Din 867 Pdf Free Download

Dimitriou Vasilis

Adjunct Professor

Vidakis Nectarios

Associate Professor

Antoniadis Aristomenis

Professor

Technological Educational Institute of Crete,

Romanou 3,

Chania 73133, Greece

Advanced Computer Aided Design

Simulation of Gear Hobbing by

Means of Three-Dimensional

Kinematics Modeling

Gear hobbing, as any cutting process based on the rolling principle, is a signally multi-

parametric 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 con-

trast 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 displace-

ments 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 con-

struction 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 sym-

metry 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 numeri-

cal and analytical models for the simulation of the gear hobbing

process, aiming to the determination of the undeformed chip ge-

ometry, cutting force components, and tool wear development

关1,2兴. The industrial weight of these three simulation data is asso-

ciated with the optimization of the efficiency per unit cost of the

gear hobbing process. The undeformed chip geometry is an essen-

tial 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

consistent software codes 关4–15兴. Despite the research and indus-

trial 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 com-

bination of work gear and hob's kinematics was selected as the

calculation region. The accuracy of the chip dimensions approxi-

mated by FRS code depends on various input parameters, such as

number of calculation planes, the discrete interpolation and pro-

jection done on the cutting section planes, and the calculated dis-

continuity 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, transi-

tional 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 re-

search 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 sys-

tem. 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

Contributed by the Manufacturing Engineering Division. Manuscript received

August 18, 2006; final manuscript received November 3, 2006. Review conducted by

Dong-Woo Cho.

Journal of Manufacturing Science and Engineering OCTOBER 2007, Vol. 129 /911

Copyright © 2007 by ASME

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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 cut-

ting scheme, the implementation of each geometric feature, kine-

matics and cutting conditions to the developed algorithm, is criti-

cal.

The process problem is basically prescribed from the geometri-

cal 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z

2

兲, outside diameter

共d

g

兲, helix angle 共 h

a

兲, gear width 共 W兲 , and pressure angle 共 a

n

兲.

The correlation of these parameters automatically yields the mod-

ule 共m 兲 of the hob tool, whereas other tool geometrical parameters

关external diameter 共 d

h

兲, number of columns 共 n

i

兲, number of ori-

gins 共 z

1

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 com-

bined 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 mo-

tions 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 h

a

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 envi-

ronment and one hob tooth rake face profile is mathematically and

visually formed. At the same moment the assembly of the effec-

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 axi-

symetric configuration. Moreover, a 3D surface is formed for ev-

ery generating position 共e.g., successive teeth penetrations兲, com-

bining 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 implementa-

tion 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 gener-

ality, the hob is considered to have one and only tooth 共numbered-

named "tooth 0"兲. The identification vector 共

v

0

兲 of "tooth 0" has

its origin 共 C

H

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

v

0

兩 = d

h

/2. The

identification vector determines the direction of the Z

h

axis of the

hob coordination system 共X

h

Y

h

Z

h

兲 and is initially placed as an

offset of the global Z axis, set at a vertical distance L

1

from the

origin of the XYZ global system. The right part of Fig. 3 illustrates

how the parameter L

1

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 simula-

tion starts is considered as time zero. At this time 共t =0 兲 the planes

YZ and Y

h

Z

h

are parallel and their horizontal distance, steady for

the whole simulation period, is set to value L

2

= d

h

/2+d

g

/2−t.

Distance L

2

practically determines the cutting depth, user defined

as an input parameter. To determine the setting angle 共

␪

s

兲, the

X

h

Y

h

Z

h

hob coordinate system is rotated about the X

h

axis, so that

the simulation process becomes completely prescribed. Using the

spatial vector

v

0

, it is easy to compute the identification vectors

v

i

of effective teeth N, with −k 艋 i 艋 n and N = n + k +1, k , n 苸 Z

+

tak-

ing 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.

Fig. 1 Intrinsic principles and features of gear hobbing process

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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

v

0

共tooth 0兲 and considering

that

v

−k

is relative to

v

0

the forward kinematics are first applied to

v

−k

and afterwards to the following vectors

v

i

, simulating pre-

cisely the real manufacturing process. For the determination of the

involved hob-gear kinematics, the following transformational ma-

trices R

x, a

共rotation about X axis with

␣

angle兲, R

y,

␸

共rotation

about Y axis with

␸

angle兲, R

z,

␪

共rotation about Z axis with

␪

angle兲, are used:

R

x,

␣

=

冤

10 0

0

cos共

␣

兲 − sin共

␣

兲

0

sin共

␣

兲 cos共

␣

兲

冥

,

R

y,

␸

=

冤

cos共

␸

兲

0

sin共

␸

兲

010

− sin共

␸

兲

0

cos共

␸

兲

冥

,

R

z,

␪

=

冤

cos共

␪

兲 − sin共

␪

兲

0

sin共

␪

兲 cos共

␪

兲

0

001

冥

共1 兲

Fig. 2 The flow chart of the developed HOB3D gear hobbing simulation code

Fig. 3 Gear hobbing simulation kinematics scheme

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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 work-

piece, 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

v

i

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 vec-

tors 共 C

H

n

1

兲

i

and 共 C

H

n

2

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 re-

volving 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 bool-

ean operations and the graphics capabilities of the CAD environ-

ment, 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 pro-

cess, their direct postprocessing is primitively enabled. Fig. 6共b兲

illustrates the chip solid geometry that is produced during a cer-

tain 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. 4 Determination of d

␪

angle in case of helical gear cutting simulation

Fig. 5 Generation principle of hob tooth path for its generating position

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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 simu-

lation 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 param-

eters, 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 option-

ally created after the completion of the simulation process.

Figure 9 illustrates the resulted solid geometry of the chip pro-

duced from the generating position gp= −4 of the above-presented

test case. The normally continuous chip solid geometry, for visu-

alization 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h

max

兲 are iden-

tified, 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 suffi-

ciently 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 subtract-

ing 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. 6 Solid chip geometry identification process

Fig. 7 Visual gear gap generation by HOB3D

Journal of Manufacturing Science and Engineering OCTOBER 2007, Vol. 129 / 915

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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 com-

putational 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 sur-

face 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 ge-

ometries, it is implicit that further manipulation and postprocess-

ing of them is effortlessly achievable, without further postprocess-

ing. The confirmation of the validity and accuracy of the proposed

method has been accomplished by comparing the produced gear

Fig. 8 HOB3D data-entry window

Fig. 9 Solid chip formation at specific generating position in gear hobbing

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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 ⫽ climb hobbing

UC ⫽ up-cut hobbing

ED ⫽ equidirectional hobbing

CD ⫽ counterdirectional hobbing

GP ⫽ generating position

RP ⫽ revolving position

f

a

⫽ axial feed 共mm/wrev兲

t ⫽ cutting depth 共 mm 兲

v

⫽ cutting speed 共 m/min 兲

m ⫽ work gear and hob tool module 共 mm 兲

n

i

⫽ number of hob columns

z

1

⫽ number of hob origins

z

2

⫽ number of work gear teeth

d

g

⫽ external work gear diameter 共 mm 兲

h

a

⫽ gear helix angle 共°兲

Fig. 11 Algorithm verification as with the aid of the calculated gear gap profile and the nomi-

nal theoretical one

Fig. 10 Typical solid chips at up-cut and climb hobbing

Journal of Manufacturing Science and Engineering OCTOBER 2007, Vol. 129 / 917

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