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1Copyright © 1999 by ASME
DETC 99/CIE-9137
1999 ASME Design Engineering Technical Conference
Las Vegas, Nevada, September 12-15, 1999
DETC99/CIE-9137
REVERSE ENGINEERING OF STANDARD MECHANICAL ELEMENTS
Hesham A. Hegazi
Assistant Lecturer of Machine Design
Cairo University, Egypt
Hahegazi@alpha1-eng.cairo.eun.eg
Sayed M. Metwalli
Professor of Machine Design
Cairo University, Egypt
Pantech@ritsec1.com.eg
ABSTRACT
This paper presents a new procedure by which standard
machine elements can be reverse engineered. Standard
elements or components need not be scanned and identified as
surface or solid model. The standard character is sufficient to
be identified by few parameters. These parameters can be
measured or deduced by calculations from certain geometric
relations. The procedure presented herein is devoted to the
reverse engineering of spur and helical gears. Other standard
machine elements can be treated the same way. A computer
aided reverse engineering (CARE) program is used to satisfy all
geometric constraints, identify other dependent geometries and
define production and inspection requirements. It allows the
further check of the design appropriateness of material selection
and loading conditions.
INTRODUCTION
It is necessary sometime to produce spare parts of standard
machine elements or components. Standard machine elements
have specially defined geometries to perform specific tasks.
They can have specific standard dimensions for some feature or
another. A screw would have specific pitch or number of teeth
per inch. The tooth form would have specific profile with
specific inclination angles and radii. A gear would have a
specific standard module or diamertral pitch. It is usually made
with involute profiled teeth. It would be more expensive to
identify a gear by scanning its surface. The teeth would be
more difficult to reproduce as free form surfaces. The mating
and meshing would not be appropriate due to inaccuracies in
identifying the surface or in reproducing it. This is similar to
reproducing a circle from a scanned image of its print. Quality
will surely be degraded. More complex yet, the performance will
be totally inferior due to variation in surface properties and
smoothness. It would be very difficult to reproduce the
involute profile from a scanned surface than to generate it by
rack cutting. In this paper, we consider the reverse engineering
of spur and helical gears. Other standard machine elements can
be treated the same way. The main geometrical parameters are
identified from few measurements. The physical relations for
each element can then define other parameters.
Toothed gears have found wide application in various
branches of mechanical engineering. It is the most common type
of drive in present-day mechanical engineering and instrument
making. In many machines, such as metal cutting machine tools,
automobiles, tractors, hoisting and transporting machinery,
rolling mills, marine engines, etc. Gear drives are intended to
transmit motion with a predetermined change of angular speed
(or torque) in both magnitude and direction. Gear geometry and
design can be found in numerous references. The list of
references at the end of the paper provides a limited tally of
what is available in the fields of machine element design,
optimization and reverse engineering. Standard textbooks
(Edwardo and McKee 1991, Metwalli 1991, Mott 1992, Shigley
1986, Spotts 1985) include design procedures of different
machine elements that include basic standard geometries and
dimensions for gears. Other references (Dippery and Echempati
1998, Dooner and Seireg 1998, Dudley 1984, Heikkinen, Korpela
and Leinonen 1997, Jog and Pande 1989, Maitra 1985, Metwalli
and El Danaf 1996) are devoted to gear design and optimization.
Some references on reverse engineering (Lefever and Wood
1996, Wang and Wang 1997, Weir, Nilroy, Bradley and Vickers
1996) provide some of the techniques that can be used and the
limitations that are present for reverse engineering of standard
components.
2Copyright © 1999 by ASME
NOMENCLATURE
aCenter Distance (mm).
cClearance (mm).
d1,2 Pitch Diameter of pinion and gear respectively (mm).
da1,2 Outside diameters in pinion and gear (mm).
FFace width (mm).
htWhole depth of teeth (mm).
mModule (mm).
SaTop land thickness (mm).
t0Width of top land (mm).
tTooth thickness (mm).
z1,2 Number of teeth in pinion and gear respectively.
áPressure angle (Degree).
áaPressure angle at the tip circle (Degree).
THEORY
1- Identification of Spur Gears:
For a set of spur gears consists of a pinion and a gear the
following parameters can be measured.
The number of teeth in pinion and gear (N1,2), the outside
diameters in pinion and gear (Do1,2), the face width (usually
equal) in pinion and gear (F), the center distance (C), the whole
depth of teeth (ht), the pressure angle (ö), and the tooth form
(standard, or standard and topping, or stub, or stub and
topping). Also the width over i teeth, and the width over i+1
teeth can be measured.
The main values of tooth dimensions for pressure angles
of 200 and 250 are given in Table 1.
Table 1: Main values of tooth dimensions for pressure angles of
200 and 250
Description Pinion Gear
Number of teeth z1z2
Pitch circle
diameter d1 = z1 m d2 = z2 m
Tip circle diameter da1 = d1 + 2 m da2 = d2 + 2 m
Root circle
diameter df1 = d1 –2(1.25 m) df2 = d2 –2(1.25 m)
Base circle
diameter db1 = d1 cos ádb2 = d2 cos á
Tooth thickness on
pitch circle S = ð m/2 S = ð m/2
Center distance a0 = (d1 + d2)/2 a0 = (d1 + d2)/2
The normal module (m) for standard tooth form can be
calculated from the following formula:
The top land thickness is given by:
Sometimes for positively corrected gears, it may be necessary to
calculate the diameter at which the tip becomes pointed or
“peaked”.
Spur gears can be classified into two types of corrected
gearing systems, S0-gearing and S-gearing. In S
0-gearing, the
two components of the mating pair of gears receives numerically
equal correction factors, but these two factors are algebraically
of opposite signs. The S0-gearing is normally meant where the
reduction ratio is large, it tends to equalize the tooth strength
and thereby reduces the susceptibility to such damage. Table 2
shows the main dimensions of S0-gearing.
Table 2: Dimensions of S0-gearing for Spur Gears
Description Pinion Gear
Number of Teeth z1z2
Pitch Diameter d1= z1md2= z2m
Tip circle diameter da1=d1+2m+2x1mda2=d2+2m-2x1m
Tooth thickness on
pitch circle S1= m(ð /2
+2x1 tan á ) S2= m(ð /2
-2x1 tan á )
Center distance a0=( d1+ d2)/2= m( z1+ z2)/2
In case of S-gearing the sum of the profile corrections of
the two mating gears is not equal to zero, it is either positive or
negative.
The main dimensions of S-gearing spur gears are shown in
Table 3.
Table 3: Dimensions of S-gearing for spur gears
Description Pinion Gear
Number of Teeth z1z2
Pitch Diameter d1= z1md2= z2m
Tip circle diameter
(With topping) da1=2(a+m-x2m)
- d2
da2=2(a+m-x1m)
- d1
Tip circle diameter
(Without topping) da1=2m+2x1m+d1da2=2m+2x2m+d2
Tooth thickness on
pitch circle
(normal section)
S1= P/2
+2mx1 tan áS2= P/2
+2mx2 tan á
Tooth thickness on
pitch circle
(transverse section)
St1= mt( ð /2
+2x1 tan á ) St2= mt( ð /2
+2x2 tan á )
Center distance
(after pushing) a = a0 ( cos á / cos áw
)
25
.
2
t
h
m=
−+= aaa invinv
d
m
dsα
π0
20
2/
(1)
(2)
3Copyright © 1999 by ASME
The topping for spur gearing can be calculated from:
The working pressure angle can be obtained from:
The top clearance can be calculated as:
The sum of the profile correction factor can then be
calculated as:
2- Identification of Helical Gears:
There are, however, some design considerations when
using helical gears. They have greater contact ratio, greater
strength, and such operational requirements, such as,
noiselessness, smoother engagement of teeth meshing, for
which the use of helical gears is preferred.
The helix angle at the pitch circle (â) can be calculated by
using the helix angle at the top land of the teeth (âa) which can
be measured, then the helix angle at the pitch circle (â) can be
calculated as:
The transverse module (mt) can be calculated from the
normal module (m) by using the following relation:
The normal pressure angle (á) and the transverse pressure
angle (á t) are related to each others by the following relation:
Helical gears can be also classified into two types of
corrected gearing systems, S
0-gearing and S-gearing. Table 4
shows the main dimensions of S0-gearing:
The main dimensions of S-gearing helical gears are shown
in Table 5.
Table 4: Dimensions of S0-gearing for helical gears
Description Pinion Gear
Number of Teeth z1z2
Pitch Diameter d1= z1m/cos âd2= z2m/cos â
Tip circle diameter da1=d1+2m(1+x1)da2=d2+2m(1+x1)
Tooth thickness on
pitch circle Sn1= m( ð /2
+2x1 tan á ) Sn2= m( ð /2
-2x1 tan á )
Center distance a0=( d1+ d2)/2=(m/cos â )( z1+ z2)/2
Table 5: Dimensions of S-gearing for helical gears
Description Pinion Gear
Number of Teeth z1z2
Pitch Diameter d1= z1m/cos âd2= z2m/cos â
Tip circle diameter
(with topping) da1=2(a+m-x2m)
- d2
da2=2(a+m-x1m)
- d1
Tip circle diameter
(without topping) da1=2m+2x1m+d1da2=2m+2x2m+d2
Tooth thickness on
pitch circle
(normal section)
Sn1= m( ð/2
+2x1 tan á ) Sn1= m( ð/2
-2x1 tan á )
Tooth thickness on
pitch circle
(transverse section)
St1= mt( ð /2
+2x1 tan á ) St2= mt( ð /2
+2x2 tan á )
Center distance a0=( d1+ d2)/2=(m/cos â )( z1+ z2)/2
The topping for helical gearing can be calculated from:
The working pressure angle can be obtained from:
The top clearance can be calculated as:
The sum of the profile correction factor can be calculated
as:
d
da
a×= ββ tantan
βcos
m
mt=
βαα costantan 1
=
(
)
amxxaym −++= 210
ttw inv
zz xx
inv ααα +
+
+
=tan2
21
21
22 1221 fafadd
a
dd
ac
+
−=
+
−=
α
α
α
tan2
)( 2121 ttw invinv
zzxx
−
+=+
(
)
amxxaym −++= 210
ααα inv
zz xx
inv w+
+
+
=tan2
21
21
)
2
(21 hym
dd
acfa−+
+
−=
αααtan2
)( 2121 invinv
zzxx w−
+=+
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
4Copyright © 1999 by ASME
SAMPLE APPLICATION
The following data are obtained by actual measurement of
the two helical gears in medium to fast running drive:
z1 = 18 teeth
z2 = 73 teeth
da1 = 102.42 mm
da2 = 380.52 mm
a = 231.5 mm
In this sample application (Maitra 1985), since the gear has
an odd number of teeth (73 teeth), its outside diameter is
measured by suitable gadget by mounting it on a mandrel and
using dial indicators. This way of measuring will be accurate.
The tip diameter of the pinion is measured by vernier calipers.
The whole depth of the tooth is measured and is found to
be (11mm), which gives a value of (5mm) of the module in case
of standard gears. It is assumed that topping has been done
since the gears appear to have been corrected.
The helix angle at the top land of the teeth (âa) can be
measured, then the helix angle at the pitch circle (â) can be
calculated from equation (7) which is reiterated as:
By measurement it was found that âa = 90. Hence, from Equation
(7) one gets:
â = 80
Therefore the pitch diameter of the pinion can be evaluated
as:
The pitch diameter of the gear is then:
The transverse pressure angle (át) is found from equation
(9) such that:
Then
át = 200 11’
From equation (11) the working pressure angle (átw) is found as:
From equation (11) the total correction factor can be
calculated from:
which gives by solving:
The individual correction factors x1 and x2 can be fond from
tip circle diameter equations in Table 3 as follows.
By solving, we get:
x1 = 0.16 and x2 = 0.36
The value of topping can be found by using equation (10):
Therefore the top clearance can be calculated from equation
(12) as:
COMPUTER PROGRAM
A computer program has been used to provide the tool to
calculate all necessary gear parameters if few measurements can
be performed. The procedure follows the process indicated in
the previous sample application section. Figure (1) shows the
main screen of the computer aided reverse engineering (CARE)
of spur and helical gears. It provides the main inputs for the
main measurable parameters. These include the number of teeth
in pinion and gear (N1,2), the outside diameters in pinion and
gear (Do1,2), the helix angle at the top land of the teeth (âa), the
face width (usually equal) in pinion and gear (F), the center
d
da
a×= ββ tantan
mm
mz
dL88.90
8cos185
cos 0
1
1=
×
== β
mmdL58.368
8cos735 0
2=
×
=
(
)
'1120
7318
20tan2
'2021 0
21
0
0inv
xx
inv +
++
=
36.0
21 =+ xx
(
)
58.3685550.231242.102 2−−+= x
(
)
88.905550.231252.380 1−−+= x
mmym 03.05.231)536.0(
258.36888.90 =−×+
+
=
mmc25.125.1103.0
252.38042.102
5.231 =
−+
+
−=
00 8costan20tan ×= t
α
tw
αcos '1120cos
27318
8cos
5
50.232 0
0+
=
5Copyright © 1999 by ASME
distance (C), the whole depth of teeth (ht), the pressure angle
(ö), and the tooth form (standard, or standard and topping, or
stub, or stub and topping). Also the width over i teeth, and the
width over i+1 teeth can be measured. These can provide a
check for the selection of the standard pressure angle.
Fig. 1: Main input screen of the computer aided reverse
engineering of spur and helical gears.
The main output parameters of the computer program are
the normal module (m), the helix angle at pitch circle, the amount
of topping, the profile shift for the pinion and the profile shift
for the gear. The output pressure angle is displayed if the width
over i teeth, and the width over i+1 teeth are entered. The
output of our sample implementation is shown in figure 2. The
program shows the outputs in red.
After satisfying all geometric constraints, the program
identifies other dependent geometries and defines production
and inspection parameters. Figure 3 shows the output geometry
parameters of the reverse engineered helical gear of the sample
application shown in red. The dependent geometry parameters
are the pitch diameter, tooth thickness, addendum and the root
diameter for both pinion and gear. For the gear set, the center
distance, the gear ratio, the transverse pressure angle, the
clearance, the minimum number of teeth and the contact ratio are
provided. The predicted accuracy is also given in the values of
the tooth to tooth composite error and the total composite
tolerance. For proper operation, the angular backlash is
provided. Virtual numbers of teeth are also given to aid in the
selection of production tools. Chordal height, thickness and the
whole depth are provided for manufacturing inspection.
Fig. 2: Main outputs of the reverse engineered helical gear of
the sample application (shown in red).
Fig. 3: Output geometry parameters of the reverse engineered
helical gear of the sample application (shown in red).
The computer aided design program allows the further
check of the design appropriateness of material selection and
loading conditions. That can be activated by the design option
of the program. Figure 4 shows the output design iteration
(shown in red) of the reverse engineered helical gear of the
sample application. The main results are the safety factors for
6Copyright © 1999 by ASME
both pinion and gear. These can only be obtained if the speeds,
power and other factors can be identified for the gear set. The
material can be identified either by iteration or by testing the
original materials of the pinion and the gear. Some software
programs are developed for material selection. The scope of
this paper, however, does not include such programs.
Fig. 4: Output design iteration of the reverse engineered helical
gear of the sample application (shown in red).
The computer aided design program also evaluates all
pertinent gear stresses. Figure 5 shows the output gear
stresses of the reverse engineered helical gear of the sample
application shown in red on top of the gear design screen. This
provides a further check of the design appropriateness of
material selection and loading conditions.
DISCUSSION AND CONCLUSION
To produce spare parts of standard machine elements such
as a spur or helical gear, it would be more expensive to identify it
by scanning its surface. It has specially defined geometries to
perform a specific task. The tooth form has a specific profile.
The gear has a specific standard module or diamertral pitch. It is
usually made with involute profiled teeth. It would be more
expensive to identify a gear by scanning its surface and the
teeth would be more difficult to reproduce as free form surfaces.
The mating and meshing would not be appropriate due to
inaccuracies in identifying the surface or in reproducing it. It
would be very difficult to reproduce the involute profile from a
scanned surface than to generate it by rack cutting.
In this paper, we considered the reverse engineering of spur
and helical gears. The main geometrical parameters have been
identified from few measurements. The physical relations for
the gears have been used to define other dependent parameters.
A computer aided reverse engineering (CARE) program has
been used effectively to satisfy all geometric constraints,
identify other dependent geometries and define production and
inspection requirements. It allowed the further check of the
design appropriateness of material selection and loading
conditions. Other standard machine elements can be treated the
same way. This would suggest the need for geometrically
identifying standard machine elements with specifically defined
parameters.
Fig. 5: Output gear stresses of the reverse engineered helical
gear of the sample application (shown in red) on top of
the gear design screen.
ACKNOWLEDGMENTS
Part of the application of this work has been supported by a
grant from USAID-FRCU ULP II Program, linkage grant number
93/02/02. The authors would like to thank all that contributed
and supported this effort.
7Copyright © 1999 by ASME
REFERENCES
Dippery, R. E., Echempati, R., 1998, “Robust Design of
Gears”, Proc of the 1998 ASME Design Eng. Tech. Conf., paper
No. DETC 98/PTG-5792.
Dooner, D. B., Seireg, A. A., 1998, “Concurrent Engineering
of Toothed Bodies for Generalized Function Transmission”,
Proc of the 1998 ASME Design Eng. Tech. Conf., paper No.
DETC 98/PTG-5780.
Dudley, D., 1984, “Handbook of Practical Gear Design”,
Technomic Publishing Co. Inc., Lancaster.
Edwardo, K. S., McKee, R. B., 1991, “Fundamentals of
Mechanical Component Design”, McGraw-Hill Corp.
Heikkinen, A., Korpela, T., and Leinonen, T., 1997, “Feature-
Based Gearbox Design”, Proc of the 1997 ASME Design Eng.
Tech, Conf., paper No. DETC 97/DAC-3738.
Jog, G. S., Pande, S.S., 1989, “Computer-Aided Design of
Compact Helical Gear Sets”, Journal of Mechanisms,
Transmissions and Automation in Design, Vol. 111, pp. 285-289.
Lefever, D.D., Wood, K. L., 1996, "Design For Assembly
Technique in Reverse Engineering and Redesign", Proc of the
1996 ASME Design Eng. Tech. Conf., paper No. 96-DETC/DTM-
1507.
Maitra, G. M., 1985, “Handbook of Gear Design”, Tata
MacGraw-Hill Publishing Co., New Delhi.
Metwalli, S.M., 1991, “Fundamentals of Machine Design,”
Cairo University.
Metwalli, S.M., El Danaf, E.A., 1996, “CAD and
Optimization of Spur and Helical Gear Sets”, Proc of the 1996
ASME Design Eng. Tech. Conf., paper No. 96-DETC/DAC-1433.
Mott, R., 1992, “Machine Elements in Mechanical Design”,
Merril Publications.
Shigley, J. E., 1986, “Mechanical Engineering Design”, First
Metric Edition, McGraw-Hill, Inc.
Spotts, M. F., 1985, “Design of Machine Elements”, Sixth
Edition, Prentice-Hall of India.
Wang, G-J, Wang, C-C, 1997, "Reconstruction of Sculptured
Surface on Reverse Engineering", Proc of the 1997 ASME
Design Eng. Tech. Conf., paper No. DETC 97/CIE-4287.
Weir, D. J., Nilroy, M.J., Bradley, C., and Vickers, G.W., 1996,
“Reverse Engineering Physical Models Employing Wrap-
Around B-Spline Surfaces and Quadrics”, Proc Inst. Mech.
Engrs, Vol. 210, pp. 147-157.