; %mm, z p o s i t i o n o f c en t r o id o f motorshaftr c=s q r t ( x c ˆ2 + z c ˆ2) ; %mm, rad iu s to c en t r o id o f
motorshaft
F c=m shaft∗ r c /1000∗w2ˆ2 ; %N, c e n t r i f u g a l r e a c t i o nf o r c e o f bear ing / counterba lance r o t a t i n g about motora x i s
c e n t r a n g l e=mod( atan2 (1∗ z c−x c ∗0 ,1∗ x c+0∗ z c ) ,2∗ pi )∗180/ p i ; %ang le between e c c e n t r i c bear ing andcen t r o i d
f o r m shaft =0 . 0 1 : 0 . 0 1 : 1 %kg , mass o f e c c e n t r i cmotorshaft , e c c e n t r i c bear ing , and motorshaftcounterba lance
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i =1;c l e a r F mag ;
f o r ang le =0:1:360 %degtheta2d ( i )=ang le ; %degtheta2 ( i )=theta2d ( i ) ∗( p i /180)−(p i /2) ; %rad , ang le o f
e c c e n t r i c and ang le o f input l ink , −90 deg tomatch NX model
i f theta2 ( i )>2∗pi %r e s o l v e theta to between 0 and 2p i
x=f l o o r ( theta2 ( i ) /(2∗ pi ) ) ;theta2 ( i )=theta2 ( i )−2∗pi ∗x ;
end
r4 ( i )=r2∗ cos ( theta2 ( i ) )+r3 ; %mm, x p o s i t i o n o fs l i d e r r e l a t i v e to ground p ivot
gamma( i )=atan ( ( r4 ( i )−r3 ) /h) ; %rad , ang le o f r o t a t i o no f blade
gammad( i )=gamma( i ) ∗180/ p i ; %deg , ang le o f r o t a t i o no f blade
disp lacement ( i )=p∗ s i n (gamma( i ) ) ; %mm, disp lacementmagnitude o f po int o f i n t e r e s t
x d i sp ( i )=disp lacement ( i )∗ s i n ( p i/2−gamma( i ) ) ; %mm, xdisp lacement o f po int o f i n t e r e s t
y d i sp ( i )=−disp lacement ( i )∗ cos ( p i/2−gamma( i ) ) ; %mm,y disp lacement o f po int o f i n t e r e s t
v4 ( i )=−r2∗ s i n ( theta2 ( i ) )∗w2 ; %mm/s , speed o f s l i d e r
v tan ( i )=v4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f v e l o c i t y to blade l e g
omega ( i )=v tan ( i ) /h ; %rad/s , angular v e l o c i t y o fa x i s o f blade r o t a t i o n
v e l o c i t y ( i )=p∗omega ( i ) ; %mm/s , v e l o c i t y magnitude atpo int o f i n t e r e s t
x v e l ( i )=v e l o c i t y ( i )∗ cos (gamma( i ) ) ; %mm/s , xv e l o c i t y at po int o f i n t e r e s t
y v e l ( i )=−v e l o c i t y ( i )∗ s i n (gamma( i ) ) ; %mm/s , yv e l o c i t y at po int o f i n t e r e s t
a c c e l 4 ( i )=−r2∗ cos ( theta2 ( i ) )∗w2ˆ2−r2∗ s i n ( theta2 ( i ) )∗a2 ; %mm/ s ˆ2 , a c c e l e r a t i o n o f s l i d e r
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a c c e l t a n ( i )=a c c e l 4 ( i )∗ cos (gamma( i ) ) ; %mm/s , tangentcomponent o f a c c e l e r a t i o n to blade l e g
alpha ( i )=a c c e l t a n ( i ) /h ; %rad/ s ˆ2 , angulara c c e l e r a t i o n o f a x i s o f blade r o t a t i o n
a c c e l ( i )=p∗alpha ( i ) ; %mm/ s ˆ2 , a c c e l e r a t i o n magnitudeat po int o f i n t e r e s t
x a c c e l ( i )=a c c e l ( i )∗ cos (gamma( i ) ) ; %mm/ s ˆ2 , xa c c e l e r a t i o n at po int o f i n t e r e s t
y a c c e l ( i )=a c c e l ( i )∗ s i n (gamma( i ) ) ; %mm/ s ˆ2 , ya c c e l e r a t i o n at po int o f i n t e r e s t
F fork ( i )=a c c e l 4 ( i ) /1000∗m fork ; %N, f o r c e o f f o rkin x
F nz ( i )=s i n ( theta2 ( i ) )∗F nc ; %N, no balanceF nx ( i )=cos ( theta2 ( i ) )∗F nc+F fork ( i ) ; %N, no
balanceF nmag( i )=s q r t ( F nz ( i )ˆ2+F nx ( i ) ˆ2) ; %N, no balance
F z ( i )=s i n ( theta2 ( i )+c e n t r a n g l e ∗ pi /180)∗F c ; %NF x ( i )=cos ( theta2 ( i )+c e n t r a n g l e ∗ pi /180)∗F c−F fork (
i ) ; %N, −F fork b/c l ook ing at r e a c t i o n f o r c e s onmotorshaft
F mag( i )=s q r t ( F z ( i )ˆ2+F x ( i ) ˆ2) ; %N
r e s u l t a n g l e ( i )=atan2 ( F z ( i ) , F x ( i ) ) ∗180/ p i ; %deg ,r e s u l t a n t f o r c e ang le with balance
i f r e s u l t a n g l e ( i )<0 %r e s o l v e r e s u l t a n t ang ler e s u l t a n g l e ( i )=r e s u l t a n g l e ( i ) +360;
endi f i <120 && r e s u l t a n g l e ( i )<100
r e s u l t a n g l e ( i )=r e s u l t a n g l e ( i ) +360;endi=i +1;
end
mass sha f t (n)=m shaft ; %record mass o f s h a f t assemblymax force (n)=max(F mag) ; %record max f o r c e f o r t h i s massn=n+1;
end
f i g u r e (1 ) ;
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p lo t ( mass shaft , max force ) ; %p lo t a l l max f o r c e s f o reach mass va lue
112