Advanced Computer Aided Design Simulation of Gear Hobbing by

Transcription

Advanced Computer Aided Design
Simulation of Gear Hobbing by
Means of Three-Dimensional
Kinematics Modeling
Dimitriou Vasilis
Adjunct Professor
Vidakis Nectarios
Associate Professor
Antoniadis Aristomenis
Professor
Technological Educational Institute of Crete,
Romanou 3,
Chania 73133, Greece
Gear hobbing, as any cutting process based on the rolling principle, is a signally multiparametric and complicated gear fabrication method. Although a variety of simulating
methods has been proposed, each of them somehow reduces the actual three-dimensional
(3D) process to planar models, primarily for simplification reasons. The paper describes
an effective and factual simulation of gear hobbing, based on virtual kinematics of solid
models representing the cutting tool and the work gear. The selected approach, in contrast to former modeling efforts, is primitively realistic, since the produced gear and
chips geometry are normal results of successive penetrations and material removal of
cutting teeth into a solid cutting piece. The algorithm has been developed and embedded
in a commercial CAD environment, by exploiting its modeling and graphics capabilities.
To generate the produced chip and gear volumes, the hobbing kinematics is directly
applied in one 3D gear gap. The cutting surface of each generating position (successive
cutting teeth) formulates a 3D spatial surface, which bounds its penetrating volume into
the workpiece. This surface is produced combining the relative rotations and displacements of the two engaged parts (hob and work gear). Such 3D surface “paths” are used
to split the subjected volume, creating concurrently the chip and the remaining work gear
solid geometries. This algorithm is supported by a universal and modular code as well as
by a user friendly graphical interface, for pre- and postprocessing user interactions. The
resulting 3D data allow the effective utilization for further research such as prediction of
the cutting forces course, tool stresses, and wear development as well as the optimization
of the gear hobbing process. 关DOI: 10.1115/1.2738947兴
Keywords: gear hobbing, simulation, 3D-Modeling, CAD
1
Introduction
In most of the demanding torque transmission systems, the key
components are premium well designed and properly fabricated
gears. The gear hobbing process is widely applied for the construction of any external tooth form developed uniformly about a
rotation center. The kinematics principle of the process is based on
three relative motions between the workpiece and the hob tool. To
produce spur or helical gears, the workpiece rotates about its symmetry axis with certain constant angular velocity, synchronized
with the relative gear hob rotation. Depending on the hobbing
machine used, the worktable or the hob may travel along the work
axis with the selected feed rate.
Many advances have been made in the development of numerical and analytical models for the simulation of the gear hobbing
process, aiming to the determination of the undeformed chip geometry, cutting force components, and tool wear development
关1,2兴. The industrial weight of these three simulation data is associated with the optimization of the efficiency per unit cost of the
gear hobbing process. The undeformed chip geometry is an essential parameter to determine the cutting force components, as well
as to predefine the tool wear development, both of them important
cost related data of the hobbing process 关3兴.
Gear hobbing is nowadays analytically understood, while data
such as chip geometry, cutting force components, mechanical
stresses, and wear performance may be determined with the aid of
Contributed by the Manufacturing Engineering Division. Manuscript received
August 18, 2006; final manuscript received November 3, 2006. Review conducted by
Dong-Woo Cho.
consistent software codes 关4–15兴. Despite the research and industrial merit of such programs in the past, a variety of restrictions
and limitations arise, owing to the simplified modeling strategies
that they imply. The basic principle, on which the FRS code was
built in the 1970s, has been the determination of common areas
between one cutting hob tooth profile and the work gear. Hereby,
a two-dimensional 共2D兲 space of a plane determined by the combination of work gear and hob’s kinematics was selected as the
calculation region. The accuracy of the chip dimensions approximated by FRS code depends on various input parameters, such as
number of calculation planes, the discrete interpolation and projection done on the cutting section planes, and the calculated discontinuity of chip thickness. Therefore it is clearly understood that
the resulting plane chip geometry does not represent exactly the
solid geometry of a real chip. As a matter of these facts, transitional thickness variations may not be traced, if they belong to an
intermediate calculation plane of the chosen ones. On the other
hand, any further postprocessing of the calculated chip and gear
geometries requires additional data processing which leads to
supplementary interpolations of the 2D extracted results.
To overcome such modeling insufficiencies, the present research work introduces a new approach, which exploits powerful
modeling capacities of up-to-date CAD software environments.
Hereby, a software module called HOB3D has been developed
from scratch and embedded to an existent commercial CAD system. The developed algorithm is built in terms of a computer
code, which is supported by a user friendly graphical interface.
HOB3D provides the extend ability to other cutting processes
based on the same cutting principle. The models output formats
provided from the “parent” CAD system 共.ipt, .sat, .iges, .dxf etc兲
Journal of Manufacturing Science and Engineering
Copyright © 2007 by ASME
OCTOBER 2007, Vol. 129 / 911
Downloaded 04 Jan 2008 to 147.27.11.190. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 1 Intrinsic principles and features of gear hobbing process
offer realistic solid parts, chips, and work gear, which can be
easily managed for further investigations individually, or as an
input to other commercial CAD, CAM, or FEA environments.
2
Gear Hobbing Features and Simulation Strategy
Gear hobbing differs significantly from conventional milling,
since its kinematics is governed by the rolling principle between
the hob and the workpiece. To optimize computationally this cutting scheme, the implementation of each geometric feature, kinematics and cutting conditions to the developed algorithm, is critical.
The process problem is basically prescribed from the geometrical characteristics of the gear to be cut, the hob that will be used,
and the involved kinematics between them. As presented at Fig. 1,
the final geometry of a resulting gear is basically described by six
parameters: module 共m兲, number of teeth 共z2兲, outside diameter
共dg兲, helix angle 共ha兲, gear width 共W兲, and pressure angle 共an兲.
The correlation of these parameters automatically yields the module 共m兲 of the hob tool, whereas other tool geometrical parameters
关external diameter 共dh兲, number of columns 共ni兲, number of origins 共z1兲, axial pitch 共⑀兲 and helix angle 共␥兲兴, are options to be
chosen. As soon as the geometrical parameters of the two combined parts are set, the kinematics chain has to be initialized. The
helix angle of the hob and the work gear prescribe the setting
angle 共␪s兲 between the parts and the way that their relative motions shall take place. Three distinct cutting motions are required:
the tool rotation about its axis, the tool axial displacement, and the
workpiece revolution about its axis. By these means, the direction
of the axial feed 共f a兲 prescribes two different hobbing strategies:
the climb 共CL兲 and the up-cut 共UC兲. In case of helical gears, two
additional variations exist, the tool helix angle 共␥兲 compared to
the helix of the gear 共ha兲. If the direction of the gear helix angle is
identical to the hob helix angle the type of the process is set to
equidirectional 共ED兲, if not to counterdirectional 共CD兲 one.
Focusing on the generation of highly precise geometric models
of undeformed solid chips, as well as on the elimination of any
loss of data occurring from the processing of the geometrical
equations involved with the gear hobbing process, the CAD based
program HOB3D has been introduced. The essential input data
concern the determination of the hob and work gear geometries
and the cutting parameters that take place for the completion of
the simulation process. When the values of the input parameters
are set, the work gear solid geometry is created in the CAD environment and one hob tooth rake face profile is mathematically and
visually formed. At the same moment the assembly of the effec912 / Vol. 129, OCTOBER 2007
tive cutting hob teeth 共N兲 is determined, as presented in Fig. 2.
The kinematics of gear hobbing process is directly applied in one
three-dimensional 共3D兲 tooth gap of the gear, considering the axisymetric configuration. Moreover, a 3D surface is formed for every generating position 共e.g., successive teeth penetrations兲, combining the allocation of the two involved parts 共hob and work
gear兲 and following a calculated spatial spline as a rail. These 3D
surface “paths” are used to identify the undeformed chip solid
geometry, to split the subjected volume, and to create finally the
chip and the remaining work gear continuous solid geometries.
3
HOB3D Simulation Process
In the developed simulation every rotation and displacement
between the hob and the work gear, for modeling enhancement
reasons, are directly transferred to the hob. By this implementation a global coordinate system that is placed on the center of the
upper gear base is always fixed and provides a steady reference
point to monitor the traveling hob.
3.1 Mathematical Formulation. Without any loss of generality, the hob is considered to have one and only tooth 共numberednamed “tooth 0”兲. The identification vector 共v0兲 of “tooth 0” has
its origin 共CH兲 on the Y h axis of the hob and its end at the middle
of the rake face, forming a module that equals to 兩v0 兩 = dh / 2. The
identification vector determines the direction of the Zh axis of the
hob coordination system 共XhY hZh兲 and is initially placed as an
offset of the global Z axis, set at a vertical distance L1 from the
origin of the XYZ global system. The right part of Fig. 3 illustrates
how the parameter L1 determines the region where the cutting
starts. Once the simulation parameters are settled, the work gear
solid model is generated and the assembly of the effective cutting
hob teeth is enabled. The moment that the gear hobbing simulation starts is considered as time zero. At this time 共t = 0兲 the planes
YZ and Y hZh are parallel and their horizontal distance, steady for
the whole simulation period, is set to value L2 = dh / 2 + dg / 2 − t.
Distance L2 practically determines the cutting depth, user defined
as an input parameter. To determine the setting angle 共␪s兲, the
XhY hZh hob coordinate system is rotated about the Xh axis, so that
the simulation process becomes completely prescribed. Using the
spatial vector v0, it is easy to compute the identification vectors vi
of effective teeth N, with −k 艋 i 艋 n and N = n + k + 1, k , n 苸 Z+ taking into account the hob geometrical input parameters, relative to
v 0.
In addition, the independent parameter ␪1 counts the rotational
angle of hob tool about its axis Y h during the cutting simulation.
Transactions of the ASME
Fig. 2 The flow chart of the developed HOB3D gear hobbing simulation code
Since ␪2 declares the rotational angle of hob tool about the work
gear and f a the axial feed of hob, their values are dependent on ␪1
and can be determined according to this angle values 共see also the
left part of Fig. 3兲.
As mentioned before, the forward kinematics of each of the N
effective cutting hob teeth occur in one gear tooth space 共gap兲.
Hereby, after the determination of v0 共tooth 0兲 and considering
that v−k is relative to v0 the forward kinematics are first applied to
v−k and afterwards to the following vectors vi, simulating precisely the real manufacturing process. For the determination of the
involved hob-gear kinematics, the following transformational matrices Rx,a 共rotation about X axis with ␣ angle兲, Ry,␸ 共rotation
about Y axis with ␸ angle兲, Rz,␪ 共rotation about Z axis with ␪
angle兲, are used:
冤
冤
冤
1
0
0
冥
冥
冥
Rx,␣ = 0 cos共␣兲 − sin共␣兲 ,
0 sin共␣兲 cos共␣兲
Ry,␸ =
cos共␸兲
0
0 sin共␸兲
1
0
− sin共␸兲 0 cos共␸兲
cos共␪兲 − sin共␪兲 0
Rz,␪ = sin共␪兲
0
cos共␪兲
0
0
1
,
共1兲
Fig. 3 Gear hobbing simulation kinematics scheme
OCTOBER 2007, Vol. 129 / 913
Fig. 4 Determination of d␪ angle in case of helical gear cutting simulation
In case of simulating helical gears fabrication, a differential
angular amount is put on the rotating system of the gear, in order
to increase or decrease the angular velocity of the rotating workpiece, ensuring the proper mechanical functionality 共meshing兲 of
the hob-cutting and gear-cut angles. Depending on the hobbing
type process a new angular parameter 共d␪兲 has to be inserted to
the whole kinematical chain for the acceleration or deceleration of
␪2. The calculation of the parametrical value of d␪ is taking place
on the pitch circle as presented in Fig. 4.
3.2 Exploitation of Modeling and Graphical Features of
the CAD Environment. Using the kinematics scheme described
in the previous paragraph, a spatial spline path is created inside
the CAD system, by interpolating the points generated from the
transformation of each vi vector of each cutting hob tooth relative
to the hob axis Y h 共using ␪1兲 and the global Z axis 共using ␪2兲, as
well as the displacement f a 共see Fig. 5共a兲兲. Thereby, the unit vectors 共CHn1兲i and 共CHn2兲i, described in Fig. 5共b兲, are rotated and
shifted for the creation of a plane properly positioned into the 3D
space, for every revolving position of a certain cutting tooth i. The
profile of the cutting hob tooth is formed on the 2D space that is
created on its rake plane. This process takes place for every revolving position, as graphically illustrated in Fig. 5共c兲.
By shifting these open cutting hob tooth profiles and using the
spatial spline path as a rail, a 3D surface path is constructed in one
gear tooth space. This path represents the generating position of
“tooth i”. With the aid of this surface path, the solid geometry of
a chip is identified for every generating position, using the boolean operations and the graphics capabilities of the CAD environment, as presented in Fig. 6共a兲. The chip geometry is restricted by
the external volume of the instantly formed working gear gap,
bounded outside the created surface. The identified solid geometry
is then subtracted from the workpiece, as it is presented in Fig.
6共b兲, generating the continuous 3D geometries of the chip and the
remaining work gear.
Bearing in mind the solid outputs form of the simulation process, their direct postprocessing is primitively enabled. Fig. 6共b兲
illustrates the chip solid geometry that is produced during a certain revolving position, whereas the detected maximum chip
thickness is presented in detail A.
To save computational time and resources, in every generating
position of each effective cutting tooth i, the overall rotation of the
hob about its symmetry axis Y h is restricted 共0 ° 艋 ␪1 艋 180° 兲. In
this way, during the entire simulation process, only motions that
affect the resulting solid geometries are taking place, without any
influence to the process sufficiency. After the completion of one
work cycle, i.e., the termination of every spatial surface path and
the subtraction of the chip solid geometries, the produced gear gap
is finally generated. Fig. 7 illustrates a cross section 共detail A兲 of
Fig. 5 Generation principle of hob tooth path for its generating position
914 / Vol. 129, OCTOBER 2007
Fig. 6 Solid chip geometry identification process
the generated gear gap, formed by the collective work of every
generating position. As can be seen in Fig. 7共c兲, the remaining
gear solid geometry holds complete geometrical information, both
for the removed and the remaining “material.”
4
Generation and Storage of Solid Chip Entities
The main data-entry window of the HOB3D graphical interface
is presented in Fig. 8. The specific data entry includes the sections
of hob and work gear data, as well as selected cutting conditions
and the user defined postdata. An up-cut 共UC兲 case for the simulation of a spur gear fabrication is used, in order to demonstrate
the program functionality.
The input data described at the previous sections are manually
inserted in their corresponding input boxes. The highlighted input
boxes, also recognized by an asterisk, refer to modeling parameters, which are automatically calculated and proposed by
HOB3D algorithm, offering simultaneously the possibility to be
user defined. The picture at the middle bottom part of the input
data window visualizes the strategy selected for each simulating
scenario. As can be observed at the output section, the program
uses a working folder, set by the user, to save in proper formats
the chosen outputs. The simulation parameters can also be saved
and reopened by selecting appropriate fields, while a simulation
report comprising the user selected output results may be optionally created after the completion of the simulation process.
Figure 9 illustrates the resulted solid geometry of the chip produced from the generating position gp= −4 of the above-presented
test case. The normally continuous chip solid geometry, for visualization purposes, has been sliced in each of the 91 assigned
revolving positions belonging to the kinematics transformations of
hob tooth “−4.” The intersecting planes of the revolving positions
along with the chip geometry are illustrated in the left part of Fig.
9, while the sliced solid geometry chip cross sections, on the same
revolving positions, are presented in the middle part. It is evident
that every revolving position generates the same profile on two
successive sliced parts of the chip. This is the reason why each
sliced part has the name of two successive generating positions.
The maximum thicknesses of each cross section 共hmax兲 are identified, as shown at the three inserted details their development is
described in the right diagram of Fig. 9.
The output chip solid geometries at 15 characteristic generating
positions of two different test cases, UC and CL, produced by the
activation of the HOB3D code, are presented in the left and right
parts of Fig. 10, respectively. Except for the direction of the axial
feed f a all other input data are identical, for both examined cases.
Examining each of the resulting chips, it is obvious that so many
of the extreme geometrical changes of the chip shapes are sufficiently determined, even if the generated chip solid is parted from
more than one domain.
The resulting 3D solid geometries for both gear gaps, generated
for the same presented cases, are ideal to verify and validate the
proposed algorithm. Since the gear gaps are generated by subtracting the solid volume of each chip from the initially cylindrical
work gear solid geometry, every cross section of them, passing
parallel to the XY plane, offers a polyline which corresponds to
the calculated profile 共see Fig. 7 right part兲. These two produced
gap profiles are compared to the standard ones introduced by Petri
关16兴 and DIN 3972 关17兴, as presented in the left part of Fig. 11. To
determine the calculated profile deviation, the dichotomy between
them and the nominal profile is calculated and the occurring error
Fig. 7 Visual gear gap generation by HOB3D
OCTOBER 2007, Vol. 129 / 915
Fig. 8 HOB3D data-entry window
is measured at its direction, as presented in detail A of Fig. 11. The
calculated error of each test case is plotted to the graphs presented
at the right part of Fig. 11, illustrating a mean error less than
10 ␮m for the working depth of produced gears, for the UC and
CL cases, respectively. Such negligible deviations satisfy the computational accuracy expectations, and verify the sufficiency of the
developed code.
5
Conclusions
The paper illustrated an advanced and validated simulation
method for gear hobbing process, based on a commercial CAD
environment. In contrast to former research attempts, in the
present investigations, the kinematics of gear hobbing is directly
applied in one tooth 3D space by the construction of spatial surface paths, for every generating position. The kinematics involves
the rotations and displacements of the two rolling parts 共hob and
work gear兲 for every possible manufacturing case of gear hobbing
process. These 3D surface “paths” are used to divide the subjected
volume and directly create the chip and the remaining work gear
continuous solid geometries. The algorithm is supported by a
computer code with the aid of a user friendly graphical interface.
Considering the quality and the format of the resulting solid geometries, it is implicit that further manipulation and postprocessing of them is effortlessly achievable, without further postprocessing. The confirmation of the validity and accuracy of the proposed
method has been accomplished by comparing the produced gear
Fig. 9 Solid chip formation at specific generating position in gear hobbing
916 / Vol. 129, OCTOBER 2007
Fig. 10 Typical solid chips at up-cut and climb hobbing
Fig. 11 Algorithm verification as with the aid of the calculated gear gap profile and the nominal theoretical one
gap profile with theoretical ones, expressing a faultless agreement
between them. The results of the present work hold significant
industrial and research interest, including the accurate prediction
of dynamic behavior and tool wear development in gear hobbing
procedure.
Nomenclature
CL
UC
ED
CD
⫽
⫽
⫽
⫽
climb hobbing
up-cut hobbing
equidirectional hobbing
counterdirectional hobbing
GP
RP
fa
t
v
m
ni
z1
z2
dg
ha
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
generating position
revolving position
axial feed 共mm/wrev兲
cutting depth 共mm兲
cutting speed 共m/min兲
work gear and hob tool module 共mm兲
number of hob columns
number of hob origins
number of work gear teeth
external work gear diameter 共mm兲
gear helix angle 共°兲
OCTOBER 2007, Vol. 129 / 917
References
关1兴 Klocke, F., and Klein, A., 2006, “Tool Life and Productivity Improvement
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关3兴 Rech, J., Djouadi, M. A., and Picot, J., 2001, “Wear Resistance of Coatings in
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关4兴 Sulzer, G., 1974, “Leistungssteigerung bei der Zylinderradherstellung durch
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关5兴 Gutman, P., 1988, “Zerspankraftberechnung beim Waelzfraesen,” dissertation,
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关8兴 Tondorf, J., 1978, “Erhoehung der Fertigungsgenauigkeit beim Waelzfraesen
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关9兴 Antoniadis, A., 1988, “Determination of the Impact Tool Stresses During Gear
Hobbing and Determination of Cutting Forces During Hobbing of Hardened
918 / Vol. 129, OCTOBER 2007
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
Gears,” dissertation, Aristoteles University of Thessaloniki, Thessaloniki,
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Antoniadis, A., Vidakis, N., and Bilalis, N., 2002, “Failure Fracture Investigation of Cemented Carbide Tools Used in Gear Hobbing—Part I: FEM Modeling of Fly Hobbing and Computational Interpretation of Experimental Results,” ASME J. Manuf. Sci. Eng., 124共4兲, pp. 784–791.
Antoniadis, A., Vidakis, N., and Bilalis, N., 2002, “Failure Fracture Investigation of Cemented Carbide Tools Used in Gear Hobbing—Part II: The Effect of
Cutting Parameters on the Level of Tool Stresses—A Quantitive Parametric
Analysis,” ASME J. Manuf. Sci. Eng., 124共4兲, pp. 792–798.
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Sinkevicius, V., 2001, “Simulation of Gear Hobbing Forces,” Kaunas University of Technology Journal “Mechanika,” 2共28兲, pp. 58–63.
Komori, M., Sumi, M., and Kubo, A., 2004, “Method of Preventing Cutting
Edge Failure of Hob due to Chip Crush,” JSME Int. J., Ser. C, 47共4兲, pp.
1140–1148.
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Analysis of Cutting Edge Failure due to Chip Crush,” Gear Technol., Sept./
Oct., pp. 64–69.
Petri, H., 1975, “Zahnhuß–Analyse bei außenverzahnten Evolventenstirnraedern : Teil III Berechnung,” Antriebstechnik, 14共5兲, pp. 289–297.
DIN 3972, 1992, Bezugsprofile von Verzahnwerkzeugen fuer EvolventenVerzahnungen nach DIN 867, Taschenbuch 106, Beuth Verlag.

Advanced Computer Aided Design Simulation of Gear Hobbing by

Transcription

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