1. Worcester Polytechnic Institute
Worcester, Massachusetts
Project 2
ES 3323: Advanced Computer Aided Design
Prof. Holly Ault
Daniel Ruiz-Cadalso
Tino Christelis
12/15/2016

2. TABLE OF CONTENTS
List of Figures................................................................................................................................. 3
Problem Statement.......................................................................................................................... 4
Background..................................................................................................................................... 5
Reverse Engineering....................................................................................................................... 8
Modelling Strategy........................................................................................................................ 10
Experimental Results .................................................................................................................... 17
Discussions ................................................................................................................................... 20
Conclusion .................................................................................................................................... 23
References..................................................................................................................................... 24
Appendices.................................................................................................................................... 25
Appendix A: Animation of Part Creation ................................................................................. 25
Appendix B: Experimental Setup Pictures ............................................................................... 26
Appendix B1: Pendulum Swing ........................................................................................... 26
Appendix B2: Bifilar Torsion Pendulum.............................................................................. 27

3. LIST OF FIGURES
Figure 1: Isometric View of the Testing Part.................................................................................. 4
Figure 2: Bifilar Torsion Pendulum Setup [1]................................................................................ 7
Figure 3: Schematic Representation of the Testing Part Geometry and Dimensions..................... 9
Figure 4: Model Tree of Completed Part and Gear Properties (Relations Tab) ........................... 11
Figure 5: Revolve Profile.............................................................................................................. 11
Figure 6: Revolve Feature............................................................................................................. 12
Figure 7: Hole Features................................................................................................................. 12
Figure 8: Round Features.............................................................................................................. 13
Figure 9: Chamfer Features .......................................................................................................... 13
Figure 10: Gear Chamfer / Bevel Feature..................................................................................... 13
Figure 11: Tooth Profile (An Involute Curve).............................................................................. 14
Figure 12: Datum Features in Preparation of Tooth Profile Design............................................. 14
Figure 13: Blend Feature Applied between Profiles, followed by Axial Pattern ......................... 15
Figure 14: Isometric View of Completed Part.............................................................................. 16
Figure 15: Schematic Diagram of the Experimental Setups......................................................... 17
Figure 16: PTC Creo Mass Properties Report .............................................................................. 21

4. PROBLEM STATEMENT
The reverse engineering design process on some components tends to be complex and
should be corroborated with experimental investigations and results. There is a wide variety of
methods to validate these results, and thus, in this experiment the computer-aided results, which
include the center of gravity and moment of inertia, will be compared to the data collected from
the pendulum swing and torsion test. The testing part is a gear-shaft and is being referred to as the
testing part throughout the report. It consists of a shaft with a step-variation in the cross-section
and four distinct gears. The whole part is initially assumed to be of the same material with
uniformly-distributed density. With the aid of some physics-derived equations relating the period,
pendulum radius, and weight of part, the center of gravity can be localized and the moment of
inertia can be calculated for comparison with the computer-aided results. See Figure 1 for a
rendered isometric view of the testing component.
Figure 1: Isometric View of the Testing Part

5. BACKGROUND
For completely describing the geometry of a gear, the following three governing
parameters were needed: number of teeth, pressure angle, and pitch diameter. With these, along
with previously-derived parametric equations that are used for defining the gear tooth profile, the
complete gear geometry can be constructed. Table 1 shows all of the gear characteristics, which
are functions of the three mentioned independent parameters.
Table 1: Gear Characteristics
Parameter Symbol Definition
Number of Teeth 𝑁 𝑁
Pitch Diameter 𝑃𝐷 𝑃𝐷
Pressure Angle 𝜙 𝜙
Base Diameter 𝐵 𝑃𝐷 ∙ cos⁡( 𝜙)
Root Diameter 𝑅 𝑃𝐷 − 𝐷𝑒
Dedendum 𝐷𝑒 1.25𝑃𝐷/𝑁
After defining all the needed characteristics of each gear, the mentioned parametric
equation that has been derived previously can be used to generate the curve for the tooth profile,
shown as follows:
𝑥(𝑡) = 𝑥𝑖(𝑡) + 𝑠(𝑡) ∙ sin(𝜃(𝑡))
𝑦(𝑡) = 𝑦𝑖(𝑡) − 𝑠(𝑡) ∙ cos(𝜃(𝑡))
𝑧(𝑡) = 0
𝑤ℎ𝑒𝑟𝑒⁡
{
𝑥𝑖(𝑡) =⁡⁡⁡⁡ 𝑟𝑖 ∙ cos(𝜃(𝑡))
𝑦𝑖(𝑡) =⁡⁡⁡⁡ 𝑟𝑖 ∙ sin(𝜃(𝑡))
𝑠(𝑡) =⁡⁡⁡⁡⁡⁡⁡⁡⁡(𝜋𝑟𝑖 𝑡)/2⁡
𝜃(𝑡) = ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑡 ∙ 90 𝑜
⁡⁡⁡𝑎𝑛𝑑⁡⁡⁡ {
𝑟𝑖 = ⁡⁡⁡⁡𝑅𝑎𝑑𝑖𝑢𝑠⁡𝑜𝑓⁡𝑏𝑎𝑠𝑒⁡𝑐𝑖𝑟𝑐𝑙𝑒
𝑡 = ⁡⁡⁡⁡⁡⁡𝐴𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦⁡𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

6. The experimental data from the pendulum swing tests, which will consist of a variation of
swing radii and their respective periods, needs to be used with the aid of some equations that
describe the relationship between these and the required output, which are the center of gravity
location and the moment of inertia. In the case where the pendulum’s total mass consists of two
parts, being the testing part and a support component, the parallel axis theorem works perfectly for
describing the moment of inertia of the part with respect to the pendulum swing axis, and is as
follows:
𝐼 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙 = 𝐼 + 𝑚𝑟2
where 𝑟 is the radius of the part’s center of gravity to the pendulum swing axis, and 𝐼 is the moment
of inertia of the part about a specified axis that is parallel to the pendulum swing axis. Therefore,
an equation was derived for the moment of inertia of the testing part, and is as follows [1]:
𝐼 𝑃𝑎𝑟𝑡(𝑟, 𝑇) = 𝐼 𝑃𝑎𝑟𝑡
′′ (𝑟, 𝑇) − 𝑚 𝑃𝑎𝑟𝑡 𝑟2
where,
𝐼 𝑃𝑎𝑟𝑡
′′ (𝑟, 𝑇) =
𝑊𝑆 𝐿 𝑆 + 𝑊𝑃𝑎𝑟𝑡 𝐿 𝑃𝑎𝑟𝑡
4𝜋2
∙ 𝑇2
− 𝐼𝑆
′′(𝑟)
Thus, with this relationship, the moment of inertia for various pendulum swing radii, which will
consist of various periods, can be calculated. The parallel axis theorem, however, is not needed for
the bifilar pendulum torsion test because, if perfectly positioned, the whole system rotates about
the axis centered through the center of gravity. In this case, the equation needed for relating the
experimental data to the required output results is the following [1]:
𝐼 𝑃𝑎𝑟𝑡(𝑇) =
𝑊𝑇𝑜𝑡𝑎𝑙 𝑑2
16𝜋2 𝐷
∙ 𝑇2
− 𝐼 𝐺𝑒𝑎𝑟
in which the moment of inertias are with respect to the rotating axis. In the above equation, d is
the spacing between bifilars, and D is the length of the bifilars; both of which are detailed below
in Figure 2.

7. Figure 2: Bifilar Torsion Pendulum Setup [1]

8. REVERSE ENGINEERING
A 2D schematic sketch (Y-Z plane) of the part was drawn in order to properly organize the
geometry, along with the sketch dimensions. Additionally, since one simple 2D sketch is not
enough to completely define the part geometry for modelling, each gear was shown individually
with a front view of each. The mentioned sketch can be seen in Figure 3. Each dimension was
measured with the use of a digital caliper, and assuming the part was originally modelled using
the drafting standard ANSI, each of these measurements were rounded to the nearest 1/32”. To
simplify the Z-coordinate dimensions, the Ordinate Dimension method was used for the sketch.
Keep in mind, the dimensions shown in the sketch were rounded to the nearest 0.01”, although the
real value is a factor of 1/32”. Chamfers were measured very carefully by distance-distance,
although noted as distance-angle in the schematic sketch. As for the fillets near the gears, the
distance from the gear front face to the point of tangency between the fillet and the outer face of
the tube was measured and used as the fillet radius.

9. Figure 3: Schematic Representation of the Testing Part Geometry and Dimensions
After having completely defined the shaft and the gear positions, the gears were
individually inspected. With the aid of the curve equation derived for gear tooth profiles (see
Background), the tooth profile for each gear was successfully determined using the following gear
parameters: number of teeth (𝑁), pressure angle (𝜙) and the pitch diameter (𝑃𝐷).
Using a simple scale available in the WPI Experimentation Lab the mass of the part was
weighed to be 2.6359𝑘𝑔, which converts to 5.811𝑙𝑏𝑓. Because the material of the part was an
unknown, a creative method had to be applied to determining what the part was made of. After the
part was modeled in PTC Creo, dividing the mass of the part by the volume of the part (calculated
in PTC Creo) equated to a part density of approximately 0.278𝑙𝑏𝑚/𝑖𝑛3
. This calculation assumes
a constant material and density throughout the part. Determining an actual material for the part
will be covered at the end of the Modelling Strategy section.

10. MODELLING STRATEGY
Good modeling strategy and design intent are essential to good engineering design.
Manufacturability, mathematics, and practicality must all be taken into consideration in the
modeling of any part. It was decided that the best way to design the part would be to group most
of the dimensions in a single revolve to create the main body, followed by hole and chamfer / fillet
features, and then completed with gear teeth modeling. For optimal visualization of this process,
an animation of the entire making of the part is available for viewing and can be found in Appendix
A.
First steps included deciding on a part
origin and setting relations for gear data. The
orientation of the part was set such that the
origin’s Z-Axis was located along the axial
center (positive Z away from the part) and
coincident with outward-facing side of the
smallest gear (positive X to the right of part).
Gear values that were calculated using methods
mentioned in the above section were all
imported in a relation tab, so that the values
could be easily accessed for later reference.

11. Figure 4: Model Tree of Completed Part and Gear Properties (Relations Tab)
The initial step that set the groundwork for the rest of the design was a revolve feature of
the entire profile of the part. Looking at the probable method of machining for the part, its axial
symmetry suggests the part was first spun and operated on with a lathe. To retain good design
intent, it follows that parts should be designed with manufacturability in mind, and so a revolve
feature was deemed optimal for the creation of the base body. Set up on a sketch on the Right
Plane, diameter and length
dimensions were all added
to the profile until it was
completely constrained.
This profile includes both
the shaft and gears so the
entire part is created in
almost a single feature. The
profile was fully revolved
about the Z Axis, forming
the base body.
Figure 5: Revolve Profile

12. Figure 6: Revolve Feature
Holes features were next to be applied. All of the hole-features were aligned with the Z-
axis to be centered along the part. The first hole created was featured on the front plane and was
set to a depth as being up to the surface of the end of the part. Two step holes were then added on
either side, each with a blind depth equal to their measured values.
Figure 7: Hole Features
Next, rounds were added to appropriate corners.

13. Figure 8: Round Features
Chamfers of various sizes and angles were applied to the step holes.
Figure 9: Chamfer Features
Because all of the gears
seem to have a slight “bevel”
along the teeth, a chamfer was
also applied to the edges of the
gears. Creating the gear teeth was
a two stage process: creating the
tooth profile and creating a
patterned blend feature. In order
for a gear tooth to transmit forceFigure 10: Gear Chamfer / Bevel Feature

14. upon other gear teeth such that the force maintains a constant direction tangential to the gear, the
geometry of a gear tooth is that of an involute circle. This mathematical definition for gear tooth
design was realized in PTC Creo by creating a new coordinate system (CS_1 below) which was
then referenced by an equation-driven curve (math for involute circle). By gear sizing standards,
the equation behind a tooth profile begins at the base diameter (bd_1). This is the only variable
needed for the involute equation.
Figure 11: Tooth Profile (An Involute Curve)
With the curve now in the PTC Creo environment, some addition datum features and copies
of the same curve needed to be made. A point was constructed to be coincident with both the prime
diameter and the involute curve; this is used later on for sketch constraints. The point and curve
were then both copied and rotated about
the Z-axis, first by (360 / N) degrees
forming one side of the tooth profile, and
then by (0.5 * 360 / N) degrees in order
to create a reference curve. A plane was
constructed through the Z-axis and the
point on this “middle” reference curve.
The “middle” curve was mirrored about
this plane, providing the other size of the
gear tooth profile.Figure 12: Datum Features in Preparation of Tooth Profile
Design

15. Given a left and right bound reference curve for the tooth profile, all that was left to do was
fill in the blanks. A 2-point tangent arc was constructed between the two reference curves, and
constrained to be tangent to a circle representing of the root diameter (rd_1). To close the profile,
and arc was created of a radius greater than the gear radius via relations, and set between the two
tooth curves. The profile created by this sketch is a “negative”, as it will be used to cut through the
main body.
The same sketch was copied to the
opposite face of the gear, and rotated by an
amount approximately equal to the angle of
the real gear teeth (computed through a
relation). A straight blend between both these
sketches was featured, and then patterned
axially about the part an amount equal to the
number of gear teeth (N_1). It should be
noted that the blend feature was chosen over
a helical sweep because a helical sweep can
only use a profile that is normal to the helical
path, which can’t apply to our teeth
“negative”s as they would be normal to a
helical path.
Figure 13: Blend Feature Applied between Profiles, followed by Axial Pattern

16. Once this process was repeated 3 more times for the other gears, the part was complete.
The only difference in values for other gears are related to prime, base, root diameters, and number
of teeth. Upon completion, the volume of the part was measured and used to produce an
approximate part density of 0.278 lbm/in3. This calculated density is very close to the density of
steel which is 0.283 lbm/in3. With only a 0.005 difference between densities, it was decided that
the material of the part would be set to the PTC Creo default values for steel.
Figure 14: Isometric View of Completed Part

17. EXPERIMENTAL RESULTS
Mathcad screenshots can be found in Appendix C
As a preparation, the part was first balanced on a straight thin ruler in order to locate its
center of gravity and confirm with the computer-aided results. The pendulum experimental setup
consisted of a rectangular lexan support part that was attached to a fluorocarbon string. The
pendulum experiments consisted of measuring the oscillating periods for a variation of pendulum
radii. Figure 15 (A and B) shows a schematic drawing of the pendulum setup for both the swing
and the torsion test, which contains the testing part, the support component, and the string, along
with the fixture where the axis takes place. Photos of the actual setups can be seen in Appendix B.
Figure 15: Schematic Diagram of the Experimental Setups (a) Pendulum Swing (b) Bifilar Torsion
The pendulum swing test was used for investigating the moment of inertia with respect to
the X and Y axis, which should be equal due to the axial symmetry of the part. The set up pictures
for this part of the experiments is shown in Appendix B1. The pendulum was first set to a radius
of 13 inches and raised to any height in which the string does not make an angle larger than 30o

18. with respect to the vertical axis. Data gathered from pendulum swings greater than 30o cannot
apply to the equations discussed in the background section. As the pendulum swung, five periods
were collected in order to minimize the error band as needed. The pendulum was then set to a
different radius and the same experiment repeated, followed by the same type of data collection.
With these results, along with the parallel axis theorem derived equations, the moment of inertias
with respect to the X and Y axes can be successfully calculated, with a significantly small error
band. Table 2 shows the pendulum swing experimental results, along with the calculated moment
of inertias for each test.
Table 2: Pendulum Swing Test Results
Period Intervals Radius = 13 in Radius = 19 in
t1 1.084s 1.332s
t2 1.088s 1.320s
t3 1.100s 1.328s
t4 1.088s 1.322
t5 1.072s 1.320s
Average Time 1.086s 1.324s
Average Moment of Inertia (X-X, Y-Y) 31.348 𝑖𝑛2
∙ 𝑙𝑏𝑚 15.056 𝑖𝑛2
∙ 𝑙𝑏𝑚
The moment of inertia with respect to the Z-axis of the part was investigated using the
Bifilar Torsion Pendulum test, in which the part was setup as shown in Appendix B2. The
pendulum was again rotated to an initial position less than 30o for compatibility with the
theoretical equations. While in motion, as soon as the pendulum achieved stable oscillations about
the vertical axis, the time for a total of ten oscillations was recorded. This was then used to
determine the average time for one oscillation. Afterwards, the moment of inertia with respect to
the Z-axis was calculated, and shown in Table 3 along with the rest of the results.

19. Table 3: Bifilar Torsion Pendulum Test Results
Period Intervals Period Time
t1 0.505s
t2 0.507s
t3 0.497
t4 0.503s
t5 0.501s
Average Time 0.503s
Average Moment of Inertia (Z-Z) 5.021 𝑖𝑛2
∙ 𝑙𝑏𝑚

20. DISCUSSIONS
With the experimental data, the test and computer-aided results can now be compared.
Figure 16 shows the Mass Properties report of the CAD part, which contains properties such as
volume, density, COG, moment of inertia, etc. Within this information, the COG is reported to be
located at coordinates (0, 0, -4.25”) from the coordinate frame system, which is located at the
leftmost face from the sketch in Figure 3. The experimentally-calculated COG was recorded as
2.75” from the right face of the rightmost gear, which in terms of the coordinate frame system
would be located at (0, 0, -4.44”). Therefore, it is safe to assume that the overall geometry of the
part was constructed correctly. However, the moment of inertia results need to be compared for
complete validation. Looking at the Mass Properties Report (Figure 16), the Inertia Tensor that is
displayed shows the same values for X-X and Y-Y, which validates our axial symmetry
assumptions for the part. The moment of inertia for X-X and Y-Y was calculated by PTC Creo to
have a value of 28 𝑖𝑛2
∙ 𝑙𝑏𝑚. Experimental results for a radius of 13” indicate an average moment
of inertia of approximately 31 𝑖𝑛2
∙ 𝑙𝑏𝑚, which compared to the computer-aided results, is very
similar.
The moment of inertia value with respect to the Z-axis is reported as 5.79 in2lbm in the
Mass Properties report generated by Creo. This compares extremely well to the moment of inertia
calculated from the Bifilar Torsion Pendulum test results, which is 5.021 in2lbm.

21. Figure 16: PTC Creo Mass Properties Report
It should be noted that while working on the calculations for the moment of inertia, the
time period for the oscillations was noticed to have a great influence on the variation of it. For
example, the X-X moment of inertia calculated with the pendulum swing test results varied by a
difference of approximately 6 in2lbm from a time of 1.084 to 1.088 seconds, and thus, the error
factor between the measured experimental time and the moment of inertia is extremely high. This
means that manual timing was critical and needed to be as precise as possible. This was, however,
compensated significantly by taking multiple time measurements of the same oscillations, which
were averaged. This concludes that the 3 in2lbm difference from the Creo results and the pendulum
swing experimental results for the small radius is not significant. This may also be the explanation
of the wide difference (approx. 13 in2lbm) between the pendulum swing experimental results for

22. the large radius and the computer-aided results. The minimal differences that are noticed in the
results exist solely due to the following: imperfect measurements, small density variations in the
distribution due to wear of the part, energy losses in the pendulum due to imperfect initial positions,
manual timing of the pendulum period oscillations, etc.

23. CONCLUSION
It is safe to conclude that the Reverse Engineering process of this project was done
correctly and with great precision. It is clear that the Creo software is extremely accurate in its
mass property calculations, and that CAD software is an essential tool for calculating the properties
about a complex object when real life experimentation is not a viable option. From this project, a
great deal was learned about specific and highly-applicable Creo features, such as creating curves
driven by equations, limitations on sweep functions, and the methodology behind angled-tooth
gear design in a CAD environment.

24. REFERENCES
[1] Gracey, W. (1948). The Experimental Determination of the Moments of Inertia of
Airplanes by a Simplified Compound-Pendulum Method. National Advisory Committee for
Aeronautics.

25. APPENDICES
Appendix A: Animation of Part Creation
Link to Animation of Part Creation: http://i.imgur.com/nCO6haF.gifv

26. Appendix B: Experimental Setup Pictures
Appendix B1: Pendulum Swing

27. Appendix B2: Bifilar Torsion Pendulum

28. Appendix C: Mathcad Files