Worcester Polytechnic Institute

Worcester, Massachusetts

Project 2

ES 3323: Advanced Computer Aided Design

Prof. Holly Ault

Daniel Ruiz-Cadalso

Tino Christelis

12/15/2016

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

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

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

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) = (𝜋𝑟𝑖𝑡)/2

𝜃(𝑡) = 𝑡 ∙ 90𝑜

𝑎𝑛𝑑 {𝑟𝑖 = 𝑅𝑎𝑑𝑖𝑢𝑠𝑜𝑓𝑏𝑎𝑠𝑒𝑐𝑖𝑟𝑐𝑙𝑒𝑡 = 𝐴𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

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:

𝐼𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙 = 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]:

𝐼𝑃𝑎𝑟𝑡(𝑟, 𝑇) = 𝐼𝑃𝑎𝑟𝑡′′ (𝑟, 𝑇) − 𝑚𝑃𝑎𝑟𝑡𝑟

2

where,

𝐼𝑃𝑎𝑟𝑡′′ (𝑟, 𝑇) =

𝑊𝑆𝐿𝑆 +𝑊𝑃𝑎𝑟𝑡𝐿𝑃𝑎𝑟𝑡4𝜋2

∙ 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]:

𝐼𝑃𝑎𝑟𝑡(𝑇) =𝑊𝑇𝑜𝑡𝑎𝑙𝑑

2

16𝜋2𝐷∙ 𝑇2 − 𝐼𝐺𝑒𝑎𝑟

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.

Figure 2: Bifilar Torsion Pendulum Setup [1]

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.

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.

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.

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

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.

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 force Figure 10: Gear Chamfer / Bevel Feature

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

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

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

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

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.

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

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.

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

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.

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.

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.

APPENDICES

Appendix A: Animation of Part Creation

Link to Animation of Part Creation: http://i.imgur.com/nCO6haF.gifv

Appendix B: Experimental Setup Pictures

Appendix B1: Pendulum Swing

Appendix B2: Bifilar Torsion Pendulum

Appendix C: Mathcad Files